Stability analysis of gyroscopic systems by matrix methods

Abstract

Understanding the behavior of complex gyroscopic systems may be enhanced by formulating a matrix description of the system's equations of motion. In particular, the application of matrix methods to stability analysis often leads to useful constraints on a system's parameter values. In this paper, a new matrix stability condition is derived for linear conservative gyroscopic systems with two degrees of freedom. The work consists first of determining matrix inequalities for the gyroscopic system model. These ineqnalities are then transformed into simple, nonlinear and algebraic relationships among the basic system parameters, thereby providing a quantitative indication of the effects of parameter changes on system stability. This method of stability analysis requires less calculation than the usual procedures of solving the eigenvalue equation or generating the system response. The new stability condition along with standard matrix stability results are applied to a model of a dynamically tuned gyroscope, and the derived regions of stability are discussed. The proposed matrix method may be applicable to the general class of complex gyroscopic systems (flexible structures and rotating devices) which includes the dynamically tuned gyroscope.