Evaluation of the stability of a two degrees-of-freedom dynamical system

• Published in: Journal of low frequency noise, vibration, and active control Vol. 42; no. 4; pp. 1578 - 1595
• Main Authors: Amer, TS, Ismail, AI, Amer, WS
• Format: Journal Article
• Language: English
• Published: London, England SAGE Publications 01.12.2023; Sage Publications Ltd; SAGE Publishing
• Subjects: Degrees of freedom; Dynamic stability; Dynamical systems; Exact solutions; Motion stability; Resonance; Runge-Kutta method; Shock absorbers; Stability analysis; Stability criteria; perturbation techniques; resonance; stability; Non-linear dynamics; vibrating motions
• ISSN: 1461-3484; 2048-4046
• DOI: 10.1177/14613484231177654

This work studies a two degrees-of-freedom (DOF) dynamical system whose governing system is solved analytically using the multiple scales approach (MSA). The solvability requirements are obtained in light of the elimination of secular terms. All resonance states are classified to understand the equilibrium of the dynamical system. Two of them are examined in parallel to get the associated equations for the system’s modulation. All probable fixed points are identified at the states of stability and instability using the criteria of Routh-Hurwitz (RH). The curves of resonance and the system’s behavior during the motion are plotted and analyzed. The numerical solutions (NS) of the governing system are obtained using the method of Runge-Kutta fourth-order, and they are compared with the analytical solutions (AS). The comparison reveals high consistency between them and proves the accuracy of the MSA. To determine the positive effects of different parameters on the motion, stability zones are studied from the perspective of their graphs. The applications of such works are very important in our daily lives and were the reason for the development of several things, including protection from earthquakes, car shock absorbers, structure vibration, human walking, television towers, high buildings, and antennas.