# Bouncing Ball System - Christine L Transform back to Ground (Lab) Reference Frame... Bouncing Ball...

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Bouncing BallSystem

Christine Lind

Introduction

Experiments &Simulations

Comparisons

Conclusions

Extras

Bouncing Ball SystemAMATH 575 Final Project

Christine Lind

Department of Applied MathematicsUniversity of Washington

June 2, 2005

Bouncing BallSystem

Christine Lind

Introduction

Experiments &Simulations

Comparisons

Conclusions

Extras

Outline

IntroductionSystem DescriptionExact SystemHigh Bounce Approximation

Experiments & SimulationsSpeaker ExperimentBouncing Ball ProgramMatlab Simulations

ComparisonsStandard MapBifurcationStrange Attractor

Conclusions

Extras

Bouncing BallSystem

Christine Lind

Introduction

System Description

Exact System

High BounceApproximation

Experiments &Simulations

Comparisons

Conclusions

Extras

Bouncing Ball System Description

Simple Physical System

Interaction between:

I Ball

I Sinusoidally Oscillating Table

Bouncing BallSystem

Christine Lind

Introduction

System Description

Exact System

High BounceApproximation

Experiments &Simulations

Comparisons

Conclusions

Extras

Bouncing Ball System Description

Initial Assumptions:

(xk , tk) - ball position and time of kth impact

I Between Impacts, ball obeys Newtons Laws:

x(t) = xk + vk(t tk)g

2(t tk)2 tk t tk+1

I Table is unaffected by impacts:

s(t) = A (sin(t + 0) + 1)

Bouncing BallSystem

Christine Lind

Introduction

System Description

Exact System

High BounceApproximation

Experiments &Simulations

Comparisons

Conclusions

Extras

Bouncing Ball System Description

Solve for Next Impact Time

d(t) = x(t) s(t) - distance between ball and tableI First t > tk where d(t) = 0 is tk+1, next impact time!

0 = d(tk+1) = xk + vk(tk+1 tk)g

2(tk+1 tk)2

A (sin(tk+1 + 0) + 1)

I Note that at time tk :

xk = s(tk) = A (sin(tk + 0) + 1)

Bouncing BallSystem

Christine Lind

Introduction

System Description

Exact System

High BounceApproximation

Experiments &Simulations

Comparisons

Conclusions

Extras

Bouncing Ball System Description

Solve for Next Impact Time

d(t) = x(t) s(t) - distance between ball and tableI Then the (Implicit) Time-Equation is:

0 = A sin(tk + 0) + vk(tk+1 tk)g

2(tk+1 tk)2

A sin(tk+1 + 0)

I Note that vk is still unknown

I Find Velocity-Equation

Bouncing BallSystem

Christine Lind

Introduction

System Description

Exact System

High BounceApproximation

Experiments &Simulations

Comparisons

Conclusions

Extras

Bouncing Ball System Description

Solve for Impact Velocity

First look at two different frames of reference

(a) Ground (Lab) Frame of Reference:I vk - ball velocity at impact kI uk - table velocity at impact k

(b) Table Frame of Reference:I vk = vk uk - ball velocity at impact k

Bouncing BallSystem

Christine Lind

Introduction

System Description

Exact System

High BounceApproximation

Experiments &Simulations

Comparisons

Conclusions

Extras

Bouncing Ball System Description

Solve for Impact Velocity

v k - velocity just before impact kvk - velocity just after impact k - coefficient of restitution (describes damping)

I vk = v kI 0 1

I = 1 - no energy loss (no damping - elastic collision)

I Transform back to Ground (Lab) Reference Frame...

Bouncing BallSystem

Christine Lind

Introduction

System Description

Exact System

High BounceApproximation

Experiments &Simulations

Comparisons

Conclusions

Extras

Bouncing Ball System Description

Solve for Impact Velocity

vk+1 = (1 + )uk+1 v k+1I Recall that for tk t tk+1 the ball position is

described by:

x(t) = xk + vk(t tk)g

2(t tk)2

v k+1 = x(tk+1) = vk g(tk+1 tk)

I The table position is given by:

s(t) = A (sin(t + 0) + 1)

uk+1 = s(tk+1) = A cos(tk+1 + 0)

I Then we can solve for the Impact Velocity

Bouncing BallSystem

Christine Lind

Introduction

System Description

Exact System

High BounceApproximation

Experiments &Simulations

Comparisons

Conclusions

Extras

Bouncing Ball System Description

Solve for Impact Velocity

I Impact Velocity Equation:

vk+1 = (1 + )A cos(tk+1 + 0)

(vk g(tk+1 tk))

Bouncing BallSystem

Christine Lind

Introduction

System Description

Exact System

High BounceApproximation

Experiments &Simulations

Comparisons

Conclusions

Extras

Bouncing Ball Exact Equations

System is described by

I Time Equation:

0 = A sin(tk + 0) + vk(tk+1 tk)

g2(tk+1 tk)2 A sin(tk+1 + 0)

I Velocity Equation:

vk+1 = (1 + )A cos(tk+1 + 0)

(vk g(tk+1 tk))

Bouncing BallSystem

Christine Lind

Introduction

System Description

Exact System

High BounceApproximation

Experiments &Simulations

Comparisons

Conclusions

Extras

Bouncing Ball Exact Equations

Non-Dimensionalization!

Too many parameters to study the system efficientlyParameters - , A, , g

I Transform system into dimensionless variables:

k = tk + 0

k =2

gvk

I New Parameter

=22(1 + )A

g

Bouncing BallSystem

Christine Lind

Introduction

System Description

Exact System

High BounceApproximation

Experiments &Simulations

Comparisons

Conclusions

Extras

Bouncing Ball Exact Equations

Dimensionless System is described by

I Phase Equation:

0 = (sin k sin k+1)+ (1 + )

(k(k+1 k) (k+1 k)2

)I Velocity Equation:

k+1 = cos k+1 (k 2(k+1 k))

I Now we can study the system simply by varying and.

Bouncing BallSystem

Christine Lind

Introduction

System Description

Exact System

High BounceApproximation

Experiments &Simulations

Comparisons

Conclusions

Extras

Bouncing Ball Exact Equations

Dimensionless System is described by

I Phase Equation:

0 = (sin k sin k+1)+ (1 + )

(k(k+1 k) (k+1 k)2

)I Velocity Equation:

k+1 = cos k+1 (k 2(k+1 k))

I Implicit Maps can be hard to analyze & simulate make an approximation that will give us an ExplicitMap...

Bouncing BallSystem

Christine Lind

Introduction

System Description

Exact System

High BounceApproximation

Experiments &Simulations

Comparisons

Conclusions

Extras

Bouncing Ball Approximation

High Bounce Approximation

Assume:change in table height maximum height of the ball

I Ball orbit symmetric about the maximum height:

xk = xk+1 vk+1 = vk

Bouncing BallSystem

Christine Lind

Introduction

System Description

Exact System

High BounceApproximation

Experiments &Simulations

Comparisons

Conclusions

Extras

Bouncing Ball Approximation

High Bounce Approximation

v k+1 = vkI Recall:

v k+1 = vk g(tk+1 tk) = vkI Explicit Time Map:

tk+1 = tk +2

gvk

I Use equation above to solve for the velocity map, andnon-dimensionalize...

Bouncing BallSystem

Christine Lind

Introduction

System Description

Exact System

High BounceApproximation

Experiments &Simulations

Comparisons

Conclusions

Extras

Bouncing Ball Approximation

High Bounce Equations

I Phase Equation:

k+1 = k + k (mod 2)

I Velocity Equation:

k+1 = k + cos(k + k)

I For = 1, this is the Standard Map!

Bouncing BallSystem

Christine Lind

Introduction

Experiments &Simulations

Speaker Experiment

Bouncing BallProgram

Matlab Simulations

Comparisons

Conclusions

Extras

Outline

IntroductionSystem DescriptionExact SystemHigh Bounce Approximation

Experiments & SimulationsSpeaker ExperimentBouncing Ball ProgramMatlab Simulations

ComparisonsStandard MapBifurcationStrange Attractor

Conclusions

Extras

Bouncing BallSystem

Christine Lind

Introduction

Experiments &Simulations

Speaker Experiment

Bouncing BallProgram

Matlab Simulations

Comparisons

Conclusions

Extras

Bouncing Ball Experiment

Speakers and Function Generators

The physical system can be explored using a setup similar tothe schematic shown below.

Bouncing BallSystem

Christine Lind

Introduction

Experiments &Simulations

Speaker Experiment

Bouncing BallProgram

Matlab Simulations

Comparisons

Conclusions

Extras

Bouncing Ball Experiment

Experimental Set-up

Nicholas B. Tufillaros experimental set-up at Bryn MawrCollege (circa 1985).

Bouncing BallSystem

Christine Lind

Introduction

Experiments &Simulations

Speaker Experiment

Bouncing BallProgram

Matlab Simulations

Comparisons

Conclusions

Extras

Bouncing Ball Experim

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