Nonminimum Phase Zeros of Multi-Degrees-of-Freedom Damped Flexible Systems

Abstract

This article investigates the nonminimum phase (NMP) zeros in the transfer function, between actuated load input and measured displacement output, of a multi-degrees-of-freedom (DoF) flexible system in the presence of proportional viscous damping. NMP zeros have a negative impact on the dynamics and control of flexible systems and therefore are generally undesirable. Viscous damping is one potential means to guarantee that no NMP zeros exist in the system. However, the impact of viscous damping on NMP zeros of multi-DoF flexible systems is not adequately studied or understood in the literature. To address this gap, a change of variable method is used to first establish a simple mathematical relationship between the zeros of a multi-DoF undamped flexible system and its proportionally damped counterpart. The “proportional” viscous damping model is used due to its practical amenability, conceptual simplicity, and ease of application. This mathematical relationship (between zeros of an undamped system and its damped counterpart) is used to derive the necessary and sufficient condition for the absence of NMP zeros in proportionally damped flexible systems. A graphical analysis of this necessary and sufficient condition is provided, which leads to the formulation of simple proportional damping strategies. A case study of a 4DoF flexible system is presented to demonstrate how a proportional viscous damping strategy can be used to simultaneously guarantee the absence of NMP zeros in multiple single-input single-output (SISO) transfer functions of a multi-DoF flexible system.