# Thermoelastic Coupling Vibration and Stability Analysis of Rotating Annular Sector Plates

Authors: Yongqiang Yang; Zhongmin Wang
Journal: Mathematical Problems in Engineering
 (2019)

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## 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.


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## Body

## 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

Figure1 shows an annular sector plate with inner radius a, outer radius b, sector angle ϕ, and thickness h. The annular sector plate in the polar coordinate (r,θ) is rotating around an axis perpendicular to its surface with a constant rotating annular speed Ω.Figure 1
Schematic diagram of the rotating annular sector plate.The strain-displacement relation in the middle surface of the rotating annular sector plate can be given by(1)εr0=∂u∂rεθ0=1ru+∂v∂θγrθ0=1r∂u∂θ-v+∂v∂rwhere u and v are the displacement field components along the radial direction r and the circumferential direction θ in the middle plane of the rotating annular sector plate, respectively.The stresses in the middle surface can be written as(2)σr0=E1-μ2εr0+μεθ0=E1-μ2∂u∂r+μru+∂v∂θσθ0=E1-μ2εθ0+μεr0=E1-μ21ru+∂v∂θ+μ∂u∂rτrθ0=E21+μγrθ0=E21+μ1r∂u∂θ-v+∂v∂rwhere E is the elastic modulus and μ denotes Poisson’s ratio.The strain-displacement relation at a distancez from the middle surface of the rotating annular sector plate can be given by(3)εr=∂u∂r-z∂2w∂r2εθ=1ru+∂v∂θ-z1r∂w∂r+1r2∂2w∂θ2γrθ=1r∂u∂θ-v+∂v∂r-2z1r∂2w∂r∂θ-1r2∂w∂θwhere w is transverse displacement of the plate.The corresponding stresses with the varying temperatureT can be written as(4)σr=E1-μ2εr+μεθ-1+μαT=σr0-zE1-μ2∂2w∂r2+μ1r∂w∂r+1r2∂2w∂θ2-EαT1-μσθ=E1-μ2εθ+μεr-1+μαT=σθ0-zE1-μ21r∂w∂r+1r2∂2w∂θ2+μ∂2w∂r2-EαT1-μτrθ=E21+μγrθ=τrθ0-zE1+μ1r∂2w∂r∂θ-1r2∂w∂θwhere α denotes the linear thermal expansion coefficient.By using (4), the membrane forces Nr,Nθ,Nrθ are given by the following.(5)Nr=∫-h/2h/2σrdz=hσr0Nθ=∫-h/2h/2σθdz=hσθ0Nrθ=∫-h/2h/2τrθdz=hτrθ0The bending and twisting moments per unit length are given by(6)Mr=∫-h/2h/2σrzdz=-D∂2w∂r2+μ1r∂w∂r+1r2∂2w∂θ2-Eα1-μMTMθ=∫-h/2h/2σθzdz=-D1r∂w∂r+1r2∂2w∂θ2+μ∂2w∂r2-Eα1-μMTMrθ=∫-h/2h/2τrθzdz=-D1-μ1r∂2w∂r∂θ-1r2∂w∂θwhere D=Eh3/12(1-μ2) is the flexural rigidity and MT=∫-h/2h/2Tzdz indicates the thermal moment.The kinetic energy is(7)T=12∫ab∫0ϕρhΩ∂w∂θ+∂w∂t2rdrdθand the potential energy is(8)U=12∫ab∫0ϕMrκr+Mθκθ+Mrθκrθrdrdθwhere κr=-∂2w/∂r2, κθ=-1/r∂w/∂r-1/r2∂2w/∂θ2, κrθ=-2/r∂2w/∂r∂θ+2/r2∂w/∂θ.Considering the symmetry of rotation, the strain produced by rotation is a function ofr, which is independent of θ, so Nrθ=0. Based on the above analyses, the force balance condition in the axisymmetric annular sector plate is given by(9)∂Nr∂r+Nr-Nθr+q=0where q=ρhΩ2r is the inertial force per unit area in the middle surface and ρ denotes the density of materials.The strain energy due to the rotating centrifugal force is reduced to the following.(10)V=12∫ab∫0ϕNr∂w∂r2+Nθ∂wr∂θ2rdrdθThe equation of motion of the rotating annular sector plate can be obtained from the following Hamilton’s principle.(11)∫t1t2δT-δU-δVdt=0Substituting (7), (8), and (10) into (11) and using (9), the differential equation of transverse vibration can be obtained.(12)D∂4w∂r4+2r∂3w∂r3-1r2∂2w∂r2-2r3∂3w∂r∂θ2+1r3∂w∂r+2r2∂4w∂r2∂θ2+4r4∂2w∂θ2+1r4∂4w∂θ4+Eα1-μ∂2∂r2+1r∂∂r+1r2∂2∂θ2MT+ρh∂2w∂t2+2Ω∂2w∂t∂θ+Ω2∂2w∂θ2-Nr∂2w∂r2-Nθ1r∂w∂r+1r2∂2w∂θ2+ρhΩ2r∂w∂r=0Given that the varying temperatureT along the lateral direction is considerably larger than that along radial direction, the thermal conduction equation can be described as follows(13)∂T∂t-kρCv∂2T∂z2+EαT01-2μρCv∂∂t-z∂2w∂r2-z1r∂w∂r-z1r2∂2w∂θ2=0where T=T(z,t) is the varying temperature, k denotes the thermal conductivity, Cv indicates the specific heat at a constant volume, and T0 represents the initial temperature of the plate.Equation (12) involves T(z,t), and (13) involves w=w(r,t). 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 ofNr and Nθ

In order to solve (12) and (13), Nr and Nθ need to be solved firstly. Considering the symmetry of rotation, the relationship between strain and displacement in the middle surface is written as follows.(14)εr0=∂u∂rεθ0=urγrθ0=0The membrane forces can be written as follows.(15)Nr=Eh1-μ2∂u∂r+μruNθ=Eh1-μ21ru+μ∂u∂rNrθ=0Based on (14), the strain compatibility equation is obtained.(16)εr0=∂rεθ0∂rConsidering that the varying temperatureT along the radial direction r is ignored, the following equation is derived by using (9) and (16).(17)r2∂2Nr∂r2+3r∂Nr∂r+3+μρhΩ2r2=0From (17), the solution of Nr can be obtained.(18)Nr=-3+μρhΩ2r28+A+Br2Substituting (18) into (9), Nθ can be obtained(19)Nθ=-1+3μρhΩ2r28+A-Br2where A and B are integral constants.Based on (15), (18), and (19), one obtains the following.(20)u=rEh-1+μ2ρhΩ2r28+1-μA+-1-μBr2The boundary conditions of clamped or simply supported edge at the inner radius (r=a) and outer radius (r=b) are given by (21)ur=a=0ur=b=0while the boundary conditions of free edge at the inner radius (r=a) and outer radius (r=b) are as follows.(22)σr0r=a=0σr0r=b=0Substituting (20) into (21), A and B can be determined by the above boundary conditions, and then, Nr and Nθ of the annular sector plate with the two radial edges having clamped or simply supported can be obtained.(23)Nr=-3+μr28+1+μa2+b28+1-μa2b28r2ρhΩ2Nθ=-1+3μr28+1+μa2+b28-1-μa2b28r2ρhΩ2Similarly,Nr and Nθ of the annular sector plate with the two free radial edges can be obtained by substituting (18) into (22).(24)Nr=-3+μr2-a28+3+μr2-a2b28r2ρhΩ2Nθ=-1+3μr28+3+μa2+b28+3+μa2b28r2ρhΩ2


### 2.3. Dimensionless Differential Equation and Boundary Conditions

The following dimensionless quantities are introduced as follows.(25)r-=rb,w-=wh,θ-=θϕ,z-=zh,τ=thb2E12ρ1-μ2,ξ=ab,c=b2ΩhρEAs a result, (12) and (13) take the form of(26)∂4w-∂r-4+2r-∂3w-∂r-3-1r-2∂2w-∂r-2-2r-3ϕ2∂3w-∂r-∂θ-2+1r-3∂w-∂r-+2r-2ϕ2∂4w-∂r-2∂θ-2+4r-4ϕ2∂2w-∂θ-2+1r-4ϕ4∂4w-∂θ-4+A1∂2M-T∂r-2+1r-∂M-T∂r-+1r-2ϕ2∂2M-T∂θ-2+∂2w-∂τ2+2η1/2cϕ∂2w-∂τ∂θ-+ηc2ϕ2∂2w-∂θ-2-ηN1∂2w-∂r-2+N21r-∂w-∂r-+N21r-2ϕ2∂2w∂θ-2-c2r-∂w-∂r-=0(27)∂2T-∂z-2-A2∂T-∂τ+A3∂∂τ∂2w-∂r-2+1r-∂w-∂r-+1r-2ϕ2∂2w-∂θ-2z-=0where M-T=∫-1/21/2T-z-dz-, η=121-μ2, A1=12(1+μ)b2αT0/h2, A2=Cvh3/kb2ρE/12(1-μ2), and A3=Eαh5/(1-2μ)kb4E/12ρ(1-μ2).For simply supported or clamped edge at the inner radius (r-=ξ) and outer radius (r-=1), N1 and N2 are as follows.(28)N1=-3+μr-28+1+μ81+ξ2+1-μ8ξ2r-2c2N2=-1+3μr-28+1+μ81+ξ2-1-μ8ξ2r-2c2For free edges at the inner radius (r-=ξ) and outer radius (r-=1), N1 and N2 are as follows.(29)N1=-3+μr-2-ξ28+3+μr-2-ξ28r-2c2N2=-1+3μr-28+3+μ1+ξ28+3+μξ28r-2c2The solution of (26) and (27) is assumed in the following form(30)w-r-,θ-,τ=Wr-,θ-ejωτT-z-,τ=T∗z-ejωτwhere j=-1, τ is the dimensionless time, and ω denotes the dimensionless complex frequency of the rotating annular sector plate.Substituting (30) into (26) and (27), one obtains the following.(31)d4Wdr-4+2r-d3Wdr-3-1r-2d2Wdr-2+1r-3dWdr--2r-3ϕ2d3Wdr-dθ-2+2r-2ϕ2d4Wdr-2dθ-2+4r-4ϕ2d2Wdθ-2+1r-4ϕ4d4Wdθ-4+A1d2dr-2+1r-ddr-+1r-2ϕ2d2dθ-2∫-1/21/2T∗z-dz-+-ω2W+2jωη1/2cϕdWdθ-+ηc2ϕ2d2Wdθ-2-ηN1d2Wdr-2+N21r-dWdr-+N21r-2ϕ2d2Wdθ-2-c2r-dWdr-=0(32)d2T∗dz-2-A2jωT∗+A3jωd2Wdr-2+1r-dWdr-+1r-2ϕ2d2Wdθ-2z-=0From (32), the solution of T∗ can be obtained(33)T∗=a1eζz-+a2e-ζz-+Eαh21-2μρCvb2d2Wdr-2+1r-dWdr-+1r-2ϕ2d2Wdθ-2z-where a1 and a2 are two integral constants and ζ=A2jω.Substituting (33) into (31) results in(34)1+ψd4Wdr-4+2r-d3Wdr-3-1r-2d2Wdr-2+1r-3dWdr--2r-3ϕ2d3Wdr-dθ-2+2r-2ϕ2d4Wdr-2dθ-2+4r-4ϕ2d2Wdθ-2+1r-4ϕ4d4Wdθ-4+-ω2W+2jωη1/2cϕdWdθ-+ηc2ϕ2d2Wdθ-2-ηN1d2Wdr-2+N21r-dWdr-+N21r-2ϕ2d2Wdθ-2-c2r-dWdr-=0where ψ=(1+μ)Eα2T0/(1-2μ)ρCv is the dimensionless thermoelastic coupling coefficient and indicates the coupling degree between the temperature and strain.Figures2(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.Figure 2
Definitions of the boundary conditions.
(a)
CC-CC (b)
SS-CC (c)
SS-FFConsidering 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.(35)CC-CC:Wr-=ξ=Wr-=1=Wθ-=0=Wθ-=1=0dWdr-r-=ξ=dWdr-r-=1=dWdθ-θ-=0=dWdθ-θ-=1=0(36)SS-CC:Wr-=ξ=Wr-=1=Wθ-=0=Wθ-=1=0dWdr-r-=ξ=dWdr-r-=1=01r-dWdr-+1r-2ϕ2d2Wdθ-2+μd2Wdr-2θ-=0=1r-dWdr-+1r-2ϕ2d2Wdθ-2+μd2Wdr-2θ-=1=0(37)SS-FF:Wθ-=0=Wθ-=1=01r-dWdr-+1r-2ϕ2d2Wdθ-2+μd2Wdr-2θ-=0=1r-dWdr-+1r-2ϕ2d2Wdθ-2+μd2Wdr-2θ-=1=0d2Wdr-2+μ1r-dWdr-+μ1r-2ϕ2d2Wdθ-2r-=ξ=d2Wdr-2+μ1r-dWdr-+μ1r-2ϕ2d2Wdθ-2r-=1=0d3Wdr-3+1r-d2Wdr-2-1r-2dWdr-+2-μr-2ϕ2d3Wdr-dθ-2-3-μr-3ϕ2d2Wdθ-2r-=ξ=d3Wdr-3+1r-d2Wdr-2-1r-2dWdr-+2-μr-2ϕ2d3Wdr-dθ-2-3-μr-3ϕ2d2Wdθ-2r-=1=0


## 2.1. Differential Equation of Transverse Vibration with Varying Temperature

Figure1 shows an annular sector plate with inner radius a, outer radius b, sector angle ϕ, and thickness h. The annular sector plate in the polar coordinate (r,θ) is rotating around an axis perpendicular to its surface with a constant rotating annular speed Ω.Figure 1
Schematic diagram of the rotating annular sector plate.The strain-displacement relation in the middle surface of the rotating annular sector plate can be given by(1)εr0=∂u∂rεθ0=1ru+∂v∂θγrθ0=1r∂u∂θ-v+∂v∂rwhere u and v are the displacement field components along the radial direction r and the circumferential direction θ in the middle plane of the rotating annular sector plate, respectively.The stresses in the middle surface can be written as(2)σr0=E1-μ2εr0+μεθ0=E1-μ2∂u∂r+μru+∂v∂θσθ0=E1-μ2εθ0+μεr0=E1-μ21ru+∂v∂θ+μ∂u∂rτrθ0=E21+μγrθ0=E21+μ1r∂u∂θ-v+∂v∂rwhere E is the elastic modulus and μ denotes Poisson’s ratio.The strain-displacement relation at a distancez from the middle surface of the rotating annular sector plate can be given by(3)εr=∂u∂r-z∂2w∂r2εθ=1ru+∂v∂θ-z1r∂w∂r+1r2∂2w∂θ2γrθ=1r∂u∂θ-v+∂v∂r-2z1r∂2w∂r∂θ-1r2∂w∂θwhere w is transverse displacement of the plate.The corresponding stresses with the varying temperatureT can be written as(4)σr=E1-μ2εr+μεθ-1+μαT=σr0-zE1-μ2∂2w∂r2+μ1r∂w∂r+1r2∂2w∂θ2-EαT1-μσθ=E1-μ2εθ+μεr-1+μαT=σθ0-zE1-μ21r∂w∂r+1r2∂2w∂θ2+μ∂2w∂r2-EαT1-μτrθ=E21+μγrθ=τrθ0-zE1+μ1r∂2w∂r∂θ-1r2∂w∂θwhere α denotes the linear thermal expansion coefficient.By using (4), the membrane forces Nr,Nθ,Nrθ are given by the following.(5)Nr=∫-h/2h/2σrdz=hσr0Nθ=∫-h/2h/2σθdz=hσθ0Nrθ=∫-h/2h/2τrθdz=hτrθ0The bending and twisting moments per unit length are given by(6)Mr=∫-h/2h/2σrzdz=-D∂2w∂r2+μ1r∂w∂r+1r2∂2w∂θ2-Eα1-μMTMθ=∫-h/2h/2σθzdz=-D1r∂w∂r+1r2∂2w∂θ2+μ∂2w∂r2-Eα1-μMTMrθ=∫-h/2h/2τrθzdz=-D1-μ1r∂2w∂r∂θ-1r2∂w∂θwhere D=Eh3/12(1-μ2) is the flexural rigidity and MT=∫-h/2h/2Tzdz indicates the thermal moment.The kinetic energy is(7)T=12∫ab∫0ϕρhΩ∂w∂θ+∂w∂t2rdrdθand the potential energy is(8)U=12∫ab∫0ϕMrκr+Mθκθ+Mrθκrθrdrdθwhere κr=-∂2w/∂r2, κθ=-1/r∂w/∂r-1/r2∂2w/∂θ2, κrθ=-2/r∂2w/∂r∂θ+2/r2∂w/∂θ.Considering the symmetry of rotation, the strain produced by rotation is a function ofr, which is independent of θ, so Nrθ=0. Based on the above analyses, the force balance condition in the axisymmetric annular sector plate is given by(9)∂Nr∂r+Nr-Nθr+q=0where q=ρhΩ2r is the inertial force per unit area in the middle surface and ρ denotes the density of materials.The strain energy due to the rotating centrifugal force is reduced to the following.(10)V=12∫ab∫0ϕNr∂w∂r2+Nθ∂wr∂θ2rdrdθThe equation of motion of the rotating annular sector plate can be obtained from the following Hamilton’s principle.(11)∫t1t2δT-δU-δVdt=0Substituting (7), (8), and (10) into (11) and using (9), the differential equation of transverse vibration can be obtained.(12)D∂4w∂r4+2r∂3w∂r3-1r2∂2w∂r2-2r3∂3w∂r∂θ2+1r3∂w∂r+2r2∂4w∂r2∂θ2+4r4∂2w∂θ2+1r4∂4w∂θ4+Eα1-μ∂2∂r2+1r∂∂r+1r2∂2∂θ2MT+ρh∂2w∂t2+2Ω∂2w∂t∂θ+Ω2∂2w∂θ2-Nr∂2w∂r2-Nθ1r∂w∂r+1r2∂2w∂θ2+ρhΩ2r∂w∂r=0Given that the varying temperatureT along the lateral direction is considerably larger than that along radial direction, the thermal conduction equation can be described as follows(13)∂T∂t-kρCv∂2T∂z2+EαT01-2μρCv∂∂t-z∂2w∂r2-z1r∂w∂r-z1r2∂2w∂θ2=0where T=T(z,t) is the varying temperature, k denotes the thermal conductivity, Cv indicates the specific heat at a constant volume, and T0 represents the initial temperature of the plate.Equation (12) involves T(z,t), and (13) involves w=w(r,t). 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 ofNr and Nθ

In order to solve (12) and (13), Nr and Nθ need to be solved firstly. Considering the symmetry of rotation, the relationship between strain and displacement in the middle surface is written as follows.(14)εr0=∂u∂rεθ0=urγrθ0=0The membrane forces can be written as follows.(15)Nr=Eh1-μ2∂u∂r+μruNθ=Eh1-μ21ru+μ∂u∂rNrθ=0Based on (14), the strain compatibility equation is obtained.(16)εr0=∂rεθ0∂rConsidering that the varying temperatureT along the radial direction r is ignored, the following equation is derived by using (9) and (16).(17)r2∂2Nr∂r2+3r∂Nr∂r+3+μρhΩ2r2=0From (17), the solution of Nr can be obtained.(18)Nr=-3+μρhΩ2r28+A+Br2Substituting (18) into (9), Nθ can be obtained(19)Nθ=-1+3μρhΩ2r28+A-Br2where A and B are integral constants.Based on (15), (18), and (19), one obtains the following.(20)u=rEh-1+μ2ρhΩ2r28+1-μA+-1-μBr2The boundary conditions of clamped or simply supported edge at the inner radius (r=a) and outer radius (r=b) are given by (21)ur=a=0ur=b=0while the boundary conditions of free edge at the inner radius (r=a) and outer radius (r=b) are as follows.(22)σr0r=a=0σr0r=b=0Substituting (20) into (21), A and B can be determined by the above boundary conditions, and then, Nr and Nθ of the annular sector plate with the two radial edges having clamped or simply supported can be obtained.(23)Nr=-3+μr28+1+μa2+b28+1-μa2b28r2ρhΩ2Nθ=-1+3μr28+1+μa2+b28-1-μa2b28r2ρhΩ2Similarly,Nr and Nθ of the annular sector plate with the two free radial edges can be obtained by substituting (18) into (22).(24)Nr=-3+μr2-a28+3+μr2-a2b28r2ρhΩ2Nθ=-1+3μr28+3+μa2+b28+3+μa2b28r2ρhΩ2


## 2.3. Dimensionless Differential Equation and Boundary Conditions

The following dimensionless quantities are introduced as follows.(25)r-=rb,w-=wh,θ-=θϕ,z-=zh,τ=thb2E12ρ1-μ2,ξ=ab,c=b2ΩhρEAs a result, (12) and (13) take the form of(26)∂4w-∂r-4+2r-∂3w-∂r-3-1r-2∂2w-∂r-2-2r-3ϕ2∂3w-∂r-∂θ-2+1r-3∂w-∂r-+2r-2ϕ2∂4w-∂r-2∂θ-2+4r-4ϕ2∂2w-∂θ-2+1r-4ϕ4∂4w-∂θ-4+A1∂2M-T∂r-2+1r-∂M-T∂r-+1r-2ϕ2∂2M-T∂θ-2+∂2w-∂τ2+2η1/2cϕ∂2w-∂τ∂θ-+ηc2ϕ2∂2w-∂θ-2-ηN1∂2w-∂r-2+N21r-∂w-∂r-+N21r-2ϕ2∂2w∂θ-2-c2r-∂w-∂r-=0(27)∂2T-∂z-2-A2∂T-∂τ+A3∂∂τ∂2w-∂r-2+1r-∂w-∂r-+1r-2ϕ2∂2w-∂θ-2z-=0where M-T=∫-1/21/2T-z-dz-, η=121-μ2, A1=12(1+μ)b2αT0/h2, A2=Cvh3/kb2ρE/12(1-μ2), and A3=Eαh5/(1-2μ)kb4E/12ρ(1-μ2).For simply supported or clamped edge at the inner radius (r-=ξ) and outer radius (r-=1), N1 and N2 are as follows.(28)N1=-3+μr-28+1+μ81+ξ2+1-μ8ξ2r-2c2N2=-1+3μr-28+1+μ81+ξ2-1-μ8ξ2r-2c2For free edges at the inner radius (r-=ξ) and outer radius (r-=1), N1 and N2 are as follows.(29)N1=-3+μr-2-ξ28+3+μr-2-ξ28r-2c2N2=-1+3μr-28+3+μ1+ξ28+3+μξ28r-2c2The solution of (26) and (27) is assumed in the following form(30)w-r-,θ-,τ=Wr-,θ-ejωτT-z-,τ=T∗z-ejωτwhere j=-1, τ is the dimensionless time, and ω denotes the dimensionless complex frequency of the rotating annular sector plate.Substituting (30) into (26) and (27), one obtains the following.(31)d4Wdr-4+2r-d3Wdr-3-1r-2d2Wdr-2+1r-3dWdr--2r-3ϕ2d3Wdr-dθ-2+2r-2ϕ2d4Wdr-2dθ-2+4r-4ϕ2d2Wdθ-2+1r-4ϕ4d4Wdθ-4+A1d2dr-2+1r-ddr-+1r-2ϕ2d2dθ-2∫-1/21/2T∗z-dz-+-ω2W+2jωη1/2cϕdWdθ-+ηc2ϕ2d2Wdθ-2-ηN1d2Wdr-2+N21r-dWdr-+N21r-2ϕ2d2Wdθ-2-c2r-dWdr-=0(32)d2T∗dz-2-A2jωT∗+A3jωd2Wdr-2+1r-dWdr-+1r-2ϕ2d2Wdθ-2z-=0From (32), the solution of T∗ can be obtained(33)T∗=a1eζz-+a2e-ζz-+Eαh21-2μρCvb2d2Wdr-2+1r-dWdr-+1r-2ϕ2d2Wdθ-2z-where a1 and a2 are two integral constants and ζ=A2jω.Substituting (33) into (31) results in(34)1+ψd4Wdr-4+2r-d3Wdr-3-1r-2d2Wdr-2+1r-3dWdr--2r-3ϕ2d3Wdr-dθ-2+2r-2ϕ2d4Wdr-2dθ-2+4r-4ϕ2d2Wdθ-2+1r-4ϕ4d4Wdθ-4+-ω2W+2jωη1/2cϕdWdθ-+ηc2ϕ2d2Wdθ-2-ηN1d2Wdr-2+N21r-dWdr-+N21r-2ϕ2d2Wdθ-2-c2r-dWdr-=0where ψ=(1+μ)Eα2T0/(1-2μ)ρCv is the dimensionless thermoelastic coupling coefficient and indicates the coupling degree between the temperature and strain.Figures2(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.Figure 2
Definitions of the boundary conditions.
(a)
CC-CC (b)
SS-CC (c)
SS-FFConsidering 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.(35)CC-CC:Wr-=ξ=Wr-=1=Wθ-=0=Wθ-=1=0dWdr-r-=ξ=dWdr-r-=1=dWdθ-θ-=0=dWdθ-θ-=1=0(36)SS-CC:Wr-=ξ=Wr-=1=Wθ-=0=Wθ-=1=0dWdr-r-=ξ=dWdr-r-=1=01r-dWdr-+1r-2ϕ2d2Wdθ-2+μd2Wdr-2θ-=0=1r-dWdr-+1r-2ϕ2d2Wdθ-2+μd2Wdr-2θ-=1=0(37)SS-FF:Wθ-=0=Wθ-=1=01r-dWdr-+1r-2ϕ2d2Wdθ-2+μd2Wdr-2θ-=0=1r-dWdr-+1r-2ϕ2d2Wdθ-2+μd2Wdr-2θ-=1=0d2Wdr-2+μ1r-dWdr-+μ1r-2ϕ2d2Wdθ-2r-=ξ=d2Wdr-2+μ1r-dWdr-+μ1r-2ϕ2d2Wdθ-2r-=1=0d3Wdr-3+1r-d2Wdr-2-1r-2dWdr-+2-μr-2ϕ2d3Wdr-dθ-2-3-μr-3ϕ2d2Wdθ-2r-=ξ=d3Wdr-3+1r-d2Wdr-2-1r-2dWdr-+2-μr-2ϕ2d3Wdr-dθ-2-3-μr-3ϕ2d2Wdθ-2r-=1=0


## 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 δ method to treat the boundary conditions. The nodes of the plate are calculated by the following formula(38)r-1=0,r-2=δ,r-N-1=1-δ,r-N=1,r-i=1-ξ21+ξ1-ξ+xii=3,…,N-2θ-1=0,θ-2=δ,θ-M-1=1-δ,θ-M=1,θ-i=121-cos⁡j-2πM-3j=3,…,M-2where xi=Si∈(-1,1)(i=3,⋯,N-2) is Gauss-Legendre integral point.Based on the Lagrange interpolation polynomial, the weight coefficients of the first derivativeAij(1) and Bkm(1) are obtained, respectively.(39)Aik1=∏ν=1ν≠i,kNr-i-r-ν∏ν=1ν≠kNr-k-r-νi,k=1,2,⋯,N;k≠i∑ν=1ν≠iN1r-i-r-νi,k=1,2,⋯,N;k=i(40)Bjm1=∏ν=1ν≠j,mMθ-j-θ-ν∏ν=1ν≠mMθ-m-θ-νj,m=1,2,⋯,M;m≠j∑ν=1ν≠jM1θ-j-θ-νj,m=1,2,⋯,M;m=jIn the case ofx=2,3,...,N-1 and y=2,3,...,M-1, the weight coefficients of the higher derivatives are as follows.(41)Aikx=xAiix-1Aik1-Aikx-1xi-xki,k=1,2,⋯,N;k≠i-∑ν=1ν≠iNAiνxi=1,2,⋯N;1≤x≤N-1(42)Bjmy=yBjjy-1Bjm1-Bjmy-1yj-ymj,m=1,2,⋯,M;m≠j-∑ν=1ν≠jMBjνyj=1,2,⋯,M;1≤y≤M-1In this paper,N=M is chosen, and (34) can be discretized into the following form by DQM.(43)1+ψ∑k=1NAik4Wkj+2r-i∑k=1NAik3Wkj-1r-i2∑k=1NAik2Wkj+1r-i3∑k=1NAik1Wkj-2r-i3ϕ2∑k=1NAik1∑m=1NBjm2Wkm+2r-i2ϕ2∑k=1NAik2∑m=1NBjm2Wkm+4r-i4ϕ2∑m=1NBjm2Wkm+1r-i4ϕ4∑m=1NBjm4Wkm+-ω2Wij+2jωη1/2cϕ∑m=1NBjm1Wkm+ηc2ϕ2∑m=1NBjm2Wkm-ηN1i∑k=1NAik2Wkj+N2i1r-i∑k=1NAik1Wkj+N2i1r-2ϕ2∑m=1NBjm2Wkm-c2r-i∑k=1NAik1Wk=0The discretization of (35)-(37) can be expressed as follows. (44)CC-CC:W1j=WNj=Wi1=WiN=0,i,j=1,2,···,N∑k=1NAik1Wkj=0,i=2,N-1;j=2,3,···,N-2∑m=1NBjm1Wim=0,j=2,N-1;i=2,3,···,N-2(45)SS-CC:W1j=WNj=Wi1=WiN=0,i,j=1,2,···,N∑k=1NAik1Wkj=0,i=2,N-1;j=2,3,···,N-21r-i∑k=1NAik1Wkj+1r-i2ϕ2∑m=1NBjm2Wim+μ∑k=1NAik2Wkj=0,j=1,N;i=1,2,···,N(46)SS-FF:Wi1=WiN=0,i,j=1,2,···,N1r-i∑k=1NAik1Wkj+1r-i2ϕ2∑m=1NBjm2Wim+μ∑k=1NAik2Wkj=0,j=1,N;i=1,2,···,N∑k=1NAik2Wkj+μ1r-i∑k=1NAik1Wkj+μ1r-i2ϕ2∑m=1NBjm2Wim=0,i=1,N;j=1,2,···,N∑k=1NAik3Wkj+1r-i∑k=1NAik2Wkj-1r-i2∑k=1NAik1Wkj+2-μr-i2ϕ2∑k=1NAik1∑m=1NBjm2Wim-3-μr-i3ϕ2∑m=1NBjm2Wim=0,i=2,N-1;j=1,2,···,NEquation (43) and one of the boundary conditions (44)-(46) can be expressed in the matrix form as(47)ω2R+ωG+KWkj=0where the matrices [R], [G], and [K] 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 Wkj 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.(48)ω2R+ωG+K=0In (48), ω 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

Whenψ=0 and c=0, (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 ξ=0.5 and μ=0.3. The frequency parameter in this study is defined by ωD=b2ω/h12ρ(1-μ2)/E, while the frequency parameter in [14, 33, 34] is defined by ωR=(b-a)2ω/h12ρ(1-μ2)/E. When ξ=a/b=0.5, ωD=4ωR. The calculation results by conversing ωD to ωR are in good agreement with those exhibited in [14, 33, 34], which can be seen in Table 1, where the node number is N=11.Table 1
First five order dimensionless natural frequencies of the non-rotating annular sector plate (ξ=0.5, μ=0.3).
Sector angle Boundary condition Methods ω 1 ω 2 ω 3 ω 4 ω 5 π / 6 SS-CC Reference[14] (ωR) 33.90 75.11 76.46 121.8 135.3 Present (ω=ωD/4) 33.9230 75.1841 76.4747 121.9212 135.2661 CC-CC Reference[14] (ωR) 48.04 85.52 104.4 142.8 151.3 Present (ω=ωD/4) 48.1164 85.6430 104.5898 142.7932 151.5388 SS-FF Reference[14] (ωR) 11.70 28.87 44.03 50.51 82.13 Present (ω=ωD/4) 11.7792 28.8884 44.0803 50.5733 82.1813  π / 4 SS-CC Reference[14] (ωR) 26.89 44.70 67.37 76.45 86.60 Reference[33] (ωR) 26.91 44.69 74.03 77.34 88.32 Present (ω=ωD/4) 26.9225 44.7239 67.4507 76.4300 86.6895 CC-CC Reference[14] (ωR) 31.39 56.85 70.22 94.54 96.73 Present (ω=ωD/4) 31.4375 56.9450 70.3175 94.3425 96.8775 SS-FF Reference[14] (ωR) 5.267 16.68 20.40 36.60 44.03 Present (ω=ωD/4) 5.2730 16.6916 20.4240 36.6488 44.0779  π / 3 SS-CC Reference[14] (ωR) 24.74 33.89 51.44 64.79 75.08 Present (ω=ωD/4) 24.7623 33.9237 51.4538 64.8695 75.1838 CC-CC Reference[14] (ωR) 26.53 39.85 61.49 65.96 79.38 Present (ω=ωD/4) 26.5729 39.9165 61.4045 66.0465 79.5028 SS-FF Reference[14] (ωR) 2.856 11.77 11.87 25.51 28.87 Present (ω=ωD/4) 2.8601 11.7788 11.8736 25.5408 28.8883  π / 2 SS-CC Reference[14] (ωR) 23.33 26.88 33.87 44.65 58.96 Present (ω=ωD/4) 23.3613 26.9207 33.9137 45.3747 56.4011 CC-CC Reference[14] (ωR) 23.80 28.73 37.63 50.33 63.30 Reference[34] (ωR) 23.83 28.77 37.69 50.42 63.30 Present (ω=ωD/4) 23.8389 28.7775 37.6349 50.4611 63.2698 SS-FF Reference[14] (ωR) 1.068 5.267 7.779 11.77 16.68 Present (ω=ωD/4) 1.0697 5.2728 7.7841 11.7778 16.6914


### 4.1. Rotating Annular Sector Plate with CC-CC

Figure3 shows the variation of the first three order dimensionless complex frequencies of the rotating annular sector plate (ϕ=π/3, ξ=0.5) with the dimensionless angular speed for the dimensionless thermoelastic coupling coefficient ψ=0. When the dimensionless angular speed c=0, the first three order dimensionless complex frequencies ω are real numbers. With the increase of the dimensionless angular speed, the real parts Re(ω) of the first three order dimensionless complex frequencies decrease, while their imaginary parts Im(ω) remain zero. When the dimensionless angular speed reaches a certain critical speed c=7.34, the real part of the first order dimensionless complex frequency becomes zero, while its imaginary part has two branches. The critical speed cd=7.34 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 7.34≤c≤7.55. When 7.55<c≤8.11, the rotating annular sector plate regains stability, and subsequently in the case of 8.11<c≤8.61, 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. cf=8.11 is called the first-second order critical flutter speed.Figure 3
First three order dimensionless complex frequencies versus dimensionless angular speed (ϕ=π/3, ξ=0.5, ψ=0).Figure4 shows the variation of the first three order dimensionless complex frequencies of the rotating annular sector plate (ϕ=π/3, ξ=0.5) with the dimensionless angular speed c for the dimensionless thermoelastic coupling coefficient ψ=0.3. In the case of thermoelastic coupling (ψ=0.3), the real parts of the first three order dimensionless complex frequencies at c=0 are greater than those in the case of uncoupling (ψ=0), 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 cd=8.45 and cf=9.23, respectively, which are greater than those in the case of uncoupling.Figure 4
First three order dimensionless complex frequencies versus dimensionless angular speed (ϕ=π/3, ξ=0.5, ψ=0.3).Figures5 and 6 show the variation of the first three order dimensionless complex frequencies ω of the rotating annular sector plate (ϕ=π/3, ξ=0.8) with the dimensionless angular speed for ψ=0 and ψ=0.3. 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 c=0. With the increase of c, the real parts of ω in the first and second modes become zero successively, and then the imaginary parts of ω 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.Figure 5
First three order dimensionless complex frequencies versus dimensionless angular speed (ϕ=π/3, ξ=0.8, ψ=0).Figure 6
First three order dimensionless complex frequencies versus dimensionless angular speed (ϕ=π/3, ξ=0.8, ψ=0.3).Figures7 and 8 show the variation of the first three order dimensionless complex frequencies of the rotating annular sector plate (ϕ=π/2, ξ=0.5) with the dimensionless angular speed for ψ=0 and ψ=0.3. 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 c=0, the first order critical divergence speed, and the first-second order critical flutter speed decrease with the increase of the sector angle ϕ.Figure 7
First three order dimensionless complex frequencies versus dimensionless angular speed (ϕ=π/2, ξ=0.5, ψ=0).Figure 8
First three order dimensionless complex frequencies versus dimensionless angular speed (ϕ=π/2, ξ=0.5, ψ=0.3).Figures9 and 10 show the variation of the first three order dimensionless complex frequencies of the rotating annular sector plate (ϕ=π/2, ξ=0.8) with the dimensionless angular speed for ψ=0 and ψ=0.3. 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.Figure 9
First three order dimensionless complex frequencies versus dimensionless angular speed (ϕ=π/2, ξ=0.8, ψ=0).Figure 10
First three order dimensionless complex frequencies versus dimensionless angular speed (ϕ=π/2, ξ=0.8, ψ=0.3).From Figures3–10, one can see that when the sector angle increases from π/3 to π/2, the type of instability does not change in the case of the ratio of inner to outer radius ξ=0.5, while it changes in the case of ξ=0.8.


### 4.2. Rotating Annular Sector Plate with SS-CC

Figures11 and 12 show the variation of the first three order dimensionless complex frequencies of the rotating annular sector plate (ϕ=π/3, ξ=0.5) with the dimensionless angular speed for the dimensionless thermoelastic coupling coefficient ψ=0 and ψ=0.3. 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 ψ=0.3, with the increase of c, 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 c=0, 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.Figure 11
First three order dimensionless complex frequencies versus dimensionless angular speed (ϕ=π/3, ξ=0.5, ψ=0).Figure 12
First three order dimensionless complex frequencies versus dimensionless angular speed (ϕ=π/3, ξ=0.5, ψ=0.3).Figures13 and 14 show the variation of the first three order dimensionless complex frequencies of the rotating annular sector plate (ϕ=π/3, ξ=0.8) with the dimensionless angular speed for ψ=0 and ψ=0.3. 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.Figure 13
First three order dimensionless complex frequencies versus dimensionless angular speed (ϕ=π/3, ξ=0.8, ψ=0).Figure 14
First three order dimensionless complex frequencies versus dimensionless angular speed (ϕ=π/3, ξ=0.8, ψ=0.3).Figures15 and 16 show the variation of the first three order dimensionless complex frequencies of the rotating annular sector plate (ϕ=π/2, ξ=0.5) with the dimensionless angular speed for ψ=0 and ψ=0.3. 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.Figure 15
First three order dimensionless complex frequencies versus dimensionless angular speed (ϕ=π/2, ξ=0.5, ψ=0).Figure 16
First three order dimensionless complex frequencies versus dimensionless angular speed (ϕ=π/2, ξ=0.5, ψ=0.3).Figures17 and 18 show the variation of the first three order dimensionless complex frequencies of the rotating annular sector plate (ϕ=π/2, ξ=0.8) with the dimensionless angular speed for ψ=0 and ψ=0.3. In the case of ξ=0.8, 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.Figure 17
First three order dimensionless complex frequencies versus dimensionless angular speed (ϕ=π/2, ξ=0.8, ψ=0).Figure 18
First three order dimensionless complex frequencies versus dimensionless angular speed (ϕ=π/2, ξ=0.8, ψ=0.3).


### 4.3. Rotating Annular Sector Plate with SS-FF

Figures19–22 show the variation of the first three order dimensionless complex frequencies of the rotating annular sector plate (ϕ=π/3, ϕ=π/2, ξ=0.5) with the dimensionless angular speed for the dimensionless thermoelastic coupling coefficient ψ=0 and ψ=0.3. 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 ξ=0.5, the corresponding critical speed deceases with the increase of the sector angle.Figure 19
First three order dimensionless complex frequencies versus dimensionless angular speed (ϕ=π/3, ξ=0.5, ψ=0).Figure 20
First three order dimensionless complex frequencies versus dimensionless angular speed (ϕ=π/3, ξ=0.5, ψ=0.3).Figure 21
First three order dimensionless complex frequencies versus dimensionless angular speed (ϕ=π/2, ξ=0.5, ψ=0).Figure 22
First three order dimensionless complex frequencies versus dimensionless angular speed (ϕ=π/2, ξ=0.5, ψ=0.3).Figures23–26 show the variation of the first three order dimensionless complex frequencies of the rotating annular sector plate (ϕ=π/3, ϕ=π/2, ξ=0.8) with the dimensionless angular speed for the dimensionless thermoelastic coupling coefficient ψ=0 and ψ=0.3. Compared with the case of ξ=0.5, 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.Figure 23
First three order dimensionless complex frequencies versus dimensionless angular speed (ϕ=π/3, ξ=0.8, ψ=0).Figure 24
First three order dimensionless complex frequencies versus dimensionless angular speed (ϕ=π/3, ξ=0.8, ψ=0.3).Figure 25
First three order dimensionless complex frequencies versus dimensionless angular speed (ϕ=π/2, ξ=0.8, ψ=0).Figure 26
First three order dimensionless complex frequencies versus dimensionless angular speed (ϕ=π/2, ξ=0.8, ψ=0.3).From Figures19–26, 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 ξ=0.5 to ξ=0.8, the first order critical divergence speed decreases, while the first-second order critical flutter speed increases.


## 4.1. Rotating Annular Sector Plate with CC-CC

Figure3 shows the variation of the first three order dimensionless complex frequencies of the rotating annular sector plate (ϕ=π/3, ξ=0.5) with the dimensionless angular speed for the dimensionless thermoelastic coupling coefficient ψ=0. When the dimensionless angular speed c=0, the first three order dimensionless complex frequencies ω are real numbers. With the increase of the dimensionless angular speed, the real parts Re(ω) of the first three order dimensionless complex frequencies decrease, while their imaginary parts Im(ω) remain zero. When the dimensionless angular speed reaches a certain critical speed c=7.34, the real part of the first order dimensionless complex frequency becomes zero, while its imaginary part has two branches. The critical speed cd=7.34 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 7.34≤c≤7.55. When 7.55<c≤8.11, the rotating annular sector plate regains stability, and subsequently in the case of 8.11<c≤8.61, 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. cf=8.11 is called the first-second order critical flutter speed.Figure 3
First three order dimensionless complex frequencies versus dimensionless angular speed (ϕ=π/3, ξ=0.5, ψ=0).Figure4 shows the variation of the first three order dimensionless complex frequencies of the rotating annular sector plate (ϕ=π/3, ξ=0.5) with the dimensionless angular speed c for the dimensionless thermoelastic coupling coefficient ψ=0.3. In the case of thermoelastic coupling (ψ=0.3), the real parts of the first three order dimensionless complex frequencies at c=0 are greater than those in the case of uncoupling (ψ=0), 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 cd=8.45 and cf=9.23, respectively, which are greater than those in the case of uncoupling.Figure 4
First three order dimensionless complex frequencies versus dimensionless angular speed (ϕ=π/3, ξ=0.5, ψ=0.3).Figures5 and 6 show the variation of the first three order dimensionless complex frequencies ω of the rotating annular sector plate (ϕ=π/3, ξ=0.8) with the dimensionless angular speed for ψ=0 and ψ=0.3. 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 c=0. With the increase of c, the real parts of ω in the first and second modes become zero successively, and then the imaginary parts of ω 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.Figure 5
First three order dimensionless complex frequencies versus dimensionless angular speed (ϕ=π/3, ξ=0.8, ψ=0).Figure 6
First three order dimensionless complex frequencies versus dimensionless angular speed (ϕ=π/3, ξ=0.8, ψ=0.3).Figures7 and 8 show the variation of the first three order dimensionless complex frequencies of the rotating annular sector plate (ϕ=π/2, ξ=0.5) with the dimensionless angular speed for ψ=0 and ψ=0.3. 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 c=0, the first order critical divergence speed, and the first-second order critical flutter speed decrease with the increase of the sector angle ϕ.Figure 7
First three order dimensionless complex frequencies versus dimensionless angular speed (ϕ=π/2, ξ=0.5, ψ=0).Figure 8
First three order dimensionless complex frequencies versus dimensionless angular speed (ϕ=π/2, ξ=0.5, ψ=0.3).Figures9 and 10 show the variation of the first three order dimensionless complex frequencies of the rotating annular sector plate (ϕ=π/2, ξ=0.8) with the dimensionless angular speed for ψ=0 and ψ=0.3. 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.Figure 9
First three order dimensionless complex frequencies versus dimensionless angular speed (ϕ=π/2, ξ=0.8, ψ=0).Figure 10
First three order dimensionless complex frequencies versus dimensionless angular speed (ϕ=π/2, ξ=0.8, ψ=0.3).From Figures3–10, one can see that when the sector angle increases from π/3 to π/2, the type of instability does not change in the case of the ratio of inner to outer radius ξ=0.5, while it changes in the case of ξ=0.8.


## 4.2. Rotating Annular Sector Plate with SS-CC

Figures11 and 12 show the variation of the first three order dimensionless complex frequencies of the rotating annular sector plate (ϕ=π/3, ξ=0.5) with the dimensionless angular speed for the dimensionless thermoelastic coupling coefficient ψ=0 and ψ=0.3. 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 ψ=0.3, with the increase of c, 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 c=0, 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.Figure 11
First three order dimensionless complex frequencies versus dimensionless angular speed (ϕ=π/3, ξ=0.5, ψ=0).Figure 12
First three order dimensionless complex frequencies versus dimensionless angular speed (ϕ=π/3, ξ=0.5, ψ=0.3).Figures13 and 14 show the variation of the first three order dimensionless complex frequencies of the rotating annular sector plate (ϕ=π/3, ξ=0.8) with the dimensionless angular speed for ψ=0 and ψ=0.3. 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.Figure 13
First three order dimensionless complex frequencies versus dimensionless angular speed (ϕ=π/3, ξ=0.8, ψ=0).Figure 14
First three order dimensionless complex frequencies versus dimensionless angular speed (ϕ=π/3, ξ=0.8, ψ=0.3).Figures15 and 16 show the variation of the first three order dimensionless complex frequencies of the rotating annular sector plate (ϕ=π/2, ξ=0.5) with the dimensionless angular speed for ψ=0 and ψ=0.3. 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.Figure 15
First three order dimensionless complex frequencies versus dimensionless angular speed (ϕ=π/2, ξ=0.5, ψ=0).Figure 16
First three order dimensionless complex frequencies versus dimensionless angular speed (ϕ=π/2, ξ=0.5, ψ=0.3).Figures17 and 18 show the variation of the first three order dimensionless complex frequencies of the rotating annular sector plate (ϕ=π/2, ξ=0.8) with the dimensionless angular speed for ψ=0 and ψ=0.3. In the case of ξ=0.8, 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.Figure 17
First three order dimensionless complex frequencies versus dimensionless angular speed (ϕ=π/2, ξ=0.8, ψ=0).Figure 18
First three order dimensionless complex frequencies versus dimensionless angular speed (ϕ=π/2, ξ=0.8, ψ=0.3).


## 4.3. Rotating Annular Sector Plate with SS-FF

Figures19–22 show the variation of the first three order dimensionless complex frequencies of the rotating annular sector plate (ϕ=π/3, ϕ=π/2, ξ=0.5) with the dimensionless angular speed for the dimensionless thermoelastic coupling coefficient ψ=0 and ψ=0.3. 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 ξ=0.5, the corresponding critical speed deceases with the increase of the sector angle.Figure 19
First three order dimensionless complex frequencies versus dimensionless angular speed (ϕ=π/3, ξ=0.5, ψ=0).Figure 20
First three order dimensionless complex frequencies versus dimensionless angular speed (ϕ=π/3, ξ=0.5, ψ=0.3).Figure 21
First three order dimensionless complex frequencies versus dimensionless angular speed (ϕ=π/2, ξ=0.5, ψ=0).Figure 22
First three order dimensionless complex frequencies versus dimensionless angular speed (ϕ=π/2, ξ=0.5, ψ=0.3).Figures23–26 show the variation of the first three order dimensionless complex frequencies of the rotating annular sector plate (ϕ=π/3, ϕ=π/2, ξ=0.8) with the dimensionless angular speed for the dimensionless thermoelastic coupling coefficient ψ=0 and ψ=0.3. Compared with the case of ξ=0.5, 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.Figure 23
First three order dimensionless complex frequencies versus dimensionless angular speed (ϕ=π/3, ξ=0.8, ψ=0).Figure 24
First three order dimensionless complex frequencies versus dimensionless angular speed (ϕ=π/3, ξ=0.8, ψ=0.3).Figure 25
First three order dimensionless complex frequencies versus dimensionless angular speed (ϕ=π/2, ξ=0.8, ψ=0).Figure 26
First three order dimensionless complex frequencies versus dimensionless angular speed (ϕ=π/2, ξ=0.8, ψ=0.3).From Figures19–26, 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 ξ=0.5 to ξ=0.8, 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 (c=0) 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.



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*Source: 8573241-2019-03-05.xml*