Predictions for the perturbed root mean square displacement of a vibrating structure using modal parameters

Abstract

A harmonically vibrating structure is considered for this study and divided into segments. The objective of the proposed research is to use knowledge from the nominal structure to make predictions as to how the root mean squared (RMS) displacement of the structure will change when either the mass or stiffness of a segment is scaled. Typically, to calculate the value for the modified RMS displacement, a linear solve is required to determine the new displacement vector. Additionally, the Neumann series may be used. However, this approach requires computations of matrix-vector products, which may become expensive for larger Degrees of Freedom (DOF) systems. Here, a method is proposed to predict the RMS displacement for the modified system from scalar values of the nominal system. Since only scalar values from the nominal system are required, predictions for the perturbed system can be made cheaply. The proposed method is based on the modal displacement equation and the assumption that the system is forced near a natural frequency. This assumption is used in Rayleigh’s quotient to approximate the new natural frequency as a function of the approximated modal stiffness and mass. The limitations and accuracy will be explored. [Work supported by ONR Grant N00014-19-1-2100.]