Asymptotic Analysis of Resonances in Nonlinear Vibrations of the 3-dof Pendulum

The dynamical response of a harmonically excited three degrees-of-freedom planar physical pendulum is studied in the paper. The investigated system may be considered as a good example for several engineering applications. The asymptotic method of multiple scales (MS) has been adopted in order to carry out the analytical computations. The solutions up to the third order have been achieved. MS method allows to identify parameters of the system being dangerous due to the resonances and yields time histories for the assumed generalised co-ordinates. Three simultaneously occurring resonance conditions have been analysed. The energy transfer from one to another mode of vibrations is illustrated and discussed. The modulation equations in an autonomous form allow obtaining the frequency response functions and drawing resonance curves.