Cumulative solutions of nonlinear longitudinal vibration in isotropic solid bars
摘要
Based on the strain invariant relationship and taking the high-order elastic energy into account, a nonlinear wave equation is derived, in which the excitation, linear damping, and the other nonlinear terms are regarded as the first-order correction to the linear wave equation. To solve the equation, the biggest challenge is that the secular terms exist not only in the fundamental wave equation but also in the harmonic wave equation(unlike the Duffing oscillator, where they exist only in the fundamental wave equation). In order to overcome this difficulty and to obtain a steady periodic solution by the perturbation technique, the following procedures are taken:(i) for the fundamental wave equation, the secular term is eliminated and therefore a frequency response equation is obtained;(ii) for the harmonics, the cumulative solutions are sought by the Lagrange variation parameter method. It is shown by the results obtained that the second- and higher-order harmonic waves exist in a vibrating bar, of which the amplitude increases linearly with the distance from the source when its length is much more than the wavelength; the shift of the resonant peak and the amplitudes of the harmonic waves depend closely on nonlinear coefficients; there are similarities to a certain extent among the amplitudes of the odd-(or even-) order harmonics, based on which the nonlinear coefficients can be determined by varying the strain and measuring the amplitudes of the harmonic waves in different locations.