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# Modeling and Analysis of a Ball and Beam System Including Impacts and Dry Friction

[+] Author Affiliations
Diego Colón

University of São Paulo, São Paulo, SP, Brazil

Átila Madureira Bueno, Ivando S. Diniz

São Paulo State University/UNESP, Sorocaba, SP, Brazil

Jose M. Balthazar

São Paulo State University/UNESP, Rio Claro, SP, Brazil

Paper No. DETC2014-34529, pp. V008T11A048; 8 pages
doi:10.1115/DETC2014-34529
From:
• ASME 2014 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference
• Volume 8: 26th Conference on Mechanical Vibration and Noise
• Buffalo, New York, USA, August 17–20, 2014
• Conference Sponsors: Design Engineering Division, Computers and Information in Engineering Division
• ISBN: 978-0-7918-4641-4

## abstract

The Ball and Beam system is a common didactical plant that presents a complex nonlinear dynamics. This comes from the fact that the ball rolls over the beam, which rotates around its barycenter. In order to deduce the system’s equations, composition of movement must be applied, using a non-inertial reference frame attached to the beam. In the Literature, a common hypothesis is to suppose that the ball rolls without slipping. If a viscous friction is supposed to be present, a simpler situation is obtained, where Lagrangean mechanics can be applied, and no contact force is known. Even then, the dynamics is very nonlinear.

However, this model does not include all the relevant phenomena, such as ball’s slipping at higher beam’s inclination angles, dry friction between the ball and the beam, and impacts between: 1) the ball and the ends of the beam, and 2) the beam and the base (ground). These additions to the model impose the necessity to calculate, in a simulation setting, the contact forces, and the Newton’s approach to determine the system’s equations becomes more convenient. Also, discontinuities in the model are introduced, and the simpler mathematical object for model such systems are the differential inclusion systems.

In this work, we deduce the Ball and Beam differential inclusion system, including dry friction and the impact between the ball and beam. We also present simulation results for the corresponding differential inclusion system in a typical situation.

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