Study of Energy Methods to Strongly Nonlinear Vibrations in a Loudspeaker Cone-Shaped Shell

H. F. Olson points out that a loudspeaker cone-shaped shell, as a nonlinear oscillation system, can be described as the Classical Duffing Equation in low frequency range. Yoshinisa, a Japanese scholar, studied the nonlinear phenomena of the loudspeaker cone-shaped shell in low frequency range driven by a stable galvanic source, including the resonance frequency changing with amplitude and leap phenomena. But their research were not taken the influence of nonlinear magnetic field into account. Its work mostly related to getting solution of nonlinear differential equation by the Numerical Calculation, but it didn’t get approximate solutions. Through research and analysis of the experiment on the loudspeaker cone-shaped shell, we obtain the Generalized Duffing Equation that’s a strongly nonlinearity system which is used to describe the loudspeaker cone-shaped shell driven by a stable voltage source, it considers the nonlinearity of mechanical resilience and the magnetic field. This paper focuses on first finding the approximate solutions (limit cycles) of strongly nonlinear oscillations and nonlinear heteronomy of the loudspeaker cone-shaped shell in low frequency range by use of energy methods. They obtained the equation relating to the forced vibration amplitude with frequency and the corresponding relation about phase versus frequency, and analysed particularly complete stability of limit cycles belonged to the strongly nonlinear systems, and drew several important conclusions. (1) As to strongly nonlinear oscillations of the loudspeaker cone-shaped shell in low frequency range, it is only likely to appear main oscillation and odd-order sub-harmonic oscillations. But it cannot appear super-harmonic vibrations and even-order sub-harmonic vibrations. (2) As to strongly nonlinear oscillations of the loudspeaker cone-shaped shell in low frequency range, two cases about main oscillation and one third sub-harmonic oscillation whose approximate solutions accord with numerical solutions very well. (3) It is worthy to study strongly nonlinear oscillations of commonly thin shell structure such as a loudspeaker cone-shaped shell by use of energy methods, and we will continue to carry out this research.Copyright © 2005 by ASME