Vibrational dynamics of a subjected system to external torque and excitation force

This work focuses on the dynamical response of a novel auto-parametric two-degrees-of-freedom (DOF) dynamical system when it is subjected to external torque and force. It consists of a forced oscillator with nonlinear stiffness that is connected to a linear spring. In accordance with the system’s DOF, an organizing system of equations of motion (EOM) is produced using Euler–Lagrange’s equations. To obtain the analytical approximate solutions (ASs) of the equations of this system, the multiple scales technique (MST) is employed. The provided solutions are being juxtaposed with the numerical solutions (NSs) for comparison to show how closely they agree, as well as the precision of the MST. In view of removing secular terms, the system’s solvability criteria are obtained. All arising resonance cases are classified, and two of them are then examined at once. The relevant system of modulation equations is solved numerically to illustrate the advantageous influence of diverse factors on the behavior of the modified phases and amplitudes. Based on the values of these parameters, the frequency response curves (FRCs) are drawn to explore different zones of stability and instability. A wide variety of values for the system’s parameters shows that its behavior is steady. The achieved outcomes are deemed novel, as the MST has been applied on a new dynamical model. The results have broad relevance and can be implemented in various engineering contexts, including the analysis of ship movements, in which nonlinear coupling between rolling and pitching motions can be found.