Asymmetric oscillations occur in numerous physical systems, in which the restoring force is often characterized by non-polynomial stiffness nonlinearities. In this study, an equivalent Helmholtz–Duffing oscillator was developed to derive approximate analytical solutions for the periodic response of asymmetric oscillators with non-polynomial stiffness nonlinearities under general initial conditions. The equivalent oscillator was formulated using quasi-static equilibrium principles, resulting in amplitude-dependent stiffness coefficients. The exact closed-form solution of the equivalent Helmholtz–Duffing oscillator was then employed to obtain an approximate analytical solution for the original asymmetric system. Numerical simulations demonstrate that the proposed approach yields highly accurate predictions, even under strongly nonlinear conditions, and significantly outperforms the conventional Taylor series expansion method in terms of accuracy.
