On the dynamics of floating four-bar linkages: bifurcations of relative equilibria

In the framework of a program to understand the geometry and dynamics of floating four-bar linkages, the authors explore the relative equilibria of an assembly that admits symmetric configurations. It is shown that a symmetric configuration is a relative equilibrium. As certain kinematic parameters which preserve the symmetry are varied, a symmetric relative equilibrium is bifurcated. The type of bifurcations can be either supercritical or subcritical pitchfork. The stability of the relative equilibria at symmetric configurations is investigated. Elementary techniques of singularity theory are applied in the analysis of the bifurcations. This investigation illustrates the possible rich dynamics in multibody systems with closed-loop structure even with only a small number of degrees of freedom. The observed global connectivity of the bifurcation diagram suggests that it might be possible to change the attitude/orientation/shape of a floating four-bar linkage by merely tuning an appropriate parameter, e.g., a mass off-set