aik Magneto-electric tensor Bi Magnetic induction vector c Speed of light
EijklC Elastic stiffness tensor
D Bending stiffness Di Displacement current eijk Piezoelectric tensor E,Y Young’s modulus Ek Electric field vector f Lorentz force h Plate thickness Hk Magnetic field vector j Current density L Plate length mi Pyromagnetic coefficients pi Pyroelectric coefficients qijk Piezomagnetic tensor ∆T Change in temperature, T – T0 αij Thermal expansion coefficients εij Strain tensor
ijεκ Permittivity/dielectric tensor
µik Magnetic permeability tensor ν Poisson’s ratio ω, Ω Vibration frequency ρ Material density σij Stress tensor
Introduction
In structural applications, multifunctional materials are usually defined as materials that possess the mechanical properties for load-carrying capability and some other functionality (e.g., thermal or electric conductance, sensitivity or indirect resistance to electromagnetic fields, field emission, wave generation or control, and micromechanical damage monitoring or structural health assessment). Multifunctional materials have quite promising applications in space structures.
44th AIAA/ASME/ASCE/AHS Structures, Structural Dynamics, and Materials Confere7-10 April 2003, Norfolk, Virginia
AIAA 2003-1624
This material is declared a work of the U.S. Government and is not subject to copyright protection in the United States.
2
The static and dynamic response of the space structures composed of multifunctional materials can be dramatically altered by the presence of electrostatic, electromagnetic or magnetic fields. In order to design such multifunctional structures, and to optimize their performance to ensure optimal operating conditions, the mutual influence of various functionalities must be understood and tailored in conjunction with the geometric and loading characteristics. The primary objective of the paper is investigation of trends in the in-field structural vibration response of multifunctional plate-like structures.
Multifunctional plate-like structures (e.g., electromagnetically conducting plates, piezoelectric polymer films) may possess strong electro-mechanical coupling; and, as a result, the mechanical vibrations of such structures can be sensitive to external electric or magnetic fields. Theoretical foundations of purely mechanical vibrations1, 2 and the electrodynamics of infinite continuum media are well established.3-6 Basic differential equations and boundary conditions governing the oscillations of linear piezoelectric plates are based on either field-theoretic approach7-9 or the lumped-circuit techniques.10 However, these studies are primarily focused on developing the mathematical framework for the analysis of idealized media involving either infinite geometry, perfectly conductive materials or high-frequency effects. The role of numerous material parameters that compose the constitutive matrices (e.g., piezoelectric coefficients11) has not been investigated in detail.
Until recently, piezoelectric effects on vibration have been studied only in ceramics and crystals for device applications.7-9 Structural applications with piezoelectric materials have been considered only recently, when the key material parameters have been characterized for a group of piezoelectric polymers.12 There are also very few studies concerning the electro-mechanical response of conducting thin-walled structures (e.g., plates, thin-walled beams and shells),13 especially, when geometric nonlinearities are present. Librescu, Hasanyan and Ambur14 have presented a geometrically nonlinear dynamic theory for conducting elastic plates in a magnetic field. For the electromagnetically simple media considered, the structural response is affected by the external magnetic field by means of ponderomotive body forces (e.g., the Lorentz force). Components of the magnetic field induced inside a thin plate depend on the transverse plate deflection and its tangential displacements caused by stretching and vibrations. Such dependence results in the bending-stretching coupling even for a symmetric and geometrically linear plate. This study has also predicted some trends for the frequency-thickness dependence for the case of perfectly conducting thin elastic plates. The present paper complements the investigation of such trends by examining the role of key material parameters on the vibration frequency. The need for simple engineering
models that can assist preliminary material selection and design considerations has been also addressed.
In this paper, the constitutive modeling of multifunctional materials and the associated multiphysics coupling are examined for electromagnetic effects on the in-field vibration of piezoelectric and conductive thin plates. Special attention is given to the critical material-field interactions and the key material parameters for electro-mechanical coupling in planar structures. Two special cases of multifunctional plates allowing modeling approximations in the simulation of electro- and magneto-mechanical coupling effects on the in-field vibration frequency are analyzed in detail. A theoretical framework for an engineering model for the vibration-frequency corrections due to the electro-mechanical coupling is presented for an electric field perpendicular to a piezoelectric plate. A phenomenological approximation for the in-field frequency shift during the conductive plate vibration is introduced for a magnetic field. Examples of vibration analysis of thin plates in electric and magnetic fields are presented and those results are compared with similar theoretical predictions14 for conductive plates.
Background: Material-Field Interactions
Interactions between electrically or magnetically conducting materials and an external electromagnetic field depend on material properties and characteristics of a particular field. Here the constitutive equations for such interactions are described along with the relevant physical assumptions (e.g., isothermal conditions and negligible electric displacement current). The material-field interactions are reviewed to discuss the electro-mechanical coupling in piezoelectric materials, the ponderomotive body forces (e.g., the Lorentz force) and the in-field surface effects. Governing Equations
The structural response of multifunctional materials placed in an electromagnetic field is governed by the constitutive equations that describe the coupling between mechanical deformation and the field components. An electromagnetic field is typically described by its electric component, Ek, and its magnetic component, Hk. The electro-mechanical coupling for linear piezoelectric materials is governed by11
kijkkijkijklEijklij T HqEeαεCσ −−∆−= , (1a)
kikkikiklikli T HaEκpεeD −−∆−= ε , (1b) where i, j, k, l = 1, 2, 3; σij and εij are the second rank tensors for mechanical stresses and strains; E
ijklC is the elastic stiffness matrix of the fourth rank tensor under the constant electric field conditions; αij is the thermal
3
expansion coefficients matrix; ∆T = T – T0 is the temperature difference from the stress-free temperature T0; eijk is the piezoelectric tensor; qijk is the piezomagnetic tensor; Di is the electric displacement current; pi is the pyroelectric vector, ij
εκ is the electric permittivity or dielectric tensor under the constant strain conditions; and aik is the magneto-electric coefficients tensor. The isothermal conditions and the absence of an electric or magnetic field reduce the number of relevant terms in Eq. (1a) and (1b).
It should be noted that if a multifunctional structure is made of an electromagnetically conductive material then an external magnetic field may induce an internal magnetic field in the material. A magneto-mechanical interaction is governed by a constitutive equation11
kikkikiklikli T HµEamεqB −−∆−= . (2) Here, i, j and k = 1, 2, 3; Bi is the magnetic induction vector for the magnetic field flux; εij is mechanical strain tensor, qijk is the piezomagnetic tensor; mi is the pyromagnetic vector, aik is the magneto-electric coefficients tensor, and µik is the 3×3 magnetic permeability tensor of material constants. Ek is the electric field vector as the negative gradient of the electric potential. Most of these terms appear in Eq. (1) for electromagnetic-mechanical coupling. Strength of Material-Field Interactions
The degree of electro-mechanical coupling in piezoelectric materials depends on the value of piezoelectric coefficients, which define the piezoelectric tensor, eijk:
=
312313323333322311
212213223233222211
112113123133122111
eeeeeeeeeeeeeeeeee
ijke , (3)
where the number of independent parameters depends on the mechanical and piezoelectric symmetries in a particular material.11 Piezomagnetic coupling is controlled by the corresponding piezomagnetic tensor,11 qijk, which has a tensor representation similar to Eq. (3). In addition to piezoelectric layers, some multifunctional structures may also contain either conducting fibers or conducting layers. In a magnetic field, B, such a structure may experience body forces, in particular, the Lorentz’s force, f:
)(1 BJf ×=c
, (4a)
where J is the current density and c is the speed of electromagnetic waves in the material. It is a volume
force caused by the energy associated with the magnetic fields generated. The magnetic field, B, results from the superposition of external or ambient magnetic field, Ba, and the current-induced field, BJ, i.e., B = Ba + BJ. The magnetic field flux, Bi, satisfies one of the Maxwell’s equations: div B = 0. Although, the applied electric field, E, may be zero, the time-dependent changes in the magnetic field may induce an electric field according to the Faraday’s law:
1c u r l c t
∂= −
∂BE . (4b)
If a multifunctional structure is a conductive plate, then an external magnetic field may also exert forces of the plate surfaces that are related to the Maxwell’s stress tensor:14
−= ij
2ji 2
141 δHHHTπij
; i, j = 1, 2, 3; (5)
where Hi are the components of the magnetic field, H, which experiences discontinuity across the surfaces that separate two media having different electromagnetic properties. The, so called, jump conditions require that the tangential components of electric and magnetic fields and the normal components of the magnetic induction vector are continuous at these interfaces.3-6 Such conditions affect the field-based Maxwell stress tensor and the resulting forces that enter into the equilibrium balance of all forces acting on a unit volume. Such discontinuity of external and internal electromagnetic fields at the deforming surfaces causes the components of the induced magnetic field to depend on both the transverse plate deflection and tangential displacements.
Modeling Considerations
Theoretical modeling of in-field deformation of multifunctional materials is inherently complicated by the fact that their constitutive properties and the resulting behavior are sensitive to the external electric and magnetic fields. Moreover, the deformation of such materials affects the field induced in a structure by the external fields and the local topology of the applied field itself. As a result, the governing equations in the mechanics of multifunctional materials are fully coupled. The field-material coupling is characterized by numerous constitutive parameters.11 To simplify the coupled governing equations, one may consider different special cases.3-6, 9, 12-15
Here the problem of material-field interactions for a multifunctional plate vibrating in electric and magnetic fields is addressed by using the linear method of superposition and geometric perturbation methodology under the assumption of linear elastic deformation and small multiphysics-based strains. First, the quasi-static linear deformation of an anisotropic piezoelectric plate in
4
a static electric field is examined. Then an engineering approximation for the in-field frequency correction is introduced for vibrating piezoelectric plates. Finally, a phenomenological model for the in-field vibration frequency shifts is presented for perfectly conductive isotropic plates in a parallel magnetic field. For this case, the modeling predictions are compared with the available theoretical results. A Piezoelectric Plate in an Electric Field
For anisotropic piezoelectric materials, the electro-mechanical terms in the constitutive equations (1) may be expressed in a matrix form as15
=
Eσ
κddS
Dε
εT
E
, (6)
where σ and ε are the second rank tensors for mechanical stresses and strains, SE is the elastic compliance matrix of the fourth rank tensor under the constant electric field conditions, E is the electric field vector, d is the 3×6 tensor of piezoelectric strain coefficients, D is the electric displacement vector, and κε is the electrical permittivity tensor under the constant strain conditions. The tensor of anisotropic piezoelectric strain coefficients, d, is the inverse of the piezoelectric tensor, e, in Eq. (1). The constitutive relationship shown in Eq. (6) is often presented in the materials literature as a part of data reduction for various characterization techniques because the strains are directly measured quantities.
The special case of thin plates allows for several assumptions. The plane stress approximation is valid, where σ3 = 0 and the shear is negligible, thus, the corresponding piezoelectric coupling can be neglected, i.e. d15 = 0. An external unidirectional electric field, E = (0, 0, E3), is such that the electric displacements D1 and D2 are zero, while D3 is given by
3332131333 )( σσσκ ε ddED +++= , (7) where d33σ3 = 0 and the two in-plane stresses can be written as
)(1 2121 νεε
νσ +
−=
Y and )(1 1222 νεε
νσ +
−=
Y . (8)
Here σ1 and σ2 are the stress components in the x- and y-direction, respectively, ε1 and ε2 are the corresponding strain components, Y is the Young’s modulus, and ν is the Poisson’s ratio. Eqs. (8) also represent the in-plane strain-stress components in the matrix Eqs (6):
3312
1
2212
1211
2
1 EdSSSS
EE
EE
+
=
σσ
εε
, (9a)
while the transverse strain component is given by
( ) 33321133 EdS E ++= σσε . (9b) Note that each strain component, εi, i = 1, 2, and 3; consists of a mechanical part, εiM, and a field-induced part, εiF, i.e. εi = εiM + εiF. From now on, we shall consider only cases when εiF << εiM, i.e., when the mechanical strains represent the main contribution and the field-induced strains have smaller effect on the total strain. The In-Field Vibration Frequency Corrections
Mechanical vibration properties of thin plates have been studied extensively by numerous authors.1, 2, 14, 16, 17 The vibration frequency of an elastic plate depends primarily on its geometry, mass and material stiffness as it is defined by
2
222
)1(12
−=
Lh
LYZ SS
ρνω , (10a)
where Y is the plate Young’s modulus, ν is the Poisson’s ratio, h and L are the plate thickness and length, ρ is the plate density and ZSS is the constant representing different boundary conditions (see Appendix). The in-field vibration frequency, ωΕ, of a thin plate exposed to an electric field, E = (0, 0, E3), depends on several geometric, material and other physical parameters:
=
31333
1,,;1,;1,1,d
dEL
hYfE ρνω . (10b)
Material parameters Y, ν, and ρ are not affected by the electric field since the property degradation is not considered. However, the geometric parameters h and L may be somewhat perturbed by the field effects transmitted through the piezoelectric strain coefficients, d33 and d31, respectively:
333hEdh =∆ and 331LEdL =∆ . (11) This phenomenon constitutes the displacement-based coupling in the electro-mechanical response of the plate.
The plate vibration frequency in Eq. (10a) then must be corrected for these field-induced geometric perturbations, i.e., ∆ω33 and ∆ω31:
2233333 )1(12 LYZ
LhEd SS
ρνω
−=∆ , (12a)
that is the thickness-dependent correction, and the length-dependent correction:
5
222331
31 )1(12)1(11
LYZ
EdLh SS
ρνω
−
+−−=∆ , (12b)
where the geometry-based corrections are calculated as
)()(33
hhh ωωω −∆+=∆ and )()(31
LLL ωωω −∆+=∆ . Finally, the in-field vibration frequency of a long thin plate is
++
−≈ 2
33133322 )1(
1)1(12 Ed
EdL
YZLh SS
E ρνω , (12c)
where the constant ZSS depends on the mechanical boundary conditions applied to the plate as specified earlier. The vibration frequency, ωΕ, can be normalized by the square root term in the Eq. (12c). A Conductive Plate in a Magnetic Field
An approach similar to the in-field geometry corrections, which have been discussed above, can be used to characterize the effective structural response of a conductive thin plate in a magnetic field. The results from such analysis can be compared to those of Librescu, Hasanyan and Ambur14 based on a geometrically nonlinear dynamic model for conducting elastic plates subjected to a magnetic field. The structural response of electromagnetically conductive plates is affected by the external magnetic fields and shows an increase in the in-field vibration frequency. In that analysis, the Kichhoff plate hypothesis is used along with the assumption of negligible free electrical charges (in the Ohm’s law) and the vanishing displacement current (in Ampere’s law). The Maxwell stress tensor is assumed without the electrical terms concerning relativistic corrections. As a result, the constitutive equation for the displacement current, Di, Eq. (1b), is not considered.
The trends predicted for the frequency-thickness dependence in the behavior of perfectly conducting thin plates14 are amenable to a pseudo-piezomagnetic analysis that is based on a constitutive relation similar to Eq. (6):
=
Hσ
µqqS
Bε
εT
E
~~
. (13)
Here, q~ is the tensor of pseudo-piezomagnetic strain coefficients that links the magnetic field, H, and the resulting strains, ε, and µik is the magnetic permeability tensor shown in Eq. (2). For anisotropic piezomagnetic materials, the degree of magneto-mechanical coupling depends on material parameters that define the piezomagnetic tensor, qijk:11
111 122 133 123 113 112
211 222 233 223 213 212
311 322 333 323 313 312
ijk
q q q q q qq q q q q qq q q q q q
=
q , (14)
which relates the field-induced stresses in a material and the magnetic field applied,11 similarly to the piezoelectric tensor, eijk, in Eq. (3). An effective “thickening” of an isotropic plate depends on the strength of the external magnetic field, H = (H1, 0, 0), and an effective pseudo-piezomagnetic strain coefficient, 31
~q . Such an effective anisotropy results from geometric “anisotropy” of a plate and the surface effects on the material-field interaction, which is geometry-dependent. The resulting in-field vibration frequency of a long thin plate is
hLHq
LYZ
Lh SS
H 13122~
)1(12+
−≈
ρνω . (15)
Here, the correction term for in-field vibration frequency is proportional to the normalized width or length of the infinitely long plate. This dependence reflects the fact that an external magnetic field may induce forces, Fs, which act on the surfaces of a conducting plate.14 These forces are related to the Maxwell stress tensor defined by Eq. (5). Moreover, the components of the magnetic field induced inside a thin plate depend on the transverse plate deflection and the surface location, where the components of the magnetic field, Hi, experience discontinuity across the surfaces that separate two media having different electromagnetic properties. These conditions on the field discontinuities affect the values of the field-based Maxwell-stress-tensor components and the resulting forces that enter into the equilibrium balance of all forces acting on a unit volume of a conducting plate.
Numerical Results
Numerical results generated by using the above analytical models illustrate the role of material parameters on the in-field geometry change and the subsequent in-field frequency shifts during the vibration of piezoelectric plates in an electric field. The frequency-thickness and the frequency-field dependence have been studied by using the newly developed approximation for the in-field vibration frequency corrections. By using a similar approach, an effective “in-field thickening” of a perfectly conducting plate has been evaluated for a magnetic field, along with the resulting in-field frequency shifts. These results are compared to the available theoretical predictions. In-Field Changes in Plate Geometry
The in-field changes of geometric parameters, which are illustrated in Fig. 1, have been routinely used for experimental characterization of piezoelectric coefficients that comprise the constitutive matrix for piezoelectric ceramics and polymers.13, 15 The degree of such changes depends on the electro-mechanical coupling, which is governed by the corresponding
6
piezoelectric coefficients. For example, the in-field changes in thickness are determined by Eq. (11a) and the value of the through-the-thickness piezoelectric coefficient, d33 (Table 1). In Fig. 2, corresponding results concerning the thickness-field dependence are shown. Since the piezoelectric coefficient, d33, shown in Table 1 links the transverse strain, ε33, with the applied electric field, (0, 0, E3), the design curves like that shown in Fig. 2 can be readily developed. Such predictions may be also used to approximate the voltage-expansion diagrams for the design of actuators and other devices.13
For anisotropic piezoelectric materials, the in-field changes in length are governed by piezoelectric coefficients that link the electric field components with the corresponding in-plane strain components. In the case of a vertically oriented electric field, (0, 0, E3), the changes in length of piezoelectric polymers are dependent on the value of coefficient, d31 (Table 2). This coefficient is determined by experimental measurements and Eq. (11b). The corresponding results concerning the length-field dependence are shown in Fig. 3. Such predictions are important for designing lateral constraints for piezoelectric structures and prevention of buckling. The importance of in-field changes in the thickness and length of plate-like multifunctional structures is further demonstrated by the vibration frequency dependence on the thickness-to-width ratio of such plates without any field effects as illustrated in Fig. 4. Table 1. Parameters affecting thickness of piezoelectrics
Type of Material Material parameters Values PVDF polymer D33 (10-12 C/N or m/V) -25a PF2V-β polymer D33 (10-12 C/N or m/V) 32b PZT ceramic D33 (10-12 C/N or m/V) 300a
a A. Preumont, Vibration Control of Active Structures, Kluwer, Dordrecht, 2002.
b B. Hilczer, J. Malecki, Electrets, Elsevier/PWN, Warszawa, p. 343, 1986.
Table 2. Parameters affecting the length of piezoelectrics
a S. Tasaka, N. Inagaki, T. Okutani, S. Miyata, Polymer, v. 30, p. 1639, 1989.
b G. T. Davis, in Polymers for Electronic and Photonic Applications, (ed. C. P. Wong), Academic Press, Boston, MA, p. 435, 1993.
c Z. Ounaies, J. A. Young, J. S. Harrison, NASA TM-1999-209359, 1999.
d A. Preumont, Vibration Control of Active Structures, Kluwer, Dordrecht, 2002.
e J. S. Harrison, Z. Ounaies, NASA CR-2001-211422, 2001.
In-Field Vibration of Piezoelectric Plates
The in-field changes of geometric parameters have been first recognized in the studies of oscillating piezoelectric crystals. Kopa18 and Tiersten19 investigated thickness vibrations in such crystals. The role of in-field thickness changes on the transverse vibration of a piezoelectric plate in a static electric field is examined here for a number of piezoelectric polymers in order to establish the trends in the frequency-geometry dependence for different materials. For moderate electric fields, the changes of geometric parameters are relatively small (Figs. 2 and 3) and the resulting frequency shifts are also small (Fig. 5). However, for higher electric fields, the vibration frequency shifts can be dramatic and may exceed 10% (Fig. 6). This effect is more significant for thicker plates that are characterized by higher thickness-to-width ratios (Fig. 7). In applications where the vibration frequency response is critical for material selection and geometric design parameters, the trends shown in Fig. 6 and Fig. 7 are especially important.
Vibration of Conductive Plates in a Magnetic Field
Vibration of multifunctional plate-like structures that are made of conductive materials can be affected by an external magnetic field. Librescu, Hasanyan and Ambur14 have presented trends for frequency-thickness dependence for the case of perfectly conducting elastic thin plates that vibrate in a magnetic field. The structural response of electromagnetically conductive media is shown to be significantly affected by the externally applied magnetic field. The transverse plate deflection and its tangential displacements caused by stretching and vibrations also influence components of the magnetic field induced inside a thin plate and, hence, its values that enter the field jump conditions3-6,14 on the plate surfaces. Here, the results of a phenomenological model for the in-filed vibration frequency corrections are compared to the theoretical predictions available for the frequency-thickness dependence for such plates.14
The results based on the pseudo-piezoelectric “in-field thickening” of a perfectly conducting plate are compared in Fig. 8 to both the mechanical frequency of the plate (frequency without any field effects) and the predictions of Librescu, et al.14 In the case of a simply supported plate in a parallel magnetic field, both theoretical predictions agree well over a wide range of thickness-to-width ratio values, h/L, that are greater than 0.06. Between the values of 0.04 and 0.06, the difference is less than 10%, while the discrepancy is below 20% for all values below 0.04. For a plate with one clamped end, the region of a quasi-linear dependence (where the results are identical) expands below the thickness-to-width ratio value of 0.05 (Fig. 9). However, the discrepancy between the two predictions increases for smaller values of the plate thickness. A similar trend
7
is observed when the second end of the plate is also clamped (Fig. 10).
The engineering model developed and its frequency results can be used to assist preliminary material selection and design considerations for wing covers, where the thickness-to-length ratio of skin panels may vary between 0.02 and 0.08.
Concluding Remarks
Constitutive modeling of multifunctional materials and the associated multiphysics coupling are examined for piezoelectric and electromagnetic effects on the in-field vibration of piezoelectric and conducting plates in electric and magnetic fields, respectively. New geometry-based corrections to the in-field vibration frequencies have been presented for the piezoelectric plate vibration in electric fields. By using this approach, the dominant trends in the frequency-thickness dependence have been examined in detail for both types of plates under different plate-end conditions. The modeling results for the in-field vibration frequency are provided for various piezoelectric polymers. For the in-field vibration of conductive plates in magnetic fields, a phenomenological model describes the vibration frequency shifts by employing an effective “in-field thickening” of conductive plates. The modeling predictions compare favorably with the available theoretical results.
Appendix: The Problem of a Vibrating Plate
Deformation of a thin elastic plate is governed by the Hooke’s law and equilibrium equations for the plane stress approximation2
)(1 2 yxx
E νεεν
σ +−
= , )(1 2 xyy
E νεεν
σ +−
= ,
and
xyxyxyEG ε
νετ
)1(2
+== ,
where σx, σy and τxy are the stress components; εx, εy and εxy are the strain components; E is the Young’s modulus, ν is the Poisson’s ratio, and G is the shear modulus. The strains are given by
2
2
xwzx ∂
∂−=ε ,
2
2
ywzy ∂
∂−=ε ,
yxwzxy ∂∂
∂−=
2
2γ ,
where z is the through-the-thickness distance and w(x, y; t) is deflection of the plate. The resulting equation of motion for a vibrating plate takes a form
04 =+∇ tthwwD ρ , where D is the plate stiffness, D = Eh3/12(1-ν2); h is the plate thickness, ρ is the plate density. When considering the time-harmonic oscillations of the plate (e.g., w = W(x,y) cos (ωt), the plate deflection, w, is such that
ww 20
4 ω=∇ , where the vibration parameter ω0 is defined as ω0 = (ρh/D)1/2ω. When the plate is supported at x = 0 and x = L, the deflection modes, W(x,y), can be described by W(x,y) = Y(y) sin (nπx/L) with n = 1, 2, 3, … . All deflection modes satisfy the boundary conditions: W(0,y) = W(L,y) = 0 for a supported plate, and additionally, for a simply supported plate,
0),(),0(2
2
2
2
=∂
∂=
∂∂
xyLW
xyW .
The equation reduces to
02 20
4
2
22
4
4
=
−
+
∂∂
−
∂∂ Y
Ln
yY
Ln
yY ωππ ,
which has the general solution16, 17
∑∞
=
+++
+−+−=
100
00
coshsinh
cossin)(
nnn
nn
yL
nDyL
nC
yL
nByL
nAyY
πωπω
πω
πω
,
where the constants An , Bn , Cn and Dn are determined by the appropriate boundary conditions for Y(y). For a simply-supported plate, the solution Y(y) also has to satisfy the above boundary conditions. After satisfying the boundary conditions and noting that ω0 = (ρh/D)1/2ω., the vibration frequencies for a simply-supported (SS) plate is given by
hD
LZ
Lh
LEZ SSSS
ρρνω 4
2
222
)1(12=
−= ,
where ZSS is a constant (6.097) for the SS boundary conditions. For clamped-supported (CS) and clamped-clamped (CC) plate, this constant is replaced by14, 16, 17 ZCS = 14.92 for the CS case or by ZCC = 31.5 for the plate clamped at x = 0, L.
The mechanical frequency, ω, may serve as a reference data for the in-field vibration of multifunctional plates.14 Note that the kinetic energy, T, of a transversely vibrating plate is2
8
∫∫
∫∫ ==
A
At
dxdyyxWth
dxdytyxwhT
),()(sin21
);,(2
222
2
ωωρ
ρ
and it is proportional to the square of the frequency, ω2.
Acknowledgement
Useful discussions with Dr. D. Hasanyan (Virginia Tech) are gratefully acknowledged.
References [1] Lord Rayleigh (J. W. Strutt), “On the Free Vibrations
of an Infinite Plate of Homogeneous Isotropic Elastic Matter,” Proc. London Math. Soc., V. 20, p. 225, 1889.
[2] S. Timoshenko, Theory of Plates and Shells, McGraw-Hill Book Co., New York, 1940.
[3] L. D. Landau, and E. M. Lifshitz, Electrodynamics of Continua I: Foundation and Solid Media. Springer-Verlag, 1984.
[4] A. C. Eringen and G. A. Maugin, Electrodynamics of Continua Media, Pergamon Press, 1990.
[5] I. T. Selezhov and L. V. Selezhova, Waves in Magnetohydroelastic Media, Naukova Dumka, Kyiv/Kiev (in Russian), 1975.
[6] S. A. Ambartsumian and G. Y. Bagdasaryan, Electroconductive Plates and Shells in a Magnetic Field, Nauka, Moscow (in Russian), 1996.
[7] R. D. Mindlin, “On the Equations of Motion of
Piezoelectric Crystals,” in Problems of Continuum Mechanics, Society of Industrial and Applied
Mathematics (SIAM), Philadelphia, pp. 282-290, 1961.
[8] H. F. Tiersten and R. D. Mindlin, “Forced Vibrations of Piezoelectric Crystal Plates,” Quart. Appl. Math., V. 20, p. 107, 1962.
[9] H. F. Tiersten, Linear Piezoelectric Plate Vibrations, Plenum Press, New York, 1969.
[10] Introduction to ANSYS for MEMS: Training Manual, ANSYS Inc., 2001.
[11] J. F. Nye, Physical Properties of Crystals, The Clarendon Press, Oxford, 1960.
[12] J. S. Harrison and Z. Ounaies, Piezoelectric Polymers, NASA/CR-2001-211422, NASA Langley Research Center, Hampton, VA, 2001.
[13] A. Preumont, Vibration Control of Active Structures: An Introduction. Kluwer Academic Publishers, Dordrecht, The Netherlands, 2002.
[14] L. Librescu, D. Hasanyan and D. R. Ambur, “Electromagnetically conducting elastic plates in a magnetic field: Modeling and dynamic implications,” Proc. of the 43rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, Paper AIAA-2002-1692, Denver, CO, April 22-25, 2001.
[15] T. L. Jordan and Z. Ounaies, Piezoelectric Ceramics Characterization, NASA/CR-2001-211225, NASA Langley Research Center, Hampton, VA, 2001.
[16] S. P. Timoshenko, D. H. Young and W. Weaver, Jr., Vibration Problems in Engineering, John Wiley & Sons, New York, 1960.
[17] A. W. Leissa, Vibration of plates, NASA SP-160, NASA, Washington, D.C., 1969.
[18] I. Kopa, “Thickness Vibrations of Piezoelectric Oscillating Crystals,” Physics, V. 3, p. 70, 1932.
[19] H. F. Tiersten, “Thickness Vibration of Piezoelectric Plates,” J. Acoust. Soc. America, V. 35, p. 53, 1963.
,
a. Thickness change
b. Length change Figure 1. Schematic of the influence of an electric field on a piezoelectric plate.
Electricfield(kV/mm)
Cha
nge
inth
ickn
ess
(m
)
0 25 50 75 1000
5
10
15
20
PF2V-
PZT
β
µ
PVDFd33<0
Figure 2. In-field changes a 5 mm thick plate made of different piezoelectric materials for various values of an electric field, E3.
Electric field(kV/mm)
Cha
nge
inle
ngth
(m
)
0 25 50 75 1000
50
100
150
200
250
µ
Nylon7
PVDFa
PTrFE
PVDFb
PVDCN-VAc
Figure 3. In-field changes in a 0.1 m long plate made of different piezoelectric materials polymers for various values of an electric field, E3.
Thickness-to-widthratio, h/L
Nor
mal
ized
Freq
uenc
y
0 0.025 0.05 0.075 0.10
5
10
15
20
25
30
35
40
45
50
55
Clampedend
Clampedends
Supportedends
Figure 4. Dependence of the normalized frequency for simply supported and clamped thin elastic plates with increasing thickness-to-width ratio.
hL ∆L
E3
hL
∆h
E3
x
y
10
Typeofpolymer
In-fi
eld
frequ
ency
chan
ge(%
)
0
0.1
0.2
0.3
0.4
0.5
PTrFE
PVDCN-VAc
Nylon7
PVDFb
Figure 5. In-field frequency reduction for a simply supported thin plate made of different piezoelectric polymers in an electric field, E3, of 100 kV/mm.
Electricfield(MV/mm)
In-fi
eld
frequ
ency
chan
ge(%
)
0 1 2 30
2
4
6
8
10
12
Nylon7
PVDFb
PTrFE
PVDCN-VAc
Figure 6. In-field frequency reduction for a simply supported thin plate (h/L = 0.1) made of different piezoelectric polymer materials.
Thickness-to-widthratio, h/L
Nor
mal
ized
vibr
atio
nfre
quen
cy
0 0.025 0.05 0.075 0.10
1
2
3
4
5
6
7
8
9
10
PVDFb
Inactiveplate
PVDCN-VAc
PTrFE
Nylon7
Figure 7. Influence of the thickness-to-width ratio on the in-field vibration frequency of a simply supported piezoelectric plate in an electric field, E3, of 2 MV/mm.
Figure 9. Influence of the thickness-to-width ratio on the in-field vibration frequency of a clamped-simply supported conductive plate in a magnetic field.
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