A Variational Approach for the Design of Spatial Four-Bar Mechanism*

Abstract

ABSTRACT An analytical formulation for computing kinematic sensitivity of the spatial four-bar mechanism is presented. The experimental code developed is used to compute an assembled configuration for the mechanism that accounts for the effect of a design variation. The mechanism is modeled using graph theory, in which a body is defined as a node and a kinematic joint is defined as an edge. The spherical joint is cut to convert the model into a tree structure by cutting an edge and introducing constraints. The effect of variation in mechanism design using concepts of virtual displacement and rotation is introduced. The variation of the spherical constraint is computed, maintaining joint-attachment vectors and orientation matrices as variables. A system of equations that has a greater number of design variables than equations is then solved using the modified Moore-Penrose pseudo inverse. A recursive formulation is introduced, which can be used to obtain the state variation of a body in terms of the state variation of a junction body and of the relative coordinates along the chain. The Jacobian matrix is then transformed from Cartesian coordinate space to joint coordinate space, using velocity transformation matrices. Kinematic sensitivity analyses that take into account changes in the joint-attachment vector and orientation matrix are presented