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Equilibrium paths of mechanical systems with unilateral constraints I. Theory

The paper shows that the behaviour of mechanical systems subject to unilateral constraints differs from that of standard systems in subtle, and yet important, ways. Therefore, a proper theoretical formulation is required for simulating their behaviour. After showing that the equilibrium equations for a multibody system subject to unilateral constraints have the same form as the standard Kuhn–Tucker conditions in optimization theory, the first–order equilibrium equations are derived and their integration is discussed. At a general integration step, one has to distinguish between constraints that are strongly active, weakly active and inactive. Whereas strongly active constraints can be treated like bilateral constraints and inactive constraints can be neglected, weakly active constraints need to be constantly re–analysed to determine if they switch to a different state. The outcome is that, in addition to the well–known limit points and bifurcation points, a new type of limit point can exist, where the path is non–smooth and the first–order equilibrium equations—after elimination of any strongly active constraints—non–singular. Such points are called corner limit points. In analogy with common limit points, the degree of instability of the system changes by one at a corner limit point.