Effects of Shear Deformation and Rotary Inertia on the Nonlinear Dynamics of Rotating Curved Beams

Abstract

In this investigation a method for the dynamic analysis of initially curved Timoshenko beams that undergo finite rotations is presented. The combined effect of rotary inertia, shear deformation, and initial curvature is examined. The kinetic energy is first developed for the curved beam and the beam mass matrix is identified. It is shown that the form of the mass matrix as well as the nonlinear inertia terms that represent the coupling between the rigid body motion and the elastic deformation can be expressed in terms of a set of invariants that depend on the assumed displacement field, rotary inertia, shear deformation, and the initial beam curvature. A nonlinear finite element formulation is then developed for Timoshenko beams that undergo finite rotations. The nonlinear formulation presented in this paper is applied to multibody dynamics where mechanical systems consist of an interconnected set of rigid and deformable bodies, each of which may undergo finite rotations. The equations of motion are developed using Lagrange’s equation and nonlinear algebraic constraint equations that mathematically describe mechanical joints and specified trajectories are adjoined to the system differential equations using the vector of Lagrange multipliers.