Application of the harmonic balance method to a nonlinear oscillator typified by a mass attached to a stretched wire

Abstract

The first-order harmonic balance method via the first Fourier coefficient is used to construct two approximate frequency–amplitude relations for a conservative nonlinear oscillatory system in which the restoring force has an irrational form. This system corresponds to the motion of a mass attached to a stretched wire. Two procedures are used to approximately solve the nonlinear differential equation. In the first, the differential equation is rewritten in a form that does not contain the square-root expression, while in the second the differential equation is solved directly. The approximate frequency obtained using the second procedure is more accurate than the frequency obtained with the first due to the fact that, in the second procedure, application of the harmonic balance method produces an infinite set of harmonics, while in the first procedure only two harmonics are produced. Both approximate frequencies are valid for the complete range of oscillation amplitudes, and excellent agreement of the approximate frequencies with the exact one are demonstrated and discussed. The discrepancy between the second approximate frequency and the exact one never exceeds 2.2%.