Volume 14, Issue 1

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Volume 14 Issue 1

The International Compendium for Crash Research

Vehicle Damage and Longitudinal Throwing Distances

When Using Biofidelic Dummies In Comparison To Conventional Dummies

Validation of the HAn-Brach Vehicle-ped Impact Mechanics Model Crash-ol-o-gy

Airbag Myths, Legends & Lore

Review of Engineering Mechanics Applied to Accident Reconstruction Parts 1-2

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BONUS MATERIAL https://spaces.hightail.com/space/YydeLCM6Kz Access Code: CMAG14120 Welcome to the Digital Download feature of Collision Magazine. Use the URL and Access Code above to gain access to a private webpage containing additional bonus material for this issue of Collision Magazine. This issue features PDF presentations from the 2020 EDR Summit held in Houston, Texas this past March. Also included are the bonus materials distributed at the Summit.

2020 EDR Summit Material March 9-11, 2020 | Houston, Texas

The annual EDR Summit focuses on Event Data Recorder research, collection and analysis for vehicle crash investigation. This is the only conference in the United States dedicated to users of EDR Tools and other in-vehicle data. The future of safety systems within vehicles on the road today demand a closer look at how they work and how the data collected can be used to assist your crash investigation.

The EDR Summit delivers the next steps in advanced EDR technology for vehicle crash analysis. Most importantly, this summit brings together industry experts from around the world to present on timely topics and case studies. Over the years, the presentations have focused on EDR data found in light trucks, passenger cars, SUVs, motorcycles, heavy commercial vehicles, active safety systems, autonomous driving, and vehicle infotainment systems. Privacy, legistation and legal aspects, including mock trials and mock depositions, have also been presented.

The EDR Summit is open anyone who has an interest in learning more about event data recorders and how they are used in vehicle safety and crash investigation. As a result, typical attendees of this Summit include law enforcement, insurance (SIU and claims), government, legal and collision reconstructionists. Therefore, if you are involved with the collection and/or analysis of EDR data from vehicle crashes or forensics you won’t want to miss the EDR Summit! www.collisionpublishing.com

Collision Magazine - Volume 14 Issue 1 1

48

Contents

Volume 14 Issue 1

inside 1

Digital Download Information

5

Collision Magazine Info and Advertiser Index

4

Letter From the Editor

6

features 6

Validation of the Han-Brach Vehicle-Pedestrian Impact Mechanics Model

by R. Matthew Brach, David Fortenbaugh and Jon Van Poppel

62

Vehicle Damages And Longitudinal Throwing Distances When Using Biofidelic Dummies In Comparison To Conventional Dummies

by Annika Kortmann and Tim Hoger

73

Crashology: Airbag Myths, Legends, and Lore

by Wesley Vandiver and Robert Anderson

34

A Review Of Engineering Mechanics Applied To Accident Reconstruction Part 1: Kinematics

by Jai Singh

88

A Review Of Engineering Mechanics Applied To Accident Reconstruction Part 2: Dynamics

by Jai Singh

80

A How-To Guide For Downloading Toyota’s Vehicle Control History Data

by Alan Moore

52

Utilizing BioMedical Engineering Physical Evidence Analysis for Motorcycle Compared to Pedestrian Collision

by Laura Liptai, and Mark Ezra

22

34

80

52

Gasoline-Powered Golf Cart Acceleration and Hard Braking Performance

by Robert D Anderson and Michael Rosenfield

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Collision Magazine - Volume 14 Issue 1 3

Letter from the editor As we begin 2020 with Volume 14 of Collision Magazine, it's a good time to think about a couple important topics related to crash reconstruction - currency and creativity.

The 2020 EDR Summit attracted over 300 attendees from all over the world gathering for the first topic: working to keep current on research and shared information. The presenters at the Summit, as well as the many other authors who develop and provide the original, cutting edge research and information found in Collision are gearing up to make the second issue of Volume 14 another not to be missed. And THAT is, in part, creativity.

The two concepts of currency and creativity are more than words, they're core components which play into everything we do as crash reconstructionists. It's no secret we need (and want) to keep current in the field. Whether it's ongoing or recurring training, involvement in research or by staying current with the latest software and hardware we use from measuring systems to the CDR Tool; it's about being and staying current. Whether in the public or private sector, we know that as soon as we fall behind, whatever we've left unpracticed, whatever we haven't updated or have come to seemingly forgot about will be that which we need most urgently one day. We are all familiar with the notion of "use it or loose it"; crash reconstruction is the very definition of that phrase. This is where, since 2006, the EDR Summit has been the top go to source of current EDR technology information anywhere in the world. While other smaller conferences have come and go, the EDR Summit has been a reliable resource for 15 years, providing reconstructionists an opportunity to stay just that much more current, that much more up to date. And that takes us back to creativity.

Some of us consider ourselves lucky to be able to pursue our passion and the presenters at the EDR Summit do just that. Not only do they educate peers with the latest research at the Summit, they also contribute to follow on issues of Collision Magazine. They go beyond the original presentation with more detailed discussions and/or examples of the material to be shared in the pages of Collision Magazine. The time the presenters and authors devote to their research and then their presentations is a testament to their passion for the work and Collision Magazine is

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grateful to them for their ongoing dedication to the science and their desire to share information so freely.

Research isn't free (perhaps this is a different way to use the word currency!) and the effort spent on a research projects, testing, development and analysis of the data takes valuable time! These authors and presenters offer their time, talent and treasure often without thanks. So, if we accomplish nothing else by putting Collision Magazine together, we'd like to make sure we acknowledge the selfless efforts of the authors and contributors to both the EDR Summit and Collision Magazine.

This year, there were more than 16 presentations on a variety of topics related to various aspects of the EDR Technology - beyond the Bosch CDR Tool. Many of those presentations will, as noted, become articles memorialized in Collision Magazine and it is our pleasure to be able to bring those to our readership. It's a way of staying current and maintaining your currency in reconstruction.

Creativity beyond the original works of the presenters an authors is, in part, left to you. You will find all of the 2020 EDR Summit presentations (in PDF format) in the digital download section located on page 1 of this issue.

While the number of articles varies from one issue of Collision to the next, the reader can always be assured the materials is original - the product of the creativity of the authors, researchers and presenters. But that doesn't or shouldn't limit the creativity of the reader. What can you do with the information? Be creative. As you read through this issue of Collision, ask yourself: how can I use the information in this article? Even if it is only one bullet point or a seemingly small portion of an article, make it something you can take with you to the next crash investigation, be creative and see how best that information can be used.

Scott Baker Managing Editor

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All rights reserved, © 2020 Crash Data Group Inc. The opinions and conclusions expressed in this publication and any associated data and throughout the articles attributed to specific authors are the opinions and conclusions of the authors noted and not necessarily the editorial staff or anyone else for that matter. While some articles have been reviewed for content, the accuracy of reprinted models or equations cannot be fully guaranteed. It is the responsibility of the reader to apply critical thinking to an individual review of the content and make their own personal judgements as to its value to them, individually.

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Collision Magazine - Volume 14 Issue 1 5

Validation of the Han-Brach Vehicle-Pedestrian Impact Mechanics Model R. Matthew Brach, David Fortenbaugh and Jon Van Poppel Engineering Systems Inc.

A

bstract When it was introduced in 2001, the Han-Brach vehicle-pedestrian impact model was fitted to various experimental data to evaluate its performance against test data to establish the empirical parameters utilized in the model. It was also compared graphically to various data sets and other pedestrian impact models to assess the ability of the model to capture the physics of pedestrian impacts. The current research used experimental data generated at the 2017 ARC-CSI conference in Las Vegas, NV, where detailed data were collected from nu-

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merous vehicle-pedestrian impact tests, to further evaluate and validate the model. These more detailed data facilitate new comparisons of results produced by the model. The analysis showed that the Han-Brach mechanics model captured the dynamics of these experimental trials. Additional insights into the ranges of the empirical parameters used in the model were generated using these new data. In addition to the analysis using the experimental data, three examples are presented in the paper that show various applications of the Han-Brach mechanics model to reconstruct a variety of vehicle-pedestrian collisions.

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Introduction The variety of algebraic formulas and physics models used for the reconstruction of pedestrian impacts can be separated into three categories: 1) empirical models, 2) mechanics models, and 3) multi-body modeling programs. These three categories have also been previously named as Type I, Type II and Type III, respectively.1 Type I models are generally based on fitting a mathematical function to experimental data resulting in some form of a regression equation that relates the speed of the vehicle to the throw distance (or vice versa). Various accounts of Type I models are available in the literature.3, 4, 5 Numerous Type I models were compiled into a software application.6 Type II models are based on the principles of engineering mechanics including impulse-momentum and rectilinear motion. These models also relate the speed of the vehicle to the throw distance of the pedestrian (or vice versa). Type III models are also based on principles of mechanics, but these models treat the pedestrian as a multibody system and allow for the geometry of the vehicle that interacts with the pedestrian to be explicitly modeled with high fidelity.15 These models may also incorporate finite element methods. Numerous publications exist for each of these three categories and deal with one or more of the generally recognized pedestrian impact configurations: wrap, forward projection, fender vault, roof vault, and somersault.2 The introduction of Type III models is more recent and, due to the complexity of the models, is largely based in dedicated, commercially available software that runs the models. The three most common software programs in the field of crash reconstruction that include these models are MADYMO,7 PC-Crash 8 and HVE.9 Various publications dealing with these programs present the theory behind the models and explore the applicability and validity of these programs.10,11,12,13 Some publications take a broader approach and assess all three types of vehicle-pedestrian accident reconstruction methods.14,15 One paper considers the use of a Type II model to evaluate the results of analysis done using a Type III model.1 The research presented here focuses specifically on the development of the Type II models (mechanics models) used for the reconstruction of vehicle-pedestrian crashes and the validation and use of one of the models specifically (Han-Brach).19 The focus on mechanics models is motivated by a need for a useful method for reconstructing vehicle-pedestrian crashes that incorporates variables

and parameters that accommodate the range of physical evidence available after a crash. For example, if the longitudinal ∆V of a vehicle involved in a crash with a pedestrian were known from imaging the event data recorder (EDR) post-crash, this physical evidence cannot be used with empirical models as empirical models do not contain this parameter. Type III multi-body models can incorporate these data, but these tools generally require considerable training and practice to master, and in some cases, significant expense. These Type III models are particularly useful when the motion of the pedestrian (arms, torso, head, etc.) or the interactions between the pedestrian and the vehicle (i.e. hood and/or windshield deformation) need to be analyzed.

Vehicle-Pedestrian Mechanics Models Examination of the literature associated with mechanics models for vehicle-pedestrian collisions shows that one of the first publications that presented this topic was the section on pedestrian throw in the book by Collins.16 In this treatment, Collins presented an equation for the throw distance with two components: one component related to the horizontal distance traveled during the airborne trajectory and a second equation for the travel of the pedestrian along the roadway prior to coming to rest. The next paper that tackled this topic was written

by John Searle and Angela Searle.17 In that paper, the authors presented a theory for the total trajectory of the pedestrian from impact to rest. The equation presented there provides the speed of the vehicle at impact as a function of the throw distance. The equation is: (1) In this equation, µ is the frictional drag coefficient between the pedestrian and the roadway, S is the throw distance (the total travel distance of the pedestrian from impact to rest), V is the initial velocity of the pedestrian at separation from the vehicle, and θ is the launch angle of the pedestrian relative to the roadway. In the derivation, the height of the center of mass of the pedestrian at impact is assumed to be level with the roadway. In this equation, the value µ applies to the total throw distance, not just the portion of the throw distance in which the pedestrian is engaging the roadway. In a later paper,18 Searle presented the derivation of an alternative formula that also relates the vehicle speed to the throw distance

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of a pedestrian. In this approach, the change in height of the center of mass of the pedestrian from the time of impact to the time at rest was included. This alternative formula is: (2) In this equation, µ is the frictional drag coefficient between the pedestrian and the roadway, θ is launch angle of the pedestrian relative to the roadway, S is the throw distance, V is the initial velocity of the pedestrian and H is the change in height of the center of mass of the pedestrian from launch to impact with the roadway (H is positive for an increase in height). In this equation, the value µ again applies to the total throw distance, not just the portion of the throw distance in which the pedestrian is engaging the roadway.

Another paper that considers the pedestrian throw problem from the perspective of classic kinematic equations is Aronberg.26 He presents a detailed analysis of the general problem of the throw distance incorporating arbitrary CG height at launch and explicitly including the apogee of the postimpact flight of the pedestrian in the equations.

This development addresses the kinematics of the pedestrian but does not include analysis of the impact between the vehicle and the pedestrian or the impact between the pedestrian and the roadway. In a subsequent paper,28 the author considers the impact between the pedestrian and the roadway including a description of testing that shows the impact with the ground does affect the stopping distance of the pedestrian. The results of that testing validate the need for a Type II pedestrian throw model to include the impact between the pedestrian and the roadway in the model.

The Han-Brach Mechanics Model The Han-Brach vehicle-pedestrian impact model,19 introduced in 2001 and built on the previous Type II models, provides the means to analyze and reconstruct vehicle-pedestrian crashes using the principals of Newtonian mechanics. The model provides an approach different from the numerous empirical models that have been the mainstay of vehicle-pedestrian impact analysis and reconstruction for the last three decades of the 20th century.

These empirical models were based on experimental data and generally provided an algebraic relationship between the measured speed of the vehicle at impact and the resulting measured throw distance of the pedestrian. Thus, these empirical models did not involve the specifics of the collision to be reconstructed (vehicle mass, etc.) or any of the physical evidence generated during the investigation 8

Collision Magazine - Volume 14 Issue 1

other than the throw distance itself (which is frequently unknown). Consequently, these Type I empirical models offered no means to utilize any available physical evidence in the reconstruction of a crash. Type II mechanics models provide this capability.

In the original publication, the Han-Brach mechanics model was fitted to the experimental data available at the time to establish the values and ranges of the various parameters utilized in the model and to assess the overall agreement between the model and the data. The result of this fitting process, with the appropriate ranges of empirical parameters determined in the analysis, showed that the model agreed reasonably well with the data. However, the experimental data used in the analysis, the best available at that time, were quite general. These data provided essentially only the speed of the vehicle and the throw distance and generally did not include parameters such as the mass of the test vehicle and the pedestrian dummy, roadwayvehicle and roadway-pedestrian drag factors, etc. These, and other quantitative parameters, are integral to the mechanics model. A distinct advantage of a Type II model that includes these parameters is in the evaluation of the uncertainty of a crash reconstruction. The uncertainty of the result of a reconstruction due to the variation in a given parameter can only be evaluated if the model incorporates that parameter. This is a very important consideration of the selection of a model for reconstructing a pedestrian crash.

For example, the uncertainty of the reconstructed preimpact speed of a vehicle due to the variation of the frictional drag between the pedestrian and the roadway (perhaps using a high-low calculation approach) can be evaluated only if the frictional drag is a parameter in the model. For the case of any of the Type I models, the value of frictional drag does not appear, so the uncertainty of those empirical models to that parameter cannot be calculated. It is very important to point out that the confidence intervals that are typically provided as part of the empirical models (the 85th percentile confidence intervals, for example) do not quantify the uncertainty for a particular crash reconstruction. These bounds associated with a given set of experimental data quantify the uncertainty of the data from all the tests in the compilation. For example, variation is introduced into the resulting data in that the masses of the different vehicles used in the tests, which were generally not recorded as they do not appear in the model, results in uncertainty in the throw distances. In aggregate, the uncertainty quantified by the percentile bounds is not the uncertainty of any given reconstruction or trial but rather the quantification of the uncertainties of all the trials in the population that were run. This uncertainty comes from the

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variety of vehicles that were used in the tests, the variety of manikins that were used in the tests, the different clothing on the manikins, the different roadway surfaces, etc. These uncertainties are independent of, and do not apply to, the uncertainties of a specific crash that is being reconstructed.

It might be argued that the uncertainty captured by the confidence intervals in a Type I model likely provides reasonable bounds for the reconstruction of any particular crash. Supporting that argument is the large range of throw distance captured by the intervals. For example, in one plot of experimental data of over 90 tests 3 the range of the 85th percentile bands is approximately ±10 kph (±6.2 mph). It may be for some reconstruction applications this range is acceptable; in other applications it may not be acceptable.

The analysis presented here utilized more complete data sets that included the details needed for direct comparison between the Han-Brach Type II mechanics model and a given test. For example, the data collected during the tests included here utilized, among other measurements, video of the impact and the post-impact trajectory of the manikin and the vehicle. The video allowed for analysis of the times associated with the various phases of the test. This level of detail was never available in prior data. Additional measurements collected during the testing were the throw distances, the pre-impact speeds of the vehicles, the accurate masses of the manikins and the vehicles at the time of the test, and the heights of the centers of mass of the manikins, among other parameters. Some parameters were

estimated from the video, including the flight distance of the manikin and the distance the vehicle moved during the contact interaction with the manikin.

The Vehicle-Pedestrian Impact (Type II) Mechanics Model Figure 1 shows various vehicle and pedestrian positions corresponding to a wrap vehicle-pedestrian event. Also included are the various physical distances and some other physical parameters that are associated with the postimpact motion of the vehicle and the pedestrian. Initial contact between the front of the vehicle and the pedestrian occurs at time τ = 0 with the velocity of the vehicle of vc0. The vehicle and pedestrian move forward at different speeds until time, τ = τ0, when a secondary impact between the torso (and possibly the head) of the pedestrian and the hood of the vehicle occurs. Between τ = 0 and τ = τ0, the center of gravity of the pedestrian has moved forward a distance xL and is at a height, h, above the ground as shown in the diagram. As the diagram implies, the derivation of the model equations is made with the assumption that the postimpact motions of both the vehicle and the pedestrian lie in the same geometric plane in which the x-axis lies along the heading of the vehicle and the y-axis is perpendicular to the roadway. In general, this assumption is reasonably met in actual collisions. The motion of the pedestrian is modeled as a point mass, i.e. the rotational inertia considerations of the torso and/or appendages is neglected. Additionally, any aerodynamic effects of the flight of the pedestrian are neglected.

Figure 1: Diagram showing coordinates and variables associated with the Han-Brach vehicle-pedestrian Type II collision model www.collisionpublishing.com

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For a forward projection collision, the user-specified value of xL is zero and there is no secondary impact between the pedestrian and the vehicle. Because of the impact, the pedestrian is projected forward instantaneously and is launched in an airborne trajectory with a velocity of vp0 and at an angle θ. In general, for a forward projection collision, the initial angle of motion of the pedestrian is parallel to the road, i.e. θ = 0°. The trajectory of the pedestrian has a range (the distance parallel to the ground) of R and reaches the ground at time τp1. At that time, the pedestrian impacts the ground with velocity components parallel and perpendicular to the ground determined by the trajectory and the initial speed. Due to the impact with the roadway, the pedestrian undergoes a change in speed (∆V) that has two components, one along the roadway (the x-axis) and another perpendicular to the roadway (the y-axis).

From the location of the center of gravity (CG) of the pedestrian at the time of the ground impact to the location of the CG of the pedestrian at rest, a distance s, the pedestrian is assumed to be uniformly decelerated with a sliding frictional drag factor, fp. The motion of the pedestrian over the distance s typically consists of a combination of rolling, sliding and tumbling. It is assumed that this motion does not include any impacts with other environmental features (such as a curb, a tree or any other vehicle). The total travel distance of the pedestrian from initial contact with the vehicle to rest is the throw distance, sp. It is assumed that from the time of separation of the pedestrian from the vehicle to the time of rest, τp, the pedestrian motion is completely independent of the vehicle. To provide a degree of generality for reconstructions, the vehicle motion is allowed an arbitrary travel distance s1 from τ = τ0 to τ= τc1 over which its velocity remains constant. Following this, the vehicle decelerates to rest over a distance s2 with drag factor, a0. Note that the distance s1 can be zero if the physical evidence dictates.

For a wrap trajectory collision, xL is not zero and a secondary impact occurs between the pedestrian and the hood and/or the windshield. Because of the impact at τ = τ0, the pedestrian “wraps” up onto the hood of the vehicle and is launched in an airborne trajectory with an initial velocity of vp0 at an angle θ to the roadway when the vehicle is decelerated. The trajectory has a range (distance parallel to the ground) of R and reaches the ground at time τp1. At that time, the pedestrian impacts the ground with velocity components perpendicular and parallel to the ground determined by the trajectory and the initial speed. As indicated in Figure 1, the model allows for a slope of the roadway along the direction of motion of the pedestrian and the vehicle. This slope is shown as ϕ and is assumed

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to be the same for the vehicle and the pedestrian over the respective ranges of motion.

Due to the impact with the roadway, the pedestrian undergoes a sudden change in speed that has two components, one along the roadway (the x-axis) and another perpendicular to the roadway (the y-axis). From the point of ground impact to the point of rest, a distance s, the pedestrian is assumed to be uniformly decelerated with an initial speed following the impact with the roadway (and the associated ∆V) and a frictional drag factor, fp. The motion of the pedestrian over the distance s typically consists of rolling, sliding and tumbling. The total travel distance of the pedestrian from initial impact to rest is the throw distance, sp. It is assumed that from the time of separation, τ0, to the time of rest, τp, the pedestrian motion is independent of the motion of the vehicle. For wrap collisions, the vehicle can be decelerated over the distance s0 at a constant deceleration of -a0, and then over the distance s2 with a constant deceleration of -a0. Allowing a0 and a2 to be different creates flexibility in the model, but generally these values are the same. For a0 ≠ 0 and a2 ≠ 0, the vehicle will decelerate from the time of initial contact with the pedestrian to rest.

Under these circumstances, the distance s1 is necessarily zero. Following separation, the vehicle decelerates to rest over a distance s2 with constant deceleration, -a0. With this information, equations can be derived to develop a Type II mechanics/analysis model. Table 1 lists the notation and definition of each of the variables used in the model.

Input Parameters a0

deceleration of vehicle over distance s0

a0

deceleration of vehicle over distance s2

s1

drag resistance coefficient of pedestrian over distance s acceleration of gravity height of pedestrian center of gravity at launch, τ0 distance of travel of vehicle at uniform speed (a0 = 0)

vc0

initial speed of vehicle

fp g h

xL

α θ μ ϕ

x-distance of pedestrian CG from initial contact to launch ratio of pedestrian speed to vehicle speed at time of launch angle of launch of pedestrian relative to x-axis impulse ratio for pedestrian-ground impact road grade angle (positive value for uphill)

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mc

mass of vehicle, weight /g

mp

mass of pedestrian, weight /g

s0

distance of travel of vehicle during pedestrian contact

(4) From the momentum exchange, the change in speed of the vehicle is calculated. This allows for the use of ∆V data from an EDR if data are imaged from the vehicle or an appropriate threshold is available.

Output Parameters v'c0

velocity of vehicle after impact with pedestrian

τp

initial speed of pedestrian following separation from the vehicle range of pedestrian throw, launch to ground impact time from impact to pedestrian initial contact with ground pedestrian ground contact distance, impact to rest pedestrian throw distance; total distance, initial contact to rest total time of travel of pedestrian, initial contact to rest

τc1

time of travel of vehicle at steady speed

s2

distance of travel of vehicle with uniform deceleration, a0

τc

vehicle travel time, initial contact to rest

vp0 R

τp1 s sp

d t0

For generality in the model, a factor, α, is introduced that relates the initial pedestrian velocity to the velocity of the vehicle such that: (5) The parameter α is an empirical parameter that can be determined from the data. It is assumed that aerodynamic drag acting on the pedestrian is negligible as the pedestrian undergoes the motion through the air over the range R. Under such a condition, the trajectory of the CG of the pedestrian is parabolic with zero horizontal acceleration and uniform vertical acceleration due to gravity. This leads to the following equations for the range, R, and trajectory time, τR: (6) and

distance between rest positions of vehicle and pedestrian time of travel of the vehicle over s0 with uniform deceleration, a0

Table 1: Notation and definition of variables and parameters used in the model

(7) The velocity components of the pedestrian at the instant before ground impact are: (8)

The Pedestrian Motion Based on the above description and Figure 1, the throw distance of the pedestrian is: (3) The mass of the pedestrian, mp, is typically negligible in comparison to the mass of the vehicle, mc. However, in cases where the momentum loss of the vehicle is important (such as an impact with a large animal), allowance is made in the model for a momentum exchange due to the collision with the pedestrian. This gives:

(9) Note that in these equations, a slope of the roadway, ϕ is included. The normal component of the impact of the pedestrian with the ground is assumed to be perfectly inelastic, restitution is zero, e = 0. This implies that the model assumes that there is a single impact between the pedestrian and the ground with no accompanying vertical bounce or rebound. According to planar impact theory 22 under these conditions, the vertical impulse due to the impact, Pn, will be Pn = -mp vpRy. The corresponding tangential impulse, Pt = μPn develops, where μ is defined by that equation. Since it is assumed that the interaction between the roadway and the pedestrian throughout the ground impact is sliding,

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tumbling and/or rolling (no multiple impacts, or bouncing), μ = -fp. This leads to a velocity at the beginning of the slide distance, s, of: (10) The source of this equation has not been previously shown and is provided in Appendix A for completeness.

With this speed of the pedestrian at the start of the tumble/ sliding phase, the tumble/slide distance s can be calculated using the following equation for uniform deceleration: (11)

In summary, the assumptions under which the model is derived are:

Vehicle Motion Following the change in momentum of the vehicle due to impact with the pedestrian, the motions of the vehicle and pedestrian are assumed to be independent of each other after separation, τ > τ0. Any incidental contact during the early part of the trajectory as the pedestrian is wrapping up onto, or separating from, the hood and windshield or secondary interaction(s) with the hood as the pedestrian is falling back to the roadway, is neglected. Over the distance, s0, and time, τ0, the initial speed of the vehicle is vc0, and the vehicle decelerates uniformly at a deceleration of a0 (which can be zero). Over the distance s1 and time interval (τc1 - τ0), the vehicle travels with the speed at the end of the distance s0 with no deceleration. The distance s0 can also be zero. Uniform deceleration of a0, occurs over the distance s2. The difference between the total travel distance of the vehicle and the pedestrian throw distance is labeled as the variable d. This distance is given by the following equation: d = s0 + s1 + s2 – sp. The default sign convention is that d is positive for the vehicle traveling further than the pedestrian. Correspondingly, d will be negative if the pedestrian travels farther then the vehicle.

Note that the model as presented above contains capability that was not part of the model as presented initially in 2001.19 This new version of the model allows for the vehicle to decelerate over the distance s0 with deceleration a0. This allows for the vehicle to decelerate over the entire distance from initial contact with the pedestrian at τ = 0 to rest if the physical evidence supports this characterization of the events. The previous edition of the model did not provide this capability. This additional capability permits brake application during the momentum exchange. This feature unfortunately confounds the speed change due to the momentum exchange with the speed change due to the braking.

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The distance between the rest positions of the pedestrian and the vehicle can be important in a speed reconstruction of a vehicle-pedestrian impact. After an incident, the location of the pedestrian at the onset of contact between the vehicle and the pedestrian is often unknown (determining the pedestrian location at impact might actually be the goal of the reconstruction). However, the distance d may be known since the rest positions of the vehicle and the pedestrian may be known from measurements or can be reasonably estimated from the scene photographs. In such cases, it is sometimes possible to reconstruct the vehicle preimpact speed and the impact location using the distance d. This technique using a Type II model is discussed elsewhere.20,21

1. The model assumes that the motion of the vehicle and the pedestrian occurs in the XZ plane, i.e. there is negligible transverse (lateral) motion of the pedestrian and/or the vehicle.

2. The preimpact speed of the pedestrian is such that assumption 1) is met.

3. Point mass impact (no rotational inertia is considered) is used to model the impact between the vehicle and the pedestrian and the vehicle and the pedestrian interact once (e.g. the vehicle does not roll over the pedestrian following the initial impact).

4. No rigid body (rotational) motion of the pedestrian is incorporated into the impact between the pedestrian and the ground.

5. None of the properties or dimensions of the vehicle, other than the mass, are explicitly used in the model although this factors into whether the interaction is categorized as a wrap trajectory of frontal projection.

6. The ground motion of the pedestrian consists of a single initial impact with an accompanying ∆V (with components along and perpendicular to the road) followed by uniform deceleration during which the pedestrian is sliding, tumbling, etc.

7. The throw model of the pedestrian (postimpact) does not permit multiple impacts (such as impacts of the pedestrian with curbs, poles, parked vehicles, or other objects).

8. Aerodynamic loading of the pedestrian during flight is not included in the model.

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9. Application of the method to wrap and forward projection collisions has been validated using experimental data; reconstruction of roof vaults and somersault crashes is possible but no data is available for validation; the model does not handle crashes where the pedestrian is carried by the vehicle or for fender vaults.

Fender vaults have been considered elsewhere.23 Analysis Based On Experimental Data A series of vehicle-pedestrian impact tests were conducted in September 2017 in Las Vegas, NV under the auspices of the ARC-CSI. These tests involved a variety of different light vehicles (sedans to full size vans) and a variety of different pedestrian manikins. Some of the tests involved a single vehicle striking two manikins. Some of the manikins were specifically fabricated to be lighter than the corresponding human counterpart to study the effects of pedestrian mass on the motion of the pedestrian (this topic is not studied here). Additionally, only tests involving manikins with realistic properties (dimensions, weights, etc.) were considered in the analysis here. Data from two “forward projections” and six “wrap trajectories” were used to study the Han-Brach mechanics model with the intent of assessing the ability of the model to 1) capture the dynamics of the pedestrian crash tests and, 2) establish ranges of the empirical parameters in the model (fp, α, θ and μ).

Consistent with the nature of the model, this assessment included the impact and the motions of the pedestrian and the vehicle following separation.

For the current testing and analysis, the input variables of the Han-Brach model were generally divided into three categories. Some variables were directly measured prior to or during the trial (e.g. mc, mp, and vc0), some variables were estimated based on video review (e.g. h, xL, and s0), and some variables were allowed to vary with upper-bound and lower-bound constraints and ultimately determined using Microsoft Excel’s Solver utility using a best fit procedure (e.g. a0, fp, α, θ and μ). The deceleration of the vehicle, a0, was constrained between 0.5 g and 0.9 g, and the pedestrian/road drag coefficient, fp, and impulse ratio of the pedestrian/road impact, μ, were both constrained between 0.3 and 1.0. Two additional variables had unique constraints based on the collision type. For “forward projections,” θ was assumed to be zero (i.e. the pedestrian center of mass is launched parallel to the roadway), and α was allowed to vary but constrained between 1.0 and 1.3 based on the original Han-Brach work.19 For “wrap” trajectories, α was assumed to be 1.0 (i.e. no “rebound effect”), and θ was allowed to vary but constrained to a range based on video review. It was also assumed that a0 was equal to -a0 (i.e. deceleration of the vehicle throughout braking at a constant

level) as this was the intent of the driver in each trial and was verified for each trial with observation of the video.

Thus, s1 was set equal to zero in all analyses. Additionally, the asphalt skid pad where the testing was conducted was essentially flat and level, so in all cases ϕ was set to zero.

To further guide the results based on the experimental data, some of the output variables of the Han-Brach model were also constrained and/or ultimately entered as parameters for which they could be solved. Based on video review, R was constrained within an appropriate 10 - 15 foot window. While it may seem counterintuitive that the value of R would not be uniquely determined for a given test using the video of the event, it was typical with all the trials that initial contact with the roadway after separation with the vehicle might occur in stages/phases. The first contact might involve one of the hands, then followed by a foot, before perhaps a hip (or some other part of the torso) contacts the ground. The main point here is that the model assumes that the impact of the pedestrian with the roadway occurs instantaneously in a singular location (thus defining R), whereas in the tests, which involve a dummy/manikin with articulated appendages, this interaction generally did not happen instantaneously. Thus, the model approximates this interaction.

The purpose of the Solver algorithm was to use the equations of the Han-Brach model to minimize a single value, Q, referred to as the quality function. The quality function, Q, is the sum of squares of the differences between the directly measured/assessed values of selected parameters and the model’s estimates (calculations) of these same parameters. The general form of the quality function, Q, used in the analysis is shown in Equation 12. With this minimization approach, the resulting fit between the model and the data is a “best fit” in the least squares sense.

(12) In this equation, n is the number of parameters to be used from the test data, wi is a weighting factor (which may be 1), ui is the parameter calculated using the pedestrian impact model and uidata is the corresponding value of the parameter determined from test measurements.

Review of the data available from the tests shows that many of the variables and parameters in the model (see Table 1) have corresponding experimental values and thus could be included in the fitting process as a term in the quality function. The general scheme associated with the analysis is to use the data acquired during testing to determine the input

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parameters that were not directly measurable (although perhaps can be bounded in the analysis).

The selection of the variables used for the quality function was based primarily on the level of uncertainty associated with the determination of the value. Values of parameters such as R and τp1 had considerable uncertainty and were deemed by the authors to be poor candidates for use in the quality function whereas values such as the distance the pedestrian moved from initial contact to rest, sp, and the time associated with this motion, τp, were physically measured post-crash (sp) or determined from the video (τp) with high accuracy and precision. These latter two parameters were deemed by the authors as good parameters for use in the quality function. Based on this rationale, four parameters were selected for use in the quality function in the analysis presented here. These four parameters were: sp, s0 + s1 + s2, τc, and τp.

The authors recognize that the use of the quality function with values of the uidata that are of different magnitudes can produce a fit that may better satisfy the values of larger magnitude. This disparity in the fitting can be mitigated by weighting factors. In the case of the selection of two times and two distances for the quality function used here, the distances have a magnitude that is roughly 10 times that of the value of the times used in the function. Thus, the value of the weighting, wi, of the time values (see Equation 12) were weighted by ten in the calculation of Q whereas the weighting of the distances was one. These weighting assignments were used throughout all the analyses.

Results Wrap Collisions The pedestrian throw distances (sp) and times (τp) calculated by the model for all the trials were all exact matches with the experimental data. The vehicle distances (s0 + s1 + s2) calculated by the model for all the trials were within one foot of the measured experimental data, and the calculated vehicle travel times (τc) were within 0.16 seconds of the measured values. The vehicle deceleration values (a0) generated through the model ranged between 0.64 and 0.75 g’s, the drag coefficients between the pedestrian and the roadway (fp) ranged between 0.38 and 0.66, and the impulse ratio (μ) ranged between 0.59 and 1.00. The launch angles (θ) ranged between 5° and 25°. The Q values for the six wrap impacts ranged between 0.59 and 3.01, considered good agreement between the Han-Brach model and the experimental data.

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Forward Projections Like the wrap collisions, the pedestrian distances (sp) and times (τp) calculated by the model were exact matches, the vehicle distances (s0 + s1 + s2) calculated by the model were within one foot of the measured experimental data, and the calculated vehicle travel times (τc) were within 0.06 seconds of the measured values. The vehicle deceleration values (a0) generated through the model were 0.69 g and 0.71 g, the drag coefficients between the pedestrian and the roadway (fp) were 0.49 and 0.64, and the impulse ratios (μ) were 0.79 and 0.80. The rebound coefficients (α) were 1.15 and 1.28. The Q values for the two forward projection impacts were 0.04 and 0.51, indicating good agreement between the Han-Brach model and the experimental data.

Comments on the values and ranges of the parameters resulting from the analysis are provided in the Discussion section following three example reconstructions. Table B.1 in Appendix B contains the numerical results of the analysis of each of the eight experimental trials analyzed.

Example Pedestrian Crash Reconstructions Three examples are presented that show the use of the Han-Brach Type II pedestrian throw model. The first example reconstruction shows the utility of the model to reconstruct the speed of the vehicle based on the rest positions of the vehicle and the pedestrian. This approach uses the motion of both the vehicle and the pedestrian, both of which are modeled in the Han-Brach Type II pedestrian throw model, to determine the preimpact speed of the vehicle. This topic has been examined elsewhere.21 The second example reconstruction employs the use of EDR data in the reconstruction of the crash. In this example, the determination of the speed of the vehicle at impact is not the goal as that is given by the data. Rather, it is the location of the pedestrian at the time of impact that is of interest. This location establishes whether the pedestrian was in the crosswalk. The third example is an application of the pedestrian throw model to determine the speed of a motorcycle approaching an intersection when it collides into the side of a minivan. Because of the crash, and the sudden deceleration of the motorcycle, the rider was launched into the air and over the vehicle. The rider subsequently hit the roadway and tumbled to rest. The authors are unaware of a complete treatment of the reconstruction of this type of crash using a pedestrian throw model in the literature.

Prior to getting into the details of the reconstruction, the authors present the assumptions to be met for this specific use of the reconstruction model. The examples presented here were analyzed using the implementation of the Han-

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Brach model with the development provided above in Equations 3 – 11.

Example 1: Pedestrian Crash Reconstruction from Rest Positions A vehicle-pedestrian wrap collision occurs on a roadway with both the speed of the vehicle and the throw distance being unknown. The point of impact is unknown but the rest positions of the vehicle and the pedestrian are known from the scene photos. Thus, it is known that the pedestrian traveled 20 feet farther than the vehicle. Additionally, it is known that the car decelerated uniformly from before impact to rest with the driver braking heavily from impact to rest. The parameters for the reconstruction are defined as: a0

0.9

deceleration of vehicle over distance s2 drag resistance coefficient of pedestrian over distance s height of pedestrian center of gravity at launch, t0 distance of travel of vehicle at uniform speed x-distance of pedestrian from initial contact to launch ratio of pedestrian speed to vehicle speed at time of launch angle of launch of pedestrian relative to road (x-axis) road grade angle impulse ratio for pedestrianground impact

fp

0.8

h

4.0 ft

s1

0.0 ft

xL

2.0 ft

α

1.0

θ

5.0°

ϕ μ

0.0° 0.8 (= fp)

mc

93.24 lb-s2/ft

mass of vehicle, weight∕g

mp

5.44 lb-s2/ft

mass of pedestrian, weight ∕g

Using these values, the speed of the vehicle that causes the distance d (see Figure 1) to go to -20 feet can be calculated iteratively or using optimization methods. For these parameter values the speed of the vehicle is 66.6 ft/s (45 mph).

Example 2: Pedestrian Crash Reconstruction Using EDR Data In this example, the EDR on the pickup truck was imaged shortly after the crash. Data associated with a Non-deployment event was stored in the module. Based on the key cycle count and the characteristics of the data, it was determined that the non-deployment event stored in the EDR

was associated with the pedestrian impact. It is interesting to note that the collision with the pedestrian did register as a non-deployment event on this module.

Two pieces of useful information were obtained from the data: 1) the speed of the vehicle at 1 second prior to algorithm enable (AE) was 29 mph, and 2) the Maximum SDM (Sensing Diagnostic Module) Recorded Velocity Change was 1.03 mph. Additionally, the data show that the speed of the vehicle at 2 seconds before AE was also 29 mph and that the brake pedal was never applied prior to 1 second before AE. Thus, for the reconstruction, the speed of the vehicle at the time of impact with the pedestrian was taken to be 29 mph. Figure 2 is a scale diagram that shows the roadway environment where the crash occurred. The rest position of the pedestrian, as shown in the diagram, is known from measurements made by the police during their scene investigation.

The reconstruction of the location of the pedestrian relative to the crosswalk at the time of the crash is a two-step process. The first step is another level of validation of the data, and that is to assess further whether the data belong to the crash using the Maximum SDM Recorded Velocity Change. Note that this value is a function of the speed of the vehicle at the time of impact (which was extrapolated from the value recorded at -1 AE by the EDR) and the masses of the vehicle and the pedestrian. In this case, the combined weight of the vehicle and the driver was 3803 pounds and the weight of the pedestrian was 135 pounds.

Using conservation of momentum and the initial speed of the vehicle of 29 mph, the change in speed of the vehicle 0.99 mph. This value compares very favorably with the 1.03 mph ∆V reported by the EDR. The second step is to model the impact of the pedestrian with the initial speed of the vehicle at 29 mph to determine the throw distance. Using a value of fp = 0.65, the throw distance sp is calculated to be 40.6 feet (see Figure 2). The reconstruction with these values shows that the pedestrian was within the crosswalk striping at the time of impact.

A check of this solution using one or more Type I models is recommended. In this case, Woods Hybrid Model is used.27 This model provides the speed of the vehicle for a given throw distance. In this case, the throw distance used in the formula is the value reconstructed using the EDR data: 40.6 feet. This reconstructed throw distance is used with the Wood Hybrid equation to give vehicle speed.

This speed can be checked with the speed imaged from the EDR.

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Figure 2: Scale diagram showing the intersection where the crash occurred and the rest position of the pedestrian. One value of the throw distance determined in the reconstruction is also shown The Wood Hybrid Equation 27 is shown in Equation 13:

speed of the motorcycle at the time of impact and its speed prior to braking.

(13)

Before proceeding to the reconstruction of the speed of the motorcycle, consideration of this application of the method is needed. The authors are not aware of any formal treatment of the use of Type I or Type II pedestrian throw models as a method for the reconstruction of the speed of a motorcycle involved in a crash. The notion of the application of a pedestrian throw model applied to a motorcycle rider has been presented previously.17,26,30 However, while the title of the first paper mentions the application to motorcyclists and mentions it tangentially in the paper, there is no formal consideration of the application separate from the primary topic of the paper which is the application to vehicle/pedestrian impact. The last paper presents a hypothetical example of a motorcycle rider without consideration of the assumptions underpinning the application.

One treatment of the subject formally considers a vaulting rider, including data references for the range of launch angle from experimental data and considers the vault of the rider including the vault of a passenger, has been presented

by Baxter.31 The treatment presented here expands on that presented by Baxter by formalizing various of the concepts and utilizing the Han-Brach Type II model rather than a simple vault equation and throw-distance equation.

The quantity sp is the pedestrian throw distance and the parameter cW is assigned three values comprising a MIN, MAX and mean values for the vehicle speed, VCOL. For a throw distance of 40.6 feet, and the three parameter values for cW presented by Wood, the MIN speed is 19.7 mph, the MAX speed is 35.4 mph, and the mean speed is 28.3 mph.

The speed of the vehicle from the EDR data is 29 mph.

This EDR speed is in the MIN/MAX range predicted by Equation 13 and differs from the mean value by a little over 2%. This shows that the results using the Type II equation match nicely with this Type I equation. Additional work at this point in the reconstruction might be to examine the effect of the uncertainties in the reconstruction. This work is not presented here.

Example 3: Motorcycle Rider Crash Reconstruction Using Pedestrian Throw Model This example shows the breadth of the applicability of this Type II model. In this example, a crash involving a motorcycle colliding into the left front wheel of a minivan is reconstructed. Because of the sudden deceleration of the motorcycle to a complete stop, the rider separated from the motorcycle and was projected over the hood of the vehicle, landing on the roadway and tumbling to rest.

Figure 3 shows a scale diagram depicting the intersection where the crash occurred with several of the positions of the motorcycle and the rider shown. These positions were measured by the police during their crash investigation. A reconstruction of the crash was needed to determine the 16

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A necessary requirement for the application of a Type I or Type II model to motorcycle rider projection is that the motorcycle stops suddenly due to the impact and is accompanied by the projecting of the rider from the seat of the motorcycle. In this way, the motion of the rider is consistent with the kinematics and kinetics addressed by Type I and Type II pedestrian throw models. Dynamics of motorcycle/vehicle impacts in which the motorcycle is laid

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Figure 3: Scale diagram showing the intersection where the crash occurred and various positions relevant to the speed reconstruction down prior to impact, or the motorcycle swipes the front, side or rear of the vehicle such that the rider is deposited onto the roadway and slides to rest, are not consistent with the application of these models.

A second consideration relative to this application deals with the potential interaction between the rider and the motorcycle and/or the rider and the vehicle as the rider is being projected. Even if the assumption that the rider is projected is met, in general, the reconstructed speed using this application will be a lower bound only. This reconstructed speed is a bound as the model assumes that, as the rider is projected, the initial speed of the motorcycle (prior to impact) is the initial speed of the rider. However, the model does not account for any part of the rider (torso, leg, foot, etc.) interacting with either the motorcycle (fuel tank, handlebars, fairing, etc.) or the vehicle (hood, roof line, fender, etc.). Note that the evidence of this interaction may be very subtle and not readily noticeable to the reconstructionist. Any interaction between the rider and either vehicle would invariably lower the initial speed of the rider (prior to the unimpeded flight) and thus decrease the throw distance. As the reconstructed speed of the rider is the surrogate for the preimpact speed of the motorcycle, the speed of the motorcycle reconstructed using this method serves only as a lower bound; i.e. the actual speed may be higher depending on whether there was interaction or not (which is generally indeterminate). Additionally, it does not appear that this method has been considered in the literature for crashes in which the motorcycle has two riders. Uncertainty therefore exists in the application of this method to crashes with motorcycles involving more than one rider.

In this example, it is assumed that the post-impact motion of the motorcycle rider was consistent with the pedestrian flight motion modeled by the Han-Brach model. The physical evidence from the crash was sufficient to establish the position of the rider at the time of launch and the rest position of the rider. These two locations provided the throw distance, sp. Using Equation 3, the throw distance is R + s = sp as xL is zero in this instance. The physical evidence showed that this distance was 76.2 feet. In addition, the physical evidence also provided the values of the components of the throw distance, R and s. The interaction of the rider and the roadway, primarily due to visible scuffing from the helmet and the clothing, discernably indicated the tumble distance of the rider as 32.4 feet. Thus, the distance R was 43.8 feet (see Figure 3). These values of R and s, along with appropriate values of ϕ, µ, and α, were sufficient for the application. Here ϕ = 0.0 (the roadway was essentially flat and level). The value α = 1.0 as the rider was not directly propelled via momentum transfer by an impact from another vehicle and μ was constrained to equal to fp also based on prior work.19 In this application of the Equations 3 through 11, the motion of the vehicle and the momentum exchange was neglected. Only the flight and tumble of the pedestrian (but also including the impact with the pavement and the associated speed change) was used to reconstruct the initial speed of the rider. In addition to the unknown initial speed of the rider, the launch angle, θ, was not known.

Moreover, the drag of the pedestrian over the tumble distance s, fp, was also not known. A traditional strategy to reconstruct under these circumstances is to compute the speed of the motorcycle for appropriate ranges of these in-

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put variables. An alternative strategy used here was to fit the data to the model using a least squares approach using a quality function similar to Equation 12. This approach to reconstructing crashes has been presented previously.24 In this instance, the data were fit to the model such that the values of vc0, fp and θ were calculated and that the model matched the values of s and R in a least squares sense using an appropriate quality function (see Equation 12).

This reconstruction process gave the following results: vc0 = 53.9 ft/s (36.8 mph), fp = 0.76, and θ = 9.9°. An initial check of these results with available data shows that these values seem realistic. The value of fp was on the high end of the range of values seen from experiments.25 An additional check of this solution using alternative approach is recommended. This result could be checked using a Type I model to assess the efficacy of this application. While this process is used frequently by the authors as shown in Example 2, a different approach was used in this situation. In addition to the physical evidence observed on the roadway, the minivan was equipped with an EDR. This EDR was imaged during the vehicle inspection and, in addition to other data, provided a lateral ∆V from the impact with the motorcycle. In this case, the impact between the motorcycle and the minivan was modeled using planar impact mechanics 22 to assess whether the ∆V for the reconstructed impact speed of the motorcycle was close to the value contained in the imaged data. The values were very close, thus providing a second method indicating confidence that the results of the reconstruction are correct.

The reconstruction of the speed at the start of the skid mark (see Figure 3) was also needed. In this case, the mark on the roadway was approximately 41.5 feet long. The mark was attributed to skidding of the rear wheel only. Assuming there was no speed loss of the motorcycle between the end of the skid mark and impact (approximately 44 feet) and that the frictional drag was f = 0.35 for rear wheel braking only, the speed of the motorcycle at the start of the skid mark was 62.4 ft/s (42.6 mph). Any contribution from the front brake over the distance from the start of the skid mark to impact or from the rear brake between the end of the skid mark and impact would increase the speed above the reconstructed value 42.6 mph.

sented in this paper, and previous papers 19,21 demonstrate that the model does indeed capture the dynamics of vehicle pedestrian impacts and their associated postimpact motions. Thus, the model can be used to reconstruct crashes, subject to the assumptions of the model.

A few observations and comments are made here related to the results of the model-fitting analysis. Comments of the range for the parameters determined in this analysis relative to previous analysis 19 are included: •

The range of launch angles for the six wrap trajectory impacts was 5° to 25°. This range closely matches the range from the prior analysis and presented in other literature.31 While the launch angles of the manikins in the tests were not measured, this range agrees well with the estimates made through observation by the authors of the videos.

•

The resulting average frictional drag of the pedestrian fp from the fitting process was fp = 0.53 with a standard deviation of 0.09. The range consisted of a minimum of 0.38 and a maximum of 0.66. The previous study constrained the value in the fitting process to between 0.7 and 0.8 with the fitting process generally taking on one of the extremes of the range. The parameter was unconstrained in the analysis presented here to assess whether the values produced via the least squares best fit process resulted in a reasonable average value.

While the average value of fp = 0.53 from the analyses here is lower than the constrained range utilized in the previous study, it is consistent with previously published data.15,25 The resulting range from the data here, 0.38 ≤ fp ≤ 0.66, is generally lower than the range of 0.55 ≤ fp ≤ 0.9 reported previously 19 and based on data from another publication.29 Additional experimental data will help refine the understanding of this parameter and its dependence on factors such as clothing and biofidelity of the manikin.

•

In previous studies, the value of μ was assumed to be the same as the pedestrian sliding drag factor, fp. In this analysis, the value of μ was allowed to vary independent from fp in the fitting process. This approach was used to gain insight into the behavior of this parameter. In all but one case, the value of μ was greater than the value of fp. The range of the ratio μ/fp is 0.9 to 2.2 with an average of 1.5 and a standard deviation of 0.5.

•

The value of α was kept at 1.0 for the wrap trajectory trials, consistent with the previous analysis.19 It was allowed to vary for the two forward projection trials.

The two values for the parameter from the two trials were 1.3 and 1.1. These values are consistent with the

Discussion The utility of the Han-Brach Type II pedestrian impact model as a reconstruction tool relies on two main points: 1) the ability of the model to capture the kinematics and kinetics of vehicle-pedestrian collisions, and 2) guidance regarding the range of each of the individual empirical parameters. These ranges are needed to apply the model to reliably reconstruct real-world crashes. The analyses pre18

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values from the previous study which ranged from 1.2 to 1.3.19 It is well-understood that vehicle crash investigators are often faced with trying to reconstruct vehicle-pedestrian accidents with limited data. Many of the models currently employed by these reconstructionists are not sufficiently robust to account for individual parameters that can significantly affect their calculations or are wholly dependent on variables that are unknown for a given crash. The HanBrach Type II mechanics model attempts to curtail some of those issues by accounting for many more variables in a vehicle-pedestrian crash than just the vehicle’s impact speed and the pedestrian throw distance as is done in the Type I models. Additionally, the model goes further than previous Type II models 18,26 that do not consider the combined motion of the vehicle and the pedestrian or does not include the impacts (vehicle-pedestrian and/or pedestrianroadway) in the development. The purpose of the current research was to further validate the Han-Brach model with a detailed data set of realistic vehicle-pedestrian impacts.

The application of the Han-Brach model to the ARCCSI vehicle-pedestrian impact test data was a successful demonstration of the model’s ability to accurately reflect real-world data. The modeled data matched extremely well with highly reliable, measured data. The analysis provided refined ranges for several other parameters such as vehicle deceleration, drag coefficients, and pedestrian launch angles that are often unknown and/or difficult to measure directly in testing. The ability to use these values along with known physical evidence and reasonable estimates of other parameters provides users with a powerful, reliable tool to perform accurate reconstructions of vehicle-pedestrian crashes. This utility was further demonstrated with the examples of real-world accident reconstructions.

The authors acknowledge the contributions of all the volunteers who participated in the pedestrian impact testing at the ARC-CSI Conference in Las Vegas in September 2017. Without their contributions to the various aspects of the testing (instrumentation, dummy preparation and repair, etc.) the data used in the analysis presented in this paper would not have been possible. Special thanks are given to a couple of individuals who contributed to this paper or the testing: Mike DiTallo, and Danny Phillips.

References 1. Brach, Raymond M., "Impulse-Momentum Analysis of Multibody Vehicle-Pedestrian Collision Simulations", presented at the 25th Annual EVU Congress, 2016, Bratislava, Slovakia.

2. Ravani, B., Brougham, D., and Mason, R., "Pedestrian Post-Impact Kinematics and Injury Patterns," SAE Technical Paper 811024, 1981, https://doi. org/10.4271/811024.

3. Happer, A., Araszewski, M., Toor, A., Overgaard, R. et al., "Comprehensive Analysis Method for Vehicle/ Pedestrian Collisions," SAE Technical Paper 2000-010846, 2000, https://doi.org/10.4271/2000-01-0846.

4. Toor, A., Araszewski, M., Johal, R., Overgaard, R. et al., "Revision and Validation of Vehicle/Pedestrian Collision Analysis Method," SAE Technical Paper 2002-01-0550, 2002, https://doi.org/10.4271/200201-0550.

5. Toor, A. and Araszewski, M., "Theoretical vs. Empirical Solutions for Vehicle/Pedestrian Collisions," SAE Technical Paper 2003-01-0883, 2003, https://doi. org/10.4271/2003-01-0883.

6. Russell, Gregory, Pedestrian Formulas for Excel, Lawyers and Judges Publishing Company, Inc.

It should be recognized that not all vehicle-pedestrian impacts can be effectively reconstructed with Type II models such as the Han-Brach model. A prime example of this is when the specific movements of the pedestrian need to be modeled. In these situations, users may wish to pursue Type III models when appropriate. Type I models, which generally require the user to know the throw distance, provide utility in checking the reconstruction results of Type II and Type III models. Lastly, the authors encourage other researchers to further study the Han-Brach model by employing different combinations of pedestrian dimensions, impact speeds and orientations, and braking responses.

Additional well-documented experimental data for the value of fp, under various conditions, will assist in the application of this method.

7. MADYMO, com/madymo

https://tass.plm.automation.siemens.

Acknowledgments

12. Moser, A., Hoschopf, H., Steffan, H., and Kasanický, G., "Validation of the PC-Crash Pedestrian Model,"

8. PC-Crash, http://www.pc-crash.com/ 9. Engineering Dynamics Corporation, http://www.edccorp.com/products/gatb.html 10. Hamacher, M., L. Eckstein and R. Paas, “Vehicle Related Influence of Post-Car Impact Pedestrian Kinematics on Secondary Impact”, IRC-12-78, IRCOBI Conference 2012.

11. Moser, A., Steffan, H., and Kasanický, G., "The Pedestrian Model in PC-Crash - The Introduction of a Multi Body System and its Validation," SAE Technical Paper 1999-01-0445, 1999, https://doi.org/10.4271/199901-0445.

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SAE Technical Paper 2000-01-0847, 2000, https:// doi.org/10.4271/2000-01-0847.

13. Day, T., "An Overview of the HVE Human Model," SAE Technical Paper 950659, 1995, https://doi. org/10.4271/950659.

14. Depriester, J-P, C. Perrin, T. Serre, S. Chalandon, “Comparison of Several Methods for Real Pedestrian Accident Reconstruction,” Proc. Int. Tech. Conf. Enhanced Safety Vehicles 2005; 14p.

15. Stevenson, T. J., Simulation of Vehicle-Pedestrian Interaction, doctoral dissertation, 2006, https:// ir.canterbury.ac.nz/handle/10092/1180.

27. Wood, Denis P. and C. K. Simms, (2000) A hybrid model for pedestrian impact and projection, International Journal of Crashworthiness, 5:4, 393-404, DOI: 10.1533/cras.2000.0150.

28. Aronberg, R. and Snider, A., "Reconstruction of Automobile/Pedestrian Accidents Using CATAPULT," SAE Technical Paper 940924, 1994, https://doi. org/10.4271/940924.

29. Hill, G. S., Calculations of Vehicle Speed from Pedestrian Throw, Impact, J. Inst. Traffic Accident Investigation, Spring 1994.

16. Collins, J. C., Accident Reconstruction, Charles C. Thomas, Springfield, IL, 1979.

30. Obenski, Kenneth S., et al., Motorcycle Accident Reconstruction and Litigation, Lawyers and Judges Publishing Company, Inc. Tucson, AZ, 2011.

17. Searle, J. and Searle, A., "The Trajectories of Pedestrians, Motorcycles, Motorcyclists, etc., Following a Road Accident," SAE Technical Paper 831622, 1983, https://doi.org/10.4271/831622.

31. Baxter, Albert T., Motorcycle Accident Investigation, 2nd Edition (revised), Institute of Police Technology and Management, Jacksonville, Florida, 1997.

Appendix A

18. Searle, J., "The Physics of Throw Distance in Accident Reconstruction," SAE Technical Paper 930659, 1993, https://doi.org/10.4271/930659.

Previously presented in this paper is the equation for the speed change along the roadway (∆Vx) of the pedestrian due to the impact between the pedestrian and the roadway.

This speed change is needed so that the velocity of the pedestrian along the roadway at the start of the slide distance can be calculated. That equation is:

19. Han, Inhwan and Raymond M. Brach, "Throw Model for Frontal Pedestrian Collisions," SAE Technical Paper 2001-01-0898, 2001, https://doi.org/10.4271/200101-0898.

20. Evans, A.K. and R. Smith, “Vehicle speed calculation from pedestrian throw distance,” IMechE, Proc Instn Mech Engrs, Vol 213, Prt D, 1999.

21. Brach, Raymond M., "Reconstruction of VehiclePedestrian Collisions Including an Unknown Point of Impact," SAE Technical Paper 2015-01-1419, 2015, https://doi.org/10.4271/2015-01-1419.

22. Brach, Raymond M. and R. Matthew Brach, Vehicle Accident Analysis and Reconstruction Methods, 2nd Edition, Publication R-397, SAE International, Warrendale, PA, 2011.

23. Kazanický, Gustav and Pavol Kohút, New Partial Overlap Pedestrian Impact Tests, presented at the 18th Annual EVU Congress, Hinckley (UK), 2009.

24. Brach, R. Matthew, Brach, Raymond M., and Mink, Richard, "Nonlinear Optimization in Vehicular Crash Reconstruction," SAE Int. J. Trans. Safety 3(1):17-27, 2015, https://doi.org/10.4271/2015-01-1433.

25. Wood, Denis, and Ciaran Simms, Coefficient of Friction in Pedestrian Throw, Impact – Journal of ATAI, vol 9 no 1, p12-14, January 2000.

26. Aronberg, R., "Airborne Trajectory Analysis Derivation for Use in Accident Reconstruction," SAE Technical Paper 900367, 1990, https://doi. org/10.4271/900367.

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(A.1) The form of this equation may seem rather terse and its source not obvious. As will be shown here, it follows directly from point mass impact dynamics.22 The equations for the velocity change in the tangential direction for mass 1 of a collision involving two point masses is: (A.2) Where:

.

In this case, m2 is the roadway, i.e. the earth. Thus, m2 ≈ ∞ and v2n = 0. Moreover, the value of can be found using the equation above and the notion of a limit for m2 → ∞.

This gives the result that = m1. Going back to Equation A.2 with v2n = 0 and = m1, and that the assumption here is the impact is inelastic (e = 0, no rebound), Equation A.2 becomes: (A.3) This equation has the same form as Equation 10 (repeated in Equation A.1).

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Appendix B S4 S8 Input Information (Knowns) Wrap Trajectory a0 0.805 0.719 a2 0.805 0.719 fp 0.517 0.548 2 g (ft/s ) 32.174 32.174 h (ft) 5.000 3.500 s1 0.000 0.000 vc0 (ft/s) 50.160 49.867 vc0 (mph) 34.200 34.000 xL 10.000 5.000 α 1.000 1.000 θ 25.075 15.996 ϕ 0.000 0.000 μ 0.635 0.684 mc 121.029 86.871 mp 4.600 1.834 s0 14.000 7.000 Output Information (Unknowns)

M4

M5

0.747 0.747 0.528 32.174 3.330 0.000 49.779 33.940 6.000 1.000 14.957 0.000 0.941 95.077 5.253 8.000

0.668 0.668 0.383 32.174 5.000 0.000 47.667 32.500 10.000 1.000 25.075 0.000 0.793 83.919 4.724 13.500

v"c0 40.764 45.544 vp0 40.764 45.544 R 32.893 43.700 tp1 1.122 1.143 s 23.407 26.200 sp 66.300 74.900 tp 2.800 2.867 tc1 0.303 0.145 s2 32.061 44.829 s0+s1+s2 46.061 51.829 tc 1.936 2.155 d -20.239 -23.071 t0 0.303 0.145 Calculated Parameter Values sp 66.300 74.900 s0+s1+s2 46.500 52.200 tp 2.800 2.867 tc 1.833 2.067 Q 1.238 0.925

43.356 43.356 38.556 1.088 17.744 62.300 2.533 0.167 39.086 47.086 2.070 -15.214 0.167 62.300 47.400 2.533 2.000 0.590

M6

M7

S2a

0.638 0.638 0.664 32.174 2.000 0.000 46.200 31.500 5.000 1.000 4.998 0.000 0.592 82.054 0.684 5.000

0.680 0.680 0.463 32.174 3.250 0.000 50.893 34.700 5.000 1.000 14.485 0.000 0.999 82.054 4.724 7.500

Forward Proj 0.706 0.689 0.706 0.689 0.486 0.638 32.174 32.174 3.000 2.333 0.000 0.000 41.844 46.875 28.530 31.960 9.000 7.500 1.281 1.145 0.000 0.000 0.000 0.000 0.786 0.797 154.068 114.533 4.600 0.684 8.000 7.500

38.936 38.936 44.804 1.574 10.296 65.100 2.867 0.304 35.253 48.753 2.217 -16.347 0.304

43.558 43.558 21.252 0.601 30.848 57.100 2.300 0.111 46.189 51.189 2.249 -5.911 0.111

44.970 44.970 40.015 1.071 21.385 66.400 2.767 0.152 46.202 53.702 2.325 -12.698 0.152

36.167 46.313 20.000 0.634 40.100 69.100 2.900 0.202 28.778 36.778 1.841 -32.322 0.202

42.927 49.143 18.716 0.547 37.784 64.000 2.467 0.167 41.576 49.078 2.115 -14.922 0.167

65.100 49.300 2.867 2.100 1.664

57.100 51.700 2.300 2.133 1.690

66.400 54.400 2.767 2.167 3.009

69.100 37.100 2.900 1.778 0.506

64.000 49.000 2.467 2.133 0.039

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Gasoline-Powered Golf Cart Acceleration and Hard Braking Performance Robert D Anderson and Michael Rosenfield

I

ntroduction Low speed vehicles (LSV) have been thought of as a low-cost, eco-friendly way to tool around in locations where there would be little interaction with larger vehicles. As of 2010, practically every state allowed LSVs on roadways with speed limits up to 35 mph.1 LSV use on public roads experienced rapid growth since the phenomena began in resort and retirement communities in the 1990’s.2 The standard golf course cart is typically limited by speed governors or gearing to a top speed of 15 mph.

Many LSVs are essentially souped-up golf carts.1 An LSV typically has a top speed between 20 and 25 mph, a limited GVWR, and has additional equipment to make it street legal, such as lights, signals, mirrors, windshield, seat belts and a horn.3 Unlike passenger motor vehicles, LSVs are not required to meet Federal Motor Vehicle Safety Standards,4 and golf cart-related injuries have steadily increased with the increased popularity and growing capabilities of these vehicles.5 A compilation of electric golf cart and scooter acceleration, hard brake, stability testing and specifications was presented in Volume 12, Issue 1 of Collision.6 This current study is intended to add to the existing published information by providing acceleration and braking performance, and specifications of gasoline-powered golf carts.

The EZ-GO Marathon is a golf cart that is representative of the type found at a golf course. The EZ-GO RXV’s used in this testing were advertised as street legal and they were equipped with headlights, tail lights, turn signals, and a horn, but they did not have a windshield, mirrors or seat belts installed. All test vehicles (Table 1) were gasolinepowered.

Model Year 1990 2009 2009

Make and Model EZ-GO Marathon EZ-GO RXV EZ-GO RXV

Wheels and Tires Stock Stock Over-size

Table 1: Test Vehicles

Instrumentation Test vehicles were instrumented with Racelogic data loggers VBOX 3i, Video VBOX Lite, and VBOX Sport (Fig22

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ure 1). These data loggers measure and record vehicle speed using the Doppler shift of GPS satellite signals.

The sample rate for the VBOX 3i, Sport and Video Lite is 100Hz, 20Hz and 10Hz, respectively. For all three data loggers, the Racelogic reported accuracy for acceleration is ±1% or better, distance traveled ± 0.05%, and speed ± 0.13 mph or better when smoothed with a moving average of 4 samples.7,8,9 The VBOX Lite has the additional ability to record video from up to two cameras allowing a low-resolution image to be viewed as a picture within a picture of a high-resolution image. The video may also be overlaid with virtual gauges, such as a speedometer. The VBOX Sport is waterproof and can be used with its internal GPS antenna or with an external GPS antenna.

Vehicle weights were measured with Intercomp Microflex (Micrflex scales) or Intercomp SW scales (SW Scales). The Microflex scales have a 4,000 pound per wheel capacity and the SW scales have a 1500 per wheel capacity. The Intercomp scales report weight in 1-pound increments with a 0.1% accuracy.

EZ-GO Marathon The 1990 EZ-GO Marathon is golf course-style golf cart.

It had been traveling at top speed in a neighborhood when it rear-ended an occupied stationary minivan parked in the shadows. It was not equipped with seat belts, turn signals, or a horn (Figure 2). The gasoline engine is under the seat, and powers the rear wheels through a CVT belt-drive transmission, with forward and reverse gear selection. Brakes are mechanical drum brakes on the rear wheels. The specifications for the EZGO Marathon are shown in Table 2.

Like the electric-powered version, the EZGO Marathon Golf Cart is equipped with rack and pinion steering. The steering wheel had approximate 1½ turns in each direction, which produced approximately 45 degrees of wheel turn in each direction so that the steering gear ratio was approximately 1:12.

The EZGO golf cart was instrumented with a Video VBOX Lite and a VBOX Sport. The weight was measured to be 794 pounds using SW scales. Performance was dem-

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Figure 1: VBOX 3i, VBOX Sport, and Video VBOX Lite (left to right) onstrated through a series of hard accelerations from a stop and hard braking (skidding) to a stop. All runs were conducted on level asphalt and unpaved clay roadways with a 200 pound driver. The engine did not idle, so the engine stops running when the accelerator is released. When depressing the accelerator pedal, there was a delay associated with the engine/CVT transmission startup that was not measured on this vehicle.

A top speed of about 14 mph was achieved in about 7 seconds and 90 feet with a 200 pound driver (Figure 3, Figure 4, and Figure 5). Acceleration peaked at about 1.4 seconds and ranged from 0.14 to 0.19 gs and then dropped quickly (Figure 6). The average acceleration to top speed was 0.075 gs.

Hard braking with rear wheel skidding on the asphalt roadway produced accelerations of about -0.30 to -0.34 gs. Hard braking with rear wheel skidding on the unpaved clay roadway produced accelerations of about -0.25 gs. Deceleration while coasting was measured to be about 0.07 gs. The brakes could overpower full accelerator pedal application so that simultaneous hard braking and full accelerator pedal application was indistinguishable from braking alone.

The 2009 EZGO RXV carts were for sale at a dealership and they were advertised as street legal. The carts had street legal equipment including head lights, horn, tail lights, brake lights, and turn signals, but did not have windshields, mirrors or seat belts installed (Figure 7). Both were equipped with rear-facing rear seats. The white cart was equipped with stock 8-inch tires and the red cart was equipped with 12-inch tires. The engine is under the seat, and powers the rear wheels through a CVT belt-drive transmission, with

EZGO RXV

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Collision Magazine - Volume 14 Issue 1 23

Figure 2: EZGO Marathon FAW RAW WT FOH WB ROH OL OW OH TW Front TW Rear Axle Ht Tire Steering Ratio Turning Diameter SN: 571337

297 497 794 8.4 65 19.6 93 44.5 66 33.5 39.5 8.4 18x8.5-8 1:12 17

wheel turn in each direction for a steering gear ratio was approximately 1:12.

pounds pounds pounds inches inches inches inches inches inches inches inches inches

The EZGO RXV golf carts were instrumented with the Racelogic VBOX 3i and IMU04 set with integrated Kalman filter. It was weighed with the Microflex scales. A tape switch was used alternately on the gas and brake pedals to mark the beginning of accelerator or brake pedal application. Performance was demonstrated through a series of hard accelerations from a stop and hard braking (skidding) to a stop. All runs were conducted on a level asphalt residential street, with a 235-pound driver and a 167-pound passenger.

feet

Table 2: 1990 EZGO Marathon Specifications forward and reverse gear selection. Brakes are mechanical drum brakes on the rear wheels. The specifications for box the EZGO RXV carts are shown in Table 3.

The EZGO Carts are equipped with rack and pinion steering. The steering wheel had approximate 1½ turns in each direction, which produced approximately 45 degrees of

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The engine does not idle. When the gas pedal is released, the engine stops. When the gas pedal is depressed, the engine starts and the CVT transmission engages. Consequently, there is a delay between depressing the pedal and the cart actually moving. This delay varied from 0.8 to 1.8 seconds, depending on how fast the engine starts, and included the time to depress the accelerator pedal. The greatest delay was with the cart equipped with the larger tires. During the testing that vehicle stopped running and could not be started because the battery died.

Hard braking produced rear wheel skidding with accelerations of -0.4 gs (stock tires) and -0.37 gs (larger tires). The brakes could overpower full accelerator pedal application so that simultaneous hard braking and full accelerator pedal application was indistinguishable from braking alone.

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Figure 3: EZGO Marathon Speed vs Distance

Figure 4: EZGO Marathon Speed vs Time www.collisionpublishing.com

Collision Magazine - Volume 14 Issue 1 25

Figure 5: EZGO Marathon Distance vs Time

Figure 6: EZGO Marathon Acceleration vs Time 26

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Figure 7: EZGO RXV with stock and larger tires (white left, red right) FAW RAW WT FOH WB ROH OL OW OH TW Front TW Rear Axle Ht Tire

264 & 280 578 & 609 842 & 889 9.5 66 38 113.5 47.5 38 35 & 35¾ 38 & 38¾ 8⅜ & 10⅜ 18x8.5-8 & 215/40-12 Steering Ratio 1:12 Turning Diameter 17.5 & 18 SN: 5048099 & 5047007

pounds pounds pounds inches inches inches inches inches inches inches inches inches

feet

Table 3: 2009 EZGO RXV Specifications The top speed was measured to be about 12 mph and 18 mph with the stock and larger tires, respectively (Figure 8 and Figure 9). With stock tires, the acceleration peaked within the range of 0.22 to 0.26 gs at about 1.5 seconds.

For the larger tires, the max acceleration was about 0.2 gs and achieved later at about 1.8 seconds, but a higher speed was obtained (Figure 8 through Figure 11). The average acceleration to top speed was about 0.1 gs for stock tires and 0.13 gs for larger tires (Figure 11).

Discussion What is street legal varies from state to state, city to city and even from community to community. In Arizona, golf carts and other LSVs are allowed on streets with speed limits less than 35 mph, they must not be manufactured or

modified to go faster than 25 mph, and they are required to have head, tail and brake lights, a windshield, reflectors, a horn, license plate light, and a mirror if the vehicle is constructed or loaded to obstruct the driver’s view to the rear.10 Some states require that LSV be able to have top speed of at least 20 mph and have additional equipment such as turn signals and seat belts.3 With the windshields installed, the RXVs in this study would be considered street legal in Arizona, but because the carts could not achieve a speed of at least 20 mph and were not equipped with seat belts or turn signals, the carts would not be legal in Florida, for example.

The acceleration, hard brake, and specifications measured for gasoline-powered golf carts are presented to expand the available body of information regarding golf cart LSVs. Compared to the electric-powered golf carts, the gasolinepowered golf carts are essentially the same vehicle with a different power source. It should not be a surprise that nearly identical brake performance on asphalt was demonstrated. As a relatively constant deceleration in all tests was observed during hard braking and/or skidding a constant deceleration can be used to represent hard braking, and skidding.

Like their electric-powered counterparts, gasoline-powered golf carts exhibited a non-constant acceleration with peak accelerations more than twice the average accelerations.

Therefore average acceleration would not represent hard accelerations as accurately as a time or distance-dependent relationship. The graphs provided may be used to represent the time dependent relationship between speed, distance, and time during hard accelerations.

A higher top speed, subtle degradation of performance during braking and lower initial peak acceleration was observed with the larger tires on the EZGO RXV (Figure 11). The effect of tire size on top speed, braking, and acceleration performance was also observed during electrical golf cart testing.6

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Collision Magazine - Volume 14 Issue 1 27

Figure 8: EZGO RXV Speed vs Distance with both tire sizes

Figure 9: EZGO RXV Speed vs Time with both tire sizes 28

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Figure 10: EZGO RXV Distance vs Time with both tire sizes

Figure 11: EZGO RXV Acceleration vs Time with both tire sizes www.collisionpublishing.com

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The brakes of the gasoline-powered golf carts mechanically over-powered the throttle so that simultaneous brake and accelerator application is similar to just braking. This was the same for the electrical golf carts, with the possible exception of the vehicles where the brake signal over-rode the accelerator signal electronically within the speed controller.

sults 6 may be used as an overall guide or indicator for the range of acceleration performance of golf carts in general, in terms of speed vs distance (Figure 15), speed vs time (Figure 16), and distance vs time (Figure 17).

One difference between electric and gasoline-powered golf carts is a perceptible additional delay between applying the accelerator pedal and the beginning of vehicle movement.

This delay is likely due to the time required for the engine to start and engage the belts of the CVT transmission. The greatest delays were measured on the RXV whose battery died during testing. A weak and/or dying battery provides a plausible explanation for the slow starting and prolonged start-up of that vehicle.

CVT FAW FOH GPS LSV OH OL OW RAW ROH TW SN WB WT

Comparison between the speed vs time relationships relative to the start of movement for the Marathon and those relative to the beginning of accelerator pedal application measured for the RXVs, differ by a quarter second. Therefore, the time to depress the accelerator pedal was about a quarter second, which leaves the other approximate half second for engine and CVT start-up. Otherwise, time-distance during acceleration for gasoline-powered golf carts is comparable to that observed for their electric-powered counterparts (Figure 15 and Figure 16).

Summary and Conclusions A golf cart’s equipment can affect performance as demonstrated by changes in top speed, braking performance, and acceleration with tire size.

Constant deceleration can be used to represent hard braking and/or skidding of golf carts. The gasoline-powered golf carts exhibited essentially the same deceleration during hard braking-skidding as their electric-powered counterparts.

The combined results for the test vehicles in this study may be used as a guide or indicator for the representative performance of gasoline-powered golf carts during acceleration in terms of speed vs distance (Figure 12), speed vs time (Figure 13), and distance vs time (Figure 14).

Compared to electric-powered golf carts, the vehicles in this study exhibited an additional delay of about half a second from accelerator pedal application to the start of vehicle movement due to the time required to start-up the engine and CVT transmission.

When little information is known about the golf cart of interest, the results for the test vehicles in this study combined with the previous electric-powered golf cart test re30

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Glossary Continuously Variable Transmission Front Axle Weight Front Overhang Global Positioning System Low speed vehicle Overall Height Overall Length Overall Width Rear Axle Weight Rear Overhang Track Width Serial Number Wheel Base Weight Total

References 1. “Low-Speed Vehicles Aren’t Crashworthy, New Tests Show,” Insurance Institute for Highway Safety, Highway Loss Data Institute, Status Report, Vol 45, No 5, May 2010.

2. Long TJ, Fugger TJ & Randles B, “Vehicle Performance Characteristics and Seat Belt Effectiveness in Low Speed Vehicles and Golf Cars,” 19th International Technical Conference on the Enhanced Safety of Vehicles (ESV), Paper No 05-0431, June 2005.

3. Florida Dept. of Safety and Motor Vehicles: Guide to Owning Low Speed Vehicles.

4. Dake J, “What is a Low Speed Vehicle (LSV)? Is it a Golf Cart?” Dirt Legal, June 18, 2018.

5. Watson DS, Mehan TJ, Smith GA & McKenzi LB, “Golf Cart – Related Injuries in the U.S.” American Journal of Preventative Medicine, Vol 35, No 1, pgs.

55-59, July 2008.

6. Anderson RD, Rosenfield MJ & Anderson RL, “Electric Golf Cart and Scooter Acceleration, Hard Brake and Stability Testing,” Collision, Vol. 12, No. 1, 2018.

7. Racelogic VBOX 3i 100Hz GPS Data Logger data sheet, 3/11/09.

8. Racelogic VBOX Sport data sheet, 4/2/14.

9. Racelogic Video VBOX Lite (RLVBVD10LT) data sheet, 3/20/2012.

10. Golf Cart Information, Town of Queen Creek, Arizona.

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Figure 12: Combined Speed vs Distance for all Gasoline Carts

Figure 13: Combined Speed vs Time for all Gasoline Carts www.collisionpublishing.com

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Figure 14: Combined Distance vs Time for all Gasoline Carts

Figure 15: Combined Speed vs Distance for Gasoline and Electric Golf Carts 32

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Figure 16: Combined Speed vs Time for Gasoline and Electric Golf Carts

Figure 17: Combined Distance vs Time for Gasoline and Electric Golf Carts www.collisionpublishing.com

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A REVIEW OF ENGINEERING MECHANICS APPLIED TO ACCIDENT RECONSTRUCTION Jai Singh, PART 1: KINEMATICS BS, MS, MA, ACTAR Biomechanical Engineering Analysis & Research, Inc.

A

bstract The subject paper is the first in a series of three papers detailing the application of engineering mechanics, specifically rigid body dynamics, to the field of accident reconstruction. The focus of the subject paper is the development and presentation of the solutions for rigid body kinematics. This presentation is within a framework that explicitly notes the expression frame for each vector quantity along with the derivative frame, when applicable. The two reference frame problem (i.e. a body fixed frame of reference for a rigid body undergoing free motion with respect to an inertial frame of reference)

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serves as the primary context. In this regard, the presentation proceeds by first stating the two point position vector problem and by defining the direction cosine matrix. The latter is one of the most general methods for representing a change in orientation kinematics. The frame referenced first time derivative is presented and contrasted against the standard non-frame referenced time derivative approach.

An important result from the presentation is the derivative transport theorem. This is followed by the presentation of the frame referenced second time derivative. Finally, the solutions for the first and second frame referenced time derivatives are presented for the multibody problem.

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Introduction Engineering mechanics, and specifically dynamics, plays a central and critical role in a large number of typically encountered tasks in the fields of accident reconstruction and biomechanics. This includes, but is not limited to, modeling and understanding vehicle dynamics 16-17, 23, 38, 42 , modeling collision phase responses 2-15, 20, 26-27, 34, 39-40, 44-46, modeling and understanding occupant kinematics 41 and using instrumentation (e.g. accelerometer) data. 1, 19, 33, 35-37, 43 The mathematics that underpins each of these broad contextual areas of application also applies to common tasks that can readily be considered as being preparatory for performing an accident reconstruction. As a simple example in regards to the last statement, consider a situation in which the start and end of a linear tire mark have been measured at the scene of an incident and with the measurements taken with respect to perpendicularly aligned curbs or curbline prolongations. The vector from the start of the tire mark (ra) to the end of the tire mark (rb) is defined as rb – ra = rb|a. The referenced curblines or their prolongations form a planar (i.e. R2) frame of reference with origin located at the intersection of the referenced curblines or their prolongations and with each curbline or prolongation representing a coordinate axis. Suppose, now, that this information is imported into a software package that has its own coordinate system. Furthermore, assume that the software coordinate system is not coincident with the measurement coordinate system and that the origins are not located at the same physical location. For the planar case, this situation is depicted in Figure 1.

Figure 1: A hypothetical linear tire mark starting at a and terminating at b.

The reference frame (oxy) represents the measurement frame (i.e. the frame of reference in which the coordinates of a and b were measured). The reference frame (OXY) denotes a second frame of interest for which the origin is not collocated with o and for which the coordinate axes have a differing orientation from xy.

If the software package that was being utilized did not have the capability for assigning new coordinate frames and/or one were asked to explain the underlying mathematics that were being implemented in the assignment of coordinate locations in the new coordinate frame, what would one say? In the first, the tire mark, as a physical entity, is unchanged by the choice of reference frame. Also, the vector rb|a, itself, is not changed by the reference frame. What does change, with a change in reference frame, are the directions along which the coordinate measurements are made or calculated and the scalar distances from each coordinate axis. Intuitively, one can reach the conclusion that the process of assigning new coordinate locations would be a function of the distance between the origins of each coordinate system (translation) and a function of a measure or metric that relates the orientations of the two coordinate systems (rotation). This relatively simple coordinate transformation problem is intrinsic to the mathematics that are employed in the field of engineering mechanics. www.collisionpublishing.com

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The use of the word mathematics, in the previous paragraphs, was not an example, collectively, of verbigeration or the employment of superfluous terminology. Fortunately or unfortunately, depending upon the bent of the reader, mathematics plays a critical and central role when it comes to understanding the details of the subject work beyond a level of mere qualitative conceptualization. The fields of mathematics, at a minimum, that are involved include vector theory, matrix theory, linear algebra and calculus. The presumption, on the part of the subject author, is that there exists a reasonable potential for a substantial variance in regards to the familiarity, on the part of the readership, with the underlying mathematical precepts.

This presents a conundrum. The presentation of the basic mathematical theory that underlines this work, within the main body of the work itself, would lead to the tendency to obscure the primary findings. Conversely, not including the relevant mathematical precepts in a concise form in association with the subject work would tend to facilitate a lack of understanding for those readers that are unfamiliar with the mathematical definitions and procedures that have been used to derive the salient relationships. In hopes of balancing these competing issues, the basic mathematical precepts that are most relevant to the subject work have been placed in an Appendix to the work (Appendix A, see note below).

A second point of introductory importance is the form of nomenclature and the form of symbols used to represent the various terms that are employed. The field of engineering mechanics traces back to at least Newton’s 1687 publication of Philosophiae Naturalis Principia Mathematica.

The intervening 333 years has seen the publication of a plethora of journal articles, technical papers and books on the topic.18, 21-22, 25, 32 The extensive corpus of work has come with a wide diversity in terms of the systems employed for nomenclature and symbolic representation of the variables involved. Some of these systems are minimalistic in that the operations and/or terms are described in prose but the variables employed lack specific designators for the frames of reference for which they are valid. This can be quite confusing when working with multiple frames of refer-

Appendix is available online at http://bearinc.org/ 36

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ence and when dealing with multiple derivatives. Even more confusing are systems that present operations such as simple addition between variables referenced in difference frames (e.g. the direct addition of a position vector in an inertial frame of reference with a position vector in a body frame of reference) or that define coordinate axes in one frame (e.g. a planar problem with XY being the coordinate axes of the inertial frame of reference) and yet depict the implementation of inertial frame laws of mechanics using variables that are inertial frame variables and yet shown as body frame variables (e.g. for a planar problem with xy being the body frame coordinates and an inertial frame translational dynamics relationship written as max rather than maX). On the other end of the spectrum are systems that read like an admixture of alphabet soup and numeric soup coupled with either a lack of specific referencing for variables in a certain frame of reference or partial dropouts. The approach taken herein is the employment of subscripts and superscripts following the work of Jazar. 28-31 However, simplifying dropouts are not employed. A consequence of this approach is that velocities are referenced by a right subscript, left subscript and a left superscript.

Accelerations are referenced by a right subscript, double left subscript and double left superscript. This befits the alphanumeric soup admixture description. However, it leaves no question about any of the reference frames that are relevant to the variables and operations employed. A listing of the symbol conventions employed in the subject work can be found under the section entitled Symbols.

A final point of introductory importance is in regard to the staving off of pointless debates over the semantics of representation of derivatives using the dot notation with respect to the underlying position vector or using the boldfaced v for the first derivative and a for the second derivative. Certain references will solely use the dot notation for time derivatives when they involve a moving frame of reference along with the adjectival modifier of apparent (e.g. apparent velocity or apparent acceleration) while reserving the boldfaced symbols for time derivatives involving the inertial frame of reference. Because the script notation explicitly indicates the frames of reference involved, the dot notation and the boldfaced symbols are equivalent and interchangeable without any loss of specificity and with no reference to a different framework that only employs each notation for variables in specific frames of reference.

The scope of the subject work has necessitated the division of the same into three separate and sequential papers.

Each subsequent paper builds upon the material presented in the previous paper or papers. The first paper in the series covers the topic of the development of rigid body kinematic relationships when working with multiple frames

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2.1

Direction cosine matrix and position vector

2.

Rigid body of reference are needed (along with proper eachan rigid body and triad their of unit vectors body frame, denoted askinematics thenotations B frame, for hasreferencing an origin o and orthonormal unit vectors {e1, e A relatively standard approach in developing the equations for the kinematics of a rigid body is by and coordinate axes).

In an accident reconstruction G frame generally taken as a clearly, ground-addition corresponding coordinate axes {x, y, context, z}. For a the problem withismultiple rigid bodies, considering the case of a single rigid body that can undergo free movement (i.e. translation and rotation) fixed frame of reference and the frame (along is generally taken as a vehicle-fixed (or another interest of reference are B needed with proper notations for referencing eachobject rigid of body and their un 2.1 Direction matrix and position vector with respect to a fixed frame ofcosine reference.

This requires the establishment of two frames of reference, both fixed) frame of reference. and coordinate axes).

In an accident reconstruction context, the G frame is generally as A relatively standard in Cartesian. developingThe thefixed equations the kinematics of as ataken rigid of which can be characterized as approach rectangular frame for of reference, denoted the Gbo fixed frame of reference and the B frame is generally taken as a vehicle-fixed (or another object of reference. The second in the series presents coincident origins axes.

There aremovement two limiting cases considering the case ofthe a single rigid body thatand undergo (i.e. translation ando frame,paper is the inertial (non-accelerating) frame with origin Ocan and with anfree orthonormal triad of unit vectors The position of awith pointrespect p can be described (i.e. expressed) in both frames of reference.

In the G frame, the fixed) frame of reference. to body. a axes fixedThe frameto ofconsider. reference. establishment two frames of refere Newton-Euler equations motion a rigid InThis the requires first, the the orientation the of respective coordinate corresponding to the unit vectors are typicallyoflabeled as {X, Y, Z}.

The {E1, Eof 2, E 3}. Thefor the B frame, the position vector fromframe o to p,ofis reference, denoted asdenoted position vector from O to p, is denoted as GrP. Inas of which can be characterized rectangular Cartesian.

The fixed third paper applies the relationships developed in the first coordinate axes remain fixed in regards to coincidence such second frame of reference is fixed to the rigid body (i.e. it translates and rotates with the rigid body). The Br . The scalar components of of each vector are(i.e. not equal, on an by axis basis, unless the The position a point p representation can be described expressed) in axis both frames of reference.

In the frame, is the (non-accelerating) with with an orthonormal of G unf bodyaccident frame, denoted as inertial the B frame, hasthat an origin and triad of represents unit vectors {e1, e2triad , e3} with two papers to pcertain reconstruction specific Ei · ei =oframe 1 Gandan Eorthonormal · eorigin = 0 {iO≠ and j}.

This a case i j two frames have{E coincident origins and axes.

There are two limiting cases to consider.

In the first, the r .

In the B frame, the position vector from o to p, is d position vector from O to p, is denoted as P E coordinate to the unit vectors areclearly, typically labeled as {X, Y 1, objective 2, E3}. The corresponding coordinate axes {x, y, z}.axes For acorresponding problem with multiple rigid bodies, additional frames problems. The advancement of the of the subject of pure translation.

The second case is where O is fixed to B  ewith 1 the and orientation of thesecond respective coordinate axesisremain inrepresentation regards to coincidence such that Eiaxis rp. Theframe scalar components of eachtofixed vector are not equal, on an axisrigid basis, i =by ofbody reference fixed the rigid body translates and reference are (along with notations for referencing each rigid body andsecond their unit vectorsbo work, that being theofdevelopment ofneeded the rigid kine-proper O. This represents a case(i.e. of it pure rotation.

Inrotates the two frames have coincident origins and axes.

There are two limiting cases to consider.

In Ei  ej = 0 {i ≠ j}. This represents a case of pure translation.

The second case is where o is fixed to O. This body frame, denoted as of the reconstruction Bcase, frame, ancontext, originof o the and anframe orthonormal triad of unit {e1the ,e and coordinate axes).

In anframes accident is generally asvectors a groundmatic relationships when working with multiple thehas relationship theofG position of point P intaken the two represents a casecorresponding of pure rotation.

In the second case, the relationship the position of point p in the two orientation of the respective coordinate axes remain fixed in regards to coincidence such that Ei  coordinate axes {x, y, z}.

For a problem with multiple rigid bodies, clearly, addition fixed frame of reference and the B frame is generally taken as a vehicle-fixed (or another object of interest reference, proceeds byoffirst definingis the position vector of frames of reference is represented as: frames reference as: represents Eiof  ereference. = 0 {i ≠are j}. needed This a case of pure translation.

The second is body whereand o istheir fixedun t jrepresented of reference (along with proper notations for referencing eachcase rigid frame a point and definingfixed) the direction cosine matrix (DCM), represents a case of pure rotation.

In the second case, the relationship of the position of point p and coordinate axes). In an accident reconstruction context, the G frame is generally taken as ai which transforms the components of a vector between difG G frames of reference represented (1) object R B B ris fixed frame of reference and therP(i.e.

Bas: frame taken as aofvehicle-fixed another P generally (1) The position point p by can beisdescribed expressed) in both frames reference. In(or the G frame, the o ferent frames of reference.

This of is afollowed the preG fixed) frame of reference. the position vector from o to p, is denoted as vector from to p, is denoted sentation of the firstposition time derivative of theOposition vector as rP. In the B frame, G G B GR is the DCM that transforms the coordinates of rP not a vector R Br . The B frame into the In equation (1),scalar B components of each vector representation are B rPexpressed equal, on an axisinbythe axis basis, unless p G and the framework of frame referenced time derivatives.

A Inbeequation (1),(i.e.Rexpressed) isofthe DCM that transforms the to co-the In B the corresponding coordinates in the G frame.

The components the DCM from the B frame G thethe The position of a point p can described in both frames of reference.

Gf two frames have coincident origins and axes. There are two limiting cases to consider. In the first, critically important outcome ofposition the section is the derivaordinates of a vector expressed in the B frame into the corG G frameorientation are: rP. In B frame, the of position vector to1the p,and isBdf vector O the to p,DCM isaxes denoted as fixed In (1),from R that transforms coordinates a vector expressed B is = of equation the respective coordinate remain inthe regards to coincidence such thatfrom Ei  eoi in tive transport formula and its relationship to standard first responding in components the are G frame.

The components . corresponding The scalar components vector representation not equal, an byto axis u coordinates in the Gcoordinates frame.

The theon DCM from the frame Ei  ej = 0 {iBthe ≠rpj}.

This represents a case of of each pure translation.

The second case isofwhere o isaxis fixed O.Bbasis, This derivatives with respect to time.two The third have section devel- oforigins the DCM from the B frame toofthe Gposition framecases are: frames coincident and axes.

There are two limiting to consider.

In the are: represents aframe case of pure Grotation.

In the second case, the relationship the of point p in the two R B  Ei  e j  Ei e j cos Ei , e j  cos Ei , e j (2) ij ops the relationships for the second time derivative.

The coordinate axes remain fixed in regards to coincidence such orientation of the respective that Ei  frames of reference is represented as: final section extends the findings sections Ei ofej the = 0 previous {i ≠ j}. This representsG R a case of pure translation.

The second case is where o is fixed to Ei  e j  Ei e j cos Ei , e j  cos Ei , e j B ij the In equation both from 1 torotation.

3 and with last Bequality being based upon the position fact(2) thatof the to problems involving more(2), than twoindices frames of reference.

G the second G represents a run case of pure In case, the relationship of the point pi (1) rP  R B rP unit vectors, by definition, are of unitislength (i.e. of as: length equal to one). There are a number of alternative frames of reference represented Rigid body kinematics ways to characterize the orientation kinematics of the rigid include, but are3 not to,upon the the fac In equation (2), both indicesInrun from 1(2), tobody.

3both andThese with the last equality being based equation indices from 1 to andlimited with the GR is the DCM that transforms the coordinates of run a vector expressed in the Ba number frame equation (1),quaternions, B use ofIn Euler angles, or spinors are and rotators.

These be directly related to into G representations G B equal can unit vectors, by definition, of unit length (i.e. of length to one).

There are of a last equality being based the DCM fact that thethe unit vecrP  upon R Bof rthe Direction cosine matrix and position vector P the corresponding coordinates in the G frame.

The components from B frame to thelimi G the DCM. The DCM is to used exclusively, in regards to the coordinate transformation matrix.

Using ways characterize theherein, orientation kinematics of the rigid body.

These include, but are not tors, by definition, are of unit length (i.e. of length equal frame are: the factapproach that the in DCM an orthogonal matrix, the inverse transformation ofThese equation (1) can be written as:directly use ofis Euler angles, quaternions, or spinors rotators. representations can be A relatively standard the equations G to one). areand a number of alternative toexpressed characRB is the DCM that There transforms the coordinates of a ways vector in the B f Indeveloping equation (1), the DCM. The DCM is used exclusively, herein, in regards to the coordinate transformation matr for the kinematics of a rigid body is by considering the terize orientation kinematics of the of rigid the corresponding coordinates in Tthe the G frame.

The components thebody.

DCMThese from the B frame G case B G G B G RisB an  E  e  E e cos E , e  cos E , e (2) the fact that the DCM orthogonal matrix, the inverse transformation of equation (1) can be w (3) r  R r  R r i jB j G j i j Pi of a single rigid body that can undergo free movement (i.e.ij P include, but are notP ilimited to, the use of Euler angles, frame are:





































translation and rotation) with respect to a fixed frame of quaternions, or spinors and rotators.

These representaB G G BR is the DCM that transforms the coordinates rPthe  Glast R BaT equality rP  B R rP based of vector expressed in the G frame into In requires equation (3),establishment G G DCM. In equation (2), both indices from 1tions toG R3 can andbe with being upon theisfact that the reference. This the of tworun frames directly related to the DCM Ei  e j  Ei e j cos  Ei , e j   cos  EThe   B components i , ej  the corresponding coordinates in the B frame.

The are: ij vectors, definition, are of unit length (i.e. of length equal one). to There are a number of alternative of reference, both ofunit which can beby characterized as rectanused exclusively, herein, intoregards the coordinate transBR is the DCM that transforms the coordinates of a vector expressed in the G f In of equation G characterize the(3), orientation ofmatrix. the rigidUsing body.the These include, but areisnot limited to, the gular Cartesian. Theways fixedtoframe reference, denoted askinematics formation fact that the DCM an orB the corresponding coordinates in the B frame.

The components are: In equation (2), both indices run from 1 to 3 and with the last equality being based fac use of Euler angles, quaternions, or spinors and rotators.

These representations can be directly related R G  with ei  E j thogonal ei E j cos (4) theto the G frame, is the inertial (non-accelerating)  frame  ei , E(i.e.  ei , Etransformation matrix, the inverse of equation upon j   cos j ij unit vectors, by definition, are of unit length of length equal to one).

There are a number of a the DCM. The DCM is used exclusively, herein, in regards to the coordinate transformation matrix.

Using origin O and with an orthonormal triad of unit vectors (1) can be written as: B ways to characterize the orientation kinematics of the rigid body.

These include, but are not limit the fact that the DCM is an orthogonal matrix, the inverse transformation of equation (1) can be written as:  RG   ei  E j  ei E j cos  ei , E j   cos  ei , E j  {E1, E2, E3}. The coordinate axes corresponding to the unit use of Euler angles, quaternions, or ijspinors and rotators. These representations can be directly vectors are typically labeled as {X, Y, Z}. The second frame B G G the DCM. The DCM is used exclusively, in regards to the coordinate transformation matr (3) (3) rP  G R B T herein, rP  B R G rP of reference is fixed to the rigid body (i.e. it translates and the fact that the DCM is an orthogonal matrix, the inverse transformation of equation (1) can be w rotates with the rigid body). The body frame, denoted as BR is the DCM that transforms the coordinates of aBvector expressed in the G frame into equation (3),an B G the B frame, has anInorigin O and orthonormal triad In equation (3), R is the rDCM that B G transforms the coorrP  GG R B T G are: P  R G rP corresponding coordinates in the B dinates frame. of The components of unit vectors {e1, ethe , e } with corresponding coordinate a vector expressed in the G frame into the cor2 3 axes {x, y, z}. For a problem with multiple rigid bodies, responding coordinates in the B frame. The components In equation (3), BRG Bis the DCM that transforms the coordinates of a vector expressed in the G f R G   eare:  E  e clearly, additional frames of reference are needed (along (4) i i E j cos  ei , E j   cos  ei , E j  ij the corresponding coordinates inj the B frame. The components are: with proper notations for referencing each rigid body and their unit vectors and coordinate axes). In an accident re B RG ij  ei  E j  ei E j cos  ei , E j   cos  ei , E j  (4) construction context, the G frame is generally taken as a ground-fixed frame of reference and the B frame is generally taken as a vehicle-fixed (or another object of interest In a more general situation, we may consider the rigid fixed) frame of reference. body as translating with respect to the with G respect to In a more general situation, we may consider the rigid and bodyrotating as translating and rotating frame.

The position of point P can be expressed as: frame.

The position of point P can be expressed as: The position of a point PGcan be described (i.e. expressed) in both frames of reference. In the G frame, the position G rP  G ro  G rP|o  G ro   G rP  G ro  vector from O to p, is denoted as Gr . In the B frame, (5) P the position vector from O to p, is denoted as Brp. The scalar components of each vector (5) representation are not Equation can be rewritten when the position of point P is initially referenced with respect to the B fra equal, on an axis by axis basis, unless the of two frames have by substitution equation (1) for the relative position vector GrP|o. www.collisionpublishing.com

G

rP  G ro  G R B B rP

Collision Magazine - Volume 14 Issue 1 37

The choice of using o as the body reference point is one that is simplifying (obviously, the position vec

Equation (5) can be rewritten when the position of point P is initially referenced with respect to the B fram (6)

G rP for rothe  Grelative R B B rP position vector GrP|o. e may consider the rigid body as translating and rotating by with respectoftoequation the G (1) substitution t P can be expressed as:

G B The choice of using o as the body reference point is one that isG rsimplifying the position vector ro  G R(obviously, (6 P  B rP of o in the B frame has zero-valued components). We may readily choose any other point affixed to the G G G G G G (5) rP  ro  rP|o  ro  rP  ro body. It is important note of that the referencing of the position of point P that withisrespect to the (obviously, B frame does The to choice using o as the body reference point is one simplifying the position vecto not imply that theofpoint Pthe is B fixed in the frame.

When P is not in may the Breadily frame, rP =a Bbody rPany (t). When P Equation (5) can be rewritten when the position of point Pframe frame of expression.

As fixed an We example, if PBchoose is point, o in hasBzero-valued components). other point affixed to th is fixed in the B frame (i.e.

P is a body point), position vector of of P, the in the B frame, remains constant and body.

It is important tovelocity notethe that theP referencing position of point P with respect to the B frame doe is initially referenced with respect to the B frame by substithe of in the B frame will be zero because the when the position of point P is initially referenced withand respect to the the velocity acceleration of B P,frame in the B frame, are zero valued. The position, velocity and acceleration not imply G that the point P is fixed in the B frame. When P is not fixed in the B frame, BrP = BrP(t). When tution of equation the relative position vector r . fixedbody to the body.

The velocity ofP,Pininthe the G frame, Gr for of P, in the G frame, even when P ispoint fixed(i.e. inisthe will not generally constant forBany motion for the relative position vector(1) .

P|o P|o is fixed in the B frame P isBa frame, point), the positionremain vector of frame, remains constant an of the rigid body. the Figure 2 depicts a conceptualization of the problem expressed in equation (6). however, of will not zero are forzero anyvalued. general of the and acceleratio velocity and acceleration P, in the be B frame, Themotion position, velocity



G

rP  G ro  G R B B rP



of P, in the G frame, even when is fixed in that the B the frame, will not generally remain constant Now,PFigure suppose position of the same point P for any motio (6)<<Insert 2 here>> (6)body. body. of the rigid Figure 2 depicts a conceptualization of the problem expressed in equation (6).

is expressed in the G frame (i.e. at time t, the coordinates

Equation (6) can be rearranged to solve position point P if2the G frame position is of Gfor r the areB frame known) andof the first time derivative isknown. taken <<Insert Figure here>>

P ody referenceThe point is one simplifying (obviously, the position choice of that usingis O as the body reference point is one vector with respect to the B Gframe.

In such a case the expresT G B G B G alued components).

We may readily choose any other point the (7) rPcan  Gto R Brearranged rP  G ro to R G for rPthe  Bro frame R Gposition rP|o  and  solve  velocity that is simplifying (obviously, the position vector ofaffixed O(6)B in Equation be of point P if the G frame sion derivative of the vector are in differentposition is known hat the referencing of the position of point P with respect to the B frame does the B frame has zero-valued components). WeB mayBreadily frames. BIn order G G theG difficulty created by = rP(t). When P xed in the B frame. When P is not fixed in the frame, (7 rP  G R B T  to rP account  G ro   B Rfor ro   B R G G rP|o G  rP  First It time derivative and velocity choose any other point affixed to2.2 theBbody. isrPimportant the use of more than one frame of reference, we introduce a body point), the position vector of P, in the B use frame, remains and necessitates the use of a mechanism that tracks the The of more than oneconstant frame of reference frame in to note that the referencing of the position of point P with thederivative concept of the frame-referenced time derivative.

The which thevelocity time derivative is taken (i.e. the derivative frame) and the frame of expression.

As an example, f P, in the B frame, are zero valued. The position, acceleration 2.2and First and velocity respect to the B frame does not ifimply thatpoint, theThe point Pmore istime P is a body theuse velocity of P the frameof will be zero necessitates because the point is of fixed toof thea body.

The Ginderivative is defined as the time quantity of than oneBframe reference thederivative use a mechanism that tracks the frame i n P is fixed in the B frame, will not generally remain constant for any motion of P in GB frame, however, will not be zero forthe any general motion of the body.

Now, suppose As an example fixed in the B frame. When P isvelocity not fixed inthe the frame, which the time derivative is taken (i.e. derivative frame) and the frame of expression. with respect to the coordinate axes of the G frame.

The picts a conceptualization of the problem expressed in equation (6). that the position of the same point P is expressed in the G frame (i.e. at time t, the coordinates of GrP are B B

the velocity of P in the B frame will be zero because the point is fixed to the body. Th rP = rP(t). When P is fixed in the B frame (i.e.ifPPtime isis aa body bodypoint, is defined the derivative of a of quantity known) and the first taken respect to the Bbe frame.

In such a case the expression velocity derivative of P in theBis Gderivative frame,with however, will notas zerotime for any general motion the and body. Now, suppos point), the position vector of P, in the B frame, remains derivative of the velocity vector are in different frames.

In order to account for the difficulty created byThe the <<Insert Figure 2 here>> with respect toP the coordinate of the B time frame. that the position of the same point is expressed in theaxes G frame (i.e. at t, the coordinates of GrP ar use of more than of B reference, wederivative introduce is the concept ofrespect the frame-referenced time derivative. constant and the velocity and acceleration of one P, frame in the known) and the first time taken with to the B frame.

In such a case the expression an functional meaning of with the ‘with respect to’ portion of each The G derivative is defined as derivative of ain quantity respect the to coordinate axes ofdifficulty the G derivative ofthe the time velocity vector are different frames.

In to order account for the created by th frame, are zero valued.

The position, velocity and acceleraG d to solve for the B frame position of point P if the G frame position is known. statement can be seen in thewith following.

For a vector axes b, time derivative frame.

The B derivative is defined as the timeofderivative of quantity respect of to the the frame-referenced coordinate use of more than one frame reference, wea introduce the concept tion of P, in the G frame, evenofwhen P is fixed in the B the B frame.

The functional meaning of the ‘with respect to’ portion of each statement can be seen in the the index and expanded representations are: The G derivative is defined as the time derivative of a quantity with respect to the coordinate axes of the G Gb, the index and expanded representations are: following. For G Ggenerally G fora vector any motion frame.

The B derivative (7) is defined as the time derivative of a quantity with respect to the coordinate axe R B T  G rPframe,  G ro  will  B Rnot ro  remain  B R G G constant rP|o P  G  rP  of the B frame. meaning of the ‘with respect to’ portion of each statement can be seen in th of the rigid body. Figure 2 depicts a conceptualization of The functional G (8) b Gb, Ei index B 1 E 1 and B 2 Eexpanded Bithe following. For a vector representations are: 2  B3E3 (8) the problem expressed in equation (6).

ve and velocity

The G derivative of equation (8) is:

G

b  Bi Ei  B 1E 1  B 2 E 2  B 3 E 3

(8

The equation (8) is: me referencewe necessitates the the use rigid of a mechanism that tracks frame inG derivative eralofsituation, may consider body as translating andthe rotating with respect toofthe G G G G G G dB d Ei G dBi dBi d b anG example, d ken (i.e. the derivative frame) and the frame of expression.

As The G derivative of equation (8) is: i position of point P can be expressed as: E i  Bi Ei  0  Ei  G b   (9)  Bi Ei   dt dt dt dt y of P in the B frame will be zero because the point is fixed todtthe body.dt The G G G G G dBi dEi G dBi dBi d b G  Gd owever, will not be zero any general motion body. Now, suppose G for G G of the G  Gthe b G frame E i the  BiG frame  and therefore Ei  0  undergo Ei (9  Bi E(5) i   fixed in Equation because the unit vectors of are rP  ro  G rP|o  G ro  rP  (9) roholds   dt dt dt dt dt dt Gr are point P is expressed in the G frame (i.e. at time t, the coordinates of time, P with respect to the G frame. For a vector Bb, the index and expanded no change in their position, with vative is taken with respect to the B frame. In such a case the expression representation are: Equation (9) holds and because the unit vectors of the G frame are fixed in the G frame and (9) therefore underg anare be rewritten when the position P isfor initially referenced with respect thetime, B frame or in different frames.

In orderof topoint account the difficulty created by the to no change in their position, with with respect to the G frame. For a vector Bb, the index and expande Gr nreference, of equation for the relative position vector representation are: P|o. we(1) introduce the concept of the frame-referenced time derivative.

Equation (9) holds because the unit vectors of the G frame the time derivative of a quantity with respect to the coordinate axes of the are G fixed in the G frame and therefore undergo no change G G ro  Gwith R B B respect rP (6) ined as the time derivative of arquantity to the coordinate in axes P  their position, with time, with respect to the G frame. meaning of the ‘with respect to’ portion of each statement can be seen inFor thea vector Bb, the index and expanded representation are: eusing indexoand expanded representations are: that is simplifying (obviously, the position vector as the body reference point is one rame has zero-valued components). We may readily choose any other point affixed to the B b  b e  b1 e 1does  b2 e2  b3 e3 (10) G (10) portant to bnote of the position of point P with respect to (8)the iB iframe  Bithat Ei  the B 1 Ereferencing 1  B2 E2  B3E3 the point P is fixed in the B frame. When P is not fixed in the B frame, BrP = BrP(t). When P B  b i ei  b1 e1  b 2 e 2  b 3 e 3 (1 Figure 2: Schematic depiction ofofaP, rigid body B derivative equation is: remains constant band B (i.e. P is a body point), theThe position vector of in the B(10) frame, The B derivative of equation (10) is: 8)frame is: asBB) undergoing motion withvelocity and acceleration nd acceleration(denoted of P, in the frame, are zero general valued. The Theposition, B derivative of equation (10) is: B B B B respect to a fixed frame of reference dbi motionB dei B dbi dbi rame, will notG dB generally constant forBany d bremain d G P is fixed in G the B frame, G G even when B dB d E dB d   b  b ei B (11)  i i i i bdy.

B B ei  0 B  B i eBi   B e i  bi B (denoted E E Ei expressed  Figure   Bi  the 0G). problem (9) db d e db dbi  Bi Ei2 depicts d b d a iconceptualization ofias in equation (6).

B i i dt dt dt  B bdt   ei  0  ei (1  bi ei   dt i ei  bdti dt dt dt dt dt dt dt dt dt dt dt

<<Insert Figure 2(11) here>> Equation holds because the unit vectors of the B frame are fixed in the B frame and therefore undergo Equation (11) holds because the unit vectors of the B frame are fixed in the B frame(11) and therefore underg e unit vectorsEquation of the G frame in the G and therefore undergo (6) canare befixed rearranged to frame solve for the B frame no change in their position, with time, with respect to the B frame.

In comparing the results shown change in expanded their position, with time, with respect to the B frame. In comparing the results shownin Bb, the no th time, with position respect to G frame.

For a vector and ofthe point P ifBthe G frame position isindex known. an be rearranged to solve for the frame position of point P ifmay the(9)Greadily frame position iswhen known. and (11),state we may readily state that when theof operand of the derivative operator a scala equations (9) and (11),equations we that the operand the derivative is aisscalar Equation (11) holds because the unit vectors of the Boperator frame that the G and reduce B derivatives both reduce simple time derivatives.

In other words, forscalar any scalar quanti that the G and B derivatives both to simple timetoderivatives.

In other words, for any quantity are fixed in the B frame and therefore undergo no change B B (7) rP  G R B T  G rP  G rox:  B R G  G rP  G ro   x: R G G rP|o (7)

in their position, with time, with respect to the B frame.

G B dx shown dx dxin equations (9) and (11), In comparing the results    x (1 G B dx dx dx dt dt First time we may readily time derivative andderivative velocityand velocity  state  that  xwhendtthe operand of the deriva(12) dtthe frame dt that the G and B derivatives both tive operator isdta scalar, re than one frame of reference necessitates the use of aInmechanism tracks in equationthat (9) and (11), for the form shown to the right of the first equality, the left superscript denot The use of more than one frame of reference both necessitates simple time derivatives. otherthe words, for any e derivative is taken (i.e. the derivative frame) and the frame of expression.

Astoleft an example, the expression framereduce and the subscript denotes the frame In in which derivative was taken. For the the use of a mechanism tracks the frame inthe which the In that both (9) and (11), for form shown tothe the derivative of the a is vector expressed in frame: point, the velocity of P in the B frame will equation be zero because point fixed toquantity the body.

The of the first equality, the left superscript denotes scalar x:B right time derivative taken (i.e. the and thezero expression frameframe) and the left subscript the frame in which the derivative was taken. For the G n the G frame, however, willisnot be for derivative any general motion ofthe the body. denotes Now, suppose

G G G G G B G db dei dbi dei B  dei d bB frame: d derivative a vector on of the same point P is expressed in the Gofframe (i.e.expressed at time t, in thethe coordinates  GB b    B b  bi bi eiGrP arei ei  bi e i  bi (1 of dt dt dt dt dt dt dt he first time derivative is taken with respect to the B frame.

In such a case the expression and 38 Collision Magazine - Volume 14 Issue 1 www.collisionpublishing.com G G G G B G dbcreated dof ei the db dei from dei bIn this d the firstGterm he velocity vector are in different frames. In order todaccount for i to the right i equality comes of equatio  GB b equation,  the bdifficulty  third  BB b the ei  bi by the e i  bi bi implementation (13) i ei   an one frame of reference, we introduce the conceptdt of the time derivative.

(12) frame-referenced and the the right of the is thedt definition that comes dt first term todt dt last equality dt dt from equation (11

se the unit vectors of the B frame are fixed in the B frame and therefore undergo B b  b ei  b1 e1  b 2 e 2  b 3 e 3 on, with time, with respect to the B frame. i In comparing the results shown in may readily state that when the operand of the derivative operator is a scalar, The B derivative of equation (10) is: es both reduce to simple time derivatives. In other words, for any scalar quantity B

(10)

B B B dbi dei B dbi dbi d B b B  Bd  Bb   e i  bi ei  0  ei  bi e i   dt dt dt dt dt dt

(11)

and (16) are apt for the case of the two frames sharing dx dx dx     x (12) the same origin. How does the result of equation (15) (12) Equation (11) holds therefore undergo dt because dt thedtunit vectors of the B frame are fixed in the B frame and no change in their position, with time, b with respect to the B frame.

In comparing the results shown(10) in against the solution from using a more commoncompare  b e  b e b e b e equations (9) and (11), we may readily state that when the operand of the derivative operator is a scalar, ly found non-frame referenced time derivative approach?

1), for the form shown to the right of the first equality, the left superscript denotes that the G and B derivatives both reduce to simple time derivatives. In other words, for any scalar quantity The B derivative of equation (10) is: in(9) both equation andthe(11), for the the left denotes the frame which derivative wasform taken.shown For theto G the Such a problem is generally presented in a form similar to x: subscriptIn essed in the B frame: right ofd the first dequality, the leftdesuperscript denotes time derivative approach?

Such athe problem is generally presented form similar the following (but the following (but without in thea subscript and to superscript db db db b  time dx dx dx  b   b   0  b e e e derivative approach?

Such aeproblem is generally presented in a form similar to the following (but (11)   expression theGdtleft subscript the frame notation):  subscript (12) without the and superscript dt G frame dt and dt x G dtdenotes dt G G notation): dtthe dt dbi de without db dedt d ei superscript notation): d i subscript and  i ei  biwas  BB b  bFor b  ei  bi the iderivative (13) of  bi ei   in which taken. the G derivative i dt dt dt dt dt dt Equation (11) holds because the unit vectors of thetime B framederivative are fixed in the B frame andapproach? therefore undergo approach?

Such aSuch problem isB generally presented in aaform formsimilar similar to follow the fo time derivative is generally presented in to the G a problem G In both equation (9) and (11), for the form shown to frame: the right of the first equality, the left superscript denotes a vector expressed in the B (17) no change in their position, with time, with respect to the B frame. In comparing the results shown in rGP  B R B rP the expression frame and the left subscript denoteswithout the frame in the which the derivative was taken.

ForGsuperscript the G notation): without the subscript and notation): B subscript and superscript (17) (17) r R r b state b i e ifrom  bwhen bthe  b 3 e 3 of theofderivative (10)B P (9) and (11), we may readily that P ermequations to the right of the third equality comes the equationoperator is a scalar, 1e 1  implementation 2 e 2operand G

B

B

B

B

i

B

B B

i

G

i

i

B

1 1

iB

2

2

B

i

3

B

i

i

3

i

B

i

i

i

derivative of a vector expressed in the B frame:

the G derivatives both to simple time In other words,(11). for any scalar quantity thethat right ofand theBlast equality isreduce the definition thatderivatives. comes from equation Taking the first time derivative of both sides yields: x: G G G G B G G frame:G ive of a vectorofexpressed in the B derivative equation (10) is: dbthe dbi derivative dei Bof both dei sides yields: d b B d Taking firstdetime i i

G

G

rP G G RBB B rP

 R B rof Taking P both sides yields: e i  bi  e i  bi  B b  bi (13) the first timerPderivative dt dt dtG dx B dx dt dx dt dt dt B B BB B B B B x B B B B   (12) dBi d b B dEi d dB i dTaking Ei i d db dEi Gdt de db dbi time derivative of both sides yields: the first dti i edt bGi b  Bi comes b  In this Ei   Bi B b to the bright (14) 0 implementation e Ei  B e i the ei (11) B  Bi Eequation, i  i i equality from of equation G both G B first time derivative of yields: dt dt the dt dt dti i of the dtTaking dtdt dtfirst term dt third dt the dt  (18) rGP(11).  GB R sides (18) B r G GP  B R B rP (12) and the first term to the right of the last equality is the definition that comes(13) from equation  P denotes  (18) r  R r  R r In both equation (9) and (11), for the form shown to the right of the first equality, the left superscript B P B P Similarly for the B derivative of a vector expressed in the G frame: the(11) expression frame andthe theunit left denotes the frame intaken whichin the derivative was taken. the G ate that forholds a vector expressed in subscript one frame, thetime derivative with respect ation because vectors of the Bterm frame are fixed the B frame and therefore undergo derivative approach?

Such aForproblem is generally presented G B a form similar to the following  B r  G R in In this equation, first to the right of the tothird  rP shown  GR derivative vector expressed in the the BThe frame: B P B rP time for DCM the methodology in Appendix A but without the frame B the Bderivative BInthe B follows B G of the B time, with result that is of theaposition, sum derivative of vector in its own frame of reference change in their with respect to the B frame. comparing the results shown in The time derivative for the DCM follows dBi derivative dEi dB d E d E d b G d The time without the subscript and superscript notation): G ifor the DCM i i  follows the methodology shown in Appendix A the but methodology without the frame G G B G B equality comes from the implementation of equation (12)        b B E E B E B b B (14)    B i iof i when iin its i i of G rate i of aations function of the scalar components the vector own frame and the (9) and (11), we may readily state that the operand the derivative operator is a scalar,   referencing.

This leads to the following result: r  R r  R r Gdt Gdt G dt G dt dt dt Gdt B G P B Pwithout B the P frame referencing. shown in Appendix Amethodology but deib eleads db dei following dei result: B db i d b Bwith d referencing.

Bderivative ib eThe This to the b  reduce  b  b e  b e  (10) time for the DCM follows the shown in Appendix Afollowing but without th of the expression respect to the derivative frame.

Because the unit andframe theboth first term to the right of the last equality is the defithe G and B derivatives to simple time derivatives.

In other words, for any scalar quantity ei i i  bderivative e3i approach?  3 bi  B b  bSuch (13) is generally presented in a form similar to the 2 2  bi ei   time G i1 1 i a problem (bu G following G B result: dt dt dt represent dt a rotation. dt dt derivationdt This leads to the r  R r

by Equations definition(13) unit length, the second term The referencing.

This leads to the following result: and (14) state that for a vector expressed in one frame, the derivative taken with respect to P B P nition that comes fromwithout equation (11).

Similarly forG vtheBG R notation): G B B the subscript and in   GB B B  the The angular Bframe derivative of equation is:is the (19) superscript rPG methodology  GR B v  B  B r in Appendix A but witho another generates a result that of the derivative of theGvector own frameDCM ofBreference ms of the velocities, is (10) detailed insum Appendix B.time Using equation (A18): The derivative for shown G rG P its G G Pthe B rBPfollows In this equation, the first term right ofexpressed the third equality comes from implementation equation  the (19) R BandBBofrthe R B BB vB P B GBP B G B BrP P derivative oftoathe vector in the G vector frame: G rP  G vown P frame P rate G B  rP  B and a second term that is a function of theG scalar components of the in its of G G G B B B G B B B dx dx dx Bequality B B comes from (12) and the first term to right of isB the thatleads equation B B B referencing.

This toBecause the  (11).  sides vPG result: R r    rP  R B B vP  G  B  rP db d ex db dbiframe.

Taking first time derivative offollowing both yields: d the b the d the last i definition the change of the unit vectors of respect to thei ederivative the G rPunit iwith B ei frame eB i ei  B bexpression  bidt   bdt (11) G (12) (17 rP  GB RBB PB rP G B b Similarly the iGB frame: i 0  B B for theB B derivative B of BaBvector expressed B  dtin dt dt dt dt dt dt of both frames are by definition unit length, the second term represent a rotation.

The derivation Substitution for the vector b in equation (15), with the vector rP:    b b e b e b b vectors   b     b    (15) G B i G B i B G B i i B G B t for the second term, in terms of theSubstitution for the vector b in equation (15), with the vector r : angular velocities, is detailed in Appendix B. Using equation (A18): P (19) B B B B G G B B G B dBi of d Ei of dB dEiB frame d EGisuperscript d (11), b because d Substitution for the b in vector  equation R B BB rP(15),  GB with  B  the rPvector  G RrPB: BB vP  GB  B  B rP i fixed Equation (11) holds the vectors the frame arethe the andleft therefore oth equation (9) and shown theBright first equality, denotes G rP  undergo G v P(14) b the Bunit Eto Ei time B  GBfor  first  Bin  GG b the form i Ei   i  Bi i Taking the derivative sides yields: Bi dtof both B B B B B B B B G B dt dt dt dt dt dt G G B G no change in their position, with time, with respect to the B frame.

In comparing the results shown in    G taken. vBP  BFor rBP theG  (20) expression frame and theGdleft denotes the frame which G b subscript BPB  rRBP B rBPv P rP B B B B derivative P was GB r B B b G rB BP R BBr  G B  in the G  G G G G  e G B b G GBb    b  b   b b e  (15)   equations (9) and (11), we may readily state that when the operand of the derivative operator is a scalar, r  v  r    r  v    r Substitution for the vector b in (15), with the (20) G B i G B i B G B i i B G B vative vector G  B B B B G b  Bof i Ba G  E i expressed G b  Bdt i Ei  G b  B G  b  G b  G B  b G P (16) G P B P G B PB v B BP r G B B  Br P equation inBthe B frame:   r   v    r G P G P B P G B P B P G B P that the G and B derivatives both reduce to simple time derivatives.

In other words, for any scalar quantity Equations (13) and (14) state that for a vector expressed in one frame, the derivative taken with respect to rP: (15), with the vector rP: Substitution for the vector b invector equation (14) x: another frameBgenerates a result that is the sum of the derivative of the that vectorthe in itsparenthetical own frame of reference be Gderivative seen term inG equation (19)B matches theAppendix result shown to the rightthe of fr GIt can readily G G G G B d Gb G The for DCM follows the methodology shown in A but G B G G is G G of the G  components G G G own G frame G db d etime db dGin eiparenthetical dband ei the d term bsubscript d  adenotes   B It that   the and the a second that function scalar of the vector its rate of can readily be seen the term in equation the result shownthe to result the without right of to (18  ectors, right the rotating frame, the left subscript denotes B b B i i i r  R B form rP (19) term Rmatches rterm     B      b E b E b b b     (16) B G  b ei B G i G  bB G i i G e It B can G G b G B seen that the parenthetical readily be in equation (19) matches shown P B P b e b b     (13)  the equality in equation (20).

When one examines the of the on the left of each equality, there isthe G B G i expression i idx referencing. i with iderivative B frame. i Because dt vectors toone the following change ofisthe unit the frame respect toi expression the the unitresult: dx dx the B (12) B formB of the B termBon the B left of B each Bequality, there is and state aThis vector expressed he rotation occurring and the left superscript denotes frame. dtEquations dtof(13) dt(14) dtequation dtleads dtexamines the equality (20).

When the  in thatdt xforthe   equality in equation (20).

When one examines the form of the term on the left of each equality, r  v  r    r  v    r vectors of both frames are by definition unit length, the second term represent a rotation.

The derivation G P G P B P G B P B P G B P a key In equation (19), the term on the left of the equality is the G derivative of the position dt difference. dt origin. dt e apt for the caseinofone the two frames sharing the same How does the result frame, the derivative taken with respect to another For thesecond angularterm, velocity vectors, the right subscript denotes theequation rotating frame, left equation subscript denotes aangular key difference.

In (19), the term on the left of the is the G equality derivative of G the position a Appendix key difference.

In equation (19), the termequality onon thethe left of is the derivative the p for the in terms of the velocities, is detailed in B. the Using (A18): The derivative for the DCM follows the methodology inBthe Appendix A without theofframe expressed in the G frame.

In (20), the of the is the G derivative vector rtime G G equation Gequation B B term B shown G left Bequality B but against the solution from using more commonly found non-frame referenced his equation, the first termthe to rotation the aright of the third comes from the implementation of P,equality theInframe about which is occurring and the left superscript denotes the expression frame.   r  v  R r    r  R v    r both equation (9) and (11), for the form shown to the right of the first equality, the left superscript denotes (20) frame generates a result that is the sum of the derivative of , expressed in the G frame.

In equation (20), the term on the left of the equality is the G de vector r , expressed in the G frame.

In equation (20), the term on the left of the equality is the G derivative vector r G P G P B B P G B P B B P G B P P Pequality is theThis referencing. leads to the following result: andEquations the first term to the right of the last definition that comes from equation (11).

(15) and (16)Gand are apt the case the two frames sharing the origin.that How does thethe result of ofBthe position vector rsame ,B seen expressed in For the BG frame. Equation (20) can (19) be equated tothe equation by to Pderivative the expression frame the for left subscript denotes frame in which the was taken.

It can readily be the parenthetical in equation matches resultto(19) shown dBb B B B  B B  of vector rB expressed in term the B (20) frame.

(20) canequation be equated equation the vector in itsof frame ofthe reference second term P, frame. own Bbframe:  ein bsolution b a Gposition Bfrom B  B b non-frame (15) the vector ri and in the referenced Equation canEquation be equated to (19) by ilarly the B(15) derivative of avector expressed the GGa frame: P,the of for equation against more G b the B i Gposition i using Bb  B  bi e B expressed derivative of acompare vector expressed in the premultiplying thecommonly latter byfound the DCM dt B. the equality in equation (20).GRWhen one examines the form of the term on the left of each equ G  G b 

 bi e i  

 

    

 























canthe readily that the parenthetical term in equaRBseen . premultiplying the DCM be GR that is a function of the scalar components the vector inlatter . GItRby premultiplying the latter byof the DCM G B (15), G Substitution for the vector r  Gbv B equation r  B with  B the r vector R rBPv:  B   B r

(19     (21) R r r r v R  r r    r r  v    r  Substitution for the vector b in equation (15), with the vector r : P of the position vector r , expressed in the B frame.

Equation (20) can be equated to equa In this equation, the first term to the right of the third equality comes from the implementation of equation B





B G Pequation P E tion B (19) B the P matches G B P the Bshown B Pthe B right P of abB dkey difference.

(19), term on left of isthe theequalG derivative of dbi dbi dB edi E BIn dGe B d d b B dB  d B dB Ediei dB d Gits bd G own the result toG equality the i of i of the unit bi ei the ei  bB bvectors  GGbB E and  i rate G b B i Gof (13) G G G G i B G i  i change i  eB G B B B  bGB dt b GBb frame (14)G Ei dtGG bE b  GG bGbiGGdt b Br G  Ei E  G b i dt Bii GB  Bi iB G dt i B GR B r G G B B B dtGi b dt Gi  B i R GP inB(16) (21) rGIn  r BP BP (20),   , expressed in G the term on the left of the equality is the vector r dt the dt dt dt dt dt R r r  R r    r BBframe.

Gity Gequation P equation G B P G theBdt B BB G(20).

B dt expression P B B B B B B When one examines the form of the G P B B P G B P frame with respect to the derivative frame.

G

B

G

G

G

G

G

B G P G PG P G

P

BB P B P G GB B

P P B

P

G

B

P

term on the left of each equality, there is a key difference.

the unit vectors of both are by definition For(12) the and angular velocity vectors, the right subscript denotesframes thedefinition rotating frame, the left taken subscript denotes ations (13) andtheBecause (14) for a vector the derivative with respectGto firststate termthat to the right of theexpressed last equalityinisone the frame, that comes from equation (11).

This exercise also shows proper manner interpreting frame referenced of vectors This exercise also shows the proper manner of interpreting frame referenced ofderivates vectors are RB(19), . premultiplying the latter by the DCM the frame about which the rotation is occurring and the left superscript denotes theown expression frame.

Inthe equation theof term on the left ofderivates the equality isshown the that unit the second term represent a rotation.

The ther frame generates result that is the sum of the derivative of the vector in its frame of reference Similarly for the Balength, derivative of aThis vector expressed in the G frame: It can readily be seen that the parenthetical inB aequation (19) matches theofresult to the rig B does Bthe B B B B B exercise shows thesame proper manner of interpreting frame referenced derivates vectors that are Equations (15) and (16) are apt for the case of the two also frames sharing the origin.

How result matrix P ofterm represented as the multiplication of DCM and a vector.

That being: r  v  r   r  v   r (20 represented as the matrix multiplication of a DCM and a vector.

That being: a second term that is a function of the scalar components of the vector in its own frame and the rate G derivative of the position vector r , expressed in the G G P G P B G B P B P G B P derivation forB the second term, ina terms the angular ve-When P on the left of each equality, the of equation (15) compare against the solution from using more commonly found(20). non-frame the inofequation one and examines the form of the term B Bequality B B B G as areferenced DCM a vector.

That being: dBi dthe Ei matrix dB i to themultiplication dEi G  dEi ofBecause d bof the d represented G  expression nge of the unit vectors frame with respect derivative frame. the unit G B G G B B B         b B E E B E B b B (14) frame.

In equation (20), the term on the left of the equal i i in Appendix B detailed i i i i equation G i R B G ron r left  of R BtheB requality is the rP G derivative of the pos locities, B. Using (A18): a the key difference.

(19), term P the P  G  dt byis dt dt dtsecond dt dt In equation dt ors of both frames are definition unit length, term represent a rotation.

Thethe derivation  d Brvector dG P  dof r  P It can readily be seen that the parenthetical term in equation (19) matches the result shown is tothe theGright of d ity is the G derivative the position rP,equality expressed    R R r  P   vector rP, expressed in B. theUsing G frame. onthe left of the deriva he second term, in terms of the angular velocities, is detailed in Appendix P Requation Rd the r  (20), (22) r  term dequationIn(A18): P





 P  P the  examines to  dt dtequated Equations (13) and (14) state that for a vector in one the derivative withone respect theexpressed equality inframe, equation (20). taken When form of the on the left of equality, is in the B frame.

Equation (20)term can be to each equation R R r  (22)there(19 dt dt    in Pthe B frame.  ofThis position vector Equation (20) canreferenced be equated to equation P, expressed another frame generates a result that is the sum of thethe derivative of the vector in itsrown frame reference exercise also shows the manner ofdtinterpreting frame derivates of vec dt ofproper

keycomponents difference.

InB equation (19), term the left of the equality is the G derivative of the position d that b is Ba function GRon and a second term of the a scalar of the vector its andthe the rate of B  B B ownB frame . of reference. theinlatter by the DCM B(15) B  G b  BB b  bi GB B frame represented  b e i premultiplying ethe B b  respect G B  b i  Bb G B frame. and denote frames asderivative matrix multiplication of a the DCM and vector. being: change of the unit with toiWhere Because the unit vector rP, expressed inthe the G frame.

Intwo equation (20), term onathe left of That the equality is the G derivative dtvectors of the expression Where  and  denote two frames of reference. vectors of both frames are by definition unit length, the second term represent a rotation. The derivation Where  and  denote two frames of reference. of the position vector r , expressed in the B frame.

Equation (20) can be equated to equation (19) by P G B G G B B B for the second term, in terms of the angular velocities, is detailed in Appendix B. Using equation (A18): R BofG particular rP  G rP  importance R B B rP  Ginthat  they rP apply for any vector such as p B G B Equations (15) and (16) are G d b G G R . by G Gpremultiplying G G  the G latter G G the G DCM G B   B b  G bG B Bi Ei and  G B  b importance b ofG bparticular (16)  that  they  P i B G  E i  G b  B G  (15) G b  (16) B G are in apply vector such as position, velocity or Bacceleration that is expressed in frame andfor for any which the derivative is taken in d B b B  B Equations dt B  B  one  P apply b  B b  bi GB B (15) e i  BB band  G Equations  GB B(16)  bi ei are  b (15) importance in that they for any vector such as position, B b of G Bparticular dt velocity or acceleration expressed in one frame and for which the derivative is taken in another frame.that Theis general forms are known as the derivative transport theorems.

G B G G B B B G

(19) by premultiplying the latter by the DCM GR .

B

  (21) d r d R R r    dt dt This exercise also shows the proper manner of interpreting frame referenced derivates of vectors (21 tha This exercise also shows the proper manner ofis interpreting    velocity or acceleration that is expressed in one frame and for which the derivative taken in another R r  r  R r    r   B G P G P B B P G B P the angular velocity vectors, the right frame. subscriptThe denotes the rotating frame, the left subscript denotes general forms are known as the derivative transport theorems. represented as the matrix multiplication of a DCM and a vector.

That being: d b frame referenced derivates ofB vectors that are represented frame. the derivative transport theorems.    are frame about which the rotation occurring leftforms superscript the frame. b  b  Bis   E The b and  general  the B E  bknown denotes b   as  Gb (16) dexpression d G B of reference.

B G G Where  band  denote two frames B B B G       a Gvector.         Band That ations (15) and (16)dtare apt for the case of the two frames sharing the same origin.

How does the result      derivates    GGof as the matrix multiplication of referenced aDCM     G B B    G B G shows the proper manner of interpreting B This exercise also frame vectors that are dt dt   d B d B B G G quation (15) compare against the solution from using a Gmore commonly found non-frame B referenced B G G G B d d   G   Gd For the angular velocity vectors, the right represented subscript denotes as the the rotating the denotes    G BG     subscript (23) Bleft (16)   being:   rGPbeing: d B  Gmultiplication  frame, G ofB a DCM and B G r a  Rimportance dt  matrix  P  vector. B  apply   for any vector  (23) such the frame about which the rotation is occurring Equations and the left superscript denotes the frame. dt(15)   (16)  expression B GBof RBinGThat that  GGthey    B particular and are G B B dt dt dtsharing From theorigin. formsHow shown inresult the twodtequations under (23), it is apparent that there are four forms Equations (15) and (16) are apt for the case of the two frames the same does the velocity orvelocity acceleration that isvector expressed one for which the derivative take of equationFor (15) compare against the solution from using a the more commonly found   and of thenon-frame position rP. Thein first twoframe forms derivatives in which the is express the angular velocity vectors, right subscript de- referenced d rP are simple d under   (23), it is apparent From the formsThe shown in theforms two equations that there are four forms for the G B R R r  frame. general are known as the derivative transport theorems.

(22)  ( G(23), P and Where  subscript and  denote two frames reference.

P r )and the that last two forms derivatives w derivative frames are the of same notes the rotating the left the Fromframe, the forms shown in thedenotes two equations under it isB rPapparent there are are fourmixed forms for the in(22 (15)

B

G

G B

G G

G i B

G

i

G G

G B

G

i

i

G G

G B

G

G

G G

G G

B

G

velocity of the position vector rP. The firstdt two forms are simpledtderivatives in which the expression and

G velocity the position vectorand rP.and The first two formsdiffer are simple derivatives in which the expression and  frame about which the of rotation is occurring thederivative expression ( GB rP and G left Bframes B rP ). Returning to the problem posed by equation P and  derivative frames are the same are mixed derivatives in which the G G ( Gr B B rP ) and the last two forms B Equations (15) and (16) of Bparticular importance in that apply for any vector such Pare P B) andB the dG rderivative: dthey superscript denotes theWhere expression frame.

Equations (15) Br Gderivatives G derivative frames are same (G last two forms are mixed in which theas posi taking the B and G and and  the denote two frames reference. denote    B A Gand isB of   B  two  reference.   G(6) differ   of  and   ).GBBReturning posed  derivative    GG isB taken    expression derivative frames (Where toframes the problem Bframe Gexpressed B the G by equation G rP and B rP one G velocity or acceleration that in and for which dtdiffer ( G rP and B rP ). Returning to thedtproblem posed by equation (6) and in ano expression and derivative frames

taking frame. the G derivative: The general forms are known as the derivative transport theorems. taking Equations the G derivative: (15) and (16) are of particular importance in that they apply for any vector such as position www.collisionpublishing.com Collision Magazine - Volume 14 Issue 1 39 From forms Gshown the twoinequations (23), it is apparent that there are four f velocity or the acceleration that is in expressed one frameunder and B for which the derivative is taken in another d d B B G G B B B G G which the exp velocity of theforms position rP.the first two forms are frame.

The general are theorems.  known transport  vector  simple   as  The  derivative    Gderivatives    GG B  in G B  G B B

 

 

 

 

(21) rthe ofthe R Bequality the rPGGderivative equality Btheposition B G P  G rPof B rP  G  difference.InInequation equation(19), (19),the theterm termR on left the derivativeofofthe ifference. on the left isBisthe P position B B R G B GG rPG GG ro B GG GB  GB ro GG rP G (27) n (20). one examines the form ofof the term onon the left ofof each equality, there is is B r ion (20).When When one examines the form the term the left each equality, there , expressedininthe theGGframe. frame.InInequation equation(20), (20),the theterm termon onthe theleft leftofofthe theequality equalityisisthe theGGderivative derivative Pr,Pexpressed     r ro  G rP  R B P G G rP  R G G ro  R G G B quation (19), ononthe is isthe ofofthe equation (19),the theterm term theleft leftofofthe theequality equality theGGderivative derivative theposition position This exercise also shows the proper interpreting frame derivates vectors that position vector expressed inthe theBmanner Bframe. frame.ofEquation Equation(20) (20) canbereferenced beequated equatedto toequation equation (19) by B of (19) Bby G are B G B G G G position vector rPr,P,expressed in can P B R G BG rP  B R G GBro  G R G GG  B Gro  rP the GG frame.

In equation (20), the term onon the left ofof the equality is is the GG derivative in the frame. equation (20), term left the equality the derivative B r Gthe GR represented as In the matrix multiplication of athe DCM and a vector.

That being: R ultiplying thelatter latter theDCM DCM tiplying the byby the B.B.     ro  rP B rP  G rP  G ro  R G G B orrPr, Pexpressed , expressedininthe theB Bframe. frame. Equation Equation(20) (20)can canbebeequated equatedtotoequation equation(19) (19)byby B GR GBR P B GB rP B GB ro  BB R G GGB  B GG ro  GG rP G . B. ter byby the DCM r atter the DCM  B GG B B GG GG B B B B B  B GPrP G rGPrPd R R (21)   r d r r (21) RR rBPrP G   ro  rP Br B G B B B B vP  G v(22) P  G vo  R G G B R rP GBRB  P P  P  B in B Substitution B B forGthe parenthetical G dt G G Equations B B G G G G (15) B B and B dt B B B (16) are of particular importance term from the rearrange ro  G rP P r (21) R BRGB rGP r R BRBBrBP rP G G BB rP rP  (21) B vP  G vP  G vo  R G G B G rG P  P  xercisealso alsoshows shows theproper proper mannerof ofinterpreting interpreting frame referenced derivates vectors thatare are ercise the frame referenced derivates ofofvectors that that they manner apply for any vector such as position, velocity or ment of equation (6): Where  and  denote two frames reference. ented the matrix multiplication aDCM DCM anda avector. vector.That Thatbeing: being: nted asasthe matrix multiplication ofofaof and Substitution for thethat parenthetical term from the rearrangement of equation (6): ws ofofinterpreting referenced of are owsthe theproper propermanner manner interpreting frame referencedderivates ofvectors vectors arewhich acceleration thatframe is expressed inderivates one frame andthatfor Substitution for the parenthetical term from the rearrangement of equation (6): trix multiplication ofof a DCM and a vector. being: matrix multiplication a DCM and a vector.That That being:    Equations (15) and are of particular they apply any vector such as position, ddinrPrthat the(16) derivative in another Thefor general forms ddis taken  importance P frame.

B B G R (22)B r RR rPrP   R (22)    B    G B   velocity or acceleration that is dt expressed the derivative is taken in another B P  G rP  G ro  R G G B R B rp     in one frame dtthe dtdtand for which are as derivative transport theorems. d r d d known d r B B B B G G B ( P Pderivative transport theorems.

(28) frame. The general forms  B G rP BG ro  B R G G  R R  are rP rPknown (22)  R R  as the (22) B B R G B rGp B B rP   dtdt dtdt B v P  G v P  G v o  R G G B R B rp (28) eand anddenote denotetwo twoframes framesofofreference. reference.

B B B B G G B  G B d B d G B B G G B v P  G v P  G v o  R G G B R B rp   G     B     GB B  B    B     G     GG B  G  (23) e two frames ofof reference. ote two frames reference. dtofofparticular dt ions (15)and and(16) (16)are are particularimportance importanceininthat thatthey they applyfor forany any vectorsuch suchasasposition, position, ns (15) apply vector Using equation (B37): Using equation (B37): tyororacceleration accelerationthat thatisisexpressed expressedininone oneframe frameand andfor forwhich whichthe thederivative derivativeisistaken taken another ininanother (23) 6)From are importance in they apply for any vector such position, Using equation (B37): (16) areof ofparticular particular importance inthat that they apply for anyit vector suchasas position, the forms shown in the two equations under (23), is apparent that there are four forms for the Thegeneral general forms areknown known thederivative derivativetransport transport theorems.

The forms are asas the theorems. nvelocity that ininone frame the is isderivatives taken ion thatis isexpressed one frame andfor forwhich which thederivative derivative takenininanother another B B B B ofexpressed the position vector rP.and The first two forms are simple in which the expression and     B B rP  G rP  G ro  G B rp rms are known asas the derivative transport theorems.

G B theorems. orms are known the derivative transport G B G frames B in   r r derivative are the same ( and ) and the last two forms are mixed derivatives in which the B B B B B From the forms shown the two equations under (23), it ( G BP dd B B B B B B GG GG G P B G rP BG ro  BG  B BB P dd G G GG  B rBp B   G G   GG  (23)B r   G G   B B  GB  (23) B  G   B B G B  G B   B B      v v v r   r r and ).

Returning to the problem posed by equation (6) and expression and derivative frames differ ( dt dt B P G P G o G B p B (29) dt dt B G P are four B P is apparent that forms for the velocity of the (29) d dG Gthere B B B B G G G G dB B B B B B   G Gthe GB derivative:    B G G GG GG BB G G  (23)  B (23) B GB GB BB B B  taking B v P  G v P  G v o  G B rp position vector first two forms are simple derivadtdt rP. BThe theforms formsshown shownininthe thetwo twoequations equationsunder under(23), (23),ititisisapparent apparentthat thatthere thereare arefour fourforms formsfor forthe the he Equations (26) inand (29) are based on the case of a non-common origin between the G frame and B fra tives which the expression and derivative frames are the and tyofofthe theposition positionvector vectorrin Thefirst firsttwo twoforms formsare are simple derivatives inwhich which the expression and simple derivatives the expression Pr.P.The n in the two equations under (23), it is apparent that there are four forms for the wn in the two equations Gunder (23), it is apparent that there are four forms for the Equations (26) and (29) are based on the caseThere of (26) a non-common between the G B frame tiveframes framesare arethe the same(( G( rGGPrPand andBB rBBPr)P ))and and the last two forms are mixed derivatives inthe which the the last two forms are mixed derivatives in which the ve same and with point P moving in B frame. areand three simpler cases on colocation of the origin same and and the last two forms are mixed derivaEquations (29) areorigin based onbased the case of frame a non-and nonvector vectorrPr. P.The Thefirst firsttwo twoforms formsare aresimple simplederivatives derivativesininwhich whichthe theexpression expressionand and B B GG B Pand and B Glast BPforms r r ssion and derivative frames differ ( and ).

Returning to the problem posed by equation (6) and r r ion and derivative frames differ ( ).

Returning to the problem posed by equation (6) and with point P moving in the B frame.

There are three simpler cases based on colocation of the origin of the frames and based on whether or not P is fixed in B.  P G P B  r ) and the last two forms are mixed derivatives in which the same ( GG(rGGP rPand r and ) and the two are mixed derivatives in which the ethe the same tives B B P Pin which the expression and derivative frames differ common origin between the G frame and B frame and theGframes Gderivative: derivative: G G he the frames and based on whether or not P is fixed in B.   r tive differ (( GB(rGBP rand ).

Returning to the problem posed by equation (6) and r vative frames differ and ).

Returning to the problem posed by equation (6) and P P ). Returning to the problem posed by equation PandB B with point P moving in the B frame. There are three sime: ve:















 



 



             

 

   

 

 

 

 

 

(6) and taking the G derivative:

pler cases based on colocation of the origin of the frames and based on whether or not P is fixed in B.

G

G G d ro G d G B d G rP G G G G G G   dRBG GB rP B G rPGG G d GG rdo dt d r P d rdt Common d rdt G GG  G G P o R BG rP B origin and P is 1. r 1.  dPCommon fixed inin B:B: origin and P is fixed in B: d r d r G P  r P GG  dt oR B 1.r   origin and P is fixed G dtG PG B B dt B Common    R r r  1.B PCommon origin and P is fixed in B: dt G rP  G ro  dt dt GRP  B B rp B G BB  dt rP B G dt G rP  o  BGrR B Bdt p  GB G Gr Gr   r r B B G G G G B B P   R r  r (24) B BBBB rpB B B P  GG  G(24) rB  GB rPBBr  G PG r  G oG   GBGGRroBB BBp rGp RGBG BRBB BrpGB PGBB  G r  B rP G rG B rG P G B PGr G G PG G roPG B rP G Pr G r (30) G B G B BP B  B rpG  G rP  G r B rP   B rp    G G G G  B r RGB GB R G G  G r B   P(24)G B B P B B P B  G BP rp BBB  R v  G rr P G G RoGG BGr G BB rP B BP  G vBP  G  BB rp B    v  GP  (24) GB rG G B r  G B G G P G P r G o r Br Br p B G Br R P  B v  B B B v r       G G G v r    R    (24) r r   R   G P G vB  P   B rp G PG G oPG BG poG BB GB pBB P B G B  B P GB vPP  GG vBP  pG  P G P G B rP G B  B  GG rro G  GB R rp GGB G BGrG BBB rP B G  GGGrrPoG R G rP B p B G B P  v v v r    R  PoG GB roPG B rp  G PG Gr B G RBBB G  PBB rP G 2.  origin and PP is 2. moving inin B:origin Common P is moving in B: G G B 2.  Common rP B2. Common origin and is moving B: inand G vP  BGr G BB Common R  GGBvvoPG BGvRPGBGB  origin and P is moving B: G vP  G vGo v B P     R  v v r G P G o B P B G B P Br from equation (7): G P B P  GB rGpG GG   G rGP G BB rGPB  GB rPBB GB  BBBB rpB B B If we substitute for r from equation (7): B G rG P rP from equation (7): or GGBG   r r r G rP  B rpr B P B P G P    from equation (7):   r   r r G GrBBP rpG  G G G P G B rG (7): e for BrP fromG equation GB p rGP GG BB BvBPG P GB vBBPB P GB  BBB rpB B B B p (31) GP  B v Gp  G  Gv G G G B B G G Gv  G      v r v v r     G P B vp  G vB  P  rPG roG  B P GB vPP  GG vBP  pG   B rp G rPG G ro G B rpG RG B G BB R G G P B p G B rP G Gr Gr Gr B  Gr  r rP  o  p B   B GB R   R (25) G G B B G G P o    r  r R R r  r G o GG G PG G o GGr p GGr PBo vp BGB GrGpRBB GGBR  BGP3.B R GBB GB R  r  G roorigins (25) rGDiffering  and PP is is fixed B:(25) (25) GB o G vPG  GG v PG Pro G G G G B B origins and fixed 3.and Differing origins and P is fixed B: G Gv  Gv  Gv B R B  G R3. G Differing   roDiffering r origins P isB:fixed B:(25) G v PG  G voG  B R pG  BGR G BB r GB rP G3.   v G G P G o v B p v  B v G B G o r  r P G P G o B p  R B G B R G P o G B B P  GG rGoG GG  GBG G rGP G G rGo G ):  B GB   BB rpB B B rB  GB rPBBr  G rG G roB r r  r  GBB G P G o G r  B rP  rorP B BG PBrorP  r  G Pr G or B36):   Bo rpG   B rp  G P G o  G B Using equation (B36): B B B B P G P G rB G G G G G  v BP  G vBP  G vBo  G  BB rp B  n (B36):     v v r r G G G G G B Bv  B B   r G Pv  G Gov  G G B PG oG  GvP  GB v G G G G G G  r r B P G o G B p  B G P G vo  G vB  P or  r      rPG ro G B vP  G vP  G vo  G B rp G P G o G B P o G rPG G ro G B rpG G B G G Gr G G G G     B rrP  ro  G rro G  B rp  G r (26)  G P  G rPG G ro GG  rB oG rGPro G ro (32) PB GG G rPBopGGBrovG pBGB rGGp (26) G vPG  GG v (26) G G G2.3 PG G (26) Second time derivative and acceleration G G G G G G r  Second  2.3 (26) time derivative and acceleration G vv PG  Gv oG Bv pG G r BG P G2.3ro G  v      v r time derivative andtime acceleration G P G o v B p v G B v The P   B o rP  ro Second ofof the solutions forfor the second derivative follow from the solutions forfor the first G P G o B p The G development Second time derivative and acceleration development the solutions the second time derivative follow from the solutions the fir The development of the solutions for the second timeisderivative follow from the solutions for f equation (7) is taken: derivative.

A double left subscript and superscript notation employed, however, to track the expre derivative.

A double left subscript and superscript notation is employed, however, to track the expr e of equation (7)the is B taken: A double leftThe subscript and superscript notation isthe employed, however, to track th derivative of equation (7)derivative isderivative. taken: frames quation (7) isIftaken: development of the(first solutions forsecond second time deand derivative operation second derivative) ininorder totolink andderivative framesfor foreach each derivative operation (firstand and derivative) order li tive of equation (7) is taken: B G and derivative frames for each derivative operation (first and second derivative) in orde B B B G frame ofof the original position vector.

For a two frame system, there were four velocity ve rivative follow from the for the first time deriva d r dexpression ro  d B rP B B frame the original position vector.

For asolutions two frame system, there were four velocity Bexpression G r B rP  B R GB  G  B dPGBr G that expression frame of the original position vector.

Forsystem, aand twosuperscript frame system, thereis were four vev  B B  d r d (27) could be defined.

For the second derivative of the same each velocity leads to two defini B B tive.

A double left subscript notation P P o d rdt d B B B r G rdt B could P  R defined.

For theFor second derivative of the same each velocity leads to two to defi P P  Bd  GB rdt d G that ro  be P rPB R could be defined. the second derivative of thesystem, same system, each velocity leads tw  BB rd  d roP that B the G acceleration vector and therefore leads to eight different accelerations.

Two ofof these accelera dt employed, however, to track the expression and derivative  G dtGdt ofof   dt B dt B  BGrP dtGR the acceleration vector and therefore leads to eight different accelerations.

Two these accele G G G  of the acceleration vector and therefore leads to eight different accelerations.

Two of these  rP are rodt   Gsimple  B  ro   G rP  G GBG dt B rP BdtR G with the expression frame and both derivative being the same.

Usin frames for each derivative operation (firstframes and second de-the G G accelerations G accelerations with the expression frame and both derivative frames being same.

Us P  BGRrG  GGG  Goare B Br Gr rP GG r BG G rGPare  simple GGGsimple   with the expression frame and both derivative frames being the sam BR G BG raccelerations oG   B r r r  B B G G    vectors b from equation (8) and b from equation (10): B PB G G GB r GG or G BGr B r Bb from  G GP rR G o from rivative) in order to link the expression frame of the origi oG   ro  (8) and   vectors equation (10): rGoP GGP roP  G rvectors  G B  Gb Bequation Gb B  (27) B rP B BRPG G P G B P from equation (8) and b from equation (10): B G G G G B B G G G r  ro GGG  nal position (27) For a two frame system, there were B rP B R B   GPGB GGG rrBoP G GGGG rPo G rP G (27) vector.

B  Gr B rG G B rP B R G B r o G G G G G   r r r r  R        B o rP G RG GPG roGo R G G B P d d d dG(27) G G G G B rP B BRPG G beE defined. second G G G velocity GG  vectors B G B G B G G G  E For  Ethe  Gfour G G B d d d d dthat b b b      Gd G GG B Br G R B r G R B r G R G Gr   dBGiBEdcould r       iE B E3E    d d d  iB G GG i  1B 1E B 2B 2E B BR PB r G GRP GB r GB GRo GG  G G r BG o PG     G Gb GG  b b     3B r r      B B G B B G G  1E 1 leads  2E 2 to B  B P  G G P R G G o R G G B Ro  P r  G GG i Ei   i i velocity 3 3 dt dt dt dt dt derivative of the same system, each two b b b E   B  B B      r r r  r r r r r    R   G GG dt i i i i 1 1 2 E2  B3 E3 B P G G P G G BGG oo GG B  o P dt dt dt dt B PB G G o P B B B G G G dt dt dt of the acceleration dt dt vector and therefore leads to    B Br Br B r G R G Gr   r      definitions B P G P G o G G B o P   G ro BB R GBBG   B GB ro GG rP GG B rPB  G rP Br   GG rvPoG rR roB   BGG   G R  rroPG G rP  B B B B vPPB o G B vPB  G B B G G accelerations. of these accelerations are dB dB dB d dB B different dB dB dB d B B Two B B eight BB B Bv  Bv  B v G R G  G   e  r  r B d B b   B v PB  G vPB  G RoB  G G B o P    db   b b bi e b e 1e  b2 e 2e  b3 e 3e  B B BB   v r  r      G G  d d d d B BB i i i 1 B P G P v G o v  G v G  B B R o G  P B B BB      b b b b e b e b b b 1 e1frame 2 2and3both    dt simple rP  BB b i  expression 3 dt dt dt bB accelerations dtwith  dt   ithe bi iei i  b B P G P G o G G B  ro  B BB bdt i i  b1 e 1  b 2 e 2  b 3 e 3 dt dt dt parenthetical term from the rearrangement of equation (6): dt dt dt dtdt he parenthetical term the from the rearrangement of equation (6): renthetical term from rearrangement ofremaining equation (6): The sixsix second are mixed derivatives.

The final forms are presented here and or the parenthetical term from theB rearrangement of equation (6): derivatives The remaining second derivatives are mixed derivatives.

The final forms are presented here anh B B B G G B remaining six second derivatives are mixed derivatives.

The final forms are presented     B step-by-step  R B rThe derivations for the first four cases shown in Appendix C. For the last two cases, the B rP B G rP B G ro B R G pB B G the Gthe G derivations for the first four cases shown in Appendix C. For the last two cases, tc (28)    Br Br B r G14 R G 1 B step-by-step  40B r Collision Magazine Volume Issue www.collisionpublishing.com    R r the derivations for the first four while cases shown in Appendix C. isFor the same last two B r PB G BG BB pstep-by-step P B G ro  B Gderivative B BvGRoGB GB  BGR B the derivative transport theorem the second derivative inin the fram B P  G P BvG  p GR  GBB GGr R  R rpemploys  (28) r  R r derivative employs the derivative transport theorem while the second derivative is the same fr B v PB  BGr G G B P PB GGrP o G o B B p (28) B B G employs the derivative transport  BB Gderivative B B B B G R G expression R B B rp Bframe.

(28)theorem while the second derivative is in the s B v P  G v P  G v oB  GB R G expression the frame.

B v P  G vBP  G v oB  R G G B the B Grp G G

G

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            

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 

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            

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BB GG G B G B G G B G B GG  GG G B B  B B rB r B B r B  B   B  G b  2 GG B  GG b  GG B   GG B  G b  (36) b  GG d Bd BB dbG G  G (31) b  b (32) B B B B B G B GG B GG G G v  G G G       v r v v r GG G G G G G G  underlying vPB PG3. vPGThe dt dt G The development of theGsolutions follow the time B psecond G B time P B derivative P vo G BB prpthe solutions the d first ddt vP  GGvo Pfor G Gfrom position vector isG b b, Gderivative for  Gfirst  Gthe dB  B   Gand B second v BG frame b  in the G frame.

G B  rP  ro   b  2 G B in BB a  GG a G G G  G G G   GG  BbG  derivative. A double left subscript and superscript notation is employed, however, to track the  (36)  GG dt expression dt dtb BB b  GG G B  b  2 G  B  G b  G B   G B  b  and derivativeorigins framesand for P each derivative operation (first and second derivative) in order theGG  GG  G G G G  G B to link G .

Differing is fixed B:  GG G dGG GGG b GG  G b  GG B GG b G G (37)   b d d G G GB B 3expression Second time derivative and acceleration 3.

The underlying position vector is b, first derivative in the B frame and second in the G frame.  bBB a    b  2 G B  G v  G B   G B  b  frame of the original position vector. For a two frame system, there were four velocity vectors a  GG B b GGG B G G G G e development of the solutions for the derivative second time follow from the solutions the first timedt dt  GG v Bb, first derivative in that could be defined.

For theframes second ofderivative the same system, each velocity leadsGfor todt two definitions GB a  GG a position G  B  b vector G B  Gis derivative the same.

Using the vectors b 4.

The underlying G G G being G G Bis employed, B B however, B B to track the GG GG  G G G G     G B G  rivative.

A double left subscript and superscript notation expression r  r   r  r     rP  G ro  G B Two rp of these of the acceleration vector therefore different accelerations  G and P G o G B leads P Bto eight o (37)         b b b b d d d B rP  G accelerations.

G GB GG G B G B G G G b, firstsecond derivative in the frame and second in the G frame.

3. (10): The underlying position vector is b(32) fromforGequation (8) and b Gfrom equation Gthe and in Gthe BG Bframe. b frame d frames each the derivative operation (first andderivative second order tothe link B B B derivative) B B in the arederivative simple accelerations with and both frames same.

Using the Bb,Bfirst GG GG derivative G G in the G expression G G frame  Bbeing G frame and second in the B frame.

4.

The underlying position vector is     v v v r dt dt dt          a a b v v v r r     B P G P G o G p GG G B G B G G (8) P and G B ob from G vector.

B equation P For a o two frame system, there were four velocity vectorsGB pression the original position b fromofequation (10): vectors Gframe GG  G G G b  B  G b  GG B  GG b (37) b  GG at could be defined. For the second derivative of the same system, each velocity leads to two definitions d Bd GB dG G  GG G B G B (38) b  GGd B bG Bb, GG G theG G frame G disdt d the acceleration vector and therefore leads to eight Two position of these accelerations G G G G G different derivative and second in the B frame.

4. accelerations.

The underlying vector B first B   in dt dt      a a b v d d d d d G G  GG GBb GG G bG B G B G   and both  E  B the Second time derivative and acceleration b expression b  Bi Eiderivative   B i Ei  Bframes simple accelerations with the frame the(33) dt G b  GG 1 E1  B being 2 2 3 E 3 same. Using dt dt  GG  G   G b  G   G b dt dt the seconddttime dt (37)  GG GB b  BGG b BG  B B G B G development of thedtsolutions (8) and Bbfor from equation (10): derivative follow from the solutionsB forGthe first ctors Gb from equation BB  Btime BB B   GG  (38) dBB  G  BGb and d dis BBb, GG Gb the G B frame G second in the B frame.

Bb BG b G  B  in first derivative G 4.

The underlying position vector  (33) a a  G  B  b  G B  G v ative. A double leftB subscript and superscript notation is employed, however, to track the expression bBB  BB GB G b  BGG B B B B B d B d B G derivative d d Bd dt dt to BG G G  the a dt and   operation BB a  G  B  b  G  B  B v derivative frames in order link d Gfor d Geach d G d  b(first b  second b d G  BB bGGBB b1 e 1   b2 ederivative) b3 e3 (34) BB b  i e i   i e i  2   BB BB B B B B  B G (33) B  b b  GG b  Bi Ei   Bi E i  B1 E1  B 2 E 2  B 3 E 3   dt dt dt dt Gdt  (38)        b b b b dBGB vectors B BBd G B in Bthe G frame and second in the B frame. ession frame of the originaldtposition vector. there wereposition four dvelocity dt dt dt dtFor a two frame4.system, first The underlying vector G B derivative b is b,GB b BBis Bb,BBfirst Bderivative B Bin theBB frame and second in the G frame: 5.

The underlying position vector dttwo dt definitions dta  G  B  b  G B  B v could be defined. For the second derivative of the same system, each velocity leads to  BB aunderlying 5. The position vector is Bb, first derivative in The remaining six second derivatives are mixed derivatives. The final forms are presented here andBG with B B B BB  GdBB b   BB B b e acceleration vector and therefore leads toB eight different accelerations.

Two of these accelerations d Bd dthe d d shown dB B B b  GB B  BB b (38) B B first BB four d      BG BB G B the step-by-step derivations for cases in Appendix C. For the last two cases, the first   in the G frame: b i e i   b i e i  b1 e 1  b 2 e 2  b 3 e 3 b and the Bis frame G(34) B B b  BB b  band GBdsecond Gb BUsing imple accelerations with the expression frame both derivative frames being the same.

BB BB the B in Bthe B Bframe and second in the G frame: d d dt dt dt dt dt 5.

The underlying position vector b, first derivative B B derivative employs the derivative transport theorem while the second derivative is in the same frame as dta dt  BBba Gdt  b  G B  B v  BB b BG Gb from equation (8) and Bb from equation (10): ors BB  dt dt BB b the expression frame.  dt   B b  GB B  BB b (38) b  GB  (34) BG BB e remaining six second derivatives are mixed derivatives. The final forms are presented here and BB  B B B  G with G BB  (39) (39) BB  Gthe  frame Bb, disB d dGBBBBb  BB BB  B bin B Bb Band second in the G frame: B first 5.

The underlying position vector derivative        a a b v G G G G G B step-by-step derivations for the first four cases shown in Appendix C. For the last two cases, the first b  b b anddboth are taken with respect to the G frame. BG BBBBB BBG B B 1. The underlying is G B B d d position dvector d derivatives G G  GG   E derivative  E  B  isE inthe  Esame frame dt dtas dt  GB a  BB a  G  B  BB v B B B b  transport b Ei  second (33) rivative employs the derivative while  Bi the G b  GGtheorem i i 1 1 2 2 3 3 BB  BB  B B  G B G dt dt dt dt dt  B  Bin The Gremaining (39) b B BBderivative b  G b the B frame and second in the G frame: G six second derivatives are mixed derivatives. expression frame. d B isBb, GBd 5. The underlying position dvector d Gd B d B b  BB first B bBB b b B B G 6. The underlying position vector is Gb,first derivative in the G frame and second in the B frame: The final forms are presented here and with the step-bydt dt dt    a a  v dt dt dt BB G B B B B B B b and both are with respect to the G frame. G B GB 1. The underlying position d B vector dBBBis BBB d BBderivatives d BB  BB B B  B  G b  Bfirst etaken  B  BB b (39) b  BB  (35) bfor bb b b2i eGBcases b    BB b  BBBB  2B  stepd derivations the  b   GG bB1 e1in bAppendix b3 e3 d d(34) d B G  GB G B  four G G B2 e B Bb BB i B b i shown i  B B  b dt dt dt dt dt B bBin theB G frame and second in the B frame: d GGb, d first d GBBdt BB derivative vector is Gdt G G BB BB cases, B B B 6.derivative BTheBunderlying B B position  dt     a a v C.G dFor the last two the first employs the dB b b G B B vector is Gb, first derivative in dGGB a  BB a  G  B  b  2 G B  B v  G B   G B  b  GB GBB 6. The underlying b dt dt dt BBposition Gb   BB b   B   B  BB b (39) dt dt dt are mixed derivative transport theorem while the second derivative is the GB b G G emaining six second derivatives derivatives.

The final forms are presented here and with GG GG  BB G B G G frameBandsecond in the  frame:  BB  BB  B B B B  B B B (40)      b b bB d d d G BB G B G G  BB GG in BG BBG G 6.

The underlying position vector is b, first derivative the frame and second in the B frame:  (35) b b b b b          2     In equation (35), the first term on the right of the final equality is typically viewed as the   GG BB four B expression G B in Bframe.

G B G C. B b  firstGGGb GB aGG BB aG  G GB  B v tep-by-step derivations for the first cases shown Appendix For the last two cases, the in the same frame as Gthe dt dt dt acceleration a point BB that isBBmoving inB the Bwhile coincident asBG a  GG a  G  B  G v B frame, B B B with B the point of interest. The ative employs the of derivative the second derivative is in the same frameGG a  BB a  GB theorem  GG transport B  b  2 G  B  B v  G  B   G B  b  G B Gfourth   GG b   G  (40) (40) second term is the tangential acceleration. The third term is the Coriolis acceleration.

The   B 6. The underlying positiondvector d G is GBBG db G B  G G  GG xpression frame. b, first derivative inb the G frame and second in the B frame: 1.

The underlying position vector is b and both derivab  GG term is the centripetal acceleration.

G b GG G G unchanged while equation (34) reduces When the point P is fixed in the body frame, equation (33) remains BG a  GG a  G  B  G v In equationtives (35),are the taken first term the righttoofthe theGfinal equality is typically vieweddtas dt the  dt withonBrespect frame.

GG  GG G to zero.

Equation (35) reduces totothe B b  B  GG b both derivatives are with takenthe with respect thefollowing: G Bframe. . The underlying vector is b and (40) b  GG acceleration of aposition point that is moving in the B frame, coincident point of interest.

The d Gd BG d  G  G b  GG GG bG GG G second term is the tangential acceleration. The third term is point the Coriolis acceleration.

Theframe, fourth When the P is fixed in the body equation (33) remains unchanged while equation (34) reduces dt dt dt     a a  v BG GG G B G G (35) to Gthe G G term is the centripetal d Gfollowing: dB d B GG  GG  G G to zero. Equation (35) reduces d G d B acceleration. d B  b (40) G b  BG b  GG b  G B  G b b Gb dt dt dt dt dt dt GG GG G G When the point P is fixed in the body frame, equation (33) remains unchanged while equation (34) reduces      a a v BB G G BB  Gthe G B BB  BB  B B  B B the fixed body frame, equation (33) (41)  PBG d point isB b GGGB in b  B d GWhen dB  B  Bb  B   G B  b  to2zero. reduces  GG b  BB b  GB   GB  G B Equation B b  G  B (35) B  b to the following: b  GGGB b(35)G B remains unchanged while equation (34) reduces to zero. dt dt dt BB B B B BB BB B point B P is fixed B  B    B b (33) remains b in Gthe bodyG frame, equation unchanged while equation (34) reduces GG a   GG a  BB a  GB  B  B b  2 GB BWhen  BB v the BB  B G BB B G B B G BB G  B  G B  b G  (41)          b b b d Equation d to d B(35)  G following:  B the B B theGG  G BreducesG to to zero. Equation (35) reduces b  following: b G BB dt dt  B  B b  GB B   GB B  B b  aas  GG  GBthe Equation (38)equality reduces is to typically the dt following: In equation (35), the first term on the right of the final viewed G G BB   B  B b  GB B   GB B  B b   dGGB b  GGB d (41) (41) d In equation (35), the first term on the right of the final acceleration of a point that is moving in the B frame, coincident with the point of interest. b  TheGB b BB B B B B B dt dt dt equality is typically viewed as the acceleration of a point (38)isreduces to theacceleration. following: The second term is the tangential acceleration. TheEquation third term the Coriolis fourth GG a  G  B  b  G  B   G  B  b  BB   B  B b  GB B   GB B  B b  (41)  GG b  GB  term is the centripetal acceleration. that is moving in the B frame, coincident with the point

2.3

2.

 Bro  P B rPB P G rP G PG ro G G  P G BB  p rP G B B prp G rP  G rG o  Second time derivative and acceleration G G G G B B B B G



G G

G G G G  r  r G r G  

 

B

B

 

 

 

BB (38) reduces to the following:  GG a  GB  B  B b  GB B   GB B  B b  of interest. The second term isEquation the tangential acceleration.

The third term is the Coriolis acceleration. The fourth term is the centripetal acceleration. Equation (38) reduces to the following: Equation (38) reduces to the following:

2. The underlying position vector is Gb and both derivatives are taken with respect to the B frame.

B B d Gd B d B  BB  B BB  B  B b  BG b b  BG b  G  a  GB  B  B b The underlying position vector is Gb and both derivatives are taken with respect to the B frame.dt G dt dt

(42)

(42)

(36) B B B (36),to (37) (40) remain unchanged. Equation (39) reduces to zero valuation. d G is Gb d G  both derivatives are takenEquations nderlying positiondvector with respect theand B frame.

B B (37) and (40) remain unchanged. Equab  and b Equations B d G d (36), d B  BB  B B B BB B B dt dt dt  b (42) G b  BG b  G  B  b  BG a  G  B  b tion (39) reduces to zero valuation. dt dt dt GG  GG  G G G G  G G G B B B B G  B  b  2 G  B  G b  G B   G B  b  (36)B d d dG d G   BB b  GG b  G  d d B B  the BB  B B BB B Following the derivation for the velocity, standard derivative ofB equation (19): b  time Bb b  taking (42) G b  BG b  G  B  b  BG a  G  B  b GG G G G G G G G dt dt dt dt remain dt the dt Equations (36), (37) and (40) unchanged.

Equation (39)velocity, reduces totaking zero valuation.  GG Following derivation for the the stanBB a  GG a  G  B  b  2 G  B  G v  G B   G  B  b  GG  GG  G G G G  G G G GG G BB G   B  b  2 G  B  G b  G B   G B  b   BB b  GG b  G  rP  (36) P  GB   B  B rPof (43) Rtime  GBequation  B  BB rP   (19): R B  BB rP  GB  B  B rP  GG dard B  BB rderivative (36), (37) and (40) remain unchanged. Equation (39) reduces to zero valuation.

GG GG G GG G G G G Equations G 3. The underlying position vector is b, first derivative in the B frame second in the G frame. b  and the BB a  GG a  G  B  b  2 G  B  G v  G BG  G  B  derivation for the velocity, taking the standard time derivative of equation (19): 3. The underlying position vector is b,Substitution firstFollowing derivative in for the rate of change of the DCM: the B frameGGdand second in the G frame.

B G GG G BB B G  d G derivative d G  in the B frame and nderlying position vector is b, first second in G frame. rP the   B  B rPthe (43) Following thethe derivation Rvelocity,  GBstandard  B  BB rP  time R B derivative  (43)(19):  BB rBP  GB BB ofB requation GGfor B  BB rP  Gtaking P b GG G BB B B B B B Bb     r  R r   r  2  r      (44) dt dt dt  G B  rP  GG P B BB P G B P G B B P G B GG  G G (37) G G GG G BB  G  (37)  GG  GG b   change rP the  B  B rP  GB  B  BB rP   G R B  BB rP  GB  B  B rP  (43) d Bd G d G   GB b R B  BB  GB  for the rate of DCM: GG b  G  B  b  G  BSubstitution GG rPof b Bb Bb to Br shows that the outer parenthetical term in equation (44) matches GG GG G G G G Changing the name of the vector Substitution for the rate of change of the DCM: P dt dt dt  GB a  GG a  G  B  b  G B  G v GG BB B B vector B expressed B B in theB B frame.

B G derivativefor ofthe therate G derivative the As a result:   GG b   G   rP position   rP  G Rof DCM: (44)   G b  G   G b theSubstitution of change the (37)  GG b GG BofBB G B  rP  2 G  B  B rP  G B   G B  rP 



GB

GG

G

B

G

B





G



GG GG G G inG the G frame and second in the B frame.

The underlying position vector is Bb, Gfirst derivative GG G BB GB a  GG a  G  B  b  G B  G v  P B parenthetical R Bthe (45) G BB B B B B GG GG router to term in equation Changing the name ofGG the  vector rP BrPGBshows   B rP Brthat R B BbBB  (44) (44) (44) matches GG rP  P  2 G  B  B rP  G B   G B  rP  B G B the G derivative of the G derivative of the position vector expressed in the B frame.

As a result: d dderivative d Bin B the G frame and second in the B frame. nderlying position vector is Bb, first  b In comparing the forms of the acceleration vectors, it can be seen that the DCM will directly change the Gb  Bb to Br shows that the outer parenthetical term in equation (44) matches dt dt dt Changingframe the name thesource vectorframe P GGboth Gof the BB expression frames of the acceleration vector expression whenofits matches  rP R B GG (45) BB  BB  B B B B  GG rP  B G B  B  b  G that the derivative of the G derivative of the position vector expressed in the B frame. As a result: (38) bG d dB d BBG b  BB b  G  B  Bit premultiplies. b b  GBB www.collisionpublishing.com Collision Magazine - Volume 14 Issue 1 41 BB dt dt dt BG a  BB a  GB  B  B b  GB B  BB v In comparing the forms of the acceleration vectors, GG Git can BB be seen that the DCM will directly change the rP  GG    BB  BB  B B B B  r  R (45) B GG  P R    (46) rPthe (38)source frame matches expression frame when its of expression frames of the acceleration vector  rP  both BG b  BB b  G  B  b  G  B  B b

4.



BB

BB

BB

B

B

B

that it premultiplies.



dt dt dt dt    again shown in Appendix C. step-by-step process is  a    rr r 2RrR  Rrvr r   r r 2r2 r2 r r  r  r r  r  a (42) R  a    r 2   v   r  rr   R   2 rv   2v  a  a R   R a  a      r  rr r2  r        r  the velocity, taking the standard time derivative of equation (19): a  expressions R  a ofthe angular r  2 kinematics   v terms.    The   r  We may rewrite this equation in terms of the B frame   r 2   v      r a  a  

remain unchanged.

Equation (39)time reduces to zeroofvaluation. the velocity, taking the standard derivative equation (19): BB B BB a P  R G B G B

G

BBB P|o G G GB P|o BGG G P|o GG B G G G GB GG B G P|o G BB GP G G GG G G GG P|o G GBP|o B P|o P|o B G GB B G BP|o P|o BB P BB GP G GG G P|oB G G BP|o G G B GG G GP|o P|oGGG G G BB G BB P|o BB B GG G G GG B GG G GG G G G BB P B P|o B GG P|oB G P|o B G P|o BB B PG GG G GGG BB P|o GG G GB GBB G BGGP|o G GP|o GB P|o BB G BBB BG BG BG BGB B P|oPB GGGBP|o P|o P|oB B G G BBP|o G P|oB BB P BB P G GGG P|oGG BB G G B P|o G G G GB B G P|o G GG BB G GG P|o P BB B GG G G G P G G BB P G GG P|o G B P|o G B G P|o G B G B P|o BB BB B B B B B B B BB P GG P|o G B P G B G P|o G B G B P Equation (39) reduces to zero valuation.

GG GG

G G

P|o

B

(4 (49

B

d dB d BB  BB  BB  B b  BG  GB  b  G  GB bB  a  GB  B  B b (43) BB B B B B   R B B rP BG  bGB  BB rP  G  B  rP  G  B  rP dtB dtB rP  dt





We may rewrite this in Appendix C. equation in terms of the B frame expressions of the angular kinematics term quations (36),DCM: (37) and (40) remainstep-by-step unchanged. process is again shown ange BB of the We may rewrite this isequation in terms ofBthe B frame expressions ofangular the angular kinematics term G We may rewrite this equation in terms ofinthe frame of the kinematics terms.

T  B  B rP  GB  B  BB rP  G R B BB rP  GB  B  B rP B step-by-step process again shown Appendix C.expressions (43) R B BB rP  GB  B



   tostep-by-step may rewrite this equation in terms theAppendix B frame of theas angular kinematics process is again shown in C. can be made Changing the name of the vector b We r step-by-step shows that the The following simplifications a function of terms. Th process is again shown in of Appendix C. expressions following a function of origin colocation and whet r  P r  process (44)      can  The risagain 2  shown  rsimplifications in Appendix C. rbe  outer  madeorasnot step-by-step R  rthe derivation  r parenthetical 2 the  r  taking   the  r   time   velocity, term  ollowing for standard derivative of equation (19):origin in equation (44) matches the G colocation and whether P is fixed in B. (50)     hange of the DCM: G

BB BB P

B

B G

B

B

B G

P

B B P

B

B G

B G

B

B

B

BB BB P

P

BB GG P|o

B G

B

B

B G

P

B G P|o

B

B G

B

B G

B

B

P





BB BB rP|o  GBB   BB rBPB  2 GBB   GBB rP|o  GBB BB  GBB   BB rP inB B. BB BBBr P  GG BB BB BB BB B BB B B B BB BB B B         r  r   r  2  r    rP          r  r   r  2  r    r a a r v r         2     BB P GG P|o G B P G B G P|o G B G B derivative of the G derivative of the position vector exBB P P|o GG B B GP B G P|o G B G B P BB P GG P|o G B P G B GGG P|o G PB BB B BBBB BBB B B G BB B Bb to Br shows GG rP  PrP|o (5 rP the rP parenthetical BP|o  B  B rPB  GBterm  (43)   2GBGfixed r  GBGBBBBB  B GrBBP Router  GBB  BB  BBin rP equation  G R B  BB rP(44)  GB  aBB vP|o  GG  BGBBr 2 GBBB rP  Gin that matches ctor 1.

Common origin and P B GG B  BB B  rP  BB GG Ga B P P|o BB: B PB Bis B B  B rP BB BB B B G BB B B B BB B B B B 1.

Common origin and P is fixed in B: in the B frame.

As a result:  pressed      R B BB rP  G   r  2  r    r (44) a a r v rP     2              a a r v r     2       (50 B P vector G B expressed B P G Bin the G B B frame.

P P P|oGG G P|o B G BP P P|o B G GB B G BP BB P BB GG G B G G B P|o G G ivative of the position As a result: BB BB B B B B  B  B rP  2and   GB vP|o  or  B rP The following simplifications can be made as a function origin isB fixed BB a P  of GG a P|o  G colocation G B whether G not B P G ubstitution for the rate of change of the DCM: GG Gcan be B made B as a B function B Borigin colocation BB G whether G G not P G The following simplifications of and in B. P  be  B R Gwhether  B as  B or GG BB R Bmade  arPfunction  aG   origin rPcolocation (51) rPnot or  colocation GG rcan G made B  of G B of BB rP and G B fix outer term in equation (44)following matches ctor Bb to BrP shows The simplifications can be as function and whether orGP not P G  thatG the B TheB following simplifications origin is R B BGG rGPd B parenthetical (45) GG rP  (45) d  d B  BB  B in B. B BB B         b b b b a b   (42) GG vector G expressed BB B Gin the B BG B frame.

B B B As B result: B simplifications can be made as a function of origin colocation and whether or not P is fixe G BGin Gin B BB. The following ivative of the position a    GG G B B B B B BB B G G G  B.     r  R r   r  2  r   r (44)  G B inP B:  BB P 1.dtG Common dt Bdt GG P B P G B B P G B R B G  B  rP  G B   G B  rP  BB a P  R G  G  B  rP  G B   GG origin and GG a P  B.P is fixed he acceleration vectors, it can be seen that the DCM will in directly change the 1. Common origin and P is fixed in B: G (40) remain BB Equations (36), (37) and unchanged.of Equation reduces 1. to zero InGG comparing the acceleration vectors, it origin can origin  vector Rof rbPthe (45) 1.valuation.

Common P is fixed B Br forms Common and Pand is fixed inG B: in B: ource frame matches the expression frames of(39) the acceleration vector GG P both B GG to outer term (44) hanging the name ofrthe P shows GG that the G B parenthetical B B B in equation B BB matches B G G G   and Common r  R   r      rP2. and Rorigin  B: BB: G rP   1.

Common origin Pris fixed GG P B G B P G B G B BB P  G in G B  PrPismoving G  B   Gin be seen that the DCM will directly change the expression he G derivative of the G derivative of the position vector expressed in the B frame.

GG As aGresult:B B B B B BB  r  BB RGG  GG    (51) r  R  r    r  G rP  GG GB   GG   GG rP       GG B G P G G B P BB BB B B G BB BBB P GP(19): BB GB G B he acceleration vectors, it for can be seen that the will directly change the GG Gmatches Btime B B BexpresB GBR BB G  BG G G R    Following the derivation the velocity, taking theDCM standard derivative ofGG equation rPR rPR arBP BGBBr  rBP GGGr frame when its source frame both of the      r    rP           r     r r      r          a R  r    r R   r     r GG B G P G B G B P BB G G P G B G B GG P B G B B G B P BB P G G B B G B P     GG P G B G B P G GG B G GB BB P B G B GG G B B GP GBB B GB B B B B B G GG BB   BP GG B R B BB G   frames aRBGBBrGGPRP ource frame matches the ofrPthe acceleration R B GG (45)  rrPP GBB  both rP expression vector (46) R of RB B BGBr GB2G GG GB GGG GrBPBB rGGPrP  (5 BBr G B rBPBBGGrGPrPB G  GG rP  P r B BB B B aGPB GGG rBP GBBGGG rBB   rP frames  GG BrP  GGR P B P B  G sion acceleration vector it G PBB P B B B B B BB B G G G GG Gof the BB B B B B G that B Bpremultiplies.

GG B B B B B B G G G G   B  rP  G  B  B rP   R B  B rP  G  B  ra R B  BB rP  G    a PRB GRBB G(43)  G GrBP  rBB rBP  Gr GG rP   aP BBBBaPRG BGRGG BBGG rBPGB GrP B GBGGBBGGBrBP B rP  (51 P  G PBB GG P P GG B  GB B GG G B B B BGG B a aPPmoving r B  vPB  GB in R B PBB 2 G  GGBB G rP B  rP      R     GR a r a rPB GB 2.

Common origin 2.

Common origin and P is moving in B: GG P rand BBis   nsed comparing the forms of the acceleration vectors, it can be seen that the DCM will directly change the GG P B G B P G B G B P BB G PB: by equation (6) and differentiating twice with respect to the G frame: source  frame Substitution forthe rate of change of the DCM: 2.

Common origin and P is moving in B: xpression frame when its matches both of the expression frames of the acceleration vector  (46) R   rP (46)  rP  2. Common origin P is moving in B: 2. BB Common origin and Pand isB moving in B: hat itGpremultiplies.

GG G B B B B BBB: B GG G G G G G G GGG    r R r   B rP and 2 GB   moving G  BB B B B B GG  B  2.

B B Common B origin P(44)  G (52) G   P BB P G B Bis B rP  G in B  G B  rP  d Gd G d d     G BB rP B R G BGG rP  BG  B  B rP  B2 G B BG rP  BG  B   G  B  rP  GG G BB GG rP B R B  BB rP  G B  rP  2 G  B  B rP  G B   G B  rP    by equation ro  (6) R B and rP      (52)     r  R r   r  2  r    r  differentiating Psed   twice with respect to the G frame: GG GP G GB B B P G GB B GBB B BBBB P BBB B BB PB B BB B P   GG equation G BB B B  dt dt rBP  Returning (6) dif-B rGG  posed by r   rGPP r    2 B BGB   dt dt to the  problem R   2GG rBP BGBGr BB GB GrBP  G G rPRB BB(46) GraB a P and vBBRrBP GBB R B BB GG P  GGB P  GG  rP   rPGG a P  GG B  B  G  B  GG P  P2 G B  Br a B GGBB  v   GGBP  G B BBG P B BBG  B  B G Bb to Br shows that the outer BB P B B R G GG B B B rP  2 B   G B  rP  GG BB B P B G B P parenthetical term in equation (44) matches theB nameBof the vector PB    (5    r  R r   r  2 r    r G Changing BB B B B B   BBB   GBP   B rP  aGPB BBGBBR GG P P B GBB aBP  G P B  GB rPB  2B G P B  G Bv B P GG ferentiating twice with respect to the G frame: o  GR    GG B (47)     r   r  2  r    r G G G B B B GG B B B B B B   BBderivative P G B of thePd B G vector B P the G derivative expressed in the B frame. As a result: a  a R Ra  a    r  2r   B ofP theGposition dB Gto d dG BbyB equation BG rBP  rP  (52 G G GG P B BB B P BB G P B G BP P  B2 G B PB v P  GGB GG GP frame: G Bv G B  eturning the problem posed (6) and differentiating twice with respect to the  r  R r GG B B B B B B B P o P  GG G G r G2   v       r  G B B B B G G RB G rBB GG BB B  P P is fixed P  B  G   aP GGGBand 3. a PDiffering rGP  B inBB:P G B G B a o  dt R Bdt BB  dtrP dt  2 GB B  BB vP  GB   B rBB GGB   G G  B BB B G Porigins G B BB a P  G  B  P rP  R G GG rP  G  B  rP  2 G   R B GG rP BB (45) BB B GG G G G G G G GG rP  (47)     (53) G r  R r    r  2   r      GG rP   G BB B G d GBd BGd Gd B B Bd G d B BB P G GG P G B P B P B B G G G G G   G G G G B  B  rPG r 2G B  B rGP r G  (47) G G G BB  G B G GG G G G ro  R B BB rP  G   a P  B R G GG rPR  rBP  G2rrGGP  BP GGGr RPGrP  rP  2 G  G B   G B  B r rBP  BrP  B R rPPBB       a r BB GG G P  G BBrP2  GGv GG GB B  B Gr B GB B  P o B BB GG P G B P G B G G B P GG GG G B B B GG G G the can be seen that the DCM will directlyBBchange B  G dt forms dt of the dtacceleration dt dt itdt rP2GGGGG   GGG   R GG rP  GG  BBBthe aBP  GGGG  GGGrPB rP  (5 RPG  GG r r itten GIn in comparing terms of the angular kinematics ofvectors, G frame expression by substitution Gv B  Rr  GG rP Bro2GGGr B B B BB rP BB a G P B P BGG BB B BGGPG GG GG G G G GB GPG B frame when of theB expression frames of the acceleration vector BB B G G B GP a  expression rPits source R B BB 2GGB B frame BBBB vP matches B GB B Bboth  BB a P  GG G  B  GG G  BB rP B      R    2     a a r v r Bvelocity B B  R    2         a a r v r (53   or ofrom equation (7), kinematic transformation of the angular vector BB P G GG P G B P G B G P G B G B BB P G (47) GG P G B GG P G B G P G B P B P      B G B BG B that it premultiplies.

BB B GG G G GGG G G GB G GG rP  GG ro  R B  BB rP  G B  rP  2 G B  B rP  G B   G  B  rP   rPB G rPB  (53) R G B B G  Bao  rPaP 2GGG 3. acceleration Differing origins and Pequation is fixed in B: G  B  rP  BB a P  R G GG a P  G  B  GG G vP B G  ematic transformation of the angular vector using (C4).

GG GG G BB B B B B B B B Differing origins and P is fixed in B: a  a  R a  r  2  r   B   G B  B vP  G  B 3.     GG P GG o B BB P G P G B P   (46) r  R r itten in of theCangular kinematics of GPframe expression by hown in terms Appendix and results in:  P 3. Differing origins P is fixed B: 3. substitution Differing origins and Pand is fixed in B: in origins GG G B B B B and P is fixed in B: BB  B GG or from equation (7), kinematic transformation of the angularGG rP  vector  3.B rDiffering B   in 3.velocity Differing origins r  G    BG rP|o BGG B   GG B  G rP|o  GG GG ro  R B G Band PPisGfixed G B: B  rP BBGrP  BR G GG GG GG B P|o B G B Returning posed equation (6)Gangular and differentiating twice with respect to the G frame: G G transformation G to the G problem Gthe Gby G acceleration G G using    quation (47) may also bewritten in terms of  the kinematics ofequation G frame expression by substitution   r  r  R  r    GB  (54) r ematic of angular vector (C4).    GG P GG o B G B P G     r   r  r 2  r   r  r G B B B B P GG B BB B B  position  P transformation GG G B GG  B B G B BR B P G B P o G B B P G B B rPr rBP GGGBr BGG  rPG rBP  rGP  BB BG  GGB G Equation (47) also(7),Gbekinematic written ino  terms ofGGofthe GGBGG BrR  anguR B GB velocity athe rGG GG GG o BB  P  G fhown the B in frame from may equation angular vector o G B r G B   Appendix Cvector and results in: GG a P  GG o (48) B  GG P      G rP|o  G BaPG BR B Ba P|o  B B B rP|o  BG  B  (54) G G G G G G BB P G GG G B GG GG B B  GGaro   R rPB G rBP GG BB   GrPB  rP  G B  G acceleration vector using equation GG G G G G (5 RaBo G  d d G G G G of d BGangular GG rP  sing equation (B35) and the  2d GGG   rGo  B G dexpression ro dkinematic vP G GG GG P  GG G lar of frame substitution of GBB B G BB BB B B BB B rP B transformation  BR  GG (C4).

BB a P  G  B   r P  kinematics B B   rPrP   ro by   rP  R BG rBP  rP   aGGP ao GG  aR (54 dtis dt dt dt dt dt o  GGB GG a P GG B G BB G rBP  G  B   Gin: he shown in C G step-by-step G G process G G G Appendix G G and results G GG GG G B B B B B   B GG G (7), kinematic  rB 2  position r   vector   from r equation r BB a P G GG a o G RB G rG B  rP  G B   G B  rP   frame B  B rP  G B   rPthe o GG rP|o2.4  B Kinematic  GG  rGG  GGG B GB P BBG B B  G B B  B P B  o  BB (47) rBP   GB RB GB rPGG(48) P|o  G  B  relationships G B P|o GG rP  GG ro  R B  BB rP  G B  rP  2 G B  B rP  G for more than two Gframes of reference BB GG ion (7) Gand differentiating twice to the BG frame:  B R G GG rGP|o  GG    GG rGP|o  GG GB   GG  (55) rP|o  of the angular equaGGG transformation GG GG G G with G respect G G velocity G G vector G using G G GG G GG G G G G BBBrP  BGG GB G ro G areG BB G  G which G situations in one may faced with arBBP|o problem that involves more  rPrPGGrGGo aP BB2GGGGrGPaoBGGBBRvB P BBBBarPGP  BGB roBGB2rPBG2BGBrBPBrPBB vPrGBBBBoaGBBPBGBGBRB GB BGG rrPPa There  BB a P  G  B GG   r  R r    be rP|o  BG   G rP|o  (55)  r  R r    r    r      r PB G  GG GP|o GG G P|o GGB BB P BB B G BP|o  G B  GG P|o G B P|o G G B P|o (48) BB B GG G G G G G tionGG(B35) andG kinematic transformation of Gthe angular B  GG  Gr Gof Gr which G rP aBBcase P|oN reference (i.e. one frame is the inertia (5 RPG  involving rP|o GG frames  P|oBreference GGGBBof GG GG G G G G G G G  a R a   r   r BB GG G B P|o G P|o        2          a a a r r v r r BB G G B G B P|o B G G using  P and  Gstep-by-step BB GG G G G GG P GG o BB P G B P o G B B P G B B o  aBare  R Ga P|oGGaGGP|o Bframes  ofrGGP|oreference).  BG   relationship rP|o  a  R   r  (55 acceleration vector equation (C4).

The Equation (47) may also be written in terms of the angular kinematics of G frame expression by substitution BB P G rBP|o GGB BP|o N-1 frames non-inertial The between the BB P G GG B tion (7) and differentiating twice with respect to the B frame: BB B GG G G G a P two R GofGG a P|o reference rP|o  GG    GG by rP|o  of the B frame position vector from (7), kinematicrelationships transformation of the angular velocity vector BB G  B  frames B  equation point P in any the reference isBgiven (6).

2.4 equation Kinematic for more than two frames of process is shown indifferentiating Appendix Ctwice and results in:tovector milarly, starting with equation (7) and with respect the Busing frame: using equation (B35) and kinematic transformation of the angular acceleration equation (C4).

2.4 Kinematic relationships for more than two frames of reference There are situations in in: which one may be faced with a problem that involves more than two frames of The step-by-step process is shown in Appendix C and results 2.4Kinematic Kinematic relationships forbemore than two frames ofinvolves reference 2.4 There relationships for more than two ofthat reference Kinematic relationships for more than two frames of referareofsituations inofwhich faced with aframes problem more than two fra reference (i.e. a case involving N frames reference whichone one may frame is the inertial frame of 2.4 Kinematic relationships for more than two frames of reference reference There are situations in which one may be faced with a problem that involves more two are situations in which one may be faced with a problem that involves more than than two frames GG GG GG G G G G G GThere G G G ence reference (i.e. a case involving N frames of reference ofposition which one frame is the inertial frame of fra re     G    BN-1   rPframes  ro   2are ro reference). non-inertial The relationship between the vector of a  GG rP  GG ro  BB rP and G B  B rP  G  B   G frames B   rP  of There are situations which one may beoffaced with of a problem that involves than two reference aincase involving N frames frames of reference). reference of The which one frame ismore the inertial frame ofvect reo reference (i.e. frames a(i.e. case involving N frames reference which one frame is the inertial frame offrames referen (48) and N-1 are non-inertial of relationship between the position GG GG GG point G G any G twoGof the G reference G G frames G G given by equation (6).

P in is a  a  a  r  r  2  v         r r (i.e. aP frames case N frames of reference of which one frame is the inertial frame of referenc P in point GG P GG o BB P G B P o G B B Preference G B N-1 G B o  involving and N-1 are non-inertial frames of reference).

The relationship between the position vect and frames are non-inertial frames of reference).

The relationship between the position vector o are situations which one may any two ofThere the reference frames isin given by equation (6).be faced with a and N-1 frames are of non-inertial frames of reference).

The relationship point P in two any two the reference frames is given by equation point P in any theofreference frames is given by equation (6). (6). between the position vector of problem that involves more than two frames of reference Similarly, starting with equation (7) and differentiating twice with respect B frame: point P to inthe any two(48) of the reference frames is given by equation (6).











 

 

        



 



  





















 



   



 





       

           







 

            

(i.e. a case involving N frames of reference of which one frame is the inertial frame of reference and N-1 frames are non-inertial frames of reference). The relationship between the position vector of a point P in any two of the reference frames is given by equation (6).

(49)

B  Bd Bd G  d Bd B rP  B R G  rP|o  dt dt dt dt  



                  

Similarly, starting with equation (7) and differentiating twice with respect to the B frame:

B B BB d Bd B d  GG B  G rP|o  2 GG B  GG rP|o  GG B   GG B  G rP|o    B d GG rP  BBB rRP G R G  GG rGP|orP|o  dt dt dt dt BB GG G G G G G G B G B

    







       

      

             

 







 

 





 rP (49) ro   R   rP

(56)

(56)

 a    r  2   v       r  r  R  r     r  2   r       r   (49) the position the reference the  frame in Where  and  denote the reference frames and Arand Where denote the of reference frames point and Brof is the o is B Ao We may rewrite this equation in terms of the B frame expressions of the angular kinematics terms. The a  R  a    r  2   v       r   position of the reference point of the A frame in the coorstep-by-step process again shown in Appendix C. in terms We ismay rewrite this equation ofthe B frame the coordinates of the frame.

Forexthe case of N = 4 (denoting the G frame as 0 and the other frames, in BB BB P

BB BB

BB

aP  R G

B

B

P

GG

G

GG GG P|o

G

GG GG

P|o

G

G G

B

G G

P|o

B

P|o

G

P|o

G G

P|o

G G

G

B

G

B

G

P|o

B

G G P|o

B

G G

G G

G

G

G G

B

G G

P|o

B

B

G G

B

G

B

P|o

P|o

G

B

P|o

dinates B frame.

For the known case of N 4 (denoting the pressions of the angular kinematics terms.

The step-bysequence, as 1, 2 and 3) where the position vectorofofthe point P is initially in=the last frame:

BB B  of  B BB frame   GG  GBthe rP  2 GB expressions  GB rP|o  GB Bof   the  B rP  y rewrite this equation BB inrP terms kinematics terms. asThe BB G B angular G frame 0 and the other frames, in sequence, as 1, 2 and step process isrP|o again shown in BAppendix C. (50) -step process is again shown inaAppendix BB   B rC. 2 B   B v  B   B   B r a  BB  B BB

P

GG

P|o

G

B

P

G

B

G

P|o

G

B



G

B

P



3) where 2 2 rP  2the r3o position R 3 3 rP vector of point P is initially known in the last frame:

(50)

     r 

BB BB B B B B B B B The following simplifications can be as colocation and BB P GG P|o G made B P a function G B G of P|oorigin G B G B P whether or not P is fixed

r 

in B.

r     r  2   r

 B  B rP  2 GB B  GB vP|o  GB B   GB B  B rP  a P|o  

(50)

rP  1 r2 o  1 R 2 2 rP

(58)

0 0 owing simplifications can be made as a function of origin colocation and whether or not rPP is  fixed r1o  0 R 1 1 rP

(59)

1.

BB BB

BB GG

aP 

1

(57)

B G

Common origin and P is fixed in B:

   r       r  r  R     r       r  (51) 42 Collision a  R    Magazine r   - Volume r   1 a  R     r   www.collisionpublishing.com     r    14 Issue Common origin and P is fixed in B:Substitution of equation (57) into equation (58) followed by the substitution of the resultant into equation GG GG P

r  G R B

GG GG

P

G

B

B G

B

B G

B

B

B

P

B G

B

B G

B

P

B G

B

B G

B

(59):

B

B

P

BB BB P

B

P

BB BB

B

P

G

G G

B

G

G G

B

G

G

P

G G

B

G G

B

P

G G

B

G G

B

G

G

P

P



     r  r  r P v  r (56)   R  r    r   R  r  r    rPPthe  position roo  RRof rthe  P r is referenceofpoint of the  frame in the (56) erence frames and o position the reference point of frame in  denote the referenceframes and ro is the       R  v    r  R    v  r    ro  the R  GrPframe as 0 and the other frames, inv  v(56) For the case of N =the 4rand P(denoting se. thereference  frame.

For case ofisNthe = 4position (denoting the reference G frame as 0 and the  other frames, in teofthe the frames of the the point of the the frame in the position of reference point of  frame in te reference frames and rroo is e2the position vector of pointvector P is initially known in the last frame: and 3) where the position of point Pthe is initially inTaking the another last frame: Gframes, derivative of e  frame.

For the the case case of of N = 4 (denoting G frame frameknown as 00 and and the other inequation (65) provides the general acceleration solution: ete frame.

For theofG as the the position the reference point ofother the frames, frame in the reference frames andNr=o 4is(denoting 3) where the the position position of last frame: frame: (66) 2 2 vector 2 3 point P is initially known in the(57) d d d vector knownasin0the 2 rpoint 2 P is2 initially 3 rP case r3o  R 3of (57) r  rin  r (57)  e3)where frame. For the of N = andlast the other frames, rP4P (denoting r3o  R 3the rP G frame dt dt dt 2 2 point 2 P 3is initially known in the last frame: 3) where the position vector of   r     r    r   a  r (57)   R    r   2 r  2 r 2 R 3 r P 3o 3 P 1 (57) 2 3 rP1 (58) rP  1 r2 o rP 1R 2 1r2r3orP  1R (58)   (58)   r  R r      r    R     r   r  r  P 2o 2 P 2 2 2 3 (57) 1rP  1r3o  1R 3 2rP 1 r 1 r 1 R 2 r (58)  0 (58) rPP0R 0r122roo  0R 22 rPP0 1 (59) rP  0 r1o  (59) 1 rP  r  R (59) P 1o 1 rP Discussion 1 1 1 2 (58) 0rP  0r2 o  0R 2 1rP 0 r  0 r 0 R 1 r (59) P  r 1o  R 1 r P (59) rPof Substitution equation (57) (58) followed into equation (58) by the substitution ofsubstitution the resultant 1o 1 P into equation The kinematic concepts presented in this paper are not equation (57) intofollowed equation (58) followed by the ofinto the equation resultant into equation 0 0 0 1 by the substitution of the resultant into equation (59): (59) r  r R r novel and instead are well-known, well-established and P followed 1o 1 P tion (57) into into equation equation (58) (58) by the substitution of of the the resultant resultant into into equation equation ion (57) followedr by the substitution generally accepted.

The specific forms of the relationships,  r  R r (56) 0 0 1 0 1 2 0 1 2 3 0 r 0 R 0 1  R 0 1 2 0 1 2 3 r  r  R  R r R R r (60) P 1o 1 2o 1 2 3o 1 2 3 P r  r (58)  Rfollowed r  Rby the R substitution r  R R ofR ther resultant (60) in symbolism, also fit this characwith variability ion (57) into equation into equation (60) albeit 0 0

P

0 0

0

0

P

0 0

P

1o

P

1



P

2o



o

1





2

N 1

0

i 1

i 1 i  1 io

i 1

i 1 i 1

i 1 0

i 1

i 1

io

i 1 0

i 1

i 1

i 1 i 1 i 1 i 1

io

i 1 0

N 1 N 1 N 1 N 1

N1 0

i 2

0 0 P

00 00 1o 0



1o

P

00 00

0

i2

N1

0

0

N 1

0 0 1o

0

io

i 1

N1

0

io

N 1 N 1 p

N 1 0

N 1

N 1 N 1

p

i 1 0

i 1

i 1 i  1 io

N 1

N 1 0

N 1

N 1

P

N1

(65)

P

00 00 P

N 1 i 2

0

p

N 1

i 1

N 1

P

i 1

io

N 1 0

N 1

N 1 N1 P

N 1 0

i 1 0

N 1

i 1

N 1

i 1

io

(66)

P

P

3o

1

2

3

P

Where  and  denote the reference frames and ro is the position of the reference point of the  frame in

terization. (60) Both of these statements are consistent with rr terms rr  on RR the rr right  RRofRR rrequality  RR can RR be RR reduced: rr (60) DCMs thethe the review nature of the subject work. The equations, efducts for for the theThe DCMs for the terms on right of the equality can be reduced: matrix products for the DCMs for the terms on the 0 0 0 1 0 1 2 0 1 2 3 fectively, speak for themselves. However, there are certain r  r  R r  R R r  R R R r (60) P 1o of 1 equality 2o 1 can 2r be 3oreduced: 2 3 Pcan be reduced: right the  the R 1 r equality (57) for the the for the 0 DCMs 1 terms 0 on 2 the right 0  r3 of for the the of 0 rterms 0 Ron r 0 right 1 R 0 the 2 equality 0 3 can be reduced: (61) rPDCMs  0 r1o for  0R  r points that are worthy 1 r2o r 2 3o (61) of note. For a problem consisting R 1 r2o 3 PR 2 r3o  R 3 rP P 1o (58) r  2r  R 0 r 3 of a total of N frames of reference, there will be a total of 0 0 terms 0 on 1 the right 0 for the DCMs 0for the of the equality can be reduced: (61) (61) (61) rrPP  0 rr1o1o  0 RR11 1 rr2o2o  0 RR22 2 rr3o3o  0 RR33 3 rrPP ation is the following: N vectors, one per frame, in which the position vector of ults of this equation is the following:r  r  R r (59) 0 0 0 1 0 2 0 3 a point may be expressed.

The number of forms, for the (61) rPthe  following: r1o  R 1 r2o  R 2 r3o  R 3 rP this equation equation is N(58)  1 equation Substitution of the equation (57) into equation followed by the substitution of the resultant into equation 3.

Discussion One of the results of this is the following: this is following: N  1 first derivative, increases to N2 and for the second deriva0 N 2 i 1 (59): NR 2 i 1 R N 1  0 R 1 1 R 20  (62) iR R N 1 R0NR11 1 R Ri 3 2 N 1   tive increases to N(62) . Therefore, for the pth order derivative i 1 this equation is the following: N 1 r1 i 1Ri  1R R 0 0 r 1 r  N R  2r  The R RNkinematic r (60) concepts presented in this paper are not novel p+1 and instead are well-known, well-establi 0 R 0 R 1 R N  2 R i  1   R (62) {p ≥ 1}, there will be N forms for the derivative. For 1   R NN11  R 11 R 22 R NNand R i i accepted. (62) (62)  1 generally The specific forms of the relationships, albeit with variability in symbolism i11 N any order derivative, if the expression frame is the same The matrix products for the DCMs for the terms on the right of the equality can be reduced: m equation (61) isequation that0 it shows the form ofi  1the general relationship with all tant result from itfitshows form of the general relationship withare allconsistent 0 1(61) isNthat 2 i  the 1 this characterization.

Both of these statements with the review nature of the subject w R  R R  R  R (62)    N 1 1 2 N 1 i nce. of reference. for the original position vector and for each derivative, the frame r  r  R The r  iequations, R 1 r  R r effectively, speak for(61) themselves. esult from from equation equation (61) (61) is is that that shows the form of of the the general general relationship relationship with all all However, there are certain points that are worthy of esult ititresult shows the form with kinematic transform of premultiplying the vector by the Another important from equation (61) is that it For a problem consisting of a total of N frames of reference, there will be a total of N vectors, one per fr of reference. reference.

N of  1 this equation is the following: One of the results of N  1 DCM that transforms from the expression frame in quesshows the form of the general relationship with all terms 0 0 iis  1 that 0it shows N  1 the form of the general relationship with all esult from 0 r  R0 0 the N 1 ini r1Pwhich position vector of a point The number of forms, for the first deriva rP  equation r1o   00(61) Rr i  (63) may be expressed.

1 io   N R 1 r r  R r (63) P 1o i 1 io N 1 P   tion, to another frame, can be applied to obtain 3. directly i  2 G frame of reference. in the N  1ofRreference.  R  R 2 and for the second derivative increases to N Therefore, for the pth order derivative {p to N (62) i  2 R R increases N  1 0 0 0 i 1 0 N 1 the expression of the vector in the second frame. When 0 r  0 r  0 R 0 R N 1 r (63)any 1 there be For order derivative, if the expression frame is the sam rPP  r1o1o   R i i11 i 1 rrioio  R NNwill rPP Np+1 forms for the derivative.

(63) 1 i 12 N the expression frames differ, for a given vector, the prove of equation (61): i 2equation Another important result from (61) is that it shows the form of the general relationship with all position vector and for each derivative, the kinematic transform of premultiplying the v irst G derivative of 0equation0 (61):i 1 the 0 0 original N 1   R i  1 rio  R N  1 rP terms in the G rframe reference.

(63) the expression of the vector in a different (63) from P  ofr1o cessthe of obtaining by the DCM that transforms expression frame in question, to another frame, can direct i2 derivative of of equation equation (61): (61): derivative frame a more involved process. this regard, the first frames diffe to obtain the expression of the is vector in the second frame. In When the expression r  r   applied R r  R r (63) 0 0 rP  0 rP derivative formula shown inunder equation (23)is a more invo a given vector, the process of obtaining the transport expression of the vector a different frame derivative ofIfequation (61):first G derivative we take the of equation (61): 0 0 first G derivative0of equation (61): process. In this regard, the first derivative is key.

The frames B and G can be replaced by any twois key. The fr transport formula shown under equation (23)  If we take the 0 r 0 0 d0 d d0 d0 3 0d P 0 1 0 d0 3 0  1  00 rP R1 d r2o R02R2 r3o R0 3R rP2 r B frames while keeping the expression and derivative frames v  r  r   R r and G can be replaced by any two frames while keeping the expression and derivative frames consis (64) 0 P 1o dt 1 2o dt 2 3o 3 P (64) (64) ddt 0 0 0 0 0dt dt dt dt  0r 0 0 0 r  r  d d d d consistent.

In other words, for any two frames A and B: 1 30 0d P1 00 other words, any two frames3  and : 1 d 00 1R 0 3 3 3 r 0 dt1 1 1r2o 0 3  p 2 03 for R 12d PR23  3 r  0220RrR3o2212r22orr3o203o2 d00dtdR2 2r03o0In rr1o1o00 r1o rr2o RR223r33o rdP  3 1  1 r2o 0  1 Rr12o 3  rp (64) d dt dt dt 1  1 2o 0 1d 0 2  r3o   R 3 3 rp  0  3  rp (64) 0 0 0 v  r  R r  dtR r  R r dt dtd dt (64) d dt 0 1d 0dt2 20 1 10 1 11 2 dt 3 0 3 3 33 02 2 3 dt 02 0d 10 0 2 3 0 R 2 r 2 2  Rr3o r2 3 r1o Rrv203o1R rR 01 r 00  1Rr21  R 2o 0  0 3 P2r  v11o1dt r3o 0r 2o21 2r  1o1o1 v rr23o3o 22dt  r2o    0R 0R 2o 2 2v 3o 3o 0 R2R003o  111r1r2o02oRR11r2o01r0112ov112ordt 23rv0pR2Rr3R3o3 0333rrr3pp0R03r03p3333rv p3 rrpp 30 3 3drp   (64) 0dt 2 2 r                 (67) 0 0 1 1 1 0 2 2 2 0 3 3 3 R 2   2r  r00 RR3  3vv3  v   2 2 r v  2 v 2  3r 3  3 r 0 11 R v0 R dt 0r 1 0 R v2o1  r2o   v1o1o00RRR1 1111r1v   r   r  R r   r   0v 0 1  r2o 2 2 3o 0 2 3o 3 3 p 0 3 p     R    R     r v r v r 0 2o 0 1 2 2 3o 0 2 3o 3 3 p 0 3 p 0 1o 1 1 2o 0 1 2o  2  2 3o 0 2 3o  3 3 p 0 3 p case:For the general case: 0 0 1 1 1 0 2 2 2 0 3 3 3  equation  Rwe vp  readily 0 v1o  R 1  1 v2o  0 1  r2o   R 2  2 v3o In 0  2  r3o (67), 3 3may 0  3  rp replace  with B and  with G. The resultAiswith an entirely In equation (67), we may readily replace B and Bvalid and acc For the general case: with such G. The result is anand entirely valid and accurate relarelationship. The difference between a relationship the second equation in equation (23) inv tionship.

The difference between such a relationship and the angular velocity term.

In both cases, the angular velocity is expressed in the G frame.

However, (65) d r  r dt positive sign is associated with the expression of the angular velocity of thethe G angular frame about the B f theGsecond equation in equation (23) involves   v  r   R  r   then  r the  r   r R (65)  negative   bevelocity sign would and is associated with the G the expression the angular term. In both cases, angular of velocity is ex- velocity of frame about the G frame. pressed in the G frame.

However, if the positive sign is   v  v   R  v    r  R  v  r  0 0 0 1 0 1 2 0 1 2 3 0 0 0  frame.

1 0 1 2 = 4 (denoting 0 1 the 2 3 the coordinates of the P 1o 1 2o For the1case 2of N3o 1 2 G frame 3 P as 0 and the other frames, in P as 1,1o 1 2othe position 1 2 3o 2 3known P in the last frame: sequence, 2 and 3) where vector of point P1 is initially 2

1

0

P

0

0

0

0

0

0 0

0 0

0

0

P

0

0

0

0

2o

0

1o

N 1

P

1

1

1

1

1

2

2

0

i 1

i 1

P

0

3o

2

2

0 0 1o

0

0 0

0

1o

P

0 0 P

P

0 0 1o

P

0 0

0

1o

1

1 1 2o

1

1 1

N 1

0

i2

N1

1o

1

0

i 2

0

1

0

2o

1 0

1

1

2o

1 0

i 1

i 1 i  1 io

i 1

i 1 i 1

0

2 2

2

i 1

i 1

i 1 0

2 2 3o

2

i 1

i 1



3

3

2 0

i 1

2 0

0

2

N1

0

io

2

N 1

P

P

P

3

0

3o

N 1 N 1 p N 1 N 1



   

0

3o 2

3

i

N 1



3

P

2

3o

io

3

3

N 1

2

2

i 1

0

3o

0

2o

i 1 0

io

0

2

0

2o

1

1

2

0

3o

N 1

io

1

1

N 1

N 2

i2

P

1

1

0

2o

2

N 1

1o

1

1

P

2

2

0

1o

3

3

1

2o

0

P

2

3o

0 0 P

P

0 0

1

0

P

0

0

0

1o

1

P

0

0

2

P

3 3 p

3

N 1 0

p

3 3

3 0

p

N 1

N 1 0

N 1

 3

3

3 0

p

3

3

N 1

p

   



 

 

P

N1

P

associated with the G expression of the angular velocity

A second point of importance, in to theabout DCM, manner a frame-referenced of regards the G frame theisBthe frame then in thewhich negative sign Taking another G derivative of equation (65) provides the general acceleration solution: derivative of a matrix multiplication involving the DCM premultiplying a vector Taking another G derivative of equation (65) provides the would be and is associated with the G expression of or theanother matr d d d mechanistically handled when compared to a standard time derivative.

For the latter, one would a r  acceleration r  r  general solution: angular velocity of the B frame about the G frame. dt dt dt the product rule of differentiation to the matrix product as shown by equation (18). For the former   r     r    r   a  r   R    r   (66) would premultiply the frame-referenced derivative of the matrix product by the DCM. Equation      shows this for the case of the first derivative.

This finding actually holds for the pth order derivative    r    r   r   R     r  1} and with the case for p = 2 shown in the derivation of the relationships in Appendix C. 0

0

0

00 00

0

P

P

0 0 P

00 00 1o 0

N 1

00 00 P

N 1

i 1

i 1 i 1 i 1 i 1

N 1 N 1 N 1 N 1

N1 0

i 2

0

p

io

N 1

i 1 0

i 1

N 1

P

i 1

io

N 1 0

i 1 0

N 1

i 1

i 1 i  1 io

N 1 N1 P

N 1 0

i 1 0

N 1

i 1

N 1

i 1

io

P

 p dp  d    r rMagazine R  R   - Volume 14 Issue 1 43   Collision P P p dt dt p

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d                   dt



(67) former is what was employed in determining the frame referenced first time derivative of the DCM and in

defining the angular velocity vectors. The second form was utilized in regards to defining a simple time readily replace  with point B andof  with G. The in result is antoentirely valid accurate thisthe A second importance, regards the DCM, isand derivative transport theorem, shownthe coordinate derivative ofReturning the DCM. It isto second form that provided the greatest utility as in defining transformations for the angular velocity and acceleration. The DCM as the scalar product of the unit vectors nce between such a relationship the second equation equation (23)byinvolves the manner in whichand a frame-referenced time in derivative of the equations under (23) and equation (67), it is imof the relevant reference frames represents one of the most general forms of representing the orientation . In both cases, the angular velocity involving is expressed the Gpremultiplyframe.

However, if the a matrix multiplication the in DCM portant to remember that the operand on the left of the as the basis transformation between the relevant reference frames. This characteristic of generality served for the choice to use the DCM. In practice, however, one would use an alternative characterization such as d with the G ing expression of the angular velocity of the G frame about the B frame a vector or another matrix is mechanistically handled equality is not limited to simply the position vector. The Euler angles or quaternions. Each of these characterizations can be directly related to the DCM. ould be and when is associated with G expression of the angular of the B is equally apt for translational velocity, transcompared to athe standard time derivative.

For thevelocity lat- relationship

Returning to the derivative transport theorem, as shown by the equations under (23) and equation (67), it ter, one would apply the product rule of differentiation to lation acceleration, angular velocity, angular acceleration, is important to remember that the operand on the left of the equality is not limited to simply the position the matrix product as shown by equation (18). For the For example: vector. Theetc. relationship is equally apt for translational velocity, translation acceleration, angular velocity, tance, in regards to the DCM, is the manner in which a frame-referenced time angular acceleration, etc. For example: former, one would premultiply the frame-referenced demultiplication involving the DCM premultiplying a vector or another matrix is rivative of the matrix product by the DCM. Equation (22)  d  d (69) r  when compared to a standard time derivative. For the latter, one woulddapply  r   r  r    r  dtd  v   a  a    v  dt  dt shows this for the case of the first derivative. This finding  dt entiation to the matrix product as thshown by equation (18). For the former, one (69) actually holds for the p order derivative {p ≥ 1} andThe with modified framework rame-referenced derivative of the matrix product by the DCM. Equation (22) of Jazar, in regards to the script nomenclature, clearly does not have the most the case for P =finding 2 shown in the holds derivation of the relationcompact appearance. This statement is apt even when compared to the notational simplifications found The modified framework of Jazar, in regards to the script the first derivative.

This actually for the pth order derivative {p ≥ within the work of Jazar. Having a consistent and systematic framework that explicitly depicts the ships in Appendix C. clearly not thereby have hopefully the most compact = 2 shown in the derivation of the relationships in Appendix C.expression nomenclature, and derivative frames for eachdoes derivative, avoiding confusion over the 







 

 

 

 



 



 

 

 

 



 

relationship between a particular formulation and the reference frames that are relevant, appears to be a fair trade for loss of compactness in appearance. Most references will show the form of the velocity and  p acceleration relationships in regards to the G derivatives of the B position vector. What is generally lacking dp  d    R  rP  R  p rP are the other forms of(68) the velocity solution and even more so for the acceleration solution. The presentation dt p dt for the total set of solutions for the two frame problem does not imply that the complete set is relevant for an accident reconstruction context. Furthermore, one may readily convert the starting parameter set into a single frame and then perform the subsequent derivative evaluations with respect to another, single, ective, the premultiplication of the DCM is distributive over addition buttheisfull not frame. Having set of solutions and the relationship between the different formulations, however, does allow for a relatively simple conversion irrespective of the approach of choice.

-matrix multiplication. In other words, for a DCM, R, and two correctly expressed

appearance. This statement is apt even when compared to the notational simplifications found within the work (68) of Jazar. Having a consistent and systematic framework   that explicitly depicts the expression and derivative frames From the algebraic perspective, the premultiplication of for each derivative, thereby hopefully avoiding confusion the DCM is distributive over addition but is not commu- over the relationship between a particular formulation Ty = xTyR = xTRy does not generally hold (in regards reference frames that are relevant, appears to be = Rx + Ry holds tativebut overRxvector-matrix multiplication. In other words, and the to regards to thefor latter, as with any case of vector-matrix multiplication, the order of a fair trade for loss of compactness in appearance. Most a DCM, R, and two correctly expressed vectors x and The proper algebraic reduction of kinematic relationships using will show the form of the velocity and accelerareferences T T Toften involves y, R(x + y) = Rx + Ry holds but Rx y = x yR = x Ry does onships of individual terms in a matrix product. Such reductions also involve not generally hold (in regards to any of the equalities). In tion relationships in regards to the G derivatives of the B tion of cross product operations between two vectors into the matrixposition productvector. What is generally lacking are the other regards to the latter, as with any case of vector-matrix mulsymmetric matrix form of the first vector multiplied by the second vector. tiplication, the order of terms is very important. The prop- forms of the velocity solution and even more so for the solution. The presentation for the total set of er algebraic reduction of kinematic relationships often in- acceleration nce in regards to the DCM is its character as both an unevaluated scalar productfor the two frame problem does not imply that solutions volves the transformation relationships of the source and using destination reference frames as wellofasindividual a scalar matrix. The terms in a matrix product. Such reductions also involve the complete set is relevant for an accident reconstruction changing the representation of cross product operations context. Furthermore, one may readily convert the startbetween two vectors into the matrix product representa- ing parameter set into a single frame and then perform the tion of the antisymmetric matrix form of the first vector subsequent derivative evaluations with respect to another, single, frame. Having the full set of solutions and the relamultiplied by the second vector. tionship between the different formulations, however, does A final point of importance in regards to the DCM is its allow for a relatively simple conversion irrespective of the character as both an unevaluated scalar product between approach of choice. the unit vectors of the source and destination reference frames as well as a scalar matrix. The former is what was References: employed in determining the frame referenced first time 1. Alem NH and GL Holstein (1977) Measurement of derivative of the DCM and in defining the angular veloc3-D motion. UMTRI Paper No. UM-HSRI-77-46, ity vectors. The second form was utilized in regards to International Workshop on Human Subjects for Biodefining a simple time derivative of the DCM. It is this mechanical Research, New Orleans, Louisiana, October 18, 1977. second form that provided the greatest utility in defining the coordinate transformations for the angular veloc- 2. Brach RM (1977) An impact moment coefficient for ity and acceleration. The DCM as the scalar product of vehicle collision analysis. SAE Technical Paper No. the unit vectors of the relevant reference frames represents 770014. one of the most general forms of representing the orienta- 3. Brach RM (1983) Impact analysis of two-vehicle collition transformation between the relevant reference frames. sions. SAE Technical Paper No. 830468.

This characteristic of generality served as the basis for the 4. Brach RM (1983) Identification of vehicle and collichoice to use the DCM. In practice, however, one would sion impact parameters from crash tests. Paper No. use an alternative characterization such as Euler angles or 83-DET-13, ASME Design and Production Technical quaternions. Each of these characterizations can be diEngineering Conference, Dearborn, Michigan, Seprectly related to the DCM. tember 11-14, 1983.

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5. Brach RM (1984) Momentum and energy analysis of automobile collisions. In: Structural Impact and Crashworthiness, Volume 2 (J Morton, Ed.). London, England: Imperial College.

6. Brach RM (1991) Vehicle dynamics model for simulation on a microcomputer. International Journal of Vehicle Design 12(4): 401-419.

7. Brach RM (1998) Formulation of rigid body impact problems using generalized coefficients. International Journal of Engineering Science 36(1): 61-71.

8. Brach RM (2007) Design of experiments and parametric sensitivity of planar impact mechanics, 16th Annual EVU Congress, Krakow, Poland, November 8-9, 2007.

9. Brach RM (2007) Mechanical Impact Dynamics, Rigid Body Collisions (Rev. ed.). New York, New York: John Wiley & Sons, pp. 147-155.

10. Brach RM and RM Brach (1987) A review of impact models for vehicle collision. SAE Technical Paper No. 870048.

11. Brach RM and RM Brach (1998) Crush energy and planar impact mechanics for accident reconstruction.

SAE Technical Paper No. 980025.

12. Brach RM and RM Brach (2007) Analysis of collisions – point mass mechanics and planar impact mechanics. Collision 2(1): 38-50.

13. Brach RM and RM Brach (2011) Vehicle Accident Analysis and Reconstruction Methods (1st ed.).

Warrendale, Pennsylvania: Society of Automotive Engineers, pp. 103-152.

14. Brach RM, KJ Welsh and RM Brach (2007) Residual crush energy partitioning: normal and tangential energy losses. SAE Technical Paper No. 2007-01-0737.

15. Carpenter N and J Welcher (2019) Inter-vehicular sliding friction and crush energy losses in impulse momentum planar collision. SAE Technical Paper No. 2019-01-0422.

16. Cliff WE and DT Montgomery (1996) Validation of PC-CRASH – a momentum-based accident reconstruction program. SAE Technical Paper No. 960885.

17. Day TD, SG Roberts and AR York (2001) SIMON: A new vehicle simulation model for vehicle design and safety research. SAE Technical Paper No. 2001-010503.

18. Daqaq MF (2019) Dynamics of Particles and Rigid Bodies, A Self-Learning Approach. Hoboken, New Jersey: John Wiley & Sons Ltd., pp. 55-90.

19. DiMasi F (1995) Transformation of nine-accelerometer-package (NAP) data for replicating headpart kinematics and dynamic loading. DOT HS 808 282, Washington, DC: US Department of Transportation, National Highway Traffic Safety Administration.

20. Emori RI (1970) Vehicle mechanics of intersection collision impact. SAE Technical Paper No. 700177.

21. Fetter AL and JD Walecka (2003) Theoretical Mechanics of Particles and Continua. Mineola, New York: Dover Publications, Inc., pp. 31-35.

22. Greenwood DT (1988) Principles of Dynamics (2nd ed.). Upper Saddle River, New Jersey: Prentice-Hall, Inc., pp. 281-354.

23. Han I and S-U Park (2001) Inverse analysis of preand post-impact dynamics for vehicle accident reconstruction. Vehicle System Dynamics 36(6): 413-433.

24. Harithuddin ASM (2014) Derivative kinematics in relatively rotating coordinate frames: investigation on the Razi acceleration (Doctoral dissertation). RMIT University, School of Aerospace, Mechanical and Manufacturing Engineering.

25. Huang L (2017) A concise introduction to the mechanics of rigid bodies (2nd ed.). Cham, Switzerland: Spring International Publishing, pp. 27-91.

26. Ishikawa H (1993) Impact model for accident reconstruction – normal and tangential restitution coefficients. SAE Technical Paper No. 93065.

27. Ishikawa H (1994) Impact center and restitution coefficients for accident reconstruction. SAE Technical Paper No. 940564.

28. Jazar RN (2007) Theory of Applied Robotics: Kinematics, Dynamics and Control (1st ed.). New York, New York: Springer, pp. 297-325.

29. Jazar RN (2008) Vehicle Dynamics, Theory and Practice. New York, New York: Springer, pp. 219-279.

30. Jazar RN (2011) Advanced Dynamics, Rigid Body, Multibody and Aerospace Applications. Hoboken, New Jersey: John Wiley & Sons, Inc., pp. 695-864.

31. Jazar RN (2012) Derivative and coordinate frames.

Nonlinear Engineering 1: 25-34.

32. Kane TD and DA Levinson (1985) Dynamics: Theory and Applications. New York, New York: McGrawHill Book Company, pp. 15-56.

33. Marine M and S Werner (1998) Delta-V analysis from crash test data for vehicles with post-impact yaw motion. SAE Technical Paper No. 980219.

34. McHenry RR (1975) A comparison of results obtained with different analytical techniques for reconstruction of highway accidents. SAE Technical Paper No. 750893.

35. McHenry BG and RR McHenry (1997) RICSAC-97 – a reevaluation of the reference set of full scale crash tests. SAE Technical Paper No. 970961.

36. Padgaonkar AJ, KW Krieger and AI King (1975) Measurement of angular acceleration of a rigid body using

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linear accelerometers. Journal of Applied Mechanics 42(3): 552-556.

37. Park S and SK Hong (2011) Angular rate estimation using a distributed set of accelerometers. Sensors 11: 10444-10457.

38. Rajamani, R (2012) Vehicle Dynamics and Control (2nd ed.) New York, New York: Springer.

39. Rose NA, SJ Fenton and RM Ziernicki (2004) An examination of the CRASH3 effective mass concept.

SAE Technical Paper No. 2004-01-1181.

40. Scurlock BJ and JR Ipser (2014) Rigorous derivations of the planar impact dynamics equations in the center of mass frame. Retrieved from: https://arxiv.org/ abs/1404.0250 41. Shea RT and DC Viano (1994) Computing body segment trajectories in the Hybrid III dummy using linear accelerometer data. Journal of Biomechanical Engineering 116: 37-43.

42. Steffan H and A Moser (1996) The collision and trajectory models of PC-CRASH. SAE Technical Paper No. 960886.

43. Takhounts E, R Eppinger, R Tannous, JQ Campbell, E Power, et al. (2003) Analysis of 3D rigid body motion using the nine accelerometer array system. Injury Biomechanics Research: 31st International Workshop.

44. Zhang J, C Liang, S Wu, R Liu and J Gao (2016) Consistency of calculation results of two typical vehicle collision models. Procedia Engineering 137: 220-224.

45. Zhou J, J Lu and H Peng (2007) Collision model for vehicle motion prediction after light impacts. 20th IAVSD Symposium, Dynamics of Vehicles on Roads and Tracks, Berkley, California, August 13-17, 2007.

46. Zou T, Y Dai, M Cai and J Liu (2012) Discussion on accuracy degree evaluation of accident velocity reconstruction model. Physics Procedia 24: 979-983.

Symbols:

Symbols Cr A Cr A|B

= =

C D A

=

EC FD A

r

=

Second time derivative of the vector CrA. The rightmost left superscript and subscript denote the expression and differentiation frames, respectively, for the first time derivative.

The leftmost superscript and subscript denote the expression and differentiation frames, EC aA . respectively, for the second time derivative. This is equivalent to FD

C B

A

=

C B

A 

=

The angular velocity vector of reference frame A with respect to reference frame B, expressed in the coordinates of reference frame C. First time derivative of the angular velocity vector (follows the same referencing

r

Position vector of point A expressed in the reference frame C. Relative position vector from point B to point A, expressed in the reference frame C. This is equal to CrA – CrB. When the point B is the origin of coordinates of reference frame C, the position vector is simply represented as CrA. First time derivative of the vector CrA where the left superscript denotes the expression frame and the left subscript denotes the frame in which the derivative was taken. This is equivalent to DC vA .

convention). This is equivalent to CB  A .

46

BR A

a

= =

G O Ei

= = =

B o ei

= = =

n i, ui

=

The antisymmetric (skew-symmetric) matrix associated with vector a.

The direction cosine matrix that transforms coordinates from a departure frame (A) to a destination frame (B).

Symbol that denotes the global (inertial) frame of reference.

Origin of coordinates for the inertial frame of reference.

Orthonormal triad of unit vectors {i: i = 1, 2, 3} for the rectangular Cartesian characterization of the inertial frame of reference.

Symbol that denotes the body frame of reference.

Origin of coordinates for the body frame of reference.

Orthonormal triad of unit vectors {i: i = 1, 2, 3} for the rectangular Cartesian characterization of the body frame of reference.

General orthonormal triad of unit vectors {i: i = 1, 2, 3} for the rectangular Cartesian characterization of a frame of reference.

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COMING JUNE 2020 psg.leica-geosystems.us

Utilizing BioMedical Engineering

Physical Evidence Analysis for Motorcycle Compared to Pedestrian Collision Laura Liptai, Ph.D., and Mark Ezra, P.E.

A

bstract The engineering investigation of trauma due to motorcycle incidents requires both a forensic analysis of the physical evidence and dynamicsbased accident reconstruction. This case study focuses on the investigation of a motorcycle trip accident on a highway undergoing resurfacing work where a surface height existed between two lanes of travel. Such trip accidents by motorcycles often lead to a secondary impact of the rider by a following or an oncoming vehicle after the motorcycle trip occurs. In this case study it was necessary to determine whether the rider was impacted while lying in the roadway or while standing. Issues of visibility of the lane height difference and angle of attack of the motorcycle toward the pavement height differential are evaluated. In the example discussed, the physical evidence of the trauma, including bone fracture mechanics, was also used to determine that the principal direction of force was consistent with axial loading of the motorcycle rider’s leg at impact. The fracture signature, combined with a dynamics-based motion analysis of motorcycle and rider masses, provided insight into whether the motorcyclist was lying down or standing at the time of vehicle impact and after the initial motorcycle trip and rider separation from the vehicle. Fracture mechanics provides a reliable methodology for unwitnessed trauma.

Keywords Biomechanics, biomedical engineering, femur fracture, bone fracture mechanics, pedestrian run over, lower extremity trauma, motorcycle accident reconstruction, vehicle accident.

Introduction The motorcycle trip accident investigated presented classic issues of a shallow angle of impact into a raised edge 52

Collision Magazine - Volume 14 Issue 1

by a motorcycle. The shape of the higher surface edge, its height above the lower surface of the construction zone, the communication of the type of hazard present, and its relevance to motorcycles were all elements in the dispute that required forensic analysis. The solution was unique and unanticipated.

At the time of the accident, the helmeted plaintiff was operating his motorcycle in a lane that had been milled in preparation for resurfacing. The resulting lane surface, open to use by traffic, was lower and rougher than that of the higher and newly resurfaced adjacent lane to the left (approximately a 1.5-2.0-inch height differential). The operator of the motorcycle was driving in the milled lane and attempted to change lane onto the higher surface at a speed consistent with the flow of traffic. However, the motorcycle hit the raised edge of the resurfaced lane at a shallow angle of approach (<20 degrees) which led to an immediate trip and fall over of the motorcycle. Due to the processional gyroscopic forces generated by the deflection of the motorcycle’s front wheel. The steering forces generated by such impacts at highway speeds have been shown to be outside the range of forces that may be likely controlled by a rider.1 The motorcycle traversed the repaved lane, slid on its side, and came to rest in the grassy median. After the motorcyclist was reported to have been run over by the left wheel of a following pickup truck traveling in the left, newly resurfaced lane, despite the pickup truck driver’s attempt to take evasive action. The plaintiff’s first medical record indicated that he was hit while walking back onto the interstate highway to retrieve parts from his motorcycle; however, all other history and plaintiff statements indicated that he was run over while on the ground. Ultimately, analysis of the evidence is consistent with the initial medical record that he was standing or walking at the time of impact.

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Mechanical Engineering Accident Reconstruction Forensic mechanical engineering reconstruction of a road accident often includes recorded measurements, roadway markings, and objects involved in an accident.2 Without this information, an investigating engineer may or may not be able to properly make assumptions for the missing data. In the subject incident, insufficient information was included in the accident report to even allow for educated

assumptions to be generated as to the point of trip of the motorcycle, the angle of slide of the motorcycle relative to the travel direction of traffic, and the total distance of slide of the motorcycle after the trip over at the edge of the resurfaced left lane. Therefore, no independent calculations of the motorcycle’s speed at trip and fall over would be reliable with the available data.

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Collision Magazine - Volume 14 Issue 1 53

However, it was possible to make other significant and reliable qualitative forensic engineering observations based upon the information available. The following forensic observations were made concerning the motorcycle’s operation at the time of the accident and issues of operation contributing to the cause of the accident: 1. The traversing of a “drop off” down onto a lower surface and the visual cues and color difference between the milled lane and the fresh blacktop surface, should have alerted the motorcyclist to the risk and danger present.

2. The motorcyclist failed to use the correct and accepted operating technique of crossing a raised ridge at an angle in excess of 20 degrees, when he tried to cross the raised lane centerline between the right lane and the left lane of the roadway.

3. Given the motorcycle’s reported speed prior to falling over and applying Newton’s first and second laws of motion, the motorcyclist should not have come to rest anywhere within the left lane of travel, yet that is where the rider reported the incident to have occurred.

The rider’s inertia alone would have carried him in the direction of, and near to, the final rest location of the motorcycle in the center median.

It was, therefore, necessary to use BioMedical forensic engineering analysis of the rider’s trauma in order to determine whether or not these injuries were consistent with a supine or a standing orientation at impact when he was hit by the following pickup truck.

Trauma Analysis After the incident, the radiologist reported multiple fractures [LL1] including the right clavicle, left femur, multiple fractures to the left fibula and the 3rd & 9th right ribs (Figure 1). Additional trauma included a subcutaneous emphysema, right knee rupture of the lateral patellar retinaculum and anterior cruciate ligament (ACL). [LL2] BioMedical Engineers were interested in the plaintiff’s right knee avulsion fracture (Figure 2). This fracture is consistent with a knee that was axially loaded with body weight at the instant of impact and to which a medially directed force was applied. The combination of the medially directed impact force applied in the area of the right knee, the inertial mass of the upper body, the axial load, and the relative fixation of the right foot on the ground likely created a large bending moment at the right knee causing the fracture.3

54

Collision Magazine - Volume 14 Issue 1

Figure 1: Trauma map of the plaintiff’s injuries Generator of Body Data (GEBOD) is a U.S. military software program that utilizes regressive statistical equations to produce human and dummy body segments based on geometric and mass properties. According to GEBOD, for a male of similar height and weight, the separation distance between the plaintiff’s knee and ankle equals 17 inches.4 The 17 inches between the two fractures, makes it unlikely that a single front and/or back tire could sequentially run over the knee and ankle given the straight-ahead roadway in the reported area of the incident.

Further, the medial fracture on the malleolus is consistent with a force with medial abduction while axially loaded (standing). The posterior malleolus fracture is consistent with a lateral force with rotation while the knee joint is axially loaded (Figure 3).

Impact forces are represented by arrows in Figure 3. Lower extremity fractures and trauma principally consisted of: left transverse tibia, left proximal and transverse mid-shaft fibula, left distal femur, right lateral patellar retinaculum, right fibular head, and the right lateral collateral ligamentous complex.

In the defendant pickup truck driver’s deposition regarding the scene of the incident, the defendant stated that the motorcycle was off to the side of the road and that the plaintiff’s point of rest was lying on the road with the plaintiff’s upper torso in the left lane and the lower body and legs in the right lane.

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Figure 2: X-ray of the plaintiff’s right knee

Figure 3: Mechanism of lower extremity fractures If the subject were run over while lying on the roadway, he would likely have sustained similar force signatures bilaterally on his lower extremities if run over by a single tire. Often, in such an impact, a type of contact trauma termed degloving occurs, especially on the ramp-up side

of the run over surface. There was no such contact trauma present and/or reported in the emergency room records or any medical records. Furthermore, the motorcyclist’s left ankle sustained fractures on the posterior and medial side of the left malleolus, but the right leg sustained fractures

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Collision Magazine - Volume 14 Issue 1 55

on the lateral aspect of the right patellar retinaculum and collateral ligamentous (Figure 4 and Figure 5). The locations of these fractures are inconsistent with a tire impacting the posterior, medial, and lateral aspect from a runover sequence. Additionally, the mechanism of injury for the plaintiff’s right anterior cruciate ligament (ACL) tear is ruptured or strained ligaments by abduction or hyperextension.5 This finding is also consistent with the plaintiff’s initial report of the history of the incident to his health

care providers and his medical records stating that he was standing at the time of impact.

The mechanism of injury for the plaintiff’s fractured distal femur conflicts in part with a crush mechanism because of the character of a distracted fracture. Additionally, plaintiff sustained a high force fracture solely on the left side which is evidence of the inequity of force to the contralateral side.

The fracture mechanics of the left lower extremity are consistent with high force trauma to the left knee while in a loaded and vertically oriented state. Similarly, the fracture mechanics at the ankle are also consistent with an axially loaded, vertically oriented leg at the instant of impact.

The right knee trauma is consistent with the right foot being planted on the ground and loaded with body weight.

The fracture on the left oblique femur and the fracture on the left fibula (Figure 6 and 7) are consistent with vehicle bumper impacts while standing.

Since the knee height of the subject is 22.54 inches (GEBOD) without shoes and the height of the top of the bumper equals 25 inches, the height of the fractures is consistent with the height of the front bumper. Collectively, the trauma is consistent with an impact while the subject was standing upright.

A femur has a maximum fracture load ranging from 15003850 lbs.6 In comparison, the static load for a single wheel of the pickup truck defendant’s vehicle is 930 lbs. The tolerance for fracture to the femur is greater than the load of Figure 4: Cross-section of the plaintiff’s right knee any one wheel. Therefore, it is less probable that the plaintiff’s femur would have fractured in a run-over sequence from only one wheel/tire. The plaintiff also sustained a 1.6 by 0.78-inch medial thigh laceration; however, if he were in the supine position, it is likely that there would have been evidence of tire width sheer force.

Inverted Pendulum Model Used to Approximate Bending Moments Generated by Bumper Impact to the Leg

Figure 5: Subject’s left knee.

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The inverted pendulum model theoretically represents a pendulum in an unstable state, with its center of gravity over the axis of rotation. By utilizing the concept of the inverted pendulum, it can be used to simply model a human being in a dynamic event. In this case, it is possible to model the injured plaintiff using the inverted pendulum model in order to understand the bending moment created by a bumper impact at any height, z above the ground, on the leg of a standing subject. This case study uses consideration of such a model in order to help understand the bending moment created at a height z after the pickup truck bumper impacts the leg at height, h. By mathematically modeling the total bending moment and estimating

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Figure 6: X-ray left knee.

Figure 7: Height of the front bumper is similar to height of the subject’s knee. www.collisionpublishing.com

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the effective mass of the leg, this information can give us a better understanding of the forces involved in this standing motorcyclist versus pickup truck bumper impact, as well as allowing estimation of the extent of anticipated trauma.7 See Appendix A for a detailed discussion of such a mathematical model.

Conclusion BioMedical evidence analysis using fracture mechanics is an evidence-based methodology that assists BioMedical engineers in accident reconstruction. This investigation and its analyses determined that it was improbable that the motorcyclist was supine in the roadway and run over by a following vehicle. The width of the vehicle’s tire compared with the location of the subject’s trauma was inconsistent with a run-over injury, and the one-sided forceful ipsilateral trauma reported in the medical records collectively point toward an impact with the front bumper of the vehicle.

Thus, it is likely that the subject was upright standing and/ or walking in the roadway when impacted by the following vehicle, possibly to recover an object after the crash. This forensic engineering analysis demonstrated that the injury causation was not from a solo motorcycle trip and fall over incident, but rather a pedestrian versus automobile incident where the motorcyclist’s medical record indicated he ambulated back into the highway after having come to rest in the median of the road. This analysis of the physical evidence is consistent with the plaintiff’s history of the incident provided to the first health care provider.

References 1. Cheng H, Obergefell L, Rizer A. The development of the GEBOD program. Paper presented at: Biomedical Engineering Conference, Proceedings of the 1996 Fifteenth Southern; Dayton, OH. 1996.

2. Collins, J. Accident Reconstruction. Springfield, Illinois: Charles C. Thomas. 1979.

3. Levine RS. Injury to the extremities. In: Nahum A, Melvin J, editors. Accidental injury – biomechanics and prevention. New York: Springer-Verlag, 1993; 474, 484.

4. Liptai LL. 2009. Trauma causation: analysis of automotive, Wiley Encycl Forensic Sci. 5. Liptai LL, Cecil J. 2009. Forensic engineering and the scientific method. J Natl Acad Forensic Eng. 26:1.

6. Metz LD. Behavior of a motorcycle after an encounter with a road irregularity parallel to its direction of travel. SAE Technical Paper 2006-01-1561. 2006.

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7. Netter F. 1989. Atlas of human anatomy. Summit (NJ): Ciba-Geigy Corporation.

8. Quyen ND, Thuyen NV. Rotary inverted pendulum and control of rotary inverted pendulum by artificial neural network. Proc Natl Conf Theor Phys.

2012;37:243-249.

9. Human Tolerance to Impact Conditions as Related to Motor Vehicle Design. SAE J885, 1986 Jul. 10. Watson-Jones R. 1962. Fractures and joint injuries.

Baltimore (MD): The Williams & Wilkins Company.

2:757.

11. Yamada H, Evans FG. 1970. Strength of biological materials. Baltimore (MD): The Williams & Wilkins Company.

Appendix A: A Mathematical Model of Bumper Injury Mechanism Resulting in the Reported Trauma and Leg Fracture Due to Bending Moments Generated by a Bumper Impact.

Inverted Pendulum Model: The inverted pendulum model is a commonly used engineering model. It theoretically represents a pendulum in an unstable state, with its center of gravity over the axis of rotation. It consists of an arm that rotates on a horizontal plane and a pendulum that freely rotates in the vertical plane. By using the inverted pendulum model, a human being in a dynamic event can be mathematically modelled.

In the presented case study, we can model the plaintiff motorcycle rider using the inverted pendulum for the forensic engineering analysis to help understand real-life scenarios, such as in the case of this motorcycle rider. This paper uses the model to help determine the bending moment at height z after the vehicle impacts the leg at height h.

By calculating total bending moment and estimating the effective mass of the leg, this information can give us a better understanding of the forces involved in pedestrian versus automobile impacts as well as the extent of anticipated trauma (Quyen 2012).

Assumptions Friction at the foot, or ground, and the distributed mass along the leg is modeled in Figure 8. The assumptions included: 1) the leg is assumed to swing like a pendulum after a side impact, and 2) the leg does not bend and remains straight throughout the impact. On the right side of Figure 8, a torso mass illustrates the movement of the hip pivot.

Translational and rotational dynamics are affected by the torso mass and the leg. The torso and the leg are not fixed

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Figure 8: Part A illustrates a lower extremity (thigh and leg) model before bumper impact. Part B illustrates the bending moments of the model at impact (Modified by Liptai from Bacon 1976) to any reference pivot, since the hip restraint correlates to the inertia of the torso mass (Bacon 1976).

Mass distribution along the leg is characterized by a linear variation:

Rotation of torso:

Translation of leg:

Translation of torso: Ly = mass or unit length at a position y from the hip pivot L1 = mass or unit length at the hip

= rotational acceleration of the lower extremity about its C.G.

Lb = mass or unit length at the ankle

= rotational acceleration of the torso about its C.G.

The bumper force is distributed over a fixed length, d, which is the depth of the bumper. A constant force distribution is also assumed.

= translational acceleration of the lower extremity = translational acceleration of the torso

Equations of Motion Rotation of leg:

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2. Solving for the bending moment due to the acceleration of the foot:

Derivation of Equations Notation: = translational acceleration of the lower extremity = translational acceleration of the torso b = bending moment due to the acceleration of the lower extremity mass below z

3. Solving for the bending moment due to the acceleration of the lower extremity:

d = depth of the bumper F = horizontal friction force at ground level g = height to the C.G. of the lower extremity

4. Solving bending moment due to the friction force (fr):

h = bumper height i = distance of torso (C.G.) from hip I1= moment of inertia of combined foot and lower extremity about their C.G. I2= moment of inertia of torso about its C.G.

5. Solving for bending moment due to the bumper force: For

K = reaction at the hip L1 = mass or unit length at the hip Lb = mass or unit length at the ankle Ly = mass or unit length at a position y from the hip pivot m1= mass of the lower extremity (total) m2= mass of torso mf = mass of foot M1 = bending moment due to the acceleration of the lower extremity Mf = bending moment due to the acceleration of the foot

6. Total bending moment at given height (Mt)

7. Effective mass at the height of the bumper:

Mfr = bending moment due to the friction force Mp = bending moment due to the bumper force Mt = total bending moment at given height mh = effective mass at the height of the bumper P = bumper force z = height of the bending moment components = rotational acceleration of the lower extremity about its C.G. = rotational acceleration of the torso about its C.G. 1. Solving for the rotational acceleration of the lower extremity:

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The equations demonstrate the origin of the left lower extremity trauma, which qualitatively substantiate the circumstance that the plaintiff was laterally impacted while in a standing position and walking back toward the median of the road. The equations can be used to calculate the bending moment at the actual height of the bumper, 25 inches (EAS). This moment may then subsequently be used to calculate the bending/impact stress on the bone and compared with the published value of 2,250 pounds (FMVSS 208; USDOT 1968) for the femur to fracture.

The bumper height can also be utilized with the equations to quantify the force in this incident. The max bending moment is expected to occur slightly above knee height, 22.54 inches (GEBOD), as illustrated by the equations.

This mathematical model supplements the basis of the ac-

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cident analysis conclusion that the plaintiff was not in a supine position during the impact to his leg and that the bumper force exceeded the tolerance level for the femur to fracture.

bending forces are significantly greater with a 12.4 or 15.4inch bumper compared to those generated with a 9-inch high bumper. The location of the maximum bending moment also changes with increasing bumper height.

Bending Moment Distribution Results As shown in Figure 9, Bacon used dummies to find the relation between the bending moment and the area of contact for various bumper heights. They found that the bending moment was the greatest between the knee and the ankle at the proximal mid-shaft. The location of greatest bending moment is shown in Figure 9. Furthermore,

Figure 9: Theoretical bending moment distribution for bumper impacts at various bumper heights.

The bumper is assumed to be a hollow bumper (Bacon 1976)

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Vehicle damages and longitudinal throwing distances when using biofidelic dummies in comparison to conventional dummies Annika Kortmann and Tim Hoger

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esides the vehicle damages, the longitudinal throwing distance is also a decisive indicator in pedestrian collisions in order to determine the collision velocity of the vehicle. Previous research has shown1 that the construction of the biofidelic dummy leads to much more realistic vehicle damages in car-pedestrian collisions in comparison to those with conventional dummies. As to whether this also results in changes to the longitudinal throwing distance has so far only been tested with the first generation of biofidelic dummies.2 In order to obtain a direct comparison of the damage differences and the longitudinal throwing distance, crash tests were carried out with the biofidelic dummy from crashtest-service.com GmbH using the same vehicle model with a velocity range from 28 kph to 80 kph and compared with existing crash tests with conventional dummies.

have a different construction. A longer bonnet, a lowered chassis or braking all lead to varying wrap around lengths of the pedestrian,3 and are among other factors to be considered for the height of the head impact.

Introduction

A crash test series at different velocities was carried out, creating a form of EES-catalogue. The crash vehicles were several VW Polo 6R. The biofidelic dummies were laterally approached in most tests and impacted at the centre of the

A difficulty with the determination of collision velocity based on vehicle damages and the comparison of corresponding crash tests is that the vehicles compared may

Similar problems arise when determining the collision velocity based on the longitudinal throw distance of the pedestrian. Furthermore, a none-braking or partially braked car, especially at lower collision speeds, can carry the dummy after the collision. This is followed by a long transport phase in which the dummy is first released from the vehicle when the car is rapidly decelerated. The subsequent occurring longitudinal throwing distances in the crash tests can thereby be extended almost arbitrary and is therefore no longer suitable for limiting the collision speed.

Crash series VW Polo 6R and Biofidelic dummies

Figure 1: Crash test vehicle VW Polo 6R (left) and impact situation with a biofidelic dummy (right) for the crash series performed 62

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Figure 2: Re-enactment (left – biofidelic) of the already available crash tests (right – conventional) – collision velocitiy here: v = 68 kph bonnet by the car, figure 1. The VW Polo was not braked during the collision and was decelerated after a defined short period of time, which was almost the same in all crash tests, to prevent “carrying” with a subsequent transport phase. The controlling, that it did not come to a carrying phase, was done over the videos. The collision speeds in the crash tests were (rounded off) 28 kph, 47 kph, 68 kph and 80 kph. In all tests, the damages and the dummy throwing distance was documented in detail. The new crash series with the biofidelic dummies build on the already existing crash test series carried out by crashtest-service.com GmbH, in which a VW Polo 6R was driven under the previously described impact constellation in six tests against a conventional dummy.4 In the velocity range between 70 kph and 80 kph, the already existing crash tests were carried out again under the same conditions but with the biofidelic dummy, figure 2. Thus, the study not only offers the possibility to investigate the increase in vehicle damage and longitudinal throwing distance with increasing collision speed, but also to make a direct comparison between the biofidelic and conventional dummy.

Formation of vehicle damages Table 1 shows an overview of the crash tests carried out. In order to later be able to carry out a complete comparison of damages even in lower velocity ranges, an additional test with a conventional dummy at a velocity of approximately 30 kph has been recorded (see feature “Opel Astra G”). The dummies are all between 1.79 m and 1.83 m in height and weigh between 74 and 90 kg.

VW Polo 6R against Biofidelic dummy Figure 3 documents the resulting damages incurred at increasing collision velocities in the range between 28 and 80 kph in a collision with a biofidelic dummy.

Both tests in the lower velocity range were carried out for financial reasons with the same vehicle, which is why the dummy was impacted in the first attempt at 27.5 kph left of the centre in www.collisionpublishing.com

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Table 1: Overview of the crash tests performed

Figure 3: Frontal and side view of the change in the damage pattern (impact with biofidelic dummy) on the VW Polo 6R with increasing collision velocity from 27.5 to 80.2 kph 64

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the direction of travel, and the second attempt with 46.7 kph to the right of the centre. In the crash attempts in the higher speed range, the biofidelic dummy was impacted in the centre of the vehicles front.

The collapsing of the front windshield at a collision speed of 46.7 kph results from the predamage occurring in the first crash attempt at 27.5 kph, which was not replaced after the crash test.

Taking into account the damages of the front windshield in the two following crash attempts, it can be assumed that also at a collision speed of approximately 50 kph the front windscreen would not have been penetrated. As the collision velocity increases, the damage to the bonnet and the windscreen also increase. By increasing the collision veloc-

ity, the head impact of the dummy also moves towards the edge of the roof. This is shown in the comparison of the motion sequences of each individual crash attempt in figure 4. At a collision speed of approximately 70 kph, for the first time the head of the biofidelic dummy contacts the front roof edge. A significant damage to the roof occurs at a collision velocity of approximately 80 kph. At this collision speed, the bonnet of the vehicle is clearly crumpled.

Damage comparison after impact with biofidelic and conventional dummy In figure 5 and figure 6 the damages of the crash vehicles at comparable collision velocities in the front and side view after an impact with a biofidelic (left) and a conventional dummy (right) are directly compared with each other. It

Figure: 4: Motion sequence of the biofidelic dummy and head impact height with increasing collision velocity www.collisionpublishing.com

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Figure 5: Comparison of the damage to the VW Polo 6R after an impact with the biofidelic dummy (left) and a conventional dummy (right) with increasing collision velocity (frontal view)

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Figure 6: Comparison of the damage to the VW Polo 6R after an impact with the biofidelic dummy (left) and a conventional dummy (right) with increasing collision velocity (side view) www.collisionpublishing.com

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should be noted that the crash attempt with the biofidelic dummy at a collision velocity of around 50 kph was approximately 13 kph slower (46.7 kph biofidelic dummy, 59.4 kph conventional dummy).

From the damage comparison between the biofidelic and conventional dummy impact in the lower speed range, it is apparent that the impact of the biofidelic dummy produces a defined crack in the windscreen, with a deeper pushed in centre but a smaller extension. The scratch marks and the slight deformation of the bonnet of the Opel (conventional dummy) does not occur to the VW Polo (biofidelic dummy). This can be explained by the wrap around behaviour of the biofidelic dummy in contrast to the rigid physique of the conventional dummy, as earlier studies have also demonstrated.5 In the transition range from medium to high collision velocities of 65 – 70 kph, the hard construction of the conventional dummy, in contrast to the biofidelic dummy, results in massive damages to the roof area when the conventional dummy contacts the rood edge area. The damage profile varies greatly. The susceptibility of the conventional

dummy in the transition range from mid to high collision velocities will be discussed separately. If the collision speed is high, at approximately 80 kph, at first sight the vehicle damages from the impact of a biofidelic or conventional dummy initially resemble. Merely the crumpled raised up positioning of the bonnet is lower when impacted by a conventional dummy.

Upon closer examination of the black VW Polo (impact against a conventional dummy) it presents more front damages. Due to the rebounding of the bumper covering after the collision, the high deformation of the crossmember is only apparent after disassembling the bumper cover.

Figure 7 shows an example of the front damages to the crash vehicle with a disassembled bumper cover from a collision at approximately 70 kph. It can be seen, that the crossmember of the VW during the impact against a conventional dummy (figure 7, right image, yellow framed) was more severely dented than in a collision against a biofidelic dummy (figure 7, left image, yellow framed). From the deformation of the front crossmember of the VW Polo in an impact with a conventional dummy, even the stride

Figure 7: Comparison of the frontal damage to the crash vehicle (bumper removed) after a collision with the biofidelic dummy (left) and a conventional dummy (right) at v = 70 kph

Figure 8: Step position for the impact with the conventional dummy (left) and the resulting crossmember deformation (centre) at v = 70 kph; comparison of the frontal damage after an impact with the biofidelic dummy (right) 68

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position of the dummy at the point of the collision can be detected, see left and middle image in figure 8. This results from the structurally rigid construction of the conventional dummy, which at this point hits the leg area. Since the feet are not pulled under in a collision with a conventional dummy, the front pedestrian under-run protection in the form of a bracket sitting just above the underbody remains completely intact (red arrows), whereas this bracket in a collision with a biofidelic dummy is severely pressed inwards, and therefore fulfilling its function (see also right image in figure 8).

The crumpling of the bonnet of the VW Polo at a collision speed of 80 kph is a result of the wrap-around movement of the biofidelic dummy (comparison of motion sequences in figure 4). Due to the rigid construction of the conventional dummy, the dummy rotates around its centre of gravity and collides then almost horizontally with the front windscreen. The bonnet remains mostly contact-free so that fewer damages to the bonnet occur (figure 5 and fig-

ure 6, bottom right images) in comparison to the impact with a biofidelic dummy colliding at the same velocity.

Susceptibility of conventional dummies in the transition range from mid to high collision velocities How susceptible the damage input on a vehicle in passenger car-pedestrian collisions with conventional dummies is, is shown in the compilation of damage pictures of the crash vehicles accordingly in figure 9.

The impact constellation (laterally approached, centre of the bonnet) and the collision velocity (66 to 68 kph) were almost identical in all the crash test attempts shown. Only the weight and the height of the conventional dummies varied slightly. It emerges that the influence of the weight between 74 and 90 kg has no clearly visible changes in the vehicle damages occurred to the presented crash vehicles.

A change in height from 1.79 m to 1.83 m, however, leads to a completely different damage pattern, as with in this

Figure 9: Comparison of the damage to the VW Polo 6R after an impact with a conventional dummy in the velocity range from 66 to 68 kph www.collisionpublishing.com

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crash test the roof edge of the vehicle was hit and massively compressed.

absorption by the vehicle can evenly be distributed over the contact surfaces between the vehicle and the dummy.

The energy absorption of the vehicle must be similar in all four crash test attempts. Three of the four attempts show very similar damages to the bonnet and the windshield.

The energy was mainly absorbed by the windscreen, bonnet and the front crossmember. The Windscreen has even been partially pierced by the conventional dummy. The slightly larger dummy in the fourth crash test (figure 9 bottom right image) results in a higher contact point of the dummies head on the vehicle.

Longitudinal throwing distance

The dummy strikes almost horizontally against the roof edge, figure 10 bottom images, so that the energy absorption mainly takes place through the compression of the roof. Simultaneously, in this position the conventional dummy has an unnaturally rigid structure. In the collision with the biofidelic dummy (figure 10 top images), the described wrap around behaviour occurs, so that the energy

The longitudinal throwing distances determined from the investigated crash test attempts are listed in table 2 and have been recorded in a graph, see figure 11. In the case of specifics present in the crash tests, this has been noted in table 2. Likewise, the throwing distances from the investigations of Hartwig et. al. 2 were also added to the graph.

By Hartwig et. al.2 investigations have been carried out on longitudinal throwing distances in crash tests with biofidelic dummies. However, it refers to the first generation of the biofidelic dummy. It is possible that the modifications to the outer structure (silicon, neoprene etc.) lead to a different rebound behaviour and thus to a change in the throwing distance.

Although the collisions with the biofidelic and conventional dummies at the same velocity level show partially

Figure 10: Deformation of the roof edge after an impact with the biofidelic dummy (top) and a conventional dummy (bottom) at approx. 70 kph 70

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Table 2: Longitudinal throw distances as a function of the collision velocity and dummy design

Figure 11: Graph with throw distances of the different dummy types as a function of the collision velocity – trend line according to Focken 6

Figure 12: Biofidelic dummy with serial rib fracture and shoulder joint injuries at a collision velocity of approx. 70 kph www.collisionpublishing.com

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major differences in the damages, there is no deviating trend determined in the longitudinal throwing distance of the biofidelic dummy compared to the conventional dummy. This is also shown by the added trend line of Focken.

6 The longitudinal throwing distances of the biofidelic dummy all lay in the range of the throwing parabola determined by Focken. The throwing distance of the biofidelic dummy therefore corresponds to that of the throwing distance of the conventional dummy. The modified structure of the dummy therefore has no significant influence on the throwing distance.

tion of the collision velocity. At velocities of 90 kph, limb separation is to be expected.

Conclusion

We thank the company crashtest-service.com GmbH for the great cooperation and the extensive documentation of the crash test attempts.

The comparison with the impact tests of conventional dummies has shown that the conventional dummy, especially in the transition range between mid to high collision velocities from 65 kph to 70 kph, has weaknesses due to its hard construction which can lead to significantly different vehicle damages caused because of the lack of the wraparound behaviour during the course of a collision.

Through the crash series with the VW Polo 6R and the biofidelic dummy, it was possible to identify that the head impact of the dummy moves nearer towards to the roof edge with increasing collision velocities and at approx. 80 kph there is significant damage to the roof. The more realistic wrapping around of the biofidelic dummy during the collision results in extensive damages, while the conventional dummy causes punctual damages. With the biofidelic dummy, due to its flexibility, it is also possible to demonstrate the functionality of the pedestrian underrun protection system designed by the automotive industry, which should reduce the undergoing of the legs under the vehicle.

When comparing the longitudinal throwing distances of biofidelic and conventional dummies no significant differences occurred, in contrast to the damage of the vehicle.

The change in structure of the dummy has no significant influence on the throwing distance.

Future work In this article, vehicle damages and the throwing distance of the dummies were analysed. In the crash tests the biofidelic dummies, in contrast to the conventional dummies, were “injured”; resulting in fractures and joint injuries, see figure 12. A subsequent “autopsy” of the dummy can then also allow a statement to be made on the expected pedestrian injuries depending on the collision speed. This connection was also analysed by Appel et. al.7 for accidents involving pedestrians and can be used as a further verifica-

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As part of an expert seminar at crashtest-service.com GmbH, in September 2018 a high speed crash test for a passenger car-pedestrian collision with a biofidelic dummy at over 100 kph was planned and carried out. In a subsequent publication, not only can a connection between vehicle damages and the pedestrian injuries be made, but it is also possible to analyse whether the expected tears also occur with the biofidelic dummy.

References 1. A. Kortmann, Crashverhalten im Crashvergleich: der neue Biofidel Dummy bei unterschiedlichen Szenarien von Pkw-Fußgängerunfällen, VKU 03/2018.

2. S. Hartwig, M. Knape, A. Kunze, M. Weyde, Interdisziplinäre Weiterentwicklung eines optimierten biofidelen Dummys als Fußgängersurrogat bei Full-ScaleCrashtests, VKU 03/18.

3. D. Otte, T. Facius, Abwickellänge WAD des Körpers von Fußgängern und Radfahrern an der Pkw-Front und Relevanz als Einflussparameter für Kopfverletzungen, VKU 02/2015.

4. Technical data oft he used dummies are obtainable at crashtest-service.com GmbH 5. A. Kortmann, Crashverhalten im Crashvergleich: der neue Biofidel Dummy, UREKO SPIEGEL 20/2018.

6. U. Focken, Experimenteller Vergleich des gebremsten und ungebremsten Anstoßes bei der Kollision zwischen Pkw und Fußgängern; Diplomarbeit, Fachhochschule Osnabrück, 1998.

7. H. Appel, U. Wanderer, S. Meißner, G. Schmidt, J. Barz, D. Kallieris, R. Mattern, F. Schüler, Mechanik und Biomechanik des Unfalls, H. J. Wagner (Hrsg.): Verkehrsmedizin, Springer-Verlag, Berlin 1984.

Authors Dipl.-Phys. Annika Kortmann, road traffic accident expert at engineering office Schimmelpfennig+Becke GmbH & Co. KG in Münster, Germany.

Dipl.-Phys. Dr. rer. nat. Tim Hoger, publicly appointed and sworn expert for road traffic accidents working at engineering office Schimmelpfennig+Becke GmbH & Co. KG in Münster, Germany.

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crash·ol·o·gy THE SCIENCE OF CRASHES Wesley Vandiver

Robert Anderson

Collision Forensics, Inc.

Biomechanics Analysis

Airbag Myths, Legends, and Lore

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xamples of airbag myths include airbags have toxic smoke or powder, negate need for seat belts, might trap you, can cause suffocation, will kill you, do more harm than good, known for killing kids, don’t inflate unless the automobile is traveling above a certain speed, may be inflated by hard braking, won’t inflate unless your seat belt is buckled, etc. Accident investigators, reconstructionists, biomechanics and others regularly address such fallacies.

In our field we have our own myths and legends surrounding airbag deployments. For example, a common myth is that deployment is based upon or related to vehicle speeds or Delta V. Airbag system deployment or non-deployment behavior is, at least in part, specified by vehicle acceleration response corridors in various barrier impact speeds and angles. The manner in which the vehicle acceleration changes, or jerk, is used to predict the magnitude of the crash early enough that safety equipment can be deployed in time to be effective in preventing or mitigating occupant injury. This is long before the crash is over and the Delta V becomes known.

Given the manner in which information surrounding the early development of airbags was introduced, and the way barrier equivalent velocities and energies are used in accident reconstruction, it is not a surprise that there would be confusion regarding the barrier equivalent vehicle acceleration corridors specified for airbag systems.

For example, TRW and NHTSA brochures indicated that an airbag will only deploy in frontal or near-fron-

tal collisions equivalent to barrier impact at about 12 mph 1 and that they offer protection supplemental to seat belts in frontal impacts comparable to or greater than a 10 to 14 mph barrier impact. 2 An Automotive Engineering article in 1993 reported that the design intent was that airbags must deploy at 14 mph BEV, not deploy under 8 mph, and between theses speeds is the gray area where they may or may not deploy. 3 An IIHS Status report indicated that actual deployment thresholds vary somewhat from one airbag design to another, but virtually all systems are made to deploy in crashes equivalent to hitting a solid barrier at 10 to 12 mph. Mercedes and BMW airbags incorporate higher deployment thresholds – more like 15 mph – for people using belts. 4 An IEEE paper in 1993 reported that normally the airbag is deployed if the collision force is equivalent to a 14 mph barrier impact or higher. 5 Accident Reconstruction Journal features declared that “virtually all systems are made to deploy in crashes equivalent to hitting a solid barrier at 10 to 12 mph”6 and that “dual deployment thresholds are already in use in Mercedes and BMW cars. When occupants don’t use belts, the airbags deploy in crashes equivalent to hitting a solid wall at 9 to 12 mph, the typical deployment threshold. Thresholds are higher – more like 16 mph – for people with belts because they are less likely to be injured in lower speed crashes.” 7 Accident Investigation Quarterly features declared that most automakers currently choose thresholds equivalent to a 10 to 12 mph barrier crash 8 and that Mercedes and BMW use dual deployment thresholds

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depending upon whether or not the occupants are seat belted. When occupants do not use seat belts, the airbag deploys in crashes equivalent to hitting a solid barrier at about 9 to 12 mph. For occupants seat belted, the thresholds are approximately 16 mph. 9 A Nissan consumer guide reported that “At a vehicle speed of about 12 mph into an immovable object, the deceleration rate is usually great enough to activate the airbag system. When a movable and collapsible object, such as a parked vehicle, is struck, the vehicle speed needed to activate the airbag system is about 24 mph.” 10 A GM pamphlet for first responders indicated that the airbag system deploys in crashes of sufficient severity in which the sensors detect a sudden deceleration comparable to a 9 to 16 mph barrier collision or above. 11 In practice, airbag behavior is more complex. In 1997, Lau et al., investigated airbag deployments in actual on-road accidents from the National Automotive Sampling System Crashworthiness Data System (NASS-CDS). When the

total reconstructed Delta V was in the 6 to 10 mph range, frontal airbags deployed in over 57 percent of the crashes, and when looking only at the longitudinal component of the crashes, the same 6 to 10 mph Delta V range resulted in frontal airbags deployments 74 percent of the time. 12 Thus, off-longitudinal-axis acceleration/Delta V was associated with an increased chance of frontal airbag deployments in those earlier systems.

As event data recorder (EDR) data became available in investigated crashes, EDR data was incorporated into the NASS. Like Lau et al, 17 years later in 2014, Wood et al researched the NASS EDR database for airbag system performance in on-road accidents. As shown in Figure 1, they found that the probability of frontal airbag deployment for a recorded Delta V of 6 to 7 mph was about 40 percent and for recorded Delta V’s of 9 to 10 mph the probability of airbag deployment increased to about 55 percent. 13 Besides including more modern airbag systems, the Wood study presumably involved higher seat belt usage.

Figure 1: Probability of airbag deployment versus Delta V 13 74

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These last two studies involve thousands of crashes from the NASS databases. In part, the approach has been to correlate the types of collisions in which airbags have or have not deployed based upon data available after the crash is over, generally a calculated or measured vehicle change in velocity (Delta V). While this information is important and valuable for describing on-road behavior of airbag systems, it is important to not confuse the concepts of correlation and causation. A colloquial example of the correlation = causation fallacy is the relationship between cholesterol and oat fiber. While there is a correlation between airbag deployment and Delta V, the vehicle’s Delta V cannot have a cause and effect relationship with deployment, because to be effective, airbag deployment algorithms must make decisions and airbags must deploy long before the total Delta V is reached. Said another way, because Delta V happens after deployment, vehicle Delta V cannot cause or result in airbag deployment. . . period.

Airbag deployment algorithms are incredibly complex, vary greatly, and are highly protected intellectual property among automobile manufacturers. However, some simplified models that were used to describe early algorithms can shed light on the fact that Delta V does not cause airbag deployments. An early model adopted in the early days of airbag development was known as the “Rule of 5 Inches Minus 30 Milliseconds.” 14 In general, the estimated distance between a fully inflated bag and the chest or head of an occupant in his/her original position was estimated to be 5 inches. Because 1) the inflating airbag should not strike the occupant, but rather the occupant should meet the airbag as he/she moved forward, and 2) the time for a typical airbag to become fully inflated was estimated at 30 milliseconds, it was reasoned that the total time available for a deployment decision was the time it would take an occupant to move forward 5 inches minus the 30-millisecond inflation time. Again, this is an aged model and was simply an approximation in its day. But, based on this

model, several approximations were made regarding sensor triggering time in various crash scenarios.

As can be seen by these reported times, the triggering time is well before the end of a typical collision which may last between 100 and 150 milliseconds. Designing an effective crash sensing system is an incredibly challenging process that must weigh numerous factors. However, in general, these systems share a few basic characteristics: 16 1. Predictive Because there is a defined period of time from the sensor triggering instant to the readiness of a fully inflated bag, the sensing system must be predictive. At the moment when the sensor triggers, the speed change experienced in the vehicle is only a fraction of its eventual magnitude. The window of observation or detection of the sensor is limited; still, sensors, individually or collectively, must make a judgement about the severity of impact at the early stage of a collision.

2. Discriminative It is extremely important that a sensing system be able to differentiate among the circumstances that call for deployment and non-deployment. Not only do unnecessary deployments become a nuisance to the users, they also represent potential hazards in situations when they are not needed. The capability of a sensing system in avoiding misfiring or false triggering is essential.

3. Real-time The task of crash sensing is a real-time application, which means that the sensing method applied for crash detection must function immediately upon the pulse being received. The sensing system cannot afford to wait until the sequence of crash pulse is com-

Sensor triggering time (milliseconds), from the onset of a frontal collision, in various types of crashes for a typical passenger car Types of Crashes

Sensor Triggering Time (milliseconds)

48-64 kph (30-40 mph) vehicle to rigid barrier 48-64 kph (30-40 mph) vehicle to angular rigid barrier 24-32 kph (15-20 mph) vehicle to rigid barrier 24-32 kph (15-20 mph) vehicle to pole 96-128 kph (60-80 mph) closing speed vehicle to vehicle head-on collision 48-64 (30-40 mph) closing speed vehicle to vehicle head-on collision

10-20 20-30 30-50 50-70 20-30 30-50

Table 1: Sensor Trigger Timing 15 www.collisionpublishing.com

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pleted before it analyzes and acts on the situation unfolding before it. The narrow window of crash data available at the time when sensor triggering is required underscores the difficulty of the problem.

These three characteristics are interconnected – each with its own advantages and disadvantages. If a sensing system is extremely sensitive, there is a higher probability that it will be able to detect crashes earlier or make better predictions. However, an extremely sensitive system that makes decisions earlier may be less discriminative. And, since these decisions are made in real time, there is no going back. Once a triggering signal is released, it cannot be undone.

A real-world example of a deployment that did not meet the previously described thresholds and serves as a reminder that deployment decisions are not based on Delta V was experienced during testing. As part of a research project conducted by the Crashology authors, it was planned that non-deployment events would be intentionally set in a Kia Soul. The vehicle is depicted in Figure 2.

Figure 3: Unplanned deployment of driver airbag As with most mistakes, there is much to be learned if the situation is closely examined. If one believed the myths that a vehicle had to be traveling at a certain speed for an airbag to deploy, this event proved that wrong. The Kia was parked. If one believed that someone must be seated in the driver seat or have a seat belt fastened for an airbag to deploy, this event proved those myths to be wrong as well. But, what about Delta V? The longitudinal Delta V was -1 kph as shown in Figure 4 and Figure 5.

The negative polarity resulted from the module bouncing off the bolts after moving forward slightly – it was stopped abruptly when the rear edge of the bolt holes reached the bolts. The lateral Delta V was zero as seen in Figure 6 and Figure 7.

Figure 2: Test vehicle The planned exercises included setting non-deployments by impacting the airbag control module (ACM) from the rear, generating acceleration in the positive-X direction.

The bolts securing the ACM were loosened slightly affording some movement forward when it was struck. A mallet was used to impact the ACM. Several impacts were made to the module resulting in no recorded events, so the impact severity was increased. After several more attempts with negative results, an impact was made to the rear of the ACM and the mallet accidentally “skipped” off the metal body of the center tunnel before striking the module. This resulted in a deployment of the driver airbag.

The engine was running, and the driver seat belt was unbuckled. The resulting unplanned deployment is shown in Figure 3.

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The key to this deployment decision was the extremely short time over which the early crash pulse happened. As can be seen by the graph in Figure 8, the peak acceleration was achieved quickly, and it was brief. As seen in Figure 9, the maximum resultant Delta V was reported at 2.5 milliseconds. This system saw the rapid change in the acceleration and predicted a severe collision that would necessitate an airbag deployment. That predicted severity never materialized, but the decision had to be made quickly.

Clearly this deployment fell outside all of the myths and lore associated with airbag deployments being tied to Delta V and the most important variable involved in this was time. This crash pulse rose at a steep angle but ended quickly. One could say the system was “fooled.” Others would say the system worked just fine, and that not all scenarios can be predicted.

Much can be learned from this as we in the collision reconstruction community encounter crashes that occur under an infinite number of circumstances. Without fur-

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Figure 4: Longitudinal Delta V table

Figure 5: Longitudinal Delta V graph www.collisionpublishing.com

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Figure 6: Lateral Delta V table

Figure 7: Lateral Delta V graph 78

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Figure 8: Longitudinal acceleration graph

Figure 9: Time for maximum resultant Delta V ther information, simply relating the fact that an airbag has deployed or not deployed in a crash as meaning the Delta V for the vehicle fell within a certain range or met a particular threshold is a dangerous assumption. In general, airbag deployments result from collisions that, when they are over, result in certain Delta Vs under very specific circumstances. The Delta V is a result of the total crash pulse, whereas the airbag deployment decision is made based upon the earliest information in the crash pulse.

References 1.

2.

3.

4.

5.

6.

TRW, “General Questions and Answers About Airbags” NHTSA, “Emergency Rescue Guidelines for Airbag Equipped Cars” DOT HS 807 579, July 1990 Airbags: Education and Experience, Automotive Engineering, September, 1993.

All Automobile Airbags Aren’t Alike – They Differ a Lot in Terms of Design, IIHS Status Report, Vol 28, No 11, October 1993.

Alrabady AI & Mahmud SM, “Development of a Decision Making Algorithm for Airbag Control, 0-7803-1229-5/93 IEEE 1993.

ARJ Vol 6 No 1, Airbags Differ in Terms of Design, Jan/ Feb 1994.

7.

ARJ Vol 7 No 2, Airbags Getting Smarter, Safer too, Mar/ Apr 1995.

8. Medical Successes Often Carry Adverse Side Effects – Airbags Are No Exception, AIQ Issue 9, Winter 1996.

9. Ziernicki et al, Automotive Airbags, AIQ Issue 16, Fall 1997.

10. Nissan 1998 Consumer Information, Safety and Warranty Guide.

11. General Motors Answers Your Questions About: Airbags and Pretensioners in GM Vehicles in Emergency Response Situations, January 2001.

12. Lau E, Ray RM and Cheng LY, “Deployment of Airbags in Traffic Accidents: Characteristics and Consequences,” 41st Annual Proceedings of the Association for the Advancement of Automotive Medicine, Nov 1997.

13. Wood M, Earnhart N & Kennett K, “Airbag Deployment Thresholds from Analysis of the NASS EDR Database,” Society of Automotive Engineers, paper number 2014-010496 (Citation SAE Int. J. Passeng. Cars – Electron. Electr.

Syst. 7(1):2014, doi:10.4271/2014-01-0496.)

14. Chan, Ching-Yao, Fundamentals of Crash Sensing in Automotive Air Bag Systems, Society of Automotive Engineers, Warrendale, Pennsylvania, Publication R-217, © 2000 15. ibid.

16. ibid.

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Collision Magazine - Volume 14 Issue 1 79

A How-To Guide

For Downloading Toyota’s

Vehicle Control History Data Alan Moore, P.E., ACTAR

S

tarting in approximately 2013, Toyota began equipping Toyota and Lexus models with a function called Vehicle Control History, or VCH. It is a recording capability, housed primarily in the airbag control module, that is triggered by a variety of events (Figure 1). The original intent of the VCH was to assist Toyota in evaluation of alleged sudden acceleration events, but it has developed into a powerful tool for general accident reconstruction use. Since the VCH capability is not well known, this guide was prepared to assist accident reconstructionists in obtaining the data.

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Figure 1: Partial List of Vehicle Control History triggers Toyota’s diagnostic tool, Techstream, is required to access the VCH data. First, an SAE J2534-compliant interface device is needed. Toyota recommends the Mongoose MFC, but the more common Nexiq USB-Link also works. The Nexiq will need an OBDII cable, either #448013 or # 493013 depending on the generation, both of which are available at diesellaptops.com. The new CDR900 interface also works, although this function may not be supported by Bosch.

Next, a Professional Diagnostic subscription to Techstream is required. An annual subscription is available for approximately $1,295. A 2-day subscription is also available for $65. To obtain the subscription, first set up an account on TIS (Technical Information System) at https://techinfo.toyota.com. This website may need to be visited first to setup the account: https://techinfo. snapon.com/TIS/termbill.aspx?action=new .

Within the TIS website, locate the Techstream download section. Click on “Full Install” (Figure 3). First install the Mongoose Pro MFC driver, even if you don’t plan to use it, as it contains components needed for the Techstream install. To do so, find the install file on your

computer, right-click on it and run as administrator. You may need, in Windows 10, to temporarily disable Virus & Threat Protection. After installation, open Techstream which is now installed on your computer.

Within Techstream, Go to Setup, VIM Select, and select your Nexiq or other interface device. Also set the user type to “Public User” (Figure 4). Click on Software Registration, then again on Software Registration. Follow the directions to create a key. After doing so, Techstream should open and allow use for the duration of the subscription.

If you opted for the much less expensive two-day subscription, it will need to be renewed for the next use.

To do so, go to techinfo.toyota.com, but do not log in.

Click on “Already have an account?” as shown in Figure 5. Then click on Upgrade/Extend my subscription. After you do this, wait for an email to re-enable your account, click on the link in it. Then you should be able to log in using the upper left login. Occasionally the account may get locked, if you try to login before receiving the email renewal, and a call to tech support is required at 877-762-7666.

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Figure 2: Subscription levels of Techstream

Figure 3: Full Install option on TIS 82

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Figure 4: Techstream Configuration

Figure 5: Renewing the Techstream subscription Once you’re running the Techstream software in a compatible vehicle with a connected interface, click on “Connect to Vehicle” (Figure 6). Then click on “Read data”, then “Vehicle Control History” (Figure 7). Several prompts will come up, including one confirming owner consent.

If successful, the window in Figure 8 will be presented, showing the various VCH events that have been recorded.

Double-clicking on each one allows viewing of the data.

After double-clicking on an event, double click on the pa-

rameters of interest to see a graph of the data (Figure 9).

The data can be viewed within Techstream, or exported as a CSV file which can be somewhat difficult to work with due to its formatting. When you’re finished, be sure to export the .TSE (Toyota Service Event) file to preserve the data. Note that it can only be opened in a current subscription of Techstream!

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Figure 6: Connect to Vehicle in Techstream

Figure 7: Click on Read Data, then Vehicle Control History 84

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In Figure 8, a column titled “Image” displayed an icon for two events. This indicates that images were recorded for the event, but they cannot be accessed from this screen.

They are accessed from a separate function known as Freeze Frame Data (FFD). After clicking on the “Read Data” function, there is a separate option for “PCS FFD Retrieval”. After a fairly long download, it will produce a window similar to Figure 10. Click on “Check” to see

the data files in a folder, and copy them to your case file.

“View” launches the PCS Data Viewer from within which you have to open the files that were previously saved (Figure 11). To save the images, go to File, Output PCS Data, and select a folder. It will dump a lot of data including the forward camera images and a detailed report that includes timestamps on the images.

Figure 8: Downloaded VCH data

Figure 9: Double click on the parameters of interest to see a graph of the data www.collisionpublishing.com

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Figure 10: Downloaded FFD data (above) Figure 11: PCS Data Viewer for Freeze Frame Data and images (right) Some things to keep in mind when using Techstream and VCH:

•

Portions of the Techstream software and subscription process are challenging, especially when renewing the two-day subscription. Some patience and troubleshooting may be required.

There are frequently discrepancies in the key cycle counters between CDR, VCH, and FFD data. Other methods, such as the odometer, must also be considered in validating the data.

•

•

Before using Techstream on an active case, definitely considering getting access to a newer Toyota to experiment with.

This guide should be considered a starting point, and does not cover all the necessary tasks to obtain and preserve the data.

•

•

This is a rapidly changing technology, especially on its use with Advanced Driver Assistance Systems, so you should expect differences between this guide and vehicles you may encounter in the future.

While most of the VCH data is contained in the airbag module, other data such as the FFD images are stored elsewhere in the vehicle. There is not currently a published method for direct-to-module download of VCH data, although this will likely change in the future.

•

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A REVIEW OF ENGINEERING MECHANICS APPLIED TO ACCIDENT RECONSTRUCTION Jai Singh, PART 2: DYNAMICS BS, MS, MA, ACTAR Biomechanical Engineering Analysis & Research, Inc.

A

bstract The subject paper is the second in a series that focuses on the role and application of rigid body engineering mechanics, specifically dynamics, to the field of accident reconstruction. Building upon the foundation of kinematics, presented previously, the focus of the subject work is the derivation of the Newton-Euler equations of motion in both the inertial frame of reference and in the body frame. The approach taken is the sequential defining and partitioning of forces that are in-

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ternal and external to a system under consideration, defining the mass center, defining the translational and angular momentum vectors and then defining the mass moment of inertia matrix in both frames of reference. These definitions are used to develop the equations of translational dynamics (Newton’s second law of motion) and rotational dynamics (Euler’s equation). The importance of frame selection and body frame origin of coordinates placement is discussed. Finally, the solution for the kinetic energy of a rigid body undergoing free motion is presented.

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Introduction The first paper in this series focused upon establishing the conceptual framework for engineering kinematics problems that involve multiple frames of reference.1 The direction cosine matrix (DCM) was introduced as a descriptor for orientation kinematics and for coordinate frame transformation. The concept of frame referenced derivatives was presented, contrasted against standard time derivatives and used for defining the derivative transport theorem. A modified nomenclatural framework, based upon the work of Jazar,2-5 was presented. This framework allowed for the inclusion, in the symbolic representation of derivative variables, of both the expression frame and derivative frame. For the two frame problem, starting with the position vector, it was shown that there are a total of four forms for the velocity solution (one form for each combination of the derivative frame and expression frame). It was further shown that there exist a total of eight forms for the acceleration solution for the two frame problem. These areas within the purview of engineering kinematics, individually and in their totality, serve as a necessary foundation for the development of the theory of engineering dynamics and its presentation.

Kinematics focuses on the study of motion of objects without specific consideration of forces and moments. Dynamics, on the other hand, focuses on the study of the motion of objects with specific consideration of mass and force as well as inertia and moment. The subsequent review proceeds by first developing the mathematical framework that relates forces and moments that are either internal or external to a system under consideration and the resultant acceleration experienced by the system. The term system, as used herein, may refer to a collection of discrete rigid bodies as well as a continuous single rigid body. This is followed by defining the mass center (i.e. center of mass) of a system under consideration. The mass center serves an important role when the origin of a body frame of reference is placed at the mass center in regards to simplifying the form of certain equations. From a notational standpoint, the single left superscript (the expression frame) and left subscript (the derivative frame) notation employed in the previous work for velocity is retained herein for terms that are direct functions of velocity. The double left superscript and double left subscript notation employed in the previous work for acceleration, with the first script pair referencing the expression and derivative frames for the first derivative and the second script pair referencing the expression and derivative frames for the second derivative, is also retained herein for direct functions of acceleration. These notational factors are first manifested in defining the translational and angular momentum, of a system, in the inertial frame of reference. This is followed by introducing the body frame mass moment of inertia matrix. The introduction of these topics is preparatory for defining the Newton-Euler equations of motion.

The primary equation for translational dynamics is Newton’s second law of motion, which is valid in an inertial frame of reference. The definition in the inertial frame of reference is followed by presenting the valid form of the relationship starting with the body frame (B frame) linear momentum and then using the derivative transport theorem to take the frame referenced derivative with respect to the inertial frame of reference (i.e. the G frame). The application of a kinematic transform (premultiplication by a DCM) to the resultant is shown to reproduce the inertial frame of reference solution. The primary equation for rotational dynamics is Euler’s equation. The relationship between the angular momentum and angular velocity, in each frame, are first presented. Euler’s equation is first derived in the inertial frame of reference. The difficulty arising from the G frame formulation of Euler’s equation, that being the time varying nature of the G frame moment of inertia matrix, is described. This issue is then resolved by developing the appropriate body frame formulation of Euler’s equation, in which the mass moment of inertia matrix is time-invariant for a rigid body, and showing that the application of a kinematic transform to the solution yields an equivalent result to the inertial frame formulation. Finally, the solutions for the kinetic energy associated with a system undergoing free motion (i.e. translation and rotation) are presented. www.collisionpublishing.com

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Equation (6) may also be written as the moment of the force, Fq, about a directional line, l, with unit vector u along the line, that passes through the origin.

Rigid body dynamics Force and moment A system of interconnected bodies, rigid or otherwise, can be subject to force developed internal to the system as well as to force applied by external sources. Tractive forces, gravitational force and forces generated during a collision are all examples of the latter. For a system consisting of N1 interconnected bodies, force acting on the ith body is the sum of the internal forces generated upon the ith body by the other N1 -1 bodies and the sum of the Ni2 external forces acting upon the ith body.

N1

Ni2

j = 1,j ≠ i

p =1

(1)

The internal force generated by the ith body of the system on the jth body of the system is equal in magnitude and opposite in direction to the force generated by the jth body on the ith body (Newton’s third law of motion).

(2)

Summing equation (1) and using equation (2) shows that the internal forces cancel and that the net force acting on the system is equal to the sum of the externally applied forces.

N1

i =1

i = 1 j = 1,j ≠ i

N1

N1 Ni2

N2

i =1 p =1

q =1

∑ Fi = ∑ ∑ Fij + ∑∑ Fip = ∑ Fq = F

(3)

In equation (3), N2 denotes the total number of unique externally applied forces that are acting on the system. The net moment generated by the externally applied forces, on the system, is equal to the sum of the moment generated by each unique externally applied force.

N2

M = ∑M q q =1

Mass center and central frames The mass center or center of mass of a body with volume V and local density ρ, is defined as a body point for which the following holds:

∫ ρ rdV = 0

(8)

V

A body frame with origin at the mass center is referred to as a central frame. If the B frame is a central frame, then the following holds for the center of mass:

∫ r dm = 0 B

(9)

dm

(4)

Where dm refers to an infinitesimal mass in the body and rdm denotes its B frame position vector. The G frame position vector of dm is: G

rdm = G ro + G RB B rdm

For a given force, Fq, acting at point P (the position vector of P being rP), the moment of the force about a point Q is: M Q = ( rP − rQ ) × Fq

(5)

When Q is the origin of coordinates of the system, rQ = 0, and equation (5) reduces to: M = rP × Fq

Collision Magazine - Volume 14 Issue 1

(6)

(10)

Multiplying equation (10) by dm and integrating the resultant:

∫ r dm = ∫ r dm + R ∫ r dm G

G

dm

B

G

o

B

B

B

(11)

dm

B

When the body frame is a central frame, the second term on the right of the equality is zero valued by the definition in equation (9).

∫ r dm = ∫ r dm = r ∫ dm = m r G

G

dm

B

90

(7)

B

∀i , j

N1

)

B

Fi = ∑ Fij + ∑ Fip

Fij = −Fji

(

M = l u ⋅ rP × Fq

G

o

B

G

o

o

B

(12)

The first and second G derivatives of equation (12) are: m GG r&o = m GG vo =

G

d G rdm dm = ∫ GG r&dm dm = ∫ GG vdm dm dt ∫B B B

GG GG && & m GG GG ro = m GG vo = m GG ao =

G

d G GG GG & G vdm dm = ∫ GG vdm dm = ∫ d ( GG Fo ) dt ∫B B B

(13)

(14)

Evaluation of the last integral results in the following:

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GG GG & m GG GG vo = m GG ao = GG Fo

(15)

This shows one of the advantages of using a central frame.

Translational and angular momentum in the inertial frame of reference The translational (i.e. linear) momentum of a body or system is defined as a vector that is equal to the mass of the body (or total mass of the system) multiplied by the translational velocity of the mass center of the body (or system). In the inertial frame of reference, the translational momentum is expressed as: G G

p = m GG v

(16)

Newton’s second of motion states that the net sum of the externally applied forces on a body or a system, in the inertial frame of reference, is equal to the global time rate of change of the translational momentum of the body or system. The G derivative of equation (16) is: G  Gd  G  Gd G  dG d G p = v = v + m m m ( )  G  G G G v dt dt  dt   dt 

G

(17)

The first term on the left is the force in the G frame. When the mass of the body or system remains constant, its time rate of change is zero and equation (17) reduces to the following: GG GG

F=

 Gd G  dG GG GG & p = m  G G v  = m GG v = m GG a dt  dt 

(18)

L = G rC × GG p

(19)

Taking the G derivative of equation (19): G G  G  dG d G G rC × GG p = ( rC × GG p ) =  dtd G rC  × GG p + G rC ×  dtd GG p  GL = dt dt    

G

(21)

The parenthetical in the second term on the right of the last equality in equation (20) is the force shown by equation (18). As a result, equation (20) can be written as: G

dG GG G rC × GG G L = GG M = GG F dt

(22)

Mass moment of inertia Most readers should be familiar with the reporting of yaw, pitch and roll moment of inertia values from commercially available vehicle specification databases and/or the salient literature.6-8 A moment of inertia, with respect to a set of axes, is simply a measure of geometric mass distribution. A standard approach to deriving the form of the components of the mass moment of inertia matrix (more aptly stated as the matrix associated with the mass moment of inertia tensor) is by starting with the kinetic energy for a rigid body B (Oxyz) rotating in G (OXYZ). The specific form of the velocity vector that is of interest is GB v . For the case in which the origins of the two frames are coincident, this velocity is: B G

v = GB ωB × B r

(23)

The angular velocity vector in equation (23), in terms of its components in the B frame, is:

G

The angular momentum (i.e. moment of momentum) of the body or system, about the mass center of the same, in the inertial frame of reference, is defined as follows, where GrC is position vector of the mass center in the inertial frame of reference: G G

 Gd G  G rC  × G p = GG vC × m GG vC = m ( GG vC × GG vC ) = 0  dt  

B G

The first term on the right of the final equality in equation (20) is zero valued:

(24)

The position vector in equation (23), representing the B frame position of a point in B, is: B

r = xe1 + y e2 + ze3

(25)

The cross product shown in equation (23), in component form, is: B G

(20)

ωB = ωx e1 + ωy e2 + ωz e3

(

)

(

)

ωB × B r = ωy z − ωz y e1 + ( ωz x − ωx z ) e2 + ωx y − ωy x e3

(26)

The kinetic energy associated with the velocity GB v is: K=

1 B 2 ( G v) dm = 21 ∫ ( GB v ⋅ GB v) dm = 21 ∫ ( GB ωB × B r ) ⋅ ( GB ωB × B r ) dm 2 ∫B B B

(27)

Substitution of equation (26) into equation (27), evaluating the dot product and rearranging the result by noting www.collisionpublishing.com

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that the components of the angular velocity are independent of the integration leads to the following result.

K=

2

ω ωx ( y 2 + z2 ) dm + 2y ∫ ( x 2 + z2 ) dm + ω2z ∫ ( x 2 + y 2 ) dm 2 ∫B B B 2

2

−ωx ωy ∫ xydm − ωy ωz ∫ yzdm − ωx ωz ∫ xzdm B

B

(28)

B

The integral terms in equation (28) can be conveniently placed into a 3 x 3 matrix, which is denoted as BI (the body frame mass moment of inertia matrix).  y 2 + z2  B I = ∫  −xy B  −xz

−xy 2 x + z2 − yz

−xz   − yz  dm x 2 + y 2 

(29)

Using the subscripts of i and j {i, j = 1 to 3} to denote the row and column of BI, equation (29) shows that when i ≠ j that BIij = BIji. If we replace the scalar components {x, y, z} with xi = {x1, x2, x3}, we may rewrite BI as: B

(

)

(

)

I ij = ∫ ( B r ) δij − x i x j dm = ∫ ( B r ) I − rr T dm B

2

B

2

(30)

I 1 I =  0  0

K=

1B T B B G ωB ( I ) G ωB 2

I xx − λ I xy I xz    I − λI = det  I yx I yy − λ I yz  = 0  I zx I zy I zz − λ  

( I − I i I ) ui = 0

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(34)

There are instances in which one may be interested in determining the mass moment of inertia matrix for a point that is not in the central frame. In such cases, the typical interest involves a translated body frame (i.e. one that is parallel to the central frame) and/or a rotated body frame.

In this regard we consider the case of a rigid body with a central frame B located at the mass center, C, as well another point, A, on the same rigid body. At the point A, we consider two additional body frames. The first, B1, is rotated with respect to the central frame while the second, B2 is parallel to the central frame. The relationship between the body frames B1 and B2 is a rotation.

I = B1 RB2

( I) R B2

B1

B2

T

= B1 RB2

( I) R B2

B2

B1

(35)

For the body position vector, BrA, we may readily define an associated antisymmetric matrix that allows for the presentation of the parallel axis theorem for the mass moment of inertia about the axes of B2 as: (36)

(32)

The three solutions for λ (the eigenvalues) are the principal mass moments and are traditionally labeled as I1. I2 and I3 with I1 ≥ I2 ≥ I3. The resultant principal mass moment of inertia matrix has the following form:

92

(33)

(31)

For a mass moment of inertia matrix in the body frame, an orientation of the body frame may be chosen such that the products of inertia vanish. The body coordinate frame associated with this orientation is called the principal coordinate frame and the corresponding mass moments of inertia are called the principal mass moments. The principal mass moments can be determined by solving the following eigenvalue problem (where I is the 3x3 identity matrix):

B

0 0  I 3 

The eigenvectors represent the principal axes, ui, that are associated with each principal mass moment.

B1

While it may not be readily apparent, the kinetic energy for the case of pure rotation with the origins co-located can be written as the following vector-matrix multiplication (the validity is born out by carrying out the multiplications):

0 I2 0

The relationship between the mass moment of inertia about the axes of B1 can now be related to the mass moment of inertia about the central frame by substitution of equation (36) into equation (35).

(37) Another point of interest is the determination of the radius of gyration. For a rigid body with mass moment of inertia I, a single point mass with the entire mass of the rigid body located at a distance of κ from the point, line or plane for which the mass moment of inertia is I, is the radius of gyration, if the following holds: m κ2 = I

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(38)

If the reference point is the mass center or if the reference line or plane passes through the mass center then the radius of gyration is the principal radius of gyration.

Rigid body translational dynamics

(44) Applying equation (44) to the G derivative of GB v :

Equation (18), repeated below, was the statement of Newton’s second law of motion in the inertial frame of reference.

(45) Applying a kinematic transform to equation (45):

(39) Looking at the first equality, we may rewrite equation (39) by multiplying through by the differential dt and integrating the resulting terms over their respective domains.

∫

GG GG

t

Fdt = ∫

G Gp

G

d p=m ∫ d v G G

G

G G

G Gv

(40)

The term on the left of the first equality, by definition, is the G frame impulse. The second equality arises from the definition of the translational momentum and by noting that the mass is constant with respect to the integration.

Evaluating the last integral leads to the well-known linear impulse-momentum relationship.

GG GG

G P = ∫ GG GG Fdt = m ∆ G v t

(41)

For a number of accident reconstruction applications, however, it is preferable to work in the body frame. This preference arises from the fact that it is generally easier to determine the forces from the driveline, suspension, tires and collisions, in the body frame. The resultant relationship is: (42) The relationship given by equation (42) is generally not a valid statement of Newton’s second law of motion simply due to the fact that the B frame is a non-inertial frame of reference. We may, however, take the G derivative of the following form of the B frame linear momentum: BB GG

F=

 Gd B  d B Gp = m  G v dt  dt 

GG GG

BB BB a = G RB GG a = G RB ( BG a + GB ωB × GB v )

(46)

Rigid body rotational dynamics The equation of motion governing the rotational dynamics of a rigid body (or system) is known as Euler’s equation.

The combination of equations governing the translational and rotational dynamics of a rigid body are known as the Newton-Euler equations of motion. In the G frame, we may define the relationship between the angular momentum possessed by a body and its angular velocity: G

L = G I GG ωB

(47)

In equation (47), GI is the mass moment of inertia matrix of the body or system with respect to the coordinate axes of the G frame. The corresponding equation in the B frame is: B

L = B I GB ωB

(48)

Only a single superscript is shown on the angular momentum because each relationship derives from the singular corresponding frame expression for the angular velocity.

If we start with the G frame expression and take the G derivative: G G

M =

G  Gd G  G  Gd G  dG d G G L = GG L = I G ωB ) =  I  G ωB + G I  ( G ωB  dt dt  dt   dt 

G

(49)

Multiplying equation (49) by the differential dt and integrating the resultant over the respective termwise domains results in the relationship between the angular impulse and angular momentum:

G

(43) (50)

Recalling the derivative transport theorem:

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Collision Magazine - Volume 14 Issue 1 93

The direct implementation of equation (49) is problematic in that GI is generally not time-invariant and therefore its G derivative is generally not zero. This can be seen by first rewriting GI in terms of BI, which is time-invariant for a rigid body, using the transformation given by equation (35) for B (Oxyz) and G (OXYZ): G

I = G RB ( B I ) B RG

M =

 Gd dG L = GG L =  dt  dt

G

The kinetic energy associated with an infinitesimal mass element of the body, dm, is: (59)

(51)

The total kinetic energy is obtained by integrating dK over the whole body:

Substitution of equation (51) into equation (49): G G

(58)

( R ( I ) R )  ω + ( R ( I ) R )  dtd ω  (52) G

B

B

B

G

G G

B

G

B

B

B

G

G

G G

B

This again shows the difficulty in using the G frame form of Euler’s equation in that the direction cosine matrices are generally not time-invariant. We may instead approach the problem by taking the G derivative of the B expression of the angular momentum vector and employ the derivative transport theorem.

(60)

For the last term, on the right of the final equality, we may substitute for the G expressions of each term:

(53) The first term on the right side of the last equality can be obtained from taking the B derivative of equation (48):

(61)

(54) The first term on the right side of the second equality reduces to zero because of BI being fixed with respect to B for a rigid body. Substitution of this result into equation (53):

The parenthetical term on the right of the last equality is a scalar and therefore:

(62) (55)

Equation (60) can thus be rewritten as:

Applying a kinematic transform to this equation: (63)

(56)

When the point P is fixed in the B frame, equation (63) reduces to:

Kinetic energy Recalling the expression of the position vector for a point P whose position is referenced with respect to the B frame: G

rP = G ro + G RB B rP

(57)

The velocity of the point, in the G frame, was shown to be:

94

Collision Magazine - Volume 14 Issue 1

(64)

When the body reference point o is located at the mass center, C, equation (64) reduces to a form in which the

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translational and rotational components of the kinetic energy are decoupled.

(65) Discussion The derivations presented in the subject work are by no means novel nor are they unknown. Instead, they are wellestablished and long-standing in the scientific literature.

In this regard, the solutions presented and the manner in which each solution was reached can be viewed as a selfcontained statement that doesn’t require any additional discussion. There are, however, a few points, in the view of the author, upon which a limited discussion is germane.

The first point is that the inclusion of all of the salient frames of reference in the explicit representation of a variable aids in both the proper interpretation of the expression frame along with the salient frames in the process of determining the derivatives. This utility can be specifically evidenced when it comes to frame referenced derivatives beyond the first. This approach for tracking each expression and derivative frame, clearly, as an underlying inception, traces to differentiating the position vector. This can be contrasted against the case in which the force or moment is specified and one seeks to determine the resultant acceleration. For a two frame problem, an applied force or moment can be expressed in terms of the vector components in the two frames of reference. This creates an apparent mismatch when there are eight potential options for the acceleration solution for a two reference frame problem. This particular issue is explored in the final paper of this series.

A second important point is that any particular kinematic variable can be transformed, in terms of its components, between different frames of reference. This is of direct relevance to the subject work due to the simple fact that the process for finding the solutions for the Newton-Euler equations can be greatly simplified by the choice of frame in which the evaluation is performed. For the translational dynamics, if one is evaluating a problem in which the forces applied to a body, with the forces expressed in the body frame, it would generally be easier to evaluate the problem in the body frame. Conversely, if the knowns for the problem are cast in the inertial frame of reference, one would reasonably expect that the problem is most easily solved in the inertial frame of reference. For the rotational dynamics, however, working in the body frame is generally the preferable choice due to the mass moment of inertia

matrix, for a rigid body, being invariant with respect to the body frame.

A third point, following the second in regards to simplifying any given analysis, is placing the origin of the body frame at the mass center of the same. The first benefit of making such a choice is that the product of inertia terms in the body frame mass moment of inertia matrix for a rigid body reduce to zero. The resultant diagonal inertia matrix reduces the number of unknown elements in the matrix and simplifies the resultant vector-matrix products. The second benefit is the decoupling of the translational and rotational rigid body modes in the determination of the system kinetic energy.

References 1. Singh J (2020) A review of engineering mechanics applied to accident reconstruction. Collision 14(1): 44-56.

2. Jazar RN (2007) Theory of Applied Robotics: Kinematics, Dynamics and Control (1st ed.). New York, New York: Springer, pp. 297-325.

3. Jazar RN (2008) Vehicle Dynamics, Theory and Practice. New York, New York: Springer, pp. 219-279.

4. Jazar RN (2011) Advanced Dynamics, Rigid Body, Multibody and Aerospace Applications. Hoboken, New Jersey: John Wiley & Sons, Inc., pp. 695-864.

5. Jazar RN (2012) Derivative and coordinate frames.

Nonlinear Engineering 1: 25-34.

6. Arndt MW, CP Dickerson, GA Mowery and SM Arndt (1995) Motor vehicle mass property envelopes.

SAE Technical Paper No. 951065.

7. Arndt MW, CP Dickerson, SM Arndt, GA Mowery and SC Shapiro (1995) Effects of passenger and cargo loading on a motor vehicle’s mass properties. SAE Technical Paper No. 952676.

8. MacInnis DD, WE Cliff and KW Ising (1997) A comparison of moment of inertia estimation techniques for vehicle dynamics simulation. SAE Technical Paper No. 970951.

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Collision Magazine - Volume 14 Issue 1 95

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