The Cartesian elastodynamics linear model of mechanical systems with flexible links

Abstract

The linearized Cartesian elastodynamics model of mechanical systems with flexible links is introduced in this paper to simplify its counterpart n-dof generalized model. The stiffness is represented by means of Lončarić’s 6 × 6 Cartesian stiffness matrix (CSM), the inertia by what von Mises termed the inertia dyad, i.e., the 6 × 6 Cartesian mass matrix (CMM). This model applies to a mechanical system with compliant components. Furthermore, the Cartesian frequency matrix (CFM) is defined as a congruent transformation of its stiffness counterpart, the transformation matrix being the inverse of the square root of the positive-definite CMM. The CFM thus defined is dimensionally-homogeneous, symmetric and at least positive-semidefinite. Upon the eigenvalue decomposition of the same matrix, the natural frequencies and the corresponding natural modes, i.e., the eigenscrews of the system, are obtained. The physical meaning of the CFM, together with that of its eigenvalues and eigenscrews, are given due interpretation in the paper, within the context of screw theory. This matrix is intended to serve as a useful tool for the elastodynamics analysis and design of a large class of multibody systems with flexible components, especially at the early design stages.