**1. Introduction**

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In this chapter we study three mechanical problems: dynamics of a pendulum of variable length, rotations of a pendulum with elliptically moving pivot and twirling of a hula-hoop presented in three subsequent sections. The dynamics of these mechanical systems is described by similar equations and is studied with the use of common methods. The material of the chapter is based on publications of the authors [1-7] with the renewed analytical and numerical results. The methodological peculiarity of this work is in the assumption of quasi-linearity of the systems which allows us to derive higher order approximations by the averaging method. All the approximate solutions are compared with the results of numerical simulation demonstrating good agreement. Supplementary, in Appendix (section 5) we briefly presented the method of averaging with higher order approximations which is used in sections 2, 3, and 4.
