Application of differential equations to model the m ti n f p p r h lic pt rmotion of a paper helicopter

Linear Velocity Angular VelocityLinear Velocity

0.96

1.2

s)

Angular Velocity

24

30

/s)

0.48

0.72

Vel

ocity

(m/s

12

18

Vel

ocity

(rad

/0 0.4 0.8 1.2 1.6 2

Time (s)

0

0.24

0 0.4 0.8 1.2 1.6 2Time (s)

0

6

Kevin J. LaTourette

Advisor Kris Green

1

Spring 2007 Science Scholars Presentation

St. John Fisher College

Outline

• Helicopter Design (previous work)

• Rotational motion

• Helicopter drop

• Analytical solutions

• Future work

2

Helicopter design can greatly change the flight p g g y g gpatterns of the helicopter

• Wing length and width have g glarge effects on rotational and linear velocities

• Larger helicopters suffer from deformity due to strength of paper

• Current model provides for the longest drop time– Same model used for all drops – Same model used for all drops

[1]

3

At equilibrium, the drag force deflects the rotors q , gupward

• Rotor deflection angle (wing Rotor deflection angle (wing flex), φ, decreases as linear velocity increases

• At terminal linear velocity, φ ≈ 70˚

φ

4

Turbulence due to the wake of rotors was ignored

• Wing blade has negligible Wing blade has negligible thickness (100 μm)

• Angular velocity is relatively low Angular velocity is relatively low (ωT ≈ 20 rad/s) [4]

5

Due to rotation of the rotors, the drag force acts , gon a larger effective area

• As the helicopter falls, it spins quickly, and the drag force acts on an area proportional to the circle traced by the wings as they rotatethey rotate

• Effective rotor area is a linear combination of the stationary com nat on of th stat onary wing area and the disc traced as the wings rotate

6

The rotational ‘blur’ effectively adds to the area ythe drag force acts on

( ) ( )( )[ ] ( )φWLtωLWφωA sinΔ2, 22 +⋅+=

• This correction adds approximately 25% to the area of the rotors for small ∆t [1]

7

Determination of angular velocity depended largely on g y p g ythe capabilities of the sensor

• Our first experiment was designed to determine whether our data Our first experiment was designed to determine whether our data collection method was possible

• Linear position was held constant and rotor rotation was controlledLinear position was held constant, and rotor rotation was controlled

– By obtaining a set of static rotational speeds, we could compare the results with the results of the actual drops

8

While the results were encouraging, it proved g g, pdifficult to apply this technique to our drops

0 09 m/s0.09 m/s

0.55

0.6

0.65

om s

enso

r (m

)

0.4

0.45

0.5

0 0.5 1 1.5 2

Time (s)

Dis

tanc

e fr

o

0.24 m/s

0 65

0.7

(m)

0.45

0.5

0.55

0.6

0.65

stan

ce fr

om s

enso

r

9

0.40 0.5 1 1.5 2

Time (s)

Dis

To reduce crosswinds and eliminate reflective noise, ,a large ‘box’ was constructed

• At just over 3 meters tall, and 1 meter wide to account for the 15˚ arc swept out by the sensor

O l i t t d i th i iti l • Only interested in the initial drop (1-2 seconds)

10

While the data gave good linear data, the angular g g , gresolution was poor

Linear Position vs Time

• Data sensor operates at ~20 Hz

0 4

0.6

0.8

1

1.2

Dis

tanc

e (m

)

Data sensor operates at ~20 Hz, too low for credible angular rotation data analysis

0

0.2

0.4

0 0.2 0.4 0.6 0.8 1Time (s)

• If helicopter path is not directly lined up with sensor during the entire fall, we lose angular data

Linear Position vs Time

0.8

1

1.2

(m)

• Chose not to use this technique for angular rotation

0

0.2

0.4

0.6

0 0.2 0.4 0.6 0.8 1

Ti ( )

Dis

tanc

e

• vt = 0.60 m/s ± 0.02 m/s

11

Time (s)

An understanding of the forces involved allow us to gcreate differential equations

Force DragLinear ),,( =φωvFd

Force DragAngular ),( =′ φωFd

12

gmF ⋅=

The resistance experienced by an object moving p y j gthrough a fluid is dependent on its velocity

• Generally • Incorrect assumption for bodies αbvF– b is constant, and depends on viscosity of the fluid, and the size and shape of the object

– α depends on shape of object

falling in Earth’s atmosphere• Experimentally, the drag force

has been approximated by [5]

D bvF −=α depends on shape of object and size of the object

• For small objects,

[5]

vFD ∝2vFD ∝j

– Raindrop falling in air– Aerosols falling in air– Objects passing through heavy

oiloil

13

Applying this linearly, we obtain our linear drag force

• Dependent on shape, b is not constant for our purposes– Incorporates effective rotor

area A(ω, φ) – Also dependent on density of air

D bvF2−=

Also dependent on density of air [7]

• We also introduce the coefficient of drag, cD

( ) DD cφωAvρF ,21

12−=

– Surface/Shape of object– Viscosity of air (fluid)

( ) DcφωAvρmgvm ),(1221−=

14

We find the rotational drag force is dependent only n th n ul r v l citon the angular velocity

• Area of the helicopter does not change with increased linear or angular velocities, only wing deflection angle (assumed constant)

D bvF2−=

constant) • Recall v = rω• Torque ( ) ( ) DD cφAωrρF 222

1−=FrωIτ ×==

( ) ( ) ( ) DD cγφAωrρcβφωAvρωI )(),( 22211221 −=

15

Our two coupled differential equations provide insight p q p ginto the flight of the helicopter• Linear equation incorporates the

s ti f th f lli b d ( )summation of the falling body and the drag associated with the fall, which varies with both linear and angular velocities

[1]

( )( ) ( ) ( ) )(),(

),(

22

21

12

21

12

21

cγφAωrρcβφωAvρωI

cφωAvρmgvm

DD

D

−=

−=

[1]

• Angular equation experiences a forcing from the liner motion forcing from the liner motion, and a separate drag force which varies only with the angular velocity ( ) ( )( )[ ] )sin(Δ2,

221 φWLtωLWφωA +⋅+=

• β and γ are scaling factors with units of length

( ) )cos(2 φAAφA rotorbody ⋅+=

16

With some careful rearrangement we arrive at our gdifferential equations

• Linear equationv = linear velocity– v = linear velocity

– ω = angular velocity– g = acceleration due to gravity (9.81m/s2) – m = mass of helicopter

ωωvv

ωvv

KγKKβ

ωvm

KvmKgv

222

22

⋅⎟⎞

⎜⎛

⋅⋅−⋅−=

– Kv & Kvω = Linear drag force coefficients• Angular equation

– β = linear drag scaling factor– I = moment of inertia

ωωvv ωI

γωvI

vI

βω 222 ⋅−⎟⎠⎞

⎜⎝⎛ ⋅⋅+⋅⋅=

I = moment of inertia– γ = angular drag scaling factor– Kω = angular drag force coefficients

• K – equations( )( ) Dωv

Dv

φctrρK

φcLWρK2 )sin(Δ

)sin(

=

=

– ρ = density of dry air– L = rotor length– W = rotor width– cD = coefficient of drag

( )Dω cArρK 2

22

1=

17

cD coefficient of drag– r = rotor radius– ∆t = change in time (~0.10 s)

Graphical solutions to our equations reflect the p qphysical results

• When dropped, the helicopter l t s if l s th is

Linear Velocity1.2

accelerates uniformly as there is no rotational motion

• When the rotation begins, the linear 0.72

0.96

ocity

(m/s

)

When the rotation begins, the linear velocity decreases, and the two settle into an equilibrium – the terminal velocities

0 0 4 0 8 1 2 1 6 20

0.24

0.48

Vel

o

• vt = 0.62 m/s– vt experimental = 0.60 m/s ± 0.02 m/s

Angular Velocity

20

25

0 0.4 0.8 1.2 1.6 2Time (s)

• ωt = 21.49 rad/s

5

10

15

Vel

ocity

(rad

/s)

18

• Numerical Solutions found using ODE Architect, implementing a fourth order Runge-Kutta Method

0 0.4 0.8 1.2 1.6 2Time (s)

0

5

Analytical solutions to our system of equations y y qprovides insight into the terminal velocities

• Obtained by setting 0== ωv22 KK

• In turn, we can solve for β and γ in terms of the angular

l iti hi h h 222

22

ωIKγωv

IKv

IKβω

ωvm

KvmKgv

ωωvv

ωvv

⋅⋅

−⎟⎠⎞

⎜⎝⎛ ⋅⋅+⋅⋅=

⋅⋅−⋅−=

velocities, which we have determined experimentally

2Kβ

III ⎠⎝

tωvvt ωKK

mgv+

=915.4≈=

mgKω

γβ ωt

t Kmgβω =

19

ωt Kγ

Our experimental determination of β and γ provide ur fin l num ric l s luti nsour final numerical solutions

• vt = 0.60 m/sLinear Velocity

1.2

– vt (old) = 0.62 m/s

• ωt = 27.56 rad/s 1 4 d/

0.72

0.96

ocity

(m/s

)

– ωt (old) = 21.49 rad/s

• As expected, the terminal linear velocity is predicted exactly0

0.24

0.48

Vel

o

velocity is predicted exactly• Terminal angular velocity is

higher than the previous prediction

0 0.4 0.8 1.2 1.6 2Time (s)

Angular Velocity

24

30

p– No reliable data

12

18

Vel

ocity

(rad

/s)

200 0.4 0.8 1.2 1.6 2

Time (s)

0

6

Futures projects would be expected to obtain better p j pexperimental data

• Other considerations include:– Accounting for the processional motion (wobble) about the z-axis as the

helicopter falls

I t i bl i fl l hi h ill id t – Incorporate a variable wing flex angle φ, which will provide a more accurate solution

– Eliminate physical distortion by using a rigid modelm p y y g g m

– Construct a better enclosure (Circular?)

– Obtain better values for coefficient of drag and the scaling factors (β, γ)

21

Application of differential equations to model the

Conclusion/Summary

pp qmotion of a paper helicopter

• Analytical solutions made significant improvements over previous modelsAnalytical solutions made significant improvements over previous models

• Data collection was only successful for linear motion, but further measures must be taken for determining the angular motionmeasures must be taken for determining the angular motion– Sonic motion sensor may not be a possible data collection source

• Several major assumptions were made for the sake of simplification, j p p ,and would help us understand the motion much better

• Future projects may wish to test different helicopter dimensions, to p j y pensure the model is accurate

22

Acknowledgements

• I would like to thank Dr. Kris Green for his assistance and helpful comments & the Mathematics Department, as well as Dr. Foek T. Hioe and the entire Physics department for ideas and suggestions and for the use of laboratories and supplies during the course of my research.

• I also must express my great appreciation to Saint John Fisher College and the Science Scholars program for support and the opportunity to conduct research at the undergraduate levelcon uct r s arch at th un rgra uat

23

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Engineering Knowledge The American Statistician 59 No 4 (pp 320-326)Engineering Knowledge. The American Statistician, 59, No. 4. (pp 320 326).

• Fowles, G. & Cassiday, G. (2005). Analytical Mechanics, Seventh Edition. CA: Brooks/Cole.

• Giancoli, D. (2000). Physics for Scientists & Engineers. Third Edition. NJ: Prentice Hall.

• Iosilevskii, A., Isoilevskii, G. & Rosen, A. (1999). Asymptotic Aerodynamic Th f O ill ti R t Wi i A i l Fli ht SIAM J l A li d Theory of Oscillating Rotary Wings in Axial Flight. SIAM Journal on Applied Mathematics, Vol. 59, No. 4. pp. 1371-1412.

• Long, L. and Weiss, H. (1999). “The Velocity Dependence of Aerodynamic Drag: A Primer for Mathematics ” The American Mathematical Monthly Vol Drag: A Primer for Mathematics. The American Mathematical Monthly. Vol 106, No. 2. p.127-135

• Nagle, R. & Staff, E. & Snider, A. (2004). Fundamentals of Differential Equations, Sixth Edition. NY: Pearson Education.

24• Weast, R. (1973). Handbook of Chemistry and Physics, 54th Edition. OH:

Chemical Rubber Publishing Company.