抄録
An analysis is made of nonlinear oscillations excited in a piezoelectric resonator by repeated impacts with a sinusoidally vibrating wall. The simulation method using an electrical equivalent circuit is employed for this analysis. This method gives better approximation to actual nonlinear oscillations than the conventional methods because the local deformation is taken into account. The simulation circuit for the impacting oscillatory system is shown in Fig. 2, where s_N is a virtual elastic component of nonlinear stiffness and s_h (a stiffiness of the local deformation) for the contact period (see Fig. 13). Using this circuit, various nonlinear oscillations are observed according to the frequency f of the vibrating wall (see Figs. 3, 4 and 12). The larger s_h/s'_1, the wider the frequency range in which these oscillations can be excited becomes, where 8'_1 is the equivalent stiffness of the fundamental mode. When s_n/s'_1 is less than about 0. 5, only stable one-half subharmonic oscillation can be excited within the frequency range f/f_1≧2, where f_1 is the natural frequency of the fundamental mode. A typical waveform and the amplitude characteristic of the l/2-harmonic oscillation are shown in Fig. 4(c) and Fig. 6, respectively. The influence of the zero mode and the 2nd mode are investigated on the stability and the excitation conditions of the 1/2-harmonic oscillation. As shown in Fig. 8, the 1/2-harmonic oscillation is apt to be unstable without the zero mode, and the excitation frequency range of the stable 1/2-harmonic oscillation becomes widest when m_0/m_1 is nearly unity, where m_0 and m_1 are the equivalent masses of the zero and the 2nd modes, respectively. The behavior of the stable 1/2-harmonic oscillation is not affected very much by the 2nd mode under the condition f_2/f_1≧4. 0, where f_2 is the natural frequency of the 2nd mode. The 2nd mode makes its excitation level very small when f_2/f_1 is a little larger than 2. 0 (see Fig. 11 (c)). In addition, it became clear that the zero mode is apt to made the higher harmonic oscillation unstable. The experimental demonstration of nonlinear oscillation was made. Experimental set-up is shown in Fig. 14 and the nonlinear oscillations observed are shown by the photos in Fig. 15. It was found that the qualitative behavior of a practical impact oscillatory device is as predicted by this simulation. The analysis in this paper will provide very useful data on the design of impact oscillatory devices such as a multi-mode piezoelectric frequency divider.
