Joint reactions in rigid body mechanisms with dependent constraints

The problem of calculating joint reaction forces in rigid body mechanisms with dependent constraints is discussed. Two essentially different reasons of constraint dependency are considered: the existence of redundant constraints and singular configuration of the mechanism. It is known that if constraint equations are dependent, the constraint reaction forces cannot be uniquely determined. In this paper it is shown that despite the fact that all of the constraint reactions cannot be uniquely determined, selected single constraint reactions or selected groups of reactions can be specified uniquely. Conditions, which must be fulfilled to obtain unique values of selected reaction forces in the presence of dependent constraints, are presented and proven. The concept of direct sum, known from linear algebra, is exploited. These purely mathematical conditions form a basis for numerical methods that enable detection of constraints with uniquely solvable reactions. Three different numerical methods are proposed. These methods are confined to constraint Jacobian matrix analysis, thus they can be easily implemented in a multibody software. Finally, two examples of detection of constraints with uniquely solvable reactions are presented.