Analysis of Subharmonic Resonance in Time-Delayed Nonlinear Systems with Asymmetric Stiffness

Due to factors such as gravity, centrifugal forces, and manufacturing deviations, mechanical structures are often not perfectly symmetrical. Consequently, traditional time-delayed Duffing oscillator systems cannot accurately describe their dynamic characteristics and behaviors. To address this, this paper introduces quadratic stiffness into the time-delayed Duffing oscillator to simulate asymmetric stiffness. This modification allows the system to accurately describe the nonlinear time-delayed structure under the influence of asymmetric stiffness. The presence of quadratic and cubic nonlinear stiffness leads to subharmonic resonance in the system, significantly impacting the system's response and stability. The paper obtains an approximate theoretical solution for one-second and one-third subharmonic resonance behaviors by using the harmonic balance method. Based on shooting method theory and parameter continuation theory, a numerical continuation method for time-delayed systems is proposed. The system is analyzed with the corresponding parameters, and the analytical solution (harmonic balance method) is compared with the numerical solution (the proposed numerical method for a time-delayed nonlinear system). The results across varying delay parameters reveal the conditions under which subharmonic resonance occurs, as well as the influence of time delay parameters on subharmonic resonance behavior. This study provides a reference for solving the dynamics of subharmonic resonance in nonlinear time-delayed structures under asymmetric stiffness and offers an effective pathway for numerical solutions and parameter tracking in time-delayed nonlinear structures.