Wind Vane Analysis & Design

Wind Vane Analysis & Design

University of Rome Tor Vergata, 2015-2018 Bachelor’s Degree in Engineering Sciences Supervisor: Prof. Vincenzo Mulone Student: Matteo Sammarco

Motivation

Skippers benefit from accurate wind readings

Introduction

A wind vane on a mast

• Apparent wind direction is a fundamental source of information for navigation and maneuvering.

• This study was motivated by the need to improve the mechanical responsiveness of direction sensors on small boats and dinghies.

• This is a small scale application where practicality and cost-effectiveness are maximized.

• This project describes the methodology and tools that were used to specify an optimal form factor.

• Finally, two optimized designs are presented and tested.

Contextualization

Wind Vane Analysis & Design, Matteo Sammarco, University of Rome Tor Vergata, page 1 of 14

Method

Design procedure for wind vanes Mathematical

Experimental

An equation of motion was derived to analytically represent the vane behavior.

A setup was arranged to generate the ideal wind conditions to place the vane.

The intrinsic properties of a particular vane were measured.

The dynamic measurements were executed for a range of wind speeds.

The calculations performed on the model were plotted for comparison.

Digital image processing was performed on the gathered data.

Validation & Design The results obtained from the two approaches were finally compared. The mathematical model was proven to be a valid reference.

Two new designs were generated and 3D printed for further comparison.

Wind Vane Analysis & Design, Matteo Sammarco, University of Rome Tor Vergata, page 2 of 14

Mathematical Approach

A second degree ODE of motion W

Derivation

!

By applying the fundamental principle of dynamics to the wind vane system, the second-order linear ODE with constant coefficients takes the form

"̈ + 2%& '"̇ + %& )" = %& )+ −

-. /

Where the characteristic parameters are

01 2304 5 '= 2 2/

2304 5 %6 = 7 2/

Fa %& = %6 1 − ')

Nomenclature " %6 %& ' + -. / 9 :

Current position of the vane Natural frequency Damped natural frequency Damping ratio Apparent wind direction Bearing dry friction Moment of inertia Wind pressure Time constant

7 3 2 04 01 5 ;. <4 =

Wind speed Wing platform area Air density Aerodynamic center Gravity center to vane pivot Lift coefficient Response time Aerodynamic force Logarithmic decrement

Wind Vane Analysis & Design, Matteo Sammarco, University of Rome Tor Vergata, page 3 of 14

System coupling Top view of a wind vane immersed in a wind W.

Mathematical Approach

Predicting the behavior of a real vane

Measurements •

The intrinsic characteristics of the vane were thoroughly measured: physical dimensions using a caliper and total mass using a precision scale.

•

The moment of inertia about its pivot can be sufficiently approximated by restricting the calculation to the effects of the fin mass !" and the counterweight mass !#. Taking into account that the vane must balance about its axis (!" $" = !#$#), we obtain

.=

/ ! 1/ 0 "

Little Hawk A small lightweight and robust wind indicator designed to be positioned at the top of the mast on dinghies up to 6m.

Characteristics

•

The aerodynamic center for any airfoil lies at approximately 3⁄4, where c is the chord length. The distance between this 3 ⁄4 line and the pivot is $5 .

•

The bearing dry friction was considered negligible compared to the aerodynamic forces.

Wind Vane Analysis & Design, Matteo Sammarco, University of Rome Tor Vergata, page 4 of 14

• • • • • •

& '( ') * + ,-

7.4 x 10-2 m 7 x 10-2 m 6.9 x 10-3 m 7 x 10-6 kg m2 1.62 x 10-3 m2 1.1 g/cm3

Mathematical Approach

Damped harmonic oscillator solution

! !0

Calculations The solution of the ODE simplifies to

! " = !$ % & '678, % -./(9: ")

1

t(s)

The damping ratio and damped natural frequency for a wind of 7 m/s are sufficient to graph the equation.

! " = !$ % & '(.*(+, % -./(18.73") Where ; = 0.165, 9? = 19.0 ABC//, 9: = 18.73 ABC//.

Response time Little Hawk Step Response Theoretical time evolution from initial angular displacement θ0 = 15° and zero velocity to fixed wind direction, wind speed 7 m/s.

An interesting remark was made on the performance and sensitivity of the wind indicator. A parameter for comparison is the response time, defined as the time for the vane to reach ±5% of the wind direction

EF = −

ln 0.05 = 950J/ ;9

Wind Vane Analysis & Design, Matteo Sammarco, University of Rome Tor Vergata, page 5 of 14

Experimental Approach

Anemometer, camera, action!

Setup Wind vanes:

Cameras:

• Little Hawk

• Sony DSC-RX10 M1

• C-Vane

• GoPro Hero 4 Session

Wind generators:

Other:

• Fan (4-7 m/s)

• Anemometer

• Hair dryer (7-10 m/s)

• Tripods and mounts

Measurements Configuration The vanes were placed in wind conditions ranging from 4 to 10 m/s with an initial displacement angle of 15° then released to return to the direction of the wind.

The Sony RX10, set to 50 frames per second, processed image data from the 1" Exmor RS CMOS sensor with full pixel readout and no line skipping. The electronic global shutter minimized distortion caused by rolling shutter phenomenon.

Wind Vane Analysis & Design, Matteo Sammarco, University of Rome Tor Vergata, page 6 of 14

The camera on the tripod was positioned above the vane pivot axis. Below, another camera captured the motion of the vane and the readings on the display of the anemometer.

Additional photos can be found in the extra materials section.

Experimental Approach

Digital image processing of the videos Processing The objective was to measure the average period and logarithmic decrement of the vane’s motion to be compared with the previous mathematical results.

Processing method: •

The video was trimmed and converted to an image sequence.

•

Every image was cropped by an area of width 1px.

•

A montage was generated by juxtaposing vertically every consecutive frame.

•

Significant points were marked along the curve.

•

An array was output containing the x , y coordinates of the significant points.

Recalling that each column of 1 pixel is equal to 1/50th of a second the average period of motion and the logarithmic decrement were determined.

. ' = ./ 0 1

2

3 4

0 567(8')

! = #$

%(') %(' + *)

+=

, -

Wind Vane Analysis & Design, Matteo Sammarco, University of Rome Tor Vergata, page 7 of 14

Montage Above: a frame extracted from the video of the test in 4 m/s wind of the Little Hawk. The cropped area of width 1px is marked in yellow.

Below: the montage.

Verification

The mathematical model is valid

! !0

Verification

1

t(s)

Testing has been repeated at least 3 times per wind speed for an averaged result.

The model represents reality up to an error of 5%.

The graphic comparison of the two approaches is shown on the left. This was test number 2 of the Little Hawk during the 7 m/s wind speed test.

Disturbance considerations

Step response comparison Theoretical time evolution versus the montage.

The time axis is valid for both curves because the graphs have been overlapped and normalized correctly for width and height.

Results Value Mathematical / Experimental Damping ratio 0.165 / 0.16 Natural frequency (rad/s) 19 / 18.4 Damped natural frequency (rad/s) 18.7 / 18.1

After the 750 ms mark the real vane behavior was influenced by turbulence. It was investigated if the turbulence was generated by the surroundings or by vortex shedding at the vane’s central cylindrical pivot.

The shedding frequency of the von Kármán vortices was compared to the natural frequency of the wind vane.

From testing it was known that the minimum wind for motion is 1.1 m/s and the maximum wind is 15 m/s. The characteristic length of the vane pivot diameter was set to ! = 8 $$ This yielded a Reynolds number in the range 10' < )* < 10+.

A good approximation for the Strouhal number was ,- = 0.19 so 01 = 34! ⁄26,- = 0.12 $/8. The wind vane was not affected by vorticity but from from the surrounding turbulent fluid motion.

Wind Vane Analysis & Design, Matteo Sammarco, University of Rome Tor Vergata, page 8 of 14

Prototype design

Designing optimized wind vanes

Prototypes Two designs are proposed to increase performance. Both have been computer designed to reduce mass and moment of inertia. This makes them more sensitive in light airs, more robust and most importantly makes them float. â&#x20AC;˘

The Falcon is an optimization of the Little Hawk. The counterweight beam has been shortened to 63mm. The gravity center of the vane is now coincident with the vane axis and the fin design brings the aerodynamic center forward.

â&#x20AC;˘

The Wingman has a lower aspect ratio and an advanced aerodynamic center.

The aerodynamic force acts along the axis because of the fin symmetrical form factor.

All of these modifications increase the damping ratio and decrease the moment of inertia. This results directly in a decrease in response time.

Optimized designs Above: Falcon. Below: Wingman.

Wind Vane Analysis & Design, Matteo Sammarco, University of Rome Tor Vergata, page 9 of 14

Prototype design

Small changes for smaller response times

Little Hawk (Commercial)

Falcon (Prototype)

Wingman (Prototype)

Dimensions

Dimensions

Dimensions

• • • • • •

! "# "$ % & '(

Performance • • •

• • • • • •

7.4 x 10-2 m 7 x 10-2 m 6.9 x 10-3 m 7 x 10-6 kg m2 1.62 x 10-3 m2 1.1 g/cm3

• • •

• • • • • •

5.5 x 10-2 m 6.5 x 10-2 m 6.1 x 10-2 m 5 x 10-6 kg m2 1.4 x 10-3 m2 0.93 g/cm3

Performance

0.165 C DE 19.0 rad/s FG 950 ms

Equation of motion

! "# "$ % & '(

• • •

C 0.16 DE 20.0 rad/s FG 890 ms

Equation of motion

) * = , -<.=1 2 345(19.7*)

) * = , -..01 2 345(24.59*)

) * = , -<.@<01 2 345(18.73*)

5.5 x 10-2 m 6 x 10-2 m 4.7 x 10-2 m 4.5 x 10-6 kg m2 1.5 x 10-3 m2 0.93 g/cm3

Performance

C 0.18 DE 25.0 rad/s FG 710 ms

Equation of motion

! "# "$ % & '(

Notes

Notes

Notes

The Little Hawk is listed for comparison.

It’s density is larger than water so it sinks.

Theoretically, the Falcon has up to a 30% decrease in response time among the competing vanes.

The Wingman trades off a lower response time for a lower density, allowing it to float in water.

Wind Vane Analysis & Design, Matteo Sammarco, University of Rome Tor Vergata, page 10 of 14

*Parameters calculated for H = 7 I/5

Prototype design

Ideal behavior comparison Ideal step response Graph of the ideal behaviors of the vanes in a wind of 7 m/s.

! !0

Legend

! " = $%&.(&)" * +,-((/. 0&") ! " = $%&.2/" * +,-(3(. 0/") ! " = $%4.)" * +,-(34. )5") Comparison

500

t(ms)

Because of the lower material density the Wingman has a lower damping ratio. The best performing vane is the Falcon.

The Falcon has a 30% decrease in response time, this gain will be higher (also for the Wingman) if the density is increased.

Because of the lower moment of inertia of the two new vanes, they have a higher frequency of motion.

Wind Vane Analysis & Design, Matteo Sammarco, University of Rome Tor Vergata, page 11 of 14

Prototype design

Theoretically, they should not break PA 2200 Nylon Plastic

Falcon

Wingman

A nylon plastic used for a variety of applications, ranging from functional to decorative products.

Von Mises stress

Von Mises stress

Material properties • • • •

Density Yield strength Ultimate tensile strength Elongation at break

0.93 g/cm3 39 MPa 48 MPa 24%

Static stress analysis Simulation of an aerodynamic force applied at the aerodynamic center of the vane. The wind pressure ! = 0.5&'(, wind speed ' = 10 */,, and lift coefficient - = 2/01, yield an aerodynamic force 23 = -45 respectively equal to 0.20N for the Falcon and 0.14N for the Wingman.

[MPa] 0

3

[MPa] 0

1.93

[mm] 0

0.87

Displacement

Displacement

Von Mises stress Maximum stress is within limits. Simulation is shown for the optimized versions of the vanes.

Displacement The wind creates a torque along the Falcons directing axis due to its fin asymmetry and raised aerodynamic center. The Wingman is not affected by this because if its fin symmetry.

Wind Vane Analysis & Design, Matteo Sammarco, University of Rome Tor Vergata, page 12 of 14

[mm] 0

1.9

Prototype testing

Prototype 3D printing & testing Rapid prototyping

Properties

The Wingman and Falcon wind indicators have been 3D printed using selective laser sintering technique in yellow nylon plastic.

The prototypes have undergone the same testing procedure as the Little Hawk.

Falcon

Step response comparison

Value Mathematical / Experimental Damping ratio 0.18 / 0.18 Natural frequency (rad/s) 25 / 23.7 Damped natural frequency (rad/s) 24.6 / 23.3 Error (%) Âą5 Response time (ms) 710

The calculated mathematical response superimposed to the experimental data for a wind speed of 7 m/s.

In blue: Falcon, in yellow: Wingman.

! !0

Wingman Value Mathematical / Experimental Damping ratio 0.16 / 0.17 Natural frequency (rad/s) 20 / 19.5 Damped natural frequency (rad/s) 19.7 / 19.2 Error (%) Âą5 Response time (ms) 900

500

t(ms)

500

t(ms)

! !0

Wind Vane Analysis & Design, Matteo Sammarco, University of Rome Tor Vergata, page 13 of 14

Summary

Thank you for your attention Analysis

Summary • Validated a mathematical model for the motion of wind vanes.

• Designed, tested and optimized two better performing wind vanes.

• Material density was found to be a crucial design parameter.

• 3D printing allows for new unconventional designs.

Graphs Video conversion Digital image processing Finite element analysis

Grapher Squared 5 ImageJ Autodesk Fusion 360

Prototyping Computed aided design 3D printing service

CATIA V5 Shapeways

Graphics Model rendering Slide design Photo editing

V-Ray MS Power Point 2019 Pixelmator Pro

References J. Wieringa “ Evaluation and Design of Wind Vanes ” Journal of Applied Meteorology 1114, 1967 Hugo & Laurent, Johann & Cerqueus, Audrey & Sevaux, Marc & Senn, Eric & Hauville, Frédéric & Coneau, Raphael. “ Methodology for optimal wind vane design ”, 1-7. 2016 https://www.shapeways.com/rrstatic/material_do cs/mds-strongflex.pdf

Wind Vane Analysis & Design, Matteo Sammarco, University of Rome Tor Vergata, page 14 of 14

Extra Materials

Extra materials

Photos

Wind Vane Analysis & Design, Matteo Sammarco, University of Rome Tor Vergata, Extra materials, page 1 of 4

Extra materials

Testing data C-Vane !

C-Vane 0.49

!

"

8.1

0.32

t

Θ0

Little Hawk !

Little 0.28 Hawk

!

"

18.4

0.16

t

Θ0

Little Hawk USD !

Little Hawk 0.12 USD

!

"

14.4

0.67

t

Θ0

Wind Vane Analysis & Design, Matteo Sammarco, University of Rome Tor Vergata, Extra materials, page 2 of 4

Extra materials

Behavior at equal material density

Form factor

Comparison at equal density Motion parameters recalculated assuming the material density equal to 1.1 g/cm3.

•

When the new designs were introduced (Falcon and Wingman), their densities were lower than the density of the nylon used for the Little Hawk. This was done in order to make them float in water.

•

It was checked if, at equal densities, the new vane shapes maintained a faster response time.

•

Using the mathematical model the motion parameters were recalculated.

•

The form factor for both wanes decreases the response time by approximately 10% compared to the Little Hawk.

•

At higher density, the Wingman performs better than the Falcon.

Little Hawk (Commercial) Damping ratio Natural frequency (rad/s) Response time (ms)

0.165 18.4 950

Falcon (Prototype) Damping ratio Natural frequency (rad/s) Response time (ms)

0.134 25 850

Wingman (Prototype) Damping ratio Natural frequency (rad/s) Response time (ms)

Wind Vane Analysis & Design, Matteo Sammarco, University of Rome Tor Vergata, Extra materials, page 3 of 4

0.14 18.4 810

*Parameters calculated for ! = 7 $/&

Extra materials

Minimum wind for motion C-Vane

1.8m/s

Little Hawk USD

Little Hawk

1.03m/s

Minimum wind for motion was measured while gradually increasing wind speed against the vane positioned at a 45° angle of attack.

Wind Vane Analysis & Design, Matteo Sammarco, University of Rome Tor Vergata, Extra materials, page 4 of 4

1.49m/s