Distribution of Real and Imaginary Zeros of Multi-DoF Undamped Flexible Systems

Abstract

This paper investigates the distribution of zeros with respect to the poles on the imaginary and real axes of the s-plane in the transfer function of a multi-DoF undamped flexible LTI system. The transfer function of a multi-DoF undamped flexible LTI system can be modally decomposed i.e. expressed as the sum of second order modes where each mode is characterized by two system parameters – modal residue and modal frequency. It is well known that when all the modal residue signs are the same, all the zeros of the multi-DoF undamped flexible LTI system are minimum phase (MP). However, the same sign for all modal residues is a sufficient condition for the elimination of non-minimum phase (NMP) zeros and not a necessary one. In order to find sufficient conditions for the elimination of NMP zeros when all modal residue signs are not the same, specific results are needed that explain the distribution of zeros with respect to the poles in the s-plane. Therefore, in this paper results are provided that elucidate the distribution of zeros with respect to the poles on the real and imaginary axes of the s-plane for any combination of modal residue signs. The real and imaginary axes are divided into four segments based on the location of the poles and the parity of the number of zeros (i.e. even or odd) in each segment is derived as a function of the system parameters. The results in this paper provide the necessary and sufficient condition for the occurrence of pole-zero flipping on the imaginary axis which is known to be detrimental to the closed-loop dynamic performance of undamped flexible LTI systems. The results from this paper will also enable the derivation of a sufficient condition for the elimination of NMP zeros in multi-DoF undamped flexible LTI systems.