A THEORY OF LONGITUDINALLY POLARISED PIEZOCOMPOSITE ROD BASED ON MINDLIN-HERRMANN MODEL

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1 Copyright 9 by ABCM th Pan-American Congress of Appied Mechanics January 4-8,, Foz do Iguaçu, PR, Brazi A THEORY OF LONGITUINALLY POLARISE PIEZOCOMPOSITE RO BASE ON MINLIN-HERRMANN MOEL M. Shataov a, b, mshatov@csir.co.za I. Fedotov b, fedotovi@tut.ac.za a Materias Sciences and Manufacturing, Counci for science and Industria research (CSIR) P.O. Box 395, Pretoria, South Africa. b epartment of Mathematics and Statistics, Tshwane University of Technoogy P.O. Box.68, Pretoria, FIN-44, South Africa HM. Tenkam b, jouosseutenkamhm@tut.ac.za b epartment of Mathematics and Statistics, Tshwane University of Technoogy P.O. Box.68, Pretoria, FIN-44, South Africa T. Fedotova c, Tatiana.Fedotova@wits.ac.za c Schoo of chemica and metaurgica engineering, University of Witwatersrand P.O. Box. 3 Wits, Johannesburg, 5, South Africa Abstract. The conventiona theory of the piezoeectric rod is based on an assumption that its atera vibrations are negigibe. In this case the rod, vibration coud be described in terms of one-dimensiona wave equation and a set of mechanica and eectric boundary conditions. However, the main assumption of the mode is ony vaid in the case of ong and reativey thin rods. As a rue, the inear dimensions of piezoeectric components used in rea transducers are comparabe with the characteristic dimensions of their cross-sections, and hence, their atera dispacements have to be taken into consideration. In the present paper, the theory of the piezoeectric rod is deveoped on the basis of the Mindin-Herrmann mode. In the frame of this theory, the ongitudina and atera dispacements are described by two independent functions and vibration of the rod is obtained in terms of a system of two partia differentia equations. The Hamiton variationa principe is used for derivation of the system of equations of motion and for obtaining the mechanica and eectric boundary conditions. On the basis of the formuated Mindin-Herrmann mode, the eectric impedance of the piezoeectric rod is cacuated. Possibe generaizations of the proposed approach are considered and concusions are formuated. An exampe of the appication of the piezoeectric rod based on the Mindin-Herrmann theory is given. Keywords: atera dispacement, Mindin-Herrmann mode, piezoeectric, thick rod.. INTROUCTION The piezoeectric transducers generate and detect utrasonic waves in continuous media such as fuids, soids, etc (Risstic, 983). They have been deveoped for many industria appications. The conventiona theory of the thickness vibrator transducer is based on an assumption that its atera vibrations are negigibe. In this case, the transducers dynamics coud be described in terms of one-dimensiona wave equation and a set of mechanica and eectric boundary conditions. However, the main assumption of the mode is ony vaid in the case of ong and reativey thin rods. As a rue, the inear dimensions of thickness vibrators are comparabe with the characteristic dimensions of their crosssections and hence, it is necessary to take into consideration the atera dispacements of these transducers. Theory of reativey thick transducers was deveoped (Shataov et a., 9b). In the present paper, the theory of the reativey thick piezocomposite rod is deveoped on the basis of the Mindin-Herrmann mode of ongitudina vibrations of rods. In the frame of this theory, it is supposed that the atera dispacements are proportiona to the product of an independent function which is subjected to determination and the distance from the neutra ine of the transducers cross-section (Shataov et a., 9a). The Hamiton variationa principe is used for derivation of the system of equation of motion and for obtaining the mechanica and eectric boundary conditions. On the basis of the obtained Mindin-Herrmann mode, the eectric impedance of the thickness vibrator is cacuated. Possibe generaizations of the proposed approach are considered and concusions are formuated. Genera soution of the probem is obtained from the variationa principe using two orthoganaities conditions of the eigenfunctions (Fedotov et a., 9).. MINLIN-HERRMANN MOEL FOR THICKNESS VIBRATOR Suppose that Oz ( 3) is the axis of poarization of a piezoeectric rod. According to the Rayeigh-Love and Bishop theories dispacements in Ox ( ), Oy ( ), Oz - directions are correspondingy u, v, and w :

2 Copyright 9 by ABCM th Pan-American Congress of Appied Mechanics January 4-8,, Foz do Iguaçu, PR, Brazi where ϕ ϕ( zt, ) (,, ) ϕ (, ) (,, ) ϕ (, ) (, ) u u xzt x zt v v yzt y zt w w zt represents the atera or transverse contraction about the z -axis and x, y - dispacements from the neutra axis of any symmetric cross-section of the bar. Note that the transverse dispacements u and v are both connected by the function ϕ and are independent of the ongitudina dispacement, w. This proves the independence of the shearing deformation thus, the entire motion can be described by two independent functions: w and ϕ. Linear strain are: u v w S ϕ S4 + yϕ x x v u w Norma strain: S ϕ and shear strain: S5 + xϕ y y w S3 w x Linear stress-strain reations are: T cs + cs + c3s3 h33 Norma stress: T cs + cs + c3s3 h33 T c S + S + c S h ( ) and shear stress: S T T T 6 c S c S i t Where cij - eastic constants at constant eectric charge density and, 3 3e ω are the eectric dispacements in which 3 const and ω π f - frequency of excitation. Expression for kinetic energy: ρ T A( z) w I pϕ dz + (3) where - thickness of the transducer, A A( z) is the cross-section area and I p ( x + y ) da is the poar ( A) moment of inertia Strain energy is as foows: P ( TS + TS + T3S 3 + T4S 4 + T5S 5 + T6S 6) dadz ( A) { ( ) ϕ ϕ ( ) } 33 3 p ϕ ϕ c + c + c w + 4c w A + I c h + h w A dz (4) where h ij is the piezoeectric coefficient and A da Eectric energy is: W ( E+ E + E33) dadz ( A) ( A) ( A) S h3s3 h33s33 + β3 3 dadz S ( β3 A3 h3a3ϕ h 33A3w ) dz s where β 3 is the dieectric constant It is aso necessary to keep in mind the foowing eectric boundary condition: i t where V( t) Ve ω ( V const) E3 dz V V ( t) (6) - excitation votage appied at the edge of the thickness vibrator and s E3 h3ϕ h33w + β3 3 is the eectric fied on the z -axis. Tota Lagrangian of the system: () () (5)

3 Copyright 9 by ABCM th Pan-American Congress of Appied Mechanics January 4-8,, Foz do Iguaçu, PR, Brazi L K P + W + λ E3 dz V ( t) { ( ) p 4 p } ( ( ) ()) S s Aw + I ϕρ c + c ϕ + c w + c w ϕ A I c ϕ + β A dz λh ϕdz λh w w + βλ (7) where λ is the Lagrange mutipier We can rewrite the pervious expression in the compact form as foow: ( ϕϕϕ λ) ( ϕϕϕ λ) ( λ ) L L ww,,,,,, Λ ww,,,,,, dz+λ,, w ( ), w( ) (7a) where ΛΛ ( ww,, ϕϕϕ,,,, λ) 3 { S Aw I p ϕ ρ ( c ) } c ϕ c 33 w c 3 w ϕ A I pc 44 ϕ β 3 A 3 λh 3 ϕ (8) is the Lagrange density and (, λ, w ( ), w() ) β s λ λh ( w ( ) w() ) Λ Λ (9) Appying the Hamitonian variationa principe to Lagrangian (7a) we obtain the system of equation of motion in the compact form: Λ Λ t w + z w Λ Λ Λ + t ϕ ϕ ϕ with the mechanica boundary conditions Λ Λ + w x w() or Λ Λ w x, + w x w () and or ϕ ϕ x, and the eectrica boundary conditions: Λ Λ Λ Λ + λ λ The expicit form of the Mindin-Herrmann mode for the poarised piezocomposite thick rod is given as foow. x, w w ϕ ρ A c 33A c 3A t z z ϕ ϕ w ρip c 44Ip + Ac ( + c ) ϕ+ c3a + λh3 t associated boundary conditions and 33 3 ϕ λ 33 x, c Aw + c A + h () or (3) or ϕ x, () w () x, ϕ (4) and the eectric boundary condition A + λ (5) 3 s 3ϕ 33 ( ) β3 3 h + h w( ) w() (6) From equation (5), we have λ A 3 x,, and substituting it into () and (), this eads to:

4 Copyright 9 by ABCM th Pan-American Congress of Appied Mechanics January 4-8,, Foz do Iguaçu, PR, Brazi w w ϕ ρ A c 33A c 3A t z z ϕ ϕ w ρi p c 44I p + A ( c + c ) ϕ+ c3a Ah33 t (7) with the eectro-mechanic boundary conditions (here we choose these boundary conditions without oosing the generaity stated in our main objective): h33 c33aw + c3aϕ 3 (8) and x, x, (9) The main probem is entirey defined by considering the initia conditions: wzt (,) gz (), (,) hz () and ϕ (,) φ (), ϕ (,) qz () () t t 3. FREE VIBRATION PROBLEM t t In this section, we are deaing with the free vibration probem, that means 3 (no eectric dispacement). Thus we appy the Fourier method to the obtained homogeneous probem. Let us assume that iωt iωt wzt (, ) Z( ze ) and ϕ( zt, ) Φ ( ze ) where i and ω is the circuar frequency. This eads to the Sturm-Liouvie probem: c33z + c3φ ω ρz () c44i pφ A( c + c ) Φ c3az ω ρi pφ With the associated boundary conditions c Z + c Φ and Φ () 33 3 z, z, The above Sturm-iouvie probem is unusua (two-dimension). So, to compiment the ack of theory in this particuar probem, we trade the difficuty by considering the case of a traveing wave. Thus we can write: ikz ikz Zz ( ) Ze and Φ ( z) Φ e (3) where Z and Φ are respectivey the ongitudina and atera non zero ampitude (unknowns) and k is the wave number and need to be determined. Hence system () becomes: ( ω ρ c33k ) Z ikc3φ (4) ikac3z + ω ρi p c44i pk A( c + c ) Φ Soving the characteristic equation of the determinant of system () for k, we obtain: β ± β + 4αγ ( ω) α, k, k where α c33c44ip, β βω ( ) Ac ( + c) c33 4( c3) ωρ( c44 + Ipc33) γ γ( ω) ω ρ ω ρip Ac ( + c) Using formua (3) we can express Z( z ) as foows: and Z Z( z, ω) a Z ( k z) + a Z ( k z) + a Z ( k z) + a Z ( k z) cos( kz j ) if Im kj where Z,3( kz j ) sin( kz j ) if Im kj and Z,4( kz j ) ( j, ) cosh( kz) orthewise sinh( kz) orthewise j and the constants a, a, a3 and a 4 are not a equa to zero. Without oss the generaity, the case Im[ k ] is considered in the discussion that foows. Hence Z Z( z, ω) acos( kz) + asin( kz) + a3cosh( kz) + a4sinh( kz) (5) Substituting (5) into the first equation of system () and soving the obtained equation for Φ : j

5 Copyright 9 by ABCM th Pan-American Congress of Appied Mechanics January 4-8,, Foz do Iguaçu, PR, Brazi ω ρ ω ρ ΦΦ ( z, ω) a 33 sin( ) 33 cos( ) kc kz a kc kz c k 3 k ω ρ ω ρ + a3 kc 33 sinh( kz ) a4 kc 33 cosh( kz ) c k 3 k (6) Substituting Eq(5) and Eq(6) into the boundary conditions (), yieds a system of four equations with four unknowns: a, a, a3 and a 4. Soving the characteristic equation of the determinant for ω using the method deveoped by Fedotov et a. (8) for soving transcendenta equation, we obtain many positive roots ω n, n,,..., so caed Z (, z ω ), Φ (, z ω ). eigenvaues corresponding to the coupe of eigenfunction ( ) 4. SOLUTION OF THE FORCE VIBRATION PROBLEM 4.. Orthogonaities of the eigenfunctions n n n n Using system () and boundary conditions () we can prove two kind of orthogonaity condition of the eigenfunctions: First orthogonaity where nm ( ) ( ) { } Z, Φ, Z, Φ ( Z, Φ ) I Z Z + AΦΦ dz δ p n m n m nm n n m m n n δ is the Kroneker s symbo and ( ) { }, Φ + Φ n n p n n Second orthogonaity Z I Z A dz is the associated square norm ( Z, Φ ),( Z, Φ ) { Ac Z Z A( c c ) c I c A( Z Z )} d ( Zz, Φ ) n n m m + + Φ Φ + Φ Φ + Φ + Φ δ 33 n m n m 44 p n m 3 n m m n n n nm Z, Φ Ac Z ( ) 4 n n 33 n A c c n c I 44 p n c AZ + + Φ + Φ + Φ 3 n n dz is the associated square norm. Where ( ) { } 4.. Soution of the probem In this section, we give the soution of probem (7)-() on the basis of the method of eigenfunction orthogonaities for vibration probem deveoped by jouosseu (8) and Fedotov et a. (9). G(, z ξ,) t G(, z ξ,) t w( z, t) Ag( ξ) + h( ξ) G ( z, ξ, t) dξ + I φ( ξ) + q( ξ) G ( z, ξ, t) dξ + p t t A + F() τ G (, z ξ, t τ) dξdτ ρ G3 (, z ξ,) t G4 (, z ξ,) t ϕ( z, t) Ag( ξ) + h( ξ) G ( z, ξ, t) dξ + I φ( ξ) + q( ξ) G ( z, ξ, t) dξ + 3 p 4 t t A + F() τ G3 (, z ξ, t τ) dξdτ ρ Zn( z) Zn( ξ)sin Ωnt Zn( z) Φn( ξ)sin Ωnt G z,, t, G( z, ξ, t ), n n ( Zn, n) Ω Φ n Ω (, ) n Zn Φn where ( ξ )

6 Copyright 9 by ABCM th Pan-American Congress of Appied Mechanics January 4-8,, Foz do Iguaçu, PR, Brazi Φ ( zz ) ( ξ)sin Ω t Φ ( z) Φ ( ξ)sin Ω t G z,, t, and G z,, t ( ξ ) n n n n n n ( ξ ) n ( Z, ) Ω Φ n Ω ( Z, Φ ) are Green s function in which 3 4 Ω n ρ ( Zn, Φn) ( ) Zn, Φn n n n n n n 4.3. Eectrica response of the Rod ωt F t Ah Ah e. is the natura frequency and ( ) The eectrica response of the rod to the excitation votage Vt () appied at the edge of its thickness is characterised by: the current through this thickness I ( ω) A 3 iωa and 3 the associated eectric impedance V ( t) Zimp ( ω) I ω 5. CONCLUSIONS ( ). The Mindin-Herrmann approach was used to buid a mode describing a thick and short piezocomposite rod ongitudinay poarised.. The Hamiton variationa principe was used to derive the system of equation of motion in the process of which the eectromechanica boundary conditions were obtained. The method of eigenfunction orthogonaies based on the variationa principe was used to obtain the exact soution of the probem in terms of the Green function. 3. The eectric impedance through the thickness of the piezocomposite rod is formuated in terms of the excitation frequency. 6. REFERENCES jouosseu Tenkam, H.M., 8. Appication of Hyperboic Equations to Vibration Theories, Master issertation, Tshwane University of Technoogy (Pretoria), South Africa. Fedotov, I., Shataov, M., Tenkam, H.M. and Fedotova, T., Juy-9. Appication of Eigenfunction Orthogonaities to Vibration Probems, Accepted for pubication in the Proceedings of the Word Congress on Engineering, London, United Kingdom Fedotov, I., Shataov, M. and Mwambakana, J.N., 8. Roots of Transcendenta Agebraic Equations: A Method of Bracketing Roots and Seecting Initia Estimations, Proceeding of the internationa conference TIME-8, Buffetspoort, South Africa. Rissitic, V.M., 983. Principe of Acoustic evices, John Wiey and Sons, New York. Shataov, M., Fedotov, I., Tenkam, H.M. and Fedotova, T., October-9b. A Theory of Piezoeectric Thickness Vibrator Transducer Based on the Rayeigh and Bishop Mode, Submitted for pubication in the proceedings of the Internationa Conference on Mathematica Science and Engineering, Venice, Itay. Shataov, M., Fedotov, I., Tenkam, H.M. and Marais, J.P., Juy-9a. Comparison of Cassica and Modern Theories of Longitudina Wave Propagation in Eastic Rods, Accepted for pubication in the Proceedings of the 6 th Internationa Congress on Sound and Vibration, Krakow, Poand. 7. RESPONSIBILITY NOTICE The authors are the ony responsibe for the printed materia incuded in this paper.