·Table of Contents ·General | Subharmonic Generation in Piezoelectrics with Cantor-Like StructureC. CHIROIU, V. CHIROIU, L. MUNTEANU, C. RUGINARomanian Academy, Institute of Solid Mechanics,Ctin Mille 15, 70701 Bucharest, Romania, E-mail: chiroiu@mecsol.ro E. RUFFINO , M. SCALERANDI Politecnico di Torino, Dipartimento di Fisica,Corso Duca degli Abruzzi 24, 10129 Torino, Italy, E-mail: scalerandi@polito.it Contact |
(2.1) |
(2.2) |
The constitutive equations of nonlinear isotropic piezoelectrics are [5]
(2.3) |
(2.4) |
(2.5) |
where e_{ij} is the strain tensor, l^{p},m^{p} are the Lame constants, A^{p},B^{p},C^{p} are the Landau constants, are the linear and nonlinear dielectric constants, , and are the linear and nonlinear coefficients of piezoelectricity and . We consider all quantities are independent with respect to x_{2} and u_{2} = 0, E_{2} = 0. We have
(2.6) |
(2.7) |
(2.8) |
(2.9) |
(2.10) |
(2.11) |
(2.12) |
(2.15) |
(2.16) |
The field of displacement u_{1}, u_{3} is expressed in the form (2.8) with
(2.17) |
with the same unknown functions and as in (2.9).
(2.18) |
(2.16) |
(2.17) |
where the bracket indicates a jump across the interface and k = 1,3.
(3.1) |
(3.2) |
(3.3) |
piezoelectric ceramics | epoxy resin | |||
l | 71.6 Gpa | 42.31 GPa | ||
m | 35.8 Gpa | 3.76 GPa | ||
A | -2000 Gpa | 2.8 GPa | ||
B | -1134 Gpa | 9.7 GPa | ||
C | -900 Gpa | -5.7 Gpa | ||
4.065 nF/m | - | |||
2.079 nF/m | - | |||
e_{1} | -0.218 nm/V | - | ||
-0.435 nm/V | - | |||
r | 7650 Kg/m^{3} | 1170Kg/m^{3} | ||
Table 1: The material constants for piezoelectric ceramics and epoxy resin |
Table 2 shows the computed frequencies and the errors obtained at the optimal solution given by the genetic algorithm.
w_{n} / 2p | 100.2 ± 0.05 | 167 ± 0.01 | 217.1 ± 0.03 | 250.5 ± 0.1 | 334 ± 0.01 | 367.4 ± 0.01 | 417.5 ± 0.1 | 501 ± 0.02 | 584.5 ± 0.03 |
617.9 ± 0.01 | 668 ± 0.03 | 835 ± 0.06 | 935.2 ± 0.06 | 1085.5 ± 0.1 | 1169 ± 0.07 | 1269.2 ± 0.02 | 1503 ± 0.05 | 1670 ± 0.4 | |
1770.2 ± 0.2 | 1987.3 ± 0.12 | 2120.9 ± 0.02 | 2250 ± 0.1 | 2471.6 ± 0.3 | 2655.3 ± 0.01 | 2672 ± 0.01 | 2972.6 ± 0.2 | 3340 ± 0.4 | |
3690.7 ± 0.01 | 3774.2 ± 0.15 | 3991.3 ± 0.24 | 4250 ± 0.03 | 4291.9 ± 0.06 | 4525.7 ± 0.2 | 4826.3 ± 0.01 | |||
Table 2: Estimation results: computed eigenfrequencies |
Resonant vibration modes are excited by applying an external electric field on both sides of the plate with w = w_{n}. The undetermined coefficients P are approximatelly determined from a genetic algorithm. In fig. 1 the admittance curve (k / rw vs w / 2p) (fig.1) in the linear regime ( @ 0.1V) marks by peaks the frequencies w = w_{n} of the modes. If is increases above a threshold value = 5.27 V the w / 2 p subharmonic generation is observed. The amplitude of waves is calculated at the surface of the plate as a function of . Figs.2-4 show the displacements of the normal modes w / 2 p = 334 kHz, 501 kHz, 835 kHz and respectively of the subharmonic modes w / 4p=167 kHz, 250.5 kHz, 417.5 kHz. Two kind of vibration regimes are found: a localised-mode (fracton) regime represented in fig.5 for w / 2p=1169 kHz, 2672 kHz and 3340 kHz and an extended-vibration (phonon) regime represented in fig. 6 for w / 2p= 4175 kHz and 4250 kHz. A sketch of the plate geometry is given on the abscissa (dashed, piezoelectric ceramic and white, epoxy resin The fracton vibrations are mostly localised on a few elements, while the phonon vibrations essentially extend to the whole plate. In the case of a periodical plate the dispersion prevents good frequency matching between the fundamental and appropriate subharmonic modes. For the homogeneous plate the mismatch w_{n} - w/2 is due to the symmetry of fundamental modes with respect to x. Only symmetric odd n can induce a subharmonic, but never w/2 coincides with a plate vibration mode. For a Cantor plate, we have obtained the same result as Craciun and Alippi [1.2]: given a normal mode w_{n}, for excitation at w = w_{n}, the value of the expected threshold E_{th} i. e. the ability of generating the w/2 subharmonic, is determined by the existence of a normal mode with: (i) small frequency mismatch w_{n} - w/2, and, (ii) large spatial overlap between the fundamental and subharmonic displacement field.
Fig 1: The admittance-frequency curve for the Cantor plate | Fig 2: The amplitudes of the surface displacement of the normal mode w /2p= 334 kHz and of the subharmonic mode w /4p= 167 kHz. |
Fig 3: The amplitudes of the surface displacement of the normal mode w /2p= 501 kHz and of the subharmonic mode w /4p= 250.5 kHz. | Fig 4: The amplitudes of the surface displacement of the normal mode w /2p= 835 kHz and of the subharmonic mode w /4p= 417.5 kHz. |
Fig 5: The normal amplitudes for two localised vibration modes (w /2p=1169 kHz, w /2p=2672 kHz and w /2p=3340 kHz) | Fig 6: The normal amplitudes for two localised vibration modes (w /2p=4175 kHz and w /2p=4250 kHz. |
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