Kinematics of Constrained Bodies

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Chapter 7 Kinematics of Constrained Bodies

In this chapter, we analyze the motion of a rigid body \(\cB\) whose mobility is restricted in some particular manner relative to some referential \(\cE\text{.}\) More generally, we consider mechanisms, that is, systems of rigid bodies \(\{\cB_1 , \cB_2 , \ldots , \cB_n \}\) interconnected in such a way that motion is transmitted from an input body to an output body. In such assemblies, rigid bodies are not free to move in all possible directions relative to one another, but rather only specific desirable motions are wanted. Certain unwanted motions between any two rigid bodies \(\cB_i\) and \(\cB_j\) are prevented by the use of mechanical constraints or joints. We will first describe the various ways in which two rigid bodies are interconnected and their resulting kinematics. Such interconnections will be called kinematic pairs. A systematic study of all possible kinematic pairs is not possible. However, special classes of kinematic pairs can be defined and studied, as was recognized long ago. We will pay particular attention to the kinematics of rigid bodies (i) in point contact and (ii) in planar motion.

Throughout this chapter, to simplify notations, we denote \(\{ \cV_{j/i} \}\) the kinematic screw of body \(\cB_j\) relative to body \(\cB_i\text{,}\) \(\bom_{j/i}\) the corresponding angular velocity, \(\vel_{P\in j/i}\) the velocity of a point \(P\) of \(\cB_j\) relative to \(\cB_i\text{,}\) etc. We may in fact simply denote a system \(\Si\) of rigid bodies as a set 1,2, ..., N in motion relative to a referential 0.