Instability and Steady-State Coupled Motions in Vibration Isolating Suspensions

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Abstract
When using non-linear spring systems for vibration isolating suspensions, motion initiated in one of the degrees of freedom may cause another degree of freedom to become unstable and commence to vibrate. Subsequently, the two motions become coupled. The paper deals with instability and coupling phenomena observed with a six degree of freedom model suspension. This is done by reducing the system to equivalent two-degree of freedom non-linear systems, and subsequently discussing the various types of solution of these. The equations of motion discussed are of the form It is shown that when the value of the coefficients n and r is n = r = 3·0 then the equations describe the instability and coupling phenomena relating to the vertical motion and the pitching motion of the mass. When n = r = 1·0 then the equations describe the conditions arising between the vertical motion and the yawing motion. The various solutions of these equations are discussed. It is shown that instability in a mode can arise only when the ratio of the linear natural frequencies of the mode initially at rest and the mode initially disturbed lies within certain limits. For n = r = 3·0 the mode with the lower natural frequency may, upon motion having been initiated in the higher mode, become unstable and commence to vibrate. On the other hand, the mode with the lower natural frequency can vibrate on its own since the higher mode remains stable at rest. For n = r = 1·0 the conditions are reversed. Thus, in both systems the coupling is non-symmetrical, but it can be reversed by changing the ratio of the natural frequencies in the appropriate way. The steady-state coupled solutions of the two types of systems are also different. For n = r = 3·0 the steady coupled solution involves a motion in which the degrees of freedom vibrate with equal or opposite phase. For n = r = 1·0 the coupled modes vibrate with a phase difference of ±90°. In both cases the amplitudes of the two degrees of freedom are related, their frequencies are equal and are determined by the amplitudes.

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