Estimation of second and third order frequency response functions using truncated models

Abstract

Given that a non-linear system consists of a series of point masses connected by springs and dampers which have polynomial force characteristics, the Fourier transform of the first order Volterra kernel between input force and the resulting motion is determined from the mass matrix and the stiffness and damping matrices formed using the linear force coefficients of the spring and damper characteristics. The second and higher order Volterra kernel transforms are functions of the non-linear force coefficients and the set of first order Volterra kernel transforms. It is shown that by writing the first order kernel transforms in pole-zero form, the second order kernel transform can be approximated within a limited frequency range by a quotient whose denominator is formed from the poles of the first order in the restricted frequency range and whose numerator is a polynomial, the maximum number of terms in the polynomial being limited by the number of poles. This formulation can be used as a basis of a curve fitting strategy: if the first order frequency response function (FRF) is measured over some frequency range and a curve fit is performed to find its poles, these can be used as a basis for curve fitting to the second order FRF, the remaining unknown parameters being the coefficients in the polynomial numerator. It is only necessary to measure the second order FRF at a limited number of points; this number being a linear function of the required spectral density and not the quadratic function if the whole FRF were to be measured. It is further shown that if the second order FRF is small, a similar approximation can be made to the form of the third order FRF.