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In this paper, we consider a fractional nonlinear oscillator for a mass attached to a stretched elastic wire. The coupling technique based on the fractional complex transform and the global residue harmonic balance method is proposed for solving this fractional oscillator. The approximated periodic solutions and frequencies are provided without complicated computation and discretization. The frequency–amplitude relationship is investigated to show its monotonic property. Numerical approximations for the nonlinear oscillator with integer or fractional orders are compared with the approximations by some existing methods. Numerical results confirm the effectiveness and robustness of the proposed method.