Application of Instantaneous Invariants to the Analysis and Synthesis of Mechanisms

Abstract

The objective of this paper is to make instantaneous invariants a more accessible tool for problem solving in the field of kinematics. We present a systematic procedure to determine the instantaneous invariants of a rigid body under geometric constraint, develop a process by which kinematic properties of rigid body motion can be expressed in terms of instantaneous invariants, and apply instantaneous invariants to solve typical kinematic synthesis problems. Four examples are given in detail for illustrative purposes.

A generalized parameterization of T1 motion and its application to the synthesis of planar mechanisms

“…(8), (12), (13) and (14), namely, b 1 , / 2 , a 3 , b 3 , / 3 , a 4 , b 4 and / 4 , are up to the fourth-order instantaneous invariants of a generally parameterized T 1 motion. These equations are more complex than the results presented in [11,13] since they allow arbitrary values of the dimensionless rotational invariants / 2 , / 3 and / 4 .…”

“…(5), (8), (12), (13) and (14), as a set, are valid only in the canonical system and are the canonical fourth-order instantaneous kinematic equations of a T 1 motion generally parameterized by l.…”

“…The general case of M 1 motion uses the rotation / as the motion parameter and a is taken to be a function of / [5,11,12]. Then a Taylor series expansion of Eq.…”

“…The constant coefficients in the canonical equations of motion are the instantaneous invariants. They are geometric quantities that define the kinematic geometry of M 1 motion and characterize the motion [5,[11][12][13].…”

“…In the general case of planar one dof motion, referred to as M 1 motion, the rotation of the moving body, /, is considered the independent motion parameter [5,11,12]. This paper considers a special case of M 1 motion known as T 1 motion, which occurs when the moving body instantaneously experiences a first-order dwell in its rotation, i.e.…”