**5. Concluding remarks**

The calculation of dynamic stable conditions and periodic vibrations of elastic mecha‐ nisms and machines is an important problem in mechanical engineering. This chapter deals with the problem of dynamic modelling and parametric vibration of transmission mecha‐ nisms with elastic components governed by linearized differential equations having timevarying coefficients.

Numerical procedures based on Runge-Kutta method and Newmark integration method are proposed and applied to find periodic solutions of linear differential equations with timeperiodic coefficients. The periodic solutions can be obtained by Newmark based procedure directly and more conveniently than the Runge-Kutta method. It is verified that the compu‐ tation time with the Newmark based procedure reduced by about 60%-65% compared to the procedure using the fourth-order Runge-Kutta method (see also Figures 7 and 15). Note that this conclusion is only true for linear systems.

The numerical methods and algorithms are demonstrated and tested by three dynamic models of elastic transmission mechanisms. In the last two examples, a good agreement is obtained between the model result and the experimental result. It is believed that the pro‐ posed approaches can be successfully applied to more complicated systems. In addition, the proposed numerical procedures can be used to estimate approximate initial values for the shooting method to find the periodic solutions of nonlinear vibration equations.
