Dynamics of mechanical systems with nonlinear nonholonomic constraints – II Differential equations of motion

Abstract

Depending on how the nonholonomic constraints have been introduced to the Lagrange‐D'Alemberts's principle, there are several differential equations of motion in the mechanics of nonholonomic systems. In this work, the most general type of differential equations of motion (fundamental to all known forms of the equations of motion for nonholonomic as well as holonomic systems) is derived. Here, the equations represent the generalization of Poincare's equation [1]. In published works [2, 3, 4, 5, 6], these have already taken into account nonlinear nonholonomic constraints and linear relations between real velocities and kinematic parameters. A method of dedication of the most generalized form of the equations of motion will be shown. It is followed by the analysis of particular cases. Then, it will be shown how to get form the generalized form to Maggi, Appell, Voronec, Chaplygin, Volterra, Ferrers, and Boltzmann‐Hamel's equations appearing in nonholonomic systems. Further, a system of material points of variable mass, where the equations of motion are derived for the most general case of reactive forces and in case of constraints depending on mass variables will be considered. All theoretical considerations are illustrated with the analysis of the relevant nonholonomic model.

Energy balance for nonholonomic nonlinear systems

“…The constant of motion (41) reads I = 1 2 (M 1 + M 2 ) ℓ 2 q2 1 + (1 + tan 2 q 1 ) q2 2 + (M 1 + M 2 g)q 3 . We incidentally remark that the model is different form the one corresponding to the requests P 1 P 2 = ℓ, | Ṗ1 | = | Ṗ2 | (equidistant points and equal intensity of velocities, see [15]): actually, in that case the restrictions yield to the nonlinear condition q1 ( q2 sin q 1 − q3 cos q 1 ) = 0.…”

“…• Equations ( 12) correspond to the ones derived in [15], as the most general form of equations of motion in Poincaré-Chetaev variables extended to nonlinear nonholonomic systems; the Voronec's equations ( 12) are the same as the Voronec's equations pointed out in [15] as the special case of Poincaré's kinematic parameters chosen as the real generalized velocities. Also the geometric approach for nonholonomic machanical systems (Lagrangian systems on fibered manifolds) performed in [10] leads to the same equations of motion as (12).…”