The Use of Zero-Mass Particles in Analytical and Multi-body Dynamics: sphere rolling on an arbitrary surface

Zero-mass particles are, as a rule, never used in analytical dynamics, because they lead to singular mass matrices. However, recent advances in the development of the explicit equations of motion of constrained mechanical systems with singular mass matrices permit their use under certain circumstances. This paper shows that the use of such particles can be very efficacious in some problems in analytical dynamics that have resisted easy, general formulations, and in obtaining the equations of motion for complex multi-body systems. We explore the ease and simplicity that suitably used zero-mass particles can provide in formulating and simulating the equations of motion of a rigid, non-homogeneous sphere rolling under gravity, without slipping, on an arbitrarily prescribed surface. Computational results comparing the significant difference in the motion of a homogeneous sphere and a non-homogeneous sphere rolling down an asymmetric arbitrarily prescribed surface are obtained, along with measures of the accuracy of the computations. While the paper shows the usefulness of zero-mass particles applied to the classical problem of a rolling sphere, the development given is described in a general enough manner to be applicable to numerous other problems in analytical and multi-body dynamics that may have much greater complexity.

ON THE POTENTIALITY OF A CLASS OF OPERATORS RELATIVE TO LOCAL BILINEAR FORMS

The inverse problem of the calculus of variations (IPCV) is solved for a second-order ordinary differential equation with the use of a local bilinear form. We apply methods of analytical dynamics, nonlinear functional analysis, and modern methods for solving the IPCV. In the paper, we obtain necessary and sufficient conditions for a given operator to be potential relative to a local bilinear form, construct the corresponding functional, i.e., found a solution to the IPCV, and define the structure of the considered equation with the potential operator. As a consequence, similar results are obtained when using a nonlocal bilinear form. Theoretical results are illustrated with some examples.

A New Approach in Analytical Dynamics of Mechanical Systems

This paper presents a new approach to the advanced dynamics of mechanical systems. It is known that in the movements corresponding to some mechanical systems (e.g., robots), accelerations of higher order are developed. Higher-order accelerations are an integral part of higher-order acceleration energies. Unlike other research papers devoted to these advanced notions, the main purpose of the paper is to present, in a matrix form, the defining expressions for the acceleration energies of a higher order. Following the differential principle in generalized form (a generalization of the Lagrange–D’Alembert principle), the equations of the dynamics of fast-moving systems include, instead of kinetic energies, the acceleration energies of higher-order. To establish the equations which characterize both the energies of accelerations and the advanced dynamics, the following input parameters are considered: matrix exponentials and higher-order differential matrices. An application of a 5 d.o.f robot structure is presented in the final part of the paper. This is used to illustrate the validity of the presented mathematical formulations.

A Vector Approach for Analytical Dynamics of a System of Rigid Bars

An analytical mechanics approach to derive equations of motion from vector kinematic description of rigid bar assemblies is developed. It is shown that various holonomic constraints governing multibody mechanical systems can be modeled using vector kinematics without using trigonometric/transcendental functions. The principle of virtual work is utilized to derive a map between the generalized coordinates associated with the vector approach and the angular velocity vector associated with the rigid bars. The utility of the vector approach is demonstrated by deriving the dynamics of tensegrity systems and a carpal wrist joint.