Nonminimal Generalized Kane's Impulse-Momentum Relations

Abstract

Introduction A methodology for deriving equations of motion for dynamical systems is Kane’s method. A close relationship between the method and those of D’Alembert and of Gibbs and Appell has been noted in the literature.1,2 The impulse–momentum approach to modeling impact was adopted by Kane3 in a general form for both holonomic and nonholonomic impulsive constraints. The basic assumption in the approach is that the duration of the impact is very short compared to the time interval of the motion, so that the impact can be considered a discrete event, and the change in the configuration of the system during impact is ignorable, although the changes in velocities of the system components can be significant.4 This allows converting the differential equations that govern the dynamics of the system to algebraic equations, by integrating the equations in velocities over the infinitesimal time period of impact. The relationships between the velocities before and after impact are given by an experimentally evaluated constant that is dependent on the material and the geometry of the collided bodies and the surfaces of collision, called the coefficient of restitution.5 The impulse–momentum approach to modeling impact was applied to different modeling methodologies. An example is the method of coordinate partitioning,6 where the acceleration form of constraint equations is utilized to eliminate dependent acceleration variables in favor of independent acceleration variables to derive minimal equations of motion. Another example is the Hamilton equations of motion.7 The impulse–momentum approach was followed successfully in the area of multibody system dynamics to model the intermittent motion of both rigid6 and flexible8 systems, but it is interesting to notice that using the approach for multibody system dynamics in the context of Kane’s equations of motion was not done until recently.9 In this Note, the impulse–momentum approach is extended to the nonminimal nonholonomic form by explicitly including the effect of nonholonomic constraints on the rapid changes of the generalized speeds. The nonminimal form of the equations of motion provides a convenient way to analyze the intermittent motion of both nonholonomic systems and complex holonomic systems with numerous configuration settings but relatively low numbers of degrees of freedom. The latter case pertains to analyses in which pseudogeneralized coordinates (i.e., additional configuration variables) are needed to facilitate the formulation, and hence more holonomic constraints are added. In the next section, nonholonomic generalized impulses and momenta are defined and are related to their holonomic counterparts by means of the constraint matrix.