Dynamics of Initially Curved Plates in the Analysis of Spatial Flexible Mechanical Systems

In this paper, the effect of the initial curvature on the nonlinear dynamics of plates in constrained flexible multibody systems is examined. A set of nonlinear differential equations that describe the motion of the elastic bodies which undergo large reference displacements is derived using the principle of virtual work in dynamics. The resulting nonlinear mass matrix is written as the sum of two nonlinear matrices. These are the conventional mass matrix that arises when the effect of initial curvature is neglected and the curvature mass matrix that represents the effect of the initial curvature. It is shown that all the components of the generalized centrifugal forces depend on the initial curvature. In particular, the generalized centrifugal forces associated with the large rigid body translation and the elastic deformation are linear functions of the initial curvature, while the centrifugal forces associated with the finite rotation are quadratic functions of the initial curvature. It is also shown that the generalized Coriolis forces associated with the finite rotation are linear functions of the initial curvature, while other Coriolis components do not depend on the initial curvature. It is demonstrated numerically that the use of the normal modes of vibration of flat plates in the analysis of the initially curved elastic components leads to a significant error in the dynamic response. Numerical results presented in this investigation are obtained using the spatial flexible multibody RSSR mechanism.