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Computer Methods in Applied Mechanics and Engineering 192 (2003) 975-1016. doi:10.1016/S0045-7825(02)00435-8Received by publisher: 2002-02-19Harvest Date: 2016-01-04 12:20:58DOI: 10.1016/S0045-7825(02)00435-8Page Range: 975-101

The numerical treatment of the methodology proposed in Part I of this paper is considered in detail. Unlike traditional approaches, a Galerkin spatial discretization of the equations of motion, now referred to the inertial frame, yields the standard form of nonlinear structural dynamics: Mq¨ + Dq˙ + P(q) = F, with M and D constant matrices. Numerical examples that involve finite vibrations coupled with large overall motions are presented. These simulations also demonstrate the capability of the present formulation in handling multibody dynamics.

Not all integrals of the motion, however, are of equal importance in mechanics. There are some whose constancy is of profound significance, deriving from the fundamental homogeneity and isotropy of space and time." L. D. Landau and E. M. Lifshitz, MechanicsDedicated to Professor Karl. S. Pister on the occasion of his 70th birthday.Abstract. In a previous work, the authors have presented a formalism for deriving systematically invariant, symmetric finite difference algorithms for nonlinear evolution differential equations that admit conserved quantities. This formalism is herein cast in the context of exact finite difference calculus. The algorithms obtained from the proposed formalism are shown to derive exactly from discrete scalar potential functions using finite difference calculus, in the same sense as that of the corresponding differential equation being derivable from its associated energy function (a conserved quantity). A clear ramification of this result is that the derived algorithms preserve certain discrete invariant quantities, which are the consistent counterpart of the invariant quantities in the continuous case. Results on the nonlinear stability of a class of algorithms that are derived using the proposed formalism, and that preserve energy or linear momentum, are discussed in the context of finite difference calculus. Some numerical experiments are presented to illustrate the conservation property of the proposed algorithms.

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The dynamic response of a flexible beam subject to large overall motions is traditionally formulated relative to a floating frame, sometimes referred to as the shadow beam. This type of formulation leads to equations of motion of the form g˜(y˙, y, t) = 0, that are implicit, nonlinear and highly coupled in the inertia terms. An alternative approach is proposed whereby all quantities are referred to the inertial frame. As a result, the inertia term enters linearly in the formulation simply as mass times acceleration. Crucial to this formulation is the use of finite strain rod theories capable of treating finite rotations. Numerical examples that involve finite vibrations coupled with large overall motions are presented in Part II of this paper.