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To achieve the performance of a mechanism to a higher degree of accuracy requires that the elastic deformations of a member in a mechanism under dynamic loading conditions be taken into account. Coupled nonlinear governing partial differential equations have been derived for transverse and longitudinal vibrations of an elastic connecting rod in a slider-crank mechanism operating at high speed conditions. The derived coupled governing nonlinear partial differential equations of motion were transformed into ordinary differential equations by use of the Kantorovich method and the method of weighted residuals. The resulting coupled ordinary differential equations were solved numerically by use of the piecewise polynomial method and the fourth-order Runge-Kutta method. The dynamic response of the system has been investigated on the basis of natural frequencies of the first mode free vibrations, the ratios of the length of crank to the length of connecting rod, viscous damping, and rotating speeds of crank. These parameters can be used by the designer to predict the vibrations of an elastic mechanism under high-speed conditions.

The problem of two normally intersecting cylindrical shells subjected to internal pressure is considered. The differential equations used for the shells are solved subject to the boundary conditions imposed along the intersection between the two cylinders. Details of a procedure for obtaining a numerical solution are given. Numerical results for a radius ratio of 1:2 are presented. Problems encountered in the numerical computation are discussed and the results of the analysis are compared with experiment.