In this paper, we apply a new perturbation technique coupled with the iteration method. This procedure is obtained by combining the iteration methods of J. H. He and Mickens into a new iteration procedure such that excellent approximate analytical solutions, valid for small as well as large values of oscillation amplitude, can be determined for nonlinear oscillations with a single degree of freedom. Four examples are given to illustrate the validity and accuracy of this procedure. We compare the approximate periods obtained by our procedure with the exact known periods. The results show that the approximations are of extreme accuracy.
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