Deformation basis and kinematic singularities of constrained systems

Abstract

This paper demonstrates that arbitrarily introducing deformation degrees of freedom does not always solve the kinematic singularities of over-constrained systems. To address this fundamental problem, this paper proposes different methods that can be used to eliminate the initial-configuration singularity of over-constrained systems when modeled using multibody system formulations such as the general nonlinear finite element floating frame of reference (FFR) formulation. It is shown that the constraint Jacobian matrix of flexible-body models, which are not topologically over-constrained, can be singular. This fundamental problem, encountered in several heavily constrained flexible multibody system applications, such as vehicles, parallel mechanisms, and industrial robots, cannot be solved by automatic elimination of redundant constraints, a feature offered by several commercial multibody system computer programs. Automatic elimination of redundant contraints in systems which are not topologically over-constrained can lead to topological changes, resulting in different solutions with different frequency contents. A simple static-force benchmark problem, with known solution, is used in this paper to shed light on the effect of using different sets of deformation basis vectors, defined using the FFR reference conditions, on the rank of the constraint Jacobian matrix. Three methods for the systematic elimination of the initial-configuration rank deficiency of the constraint Jacobian matrix are proposed. The first method is the modal perturbation method in which selected elements of the modal transformation matrix are perturbed to preserve the system topology and the solution accuracy. The second method defines the deformation basis vectors using the new geometrically accurate absolute nodal coordinate formulation (ANCF)/FFR finite elements. These elements can lead to coupled axial and bending deformation basis vectors, which cannot be obtained using conventional finite elements. In the third method, an equivalent system that has the same solution as the original problem is developed by using a proper set of multibody system constraint equations. The results obtained in this investigation confirm the fact that the choice of the deformation basis vectors should not be restricted to one set, such as the free-free modes, to avoid obtaining incorrect results.