Abstract: Taking the wrecking ball demolition machine as an engineering prototype, its working principle is analyzed, and the last section of its working process is equivalent to a spring rigid rod model.Through dynamic analysis of this equivalent model, a Mathieu equation containing periodically time-varying coefficients is derived and its parametric resonance characteristics are investigated.Using perturbation methods, the critical dynamic mechanism governing the transition is systematically revealed from steady periodic solutions to unstable periodic solutions in the dual-parameter excitation frequency plane.Detailed discussions are presented regarding the approximate analytical expressions of transition curves under different correlation parameters, accompanied by simulation results demonstrating critical boundaries between stable and unstable regimes.Subsequently, harmonic balance method is employed to determine stability boundaries, establishing the frequency ratio between vertical oscillation of the spring-mounted mass and angular motion of the pendulum.Through constructing Hill’s infinite determinant, stability charts under various frequency ratios are generated to identify parametric resonance regions in the dual-parameter plane.Finally, numerical validation via phase portraits and time series diagrams confirms the effectiveness and accuracy of the proposed methodology.