Symbolic Approach in Modeling of Flexible Multi-body Systems Using Absolute Nodal Coordinate Formulation (ANCF) Cable Element

The simulation of multibody systems (MBS) is crucial in product design and development, especially for complex assemblies with joints and links that exhibit both rigid and deformable characteristics, leading to significant deflections. This work focuses on solving a flexible MBS simulation problem that involves a set of nonlinear equations using a symbolic approach. This way, we directly compute the Jacobian matrix of generalized forces, avoiding the step of computing the stiffness matrix, which can be a cumbersome task when dealing with exact curvature expressions. Symbolic approach allows us to apply the Newton-Raphson (N-R) method directly to solve the set of nonlinear equations by using Jacobian matrix. We may compute generalized elastic force from elastic strain energy by taking the derivative with respect to generalized nodal coordinate. Compute the expression symbolically for the generalized elastic force vector and it’s Jacobian with respect to generalized nodal coordinate. After computing the symbolic solution, make functions from the symbolic solution to apply the N-R method. These functions avoid the substitution of symbolic variables to numeric values and in this way speed up the simulation time.