1978 Volume 34 Issue 4 Pages 223-233
The dynamic depolarizing-field effect in a long piezoelectric bar under the longitudinal-effect extensional virbration has been investigated. Fundamentals are described in the paragraph 2. The depolarizing field is caused by the space and surface free charges due to the spatial distribution of the polarization associated with sound waves. The present analysis is based on the following four assumptions:[1] The physical property of the bar is axially symmetric and dielectrically isotropic. [2] The strain S_3 depends only on z (lengthwise direction). [3] The vibration is excited by alternating true charge on the end electrodes or driven by the alternating stress applied upon the end faces. [4] The field produced by the charge is approximated by that due to the changed disc or ring. In the paragraph 3, fundamental equations are established to determine the distributions of polarization and strain. The space free charge in the interior and the surface charge on the side face are derived from (3. 6) and (3. 9). Then the depolarizing field is given by (3. 13) or (3. 13'), which leads to an integral equation [A] (3. 15), the 1st fundamental equation. The wave equation brings the 2nd equation [B] (3. 20), and the 3rd one [C] (3. 25) is obtained from the boundary condition at the end faces. In the case of sound propagation along an ifinitely long rod (paragraph 4), simulatneous equations [A] and [B] can be exactly solved for L→∞, and the dispersion relation is given by (4. 11) or (4. 13)〜(4. 15), where an effective compliance s_eff and an effective electromechanical coupling coefficient K are introduced. The problem for the vibration of a finite bar (in paragraph 5) can be approximately solved by replacing K(y) with K_a(y) (eq. (5. 1)), where the parameters are chosen as mentioned in Appendix A. 1. (All of the appendices will be described at the end of the succeeding paper. ) The solutions take the forms of (5. 3) and (5. 4), where k and λ_i's should satify eq. (5. 19) with respect to Λ. The P'_k and P'_i's are given in (5. 22) and (5. 23), respectively. (Details of derivartion are described in A. 2. ) It is assumed that the wave number k(x) defined in eq. (6. 2) is nearly equal to k. Then the dispersion relation is given by (6. 4) or eqs. (6. 5)〜(6. 7) in the same forms as those in the case of an infinitely ong rod. The admittance of vibrator is obtained as shown in (6. 8), where the parameters U etc. implicitly involve χ, k^^^〜^2, k and L. In order to get more simple expressions convenient for use, various quantities are given in the alternative forms by (6. 12), (6, 13) replacing U etc. by u etc. as shown in (6. 11). The admittance is thus expressed by (6. 17). The constants of the vibrator, , k^^^〜^2 and v, are evaluated by (6. 24) as described in 6. 2 using ω_R and ω_A obtained by the resonance-antiresonance method. In the case of small w, the admittance is expressed in the simple form of (6. 25). This expression is characteristic of the admittance for the electromechanical transducer with logitudinal-effect coupling. The antiresonance occurs at a frequency near the half wavelength frequency 1/(2l√ρs_<eff>). The L-dependence of resonant and antiresonant frequencies are shown in Fig. 2. Numerical values of the various parameters Φ's and ψ's in the foregoing formulae are shown in Fig. 3, which serves for the experimantal determination of electoromechanical constants of the bar. The present result improves the Ogawa's result reported early.
