Engineering Quarterly Summer 2022

Engineering Quarterly VOL.IV. . . No.4

SUMMER 2022

W ELCOME !

The Engineering Quarterly is published four times each year by the Hamm School of Engineering at the University of Mary. In each issue, we try to include articles from each of the five majors we offer: Electrical Engineering, Mechanical Engineering, Civil Engineering, Computer Science, and Construction Management. The intended audience of the Engineering Quarterly includes engineers, engineering students, engineering alumni, future engineering students, and people working in fields related to engineering such as science, computing, and mathematics.

In this issue we describe our student design projects, new faculty, new building, scholarships, and service activities. As usual, we also include some problems from our courses that you might like to try. Enjoy!

Dr. Terry Pilling Dean

O RDINATION OF FATHER ATHANASIUS On June 12, 2022, at St. George Antiochian Orthodox Christian Cathedral, our electrical engineering professor Deacon Damian Oweis was ordained to the Holy Priesthood. His name is now Father Athanasius Oweis.

It is very gratifying for the Hamm School of Engineering to have a professor who has also been ordained. We are very proud of Father Oweis and hope that his presence as a priest will further integrate the religious and cultural education of our students as well as provide an inspiration to his fellow faculty members.

N EW FACULTY Assal Haddad joins our faculty this fall as Professor of Civil and Environmental Engineering. Assal received his Ph.D. in EnvironmenAssal tal Engineering at the University of Central Florida in 2011 with thesis Assessment of the Use of Vegetative Mulch as Daily and Intermediate Alternative Landfill Cover; his M.S. in Civil Engineering at the Jordan University of Science and Technology in 2004; and his B.S. in Civil Engineering at the Jordan University of Science and Technology in 2001.

His current research involves Solid Waste Management and local environmental issues such as waste open burning, evaluating local landfills that are not properly engineered, and social environmental awareness and behavior.

Before joining the Hamm School of Engineering, Assal was a professor in the Applied Science University in the Civil Engineering Department. His hobbies include playing the guitar, the piano, and learning to play the drums, and “denying that now that I am over 44 I will never be a rockstar.” Assal is interested in the progress achieved in Artificial Intelligence. “I am always amazed about how close they got to imitating and predicting human behavior. I am obsessed with how the Facebook algorithm works, that I can never really enjoy the newsfeed on FB because I am too busy analyzing why the algorithm chose to show me this specific ad or news bit at this specific time.”

Jerika Cleveland joins our faculty this fall as Professor of Electrical Engineering and Computer Science. Jerika received her Master’s degree (2017) and her Ph.D. (2022) at North Dakota State University. She was a Hap Jerika Arnold Scholar during a summer internship in 2017 for the Air Force Research Labs down in Eglin, FL. Her research has been in applied electromagnetics with biomedical applications. Her master’s work, entitled The Design and Analysis of a Microstrip Line Which Utilizes Capacitive Gaps and Magnetic Responsive Particles to Vary the Reactance of the Surface Impedance. was on the theory of using magnetite core, silver-coated particles to change the signal direction of an antenna; and her Ph.D. work is designing and creating this antenna. She has continued to do additional work with undergraduate researchers in electromagnetics and programming a simulation of a human

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neuron firing. “Art has always been a hobby of mine. I paint, sketch, and create digital art on a tablet, usually making portraits or abstracts. You may see me doodling while thinking about technical ideas as it helps me focus. I have also been doing CrossFit for two years now and will more than likely work it into some word problems for my classes. I deadlift 205lb and overhead squat 110lb, so ask me about my gains – they are not just for amplifiers!”

Dayakar Lavadiya joins our faculty this fall as Professor of Civil and Mechanical Engineering. Dayakar received his PhD in Civil Engineering (structural mechanics) from Dayakar North Dakota State University in 2021 with research on artificial intelligence for damage detection in infrastructure. His thesis was entitled Neural Networks and Sensitivity Analysis for Detection and Interpretation of Structural Damage. Prior to this accomplishment Dayakar received his first doctorate in Mechanical Engineering from Utah State University with emphasis on composite materials with thesis Effective Properties of Randomly Oriented Kenaf Short Fiber Reinforced Epoxy Composite. His current research interest includes sustainable materials for infrastructure, solid mechanics, and application of artificial intelligence in engineering. Prior to this appointment, Dayakar was a post-doctoral research fellow in the Sustainable Infrastructure Research Initiative (SIRI) group at the University of North Dakota. Dayakar’s hobbies include playing sports (volleyball, cricket), watching movies, cooking, hiking, kayaking and camping with his wife Jamie and three stepchildren (Joshua, Adele and James).

Surendra Batukdeo joins our faculty this fall as Professor of Electrical Engineering and Computer Science.

Surendra earned his M.S. in ElecSurendra trical Engineering from the California State University in Northridge in 1999 and his B.S. in Electrical Engineering from Wichita State University in 1997. Since then Surendra has over 20 years of industry experience in software engineering and artificial intelligence as an educator and mentor in emerging and disruptive technologies. He is the proud father of one 14 year old son, he is passionate about flying Cessna, swimming and rowing, and enjoying baroque music, and one of his seven habits is service to mankind.

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2022 S ENIOR D ESIGN P ROJECTS

On Friday, April 22, the Seniors presented their design projects from Professor Tom Volkman’s Senior Design course. We have our senior design presentations at the same time each year as the meeting of our Industry Advisory Board so there were a number of industry engineers who attended and helped the faculty judge the presentations. At the Hamm School of Engineering, our senior design projects are multidisciplinary projects that involve engineering students from different departments. Many projects that are designed in industry are multi-disciplinary ones and it is therefore important for our students to experience this aspect of the process. It involves learning how to delegate the best engineer for a given part of the design and also how to communicate effectively across disciplines. It also allows our students to choose more interesting projects for their capstone senior design course.

Robotic Chess Board

William Turner (ME) designed an electronic sensor system to notify how many spots are open on each level of the structure. A unique detection system provides a roof level indicator to notify drivers where the open spots are located so driver doesn’t have to search for a parking space.

Ethan Melder, Rami Douri, Michael Storick, and Lucas Meyer designed and built a robotic chess board for their senior design project. The goal of this project was to develop a robotic chess game in which you play against artificial intelligence (AI) controlled by a computer. A robotic arm controls the piece movement as commanded by the AI. Rami Douri (EE) developed the software for the chess engine and to control the robotic arm. The arm is controlled by servo motors.

Rami also assisted in building the chess board.

Parking Garage

William Blewett, Davids Atelbauers, Blake Rubio, Daniel Turner, and Izaiah Bitz designed a parking garage for the University of Mary campus for their senior design project.

Izaiah Bitz (CE) was responsible for defining the overall scope and purpose of the parking structure, as well as doing mathematical calculations for both the structural design of the reinforced concrete on the main parking levels and on the ramps/entrances/exits. Davids Atelbauers (CE) performed an analysis on the existing parking lot structures as well as design the foundation for the new structure. He also designed the drainage system and performed an economic analysis of the design.

William Blewett (CE) concentrated on the design of the steel structure, including the sizing of the columns and beams. William was also responsible for sustainability issues incorporating solar panels for energy and green walls to offset CO2 emissions. Blake Rubio (CE) was responsible for performing site surveys, soil analysis and geotechnical reports associated with the structure. He also work on the foundation and steel designs.

Electric Hydrofoil Karsen Meyer, Chandler Olson, Mark Sammons, and James Jurek designed an Electric Hydrofoil for their senior design project.

The goal of this project was to prototype an electric hydrofoil that is significantly cheaper than ones currently on the market, while operating with the same or very similar performance specifications.

Karsen Meyer (ME) performed the overall design of the board, the hydrofoil and the mast. He also designed the waterproof housing for the battery.

Lucas Mayer (ME) was responsible for designing the claw for picking up the chess pieces and for designing the base and rail which is connected to the robotic arm. Ethan Melder (ME) designed the mechanics for the robotic arm and performed analysis on material strength to ensure that it was strong enough to pick up and move the chess pieces.

Michael Storick (EE) designed a circuit board which attaches to the bottom of the chess board and detects the movement of chess pieces. He also assisted in the building the board and developing the graphical interface for the AI.0 Biomemtic Arm Daniel Minor, Peter Smith, and Matthew Bitner, designed a biomimetic arm for their senior design project. The goal of this project was to create a semi-biomimetic robotic prosthetic hand, for transradial amputees, that balances functionality, modularity and cost effectiveness.

Peter Smith (ME) performed analysis of forward displacement and inverse kinematics, as well as 3D modeling of the phalanges and quality control of all joints.

James Jurek (ME) was responsible for developing a communication system between a hand transmitter and a receiver in the board to control the speed. Mark Sammons (ME) designed a cooling system for the cooling system to ensure that the electronics, battery pack and battery management system don’t overheat. Chandler Olson (EE) designed a control system which determines the speed of the hydrofoil.

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4 for a 4 × 4 grid, 1 through 5 for a 5 × 5, 1 through 6 for a 6 × 6, etc. –– so that no digit appears more than once in any row or any column. The grids are divided into heavily outlined groups of cells called cages and the numbers in the cells of each cage must produce a certain ‘target’ number when combined using the specified mathematical operation. Digits may be repeated within a cage, as long as they are not in the same row or column and a single-cell cage is a free space.

Daniel Minor (ME) did 3D modeling of the palm and thumb, as well as did 3D printing of the fingers, thumb, palm and forearm.

Daniel also created assembly drawings for all 3D parts.

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Matthew Bittner (ME) did modeling of the forearm and the wrist universal joint.

Matthew concentrated on the slave controller; determining the power requirements and writ- Having our own facilities attached to the ening Arduino code to run all of the servos. gineering school will make the logistics much easier for the students. It will also allow us to use the facilities and equipment at our discretion which is necessary for students to work on projects around the clock as we do with our other laboratories in the Engineering School.

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N EW C IVIL E NGINEERING AND C ONSTRUCTION M ANAGEMENT L ABORATORY M ISSION TO P ERU

By JAMES CARRICO

Construction of the new addition for our Construction Management and Civil Engineering programs began in summer, 2021 and was completed this fall. The Construction Materials Laboratory was sponsored by Miller Insulation and contains our wet labs for concrete and asphalt work as well as a crane and testing equipment.

K EN K EN P UZZLE

By NANCY KELLY

We have been fortunate until now that local companies have allowed us to use their laboratory and testing facilities to teach our students while we were designing and constructing our own facilities.

KenKen and KenDoku are a style of arithmetic logic puzzle invented in 2004 by Japanese math teacher Tetsuya Miyamoto.

KenKen derives from the Japanese word for cleverness. As in Sudoku, the goal of each puzzle is to fill a grid with digits –– 1 through

This Spring engineering students participated in the annual University of Mary service trip to Ayaviri, Peru. Ayaviri is a remote town high in a mountainous district of southern Peru. The local population is roughly 20,000 and though public services do exist they are limited. Students in the St. Gianna School of Health Sciences have regularly gone on this trip to serve in a clinic and school run by Salesian Sisters in the area. Students and faculty in the Hamm School of Engineering are now participating in this trip to help in the clinic and extend the scope of services offered by the school. On this first year of engineering involvement, the two engineering students and faculty members were asked to build a shelter for the sisters’ chickens to provide protection during the rainy season. The construction of the chicken shelter is outlined below. In addition, we also participated in other repairs around the sisters’ facility with the other student participants; including repairing a road, facilitating repairs to a greenhouse, and painting. One of the engineering student also helped out at the clinic by repairing exercise equipment and by fabricating assistive devices such a crutches.

Here we see our two engineering students setting out on their first day ready to work in the low oxygen levels and high UV exposure at 12,000 ft. At this elevation hydration is essential, sun screen is a must and over exertion could lead to serious illness.

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H AAS S CHOLARSHIP

One of the first tasks was to dig post holes for the frame of the shelter. The chickens saw an opportunity in the digging to find new areas to nest and look for food.

Next we assembled and put up the frames. We used the cinder blocks as make shift forms so that we could assemble four identical sides without having to make repeated measurements.

a. Civil Engineers: Infrastructure Project – compare the use of steel vs. concrete for a building, the use of asphalt vs.

Portland cement for a road, Installing Ben Pihl is the recipiOzone/Blacklight/Chemical treatment ent of the 2022 HAAS for a water treatment plant, etc. scholarship.

Ben is seen here next to the b. Construction Managers: 10-house subdiviinjection molder that he sion build using wood siding vs. aluand his class mates deminum siding, construction of an insigned and fabricated frastructure project within your interthis year, in EME 323 est area (road, structure, pipeline, etc.), - Design of Machine starting your own construction contracComponents. Ben did Ben Pihl tor company, etc. much of the machining and all of the welding for the fabrication of the injection molder. c. Electrical Engineering and Computer SciThe HAAS scholarship is intended to support ence: start a software company, a students in learning fabrication skills such factory building electrical components, as welding and machining. The mechanical power stations, use of automate proengineering faculty award the HAAS scholduction vs. hand-made, comparison of arship annually to a student who incorporates different operating systems for a commachined and welded components into their pany (i.e. Linux, Windows, Apple, design projects.

Google, etc. all aspects of the company run on that system – employees have phones/laptops/other devices exclusive to that company), etc.

ENR 460: S PRING P ROJECT

By ANTHONY WALDENMAIER d. Mechanical Engineers: Selection of equipment for a manufacturing process, opening a new automotive plant, autoENR 460: Engineering Economy mated vs. human labor in a producSpring 2022 tion plant, different materials to build Course Project the same thing (i.e. carbon fiber vs. alu100 points possible. minum vs. steel), etc.

This assignment is subject to change, pend- e. Senior Design Students: Economical Analing majority agreement among the students ysis of your project. Do a LCCA of of ENR 460. Due to the diverse nature of you project in an appropriate context (such as mass-scale production or conThen we attached the metal siding to function this course, this project will be graded holistically, on the premise of completeness, thorstruction). Your second alternative may as both the roof and flooring of the shelter. oughness, and depth of completed work. Subconsider a change in materials, ways jective opinions will apply, but must be reato build/manufacture, aesthetic changes sonably justified with valid and relevant arguwith potential benefits, use of better vs. ments. worse components, etc.

Due Dates: f. More Abstract (suggested for students with unsure area of interest): Start a company, upgrade a facility, roll out a new a. Initial Proposal: 3/25/2022 bring to class product (i.e. the PlayStation 6, the next as a hard copy. This is an opportunity iPhone/Galaxy, etc.), etc. to form groups for background research based on similar projects, time for quesYou will need to research costs and time tions, and class-wide discussion (brain frames associated with your project, looking We were given a hand with some of the finstorming) ishing touches by some helpful occupational at relevant real-world knowledge. You will therapy students and we handed the shelter need to research and generate at least six over to the sisters and the chickens on the b. Progress Check: 4/13/2022 Submit, at types of costs, which may include, but are minimum, the 6 (or more) researched fourth and final day of our service projects. not limited to the list below, and identify time costs frames associated (project lifespan? Monthly c. Final Report: 4/27/2022 Submit all project vs. yearly costs? One-time costs? Etc.) documents by 3:00pm, the end of the scheduled Final Exam Period. Early a. Initial Costs: construction costs, materials, start-up costs, buying vs. renting, etc. submissions will receive “Early Submission Credit” equivalent to 1 point b. Annual (or periodic) costs: operation and (1%) per day (maximum of 10% maintenance, overhead, labor, etc. bonus). This will include a submitted packet of the “Deliverables” addressed c. Lifespan or project life (this should cover at least 10 periods or sub periods (such later in this document. as a 10-month construction project, a Responsibilities: 50-year design life, 5 years service life Service projects like the one described with replacement, etc.) here are a core part of the identity of the UniYou are in charge of generating a lifeversity of Mary and of the education we in- cycle cost analysis (LCCA) relevant to your d. Labor Costs associated tend to impart. In the future we hope to in- field of study. You will need to generate at crease the level of engineering participation in least 2 alternatives where you will compare e. Salvage Value (direct re-sale values, trade in values, in-direct social costs/benefits, the annual Peru trip and expand the scope of pricing, performance, social impacts, environetc.) the projects we take on. mental impacts, economical impacts, and ethical dilemmas between the two options. Sug- Deliverables: gested areas of study include:

VOL.IV. . . No.4

1. Initial Proposal (3/25/22) – provide a copy of your project idea and some initial thoughts on what costs and considerations to research.

2. Progress Check (4/13/22) – Submit, at minimum, the 6 (or more) researched costs 3. Research Summary (4/27/22) – Provide a list or table (situational) with itemized costs and lifespans you have researched. Provide the source of your costs for each (this will likely be from a website; provide the overall organization, a hyperlink, and any extra detail that may be situational).

Example: Road Maintenance and Materials Costs (this example is for the sand/gravel material used as a base layer for roads) i. Organization: North Dakota Department of Transportation (NDDOT) ii. www.dot.nd.gov/pacer/AABP2022E.pdf iii. Extra Detail: 2022 NDDOT Bid Pricing, Spec 302, Code 120, Aggregate Base Course Class 5 iv. Price: $27.22 per ton 4. Approach Summary (4/27/22) – in this summary, explain how you identified the problem (situational) and approached your solution. Explain or justify why you selected your problem, why you chose specific pricing, etc.

This summary should walk the reader through your though process.

5. Cost Analysis (4/27/22) – Show the cost analysis of your multiple alternatives.

This may be presented in the form of a table, bar graph, cumulative cost plot (costs vs. time), or any other situational and appropriate summary. Calculations may be done on spreadsheet software or by hand (penmanship and neatness count, be professional).

6. Summary of Results (4/27/22) (HLC GS 1.3, 1.4, 1.5, 1.6) – clearly and concisely present your alternatives, final costs (in terms of Present worth, Future worth, Equivalent Annual worth, or whatever cost model is most appropriate). This section should simply present the name of the alternative, the cost, and the final decision. Here, you can also identify intangible factors that lead you to selecting the less economic option.

Example: Samsung TV vs. Vizio TV (same size and similar specs) i. Samsung: Present Cost $1699 ii. Vizio: Present Cost $1599 iii. Economically, the Vizio will cost less and would be the better economic choice. However, the remote of the Samsung is more familiar to the user, and the user’s history with Samsung has led to the choice of selecting the Samsung. The user is willing to pay the $100 difference based on the personal track record with Samsung, the feel of the remote control, and brand name. It was

Engineering Quarterly SUMMER 2022 determined, subjectively, that the Vizio TV has too many unknowns, and the $100 difference was not enough to select the Vizio over the Samsung.

7. Reflection (4/27/22) – After all research, calculations, and work is done, reflect on your work. You should address each of the following points with a brief paragraph (about five sentences, but feel free to elaborate and reflect further). Some of these will be more abstract, but they are intended to help you see the project on a holistic or global level. You are asked to take a step back, take a deep breath, and think about the big picture. How does your final decision impact other things? a. (ABET: 4C, 4D; HLC: GS 1.1, 1.3, 1.7, 1.8) Identify and assess any environmental impacts of your selected project (i.e. what affects, positive or negative, would this project have on the environment?) b. (ABET: 4C, 4D; HLC: GS 1.2, 1.3, 1.7, 1.8, 2.2, 2.3, 2.4) Identify and asses any social impacts of your project (i.e. how would this project affect society? This portion may also address any performance models you used). c. (ABET: 4C, 4D; HLC: GS 1.3, 1.7, 1.8) Identify and assess economic issues beyond the project itself (i.e. would you selection have an impact on the great economy; locally, regionally, nationally, etc.) d. (HLC: SE 1, SE 3; GS 2.1) Identify potential sources of ethical conflict within your project (you may directly reflect using the data and research you have collected, or you may reflect on an indirect/hypothetical situation). example: You sought operating costs from a similar manufacturing process, knowing that the other company will be a competitor in the market. Though this may be the best data you can find, there is a gray area in terms of conflict of interest in comparing your setup to a competitor. You may elaborate on what future ethical conflicts could arise, and how these may conflict with the Benedictine Values you have learned in your UMary education. e. (HLC: SE 2.1, 2.2) Identify and assess any potential conflicts that may have developed that conflict faith versus reason (was there an instance where faith was compromised for the sake of reason?

Was reason comprised based on faith? In this context, “faith” may be interpreted as religious faith, faith in your knowledge or skill set, faith in the industry or service your project addresses, or other interpretations.

Examples:

5 i. Your project may have some religious implications that directly (or indirectly) conflict with your spiritual background, such as buying less expensive parts from a company that does not have the same values as you have.

The conflict may arise that you are compromising your faith to save money on the parts. In a real world context, Chick-Fil-A and Hobby Lobby have both come under fire in recent years for staying true to their ethical/moral/religious beliefs – which has compromised profitability of the company. ii. Refer to the Samsung vs. Vizio example in the Summary of Results description.

The user’s faith in the Samsung brad lead them to undermine the economic decision. This would be a conflict between faith and reason that can be addressed.

E XPLORING THE S OLAR S YSTEM

By TERRY PILLING

The motions of the planets, moons, asteroids, and other objects in the solar system are governed by the gravity. The gravitational force is described by Einstein’s General Theory of Relativity. In the case of weak gravitational fields (as is the case in most regions of the galaxy except near exotic objects such as black holes) we can use either Newton’s law, or the first order correction to Newton’s law called the Post-Newtonian Approximation to calculate the motions to a very good degree of accuracy. Engineers and scientists have used the laws of gravity to send rockets and space probes across the solar system, landed them on other planets, and explored moons, asteroids, and many other things. We can even use Newton’s laws to simulate the evolution of the solar system over time and predict what would happen in various interesting circumstances such as if another star or a black hole passed nearby or if an asteroid or comet got knocked out of the Keiper belt and entered the inner solar system.

In this article we will create a computer model of the solar system by implementing the gravity between between each pair of particles, placing the system in an initial configuration, and then letting it evolve over time to see what happens. A good reference for information on the solar system is the website www.nineplanets.org and further links to sites you will find there.

Since we will be tracking the orbits of various objects in a three-dimensional universe we need to describe them in terms of vectors.

So let us first write a program to define and manipulate vectors. Examine the following header file called vector.h

VOL.IV. . . No.4 vector.h # i f n d e f VECTOR_H # d e f i n e VECTOR_H # i n c l u d e < s t d i o . h> class vector { d o u b l e _x ; d o u b l e _y ; d o u b l e _z ; public : v e c t o r ( v o i d ) { _x = 0 ; _y = 0 ; _z = 0 ; } void s e t ( double x , double y , double z ) { _x = x ; _y = y ; _z = z ; } d o u b l e x ( v o i d ) { r e t u r n ( _x ) ; } d o u b l e y ( v o i d ) { r e t u r n ( _y ) ; } d o u b l e z ( v o i d ) { r e t u r n ( _z ) ; } d o u b l e mag ( v o i d ) { double r0 = 1 . 0 , a = 1 . 0 ; d o u b l e r = _x * _x + _y * _y + _z * _z ; while ( a >0.0000001) { a = r0 ; r0 = ( r0 + r / r0 ) / 2 ; a = ( a > r 0 ) ? a−r 0 : r0−a ; } return r0 ; } f r i e n d v e c t o r o p e r a t o r +( v e c t o r a , vector b) ; f r i e n d v e c t o r o p e r a t o r −( v e c t o r a , vector b) ; f r i e n d v e c t o r o p e r a t o r −( v e c t o r a ) ; f r i e n d v e c t o r o p e r a t o r *( double a , vector b) ; friend vector operator /( vector b , double a ) ; v o i d p r i n t ( v o i d ) { p r i n t f ( " %.2 l f \ t %.2 l f \ t %.2 l f \ t " , _x , _y , _z ) ; } }; v e c t o r o p e r a t o r +( v e c t o r a , v e c t o r b ) { v e c t o r temp ; temp . _x = a . _x + b . _x ; temp . _y = a . _y + b . _y ; temp . _z = a . _z + b . _z ; r e t u r n ( temp ) ; } v e c t o r o p e r a t o r −( v e c t o r a , v e c t o r b ) { v e c t o r temp ; temp . _x = a . _x − b . _x ; temp . _y = a . _y − b . _y ; temp . _z = a . _z − b . _z ; r e t u r n ( temp ) ; } v e c t o r o p e r a t o r −( v e c t o r a ) { v e c t o r temp ; temp . _x = −a . _x ; temp . _y = −a . _y ; temp . _z = −a . _z ; r e t u r n ( temp ) ; } v e c t o r o p e r a t o r *( double a , v e c t o r b ) { v e c t o r temp ; temp . _x = a * b . _x ; temp . _y = a * b . _y ; temp . _z = a * b . _z ; r e t u r n ( temp ) ; } v e c t o r o p e r a t o r / ( v e c t o r b , double a ) { v e c t o r temp ; temp . _x = b . _x / a ; temp . _y = b . _y / a ; temp . _z = b . _z / a ; r e t u r n ( temp ) ; } # endif

It defines a class called vector which contains three data elements of type double.

These will be the three components of a 3dimensional vector. Then there is a public section of the class which contains several functions, or methods as they are called in the context of object oriented programming.

The first method is called the ‘constructor’ and simply initializes the components to zero when the vector is first defined. Notice that we use the convention that the numerical components of the vector, or in other words, the numbers that the computer is storing in each of the x, y, and x components have an underscore in front of them to distinguish them

Engineering Quarterly SUMMER 2022 from the corresponding variable name. For example the x component of the vector has the numerical value _x. The second method allows us to set the components to whatever values we like. The next three methods, when called, simply return the components of the vector and the magnitude method returns the length of the vector as can be seen by its implementation. The next five methods are the reason we choose to do this in C++.

Each of them defines a mathematical operation which can be performed on vectors. They are addition, subtraction, reflection through the origin, multiplication by a number and division by a number respectively. The ‘friend’ attribute means that these functions are defined outside the class scope but yet still has access to all of the private and protected members of the class.

The next thing that we need to do is to write another header file wherein we will define a new class called ‘particle’. This way we can lump all of the properties we would like our particles to have into the definition of the class and it will be very easy to add new things when we need them later on. particle.h # i f n d e f PARTICLE_H # d e f i n e PARTICLE_H # include " vector . h" class particle { d o u b l e _mass ; double _ r a d i u s ; v e c t o r _pos ; vector _vel ; vector _accel ; public : p a r t i c l e ( v o i d ) { _mass = 0 ; _ r a d i u s = 0;} v o i d s e t _ s t a t i c s ( d o u b l e m, d o u b l e r ) { _mass = m; _ r a d i u s = r ; } v o i d s e t _ m a s s ( d o u b l e m) { _mass = m; } void s e t _ r a d i u s ( double r ) { _ r a d i u s = r ; } void s e t _ p o s ( double x , double y , double z ) { _pos . s e t ( x , y , z ) ; } void s e t _ v e l ( double x , double y , double z ) { _ v e l . s e t ( x , y , z ) ; } void s e t _ a c c e l ( double x , double y , double z ) { _ a c c e l . s e t ( x , y , z ) ; } v e c t o r pos ( void ) { return ( _pos ) ; } v e c t o r vel ( void ) { return ( _vel ) ; } v e c t o r a c c e l ( void ) { return ( _ a c c e l ) ; } d o u b l e mass ( v o i d ) { r e t u r n ( _mass ) ; } double r a d i u s ( void ) { return ( _ r a d i u s ) ; } v o i d new_pos ( v e c t o r a ) { _ p o s . s e t ( a . x ( ) , a .y() , a . z () ) ; } void new_vel ( v e c t o r a ) { _ v e l . s e t ( a . x ( ) , a .y() , a . z () ) ; } void new_accel ( v e c t o r a ) { _ a c c e l . s e t ( a .x() , a .y() , a . z () ) ; } }; # endif

The first section of the class (called the ‘private’ section) defines various quantities that we think may be useful properties of particles in our models. These are the particles mass, radius, position, velocity and acceleration. Notice that the position, velocity and acceleration are of type vector as defined by our previous class file. This is why we need to have the #include "vector.h" statement at the top. The public section of the class defines a number of methods. They are mainly designed for ease in setting values and getting values from the private section. We will leave it to the reader to figure out what each of them is doing. It will become clear momentarily when we use this class in a program.

Now we want to write a simulation of the solar system. The way that will work is to take some initial data about the positions, ve-

6

locities and masses of the planets and then compute the net gravitational acceleration that each planet would feel from all of the other ones. This is accomplished using Newton’s law of universal gravitation which says that ~a =

GM ~r r3

(1)

So at each time step, planet i which is at distance r from planet j will contribute an acceleration given by the above equation to the new acceleration of planet j. So the new acceleration of planet j will be given by ~anew = ~aold +

GMi ~r r3

(2)

In C++ code this will become planet[j].new_accel( planet[j].accel() + newton*planet[i].mass() *r/pow(r.magnitude(),3));

and then the program would loop through all planets i to get the total new acceleration of planet j. Once this is calculated the program runs the clock forward 1 time step of length ∆t using the calculated accelerations to find the new positions and velocities using the new acceleration and the following formulas coming from classical Newtonian mechanics: ~xnew = ~xold + ~vold ∆t ~vnew = ~vold + ~a∆t

(3)

and these translate into the following lines of computer code: planet[i].new_position( planet[i].position() + dt*planet[i].veloc()); planet[i].new_veloc( planet[i].veloc() + dt*planet[i].accel());

Now the program can print out the new positions (and velocities if you like) and then proceed to recalculate the accelerations and continue in this fashion for as many time steps (days) as you specify. Here is the program. solar-system.cc / / We a r e u s i n g u n i t s s o t h a t mass i s measured i n / / 10^24 kg , d i s t a n c e i s i n 10^9 m and time i s / / m e a s u r e d i n 86400 s e c o n d s ( i . e . d a y s ). // / / I n t h e s e u n i t s we h a v e t h e f o l l o w i n g constants : // / / Newton ’ s c o n s t a n t = 0.0004981367808 / / Mass Sun = 1988900 / / Mass E a r t h = 5.974 / / Mass Moon = 0.0735 // R a d i u s Sun = 0.6961 // Radius Earth = 0.006378140 // R a d i u s Moon = 0.001738 // E a r t h−Sun d i s t a n c e = 1 4 9 . 5 9 7 8 7 0 6 6 0 / / Moon−Sun d i s t a n c e = 149.2134707 // E a r t h−Moon d i s t a n c e = . 3 8 4 4 0 0 // Earth Speed = 2.573501006 / / Moon S p e e d = 2.6734 # include # include # include # include

< s t d i o . h> < s t d l i b . h> " vector . h" " p a r t i c l e . h"

s t a t i c d o u b l e newton = 0 . 0 0 0 4 9 8 1 3 6 7 8 0 8 ; c o n s t char * h e l p [ ] = { " \ nUsage : . / a . o u t <param f i l e > < t i m e > <timestep >\ n" , " \ n< t i m e > i s t h e number o f s t e p s o f < t i m e s t e p > days , \ n " , " <param f i l e > c o n t a i n s t h e p o s i t i o n s ,

VOL.IV. . . No.4

velocities , \ n" , " masses , and r a d i i o f any p a r t i c l e s you want t o u s e . \ n " , " < p a r a m e t e r f i l e > c o n t a i n s one l i n e f o r each p a r t i c l e \ n" , " and t h e f o r m a t o f e a c h l i n e i s : \ n " , " \ nx y z v_x v_y v_z mass radius \ n" , " \ nThe o u t p u t i s t h e t i m e i n t h e f i r s t column \ n " , " f o l l o w e d by c o l u m n s c o n t a i n i n g t h e x , y , and z \ n " , " c o o r d i n a t e s of each of the p a r t i c l e s . \ n \ n" , "" }; i n t main ( i n t a r g c , char ** a r g v ) { i f ( a r g c != 4) { f o r ( i n t i = 0 ; * h e l p [ i ] ; i ++) f p u t s ( help [ i ] , s t d e r r ) ; return 1; } i n t TIME = a t o i ( a r g v [ 2 ] ) , t i m e = 0 , d t = a t o i ( argv [ 3 ] ) , i , j ; i n t NUM_PARTICLES = 0 ; FILE * i n f i l e ; char l i n e [ 2 5 5 ] ; i f ( ( i n f i l e = fopen ( argv [ 1 ] , " r " ) ) != 0) { while ( f g e t s ( line , s i z e o f ( l i n e ) , i n f i l e ) ! = 0 ) NUM_PARTICLES++; } fclose ( infile ) ; d o u b l e x , y , z , v_x , v_y , v_z , mass , r a d i u s , R ; p a r t i c l e p l a n e t [ NUM_PARTICLES ] ; v e c t o r zero , r ; i f ( ( i n f i l e = fopen ( argv [ 1 ] , " r " ) ) != 0) { f o r ( i = 0 ; i < NUM_PARTICLES ; i ++) { fgets ( line , sizeof ( line ) , i n f i l e ) ; s s c a n f ( l i n e , "%l f%l f%l f%l f%l f%l f%l f% l f " ,&x ,&y ,& z ,& v_x ,& v_y ,& v_z ,& mass ,& r a d i u s ) ; planet [ i ] . set_pos (x , y , z ) ; p l a n e t [ i ] . s e t _ v e l ( v_x , v_y , v_z ) ; p l a n e t [ i ] . s e t _ s t a t i c s ( mass , r a d i u s ) ; } fclose ( infile ) ; } w h i l e ( t i m e <= TIME ) { / / p r i n t out the time p r i n t f ( "%d \ t " , t i m e ) ; / / c a l c u l a t e t h e new p o s i t i o n s and velocities / / using the accelerations f o r ( i = 0 ; i < NUM_PARTICLES ; i ++) { p l a n e t [ i ] . new_vel ( p l a n e t [ i ] . v e l ( ) + d t * planet [ i ] . accel () ) ; p l a n e t [ i ] . new_pos ( p l a n e t [ i ] . p o s ( ) + d t * planet [ i ] . vel () ) ; p l a n e t [ i ] . pos ( ) . p r i n t ( ) ; } p r i n t f ( " \ n" ) ; / / c a l c u l a t e t h e new a c c e l e r a t i o n s f o r ( i = 0 ; i < NUM_PARTICLES ; i ++) { p l a n e t [ i ] . new_accel ( zero ) ; f o r ( j = 0 ; j < i ; j ++) { r = p l a n e t [ i ] . pos ( ) − p l a n e t [ j ] . pos () ; R = r . mag ( ) ; p l a n e t [ j ] . new_accel ( p l a n e t [ j ] . a c c e l ( ) + newton * p l a n e t [ i ] . mass ( ) * r / ( R*R*R ) ) ; p l a n e t [ i ] . new_accel ( p l a n e t [ i ] . a c c e l ( ) − newton * p l a n e t [ j ] . mass ( ) * r / ( R*R*R ) ) ; } } t i m e += d t ; } return 0;

Engineering Quarterly SUMMER 2022

7

you recognize that we are using huge units!)

We have also positioned the earth and the moon and given them a small initial velocity in the y-direction so that they will orbit rather than merely fall into the sun. We have designed the units so that one unit of time is equal to a day and so when you run the program by typing ./a.out initfile 365 1 > datafile

you will get a file containing the output of 365 days of orbital data.

Now let us show how to use the gravity of Jupiter to slingshot a space probe to Pluto.

Start with the following initialization file containing the data for the Sun, the Earth, Jupiter, Pluto, and an artificial satellite (which we approximate as massless) slingshot-pluto.init 0 0 0 0 0 0 1988900 0 . 6 9 6 1 149.5978707 0 0 0 2.573501 0 5.97400 0 150 0 0 2 . 9 2 4 1 . 6 0 4 0 0 0 778.330 0 0 0 1.128384 0 1900.0 0 4504.330 0 0 0 0.470016 0 102.0 0

We next run our simulation for 600 days which gives the following trajectory. It is interesting since it shows the space probe scattering gravitationally off of Jupiter resulting in a slingshot boost which sends it in a new direction at a greater speed. Next we run the simulation for 33,000 days. The resulting trajectory shows the probe reaching farther out in the solar system than Pluto but then visiting Pluto on the way back.

This is, in fact, exactly what NASA scientists and engineers do in order to reach farther into the solar system using less spacecraft fuel. They get a gravity boost from the planets along the way.

The following table gives the data for the major planets in the solar system. The dis} tances are in units of a million meters, the velocities of the planets are given in millions km The program takes a file name and an integer per day, and the masses are in kilograms. as command line arguments. The file name is one which contains the initial data regarding the particles. The first integer is the number Name Dist Veloc Mass of time steps you would like to have and the Sun 1.99e30 second integer is the number of days between Mercury 57910 4.359680 3.30e23 each time step. For example, for the sun, the Venus 108200 3.025728 4.87e24 earth and the moon you would want a file containing the following three lines.

Earth 149600 2.738560 5.97e24 Moon 384 0.050000 7.35e22 0 0 0 0 0 0 19889 .6961 Mars 227940 2.084832 6.42e23 149.5978 0 0 0 2.5735 0 5.974 .0000 149.2134 0 0 0 2.6734 0 0.073 .0000 Jupiter 778330 1.128384 1.90e27 Saturn 1429400 0.834624 5.68e26 The first line tells us that we have chosen Uranus 2870990 0.587520 8.68e25 this object to reside at the origin of our 3Neptune 4504300 0.470016 1.02e26 dimensional coordinate system. It has a huge mass and radius (as is even more clear when Pluto 5913520 0.409536 1.27e22

You may like to spend some time playing around with this program by throwing a few comets or Jupiter sized planets into the mix and see what havoc they can play on the solar system. It is quite surprising how easily perturbations can ruin everything and end up throwing the earth and the moon clear out of the solar system! For example, here is an initialization file containing a small comet and a Jupiter in addition to the sun, earth and moon.

0 0 0 0 0 0 198890 0.696 149.5978 0 0 0 2.5735 0 5.9740 0 149.2134 0 0 0 2.6744 0 0.0735 0 1000 0 0 0 0 .2 0.04 0 2000 0 0 0 0 .1 19880 0

We have given the comet and Jupiter some initial motion in the z-direction. You will find it very interesting to increase the time scale and see how the system evolves. It is remarkable how stable this system is as you can see if you run it 20000 days or so. You will see that the earth and moon manage to stay together even with the extra perturbations of their orbits.

Now, try increasing the mass of the Jupiter a little if you want to see some real havoc created! I have even found certain configurations which will send the earth or the moon out to twice its usual distance from the sun for a single cycle and then it returns again. I wonder if this sort of thing could have been the cause of the ice-ages we have had in the ancient past?

Here are some further interesting avenues you might like to explore: orbital precession: Run the Earth-Moon-Sun simulation for 366 days. Then adjust the view so that you can see the spot where the earth meets up with its original position a year before. You should notice that the Earth exactly lines up with its previous path so that the orbit closes into a circle. However, what do you notice about the orbit of the moon? Does the earth’s orbit line up properly when all of the planets are included in the simulation rather than only the earth, the moon and the sun? black holes: What would happen if a star or a black hole flew by our solar system just outside the orbit of Pluto? Hint: Just add another sun, starting at about the same place as Pluto and with some fairly large initial velocity which would carry it straight past without deflecting it noticeably. Now adjust these parameters and watch what havoc is created in the solar system. This situation is possibly what will happen if the Andromeda galaxy collides with our own galaxy, The Milky Way. It looks as though Andromeda is on a collision course and, depending on our sideways motion, it will collide with us in 7 billion years.

Galilean Satellites: Use the data from the internet to model the Jovian system. In other words, have Jupiter in place of the sun, and the moons of Jupiter orbiting it. Note that in order to get better accuracy you may want to go to the internet and find the exact positions

VOL.IV. . . No.4

and orbital velocities of the Galilean satellites at some particular date and then see how they evolve from there. Can you predict with any accuracy where they will be several days later? You could even look at Jupiter with a telescope and check your answer since you can easily see the four Galilean satellites orbiting Jupiter with a moderate sized telescope.

Oort Cloud: For a long time it was known that although everything in the solar system including short period comets travel in the same direction around the sun, the long period comets (which take millions of years to return) have orbits that are random. They can come in from above or below the plane of the orbits of the rest of the solar system and they can move in both prograde or retrograde motion. This was a fact that nobody could explain. Then in 1950 Dutch Astronomer Jan Oort wrote an article in which he argued that if there were a huge number of comets moving in slow circular orbits in the prograde direction and in the plane of the solar system but at a very large distance from the sun (from 10,000 to 100,000 AU), then the passing of several stars parallel (above and below) the plane of the solar system would be enough to completely randomize the orbits of the comets until they formed a cloud around the solar system (now called the Oort cloud) and that this cloud would explain the strange orbital characteristics of the long period comets. Try to model this with a computer simulation. Start

Engineering Quarterly SUMMER 2022 with a number of comets at, say, 50,000 AU in circular orbits in the plane of the solar system.

Then have a few stars fly by at say 75,000 AU or more, both above and below the plane and at random times. Does this cause the original orbits to become a randomized cloud as Oort hypothesized? What about the origin of the long period disk of comets? If you start your simulation with a few inner planets as well as a multitude of kilometer-sized comets on short period orbits, will the planets eventually throw the comets out to long period circular orbits so that the Oort cloud could then form via the passing of stars?

Mass Extinction: Another Dutch astronomer, named Piet Hut, observed that extinction events due to comets happen roughly every 26 million years. This led him to hypothesize that the sun is actually part of a binary star system with another star he dubbed ‘Nemesis’. The hypothesis is that every 26 million years the other star comes close to the sun in its elliptical orbit and disturbs comets from the Oort cloud, some of which then attain collision trajectories with the earth. Other astronomers have said that Hut’s idea would not work because over the course of 26 million years, other stars would pass in between the other star and the sun due to the motion of the sun around the galactic center. Hut then did simulations which seem to show that the hypothetical binary system of the sun and Nemesis is extremely stable under such per-

8

turbations and would continue to cause the effect on the Oort cloud even if other stars pass in between once in a while. Your task is to try and repeat Hut’s simulation. Set up a binary system between the sun and another star with a large time scale such that it takes 26 million years to orbit the sun. Then throw some stray stars through the middle and see if this disrupts the system and ruins the 26 million year period. If it does not, then you have verified the possibility that the sun is part of a binary system and that this may indeed be responsible for the periodic mass extinctions. Can you think of any other possible reasons for the periodic cycle and test them as well?

Chaos: Starting with only two planets, Jupiter and Saturn for example, the system is stable. Show this by running a simulation for several billion years. Then add Uranus into the mix and so the same thing and you will see that the system is chaotic. I.e. In each case, run two simulations each differing by some very small perturbation. The two-body system will be stable, whereas the three-body system will be chaotic.

Contact Information For more information about our engineering program email: Engineering@umary.edu Next issue: November, 2022