The dynamic stability of an unbalanced mass exciter

Abstract

The dynamic stability of single- and multi-degree-of-freedom unbalanced mass exciter systems is discussed. Previous work concerning this subject by A. Sommerfield, Y. Rocard, R. Mazet, V.0. Kononenko, Y.G. Panovko and I.I. Gubanova is summarized. A single-degree-of-freedom system consisting of a linear mechanical oscillator with a rotating unbalanced mass connected rigidly to it is defined as the basic single-degree-of-freedom system. This system is mathematically equivalent to the one used by Rocard in his analysis. The differential equations of motion for the system are obtained by using Lagrange's Equations. Global stability, stability in the sense of Laplace, is proved using Liapunov's second method. Four separate local stability analyses of this system are developed, two of which assume a constant angular velocity, [omega], of the unbalanced mass and two which allow for periodic variations in [omega]. These analyses are termed zero and first order respectively. The first zero order analysis is based directly on the differential equations of motion and the zero order steady state solution. The steady state torque output of the vibration exciter motor and the steady state torque requirements of the oscillator are obtained as functions of the operating frequency. Stability is determined by examining the behavior of the system in the vicinity of the intersection points of these two functions. The second zero order analysis examines the behavior of small perturbations added to the steady state solution. The system is considered stable if these perturbations disappear with time. The first first order analysis is a perturbation type, but is based on a steady state solution which allows for periodic variations in [omega]. The second first order analysis is also based on the first order perturbed equations of motion but is a Floquet type analysis. Validity criteria for the zero and first order analyses are obtained, and the zero order region of validity is plotted graphically. A representative set of systems is analyzed numerically, and the results are presented in a figure showing the stability boundary as a function of the system parameters in non-dimensional form. Two distinct types of muIti-degree-of-freedom systems are discussed. The first consists of a single oscillator mass that is free to perform planar motion. It is shown that when an unbalanced mass exciter with a uniaxial force output is mounted on the oscillator in such a way that only one mode is excited, the problem reduces to the sing le-degree-of-freedom problem. The second system consists of a series of linear sing I e-degree -of -freedom oscillators with an unbalanced mass exciter mounted on one of them. The special case of a three oscillator system with equal masses is used to demonstrate that, for systems with widely separated resonances, the "equivalent" singIe-degree-of-freedom analysis presented by Kononenko is valid. From these results it is concluded that, in any multi-degree-of-freedom system, an "equivalent" single-degree-of-freedom analysis may be used to examine the stability of the system near any resonance as long as that particular mode is the only one which is being significantly excited. Appendices covering the details of Rocard's analysis, and of the unbalanced mass exciters designed and built at the California Institute of Technology are included.

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