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A dynamic solution is presented for the propagation of waves in an electric-magneto-elastic plate composed of piezoelectric, piezomagnetic materials and elastic matrix. The electric-magneto-elastic plate is polarized along the thickness direction. The generalized displacements are expressed as the sum of the gradient of a scalar (dilatation wave) and the curl of a vector (shear wave). With the help of dynamic equilibrium equations and geometric equations, we can obtain dynamic equations of the dilatation wave and the shear wave. The conclusion that the types of the dilatation waves and the shear waves remain unchanged after being reflected by the boundary can be obtained through the analysis of these kinetic equations. The dispersion properties and phase velocity surface of the dilatation and shear wave can be obtained by solutions of dynamic equilibrium equations. Influences of the piezoelectric and piezomagnetic parameters on wave characteristics are discussed.

The electric-magneto-elastic structures or materials have the capacity of converting energy from one form to the other (among mechanical, magnetic and electric energy) [_{3}(B) and magnetostrictive CoFe_{2}O_{4}(F) material. Xiao Guo et al. [

The above existing works of literature are mainly researches on surface waves in electric-magneto-elastic plates. Hence in the present study, characteristics of bulk wave in an electric-magneto-elastic plate have been studied, the physical of waves and the geometric dispersion of waves along the thickness of an infinite plate are simultaneously studied. The elastic displacement is expressed as the sum of a scalar gradient and a vector curl. Based on dynamic equilibrium equations and geometric equation, we can obtain dynamic equations of the dilatation wave and shear wave, and the dispersion properties and phase velocity surface of the dilatation and shear wave can be obtained by solutions of dynamic equilibrium equations. The main aim of the study is to evaluate the influence of piezoelectric, piezomagnetic and elastic parameters on characteristics of bulk wave in an electric-magneto-elastic plate.

The electric-magneto-elastic material possess characteristic of the magnetic-electric-mechanical coupling. In this paper, only wave of magnetic potential and electric one and displacements are considered, so that the displacement vector u , electric potential φ , and magnetic potential ψ are chosen as basic variables. The basic equations are listed as follows.

The constitutive equations of the electric-magneto-elastic material are given by

σ i j = C i j k l ε k l − e i j n E n − q i j n H n (1)

D n = e k l n ε k l + g m n E m + α m n H m (2)

B n = q k l n ε k l + α m n E m + μ m n H m (3)

where σ i j , ε k l , D m and B m are, respectively, stresses, strains, electric displacements and magnetic induction tensors, C i j k l are the elastic constants, e i j n the piezoelectric tensors, g m n the dielectric tensors, α m n the magnetic-electric mutual inductance tensors, q i j n the piezomagnetic tensors, μ m n the magnetic permeability tensors, all the subscripts i , j , k , l , m , n = 1 , 2 , 3 , two identical subscripts in an integral expression denote the sum of Einstein.

The generalized dynamic equilibrium equations absent of the mechanical body force are given by

σ i j , j = ρ u ¨ i (4)

D n , n = 0 (5)

B n , n = 0 (6)

where ρ the mass density, the comma in subscript indicates partial derivative.

The generalized geometrical equations are

ε k l = 1 2 ( u k , l + u l , k ) (7)

E m = − φ , m (8)

H m = − ψ , m (9)

Combing of Equations (1)-(9), one easily obtains

1 2 C i j k l ( u k , l j + u l , k j ) − e i j n φ , n j − q i j n ψ , n j = ρ u ¨ i (10a)

1 2 e k l n ( u k , l n + u l , k n ) + g m n φ , m n + α m n ψ , m n = 0 (10b)

1 2 q k l n ( u k , l n + u l , k n ) + α m n φ , m n + μ m n ψ , m n = 0 (10c)

Let us consider an anisotropic electric-magneto-elastic plate as shown in

The angle between the incident direction of the wave and the x_{1}-axis is assumed to be θ , so that the following relationships can be obtained as

k 1 = k cos θ k 2 = k sin θ } (11)

where k is wave number in the plane x 1 0 x 2 .

The elastic wave tensor can be expressed as the following form

u i = Φ , i + ε i j k Ψ j , k (12)

where ε i j k is a permutation tensor, the scalar field Φ and the vector field Ψ j are assumed to be

Φ = f ( x 3 ) e i ( k 1 x 1 + k 2 x 2 − ω t ) (13)

Ψ j = g j ( x 3 ) e i ( k 1 x 1 + k 2 x 2 − ω t ) (14)

In Equation (13), f ( x 3 ) is tried to take the following form

f ( x 3 ) = A m sin ( η m d x 3 ) (15)

where η m d is the mth wave number of the dilatation wave in the thickness direction x 3 .

In Equation (14), g j ( x 3 ) is tried to take the following form

g j ( x 3 ) = F j n sin ( η n s x 3 ) (16)

where η m s is the mth wave number of the shear wave in the thickness direction.

The electric-magneto-elastic plate is assumed to be polarized in the direction x 3 , the static electric-potential and magneto-potential is tried to take the following forms

φ = [ f ′ φ ( x 3 ) + i ( k 1 g 2 φ ( x 3 ) − k 2 g 1 φ ( x 3 ) ) ] e i ( k 1 x 1 + k 2 x 2 − ω t ) (17)

ψ = [ f ′ ψ ( x 3 ) + i ( k 1 g 2 ψ ( x 3 ) − k 2 g 1 ψ ( x 3 ) ) ] e i ( k 1 x 1 + k 2 x 2 − ω t ) (18)

where

{ f φ ( x 3 ) = A m φ sin ( η m d x 3 ) f ψ ( x 3 ) = A m ψ sin ( η m d x 3 ) (19)

{ g j φ ( x 3 ) = F j n φ sin ( η n s x 3 ) g j ψ ( x 3 ) = F j n ψ sin ( η n s x 3 ) (20)

In the subsequent numeric example, the electric-magnetic-elastic plate is assumed to be polarized in the direction x 3 , the thickness of the plate is 2h, the mass density ρ = 7454 kg / m 3 , the material parameters are given by Tables 1-3.

Except these parameters given by Tables 1-3, the others are assumed to be zeros.

For smooth rigid boundary conditions, namely, u 3 | x 3 = ± h = 0 , σ 23 | x 3 = ± h = σ 31 | x 3 = ± h = 0 , φ | x 3 = ± h = 0 , D 1 | x 3 = ± h = D 2 | x 3 = ± h = 0 , ψ | x 3 = ± h = 0 , B 1 | x 3 = ± h = B 2 | x 3 = ± h = 0 , η m d and η m s which satisfy the boundary condition are taken as the following forms

η m d = m π 2 h , m = 1 , 3 , 5 , ⋯ (21)

η n s = n π 2 h , n = 0 , 2 , 4 , 6 , ⋯ (22)

For the numeric example, Equations (10a)-(10c) can be rewritten as

C 1111 u 1 , 11 + C 1122 u 2 , 21 + C 1133 u 3 , 31 + 1 2 C 1212 ( u 2 , 12 + u 1 , 22 ) + 1 2 C 3131 ( u 3 , 13 + u 1 , 33 ) − e 111 φ , 11 − e 311 φ , 31 − q 311 ψ , 31 − q 111 ψ , 11 = ρ u ¨ 1 (23)

C 2211 u 1 , 12 + C 2222 u 2 , 22 + C 2233 u 3 , 32 + 1 2 C 2121 ( u 2 , 11 + u 1 , 21 ) + 1 2 C 2323 ( u 2 , 33 + u 3 , 23 ) − e 222 φ , 22 − e 223 φ , 23 − q 222 ψ , 22 − q 223 ψ , 23 = ρ u ¨ 2 (24)

C_{1111} = C_{2222} | C_{1122} = C_{1133} | C_{2211} = C_{2233} | C_{3311} = C_{3322} | C_{3333} | C_{1212} = C_{2121} | C_{2323} = C_{3131} |
---|---|---|---|---|---|---|

79.7 | 35.8 | 35.8 | 35.8 | 66.8 | 17.2 | 14.4 |

Piezoelectric parameters, unit (C/m^{2}) | Piezomagnetic parameters, unit (Vs/m^{2}) | ||||
---|---|---|---|---|---|

e_{111} = e_{222} | e_{333} | e_{232} = e_{311} | q_{111} = q_{222} | q_{333} | q_{232} = q_{311} |

−5.9 | 15.2 | 10.5 | −60.9 | 156.8 | 108.3 |

Dielectric parameters, unit (As/Am) | Magnetic permeability parameters, unit (Vs/Am) | ||||
---|---|---|---|---|---|

g_{111} | g_{222} | g_{333} | μ_{111} | μ_{222} | μ_{333} |

3.8 × 10^{−}^{10} | 3.8 × 10^{−10} | 3.8 × 10^{−10} | 5.4 × 10^{−6} | 5.4 × 10^{−6} | 5.4 × 10^{−6} |

C 3311 u 1 , 13 + C 3322 u 2 , 23 + C 3333 u 3 , 33 + 1 2 C 3131 ( u 3 , 11 + u 1 , 31 ) + 1 2 C 3232 ( u 3 , 22 + u 2 , 32 ) − e 333 φ , 33 − e 232 φ , 23 − e 311 φ , 13 − q 333 ψ , 33 − q 232 ψ , 23 − q 311 ψ , 13 = ρ u ¨ 3 (25)

1 2 e 311 ( u 3 , 11 + u 1 , 31 ) + 1 2 e 232 ( u 2 , 32 + u 3 , 22 ) + e 111 u 1 , 13 + e 222 u 2 , 23 + e 333 u 3 , 33 + g 111 φ , 11 + g 222 φ , 22 + g 333 φ , 33 = 0 (26)

1 2 q 311 ( u 3 , 11 + u 1 , 31 ) + 1 2 q 232 ( u 2 , 32 + u 3 , 22 ) + q 111 u 1 , 13 + q 222 u 2 , 23 + q 333 u 3 , 33 + μ 111 ψ , 11 + μ 222 ψ , 22 + μ 333 ψ , 33 = 0 (27)

Substituting of Equations (12), (17) and (18) into Equations (23)-(27), gives

− C 1111 k 1 2 u 1 − C 1122 k 1 k 2 u 2 + C 1133 i k 1 u 3 , 3 − 1 2 C 1212 ( k 1 2 u 2 + k 1 k 2 u 1 ) − 1 2 C 3131 ( k 1 2 u 3 − i k 1 u 1 , 3 ) − e 111 i k 1 φ , 3 + e 311 k 1 2 φ + q 311 k 1 2 ψ − q 111 i k 1 ψ , 3 = − ρ ω 2 u 1 (28)

− C 2211 k 1 k 2 u 1 − C 2222 k 2 2 u 2 + C 2233 i k 2 u 3 , 3 − 1 2 C 2121 ( k 1 k 2 u 2 + k 2 2 u 1 ) − 1 2 C 2323 ( − i k 2 u 2 , 3 + k 2 2 u 3 ) − e 222 i k 2 φ , 3 + e 232 k 2 2 φ + q 232 k 2 2 ψ − q 222 i k 2 ψ , 3 = − ω 2 ρ u 2 (29)

C 3311 i k 1 u 1 , 3 + C 3322 i k 2 u 2 , 3 + C 3333 u 3 , 33 + 1 2 C 3131 ( i k 1 u 3 , 3 + u 1 , 33 ) + 1 2 C 2323 ( i k 2 u 3 , 3 + u 2 , 33 ) − e 333 φ , 33 − e 232 i k 2 φ , 3 − e 311 i k 1 φ , 3 − q 333 ψ , 33 − q 232 i k 2 ψ , 3 − q 311 i k 1 ψ , 3 = − ρ ω 2 u 3 (30)

− 1 2 e 311 ( k 1 2 u 3 − i k 1 u 1 , 3 ) − 1 2 e 232 ( − i k 2 u 2 , 3 + k 2 2 u 3 ) + e 111 i k 1 u 1 , 3 + e 222 i k 2 u 2 , 3 + e 333 u 3 , 33 − g 111 k 1 2 φ − g 222 k 2 2 φ + g 333 φ , 33 = 0 (31)

− 1 2 q 311 ( k 1 2 u 3 − i k 1 u 1 , 3 ) − 1 2 q 232 ( − i k 2 u 2 , 3 + k 2 2 u 3 ) + q 111 i k 1 u 1 , 3 + q 222 i k 2 u 2 , 3 + q 333 u 3 , 33 − μ 111 k 1 2 ψ − μ 222 k 2 2 ψ + μ 333 ψ , 33 = 0 (32)

The following equations can be obtained by using the orthogonality of the trigonometric function for Equations (28)-(32) and by dealing with the real and imaginary parts of Equations (28)-(32).

C 3131 A m cos m π x 3 2 h = e 311 A m φ cos m π x 3 2 h + q 311 A m ψ cos m π x 3 2 h (33)

C 2323 A m cos m π x 3 2 h = e 232 A m φ cos m π x 3 2 h + q 232 A m ψ cos m π x 3 2 h (34)

( C 3131 k 1 + C 2323 k 2 ) A m cos m π x 3 2 h = e 311 k 1 A m φ cos m π x 3 2 h + q 311 k 1 A m ψ cos m π x 3 2 h (35)

[ ( e 311 + e 111 ) k 1 2 + ( e 232 + e 222 ) k 2 2 + e 333 ( η m d ) 2 ] A m cos m π x 3 2 h = − [ g 111 k 1 2 + g 222 k 2 2 + g 333 ( η m d ) 2 ] A m φ cos m π x 3 2 h (36)

[ ( q 311 + q 111 ) k 1 2 + ( q 232 + q 222 ) k 2 2 + e 333 ( η m d ) 2 ] A m cos m π x 3 2 h = − [ μ 111 k 1 2 + μ 222 k 2 2 + μ 333 ( η m d ) 2 ] A m ψ cos m π x 3 2 h (37)

( − C 1111 k 1 2 − C 1122 k 2 2 − C 1133 ( η m d ) 2 − C 1212 k 1 k 2 ) A m + e 111 ( η m d ) 2 A m φ + q 111 ( η m d ) 2 A m ψ = − ρ ω 2 A m (38)

( − C 2211 k 1 2 − C 2222 k 2 2 − C 2233 ( η m d ) 2 − C 2121 k 1 k 2 ) A m + e 222 ( η m d ) 2 A m φ + q 222 ( η m d ) 2 A m ψ = − ρ ω 2 A m (39)

( − C 3311 k 1 2 − C 3322 k 2 2 − C 3333 ( η m d ) 2 ) A m + e 333 ( η m d ) 2 A m φ + q 333 ( η m d ) 2 A m ψ = − ρ ω 2 A m (40)

a 11 F 1 n + a 12 F 2 n + a 13 F 3 n + a 14 F 1 n φ + a 15 F 2 n φ + a 16 F 1 n ψ + a 17 F 2 n ψ = 0 (41)

a 21 F 1 n + a 22 F 2 n + a 23 F 3 n + a 24 F 1 n φ + a 25 F 2 n φ + a 26 F 1 n ψ + a 27 F 2 n ψ = 0 (42)

a 31 F 1 n + a 32 F 2 n + a 33 F 3 n + a 34 F 1 n φ + a 35 F 2 n φ + a 36 F 1 n ψ + a 37 F 2 n ψ = 0 (43)

a 41 F 1 n + a 42 F 2 n + a 43 F 3 n + a 44 F 1 n φ + a 45 F 2 n φ + a 46 F 1 n ψ + a 47 F 2 n ψ = 0 (44)

a 51 F 1 n + a 52 F 2 n + a 53 F 3 n + a 54 F 1 n φ + a 55 F 2 n φ + a 56 F 1 n ψ + a 57 F 2 n ψ = 0 (45)

a 61 F 1 n + a 62 F 2 n + a 63 F 3 n + a 64 F 1 n φ + a 65 F 2 n φ + a 66 F 1 n ψ + a 67 F 2 n ψ = 0 (46)

a 71 F 1 n + a 72 F 2 n + a 73 F 3 n + a 74 F 1 n φ + a 75 F 2 n φ + a 76 F 1 n ψ + a 77 F 2 n ψ = 0 (47)

a 81 F 1 n ( η n s ) cos ( η n s x 3 ) + a 82 F 2 n ( η n s ) cos ( η n s x 3 ) + a 83 F 3 n ( η n s ) cos ( η n s x 3 ) + a 84 F 1 m φ ( η m s ) cos ( η m s x 3 ) + a 85 F 2 m φ ( η m s ) cos ( η n s x 3 ) + a 86 F 1 n ψ ( η n s ) cos ( η n s x 3 ) + a 87 F 2 n ψ ( η n s ) cos ( η n s x 3 ) = 0 (48)

The above Equations (32)-(40) do not contain the shear term η n s , so they are P-wave equations. Equations (41)-(48) are not related to the dilatation wave term η m d , so they are S-wave equations. This indicates that the kind of wave is invariable after reflection. Equations (33)-(37) and Equations (48) are true according to Saint-Venant’s principle. a i j in Equations (41)-(48) are given by Appendix A.

Equations (38)-(40) about the longitudinal wave can be written in the following matrix form

[ b 11 − ρ ω 2 − b 12 − b 13 b 21 − ρ ω 2 − b 22 − b 23 b 31 − ρ ω 2 − b 32 − b 33 ] { A m A m φ A m ψ } = { 0 0 0 } (49)

For Equation (49) to have a nontrivial solution, it is required that

| b 11 − ρ ω 2 − b 12 − b 13 b 21 − ρ ω 2 − b 22 − b 23 b 31 − ρ ω 2 − b 32 − b 33 | = 0 (50)

where b i j ( i , j = 1 , 2 , 3 ) are given by Appendix B.

The relationship of wave number p = k 2 + m 2 and circle frequencies ω can be obtained by solving Equation (50), and then phase velocity is defined as

c = ω p (51)

and phase velocity surface

f = c ( θ , ω , p ) = 0 (52)

can be obtained.

Before plotting the figures, we introduce the dimensionless frequency ω ¯ = ω / c p , phase wave velocity c ¯ = c / c p , where c p = c 1111 / ρ .

velocity curves in the different incident direction θ . It can be found from the four Figures that with the increase of m, the dispersion of wave becomes more and more serious. The larger m is, the closer the direction of wave propagation is to the thickness of the plate, this means that the closer the wave propagation direction is to the direction x 3 , the more dispersive it is. This can be explained as that the geometrical and material dispersion of longitudinal wave propagating along the thickness direction is more serious than propagating in middle plane.

can be seen from comparison of the left Figures of Figures 3(a)-(c) that the phase velocity surface changes from an approximate circular to an approximate ellipse as the wave number k changes from 2 to 200. This is due to the anisotropy in the x 1 0 x 2 plane caused by the polarization of the electromagnetic field along thex_{3}-axis. It can be found from right Figures of

To further clarify the conclusions obtained from

The matrix form of Equations (41)-(47) about the shear wave can be written as

[ a 11 a 12 a 13 a 14 a 15 a 16 a 17 a 21 a 22 a 23 a 24 a 25 a 26 a 27 a 31 a 32 a 33 a 34 a 35 a 36 a 37 a 41 a 42 a 43 a 44 a 45 a 46 a 47 a 51 a 52 a 53 a 54 a 55 a 56 a 57 a 61 a 62 a 63 a 64 a 65 a 66 a 67 a 71 a 72 a 73 a 74 a 75 a 76 a 77 ] { F 1 n F 2 n F 3 n F 1 n φ F 2 n φ F 1 n ψ F 2 n ψ } = { 0 0 0 0 0 0 0 } (53)

The condition of the nontrivial solution of Equation (53) is that the coefficient determinant of Equation (53) must be zero.

| a 11 a 12 a 13 a 14 a 15 a 15 a 17 a 21 a 22 a 23 a 24 a 25 a 26 a 27 a 31 a 32 a 33 a 34 a 35 a 36 a 37 a 41 a 42 a 43 a 44 a 45 a 46 a 47 a 51 a 52 a 53 a 54 a 55 a 56 a 57 a 61 a 62 a 63 a 64 a 65 a 66 a 67 a 71 a 72 a 73 a 74 a 75 a 76 a 77 | = 0 (54)

Before plotting the Figures, we introduce the dimensionless frequency ω ¯ = ω / c s , the phase wave velocity c ¯ = c / c s and the group velocity C g = c g / c s , where c s = c 1313 / ρ . The symbols of EMEP and EP denote the electric-magnetic-elastic plate and the elastic plate, respectively.

In the subsequent example, let n = 2 , θ = 0 ∘ , 30 ∘ , 60 ∘ and 90˚ in Equations (54) and (56). The dimensionless circle frequency and the phase velocity corresponding to the different wave numbers k can be obtained from the solution of Equation (54) and Equation (56), respectively. For the purpose of investigation to the effects of the piezoelectric and the piezomagnetic parameters on the dispersion characteristics of the shear waves, we consider the EMEP and the elastic plate EP with the same modulus of elasticity as the former. The dispersion curves of shear waves with the different incident angles θ, namely, the relationship curves of the dimensionless circle frequency ωand the wave number k, are

plotted in Figures 5(a)-8(a). The relationship curves of the dimensionless phase velocity C and the wave numberk are plotted in Figures 5(b)-8(b).

When the incident angle θ is 0˚, the relationship curves between the dimensionless circle frequencies and the wave numbers for EP and EMEP are shown in

curves of waves in the two plates are different, especially that of the second wave mode. This indicates that the dispersion characteristics of wave are affected by the piezoelectric and piezomagnetic parameters. It also can be found from

When θ is 30˚, the dispersion curves of the different wave modes for EP and EMEP are shown in

When the incident angle θ is 60˚, the dispersion curves of the different wave modes for EP and EMEP are shown in

When the incident angle θ is 90˚, the dispersion curves of the different shear wave modes for EP and EMEP are shown in

The phase velocity surface of the third shear wave mode for EMEP is shown in

changes from 2 to 40, this means that the closer the direction of wave propagating is the middle plane x 1 0 x 2 , the more different the phase velocities of the wave propagating along the different directions are. It can be found from comparison of

To further clarify the conclusions obtained from

The bulk wave in the electromagnetic elastic plate is studied, and the following conclusions are obtained by the numerical examples:

1) The types of P-wave and S-wave reflected by the boundary of the plate remain unchanged.

2) The dispersion of the wave propagating along the mid-plane is less than that of the wave propagating along the plate thickness.

3) The closer the propagating direction of the longitudinal and the shear waves are to the middle plane, the more different their phase velocities in different directions are.

4) The phase velocity surface of the shear wave is anti-symmetry about the coordinate origin.

5) The phase velocity surface of a longitudinal wave is symmetrical about θ = 75 ∘ and θ = 165 ∘ , this means that EMEP is approximately orthotropic.

This work is supported by National Natural Science Foundation of China under the Grant Number 11372109.

The authors declare no conflicts of interest regarding the publication of this paper.

Xie, G.Q., Song, X.P. and Xiao, X.D. (2021) Bulk Waves in the Infinite Electric-Magnetic-Elastic Plate with Mixed Boundary Conditions. Journal of Electromagnetic Analysis and Applications, 13, 21-39. https://doi.org/10.4236/jemaa.2021.132002

a 11 = ( C 1122 − C 1133 ) k 1 k 2 + 0.5 C 1212 k 1 2 ,

a 12 = ( − C 1111 + C 1133 ) k 1 2 − 0.5 C 1212 k 1 k 2 − ρ ω 2

a 13 = − 0.5 C 3131 k 1 k 2 ,

a 14 = − e 111 k 1 k 2 ,

a 15 = e 111 k 1 2 ,

a 16 = − q 111 k 1 k 2 ,

a 17 = q 111 k 1 2

a 21 = − 0.5 C 3131 k 1 2 k 2 ,

a 22 = 0.5 C 3131 k 1 3 − 0.5 C 3131 k 1 ( η m s ) 2 ,

a 23 = ( C 1111 + C 1122 ) k 1 2 k 2 + 0.5 C 1212 k 1 3 + 0.5 C 1212 k 1 k 2 2 − ρ ω 2 k 2 ,

a 24 = e 311 k 1 2 k 2 ,

a 25 = − e 311 k 1 3

a 26 = q 311 k 1 2 k 2 ,

a 27 = − q 311 k 1 3 ,

a 31 = ( C 2222 − C 2233 ) k 2 2 − 0.5 C 2121 k 1 k 2 − ρ ω 2

a 32 = ( − C 2211 + C 2233 ) k 1 k 2 − 0.5 C 2121 k 2 2 ,

a 33 = − 0.5 C 2323 k 1 k 2 ,

a 34 = e 222 k 2 2 ,

a 35 = − e 222 k 1 k 2 ,

a 36 = q 222 k 2 2 ,

a 37 = − q 222 k 1 k 2 ,

a 41 = − 0.5 C 2323 k 2 3 + 0.5 C 2323 k 2 ( η m s ) 2 ,

a 42 = 0.5 C 2323 k 1 k 2 2 ,

a 43 = ( C 2211 − C 2222 ) k 1 k 2 2 + 0.5 C 2121 ( k 2 k 1 2 + k 2 3 ) − ρ ω 2 k 1 ,

a 44 = e 232 k 2 3 ,

a 45 = − e 232 k 1 k 2 2

a 46 = q 232 k 2 3 ,

a 47 = − q 232 k 1 k 2 2 ,

a 51 = ( C 3322 + C 3333 ) k 2 ( η m s ) 2 − ρ ω 2 k 2 ,

a 52 = ( C 3311 − C 3333 ) k 1 ( η m s ) 2 + ρ ω 2 k 1 ,

a 53 = 0.5 ( C 3131 k 2 − C 2323 k 1 ) ( η m s ) 2 ,

a 54 = e 333 k 2 ( η m s ) 2 ,

a 55 = − e 333 k 1 ( η m s ) 2 ,

a 56 = q 333 k 2 ( η m s ) 2 ,

a 57 = − q 333 k 1 ( η m s ) 2 ,

a 61 = 0.5 e 311 k 1 2 k 2 + 0.5 e 232 k 2 [ k 2 2 + ( η m s ) 2 ] + ( e 222 + e 333 ) k 2 ( η m s ) 2 ,

a 62 = 0.5 e 311 k 1 [ k 1 2 + ( η m s ) 2 ] − 0.5 e 232 k 1 k 2 2 + ( e 111 − e 333 ) k 1 ( η m s ) 2 ,

a 63 = 0 ,

a 64 = g 111 k 1 2 k 2 + g 222 k 2 3 + g 333 k 2 ( η m s ) 2 ,

a 65 = − g 111 k 1 3 − g 222 k 1 k 2 2 − g 333 k 1 ( η m s ) 2 ,

a 66 = a 67 = 0

a 71 = 0.5 q 311 k 1 2 k 2 + 0.5 q 232 k 2 [ k 2 2 + ( η m s ) 2 ] + ( q 222 + q 333 ) k 2 ( η m s ) 2

a 72 = 0.5 q 311 k 1 [ k 1 2 + ( η m s ) 2 ] − 0.5 q 232 k 1 k 2 2 + ( q 111 − q 333 ) k 1 ( η m s ) 2 ,

a 73 = a 74 = a 75 = 0 ,

a 76 = μ 111 k 1 2 k 2 + μ 222 k 2 3 + μ 333 k 2 ( η m s ) 2 ,

a 77 = − μ 111 k 1 3 − μ 222 k 1 k 2 2 − μ 333 k 1 ( η m s ) 2 ,

a 81 = 0.5 C 3131 k 1 k 2 + 0.5 C 2323 [ k 2 2 − ( η m s ) 2 ] ,

a 82 = − 0.5 C 2323 k 1 k 2 − 0.5 C 3131 [ k 1 2 − ( η m s ) 2 ]

a 83 = − ( C 3311 + C 3322 ) k 1 k 2 ,

a 84 = − e 311 k 1 k 2 ,

a 85 = e 311 k 1 2 ,

a 86 = − q 311 k 1 k 2 ,

a 87 = q 311 k 1 2

b 11 = C 1111 k 1 2 + C 1122 k 2 2 + C 1133 ( η m d ) 2 + C 1212 k 1 k 2

b 12 = e 111 ( η m d ) 2

b 13 = q 111 ( η m d ) 2

b 21 = C 2211 k 1 2 + C 2222 k 2 2 + C 2233 ( η m d ) 2 + C 2121 k 1 k 2

b 22 = e 222 ( η m d ) 2

b 23 = q 222 ( η m d ) 2

b 31 = C 3311 k 1 2 + C 3322 k 2 2 + C 3333 ( η m d ) 2

b 32 = e 333 ( η m d ) 2

b 33 = q 333 ( η m d ) 2