Approximate analytical solutions of autonomous conservative oscillators via the parameter expansion method

Large-amplitude vibrations of mechanical systems, which are commonly encountered in engineering applications, are examined in this paper using thin cantilever beams with flexible roots and intermediate lumped masses as models. Classical perturbation and linearization methods are insufficient for such systems because they are controlled by a severely nonlinear autonomous conservative oscillator with a fifth-order restoring force. Using the parameter expansion method (PEM), an approximate analytical solution to a generalized nonlinear governing equation is obtained. The universality of the formulation is then demonstrated by specializing the resulting solution to a number of physically relevant anharmonic oscillators. Quantitative comparisons with numerical simulations and established analytical methods show that PEM yields improved accuracy over a wide range of vibration amplitudes while maintaining a compact analytical structure. Unlike conventional perturbation-based approaches, the proposed formulation avoids small-parameter assumptions and enables efficient generation of higher-order terms without algebraic complexity. The results confirm that PEM provides a reliable and computationally efficient framework for analyzing strongly nonlinear oscillatory systems encountered in realistic engineering configurations.