1 Introduction

As the basic elements, rotating structures such as beams, disks or plates, and shells are widely utilized in various machines such as floppy disks, VCDs/DVDs, hard disks, turbines, gyroscopes, and annular saw-blades. With the new generation of hard disk drives (HDDs) of high productivity and high capacity, and the high-speed annular saw-blades of precision machining being expected, the rotation speeds of the annular plates tend to be higher and higher so that the dynamic stability of them has become an important research topic^[1].

The rotating disks may lose stability by buckling at a high speed, which is related to the natural frequency of backward traveling wave (BTW) of a rotating disk. Resonance will be excited under a stationary transverse load at a critical speed where the frequency of BTW is vanished. Lamb and Southwell^[2] were the first to investigate the free vibration of a rotating disk. Since then, the free and forced vibrations, as well as the buckling stability related to the rotating thin-walled structures like shells, plates or disks, and beams, have attracted attention from many authors^{[3, 4, 5, 6, 7, 8, 9]}. Lee and Waas^[10] and Lee et al.^[11] studied the vibration characteristics and dynamic instability of a rotating laminated annular plate by considering the effect of a frictional loading. Furthermore, many investigations on the three-layer sandwich structures with the viscoelastic core embedded between two elastic face layers have been performed^{[12, 13, 14, 15, 16]}. Wang and Chen^[17] studied the dynamic characters of a sandwich plate with two laminated face layers, in which the extensional and shear modules of viscoelastic central layer are described by complex constants. Chen and Chen^[18] studied dynamic instability of the orthotropic sandwich circular plate subjected to a radial loading by Bolotin’s method. Based on Reddy’s refined highorder shear deformation theory, Meunier and Shenoi^[19] analyzed the forced vibration of a fiber reinforced plastic sandwich plate under a transverse force. There are also a few works carried out on the subject of the dynamic characteristic of a rotating sandwich plate. Chen et al.^[20] and Chen and Chen^[21] studied dynamic instability and critical speeds of a rotating orthotropic sandwich circular plate undergoing a periodic radial loading by Bolotin’s method. In their investigations, the simple model with complex constants for extensional and shear modules of the viscoelastic core was used. The authors recently investigated the dynamic characteristics of frequencies and damping of the traveling waves for the rotating laminated plate with the viscoelastic core. In the above works, the extensional and shear modules of the viscoelastic central layer were described by complex constants, which was not reasonable for a rotating structure. Because the behavior of viscoelastic material strongly depends on the frequency as the rotating speed increases, in our research, we introduce the frequency-dependent complex modulus for the viscoelastic layer^[22].

There is another kind of aeroelastic instability, besides the buckling instability, which is called disk flutter for a disk rotating in air at a higher speed beyond the critical one. The rotating disk may be unstable aerodynamically arising from the coupling with surrounding airflow to induce the disk vibration with the larger amplitude. Some empirical models which are related to the air pressure loadings acting on the disk vibration and the flutter have been proposed. Hosaka and Crandall^[23] and Hansen^[24] predicted flutter characters in enclosed and unenclosed disks by a rotating damping model. The acoustic coupling in disk stacks was modeled by some discrete springs^{[25, 26]}. These models may explain certain instability and vibration coupling characters in the rotating disks, but some coefficients in these models have to be determined by experiment. Kim et al.^[27] and Hansen et al.^[28] suggested an experimental method to predict the disk flutter based on the generalized rotating damping model in which the damping force varying with the speed of the disk was estimated experimentally. In these aeroelastic models, the parameters are dependent on the disk rotation speed, the viscosity of the fluid, the mode frequencies, and the disk-enclosure configuration, which may limit the use of the model. Wang and Huang^[29] further simplified the above aeroelastic model by some nondimensional constants instead of the coefficient operators according to the experimental data. The results showed that the predicted values of the mode frequencies, critical speeds, and flutter speeds agree well with the experimental ones. Based on the rotating damping model, Wang and Huang^[30] proposed the acoustic feedback control method to decrease the flutter speeds of a rotating disk in an enclosure and studied the effect of the sizes and placements of the distributed piezoelectric actuators on the performance of feedback control. In the present work, the sandwich plate with a viscoelastic core layer is introduced to increase the aeroelastic stability due to the high energy dissipation performance of viscoelastic materials.

In this paper, we suggest a traveling wave dynamic model for a rotating sandwich annular plate with a viscoelastic core layer taking into account the surrounding airflow. The all governing and boundary condition equations of the aeroelastic system are obtained based on Hamilton’s principle. In the modeling, the viscoelastic behavior is simulated by the frequencydependent complex modulus model^[31], and the aerodynamic loading exerted to the rotating annular plate is described by the simplified rotating damping model^{[29, 30]}. Galerkin’s methodis used to obtain the solutions of fundamental equations for the dynamics and stability of the rotating sandwich plate. The frequencies and damping of forward and backward traveling waves of the rotating plate, which are dependent upon the geometrical and material parameters of the plate, are obtained numerically. The critical speed and the flutter speed varying with the geometrical and material parameters of the sandwich plate are discussed.

2 Fundamental equations

A three-layer symmetrical sandwich annular plate is sketched in Fig. 1 with the inner and outer radii ri and ro, respectively, rotating in air with a constant rotation speed Ω. The sandwich plate is composed of the constraining layer and the host plate layer, which are assumed to be homogeneous, isotropically elastic, and viscoelastic layers. A space-fixed coordinate system (r, θ, z) is introduced in the modeling. The rotating annular plate with small transverse motion is modeled with the linear Kirchhoff’s plate theory. The displacements of the constraining and host plate layers can be expressed by^[32]

where the superscripts c and p denote the constraining layer and the host plate layer, respectively. For the viscoelastic layer, we have Here, the superscript v denotes the viscoelastic core layer of the plate. u_r0 ^χand u_θ0^χ are, respectively, the radial and circumferential displacement components on the neutral surface of layer χ (χ=c, p, v), and β_r^v and β _θ^v are, respectively, the angular displacements of the viscoelastic layer. w(r, θ, t) is the transverse displacement of the plate. h^χ is the thickness of the layerχ (χ=c, p, v).

To comply with no-slip constrains between layers^[32], one can get the displacements of the viscoelastic layer by applying the displacement continuity condition at the interface of layers to Eqs. (1)-(3),

With the consideration of symmetry of the annular plate deformation, we have u_θ0^v = 0 and u^r0_v = u^r0_v(r). The additional notations are introduced to simplify the following analyses. Furthermore, with the assumptions that the constraining and host layers have the same thickness and the same material, the simpler forms for the viscoelastic layer of Eq. (4) can be obtained by The strain-displacement relations for the laminated plate are written as follows: where z^c =z - $\frac{{{h}^{v}}+{{h}^{c}}}{2}$ , and z^p =z+$\frac{{{h}^{v}}+{{h}^{P}}}{2}$ . The strain-stress relationship for constraining and host layers by Hooker’s law is given as where α₁₁^χ = α₂₂^χ =$\frac{{{E}^{x}}}{1-{{({{v}^{x}})}^{2}}}$ , α₃₃^χ =$\frac{{{E}^{x}}}{2{{(1+{{v}^{x}})}^{{}}}}$ , α₁₂^χ = α₂₁^χ =$\frac{{{E}^{x}}{{v}^{x}}}{1-{{({{v}^{x}})}^{2}}}$ , and E^χ and ν^χ (χ =c, p)denote Young’s modulus and Poisson’s ratio of the layer χ, respectively.

For the middle viscoelastic layer, the frequency-dependent constitutive relationship with complex modulus is adopted^[15],

in which α₁₁^v=α₂₂^v= $\frac{{{E}^{v}}(i\omega )}{1-{{({{v}^{v}})}^{2}}}$ , α₁₂^v = α₂₁^v= $\frac{{{E}^{v}}(i\omega ){{v}^{v}}}{1-{{({{v}^{v}})}^{2}}}$ , α₃₃^v= $\frac{{{E}^{v}}(i\omega )}{2(1+{{v}^{v}})}$ , α₄₄^v = α₅₅^v = G^v(iω), and ω isthe natural frequency of the plate. E^v(iω)=E₁^v (ω)+ iE₂^v (ω) and G^v(iω)=G₁^v(ω)+ iG₂^v(ω) are, respectively, the complex Young’s modulus and shearing modulus, and the relationship G^v(iω)=$\frac{{{E}^{v}}(i\omega )}{2(1+{{v}^{v}})}$holds. E₁^v (ω) and G₁^v(ω) are, respectively, the storage Young’s modulus and the shearingmodulus. E₂^v (ω) and G₂^v(ω) are, respectively, the loss Young’s modulus and the shearingmodulus. ν^v is Poisson’s ratio of the viscoelastic layer, and i denotes the imaginary unit.

For the rotating sandwich annular plate, the variation of the kinetic energy of K gives

where V χ = (−Ωuχθ0)er +Ω(r + uχr0)eθ +($\frac{\partial w}{\partial t}+\Omega \frac{\partial w}{\partial \theta }$)e_z (χ =c, v, p) is the velocity field of thesandwich plate. e_r, eθ, and e_z denote the unit vectors along, r-, θ-, and z-axis, respectively. The variations of the strain energy of U and the work done by internal forces of W for thesandwich plate are expressed by where σ_r0^χ, σ_θ0^χ, and τ_rθ0^χ (χ =c, p, v) are, respectively, the membrane stresses of the constraining layer, host plate layers, and middle layer, The variation of the work done by the aerodynamic force of Wa for the sandwich plate can be given by where the equivalent aerodynamic force q_a(r, θ, t) acting on the mid-plane of the rotating plate is utilized the rotating damping model^{[29, 30]}, i.e., where C_d and Ω_d are, respectively, the damping coefficient and the rotation speed of the distributed viscous damping force relative to the plate.

For convenience of analysis, the following nondimensional variables are introduced:

We drop the bars on the variables for convenience. Based on Hamilton’s principle, we can get the fundamental equations for the sandwich annular plate, and the boundary conditions for the sandwich plate clamped at r = r_i and simply supported at r = r_o are given as where the coefficients aj (j = 1, 2, · · · , 12) are the same as in Ref. ^[22]. ∇⁴ =${{(\frac{{{\partial }^{2}}}{\partial {{r}^{2}}}+\frac{\partial }{r\partial r}+\frac{{{\partial }^{2}}}{{{r}^{2}}\partial {{\theta }^{2}}})}^{2}}$is the biharmonic differential operator. h =$\underset{x=c, P, v}{\mathop{\Sigma }}\, $h^χ is the total thickness of the laminated annular plate. C =$\frac{{{C}_{d}}}{(2{{\rho }^{c}}{{h}^{c}}+{{\rho }^{v}}{{h}^{v}})}$is the nondimensional ratio of aerodynamic damping, and Ω_d/Ω is the damping speed ratio.

3 Method of solution

To solve the dynamic equations, we use an approximation method and assume that the displacements are in the forms of

where λ is the complex eigenvalue whose real part and imaginary part determine the vibration frequency and damping character of the system, respectively. m and n represent the numbers of nodal circle and diameter of the mode (m, n). C_rl⁺, C⁻_rmn, C_θmn⁻, and C_wmn are the undetermined coefficients. In order to satisfy the boundary conditions of Eqs. (23)-(24), the functionsV _rl⁺ (r), V _rmn⁻(r), V_θmn⁻(r), and W_wmn(r) are taken by where d_rl = 1+$\frac{1-k}{(l+1)+{{a}_{13}}(1-k)}$

With Galerkin’s method, we substitute the first equation of Eq. (25) into the nondimensional Eq. (19) and calculate the inner product withV_r$\hat{l}$⁺(r) ($\hat{l}$ = 0, 1, · · · , L)to get AC_L = D, in which C_L = [C_r0⁺, C_r1⁺, · · · , C_rL⁺]^T.A and D are, respectively, an (L+1)×(L+1) matrix and an (L+1) column matrix whose entries are the same as those in Ref. ^[22]. Upon substituting the last three equations of Eq. (25) into Eqs. (20)-(22) and calculating the inner products withV_{r $hat{m}$n}⁻(r)e^{−i(nθ+λt)}, V_{θ $hat{m}$n}⁻(r)e^{−i(π2+nθ+λt)}, andW_$hat{m}$n(r)e^{−i(nθ+λt)}( $hat{m}$ = 0, 1, · · · , M), respectively, as well as eliminating the coefficients C_rmn⁻ and C_θmn, ⁻ one can get a matrix equation forthe coefficients C_wmn as

Here, C = [C_w0n, C_w1n, · · · , C_wMn]^T. M is an (M + 1) × (M + 1) real matrix, and N and Q(iω) are (M + 1) × (M + 1) complex-valued matrices whose entries can be referred to Appendix A.

The characteristic equation for the condition of nontrivial solutions for Eq. (27) is

The iterative method is introduced to solve the nonlinear characteristic equation with the natural frequency related stiffness matrices. The initial value of the natural frequency of the rotating plate ω₀ is chosen to determine the complex Young’s modulus E^v(iω₀) and the shearing modulus of G^v(iω₀). The eigenvalue λ is obtained by the roots of the nonlinear characteristic equation. These roots generate M + 1 pairs eigenvalues for each nodal diameter n. In each pair, eigenvalues denoted by λ^FTW and λ^BTW are related to the forward traveling wave (FTW) and BTW, respectively, along and against the rotation direction of the plate. The real parts of the eigenvalues, Re(λ), and the imaginary parts, Im(λ), are related to mode frequencies and damping of the plate vibration. The natural frequency of the rotating plate is ω = (Re(λ^FTW) + Re(λ^BTW))/2^[28] which is the same as the frequency observed in the corotating system. Replace ω₀ by ω and repeat the above steps until the relative error of ω and ω₀ is less than 10−6.

4 Simulation results and discussion

Case studies are conducted to show the dynamic characteristics of a sandwich annular plate rotating in air. First, we take all layers of the plate the same material so that it can be compared with the results of a monolayer plate for verification. The natural frequencies of the stationary plate (i.e., Ω = 0 r·min^-1) and rotating plate (i.e., Ω = 1 000 r·min^-1) with and without surrounding airflow are obtained to compare with the results in Refs. ^[30] and ^[31]. There shows good agreement between them even as the truncation number takes small ones like L = 5 and M = 5. Then, we study the dynamic character of a sandwich annular plate with the viscoelastic core layer. Young’s modulus, Poisson’s ratio, and the density of the constraining layer and the host layer are E^c = 200 GPa, ν^c = 0.3, and ρ^c = 7 840 kg·m⁻³, respectively.The density and Poisson’s ratio of the viscoelastic core layer are chosen as ρ^v = 500 kg·m⁻³ and ν^v =0.3, respectively. The frequency-dependent complex Young’s modulus and shearing modulus of the viscoelastic core are used by^[31]

where E₀ = 19.6 MPa, E_∞ = 98 MPa, ξ = 0.75, τ = 2.24 s, E'₀ = E₀, and E'₁ = E_∞ − E₀. In the simulation, the geometrical material parameters are ri = 30 mm, ro = 120 mm, and h = 0.5 mm, respectively. The parameters of the aerodynamic force are set as C = 0.01 and Ω_d/Ω = 0.85^[29].

The frequencies for all FTW modes increase with the rotation speed of the sandwich annular plate. The frequencies of BTWs decrease at first and vanish at a critical speed where the buckling occurs, finally increase beyond that speed, as shown in Fig. 2(a). In the simulation, the thickness ratio δ = h^v/h = 0.6. A stationary transverse force may excite resonance of the plate at critical speeds. The dampings for some higher BTW modes, as shown in Fig. 2(b), are initially positive and increase and then decrease to reach a negative value as the rotation speed increases. The negative value of the damping for BTW modes, Im(λ)<0, means that the plate loses stability aeroelastically, and flutter occurs. The speed at which Im(λ) equals zero is defined as the flutter speed. The plate is stable at modes (m = 0 and n = 0, 1, 2) due to the positive values of the frequencies and damping at these modes. Furthermore, Fig. 3 shows that the critical speed and the flutter speed depend on the vibration modes. We can see that the critical and flutter speeds decrease with the nodal diameter n and increase thereafter. They reach the minimum values of the critical speed for the mode (0, 22) and the flutter speed for the mode (0, 14), respectively. The modes, at which buckling and flutter take place and the corresponding speeds are the minimums, are named as the critical and flutter modes of the rotating plate, respectively. The critical mode and the flutter mode are also found to vary with the geometrical and material parameters of the sandwich annular plate.

There shows obvious influence of the thickness and material parameters on the frequencies and damping and the stability character of the rotating sandwich plate. The mode frequency and the critical speed varying with the thickness ratio δ are shown in Fig. 4. The mode frequencies increase with the thickness of the viscoelastic core layer, reach the maximum values at δ = 0.85, and then decrease lightly, as shown in Fig. 4(a). The results of the critical speeds are plotted in Fig. 4(b), in which the minimum values for different modes are obtained at about δ = 0.5. By comparisons on all the modes, we can see that for different thickness ratios δ, it shows different critical modes. For example, at the ranges of 0.1 < δ < 0.4 and 0.82 < δ < 0.9, the mode (0, 10) is the critical mode, and at 0.4 < δ < 0.6 and 0.6 < δ < 0.82, the critical modes are (0, 20) and (0, 30), respectively. The minimum rotation speeds of buckling and flutter instability for the sandwich plate are plotted in Fig. 4(c). One can see that, for the buckling instability, the critical speed reaches the minimum value at δ = 0.45 and the maximum value at δ = 0.9. As for the flutter instability, the minimum flutter speed reaches at δ = 0.3, and the maximum value reaches at δ = 0.88. With the increase of δ, the critical and flutter speeds are improved obviously. The flutter mode and the flutter frequency where flutter occurs varying with the thickness ratio are listed in Table 1. It can be seen that, different thickness ratios δ for the sandwich plate are always associated with different flutter modes and flutter frequencies, and the frequency gets the maximum value at about δ = 0.6. Figures 5(a) and 5(b) show the mode frequencies and damping of the rotating sandwich plate for different Young’s modulus ratios. In the simulation, we choose δ = 0.6 and E_c = 200 GPa. From the figure, the frequencies of the FTW and BTW modes increase with the ratio of Young’s modulus R_E =E' ₀/E^c, while the mode damping decreases. The critical and flutter speeds depending on the Young’s modulus ratio are shown in Fig. 5(c). It shows that the critical and flutter speeds increase with the ratio. These may be explained by the enhanced bending stiffness. Furthermore, the flutter mode and the flutter frequency for different ratios of Young’s modulus are listed in Table 2. The effect of the loss factor coefficient is obtained in Fig. 6. Figures 6(a) and6(b) indicate that the mode frequencies and damping increase with the loss factor coefficientE'₁. Figures 6(c) and6(d) plot the critical speed and the flutter speed varying with the loss factor coefficient E'₁, respectively. It is shown that the critical and flutter speeds are improved as the loss factor coefficient increases due to the increase of damping of the rotating plate system, which improves the dynamic stability of the plate. More details on the flutter modes and flutter frequencies with the loss factor coefficient are listed in Table 3.

5 Conclusions

The dynamic and stabilization characteristics of a rotating sandwich plate with a frequencydependent viscoelastic core are analyzed quantificationally. The effect of the thickness and material parameters on frequencies and damping of FTWs and BTWs of the rotating plate and the dynamic stability are discussed. The numerical results show that the dynamic stability can be enhanced due to the increased critical and flutter speeds at some particular values of the thickness ratio as well as Young’s modulus ratio and the loss factor coefficient.

Appendix A

The entries for matrices M, N, and Q are

in which