Numerical Determination of the Response of a Linear Vibration System With a Singular Mass Matrix

Abstract

In numerically determining the response of a linear second-order multidegree-of-freedom vibrational system subjected to a general excitation, the common approach of applying one of the many multistep methods of numerical analysis (e.g., Milne-Simpson, Adams-Bashforth, etc.) leads ultimately to the solution of a system of linear equations. However, when the mass matrix of the original vibrational system is singular, the coefficient matrix of the system of equations also becomes singular and thus the response cannot be determined. Presented is a means of applying these multistep methods to vibrational systems which results in a method that is capable of obtaining the response independent of the singularity of the mass matrix. This technique is particularly useful in optimization where the values of the parameters of the system are unknown in advance, and thus the method of determining the response must be applicable for a wide range of values of the parameters. In the development and investigation of this technique, the causes of the stability problems which develop from the application of multistep methods to systems with nearly singular mass matrices become apparent.