Analysis of spatial vibrations of piezoceramic eccentric cylindrical shells interacting with an annular fluid layer

The work is devoted to a numerical study of the behavior of vertically oriented eccentric (non-coaxial) electroelastic shells, containing a fluid in the gap between them. The solution of the problem is found in the framework of a three-dimensional formulation using the finite element method. The shells are made of the piezoelectric material polarized in the radial direction. Thin-walled structures are considered in the framework of the classical theory based on the Kirchhoff – Love hypotheses, as well as the equations of linear electroelasticity. The distribution of the electric potential through the thickness is assumed linear. The fluid is considered within the framework of a potential theory. The study of the low natural frequencies and vibration modes is carried out for different variants of the boundary conditions at the edges of the shells, the level of fluid in the gap and the eccentricity of the inner shell. The influence of the electric boundary conditions specified on the parts of the shells surfaces covered with electrodes is estimated.


INTRODUCTION
n the last few decades, the wide-spread use of structural elements in the form of nested shells with parallel axes, interacting with a fluid or gas contained in the gap between them, gave an impetus to the appearance of many publications, in which such systems are the subject of research. An extensive bibliography, covering recent studies into the dynamic behavior of coaxial shells interacting with both stationary and flowing fluids, is given in review [1] and monograph [2]. Smaller reviews extending this list of cited papers and including some unlisted pioneering studies are given in recent articles of the authors [3,4]. However, in the context of this study, the matter of particular current interest is a few works that take into account the axis misalignment of the inner and outer shells [5][6][7][8][9]. In [5], it was shown that the I axial eccentricity affects the vibration process only in the case of a narrow annular gap between infinite shells. The analysis of the added liquid masses for two infinite cylinders with eccentricity, containing either an ideal or a viscous fluid in the gap between them, was performed in [6] and [7]. As noted in [7], the added masses and viscous damping coefficients, which grow with increasing eccentricity, are of greater practical interest, because they are widely used in studying hydroelastic vibrations and stability of structures in various engineering applications. The analysis of natural frequencies of cylindrical shells with an eccentricity immersed in a rigid cylindrical container was performed in works [8,9]. The first of these papers considers the system, in which the non-viscous compressible fluid is contained only in the annular gap between the container and the shell, whereas the second paper examines the situation when the inner shell also contains a fluid. It has been demonstrated that with increasing eccentricity, the frequency of vibrations decreases, which is most significant for low circumferential and axial modes. Particular attention should also be paid to publications dealing with the analysis of coaxial shells partially filled with a fluid [10][11][12][13][14][15]. In all these papers, the studies of the dynamic behavior of coaxial shells are made with account of the free-surface sloshing effect, but do not include numerical simulation, which would allow one to estimate the effect of this boundary condition. Note that here we cite only those works which analyze the effect of partial filling of shells on the dynamic characteristics of the system. It is shown that an increase in the fluid level causes a decrease in the natural vibration frequencies. A more general review of papers aimed at studying hydroelastic vibrations of partially filled coaxial cylindrical shells is given in monograph [16]. In the above mentioned works, the solution of the problem is searched for in the context of a two-dimensional (flat) or axisymmetric problem formulation. In the case when non-coaxial structures are set in the vertical or horizontal position and partial filled with fluid, their symmetry with respect to the circumferential coordinate is disturbed, which necessitates the use of more complex spatial models. As far as the authors know, such studies, including those that take into account the elasticity of both shells, are not presented in the literature. Electroelastic materials embedded in or attached to engineering structures have been used for rather long time in various fields of technology in order to improve the performance characteristics of a product. Intellectual systems of passive or active control of dynamic behavior built on the basis of their peculiar properties serve to reduce the level of mechanical vibrations or acoustic noise. The bibliography of basic works dealing with the analysis of various approaches used in modeling the structures with piezoelectric elements, and presenting examples of their practical use can be found in monograph [17]. In [18], the authors come to a conclusion that successful practical application of piezoelectric elements coming in contact with a fluid or gaseous medium is possible only with an adequate description of their coupled response. In the case of thin-walled structures this can be achieved with the model proposed in [19]. The model is based on a singlelayer representation of the elastic body displacements and a layered description of piezoelectric properties. This approach was used for modeling the plates and single shells, including those interacting with a quiescent fluid, and also the elastic structures with external patches made completely or partially from the electroelastic material [20][21][22][23]. For a linear distribution of electrical properties through the body thickness, this method, when used in the framework of finite element modeling, allows us to simplify the solution by eliminating the electrical components at the element level. In the literature, there are no papers describing the application of the above method to coaxial shells. Thus, the purpose of this work is to analyze the natural frequencies and the corresponding modes of vibrations of elastic and piezoelastic eccentric (non-coaxial) cylindrical shells at different values of the annular gap width and different levels of ideal compressible fluid contained in this gap. The solution of the problem is found using a modified version of the finite element algorithm developed in the previous works for evaluating the hydroelastic interaction of single shell structures with arbitrary cross section completely or partially filled with fluid [24][25][26].

STATEMENT OF THE PROBLEM AND CONSTITUTIVE RELATIONS
e consider a structure consisting of two vertically located eccentric (non-coaxial) shells of length L, radiuses (1) R and (2) R , and thicknesses (1) h and ( 2 ) h ( Fig. 1) made of electroelastic material (piezoceramics). Hereinafter, the superscripts "(1)" and "(2)" characterize the inner and outer shells, respectively. The axis of rotation of the inner shell is displaced laterally with respect to rotation axis of the outer shell by the value a   ( ) a R R , and the space between them is filled with an ideal compressible fluid to the level H   (0 ) H L . It is necessary to investigate the natural frequencies and modes of vibrations of the system under different kinematic and electric boundary conditions, and geometric parameters (the size of the annular gap, the height of the fluid, eccentricity). An ideal compressible fluid is considered in the framework of a potential theory, in which the wave equation in the case of small perturbations takes the following form [27]: where  is the velocity potential, c is the speed of sound in the liquid,  is the differential Nabla operator,  t is the time derivative. Using the perturbation velocity potential as an unknown function allows to take into account both the flow or the rotation of the inviscid compressible fluid with a slight modification of the numerical algorithm.
In the case of complete filling of the shell with a fluid, the following boundary conditions are prescribed for the velocity potential: In the case of partial filling, it is assumed that the free surface of the fluid free S remains stationary and is free from the dynamic pressure and surface tension. The corresponding boundary condition is given as [28]   : 0 On the wetted surfaces where ( ) where ˆ and ( ) i w are the approximations of the velocity potential and the normal components of the vectors of shell displacements; n F and f m are the basis functions and their number, respectively.
In the general case, the behavior of the electroelastic body is described by the equations of state, piezoelectric effect, and the electrostatic relations [29,30]    T cε e E, (6) Here: , ε  are the stress and linear strain tensors; E and D are the vectors of electric field intensity and electrical induction; c, e, d are the matrices of elastic constants, piezoelectric and dielectric coefficients;  is the electrostatic potential. In the case when a thin-walled structure with a radially polarized field is in the plane stress state, Eqns. (6)- (9) can be simplified. In particular, of all components the vectors of the electric field strength and nonzero electrical induction retain only 3 E and 3 D , and Eqns. (6) and (7) are written as follows [31,32]: where 3 3 33  66  66  3  3  33 3 33  33  33  33 33 , , , Here we proceed from the assumption that the shell surface is covered with thin weightless electrodes, which can either be shorted, which corresponds to the boundary condition  open circuited). From Eqns. (6)-(9) the following integral relation can be obtained [21] In the shell simulation, it is assumed that their curved surfaces can be represented with a reasonable degree of accuracy as a set of flat segments [33]. The strains in each segment are determined according to the classical theory of thin plates [34].
The corresponding relations in the Cartesian coordinates   , , x y z , which are tied in with the lateral surfaces of thinwalled structures, are written as Hereinafter, the over-line means that the quantity is defined in the coordinates   where the coefficients entering into the stiffness matrix  ( ) and the structure of the matrix ( ) i G will be presented below. A mathematical formulation of the problem of the dynamics of elastic shells is based on the variational principle of virtual displacements, which, with account of relation (4) and the work made by the inertial forces, can be written in the matrix form as where ( )

NUMERICAL IMPLEMENTATION
ollowing [21], let us decompose the shells through its thickness into N layers and represent the component of the electric field 3 E for each layer as is the coordinate measured from the middle surface of the shell; is the difference in the electrostatic potentials between the upper and lower surfaces of the layer, which, as well as the components of the vectors of shell displacements ( ) i u and the velocity potential  , turns to be an unknown variable.
The numerical solution of the problem is found using the finite element method (FEM) [33]. To describe the velocity potential ˆ,  the basic functions n F , and membrane displacements of shells ( ) ( ) ( , ) i i u v we use the Lagrange shape functions with linear approximation, and for bending deflections of shells ( ) i w we use the Hermite non-conformal shape functions The discretization of the fluid and shell regions is made using spatial prismatic (8-node brick) and flat quadrangular (4-node plane) finite elements, respectively. Expressing the unknown variables in terms of their nodal values, we arrive at the following matrix relations where Using the introduced relations the matrix ( ) i G in Eqn. (15) is generated in the following way where From Eqns. (5), (13) and (17) taking into account (18)- (20), we derive with the aid of the standard FEM operations a coupled system of equations, which can be used to describe the interaction of electroelastic shells with an ideal compressible fluid. The system can be represented in a matrix form as (1) Here ( ) i w N are the shape functions for the normal component of the nodal displacement vector of the shells.
Since some matrices in (23) are written in the Cartesian coordinates   , , x y z , the derivation of the solution in the global coordinates, which have been used to specify the problem geometry and to write the constitutive equations for the fluid, involves the transformation of the displacements and rotation angles using the matrix of the directional cosines γ of size  3 3 in the following way: During the numerical implementation, at each node of the finite element of the shell it is necessary to determine six unknowns (three displacements and three angles of rotation). Therefore, the transformation of conventional matrices is performed using the diagonal matrix            diag , , , , , , L : , .
Expressing the quantities associated with the electrostatic potential in terms of the shell displacements yields, instead of (22), the system of equations with respect to the displacements of electroelastic bodies and the velocity potential of the fluid (1) which can be written in a more compact form Representing the motion of shells and fluid in the exponential form we obtain instead of (28) the following expression Here,  ( ) i u and  ( ) i  are some functions of the coordinates;   i    is the characteristic coefficient, in which  is the natural frequency of vibrations,  is the quantity responsible for the system damping, and   i 1 is the imaginary unit. The system of Eqns. (30) is reduced to an asymmetric generalized eigenvalue problem where I is the unit matrix. The calculation of the eigenvalues of the system of Eqns. (31) is carried out using the ARPACK procedures, which are based on the implicitly restarted Arnoldi method [35].

RESULTS
few numerical examples were considered to investigated the behavior of vertically oriented coaxial cylindrical shells, containing an ideal compressible fluid in the annular gap between them. The system parameters are listed in Tab. 1. The shells were made of PZT-5H piezoceramics, the physical and mechanical characteristics of which are given in Tab

Verification of the numerical model
The assessment of reliability of the constructed algorithm for the numerical investigation of an electroelastic thin-walled body was made by comparing its performance with the results of work [21]. Here the simply supported at both ends   ( 0) v w shell made of PZT-5H piezoceramics was investigated (Tab. 3). The boundary conditions for the fluid were specified as follows: 0  Table 3: Geometrical parameters of electroelastic shells and physical properties of the fluid [21].
A comparison between the natural frequencies of the empty shell and the shell filled with an ideal compressible fluid is given in Tab. 4. The number of half-waves in the circumferential and meridional directions is denoted by j and m, respectively. According to the data presented in the table, there is good agreement between the results obtained by analytical and numerical solution of the problem.
The assessment of the reliability of the results obtained for coaxial shells   ( 0) made of isotropic elastic material and containing fluid in the gap between them was carried out by comparison with the results of analytical solution presented in [36]. In this case, the matrices ( )

Without fluid With fluid
h h h, m 0.002  Note that in [36], it was first shown that for two elastic coaxial shells, along with the in-phase (the direction and number of meridional half-waves m coincide) and out-of-phase (opposite directions) vibration modes there also exist mixed modes (the number of half-waves varies). From the data presented in Tab. 6, we may conclude that the results obtained in the framework of the model used in this paper agree fairly well with the data of the analytical solution presented in [36]. Fig. 2 shows the graphs of the lowest natural frequencies of vibrations  as a function of the dimensionless displacement of the inner shell  at different filling levels  for the gap  1 10.

Natural frequencies of electroelastic shells
k The results were obtained for two configurations of electroelastic bodies under different boundary conditions. As is evident from the presented data, in the presence of eccentricity the frequencies decrease regardless of the choice of electrical and kinematic boundary conditions at the edges of the shells. According to Refs. [6,7,9], the reason for such a behavior is the growth of the added fluid mass. In this case, a decrease in the level of a fluid and, consequently, a decrease in its the added mass contributes to a stronger dependence of the lowest vibration frequency on the axial misalignment of the shells and eventually leads to their greater relative decrease. The cantilevered structures are less sensitive to the eccentricity of the inner shell, so that over a fairly wide range of the parameter  a decrease in the frequencies is inessential. Similar dependencies for other values of the annular gap k do not demonstrate qualitative differences, which is supported by the data given in Tab. 7. It can be seen from the above data that the curves are of a qualitatively different character, which is associated with the boundary conditions at the edges of the shells, the size of the annular gap and the level of filling with a liquid. The higher is the fluid level in the annular gap between the shells, the larger is the relative difference in frequencies  . Fig. 6 shows the dependences, which allow us to estimate the variation in the lowest natural frequencies of vibrations with the level of the fluid in the gap at  1 10 k and various kinematic boundary conditions. The curves presented in the figure reflect the growth of frequencies with a decrease in the filling level. It can be seen that they are of qualitatively similar character for shells with different boundary conditions. However, in the case of rigid clamping at both ends (Fig. 6a), there is a range of the fluid levels, at which the frequencies practically do not change regardless of the displacement of the inner shell .
 For cantilevered shells (Fig. 6b), such a behavior is not observed. In this case, the differences in the frequencies obtained at different electrical boundary conditions are insignificant.    k From the presented data it is clear that the amplitude of the displacements on the wetted areas of the shell surfaces exceeds the displacement amplitude on the areas not contacting with the fluid. A decrease in the level of the fluid also leads to a decrease in the maximum amplitudes of the displacements occurring in the shells. Moreover, in the case of the eccentricity, the displacement maximum occurs in the areas where the distance between the thin-walled bodies is minimal   ( 2) (1) ( ) R R a and gradually decreases towards the region where the spacing between them is maximum  

CONCLUSION
he dynamic behavior of eccentric (non-coaxial) electroelastic cylindrical shells, containing a quiescent fluid in the annular gap between them, was studied in a three-dimensional formulation with the use of the proposed mathematical model, the numerical implementation of which was based on the finite element method. The analysis of the obtained results allowed us to estimate the influence of the size of the annular gap between the shell, the level of the fluid in it, the eccentricity of the inner shell, and the electrical and kinematic boundary conditions on the lowest natural frequencies of vibrations and the corresponding mode shapes. It was demonstrated that both an increase in the misalignment of the shells and the fluid level in the annular gap and a decrease in the size of the gap itself are responsible for a decay of the lowest vibration frequencies. The growth of the lowest frequencies is provided by setting on the electrode covered surfaces of the shells the electrical boundary condition peculiar to the open-circuited configuration.