The stability analysis for the motion of a nonlinear damped vibrating dynamical system with three-degrees-of-freedom

This article investigates the motion of double nonlinear damped spring pendulums with 3DOF in which its pivot point moves in an elliptic trajectory. The controlling system of motion is derived applying the Lagrange's equations and solved asymptotically up to the third approximation utilizing the multiple scales technique. The arising resonance cases are categorized in view of the solvability conditions in which the equations of modulation are obtained. The steady-state solutions in accordance with the stability and instability criteria are examined. The novelty of this work lies in obtaining high accuracy solutions compared with the previous works and in studying of the nonlinear stability analysis to identify the stability and instability areas. The time histories of the attained results, the response resonance curves, and the regions of stability are discussed and represented graphically to explore the good influence of the distinct parameters on the dynamical behaviour of the investigated system. The importance of this work is due to its great uses in various engineering applications.