944. The Formulation of Mass Matrices for Dynamic Analysis

Paper

Abstract

Mathematical modeling of structural mass properties is necessary to improve the accuracy and speed of analyzing the characteristics of structures to ensure structural integrity under design static and dynamic loads. This calls for a closer interface between the mass property specialist and dynamist. This paper presents one way in which communication between two analytical areas has been improved.
This paper discusses the use of mass property data in the dynamic analyses of structures. In particular, the need for formulation of mass representations to be used in conjunction with stiffness representations developed from finite element models is discussed. Matrix notation is used to describe the mathematical representations, and the basic differential equation using these matrices is examined. The normal modes obtained from the eigenvalue / eigenvector problem are related to the mass matrix. The Lagrangian formulation using the mass matrix for forming kinetic energy is noted.
Methods of representing mass distribution are reviewed, and a technique for generating mass matrices is shown in detail. The least-squares method is developed from the geometric properties of the structure. A point center-of-gravity location, mass, and mass moments of inertia are assumed as typical of the mass property data available. The least-squares fit acts as an interpolation technique for spreading mass properties to collocation points common to the stiffness model. The limitations and assumptions in the method are noted.
After a mass matrix has been developed from a series of component masses, further information is available about the total properties of the structure. The geometric properties are used to calculate the rigid-body transformation. This transformation applied to the mass matrix yields the total mass properties of the structure (i.e., cg location, mass, mass moments, and products of inertia). This presents an appropriate check of the assembly of masses into a matrix format.
An example is presented to illustrate the use of the least-squares distribution and the lumped-mass technique is compared with the least-squares method.
Two appendices provide further information concerning the subject of this paper.