Energy decaying scheme for non-linear beam models

This paper is concerned with the time integration of the equations of motion describing the non-linear dynamic response of beams. Desirable characteristics of integration schemes for structural dynamics problems include unconditional stability, accuracy, and high frequency numerical dissipation. Several schemes exist that present all these features when applied to linear problems. Though the application of those schemes to non-linear problems is often successful, proofs of unconditional stability are rarely derived. A finite difference integration scheme is derived in this paper for the non-linear dynamic response of beams. Though of a finite difference nature, the proposed scheme mimics the integration scheme obtained by applying the time discontinuous Galerkin methodology to a single degree of freedom linear oscillator. A formal proof of unconditional stability for the non-linear problem is derived based on an energy decay argument. Numerical examples using the proposed scheme are given, and the results are compared with the predictions of other available schemes. The accuracy of the proposed scheme and its high frequency numerical dissipation characteristics are demonstrated in these examples.