A unified analytical expression of the tangent stiffness matrix of holonomic constraints

While an accurate computation of the tangent stiffness matrix of multibody systems is usually of critical importance in various numerical analyses, one contribution often neglected in dynamics is represented by the tangent stiffness matrix of constraints. This term, resulting from the change of joints reactions with respect to the coordinates of the connected bodies, is indeed easily discarded for the sake of performance due to its slight effect on dynamic simulation results. On the contrary, it plays a critical role when static or eigenvalue analyses are required, especially for those cases featuring free motions. While the topic of holonomic constraint equations and their respective Jacobian matrix is not new in literature, this article aims to provide a general and unified formulation based on quaternion parametrization together with a consistent analytical expression of the tangent stiffness matrix derived through linearization. The article includes also the rheonomic contribution to holonomic constraints. The formulations presented in this work are built on a mixed-basis formulation, in use in many engineering applications, and allow to easily derive specialized versions (e.g. revolute, cylindrical, prismatic joints, etc.) from the same equation set. Examples demonstrate the doubly-positive effect of this additional stiffness term: first, static analyses are able to converge to the actual equilibrium position and second, the eigenvalues analyses proved to be more consistent. For this latter set of tests the non-linear dynamic results, observed in the frequency domain, are compared against those coming from eigenvalue analysis, in order to prove the augmented accuracy of the results.

Automated summary

This article discusses the importance of the tangent stiffness matrix of constraints in multibody systems and provides a general formulation based on quaternion parametrization. The article also presents the analytical expression of the tangent stiffness matrix derived through linearization. Examples demonstrate the positive effect of this additional stiffness term on static and eigenvalue analyses.