1 Introduction

Rotating shells are frequently used in different industries such as aerospace and oil industries. As any shell rotates, unbalancing would cause harmonic force. If the frequency of this harmonic force equals the natural frequencies of the shell, the resonance destructive phenomenon happens. Therefore, investigating the vibration of a rotating shell has been of interest to researchers. Moreover, sandwich structures and functionally graded materials (FGMs) have received much more attention recently. Integrating piezoelectric materials as sensor and actuator with different structures in order to measure and control the vibrations of the systems, FGMs would make a sandwich structure. Moreover, FGMs have been studied more since they could tolerate high-temperature gradient simultaneously with the high pressure.

Li and Lam^[1] studied the free vibration analysis of a single layer thin rotating cylindrical shell with different boundary conditions with the generalized differential quadrature (GDQ) method. In order to drive the equations of motion, Love's shell theory was applied. The effects of different parameters on the natural frequencies such as L/R ratio, rotating speeds, and circumferential wave numbers were investigated.

Liew et al.^[2] investigated the free vibration analysis of a single layer rotating cylindrical shell using the harmonic reproducing kernel particle method based on Love's thin shell theory. The shell was considered with different boundary conditions, and the natural frequency was investigated in terms of different parameters.

According to Love's first approximation theory, the free vibration analysis of rotating truncated conical shells was investigated by Civalek^[3]. Discrete singular convolution (DSC) was used to calculate the natural frequency subject to the boundary conditions, the h/R ratio, cone angles, and the circumferential wave number.

Civalek and Gurses^[4] studied the free vibration analysis of rotating cylindrical shells. Based on Love's shell theory, the governing equations were obtained, and then with the DSC method, discretization was applied. Finally, effects of the boundary condition and geometric parameters on the natural frequencies were discussed.

Malekzadeh and Heydarpour^[5] investigated the free vibration behavior of the functionally graded (FG) cylindrical shell with different boundary conditions under thermal environment. The equations of motion were obtained based on the first-order shear deformation theory (FSDT) and Hamilton's principle. To discretize the governing equations, the differential quadrature method (DQM) was used. Finally, the distribution of natural frequency in terms of different geometrical and material properties was discussed.

The free vibration analysis of thin rotating cylindrical shells subject to various boundary conditions was illustrated by Sun et al.^[6]. Sanders' shell theory was used to derive the equations of motion, and with Fourier series expressions, the governing equations were discretized. The Campbell diagram and effects of different parameters on the natural frequency were discussed.

With the constant-gain negative velocity feedback control strategy, nonlinear vibration control of FG sandwich cylindrical shells was illustrated by Sheng and Wang^[7]. The sandwich structure consisted of an FG layer at the middle and two piezoelectric layers at the bottom and top. Hamilton's principle was used to derive the equations of motion according to the FSDT. They derived equations of motion as the coupled nonlinear partial differential equations (PDEs). The Galerkin method was used to discretize the governing equations and then solved with the Runge-Kutta method. Eventually, the forced vibration response of the system was investigated subject to different parameters.

The vibration analysis was investigated for FG cylindrical microshells using the modified couple stress theory (MCST) under a temperature-dependent condition^[8]. The FGM scale parameter was considered to be changed along the thickness. Heydarpour et al.^[9] explained the free vibration behavior of the rotating FG carbon nanotube-reinforced composite (CNTRC) truncated conical shell. Using the FSDT, the equations of motion were derived, and with the DQM, discretization was used. Eventually, effects of vertex angle and angular velocity on the natural frequencies were surveyed.

Duc et al.^[10] studied nonlinear vibration of functionally graded piezoelectric material (FGPM) double curved shallow shells subject to various loads. Reddy's higher-order shear deformation theory was used to achieve the equations of motion. With the Galerkin method, discretization was done, and then the fourth-order Runge-Kutta method was used to solve the equations. Eventually, the natural frequency and nonlinear dynamic response were investigated in terms of different parameters.

Sofiyev and Kuruoglu^[11] studied the dynamic instability of laminated cylindrical shells containing an FG layer at the middle and two isotropic layers at its inner and outer surfaces. The governing equations were derived based on the Donnell shell theory and Galerkin method. Eventually, the unstable regions of the system were computed in terms of the non-homogeneous index of the FG layer.

Mercan et al.^[12] investigated the free vibration of cylindrical FGM with the DSC method. Love's shell theory was used to obtain governing equations. Finally, the natural frequency was computed in terms of different boundary conditions, materials, and geometric parameters.

The non-linear vibration of rotating FGM shells was investigated by Sheng and Wang^[13]. The FSDT was used to derive the equations of motion with Hamilton's principle by considering von Kármán terms. The natural frequency and chaotic response of the system were calculated.

Razavi et al.^[14] investigated the free vibration analysis of functionally graded piezoelectric (FGP) cylindrical nanoshells. Based on the couple stress theory, Love's thin shell theory, and the size-dependent piezoelectric theory, the governing equations were derived with Hamilton's principle. To discretize the equations of motion, the Galerkin method was used. Eventually, the natural frequency was discussed in terms of various geometric parameters and mechanical properties.

In this paper, the three-dimensional free vibration analysis of rotating shells is investigated. The shell is considered as a sandwich structure. The core layer is an FGM shell, and two piezoelectric layers are integrated on its inner and outer surfaces. One main innovation of this study is to consider piezoelectric layers as FGMs called FGPMs. The FSDT is used to explain the displacement field. With Hamilton's principle, the equations of motion are derived as PDEs. Then, the Galerkin method is used to discretize governing equations. To validate this study, the results for both rotating and non-rotating shells are compared with the previously published results, and good agreement is observed. Finally, the distributions of natural frequencies in terms of geometrical parameters (thickness ratio and L/R ratio), axial and circumferential wave numbers and the non-homogeneous index of both FG and FGP layers are investigated. Moreover, the effect of the rotational speed on the natural frequency (Campbell diagram) is discussed according to the solving procedure.

2 Structural dynamic modeling

In this section, the equations of motion of the rotating sandwich cylindrical shell are presented. Figure 1 shows the schematic of the system. The middle layer, inner layer, and outer layer are named as FG layer, sensor, and actuator, respectively. The orthogonal curvilinear coordinates (x, θ, z) are considered at the mid-surface of the core. The subscripts f, a, s, and p represent the FG core layer, the actuator, the sensor, and the piezoelectric layer, respectively.

The FSDT is used to describe the displacement field as^{[7, 15]}

(1)

Displacements of a general point in the rotating laminated shell are denoted by (u, v, w) which contain mid-surface displacements along the x-, θ -, and z-directions, represented by (u₀, v₀, w₀), and the rotations of the normal vector of the mid-surface along the x- and θ -directions are expressed by φ_x and φ_θ, respectively.

The strain vectors are described as

(2)

The strain vector ε consists of mid-surface strain e and mid-surface curvature K, and their elements are obtained as^[16]

(3)

The FGMs are one type of composites where the components mix continuously. Using ceramic and metal for FGM components can result in enduring thermal stress simultaneously with mechanical pressure. Because of this benefit, piezoelectric layers have been considered to be FG which are called FGPMs. Material properties of FGM^{[5, 10]} and FGPM^[17-18] vary along the thickness direction z according to a power law,

(4)

The first two equations express variations of properties in the FGM, and the third one is used for the FGPM. The Young's modulus and density of the FG core are represented by E_f and ρ_f, respectively, and stiffness matrix elements of the piezoelectric layers are expressed by Y_i, where Y_b is the material property of the FGPM layer at the inner radius. The Poisson's ratio of the FGM is assumed to remain constant. Moreover, The subscripts m and c refer to metal and ceramic, respectively.

Considering the electric field for FGP layers, the stress-strain relations for both FG and FGP sections are presented as follows^[7]:

(5a)

(5b)

where the stiffness matrix is demonstrated by C_i, and e_i, ζ_i, D_i, and E_i represent the effective piezoelectric constant, effective permittivity constant, electrical displacement, and electrical field, respectively, which are described as follows^[19-21]:

(6)

Electrical field vectors of piezoelectric sections are obtained by negative gradients of electric potentials as^[7]

(7)

where φ_a and φ_s express electrical potentials of actuator and sensor layers, respectively.

The distribution of the electric potential for the sensor section is expressed as

(8)

where and ψ_s expresses the distribution of the electric potential along the planar coordinate of sensor section.

By considering an applied voltage in the piezoelectric section, the electrical potential function for the actuator layer is represented as^[19]

(9)

in which and the applied electrical potential is demonstrated by U(x, θ, t) and is the same as the sensor layer, and ψ_a indicates the distribution of the electric potential along the planar coordinate of the actuator layer.

For the sandwich cylindrical shell, which rotates about its longitudinal axis at the rotational speed Ω, the velocity vector can be expressed as^{[2, 22]}

(10)

By substituting the displacement field terms defined in Eq. (1) into Eq. (10), the final velocity can be calculated as

(11)

By considering the middle FGM layer and both two FGP layers, the total kinetic energy of the system is expressed as^[23-24]

(12)

Moreover, for the sandwich cylindrical shell, the potential energy is described as^[25]

(13)

where H_f, H_a, and H_s represent the potential energy terms of the middle layer (FGM), actuator layer (FGP), and sensor layer (FGP), respectively.

The equations of motion of the rotating sandwich cylindrical shell are obtained by using Hamilton's principle as^[25-27]

(14)

For the rotating sandwich cylindrical shell, the total kinetic energy and total potential energy of the system are denoted by K, H, respectively, and the external axial loading work is represented by V_N.

Substituting the required terms into Eq. (12) and Eq. (13), then substituting the kinetic energy and potential energy into Eq. (14) and applying Hamilton's principle, lead to the final governing equations of the rotating sandwich cylindrical shell in the PDEs,

(15)

where I_i and c_i represent mass terms and rotational matrix terms of the rotating sandwich cylindrical shell, and k_ij expresses linear operators described in Appendix A.

3 Solution procedure

The boundary conditions are considered to be simply-supported. Therefore, the solutions to Eq. (15) may be described as^{[19, 25]}

(16)

where and the number of axial half-waves and the number of circumferential waves are represented by m and n, respectively.

Because of the rotational matrix, using Eq. (16) cannot convert the PDEs in Eq. (15) into the ordinary differential equations (ODEs). Therefore, in order to discretize Eq. (15), the Galerkin method is applied as follows^[28]:

(17a)

(17b)

(17c)

(17d)

(17e)

(17f)

(17g)

(17h)

With the Galerkin method as described in Eq. (17), the governing equations which are derived as PDEs demonstrated in Eq. (15), are converted to the ODEs. Therefore, the final equations of motion of the rotating sandwich cylindrical shell are described as follows:

(18)

In order to calculate T_ij coefficients, solution functions described in Eq. (16) should be substituted into k_ij operators which are presented in Appendix A.

The last two equations of Eq. (15) are algebraic equations which can be solved as follows:

(19)

By substituting two electric potential functions as described in Eq. (19) into the first five rows of Eq. (18), the final governing equations of the rotating sandwich cylindrical shell can be obtained.

4 Simulation and results

With the dynamics matrix, the natural frequencies of the rotating sandwich cylindrical shell could be calculated as follows:

(20)

where "eig" represents the eigen values of the dynamics matrix D'.

4.1 Validation

To verify this study, for a non-rotating FG shell, the natural frequencies achieved in this study have been compared with the previously published results by Loy et al.^[29] as shown in Table 1. The FG shell consists of stainless steel and nickel, where the non-homogeneous index is 2, and the other material properties are presented by Loy et al.^[29]. The natural frequency could be found in terms of different values of the circumferential wave number n in Table 1. To simplify the sandwich structure into a single layer shell, the thicknesses of inner and outer layers h_s and h_a are considered to be zero.

According to Table 1, good agreement is observed with the previously published results presented by Loy et al.^[29].

Moreover, for verifying rotating terms, based on Liew et al.^[2], the variations of non- dimensional frequency parameter versus the L/R ratio for a rotating single layer shell with the simply-supported boundary condition are studied (see Fig. 2). The parameters are as follows:

As observed in Fig. 2, variations of frequency parameter with the L/R ratio for this study based on the FSDT are very similar to those presented by Liew et al.^[2] according to Love's thin shell theory.

4.2 Vibration analysis of rotating sandwich cylindrical shell

To investigate the effect of different parameters on the natural frequencies of the rotating sandwich cylindrical shell, the core FG layer is considered to consist of aluminum (Al) as the metal part and alumina (Al₂O₃) as the ceramic sector. Furthermore, both sensor and actuator layers which are considered as FGPMs are made of PZT-4. Geometric parameters and material properties of both piezoelectric layers are assumed to be identical. The material properties of the middle FG layer are as follows^[30]:

The material properties of piezoelectric layers are as follows^{[18, 31-32]}:

Figures 3-5 illustrate the effects of the non-homogeneous index of FGMs on the first three natural frequencies. As observed in Figs. 3-5, the natural frequencies decrease as the non-homogeneous index of FG layer (g_f) increases despite of the non-homogeneous index of FGP layers (g_p) which causes the increasing natural frequencies. The simulation data are

The ratio of length to radius L/R is an important parameter in shells. The effects of L/R on the natural frequencies are investigated in Fig. 6. The natural frequency decreases as the L/R ratio increases. This behavior is because of stiffness decreasing as a result of increasing the L/R ratio. Also, convergence of the natural frequency for various values of non-homogeneous index of FG layer (g_f) is observed. The calculation data used are as follows:

The thickness of the sandwich shell is the other important parameter which affects the natural frequencies. To investigate the effect of the ratio of the FG layer thickness (h_f) to the FGP layers thickness (h_p), it is assumed that the total thickness is a constant. According to Fig. 7, as the middle layer (FG layer) gets thicker than the FGP layers, the natural frequency increases. The simulation data are as follows:

Furthermore, the effects of the axial half-waves number (m) and the number of circumferential waves (n) on the natural frequency of the rotating sandwich cylindrical shell are discussed. As shown in Fig. 8, by increasing the circumferential wave number, the natural frequency decreases at first and then increases. Also, it is illustrated that, the effect of non-homogeneous index of FG layer (g_f) on the natural frequency of the rotating sandwich cylindrical shell is insignificant for different values of the circumferential wave number. The calculation data are as follows:

Additionally, variations of the natural frequency with the axial half-wave number m are investigated. It is observed in Fig. 9 that, increasing axial half-wave number leads to an increase in the natural frequency. Furthermore, similar to Fig. 2, increasing the non-homogeneous index of FGP layers (g_p) increases the natural frequency. The computing data are as follows:

The Campbell diagram is an important figure in the rotor dynamics scope. As the spin speed increases, the natural frequencies change. When the spin speed is equal to the natural frequency, it is called the critical speed. Considering a rotor dynamics problem, the critical speeds should be avoided. Figure 9 demonstrates the variation of natural frequencies in terms of the spin speed. According to the rotor dynamics principles, as the spin speed changes, the natural frequency is divided into two frequencies, one is called the forward frequency, and the other is called the backward frequency. As the spin speed increases, the forward frequency decreases, despite of the backward frequency which increases the spin speed. In Fig. 9, only the forward frequency is observed, and the backward frequency is not plotted. The reason is the Galerkin method which is used to discretize the PDEs. Comparing Eq. (18) with Eq. (19) illustrates that using the Galerkin method according to harmonic functions leads to omission of the coriolis matrix (C'), since sine and cosine are orthogonal functions. The calculation data are as follows:

5 Conclusions

The three-dimensional free vibration analysis of a rotating sandwich cylindrical shell is studied. The middle layer is an FGM shell, and two piezoelectric layers are integrated on its inner and outer surfaces. One innovation of this paper is to consider piezoelectric layers as FGP layers because of ability to tolerate thermal stress simultaneously with pressure. The displacement field is described using the FSDT. The governing equations are derived applying Hamilton's principle in the form of PDEs. With the Galerkin procedure, the discretization is done. To validate this study, the sandwich structure is simplified to a single layer shell, and the results of this study are compared with the previously published results, and good agreement is observed. Eventually, the effects of different geometric parameters such as the L/R ratio and h_f /h_p ratio, mechanical properties such as the non-homogeneous index of both FG and FGP layers, the axial half-wave number, the circumferential wave number, and the spin speed (Campbell diagram) on the natural frequency are investigated.

Some important conclusions are obtained.

(ⅰ) Investigation on geometric parameters of the sandwich rotating shell demonstrates that, as the FG layer gets thicker than the FGP layers, the natural frequency increases.

(ⅱ) Increasing the non-homogeneous index of FG layer (g_f) leads to the decrease in the natural frequencies since the mass of cylinder increases rather than the bending stiffness of cylinder increases. Despite of the effect of non-homogeneous index of FG core, increasing the non-homogeneous index of FGP layers (g_p), causes the increase in the natural frequencies of the system.

(ⅲ) As the circumferential wave number increases, the natural frequency of the system decreases at first and then increases. It is observed that, for different values of the circumferential wave number, the effect of non-homogeneous index of FG layer (g_f) on the natural frequency is insignificant (except for g_f =0, which is homogeneous). Moreover, increasing the axial half wave number tends to increase the natural frequency.

(ⅵ) Using the Galerkin method to discretize the equations results in omission of the coriolis effects. Therefore, only the just forward frequencies would be calculated.

(ⅴ) Investigation on the non-homogeneous index of FG and FGP layers indicates that, for all values of (g_f), all natural frequencies are real. However, as the non-homogeneous index of FGP layers changes, the natural frequency sometimes increases and sometimes decreases. Considering 0 < g_p < 1 causes an increase in the natural frequency. For all values of non-homogeneous index of FGP layers in the range of 0 < g_p < 1, the natural frequency gets complex, except for g_p =0, which leads to a homogeneous material. The decay rate for all values is negative which tends to a stable structure except for g_p =1 where the decay rate gets positive showing an unstable structure. Since most of the mechanical structures have a stable characteristic, it can be concluded that g_p ≠ 1. Therefore, investigation on the effects of non-homogeneous index of FG and FGP layers illustrates that, the non-homogeneous index of FG layers could be considered as any value despite of the non-homogeneous index of FGP layers which would be described as g_p ∈ [0, 1).

Appendix A