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Absorber control of the finite amplitude, nonlinear vibrations of a simple shear suspension system
Abstract
A constitutive equation for simple shearing of an incompressible, isotropic, visco-hyperelastic quadratic material, a class which includes the viscoelastic Mooney-Rivlin and neo-Hookean materials, is applied to study the absorber control of the finite amplitude, nonlinear forced vibrations of a rigid body supported symmetrically by a simple shear suspension system. For the special class of materials whose shear response function is a constant, the motion of the system is governed by a system of equations whose exact solutions are briefly discussed. For a broader class of viscoelastic quadratic materials, the motion of the nonlinear system is characterized by a coupled pair of differential equations whose exact solutions are not known, however, their approximate solutions are derived by applying the method of harmonic balance. It is found that the finite amplitude motion control of the load with a linear vibration absorber not only is based on the suppression bandwidth value but also on the stability analysis of the motion. When the absorber is removed from the load, the equation of motion for the load reduces to the forced, damped Duffing equation whose approximate solution for both horizontal and inclined motions is examined by applying the method of harmonic balance. General equations that determine the instability behavior of the load in either damped or undamped vibrational motion are derived, and some illustrative examples are presented as well. Also, the exact solutions for both the horizontal and inclined, free, undamped vibrational motion of the load are briefly reviewed. Toward the end of this work, a new method based on Jacobian elliptic functions is applied to obtain the approximate solution of a two degree of freedom system whose equations of motion are characterized by a set of second order ordinary differential equations with cubic nonlinearities. General equations for the approximate solutions for free and forced vibrations of the system also are examined.
Subject Area
Mechanical engineering|Mechanics|Mathematics
Recommended Citation
Elias Zuniga, Alex, "Absorber control of the finite amplitude, nonlinear vibrations of a simple shear suspension system" (1994). ETD collection for University of Nebraska-Lincoln. AAI9425280.
https://digitalcommons.unl.edu/dissertations/AAI9425280