Stability of assemblies

right angled component C 1. For the moment, let us ignore frictional forces. C 1 is firmly attached to a gripper, as indicated in the figure by the ground symbol. The vector g indicates the direction of a uniform gravity field. In the orientation shown in Figure 1(a), the assembly is unstable under the influence of gravity. Gravitational forces produce a downward translational motion on component C 2 which causes component C 3 to translate towards the right. Is there an orientation of the assembly in which it is gravitationally stable? The orientation shown in Figure 1(b) is a stable one. The gravitational force on component C 2 is balanced by the internal contact forces exerted by the other two components. The same is true for component C 3. In this paper, we address the problem of the stability of a collection of contacting objects, acted upon by a uniform external gravity field. One (or more) of the objects is assumed to be fixed in a gripper. All the objects are assumed to be rigid and frictionless. Under these assumptions, we solve the following problems: Problem 1: (Stability) Given an assembly in a given orientation , is the assembly stable? Problem 2: (Stable Orientation) Given an assembly, determine an orientation (if one exists) that makes the assembly stable under the gravity field. In the first problem, if we find that the assembly is unstable , we would also like to know which parts in the assembly will have motion and in what direction. This can be used either to redesign parts or provide fixtures during assembly. Figure 1: (a) An assembly that is gravitationally unstable. (b) The assembly in an orientation that makes it gravitationally stable.