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Research Papers

Energy Dissipation in Dynamical Systems Through Sequential Application and Removal of Constraints

[+] Author and Article Information
Jimmy Issa, Alejandro R. Diaz

Department of Mechanical Engineering, Michigan State University, East Lansing, MI 48824-1226

Ranjan Mukherjee1

Department of Mechanical Engineering, Michigan State University, East Lansing, MI 48824-1226mukherji@egr.msu.edu

An example of a special situation is where the system of masses move in a manner that maintains x2(t)0; this will occur if one of the modes have a zero displacement for mass m2 and the system vibrates in that mode.

The constrained system can also be described by differential algebraic equations comprised of the original ordinary differential equations in Eq. 4 and the algebraic constraint equation—see Sec. 1.3.4 of Ref. 11, for example.

Alternatively, Eq. 20 can be viewed as a complementarity between the velocity after impact and the impulse.

Such impulsive moments can be generated in experiments by an electromagnetic brake, as in Ref. 10.

1

Corresponding author.

J. Dyn. Sys., Meas., Control 131(2), 021011 (Feb 09, 2009) (9 pages) doi:10.1115/1.3023126 History: Received December 18, 2007; Revised June 19, 2008; Published February 09, 2009

A strategy to remove energy from finite-dimensional elastic systems is presented. The strategy is based on the cyclic application and removal of constraints that effectively remove and restore degrees of freedom of the system. In general, application of a constraint removes kinetic energy from the system, while removal of the constraint resets the system for a new cycle of constraint application. Conditions that lead to a net loss in kinetic energy per cycle and bounds on the amount of energy removed are presented. In linear systems, these bounds are related to the modes of the system in its two states, namely, with and without constraints. It is shown that energy removal is always possible, even using a random switching schedule, except in one scenario, when energy is trapped in modes that span an invariant subspace with special orthogonality properties. Applications to nonlinear systems are discussed. Examples illustrate the process of energy removal in both linear and nonlinear systems.

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Copyright © 2009 by American Society of Mechanical Engineers
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References

Figures

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Figure 1

A three degree-of-freedom mass-spring system

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Figure 2

Scenario depicting two masses that stay connected after impact

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Figure 3

A linear system with 3DOF in state α and 2DOF in state β

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Figure 4

Action-reaction pair of impulsive moments

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Figure 5

Plot of displacements of the bars and total energy of the system

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Figure 6

A modified version of the system in Fig. 3 in states α and β

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Figure 7

Plot of modal amplitudes in state α and total energy of the system

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Figure 8

Second mode of vibration of the system in Fig. 6

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Figure 9

An example nonlinear system

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Figure 10

Plot of generalized coordinates and total energy of the nonlinear system

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