This paper aims to investigate nonlinear oscillations of an elevator cable in a drum drive. The governing equation of motion of the objective system is developed by virtue of Lagrangian's method. A complicated term is broached in the governing equation of the motion of the system owing to existence of multiplication of a quadratic function of velocity with a sinusoidal function of displacement in the kinetic energy of the system. The obtained equation is an example of a well-known category of nonlinear oscillators, namely, non-natural systems. Due to the complex terms in the governing equation, perturbation methods cannot directly extract any closed form expressions for the natural frequency.  Unavoidably, different non-perturbative approaches are employed to solve the problem and to elicit a closed-form expression for the natural frequency. Energy balance method, modified energy balance method and variational approach are utilized for frequency analyzing of the system. Frequency-amplitude relationships are analytically obtained for nonlinear vibration of the elevator's drum. In order to examine accuracy of the obtained results, exact solutions are numerically obtained and then compared with those obtained from approximate closed-form solutions for several cases. In a parametric study for different nonlinear parameters, variation of the natural frequencies against the initial amplitude is investigated. Accuracy of the three different approaches is then discussed for both small and large amplitudes of the oscillations.