A closed-form solution for nonlinear oscillation and stability analyses of the elevator cable in a drum drive elevator system experiencing free vibration

Responses of the dynamical systems to some extent are affected by the natural frequencies. In the present paper, the parameter expansion method (PEM) is employed to investigate nonlinear oscillation and stability of the elevator’s drum as a single-degree-of-freedom (SDOF) swing system. A sensitivity analysis to observe the influence of various parameters on the nonlinear dynamic response, stability and natural frequency is performed. Comparing the results of the proposed closed-form analytical solution, the traditional numerical iterative time integration solution, and the linearized governing equations confirm the accuracy and efficiency of the proposed approach. Based on the results of the proposed closed-form solution, the linearization errors in calculating the natural frequencies in different cases are discussed as well. In contrast to the available numerical methods, the proposed method is free from the numerical damping and the time integration accumulated errors. Moreover, in comparison with the traditional multistep numerical iterative time integration methods, a much less computational time is required for the method in this research. Results reveal that for nonlinear systems, the natural frequency is remarkably affected by the initial conditions. Furthermore, the stability decreases as the dimensions of the mechanism increase.