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TECHNICAL PAPERS

Order Reduction of Parametrically Excited Linear and Nonlinear Structural Systems

[+] Author and Article Information
Venkatesh Deshmukh1

Department of Mechanical Engineering, University of Alaska Fairbanks, Fairbanks, AK 99775

Eric A. Butcher2

Department of Mechanical Engineering, University of Alaska Fairbanks, Fairbanks, AK 99775ffeab@uaf.ediu

S. C. Sinha

Nonlinear Systems Research Laboratory, Department of Mechanical Engineering, Auburn University, Auburn, AL 36849

1

Current address: Department of Mechanical Engineering, Villanova University, Villanova, PA 19085

2

Corresponding author.

J. Vib. Acoust 128(4), 458-468 (Dec 21, 2005) (11 pages) doi:10.1115/1.2202151 History: Received January 26, 2004; Revised December 21, 2005

Order reduction of parametrically excited linear and nonlinear structural systems represented by a set of second order equations is considered. First, the system is converted into a second order system with time invariant linear system matrices and (for nonlinear systems) periodically modulated nonlinearities via the Lyapunov-Floquet transformation. Then a master-slave separation of degrees of freedom is used and a relation between the slave coordinates and the master coordinates is constructed. Two possible order reduction techniques are suggested. In the first approach a constant Guyan-like linear kernel which accounts for both stiffness and inertia is employed with a possible periodically modulated nonlinear part for nonlinear systems. The second method for nonlinear systems reduces to finding a time-periodic nonlinear invariant manifold relation in the modal coordinates. In the process, closed form expressions for “true internal” and “true combination” resonances are obtained for various nonlinearities which are generalizations of those previously reported for time-invariant systems. No limits are placed on the size of the time-periodic terms thus making this method extremely general even for strongly excited systems. A four degree-of-freedom mass- spring-damper system with periodic stiffness and damping as well as two and five degree-of-freedom inverted pendula with periodic follower forces are used as illustrative examples. The nonlinear-based reduced models are compared with linear-based reduced models in the presence and absence of nonlinear resonances.

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Copyright © 2006 by American Society of Mechanical Engineers
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Figures

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Figure 1

Reduced order trajectories for periodic mass-spring-damper system with initial conditions on the first eigenmode

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Figure 2

Reduced order trajectories for periodic mass-spring-damper system with initial conditions on the second eigenmode

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Figure 3

Five mass inverted pendulum with periodic follower force

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Figure 4

Reduced order trajectories for five mass inverted pendulum with initial conditions on the first eigenmode

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Figure 5

Reduced order trajectories for five mass inverted pendulum with initial conditions on the second eigenmode

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Figure 6

Double inverted pendulum with periodic follower force

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Figure 7

(a) Comparison of the actual, nonlinear reduced and linear reduced model trajectories for double inverted pendulum away from resonance. (b) Integral square error for gradually increasing initial conditions for double inverted pendulum away from resonance.

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Figure 8

The projection of the invariant manifold onto (ym,ys) space. The parametric forcing period of normalized transformed system is 2.0

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Figure 9

(a) Comparison of the actual, nonlinear reduced and linear reduced model trajectories for double inverted pendulum in 1:1 true internal resonance. (b) Integral square error for gradually increasing initial conditions for double inverted pendulum in 1:1 true internal resonance.

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Figure 10

(a) Comparison of the actual, nonlinear reduced, and linear reduced model trajectories for double inverted pendulum in true combination resonance. (b) Integral square error for gradually increasing initial conditions for double inverted pendulum in true combination resonance.

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