On the modal analysis of flexible multibody systems with singular mass and stiffness matrices

Abstract

AbstractA complex mechanical system can be described as a combination of individual rigid and/or flexible components and a set of kinematical constraints. The restrictions can be either internal, which come on the scene in the case of parameterizing the configuration manifold of a system with a number of parameters larger than its intrinsic dimension, e.g. by employing three directors to describe the Lie Group SO(3) (basically for rigid bodies and beams) or by employing a single unit director to describe the two dimensional sphere with unit radius embedded in the three‐dimensional Euclidean space, i.e. S2 1 , (basically for inextensible shells, in which the component of the Green‐Lagrange strain tensor E33 is set to be zero at every time), or external ones, which come on the scene in the case of constraining bodies by means of joints, connections or supports. The linearized form of the resulting governing equations possesses usually singular mass and stiffness matrices. This fact represents an issue when the calculation of natural frequencies and natural modes for the whole system is needed. The classical modal analysis (solving directly the generalized eigenvalue problem from the stiffness and mass matrices) is no longer applicable. To overcome this difficulty, it is firstly necessary to utilize the linearized constraint equations that complete the missing information at the level of the mass and stiffness matrices and secondly, to redefine the eigenvalue and ‐vector problem within a suitable framework. In the current work, we set the focus on the development framework for the modal analysis able to deal with singular matrices.