Dynamics of a flexible body: a two-field formulation

Abstract

A two-field formulation of the nonlinear dynamics of an elastic body is presented in which positions/orientations and the resulting velocity field are treated as independent. Combining a nonclassical description of elastic velocity that includes the convection velocity due to elastic deformation with floating reference axes minimizing the relative kinetic energy due to elastic deformation provides a fully uncoupled expression of kinetic energy. A transformation inspired by the classical Legendre transformation concept is introduced to develop the motion equations in canonical form. Finite element discretization is achieved using the same shape function sets for elastic displacements and velocities. Specific attention is brought to the discretization of the gyroscopic forces induced by elastic deformation. A model reduction strategy to construct superelement models suitable for flexible multibody dynamics applications is proposed, which fulfills the essential condition of orthogonality between a rigid body and elastic motions. The problem of expressing kinematic connections at superelement boundaries is briefly addressed. Two academic examples have been developed to illustrate some of the concepts presented.

A Two-field Approach to Multibody Dynamics of Rotating Flexible Bodies

“…The only difference lies in adding to the material expression of the body frame velocity field r W r , the material expression r ω m of the driving angular angular velocity. It makes thus make it possible and easy to generalize the two-field formulation proposed in references [24,25].…”

“…The superelement implementation of the floating system of equations of Table 1 can be achieved by adopting a modal reduction basis Y which applies equally to displacements and velocities [24,25]…”

“…It is shown that its full decoupling results from the definition of a compound elastic velocity field that includes a convective term due to angular rotation and assumed orthogonal to rigid body motion. In Section 4, a canonical expression of kinetic energy is introduced which allows to treat displacement and velocity fields independently, allowing thus the generalization of the two-field floating frame of reference approach introduced in [24,25] to elastic motion in an reference frame rotating at uniform speed. Finite element discretization is discussed in Section 6.…”