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.

本文研究了具有 3DOF 的双非线性阻尼弹簧摆的运动，其中其枢轴点沿椭圆轨迹移动。运动控制系统是应用拉格朗日方程推导出来的，并使用多尺度技术渐近求解到第三个近似值。根据获得调制方程的可解性条件对产生的共振情况进行分类。检查了符合稳定性和不稳定性准则的稳态解。与以前的工作相比，这项工作的新颖之处在于获得了高精度的解决方案，并研究了非线性稳定性分析以识别稳定和不稳定区域。所获得结果的时间历程、响应共振曲线、并讨论了稳定区域并以图形方式表示，以探索不同参数对所研究系统动力学行为的良好影响。这项工作的重要性在于它在各种工程应用中的广泛用途。