Stability of three degrees-of-freedom auto-parametric system

This paper focuses on studying the dynamical response of a vibrating three degrees-of-freedom (DOF)auto-parametric system near resonance. The structure of this system is composed of an attached damped oscillator with a damped spring pendulum. The governing equations of motion are derived using Lagrange’s equations of second kind. They are asymptotically solved using the multiple scales approach to obtain the analytic solutions up to the third approximations as new and accurate results. The resonance cases are classified and the effect of the different parameters of considered system is analysed. The stability and instability regions are examined in which the behavior of the system is found to be stable for a wide range of parameters. The achieved results reveal that we can use the pendulum as a dynamic absorber. The significance impact of this work is due to its great engineering applications in the high towers, buildings and bridges.