Abstract Summary
Models of multibody mechanical systems subject to motion constraints are employed in several areas of large engineering importance, like automotive, railway, marine and aerospace structures. A better understanding of the dynamics of these models leads to better designs. However, investigation of their dynamics is still a quite challenging engineering task. This is mainly due to the fact that the equations of motion governing the behavior of such systems are represented by a set of differential algebraic equations (DAEs) of high index. Since the treatment of these equations is a delicate task, much research effort has been devoted to the subject, in order to cure the associated problems, like constraint violation, leading to a gradual drift of the numerical solution from the exact response. Essentially, all the previous efforts at tempt to overcome these problems by application of index reduction or coordinate partitioning techniques. The analysis performed in the present work is characterized by three distinct features. First, the equations of motion before enforcement of the motion constraints are linear second order ordinary differential equations (ODEs). This is in contrast to most of the previous studies on the subject, where the underlying set of equations is strongly nonlinear, due to consideration of large rigid body rotations. Moreover, the motion constraints are also expressed in a linear form with respect to the coordinates. Finally, the present analysis is based on an appropriate set of equations of motion, which are expressed as a system of second order ODEs in both the original generalized coordinates and the Lagrange multipliers related to the constraint action. This was achieved in some recent work of the authors, by combining some fundamental concepts of Analytical Dynamics and Differential Geometry. Specifically, this causes a natural elimination of singularities associated with DAE formulations and leads to major advantages compared to previous work in computational Multibody Dynamics. Due to the linear nature of the class of systems examined, a modal analysis becomes applicable. Therefore, the analysis of the eigenvalue problem arising from the set of the equations of motions employed is of vital importance. First, it is shown analytically that this problem possesses two separate sets of eigenvalues. The first corresponds to a set of single eigenvalues, which are shown to coincide with the set of eigenvalues of the reduced system, resulting after elimination of the motion constraints. The second set of eigenvalues is related to the motion constraints directly. In this set, all the eigenvalues are double and possess a single eigenvector (that is, they have algebraic multiplicity two and geometric multiplicity one). In sharp contrast to previous work, all these eigenvalues are bounded, due to the inertia assigned to the La grange multipliers. Based on these results, the solution of the undamped problem is performed first, using a generalized modal analysis. Then, the solution of the corresponding problem with classical damping is also determined. Finally, the accuracy of the results obtained is checked and verified by comparison with results obtained by application of classical numerical methods.
