On the Zeros of an Undamped Three Degrees-of-Freedom Flexible System

Abstract

This paper presents an investigation of zeros in the SISO dynamics of an undamped three degrees-of-freedom (3DOF) linear time invariant (LTI) flexible system. Of particular interest are non-minimum phase zeros, which severely impact closed-loop performance. This study uses modal decomposition and zero loci to reveal all types of zeros—marginal minimum phase (MMP), real minimum phase (RMP), real non-minimum phase (RNMP), complex minimum phase (CMP), and complex non-minimum phase (CNMP)—that can exist in the system under various parametric conditions. It is shown that if CNMP zeros occur in the dynamics of an undamped LTI flexible system, they will always occur in a quartet of CMP-CNMP zeros. Consequently, the simplest undamped LTI flexible system that can exhibit CNMP zeros in its dynamics is a 3DOF system. Motivated by practical examples of flexible systems that exhibit CNMP zeros, the undamped 3DOF system considered in this paper comprised one rigid-body mode and two flexible modes. For this system, the following conclusions are mathematically established: (1) This system exhibits all possible types of zeros, (2) The precise conditions on modal frequencies and modal residues associated with every possible zero provide a mathematical formulation of the necessary and sufficient conditions for the existence of each type of zero, and (3) Alternating signs of modal residues is a necessary condition for the presence of CNMP zeros in the dynamics of this system. Conversely, avoiding alternating signs of modal residues is a sufficient condition to guarantee the absence of CNMP zeros in this system.