A study of the geometry and the kinematics of instantaneous spatial motion

Abstract

The geometry and the kinematics of instantaneous general spatial motion is presented in a new way using elementary methods in vector algebra. Known results are derived together with some new results. Special cases of three dimensional motion, including that of planar motion, are discussed. The geometry and the kinematics of the relative motion between two bodies are used as alternative means of deriving expressions for the properties of the paths instantaneously traced in the fixed body by points attached to the moving body. These expressions are used to investigate special surfaces in the moving body which are characterised by one or other property of the tangential or normal component of acceleration of the points contained in them. In a body undergoing what has been called “a general spatial motion” it is found that the centre of acceleration is a single point, which is unique at each instant. Under special conditions it is found that there may be no centre of acceleration or that there may be an axis of zero acceleration in the moving body. New results, relevant at an instant of the motion, include: (i) an expression for a family of ellipsoids characterised by the magnitudes of the total accelerations of points in the moving body; and (ii) expressions for the first curvature of the paths in the fixed body traced by points attached to the moving body. The significance of the axodes and their possible use in the synthesis of mechanism is discussed.