Vibration Control in Rotating or Translating Elastic Systems

A. Galip Ulsoy Assistant Professor, Department of Mechanical Engineering and Applied Mechanics and The Center for Robotics and Integrated Manufacturing, University of Michigan, Ann Arbor, Mich. 48109

Vibration Control in Rotating or Translating Elastic Systems The reduction of vibration in rotating or translating elastic systems (e.g., shafts, circular saws, belts, handsaws) is an important engineering problem. This paper presents the characteristics of rotating or translating elastic system vibration problems which are significant for the design of active controllers. The effect of the rotation or translation velocity on the controller design, and the effects of observation and control spillover are discussed. Simulation results for two example problems, a rotating cantilever shaft and an axially moving string, are used to illustrate the design and performance of active vibration controllers for rotating or translating elastic systems.

Introduction Reduction of vibration in rotating elastic systems (e.g., rotors [1], circular saws [2]), and translating systems (e.g., power transmission belts [3], and band saws [4]) is an important engineering objective in many technological areas. Vibration reduction is generally achieved through design modifications to increase effective stiffness or damping [4-6]. However, in recent years there has been increased interest in active vibration control methods [1-2, 7-10] for rotating or translating elastic systems. Radcliffe and Mote [2, 9] have developed a laboratory active control system for circular saws which uses electromagnets to increase the effective damping and stiffness of transverse vibrations. This is an extension of the work done previously by Ellis and Mote [8]. Mote and Holoyen [10] have also developed an active control system for circular saw temperature. Contolling saw temperature reduces saw vibration indirectly through effective stiffness modifications. Gonhalekar, et al. [7] also used electomagnets in laboratory tests to show that increased damping of shaft vibrations could be realized through active control. Stan way and Burrows [1], have presented an analysis showing that rotor vibrations are controllable by forces applied through flexible supports. Although the dynamic behavior of systems subject to constant transport velocity are well known (e.g., [11-14] these dynamic characteristics have not been systematically accounted for in the design of active control systems. The purpose of this paper is to present and discuss the important considerations in the design of active vibration contollers for rotating or translating elastic systems. First the general form of the equations of motion for such systems is presented, and their dynamic characteristics are examined. Next the design of an active controller, considering the effects of transport (i.e., rotation or translation) velocity and observation and control spillover, is discussed. Simulation results are presented for two representative examples; a rotating cantilever shaft and an axially moving string. Contributed by the Dynamic Systems and Control Division for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript

received by the Dynamic Systems and Control Division, February 23, 1983.

Equations of Motion For purposes of control system analysis and design, the vibration of rotating or translating undamped elastic systems is described by a system of ordinary differential equations of the form, Mq + Cq + Kq = f (1) Equations of the form (1) are obtained by direct lumping of the mass and stiffness properties of the elastic system, or by discretization of the partial differential equations derived based on distributed properties. The important features of these systems is that the matrices C and K depend on the constant velocity of rotation and translation of the elastic system. These velocity dependent matrices arise due to coriolis and centripetal acceleration effects. The matrix C is skew symmetric and has the property, qrCq = 0 (2) Thus, C does not represent energy dissipation, but rather transfer of energy from one vibration mode to another. The matrix K contains terms which decrease with the square of the velocity, thus some eigenvalues of the system are reduced with increasing velocity. These features, as well as the effect of damping, are illustrated in the examples below. Rotating Cantilever Shaft. The equations of transverse vibration of a rotating cantilever shaft, based on a single lumped-mass model as shown in Fig. 1, are m

0

0

m

ImQ, 2

(k-mQ ) ft,Q

2/wfi

"0i

7i

-/3 2 fi 2

(k-mQ )

(3)

where we have included a damping force of the form, - *«/ = dMi-ftOftfei +(0i Q2+ /32Q\{Zal)

4>l(Za2)

4>i(zai)

2(za2)

(29)

l(Zai) htial)