Kane’s equations for nonholonomic systems in bond-graph-compatible velocity and momentum forms

Bond-graph Representation of Ideal Constraints Via the Principle of Virtual Power

“…This allows the physical source of power delivered by a rheonomic constraint to be identified, enabling a calculation of the energy cost of the constraint over a given time interval. An alternative to adjoining rheonomic constraints with Lagrange multipliers is to employ constraint-adapted generalized velocities in the f matrix, together with the method of constraint relaxation [1,15]. The constraint reactions, formerly supplied by the Lagrange multipliers, then become available at the end of the analysis, when the constraints are enforced.…”

“…The details will be illustrated in several examples. (A general approach is presented in [1], although this is limited to systems without adjoined constraints. )…”

“…With the matrices A and D in hand, one may also state the matrix form of Hamilton's equations for this problem as p = D T q + e, where p = A q [1].…”

“…In a recent journal paper [1], the authors reported (but did not prove) that for a general system of constrained particles, having possible velocities represented by a partial velocity matrix, the partial velocity matrix is the modulus of a multibond graph [2] modulated transformer element, which represents the subsystem of embedded constraints described by the partial velocity matrix, in the sense that the effort outputs of the transformer element are the particlelevel constraint forces for the embedded constraints. Although it is not the primary contribution of the paper, this fact is previously unreported in the literature.…”