Dynamics of constrained mechanical systems around an equilibrium position

Abstract

This study focuses on a special class of multibody dynamic systems, involving bilateral motion constraints. Specifically, both the equations of motion and the equations of the motion constraints appear in a linear equality form. First, following an appropriate Analytical Dynamics approach, the dominant dynamics of the systems examined is eventually represented by a coupled set of second order linear ordinary differential equations for both the system coordinates and the Lagrange multipliers corresponding to the motion constraints. This allows a thorough investigation of the dynamics, based on classical linear procedures. For this, the structure of the corresponding undamped eigenvalue problem is revealed first in a complete form. This opens the way to determine the response of the undamped system by applying an appropriate modal analysis technique. Next, damping terms leading to classical modes of vibration are also included in the governing equations and their effects are studied in detail. Finally, the analytical results are complemented and illustrated further by considering a typical set of mechanical examples. Besides predicting the dynamics of linear constrained systems, the new methodology provides a strong basis for investigating in depth several related theoretical and technical issues, like the stability properties of equilibrium solutions of mechanical systems or of numerical schemes applied to the solution of multibody systems with nonlinear characteristics.