The energy entering the resonant region of a system can be significantly reduced by introducing designed nonlinearities into the system. The basic choice of the nonlinearity can be either a nonlinear spring element or a nonlinear damping element. A numerical algorithm to compute and compare the energy reduction produced by these two types of designed elements is proposed in this study. Analytical results are used to demonstrate the procedure. The numerical results indicate that the designed nonlinear damping element produces low levels of energy at the higher order harmonics and no bifurcations in the system output response. In contrast the nonlinear spring based designs induce significant energy at the harmonics and can produce bifurcation behaviour. The conclusions provide an important basis for the design of nonlinear materials and nonlinear engineering systems.

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A novel approach to suppress resonant vibration is presented by employing a single degree of freedom transmissibility system which utilizes a nonlinear damping element. Studies have shown that the nonlinear damping element can reduce the output energy at the driving frequency and at the same time spread the output signal energy over a wider range of harmonics. It will also be shown that the reduction becomes larger as the nonlinear damping characteristic gets stronger and in most cases, the power at the harmonics in the output spectrum will be much less if the nonlinear damping characteristic is an odd function. Hence, an odd polynomial nonlinear damping element can be introduced between the incoming signal and the structure of interest to suppress resonant vibration. An expression is derived to express the transmitted force spectrum in terms of the nonlinear generalized frequency response functions, to clearly show how the energy, at the excitation frequency, is modified by the nonlinearity.

Existing algorithms for computing the th-order frequency response functions of the Duffing oscillator have helped promote frequency domain analysis of nonlinear systems but still have a number of practical difficulties. A very efficient algorithm to enable the th-order symmetric generalized frequency response functions (GFRFs) to be written down directly in terms of the coefficients of the generalized higher order Duffing oscillator model is developed in this paper. The analytical expression derived in this paper shows that the key procedure in the determination of the higher order GFRFs is the computation of the Stirling set of the second kind. This enables the structure of the higher order GFRFs to be seen more clearly.Index Terms-Higher order frequency response functions, nonlinear systems, Volterra series.