A Recursive Algorithm for Solving the Generalized Velocities From the Momenta of Flexible Multibody Systems

Hamilton’s equations can be used to define the dynamics of a tree configured flexible multibody system. Their states are the generalized coordinates and momenta $(p,q)$⁠. Numerical solution of these equations requires the time derivatives of the states be defined. Hamilton’s equations have the benefit that the time derivative of the system momenta are easy to compute. However, the generalized velocities $q̇$ need be solved from the system momenta as defined by $p=J(q)q̇$ to support the computation of $ṗ$ and the propagation of $q$⁠. Because of the size of $J$⁠, the determination of $q̇$ by linear equation solution schemes requires order $([N+∑i=1Nni]3)$ arithmetic operations, where $N$ is the number of bodies and $ni$ is the number of mode shape functions used to model the $ith$ body deformations. It has been shown that $q̇$ can be solved recursively from the momentum equations for rigid multibody systems (Naudet, Lefeber, and Terze, 2003, “Forward Dynamics of Open-Loop Multibody Mechanisms Using An Efficient Recursive Algorithm Based On Canonical Momenta,” Multibody Syst. Dyn., 10, pp. 45–59). This paper extends that result to flexible multibody systems. The overall arithmetic operations to solve for $q̇$ in this case is proportional to $N$ if the effort to solve for the flexible coordinate rates for each body is weighted the same as that for the joint rate. However, each time the flexible coordinates rate of a body is solved an order $(ni3)$ operations is incurred. Thus, the total computational effort for flexible multibody systems includes an additional order $(∑i=1Nni3)$ operations.