In this paper, period-1 motions varying with excitation frequency in a periodically forced, nonlinear spring pendulum system are predicted by a semi-analytic method. The harmonic frequency-amplitude for periodical motions are analyzed from the finite discrete Fourier series. The stability of the periodical solutions are shown on the bifurcation trees as well. From the analytical prediction, numerical illustrations of periodic motions are given, the comparison of numerical solution and analytical solution are given.

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