Abstract
An analysis is presented for determining exact steady state response for a class of strongly non-linear multiple-degree-of-freedom oscillators. These oscillators consist of a linear component, with an arbitrary number of degrees of freedom and configuration, incorporating a component with a geometric non-linearity. Their behavior is analyzed by generalizing previous work on single- and two-degree-of-freedom piecewise linear systems under harmonic excitation. First, a methodology is developed for locating harmonic and subharmonic response characterized by one abrupt change of the parameters of the non-linear element per response cycle. Then, a stability analysis is presented, which is appropriate for general piecewise linear systems. The present analysis is applicable to a variety of engineering areas. Here, its validity and effectiveness is demonstrated by applying it to two classes of problems with large practical significance and rich dynamics. Performance of vibration absorbers with elastic stops is investigated first, while in the second category of examples the oscillators exhibit 2:1 and 3:1 internal resonance. Regular and irregular motions are encountered and analyzed. The response characteristics are compared with those of systems with continuous non-linearities.
Highlights
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Note that if the damping coefficients are non-negative, the Hopf bifurcation is excluded for single degree of freedom oscillators [7, 12], but it occurs for systems with N';?!:
The non-linearity of the system is modeled as a component with piecewise linear damping and stiffness properties
Summary
The system examined is modeled as a general linear multiple-degree-of-freedom oscillator, including a component carrying elastic stops with damping and restoring forces, expressed by. In equation (1), x is the relative displacement of the oscillator carrying the stops with respect to the component of the system these stops collide with. The forcing vector f( r) is split in the form : f c cos ( r + ) + fs sin ( r + ), and the particular solution of equation (2) is determined in the form from.
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