A new C1-continuous variational integration scheme for mechanical systems subjected to acceleration-dependent forces

Abstract

The C¹-continuity of calculated state curves is crucial for engineering problems subject to acceleration-dependent forces, such as hydrodynamic loads and control forces. Conventional variational integration schemes with prominent energy and momentum preserving properties are favored to calculate the dynamics of mechanical systems, however, can sometimes lose their efficacy due to a lack of continuity. In this work, a novel C¹-continuous variational integration scheme is developed within a simple and general construction framework, ensuring the continuity of the generalized coordinates and their first derivative at the discrete-time points. This scheme is constructed by approximating generalized coordinates and velocities using Hermite polynomials within a certain time span with the action integral computed numerically. This framework greatly simplifies the derivation and implementation by avoiding the summation of discrete node variations, and it is also suitable for constructing other variational schemes based on Lagrangian polynomials of various orders. The algorithmic characteristics, including stability, dissipation, period elongation, and convergence order, are theoretically analyzed. The momentum-preserving and nearly energy-preserving properties are numerically demonstrated. Moreover, practical engineering problems subject to acceleration-dependent forces are investigated, which have well confirmed the feasibility of the proposed C¹-continuous variational scheme in practical dynamic analyses.