Positive Real Zeros in Flexible Beams

Eric H Maslen

doi:10.1155/1995/395365

Abstract

Feedback control of flexible structures naturally involves actuators and sensors that often cannot be placed at the same point in the structure. It has been widely recognized that this noncollocation can lead to difficult control problems and, in particular, difficulty in achieving high robustness to variation in the dynamic properties of the structure. This problem has previously been traced to transmission zeros in the dynamic transfer function between sensor location and actuator location, especially those lying on the positive real axis in the complex plane. In this artie/e, the physical significance of these zeros is explored and the dynamic properties of beams that give rise to real positive zeros are contrasted to those of torsional and compressive systems that do not.

Highlights

In considering active control of flexible structures, interest has been focused on the elemental problems of vibration dissipation in uniform beams in tension, torsion, or lateral shear (Rosenthal, 1984; Spector and Flashner, 1989)
These three problems are similar in that they all arise in the same physical structure; they are all conveniently studied as nondissipative; and they all present a difficult control problem when the sensor and actuator are not placed at the same point along the beam
Having established a necessary condition for real zeros, the existence of such zeros and the nature of the associated forced response are demonstrated by a simple example

Summary

INTRODUCTION

In considering active control of flexible structures, interest has been focused on the elemental problems of vibration dissipation in uniform beams in tension, torsion, or lateral shear (Rosenthal, 1984; Spector and Flashner, 1989). Perhaps the most important manifestation of this difference is that noncollocated actuator/sensor transfer functions for transverse beams exhibit positive (and negative) real zeros (Spector and Flashner, 1989; Lefante, 1992) whereas the torsional and tensile transfer functions do not (Rosenthal, 1984). These positive real zeros naturally attract root loci into the right half of the complex plane, producing systems with very limited stability margins. Having established a necessary (but not sufficient) condition for real zeros, the existence of such zeros and the nature of the associated forced response are demonstrated by a simple example

Lateral Beam

SINGLY CONNECTED SYSTEMS

UNIFORM TRANSVERSE BEAM

CONCLUSIONS