A new algorithm for the frequency response analysis of multi-input multi-output systems by using poliharmonic balance equations

Harmonic balance method is a practical tool which is used widely for the analysis of frequency response characteristics. In the classical application,the method become complex because of the increase in the number of terms as the number of harmonics or the degree of nonlinear terms are increased. As a solution to this problem symbolic algorithms are developed which performs automated computations for systems with polynomial nonlinearities has been developed. Though the algorithms have polyharmonic waveform, commensurate type signal forms where the harmonics are selected as integer multiples of the main harmonic are considered. However, the dependence of the frequency response of nonlinear systems to the frequency content of the input sometimes requires the analysis to be realized for incommensurate frequency components which called multitone signals. In this study a new algorithm including multitone signal forms is presented for the harmonic balance analysis of nonlinear systems with polynomial nonlinearities. For this purpose a polyharmonic signal form which includes multitone interactions is defined. Generalized expressions are derived for the assumed system structure and signal forms and the symbolic algorithm for obtaining the balance equations is presented. The method is validated using time domain simulations and the effect of the amplitudes of the input components on the output spectrum is illustrated by examples. In addition, the application area of the method is extended to include multi-input multi-output (MIMO) systems. For this purpose after an expression is defined for the representation of MIMO systems, the algorithm structure for SISO is adapted to include MIMO systems. The method is illustrated on an electrical system described by two-input two-output Duffing model. In addition to time domain results, frequency domain characteristics are investigated and the results are validated by numerical simulations. The effects of the parameters of the input such as amplitude and frequency on the frequency response characteristics are also investigated.