transversely isotropic in the x 1 –x 2 plane. Therefore it is sufficient to study the u 1
displacement only. In this case, Eq. (5.86) 1 reduces to
À3h
À2
κ
2 c 44 u
1
ð Þ
1 ¼ ρ€ u
1
ð Þ
1 :
ð5:90Þ
For time-harmonic motions with a frequency ω, Eq. (5.90) determines
ω
2
¼
3κ
2 c 44
ρh
2
:
ð5:91Þ
The exact fundamental thickness-shear frequency determined from the three-dimensional equations is [6].
ω
2
¼ ω
2
1 ¼
π
2 c 44
4ρh
2
:
ð5:92Þ
Setting the frequencies in Eqs. (5.90) and (5.91) equal, we obtain the fundamental
thickness-shear correction factor as
κ
2
¼
π
2
12
:
ð5:93Þ
5.9 Thickness-Shear Approximation
Thickness-shear vibration modes and standing as well as propagating waves are
widely used in plate piezoelectric devices. These motions are described by u
1
ð Þ
a and
are usually accompanied by small flexural deformations described by u
0
ð Þ
3 . The small
flexural deformation can be eliminated by the so-called thickness-shear approximation [7] which simplifies the equations significantly. In this section we perform this
approximation for unelectroded plates of crystals of class (6mm). We consider the
case when there are no surface electromechanical loads. The electric field in the
surrounding free space is neglected. We proceed as follows. From Eqs. (5.85) and
(5.87) 1 , we obtain, by eliminating φ
(0) ,
κ
2 c 44 þ
e
2
15
ε 11
u
0
ð Þ
3,aa þ u
1
ð Þ
a,a
À
κe 15 q
ε 11
p
0
ð Þ
À n
0
ð Þ
¼ ρ€ u
0
ð Þ
3 :
ð5:94Þ
Consider the following waves:
5.9 Thickness-Shear Approximation
129
displacement only. In this case, Eq. (5.86) 1 reduces to
À3h
À2
κ
2 c 44 u
1
ð Þ
1 ¼ ρ€ u
1
ð Þ
1 :
ð5:90Þ
For time-harmonic motions with a frequency ω, Eq. (5.90) determines
ω
2
¼
3κ
2 c 44
ρh
2
:
ð5:91Þ
The exact fundamental thickness-shear frequency determined from the three-dimensional equations is [6].
ω
2
¼ ω
2
1 ¼
π
2 c 44
4ρh
2
:
ð5:92Þ
Setting the frequencies in Eqs. (5.90) and (5.91) equal, we obtain the fundamental
thickness-shear correction factor as
κ
2
¼
π
2
12
:
ð5:93Þ
5.9 Thickness-Shear Approximation
Thickness-shear vibration modes and standing as well as propagating waves are
widely used in plate piezoelectric devices. These motions are described by u
1
ð Þ
a and
are usually accompanied by small flexural deformations described by u
0
ð Þ
3 . The small
flexural deformation can be eliminated by the so-called thickness-shear approximation [7] which simplifies the equations significantly. In this section we perform this
approximation for unelectroded plates of crystals of class (6mm). We consider the
case when there are no surface electromechanical loads. The electric field in the
surrounding free space is neglected. We proceed as follows. From Eqs. (5.85) and
(5.87) 1 , we obtain, by eliminating φ
(0) ,
κ
2 c 44 þ
e
2
15
ε 11
u
0
ð Þ
3,aa þ u
1
ð Þ
a,a
À
κe 15 q
ε 11
p
0
ð Þ
À n
0
ð Þ
¼ ρ€ u
0
ð Þ
3 :
ð5:94Þ
Consider the following waves:
5.9 Thickness-Shear Approximation
129