A procedure based on finite elements for the solution of nonlinear problems in the kinematic analysis of mechanisms

Abstract

In the present paper the kinematic analysis of mechanisms is based on the application of finite elements is discussed. It is shown how the kinematic properties of the rigid-body motions of a mechanism can be obtained from an analysis of the stiffness matrix of a simple model comprising rod-type elements in the case of planar mechanisms. In the event that there is also a more complex finite element model of the mechanism, onemay in addition obtain thenode values from the results achieved with the simple model. Special attention is given to nonlinear position problems, i.e. initial, successive, deformed, and static equilibrium. An error function is provided that is valid in each case. This function is derived from the elastic potential function, and uses Laggrange multipliers and penalty functions. The result is an application of the primal-dual method, or augmented Lagraange multipliers (ALM) method. This function is minimized by means of Newtons' method, which leads in simple form to the vector gradient as a force vector. The second-derivative matrix is derived from the stifness matrix, to which a complementary matrix owing toe nonlinearity introduced by the large displacements is added. This method can be easily implemented on a computer. The computer program will be able to perform a wide variety of kinematic analyses of any planar mechanism with lower pairs. The models of the mechanism are very simple, and need only a few tens of degrees of freedom even for the most complex mechanisms. The CPU time is also very low due to the simplicity of the method and its good convergence properties.