Approximation of the Duffing oscillator frequency response function using the FPK equation

Although a great deal of work has been carried out on nonlinear structural dynamic systems under random excitation, there has been a comparatively small amount of this work concentrating on the calculation of the quantities commonly measured in structural dynamic tests. Perhaps the most fundamental of these quantities is the frequency response function (FRF). A number of years ago, Yar and Hammond took an interesting approach to estimating the FRF of a Duffing oscillator system which was based on an approximate solution of the Fokker–Planck–Kolmogorov equation. Despite reproducing the general features of the statistical linearisation estimate, the approximation failed to show the presence of the poles at odd multiples of the primary resonance which are known to occur experimentally. The current paper simply extends the work of Yar and Hammond to a higher-order of approximation and is thus able to show the existence of a third ‘harmonic’ in the FRF. A comparison is made with previous work where an approximation to the FRF was computed using the Volterra series.