Dynamics of mechanical systems with nonlinear nonholonomic constraints – III Analysis of motion

Abstract

The paper analyzes the motion of nonholonomic mechanical systems composed of two particles with imposition of various nonlinear limitations to the velocities of the particles – parallelism of velocities, equality of the intensities of velocities and perpendicularity of velocities. The analysis for such systems includes: equations of constraints, reactions of constraints, i.e. the mode of variations of such constraints, trajectories of the points of the systems, linear integrals for generalized velocities, i.e. cyclical coordinates. It is clearly demonstrated on these models that in the case of nonlinear nonholonomic constraints the Hamiltonian effect, in the general case, has no stationary value. Lastly, the equations of brachistochronic motion of described systems are derived and the brachitochrones of specified points are determined.

Energy balance for nonholonomic nonlinear systems

“…Example 9 We implement (41) for a frequently proposed system ( [16], [2]), consisting in two material points (P 1 , M 1 ) and (P 2 , M 2 ) on a plane whose velocities are both orthogonal to the line joining P 1 with P 2 . The linear kinematic constraints for P 1 ≡ (x 1 , y 1 ) and P 2 ≡ (x 2 , y 2 ) are ẋ1 (x 2 − x 1 ) + ẏ1 (y 2 − y 1 ) = 0, ẋ2 (x 2 − x 1 ) + ẏ2 (y 2 − y 1 ) = 0.…”