Contact-constraint forces associated with the non-generalized coordinates and interpretation of their Lagrange multipliers

Non-conformal contact constraints, widely used in engineering applications, are formulated in terms of surface parameters, referred to as non-generalized coordinates. For each contact between two rigid bodies, five contact constraints, written in terms of generalized and non-generalized coordinates, are enforced. This approach leads to five Lagrange multipliers related by nonlinear algebraic equations, making some multipliers dependent. Because four surface parameters are introduced and five constraints are imposed, such a contact formulation eliminates only one degree of freedom, which is the relative motion along the normal to the contact surfaces. This paper establishes a general proof that four of the Lagrange multipliers associated with the five contact constraints are zero in case of scleronomic contact constraints but they may differ from zero in case of rheonomic contact constraints. To this end, the contact constraints are divided into two groups: tangent-plane geometric constraints (TPGC) and normal-vector geometric constraint (NVGC). It is shown, in the general case of three-dimensional free rolling and sliding contact, that the scleronomic TPGC Lagrange multipliers are always zeros, while the NVGC Lagrange multiplier defines the normal contact force. The rheonomic TPGC Lagrange multipliers, on the other hand, can differ from zeros. It is shown that the power of the contact-constraint forces associated with the non-generalized coordinates is always zero regardless of whether the constraints are scleronomic or rheonomic. The details of an example that can be used as benchmark problem in future investigations are worked out.