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Autoparametric vibrations, which may occur within mechanical structures and machinery, are studied by several researches [1, 2]. An oscillator with an attached pendulum is an example of the autoparametric system which is used in many mechanical and civil engineering applications. Gantry cranes, lifts or special dynamical absorbers, mounted in buildings and working as dynamical dampers against earthquake, are classical examples where interactions between the support and the pendulum occurs [3]. The phenomenon of vibrations absorption of the mass—spring oscillator can be achieved due to proper pendulum swinging. However, for some parameters the situation may worsen and pendulum vibrations may increase dramatically, and then the protection of the structure (modelled as a mass-spring oscillator) is lost. Response of the system can be regular or, under some circumstances, may become chaotic [4]. To avoid such dangerous situations a special suspension of the system is analysed in this paper.
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