抄録
The paper describes an analysis on the fundamental resonant frequency and the vibration mode of the plate tuning fork resonator and gives a design guide of the resonator. There are two types in the flat plate type resonator, the uniform arm type and the stepped arm type, as shown in Fig. 1 (a)〜(c). The stepped arm type resonator in Fig. 1 (b) is analyzed, since the other type resonators are asumed as the modifications of that of Fig. 1 (b). The half section of the resonator drawn by thick line in Fig. 2 can be expressed with the mechanical equivalent network shown in Fig. 3. If the external force P_<a1> is applied at a tree end of the stepped arm in Fig. 2, admittance matrices of the parts A_B, B_B, C_B are given by Eqs. (1)〜(3) respectively, and admittance Y_s and Y_f are given by Eqs. (6) and (7). Therefore, the driving point impedance z_i(=P_<a1>/V_<a1>) is derived as Eq. (14), and the equation to determine resonant frequency is given as Eq. (15) from resonant condition z_i=0. Similaly, the resonant frequency equetion of the uniform arm type resonator in Fig. 1 (a) is given as Eq. (17). The calculated and the experimental values of resonant frequencies against base width ω_c of the resonator are shown in Fig. 4. The differeces between both values are less than 3. 4% in the range of 2<ω_c<8 mm. The resonant frequency for the arm length 1_b is shown in Fig. 5, in which the resonant frequency of a cantilever with same arm dimensions as one of the plate tuning fork is shown by doted line for reference. The relation of the normalized resonant frequency ratios versus the rength ratio (l_a/l_<ab>) for the stepped arm type is shown in Fig. 6. The differences in the experimental values between the symmetric type and the unsymmetric type are less than 1. 3% as shown in Fig. 6; and the differences between the calculated and the experimental values inn two types mentioned above are within 1. 5%. It is clear that the stepped arm type resonator gives lower resonant frequency by about 35% as compared to the uniform type one. The resonant vibration mode of the resonator is determined by the combination of both flexural and tosional vibration displacements. The respective velocities (v_m and φ_m) at an arbitral point of the resonator are determined from Eqs. (18) and (19), and then the resultant displacement η_m is calculated by substituting the v_m and φ_m into Eq. (21), in which plus sign in the right side of Eq. (21) indicates that it lies outward by (ω'_n/2) from neutral line, and the minus sign means it lies inward by (ω'_n/2). The vibrational displacement on the neutral axis of the uniform arm type is shown in Fig. 7, and the location of nodal line against the base width ω_c is shown in Figs. 8 and 9. The displacement on the neutral axis against the length ratio (l_a/l_<ab>) of the stepped arm type is shown in Fig. 10, and the location of nodal line shifts, as shown in Fig. 11, as l_a increase. The analytical and experimental results mentioned above are useful for the design of the plate tuning fork in low frequencies below several hundred Hz.
