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

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.

Automated summary

This work analytically studies a two DOF dynamical system using the MSA. Solvability requirements and resonance states are obtained, and stability and instability states are identified using RH criteria. The curves of resonance and system behavior during motion are plotted and analyzed. Numerical solutions using the Runge-Kutta fourth-order method are compared with analytical solutions, proving the accuracy of the MSA. Stability zones are studied to determine the positive effects of different parameters on motion.