Abstract: A nonlinear dynamic equation of paper vibration is established by the perturbation theory method. The relationship between the bass and the treble phenomena is predicted and verified by experiments. There is a fractional octave solution related to the bass solution in nonlinear dynamic equation of the model, which is consistent with the fractional octave vibration phenomenon in experiments. The intrinsic characteristics of integer multiple sizes of harmonic, as well as the derivative of fractional bass and harmonic generation are explained by the model. The integer and fractional vibration in the experiment satisfy the theoretical prediction.

Key words: paper, the nonlinear dynamic equation, fractional vibration, integer vibration, tuning fork