Implementation of the Boundary Element Method in the Dynamics of Flexible Bodies

Abstract

SUMMARYThis paper presents the implementation of the Boundary Element Method in the dynamics of flexible multibody systems. Kane's equations are used to formulate the governing boundary initial value problem for an arbitrary three-dimensional elastic body subjected to large overall base motion. Using continuum mechanics principles, direct boundary element incremental formulations are derived. The Galerkin approach was employed to generate the weighted residual statement which serves as a transitory point between continuum mechanics and boundary integral equations. By adapting the updated Langrangian formulation for large displacements analysis and using the Maxwell-Betti reciprocal theorem, integral representations for geometric stiffening were also derived. The non-linear terms were found to be functions of the time-variant stresses associated with the inertial forces at the reference configuration. The domain integrals arising from body forces (such as gravitational loads, inertia loads and thermal loads, etc.) are presented as DRM integrals (Dual-Reciprocity Method). Using the substructuring technique the elastic body is divided into several regions leading to a system of equations whose matrices are sparse (block-banded). The linearized equations of motion were discretized along the boundary of the body, and an algorithm for the integration involving the Houbolt method was used to establish an algebraic system of pseudo-static equilibrium equations. A Newton-Raphson-type iteration scheme was used to solve these discretized balance equations. To take advantage of the sparsity of the matrices, special routines were used to decompose and solve the resulting linear system of equations.An illustrative example is presented to demonstrate the validity of the method as well as how the effects of geometric stiffening effects are captured. The example consists of spin-up manoeuvre of a tapered beam attached to a moving base. The beam was modelled as two-dimensional plane strain problem divided into a number of substructures. Numerical simulation results show how the phenomenon of dynamic stiffening is captured by the present approach.

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