The Bouncing Ball Apparatus as an Experimental Tool

[+] Author and Article Information
Ananth Kini

Aerospace and Mechanical Engineering, University of Arizona, Tucson, AZ 85721anantẖkini@rediffmail.com

Thomas L. Vincent1

Aerospace and Mechanical Engineering, University of Arizona, Tucson, AZ 85721vincent@u.arizona.edu

Brad Paden

Mechanical and Environmental Engineering, University of California, Santa Barbara, CA 93106paden@engineering.ucsb.edu


Author for all correspondence.

J. Dyn. Sys., Meas., Control 128(2), 330-340 (Jun 01, 2005) (11 pages) doi:10.1115/1.2194069 History: Received February 20, 2004; Revised June 01, 2005

The bouncing ball on a sinusoidally vibrating plate exhibits a rich variety of nonlinear dynamical behavior and is one of the simplest mechanical systems to produce chaotic behavior. A computer control system is designed for output calibration, state determination, system identification, and control of a new bouncing ball apparatus designed in collaboration with Magnetic Moments. The experiments described here constitute the first research performed with the apparatus. Experimental methods are used to determine the coefficient of restitution of the ball, an extremely sensitive parameter needed for modeling and control. The coefficient of restitution is estimated using data from a stable one-cycle orbit both with and without using corresponding data from a ball map. For control purposes, two methods are used to construct linear maps. The first map is determined by collecting data directly from the apparatus. The second map is derived analytically using a high bounce approximation. The maps are used to estimate the domains of attraction to a stable one-cycle orbit. These domains of attraction are used to construct a chaotic control algorithm for driving the ball to a stable one-cycle from any initial state. Experimental results based on the chaotic control algorithm are compared and it is found that the linear map obtained directly from the data not only gives a more accurate representation of the domain of attraction, but also results in more robust control of the ball to the stable one-cycle.

Copyright © 2006 by American Society of Mechanical Engineers
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Figure 1

(a) Photograph of the bouncing ball apparatus (BBA) and (b) a front section drawing of the BBA

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Figure 2

Internal control system loop in the BBA

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Figure 3

Outer computer control system loop for BBA

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Figure 4

Schematic of the laboratory setup for the bouncing ball experiment

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Figure 5

Plot of the ball driven chaotically in state space before being captured within domain of attraction estimated using the linear map obtained from experimental data

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Figure 6

Plot of motion of ball within domain of attraction showing number of stable periodic one-cycle bounces

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Figure 7

A number of stable one-cycle periodic bounce heights within the domain of attraction

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Figure 8

Plot of the ball driven chaotically in state space with domain of attraction obtained using linear map obtained from the linearization of the high-bounce approximation

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Figure 9

Data collected after the ball had entered the domain of attraction—ball is seen to leave the domain of attraction and return to the fixed point a few times

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Figure 10

Piston position sensor calibration plot

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Figure 11

Frequency response of piston: (a) piston gain characteristics and (b) piston phase characteristics

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Figure 12

(a) Bounce output signal recorder for a one-cycle periodic bounce and (b) bounce output signal magnified to show two consecutive pulses

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Figure 13

(a) Bode plots of the continuous and discrete-time pseudoderivative filters and (b) Simulink® block implementation of the discrete-time pseudoderivative filter for velocity estimation




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