On the solutions and stability for an auto-parametric dynamical system

The main goal of this study is to look at the motion of a damped two degrees-of-freedom (DOF) auto-parametric dynamical system. Lagrange's equations are used to derive the governing equations of motion (EOM). Up to a good desired order, the approximate solutions are achieved utilizing the method of multiple scales (MMS). Two cases of resonance, namely; internal and primary external one are examined simultaneously to explore the solvability conditions of the motion and the corresponding modulation equations (ME). These equations are reduced to two algebraic equations, through the elimination of the modified phases, in terms of the detuning parameters and the amplitudes. The kind of stable or unstable fixed point is estimated. In certain plots, the time histories graphs of the achieved solutions, as well as the adjusted phases and amplitudes are used to depict the motion of the system at any instant. The conditions of Routh-Hurwitz are used to study the various stability zones and their analysis. The achieved outcomes are considered to be novel and original, in which the used strategy is applied on a particular dynamical system. The significance of the studied system can be observed in its applications in a number of disciplines, such as swaying structures and rotor dynamics.

Automated summary

The main goal of this study is to investigate the motion of a damped two degrees-of-freedom auto-parametric dynamical system. Lagrange's equations are used to derive the equations of motion, and the method of multiple scales is employed to obtain approximate solutions. The study examines two cases of resonance and explores the solvability conditions and modulation equations. The stability of fixed points is estimated, and plots are used to depict the motion of the system. Routh-Hurwitz conditions are used to analyze stability zones. The outcomes of this study are considered novel and original, with applications in various disciplines.