Equations of motion for singular systems of massed and massless bodies

Abstract

In many cases, mechanical systems include elements that differ greatly in inertia characteristics. It seems to be quite natural for a researcher who has to deal with such a system to have the desire to neglect its comparatively small inertia characteristics by putting them equal to zero. On such a simplification, the researcher has to do with a mechanical system that, along with 'massed' bodies (all inertia characteristics of which are distinct from zero), also includes 'massless' bodies (some inertia characteristics of which are zero). An important feature of systems of massed and massless bodies is that they may turn out to be singular. The analysis of systems of this type, in comparison with regular ones, involves some additional problems, whose solution, despite currently available methods of study of singular and singularly perturbed equations, may present a considerable challenge.In the current paper, the notion of the 'rank of a system of massed and massless bodies' is introduced, and an approach to the solution of the above problems based on this notion is proposed. This approach makes it possible to write generalized Lagrange equations for singular mechanical systems, to identify conditions for going from singularly perturbed equations of motion to singular ones to be correct, to identify conditions for the unique existence of the solution of the resulting singular equations of motion and to write them in normal form.

Stabilization of singularly perturbed systems using acceleration-dependent forces

“…Let the singular perturbance of the system be due to the smallness of the parameter μ. Then, as shown in [2], the position of the heavy bodies can be described using r = n − e generalized coordinates, where e is the excess of the inertia matrix at μ = 0. Since by virtue of the singular perturbance of the system e > 1 [15], r < n. Let the vector x = (x 1 , x 2 , .…”

“…The preceding reasoning is based on the fact that the excess e of the inertia matrix at μ = 0 is a constant that is different from zero in some vicinity of q, i.e. a 'constant-rank' system is considered [2]. However, the proposed approach remains valid for 'variable-rank' systems too [2], where the excess of the inertia matrix in the position q is less than in its vicinity.…”

“…Although the Lagrange equations of holonomic systems are always regular, their inertia matrix may be ill-conditioned, and the equations themselves may be singularly perturbed [2], and thus stiff. As a consequence, even relatively small errors in the parameters of the control actuators and moderate perturbing actions (permanent perturbations) may result in a significant difference between the calculated and the actual motions of the system.…”

“…To accomplish this, add an acceleration-dependent term to the right-hand side of equation (2), that is, let…”

“…a 'constant-rank' system is considered [2]. However, the proposed approach remains valid for 'variable-rank' systems too [2], where the excess of the inertia matrix in the position q is less than in its vicinity. For the first part of statement 4 to be true it is necessary that the parameters τ and ε be in a certain ratio.…”