Analyzing the motion of a forced oscillating system on the verge of resonance

In this paper, a nonlinear vibrating dynamical system of two degrees-of-freedom is examined. By virtue of the generalized coordinates, the controlling system of motion is constructed using Lagrange's equations. Among other perturbation approaches, the technique of multiple scales (TMS) is utilized to get the required solutions analytically of this system till the third approximation. These solutions allow us to discuss the system's behavior and realize both the solvability requirements and the modulation equations in the context of the system's resonance scenarios. The resonance curves and solution diagrams are drawn to demonstrate the influence of the various parameters of the behavior of the considered system. The stability and instability of the gained possible fixed points are examined. The used method is considered traditional, but it is applied to a certain specified model. Therefore, the obtained outcomes are considered novel. The importance of this work is represented in its various applications in practical life, such as the motion of railway vehicles, for transporting goods in ports, and for facilitating the movement of passengers' baggage at airports.

Automated summary

This paper examines a nonlinear vibrating dynamical system of two degrees-of-freedom, constructs the controlling system of motion using Lagrange's equations by virtue of the generalized coordinates. The technique of multiple scales (TMS) is utilized to analytically obtain the required solutions of the system till the third approximation. The obtained outcomes, although using a traditional method, are considered novel as they are applied to a specific model. The importance of this work lies in its various applications in practical life, such as the motion of railway vehicles, transportation of goods in ports, and facilitating the movement of passengers' baggage at airports.