Abstract
This study investigates the thermoelastic coupling vibration and stability of rotating annular sector plates. Based on Hamilton’s principle and thermal conduction equation with deformation effect, the differential equation of transverse vibration for a rotating annular sector plate is established. The differential equation of vibration and corresponding boundary conditions are discretized by the differential quadrature method. Then, the thermoelastic coupling transverse vibrations under three different boundary conditions are calculated. The change curve of the first three order dimensionless complex frequencies of the rotating annular sector plate with the dimensionless angular speed are analyzed in the case of the thermoelastic coupling and uncoupling. The effects of the dimensionless angular speed, the ratio of inner to outer radius, the sector angle, and the dimensionless thermoelastic coupling coefficient on transverse vibration and stability of the annular sector plate are discussed. Finally, we obtained the type of instability and corresponding critical speed of the rotating annular sector plate in the case of the thermoelastic coupling and uncoupling.
1. Introduction
As a basic structure, the annular sector plate has been widely used in practical engineering, such as missiles, ships, instruments, and machine structures. The behavior of annular sector plate is very important for these structures, which has attracted great attention from many researchers. Some research work has been done on the bending behavior of the annular sector plate. For example, Jomehzadeh et al. and Sahraee [1, 2] analyzed the bending of functionally graded annular sector plates based on the Levinson plate theory and the first order shear deformation plate theory. Fallah and Nosier [3] reformulated the governing equations of the first order theory into the interior and edge-zone problems of the circular sector plate and analyzed the bending of functionally graded circular sector plates subjected to transverse loading. Qian and Yan [4] studied the bending problems of thin elastic annular sector plate with simply supported along radial edges and free along circular edges by a solution of deflection in the form of Fourier-Bessel double series.
Recently, more and more researchers have realized the importance of vibration analysis of the annular sector plate and have done considerable researches on this topic. On the one hand, some researchers have been devoted to unifying the vibration modeling of circular, annular, and sector plates. For example, Shi et al. and Wang et al. [5–7] established a unified vibration model of circular, annular, and sector plates with arbitrary boundary conditions and presented a unified method based on a new form of trigonometric series expansion for free vibration analysis of circular/annular sector plates. Later, Zhong et al. and Guan et al. [8, 9] applied the models and methods presented above to analyze free vibration of sector-like thin plate with various boundary conditions. Zhao et al. [10] constructed the unified theoretical model of functionally graded porous (FGP) circular, annular, and sector plates with general elastic restraints based on the first order shear deformation theory and studied free vibration and force vibration of FGP plates. On the other hand, some researchers have studied the effects of the structure and boundary condition on free vibration of the annular sector plate. Rezaei and Saidi [11] studied free vibration response of the fluid-saturated porous annular sector plates by building relative motion model between fluid and solid skeleton of the porous medium. By applying the Ritz method, Zhou et al. [12] obtained natural frequencies of free vibration of annular sector plates based on the three-dimensional elasticity theory. Belalia and Houmat [13] analyzed the nonlinear free vibration of moderately thick functionally graded sector plates by the p-version of the finite element method. Mizusawa [14] investigated free vibration of isotropic annular sector plates with arbitrary boundary conditions by using the spline element method. However, some sector plates rotate at the certain angular speed in many fields of engineering, such as sector mechanism in vibration mill and sector gear in high-speed rapier loom. The researches mentioned above have not involved the effect of the rotating angular speed on vibration characteristics of the annular sector plate.
Furthermore, the annular sector plate is under varying temperature environment in actual engineering applications, which needs to be considered. Behzad et al. [15] used 3D elasticity theory to analyze the thermal buckling of functionally graded perforated annular sector plates. Shaterzadeh et al. [16] used the finite element method to discuss the stability of composite perforated annular sector plates under thermomechanical loading. Mirtalaie [17] studied free vibration of functionally graded sector plates in thermal environment and examined the effects of temperature field, volume fraction exponent, radius ratio, and sector angle on free vibrations of the sector plate. The above researches have not involved the interaction between temperature field and strain field. In fact, the temperature field and the strain filed can affect each other, so the thermoelastic coupling should be taken into account in the vibration characteristic analysis of the rotating annular sector plate.
The differential equation of thermoelastic coupling transverse vibration of the rotating annular sector plate is the fourth-order partial differential equation with variable coefficients. The high-order partial differential equation has been solved by the finite element method and Galerkin method in some literature [13, 16, 18–20]. The differential quadrature method (DQM) was firstly proposed by Bellman and Casti in early 1970s [21]. Later, some researchers continued to improve this method and applied it to numerical solution of problems in engineering [22–25]. In recent years, as a numerically accurate and computationally efficient technique, the differential quadrature method (DQM) has also been applied in vibration analysis. Tornabene et al. [26, 27] used the generalized differential quadrature method to study free vibration of functionally graded conical, cylindrical shells and annular plates, and they compared numerical solutions using the DQM with those obtained using commercial programs which show the DQM can provide accurate and computationally efficient results. Fantuzzi [28] used the DQM to analyze free vibration analysis of arbitrarily shaped functionally graded carbon nanotube-reinforced plates. Shao et al. [29] discretized the transverse vibration differential equations of moving membrane by using the DQM and investigated the effects of the density coefficient and the tension ratio on transverse vibration characteristics of the membrane. However, few papers have focused on the fourth-order partial differential equation of thermoelastic coupling transverse vibration by DQM. In this paper, we use the DQM to solve the differential equation of thermoelastic coupling transverse vibration of the rotating annular sector plate.
This study aims to construct the differential equation of thermoelastic coupling transverse vibration of the rotating annular sector plate based on Hamilton’s principle and the thermal conduction equation. The dimensionless complex frequencies of the rotating annular sector plate in the case of the thermoelastic coupling and uncoupling are analyzed by the differential quadrature method. The change curve of the first three order dimensionless complex frequencies of the rotating annular sector plate with the dimensionless angular speed is analyzed under different boundary conditions. The effects of the ratio of inner to outer radius, the sector angle, and the dimensionless thermoelastic coupling coefficient on dynamic stability of the rotating annular sector plate are analyzed.
2. Differential Equation of Thermoelastic Coupling Vibration
2.1. Differential Equation of Transverse Vibration with Varying Temperature
Figure 1 shows an annular sector plate with inner radius , outer radius , sector angle , and thickness . The annular sector plate in the polar coordinate is rotating around an axis perpendicular to its surface with a constant rotating annular speed .

The strain-displacement relation in the middle surface of the rotating annular sector plate can be given bywhere and are the displacement field components along the radial direction and the circumferential direction in the middle plane of the rotating annular sector plate, respectively.
The stresses in the middle surface can be written aswhere is the elastic modulus and denotes Poisson’s ratio.
The strain-displacement relation at a distance from the middle surface of the rotating annular sector plate can be given bywhere is transverse displacement of the plate.
The corresponding stresses with the varying temperature can be written aswhere denotes the linear thermal expansion coefficient.
By using (4), the membrane forces are given by the following.
The bending and twisting moments per unit length are given bywhere is the flexural rigidity and indicates the thermal moment.
The kinetic energy is
and the potential energy is where , , .
Considering the symmetry of rotation, the strain produced by rotation is a function of , which is independent of , so . Based on the above analyses, the force balance condition in the axisymmetric annular sector plate is given bywhere is the inertial force per unit area in the middle surface and denotes the density of materials.
The strain energy due to the rotating centrifugal force is reduced to the following.
The equation of motion of the rotating annular sector plate can be obtained from the following Hamilton’s principle.
Substituting (7), (8), and (10) into (11) and using (9), the differential equation of transverse vibration can be obtained.
Given that the varying temperature along the lateral direction is considerably larger than that along radial direction, the thermal conduction equation can be described as followswhere is the varying temperature, denotes the thermal conductivity, indicates the specific heat at a constant volume, and represents the initial temperature of the plate.
Equation (12) involves , and (13) involves . In this way, the temperature and deflection fields are coupled together, and the coupling method must be used to solve both equations.
2.2. Solution of and
In order to solve (12) and (13), and need to be solved firstly. Considering the symmetry of rotation, the relationship between strain and displacement in the middle surface is written as follows.
The membrane forces can be written as follows.
Based on (14), the strain compatibility equation is obtained.
Considering that the varying temperature along the radial direction is ignored, the following equation is derived by using (9) and (16).
From (17), the solution of can be obtained.
Substituting (18) into (9), can be obtainedwhere and are integral constants.
Based on (15), (18), and (19), one obtains the following.
The boundary conditions of clamped or simply supported edge at the inner radius () and outer radius () are given by
while the boundary conditions of free edge at the inner radius () and outer radius () are as follows.
Substituting (20) into (21), and can be determined by the above boundary conditions, and then, and of the annular sector plate with the two radial edges having clamped or simply supported can be obtained.
Similarly, and of the annular sector plate with the two free radial edges can be obtained by substituting (18) into (22).
2.3. Dimensionless Differential Equation and Boundary Conditions
The following dimensionless quantities are introduced as follows.
As a result, (12) and (13) take the form ofwhere , , , , and .
For simply supported or clamped edge at the inner radius () and outer radius (), and are as follows.
For free edges at the inner radius () and outer radius (), and are as follows.
The solution of (26) and (27) is assumed in the following formwhere , is the dimensionless time, and denotes the dimensionless complex frequency of the rotating annular sector plate.
Substituting (30) into (26) and (27), one obtains the following.
From (32), the solution of can be obtainedwhere and are two integral constants and .
Substituting (33) into (31) results inwhere is the dimensionless thermoelastic coupling coefficient and indicates the coupling degree between the temperature and strain.
Figures 2(a), 2(b), and 2(c) present the definitions of the boundary conditions. The clamped, simply supported, and free boundary conditions are abbreviated as letters C, S, and F, respectively. For example, the symbolism SS-CC identifies an annular sector plate with the two radial edges having simply supported boundary conditions and the two circular edges having clamped boundary conditions, respectively.

(a) CC-CC

(b) SS-CC

(c) SS-FF
Considering that the edge of plate is held at a constant temperature, the dimensionless boundary conditions of CC-CC, SS-CC, and SS-FF are given as follows, respectively.
3. Discretization Method of Vibration Equation
The differential quadrature method (DQM) is used to solve (34). The DQM [30–32] approximates the derivatives of the function at the given nodes by weighted sums of the function at the total nodes. According to DQM, the annular sector plate adopts the method to treat the boundary conditions. The nodes of the plate are calculated by the following formulawhere is Gauss-Legendre integral point.
Based on the Lagrange interpolation polynomial, the weight coefficients of the first derivative and are obtained, respectively.
In the case of and , the weight coefficients of the higher derivatives are as follows.
In this paper, is chosen, and (34) can be discretized into the following form by DQM.
The discretization of (35)-(37) can be expressed as follows.
Equation (43) and one of the boundary conditions (44)-(46) can be expressed in the matrix form aswhere the matrices , , and involve the dimensionless angular speed, the ratio of inner to outer radius, the sector angle, and the dimensionless thermoelastic coupling coefficient. The necessary and sufficient condition when has nonzero solution is that coefficient determinant is equal to zero. Then, the eigenvalue equation of the thermoelastic coupling vibration of rotating annular sector plate is as follows.
In (48), is a complex eigenvalue. Therefore, one can obtain the complex frequency of the annular sector plate with various parameter values by solving the eigenvalue equation.
4. Numerical Results and Discussion
When and , (34) can be reduced to the differential equation of transverse vibration of the nonrotating annular sector plate. The first five order natural frequencies of the nonrotating annular sector plate with three different boundary conditions are calculated in the case of and . The frequency parameter in this study is defined by , while the frequency parameter in [14, 33, 34] is defined by . When , . The calculation results by conversing to are in good agreement with those exhibited in [14, 33, 34], which can be seen in Table 1, where the node number is .
4.1. Rotating Annular Sector Plate with CC-CC
Figure 3 shows the variation of the first three order dimensionless complex frequencies of the rotating annular sector plate (, ) with the dimensionless angular speed for the dimensionless thermoelastic coupling coefficient . When the dimensionless angular speed , the first three order dimensionless complex frequencies are real numbers. With the increase of the dimensionless angular speed, the real parts of the first three order dimensionless complex frequencies decrease, while their imaginary parts remain zero. When the dimensionless angular speed reaches a certain critical speed , the real part of the first order dimensionless complex frequency becomes zero, while its imaginary part has two branches. The critical speed is called the first order critical divergence speed. The divergence instability appears in the first order mode of the rotating annular sector plate in the region of . When , the rotating annular sector plate regains stability, and subsequently in the case of , the real parts of the first and second order complex frequencies merge with each other and keep positive, while their imaginary parts become two branches with positive and negative values. The result shows that the annular sector plate undergoes a coupled-mode flutter instability of the first and second order modes. is called the first-second order critical flutter speed.

Figure 4 shows the variation of the first three order dimensionless complex frequencies of the rotating annular sector plate (, ) with the dimensionless angular speed for the dimensionless thermoelastic coupling coefficient . In the case of thermoelastic coupling (), the real parts of the first three order dimensionless complex frequencies at are greater than those in the case of uncoupling (), and the annular sector plate undergoes the divergence instability in the first order mode and the coupled-mode flutter instability of the first and second order modes. The first order critical divergence speed and the first-second order critical flutter speed in the case of thermoelastic coupling are and , respectively, which are greater than those in the case of uncoupling.

Figures 5 and 6 show the variation of the first three order dimensionless complex frequencies of the rotating annular sector plate (, ) with the dimensionless angular speed for and . In comparison with Figures 3 and 4, we can see that when the ratio of inner to outer radius increases from 0.5 to 0.8, the real parts of the first three order dimensionless complex frequencies increase at the dimensionless angular speed . With the increase of , the real parts of in the first and second modes become zero successively, and then the imaginary parts of have two branches with positive and negative values. This shows that the annular sector plate undergoes the divergence instability of the first and second modes, and the corresponding first order critical divergence speed increases with the increase of the ratio of inner to outer radius. When the dimensionless angular speed further increases, the annular sector plate undergoes the coupled-mode flutter instability of the second and third order modes. By comparing Figure 5 with Figure 6, it is found that the corresponding critical speed in the thermoelastic coupling case is greater than that in the case of uncoupling.


Figures 7 and 8 show the variation of the first three order dimensionless complex frequencies of the rotating annular sector plate (, ) with the dimensionless angular speed for and . The annular sector plate undergoes the divergence instability in the first order mode and the coupled-mode flutter instability of the first and second order modes, and the first order critical divergence speed and the first-second order critical flutter speed increase when the dimensionless thermoelastic coupling coefficient increases from 0 to 0.3. From Figures 3, 4, 7, and 8, it is obtained that when other parameters are invariable, the real parts of the first three order dimensionless complex frequencies at , the first order critical divergence speed, and the first-second order critical flutter speed decrease with the increase of the sector angle .


Figures 9 and 10 show the variation of the first three order dimensionless complex frequencies of the rotating annular sector plate (, ) with the dimensionless angular speed for and . In comparison with Figures 7 and 8, it is observed that when the ratio of inner to outer radius increases from 0.5 to 0.8, the divergence instability of the first order mode does not occur, while the coupled-mode flutter instability of the first and second order modes appears, and the corresponding first-second order critical flutter speed increases with the increase of the ratio of inner to outer radius. By comparing Figure 9 with Figure 10, it can be seen that the first-second order critical flutter speed increases with the increase of the dimensionless thermoelastic coupling coefficient when other parameters are invariable.


From Figures 3–10, one can see that when the sector angle increases from to , the type of instability does not change in the case of the ratio of inner to outer radius , while it changes in the case of .
4.2. Rotating Annular Sector Plate with SS-CC
Figures 11 and 12 show the variation of the first three order dimensionless complex frequencies of the rotating annular sector plate (, ) with the dimensionless angular speed for the dimensionless thermoelastic coupling coefficient and . As can be seen from Figure 11, the plate undergoes the divergence instability of the first order mode and the coupled-mode flutter instability of the first and second order modes in the case of thermoelastic uncoupling. However, for , with the increase of , the real part of the first order dimensionless complex frequency remains a positive value, which decreases firstly and then increases. The annular sector plate does not undergo the divergence instability of the first order mode while it only undergoes the coupled-mode flutter instability of the first and second order modes. It shows that the dimensionless thermoelastic coupling coefficient can change the type of instability of the annular sector plate. By comparing Figure 11 with Figure 12, it is found that when , the real parts of the first three order dimensionless complex frequencies in the thermoelastic coupling case are greater than that in the case of uncoupling.


Figures 13 and 14 show the variation of the first three order dimensionless complex frequencies of the rotating annular sector plate (, ) with the dimensionless angular speed for and . The values of the real parts of the second and third order dimensionless complex frequencies of the annular sector plate present declining, rising, and declining tendency. The annular sector plate undergoes the divergence instability of the first order mode and the coupled-mode flutter instability of the first and second order modes, and the corresponding critical speed in the thermoelastic coupling case is greater than that in the case of uncoupling.


Figures 15 and 16 show the variation of the first three order dimensionless complex frequencies of the rotating annular sector plate (, ) with the dimensionless angular speed for and . we can see that the annular sector plate undergoes the divergence instability of the first order mode firstly, then it undergoes the coupled-mode flutter instability of the first and second order modes in the thermoelastic uncoupling case. However, it undergoes the divergence instability of the first and second order mode and the coupled-mode flutter instability of the first and third order modes in the thermoelastic coupling case.


Figures 17 and 18 show the variation of the first three order dimensionless complex frequencies of the rotating annular sector plate (, ) with the dimensionless angular speed for and . In the case of , the annular sector plate only undergoes the coupled-mode flutter instability in the first and second order modes. The corresponding first-second order critical flutter speed in the thermoelastic coupling case is greater than that in the case of uncoupling.


4.3. Rotating Annular Sector Plate with SS-FF
Figures 19–22 show the variation of the first three order dimensionless complex frequencies of the rotating annular sector plate (, , ) with the dimensionless angular speed for the dimensionless thermoelastic coupling coefficient and . We can see that rotating annular sector plate with SS-FF undergoes the divergence instability in the first order mode and the coupled-mode flutter instability of the first and second order modes. The corresponding critical speed in the thermoelastic coupling case is greater than that in the case of uncoupling. When the ratio of inner to outer radius , the corresponding critical speed deceases with the increase of the sector angle.




Figures 23–26 show the variation of the first three order dimensionless complex frequencies of the rotating annular sector plate (, , ) with the dimensionless angular speed for the dimensionless thermoelastic coupling coefficient and . Compared with the case of , besides the divergence instability in the first order mode and the coupled-mode flutter instability of the first and second order modes, the rotating annular sector plate with SS-FF also undergoes the divergence instability in the second order mode. When other parameters are invariable, the corresponding critical speed deceases with the increase of the sector angle, but increases with the increase of the dimensionless thermoelastic coupling coefficient.




From Figures 19–26, we can see that if the sector angle and dimensionless thermoelastic coupling coefficient are constant, when the ratio of inner to outer radius increases from to , the first order critical divergence speed decreases, while the first-second order critical flutter speed increases.
5. Conclusions
The thermoelastic coupling transverse vibration and stability of the rotating annular sector plate with three boundaries are investigated by DQM. The effects of the dimensionless angular speed, the ratio of inner to outer radius, the sector angle, the dimensionless thermoelastic coupling coefficient, and the boundary condition on transverse vibration and stability are discussed. The results are listed as follows.
(1) When other parameters are invariable, the real parts of the first three order dimensionless complex frequencies of the nonrotating annular sector plate () decrease with the increase of the sector angle. And their values in the case of thermoelastic coupling are greater than that in the case of uncoupling.
(2) The increase of the dimensionless thermoelastic coupling coefficient can change the type of instability of the annular sector plate under CC-CC and SS-CC boundary conditions, but it does not change under SS-FF boundary condition. The corresponding critical speed in the case of thermoelastic coupling is greater than that in the case of uncoupling when other parameters and the type of instability are invariable.
(3) The annular sector plate undergoes the divergence instability and the coupled-mode flutter instability under the three boundary conditions; however, the modes undergoing these two kinds of instability are different, which depend on the ratio of inner to outer radius, the sector angle, the dimensionless thermoelastic coupling coefficient, and the boundary condition.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
The authors gratefully acknowledge the support of the National Natural Science Foundation of China (No. 11472211).