Dynamic and Vibration Analysis of Nano-Composite Hollow Cylindrical with Piezoelectric Layers in thermal load by DQM Method

Cylindrical structures in many engineering constructions are used, so proper analysis of cylinder behavior under various conditions is important. In this paper we investigate the dynamic and vibrational response of carbon nanotube-reinforced carbon nanotube reinforced composite cylinder shell with two piezoelectric layers. The equations of motion are extracted by assuming Sanders shell theory using the Hamiltonian principle. The Quadrature Differential Method (DQM) is used to solve the equations of motion. Frequency changes and dynamic response (middle layer displacement) have been studied by varying geometric and piezoelectric parameters. Among piezoelectric parameters, the parameter of C11 has a lower effect than the effective transverse coefficient of e31 in the frequency response. Other piezoelectric parameters have very little effect on frequency. A mild initial heat field increases the displacement amplitude by decreasing the strength and brittle of the material. But the heat field reduces the hardness of the matrix to a greater extent and increases the frequency and amplitude of the displacement.


Introduction
Composite cylinders are one of the most used structures in engineering structures.
Among the materials used in the last two decades to improve the properties of composites are nanoparticles and piezoelectric layers. Piezoelectricity is a linear variable related to the microscopic structure of solids. Some ceramics become polarized when subjected to pressure. This linear and apparent phenomenon is attributed to the direct piezoelectric effect. The direct piezoelectric effect is always accompanied by the reverse piezoelectric effect and occurs when a piezoelectric component is placed in an electric field. When external pressure forces neutralize the dipole moment. When an external stress is applied to the piezoelectric component, the charges are shifted in such a way that the polarity of the dipole disappears.
Accordingly, a polarized grid is created and the result is an electric field. Thus the definition of piezoelectricity is the generation of electric charge in the material by mechanical pressure (direct effect), and, conversely, the creation of a mechanical stress in response to an electric field. By changing the direction of the electric field it can cause pressure or tensile mechanical stress [1]. Piezoelectric materials are divided into two types of piezo-ceramics and piezo-polymers, such as Polyvinylidene fluoride [2]. The major applications of piezoelectric materials are briefly switching applications, keyboard production applications, audio applications as both sensors and actuators, Is live systems and etc. One of the studies in the field of composite cylinders with piezoelectric layer is Chen et al. [3] in 2007 who investigated three-dimensional free vibrations of a piezoelectric cylindrical tank containing compressible fluid and its governing equations by space method computed. Santos et al. [4] in 2007 a finite element model for flexural and vibration analysis of composite shell with piezoelectric sensor and actuator layers proposed. In 2008, Beigloo and Kani [5] the static 3 composite sheet with piezoelectric layers on two lateral surfaces studied. In 2010, Beigloo and Kani [6] vibrations of a multi-layered piezoelectric cylindrical shell and obtained the governing equations by the space-state method investigated. Eftekhar et al [7] Vibration smart control analysis of a temperature-dependent functionally graded (FG)-carbon nanotubes (CNT)-reinforced piezoelectric cylindrical shell embedded in an orthotropic elastic medium is investigated. Jie Xu and Shuyu Lin [8] the threedimensional coupled vibration of composite cylindrical piezoelectric transducers, the cylindrical piezoelectric transducer consists of an inner axially polarized piezoelectric ceramic cylinder and an outer metal cylinder with the same height studied. Wang et al [9] the exact theoretical models for radially polarized multilayer piezoelectric cylindrical transducers by taking into account the electrodes and electrical connections of piezoelectric layers studied. Yanqing WANG et al [10] the nonlinear free vibration of piezoelectric cylindrical Nano shells studied. Given that heat load is one of the common loads in the industry and affects components at various positions, research has also been conducted on the thermal loading on composite cylinders that can be found in Mousavi et al. [11] the dynamic and vibrational response of a thermo-elastic coupler in a multilayer cylindrical shell that first undergoes a low-temperature heat field and then enters a thermal shock using the 4th order Runge-Kutta method investigated.
Heydarpour et al [12] a coupled thermo-elastic approach based on the Lord-Shulman (L-S) and Maxwell's formulations to study the wave propagation in functionally graded (FG) cylindrical panels with piezoelectric layers under a thermal shock loading studied.
Rahmani and Moslemi Petrudi [13] the vibrational and dynamic response of the cylindrical shell of a nanocomposite under heat shock using the DQM method investigated. In this paper, the vibrational and dynamic response of composite cylinders reinforced with carbon nanotubes and piezoelectric bilayers under thermal load is investigated.

Statement of the problem
This paper a cylindrical shell with length L and the number of n composite layers of total thickness H and an outer layer of layer thickness of ha and a sensor layer of inner surface of shell of thickness hs investigates. The radius from the center of the shell to the middle of the composite layers is R and the radius from the center of the shell to the middle of the sensor layer is R1 and the radius from the center of the shell to the operating layer is R2. The middle surface of the cylinder is inserted into the cylindrical coordinate system x and θ and z and the distance from the middle surface is measured by the z coordinates whose positive direction is inside the cylinder and the

Extraction of Equations of Motion
To derive the equations of motion, the Sanders theory of thin shells is used. Thus the displacement field in the crust can be expressed by the following equations: In these relations u, v and w are the displacement components of the crust's middle layer in the order of x, θ, and z, respectively, and ψx and ψθ are rotations perpendicular to the middle surface around x and θ. Sanders developed an 8th-order shell theory from the principle of virtual work that the stress-strain relations for a circular cylindrical shell can be expressed as follows [14]: In these equations, ԑx, ԑθ and ԑ xθ are the middle layer membranes and κx, κθ and κxθ are the bending strains, and the ԑxz and ԑθz transverse shear strains. Assuming axial symmetry and force, the deformations in the lateral direction are small and neglected, and the relation (2) is simplified as follows [15]: The equations of motion for the dynamic behavior of a circular cylindrical shell, assuming axial symmetry can be written from the general form of Sanders shell theory, as follows: [16]: The result of the moment and the coefficients (I ( =1 and 2) ) in equations (4) are defined as follows:

Composite layer relations
In shell and plate theory, it is appropriate to integrate the stresses along the shell thickness to introduce the resulting force and moment. The basic equations of a shell for a non-isotropic material are assumed as follows [18]: Where the strains are inserted from relation (3)  Now by inserting the relation (10) in the relation (9) and then the relation (9) in the relation (8) and finally by the relation (7), the equations of force and moment for the composite layer are obtained. To obtain the equations of motion, Hamilton's principle is used for these equations and the general form of this principle is as follows: To solve the thermos-elastic coupling problems, it is necessary to solve the equations of motion and energy equations simultaneously.
In this respect, T1 and T0 are functions to be obtained from the equation system and Cv is the specific heat capacity per unit volume. The Galerkin method is used to obtain two non-dependent equations of thermal conductivity of the shell from Equation (12) and with its average at z-shell thickness, assuming a linear distribution in the shell thickness given by Equation (13). The two variables T1 and T0 fall into the energy equations. For the multilayer cylindrical shell under heat shock with axial symmetry and uniform distribution along x, the energy equation 8 in terms of displacement terms is summarized [19]: The following two integrals give two independent energy equations using two independent functions T1 and T0:

Sensor layer relationships
Since there is no external electrical charge in the sensor, the electrical displacement of this layer will be zero along the radius, so for the sensor layer: Accordingly, by equating the electric field intensity in the sensor and the experimental relation E i = φ i [21] integral with z: According to the second-order shell theory, the power-moment relationship of the sensor layer is defined as follows [22]:

Operator Layer Relations
To obtain the equations governing the operator, it can be assumed that the distribution of the electrical potential within the operator is as a first-order function [23] in the following relation: 12 Now that the potential difference is required in the operator, then the following electrical boundary conditions are considered: Using the above boundary conditions and using the Maxwell [23] relation, the following relation will be obtained for the distribution of the electric potential: Accordingly, Ez [23]: Accordingly, the operator stress relation will be obtained as follows: According to the second-order shell theory, the force-moment relations of the operating layer are defined as follows [22]:

The equations of motion of the cylindrical shell
By inserting equations (7) and (25) and (32) in relation (5) and then inserting relation (5) in relation (4), the equations of motion of the cylindrical shell will be obtained as follows: ( The coefficients A i (i=1 to 11) and I i (i=1 to 2) are defined as follows: In the energy equations and motion equations some parameters are omitted because of axial symmetry = 0 So the three equations of motion and the two energy equations remain as partial derivatives which must be written in the matrix ̈+̇+ = and the final form of the ordinary differential equations is as follows:

Solution by DQM Method
DQM is one of the numerical methods in which the weighted coefficients of the governing differential equations are converted into first-order algebraic equations.
Thus, at each point, the derivative will be expressed as a linear sum of the weighting coefficients and the function values at that point and the other points in the domain and in the coordinate axis. In general, in these methods, the one-dimensional function derivative is defined as follows [24][25]: ,also L1(x) is a derivative of Lagrange's orthogonal polynomial function of order N and is defined as ( ) = ∏ ( − ) =1 , By substituting Eq (38).
(39) For ij ,i,j=1,2,…,N where Aij represents the weighting coefficient of order r corresponding to it grid point. The boundary conditions for a simply-supported shell are given as: (42) at x =0 and x = L u = v = w = M xx = Ѱ = 0 clamped-free circular cylindrical shell. The boundary conditions for this case are given as: And for clamped-clamped boundary conditions:

Results and Discussion
The properties of carbon nanotubes are presented in Table 1 and the values of carbon nanotube performance parameters are presented in Table 2. Table 1 Material properties of single-walled carbon nanotube(10,10)( 12 = 0.175) [17] .  Figure 2 shows a comparison of the results of the main frequency changes at different L / R ratios with the reference paper [27]. The cylindrical shell specifications of this article are presented in Table (3). A comparison of the dimensionless frequency for the simple supporting state at the two ends of the cylinder is shown in Figure 2.   Table 4 presented.    21 Figures 6 to 8 shows the frequency variations of these geometrical parameters for the two support modes with two end-fixed and two end-simply supporting. Figure 6 shows that as the cylinder radius increases, the frequency decreases. The ratio of the radius to the thickness of the shell is one of the parameters commonly studied in the studies. As the ratio increased, the frequency increased.

22
The effect of the cylindrical shell length on the frequency is shown in Figure 8. In the following, the effects of piezoelectric coefficients on the main frequencies are discussed and the results for the different piezoelectric coefficients are given in Figures   9 to 11. The effect of some coefficients is very small. The effect of piezoelectric coefficients with increasing L / R ratio increased. With the natural frequency C given the diagram having a direct relation to the value 11 and also the longer the ratio of the length of the beam thickness to the natural frequency the lower the C, the effect of 11 is the influence of the effective piezoelectric coefficient in Figure 7. As the effective piezoelectric transverse coefficient increases, the natural frequency will increase. To verify the boundary conditions and the support, the variations are investigated and finally the dimensional frequency changes of the first 4 modes relative to the composite density are presented in Figures 12 and 13. According to the graphs, as the composite density increases, the mass of the beam increases and its natural frequency decreases for all boundary conditions.    The effect of other piezoelectric coefficients on the main frequency changes is negligible, so their variation is neglected. According to the diagrams of the piezoelectric coefficients it is clear that most of the changes in frequency are caused by the change of the coefficient C11. Figure 14 shows the effect of the temperature change on the natural frequency, according to the results, as the temperature changes, the natural frequency decreases. The thermal field does not have much effect on the original frequency (first frequency) but increases the frequency of the subsequent frequencies and reduces the frequency. Figure 15 shows the effect of increasing the thermal load on the displacement of the composite middle layer. The presence of a mild initial heat field increases the displacement amplitude by decreasing the strength and brittle of the 24 material. But the heat field reduces the hardness of the matrix to a greater extent and increases the frequency and amplitude of the displacement.

Conclusion
In this study, the effect of geometrical parameters on the vibrational and dynamic response of carbon nanotube-reinforced composite cylinder shell with piezo layers was investigated. The effects of geometrical parameters such as the angle of the fibers, the change of cross-section dimensions and the change in the support as well as the change of the piezoelectric coefficients were investigated. The results show that among the piezoelectric parameters the parameter C11 has less effect than the transverse effective coefficient e31 on the frequency response. The effect of other piezoelectric parameters on these two parameters in the low frequency response is also evaluated. The value of C11 is directly related to the natural frequency, and the 25 lower the length to beam ratio, the greater the effect of C11 on the natural frequency.
As the effective piezoelectric transverse coefficient increases, the natural frequency will increase. The more boundary conditions are, the greater the natural frequency due to the increased rigidity of the structure. Also, as the density of the composite layer's increases, the natural frequency decreases. The presence of a mild initial heat field increases the displacement amplitude by reducing the strength and brittle of the material. But the heat field reduces the hardness of the matrix to a greater extent and increases the frequency and amplitude of the displacement.