The response of nonlinear parametrically excited systems is investigated. It is shown that perturbation methods including the method of multiple scales and the averaging method using the Krylov–Bogoliubov technique can provide precise predictions of the behavior of the system in the neighborhood of the principal parametric resonance. However, increasing the excitation frequency, they imply an infinite growth of the vibration amplitudes, which contradicts numerical and practical findings. To tackle this problem, the method of varying amplitudes (MVA) is used. Employing the MVA, an expression for the upper bound to the displacement response of the system is derived. The parametric excitation frequency, at which this response is attained, is defined explicitly. MVA results considering the first harmonic are identical to the results obtained by the first approximation of the Mitropolskii technique and show good agreement with numerical results obtained from direct integration of the equation of motion. Taking the first and third harmonics into account, MVA shows an excellent capability to capture the frequency at which the upper bound response is attained: the difference between the upper bound to the displacement response obtained from the MVA and the one obtained from numerical integration is less than 0.1%.

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