Multibody Dynamics Problems in Natural Coordinates: Theory, Implementation and Simulation

Abstract

We present a framework for modeling multibody systems based on the method of natural coordinates and Lagrange's equation of the first kind, resulting in a system of Differential-Algebraic Equations (DAEs). The C++ package DAETS (DAEs by Taylor Series), a robust high-index DAE solver, is utilized to solve the models. The simulation process is straightforward, with no need to derive equations of motion directly. Instead, the user supplies a Lagrangian, kinematic constraints, and if applicable, a dissipation function and external forces. A corresponding system of DAEs is formed by computing the required derivatives via automatic differentiation. DAETS primarily uses Cartesian coordinates as variables, eliminating angles and the associated trigonometric functions, which results in simplified models. Furthermore, DAETS provides direct access to the position/velocity data of any desired points or vectors as output, facilitating post-processing tasks, such as visualization. The main focus of this thesis is on establishing the viability of our framework through case studies. We simulate seven multibody systems and compare our results with those of reference models developed in the Simulink environment of MATLAB. A detailed account of the modeling process is given for each system, demonstrating the ease and intuitiveness of our approach. We also provide, from both DAETS and Simulink, the time history plots of several position coordinates to allow for direct comparison. Finally, we compute two types of errors over time. Our findings show that the results of DAETS match those of the reference models under different error tolerances for the studied systems, indicating that our framework is capable of simulating a wide variety of mechanisms with a superb degree of accuracy.