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**Author details** 

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**1. Introduction**

http://dx.doi.org/10.5772/48168

**and Similar Problems** 

A. O. Belyakov and A. P. Seyranian

Additional information is available at the end of the chapter

sections 2, 3, and 4.

presented.

system.

**2. Pendulum with periodically variable length**

work is properly cited.

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

**Dynamics of a Pendulum of Variable Length** 

**Chapter 4**

Oscillations of a pendulum with periodically variable length (PPVL) is the classical problem of mechanics. Usually, the PPVL is associated with a child's swing, see Fig. 1. Everyone can remember that to swing a swing one must crouch when passing through the middle vertical position and straighten up at the extreme positions, i.e. perform oscillations with a frequency which is approximately twice the natural frequency of the swing. Among previous works we cite [8–15] in which analytical and numerical results on dynamic behavior of the PPVL were

The present section is devoted to the study of regular and chaotic motions of the PPVL. Asymptotic expressions for boundaries of instability domains near resonance frequencies are derived. Domains for oscillation, rotation, and oscillation-rotation motions in parameter space are found analytically and compared with numerical study. Chaotic motions of the pendulum depending on problem parameters are investigated numerically. Here we extend our results published in [1–4] in investigating dynamics of this rather simple but interesting mechanical

and reproduction in any medium, provided the original work is properly cited.

©2012 Seyranian and Belyakov, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original

© 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,

