Multiple internal resonances of rotating composite cylindrical shells under varying temperature fields

Composite cylindrical shells, as key components, are widely employed in large rotating machines. However, due to the frequency bifurcations and dense frequency spectra caused by rotation, the nonlinear vibration usually has the behavior of complex multiple internal resonances. In addition, the varying temperature fields make the responses of the system further difficult to obtain. Therefore, the multiple internal resonances of composite cylindrical shells with porosities induced by rotation with varying temperature fields are studied in this paper. Three different types of the temperature fields, the Coriolis forces, and the centrifugal force are considered here. The Hamilton principle and the modified Donnell nonlinear shell theory are used to obtain the equilibrium equations of the system, which are transformed into the ordinary differential equations (ODEs) by the multi-mode Galerkin technique. Thereafter, the pseudo-arclength continuation method, which can identify the regions of instability, is introduced to obtain the numerical results. The detailed parametric analysis of the rotating composite shells is performed. Multiple internal resonances caused by the interaction between backward and forward wave modes and the energy transfer phenomenon are detected. Besides, the nonlinear amplitude-frequency response curves are different under different temperature fields.


Introduction
Composite materials have the advantages of high specific stiffness, specific strength, and good vibration reduction performance, which are widely used in various engineering fields [1][2][3][4][5] . Rotating thin structures, as a key basic structure, have been employed in mechanical, civil, and aerospace engineering [6][7][8][9] . Due to the anisotropy, Coriolis, and centrifugal effects, the vibration behaviors of rotating composite structures must be different from those of homogeneous stationary structures. Hence, it is of great significance to carry out analysis and research on the dynamic responses of rotating composite thin structures for guiding the corresponding engineering applications. To date, the vibration analysis of rotating composite structures has been investigated by many scholars [10][11][12] .
When mechanical systems such as rotor systems generate vibrations, the resonances of the structures such as rotating shells can be caused by appropriate geometrical parameters or rotation effect, and thus the large amplitude vibrations of these structures may occur under the transverse loads [13][14][15] . It affects the normal operation of the rotating structures seriously. Therefore, the study on nonlinear large amplitude vibrations is very important for the reliability and safety of the equipment, and the nonlinear vibration cannot be ignored in this case [16][17][18][19][20] .
There are some investigations on the nonlinear dynamic responses of rotating composite structures. A simplified mathematical model to study the nonlinear vibrations of rotating composite beams was established by Rafiee et al. [21] . Gu et al. [22] performed the nonlinear vibration of graphene platelets reinforced pre-twisted composite blades. Li et al. [23] reported the nonlinear vibrations of rotating composite shells under arbitrary boundary conditions. Moreover, it is worth noting that all the above studies did not consider the influence of the temperature. Actually, the working environment of these rotating structures is usually with varying temperature fields, and the temperature will greatly affect the mechanical properties. Considering the thermal effect, Khosravi et al. [24] analyzed the vibration behavior of carbon nanotube reinforced rotating composite beams, and found that the instability of the rotating composite beam under the hinged-clamped boundary conditions could be caused by a decrease in the temperature. Considering the hygro-thermal effect, Li et al. [25] proposed a general approach for dealing with the thermal vibration characteristics of functionally graded porous stepped cylindrical shells. Liu et al. [26] demonstrated that the nonlinear vibrations of rotating composite eccentric shells were significantly influenced by the external loads, the rotating speed, and the temperature.
Based on the literature review, there are just a few studies on the dynamic response of rotating composite structures under the thermal load. There is no literature available to deal with the multiple internal resonances of composite shells with porosities induced by rotation under various temperature fields. Hence, this paper aims to reveal the complex mechanisms of multiple internal resonances and the energy transfer phenomena caused by rotation, and to conduct the effect of varying temperature fields on the nonlinear vibration characteristics. The Hamilton principle and modified Donnell nonlinear shell theory are used to gain the equilibrium equations, which are then transformed into nonlinear ordinary differential equations (ODEs) by utilizing the multi-mode Galerkin technique. The pseudo-arclength continuation method is introduced to obtain accurate results. The effects of key parameters on multiple internal resonances of rotating composite shells are studied.

Rotating composite shell model under various thermal loads
In this study, a rotating composite thin shell with a constant rotating speed Ω is made of ceramic and metal, and the geometry of the shell is presented in Fig. 1. The material properties vary by gradient from inside to outside, in which the inner and outer surfaces are metal and ceramic, respectively. Porosities are considered here due to the manufacturing process. The rotating composite thin shell is under external harmonic excitation and various temperature fields, wherein three different types of temperature field distributions, namely, uniform temperature (UT) variation, linear temperature (LT) variation, and nonlinear linear temperature (NLT) variation, are studied here as representative temperature fields.
Case 1 UT variation At the initial reference Kelvin temperature T 0 , the temperature of the entire composite thin shell is approximated to rise uniformly to a final value T , and the process can be given by in which ∆T denotes the temperature difference between the initial state and the final state. Case 2 LT variation Assume that the temperature difference changes linearly along the thickness. Then, the temperature expression can be given by Case 3 NLT variation The temperature is considered to be a nonlinear change along the shell thickness direction. Due to the lack of a heat source, a one-dimensional steady-state heat transfer equation is satisfied as follows: and the boundary condition is where c and m denote ceramic and metal, respectively, κ(z) denotes the thermal conductivity, and the non-uniformity of thermal conductivity is given in a power form as where N ∈ [0, ∞) denotes the power-law index. By solving Eqs. (3)-(5), the solution of temperature variation T as a nonlinear function of the thickness z can be obtained.
The temperature-dependent material parameters are in which P 3 , P 2 , P 1 , P −1 , and P 0 stand for the temperature coefficients of the constituent materials. Due to manufacturing defects, porosities are usually not evenly distributed in the material, and they are abundant in the middle plane and dissipate gradually towards both ends [27] . Therefore, the effective material parameter is modified as where V m and P m denote the volume fraction and material property of metal, respectively, V c and P c denote those of ceramic, respectively, and α (α 1) denotes the porosity volume fraction.
The sum of the two volume fractions is V m (z) + V c (z) = 1. The volume fraction of the rotating composite shell, which obeys the power-law distribution, changes along the thickness direction as As for the rotating composite thin shell with uneven porosity distribution, the effective material parameter can be rewritten as where the material property P denotes the general mass density ρ, the elastic modulus E, Poisson's ratio ν, or the thermal expansion coefficient α T . An improved Donnell nonlinear shell theory, which has better accuracy under small circumferential wave numbers, is adopted, where the strain-displacement relation is The stress-strain relation considering the thermal effect is [28] where By using the Hamilton principle, the partial differential equations considering the structure damping coefficient c d are obtained as where in which f represents the amplitude of the radial force, (x 0 , θ 0 ) denotes the point at which the radial force is applied, δ represents the Dirac delta function, and ω f denotes the excitation frequency.

Solution method
For a cylindrical shell simply supported at both ends, to discretize this system, the displacement functions are [29] u where u m,n (t), v m,n (t), and w m,n (t) denote the displacement amplitude components, and M 1 and N 1 represent the truncated coefficients. The subscript c represents the driving mode, and the subscript s represents the companion mode. Substitute Eqs. (17)- (19) into Eqs. (13)- (15) and then transform the motion equations into nonlinear ODEs by using the multi-mode Galerkin technique. The detailed derivation process can be found in Refs. [30] and [31].
The system of nonlinear coupled ODEs is expressed in a matrix form as where X = [u, v, w] T , M , C, and K indicate the mass, the damping, and the stiffness matrices, respectively, Γ denotes the other nonlinear parts such as the temperature non-linearity, G represents the gyroscopic matrix caused by the rotation, and F denotes the vector of external force. Since these matrices all have very tedious expressions, they are not stated here.
Since the moments of inertia of the displacements u and v have little impact on the radial response, they are not considered here. Therefore, the results can be substituted into Eq. (15) to obtain the nonlinear ODEs only related to w.
Then, the pseudo-arclength continuation technique can be applied to solve the nonlinear ODEs on the software MATCONT [32] . This method can be extended to analyze shell structures with different geometric shapes and boundary conditions by establishing the corresponding shell displacement functions.

Results and discussion
First, the dimensionless frequency of a rotating infinitely long thin shell is compared with that obtained by Chen et al. [33] as shown in Table 1. The dimensionless forward and backward traveling wave frequencies in the table are, respectively, expressed by ω * ft and ω * bt . The comparison shows that the derivation in this paper is correct, and the proposed method can effectively analyze the traveling wave vibrations of rotating shells. The nonlinear vibration response curves of a stationary thin shell are further compared as presented in Fig. 2  The material properties of the composite shell subject to various temperature fields are presented in Table 2. If not specified, the other parameters adopted are L = 0.5, R = 1, h = 0.02, Ω * = 0.02, α = 0.1, N = 1, T m = T 0 = 300 K, ∆T = 300 K, (x 0 , θ 0 ) = (L/2, 0), 2ξ 1,n = 0.003, and f = f 0 I 0 hω 2 0,(m=1,n=4) , where f 0 represents the modal excitation amplitude, and its value is 0.002. ω 0,(m=1,n=4) is the natural frequency of the mode (m = 1, n = 4) of a cylindrical shell without rotation. For the sake of simplicity, the subscript "(m = 1, n = 4)" is ignored. In Fig. 3, with the rotating speed increasing, the forward wave frequency decreases slightly and increases subsequently, while the backward wave frequency keeps increasing. The fundamental frequency occurs at (m = 1, n = 4). Additionally, many intersection points are observed on the Campbell diagram at a specific wave number. As illustrated in Fig. 4, the forward wave frequency in the mode (m = 1, n = 4) is the same as the backward wave frequency in the mode (m = 1, n = 5) at a specific rotating speed. Therefore, due to the rotation effect, the energy transfer between these two fundamental modes is likely to happen, and further nonlinear multiple resonance phenomena occur. In order to capture the peculiar phenomenon correctly, 50 degrees of freedom (DOFs) are considered here as follows: u 1,4,c ,  u 1,5,c , u 2,4,c , u 2,5,c , u 3,4,c , u 3,5,c , v 1,4,c , v 1,8,c , v 1,5,c , v 1,10,c , v 2,4,c , v 2,8,c , v 2,5,c , v 2,10,c , v 3,4,c ,  v 3,8,c , v 3,5,c , v 3,10,c , w 1,4,c , w 1,5,c , w 2,4,c , w 2,5,c , u 1,4,s , u 1,5,s , u 2,4,s , u 2,5,s , u 3,4,s , u 3,5,s , v 1,4,s ,  v 1,8,s , v 1,5,s , v 1,10,s , v 2,4,s , v 2,8,s , v 2,5,s , v 2,10,s , v 3,4,s , v 3,8,s , v 3,5,s , v 3,10,s , w 1,4,s , w 1,5,s , w 2,4,s , w 2,5,s , u 1,0 , u 3,0 , u 5,0 , w 1,0 , w 3,0 , and w 5,0 .   Figure 5 presents the nonlinear dynamic responses of the rotating composite thin shell subject to various temperature fields. There are four coupled peaks in the amplitude-frequency response curves due to the multiple internal resonances and rotating effect. The first peak is bending to the left, and the rest peaks turn to the right as the nonlinear frequency ratio increases, and the amplitude of the second peak is larger than that of the others. Besides, analysis shows that the bending of the dynamic response curves results in a jump phenomenon. The jump phenomenon implies a sudden change in amplitude, and the multiple amplitudes mean several resonance responses. In addition, the locations of the Neimark-Sacker (NS) bifurcation points and branch point of cycles (BPC) on the response curves vary with different temperature fields. However, the response curves under the NLT field and those under the LT field have little difference.
In Fig. 6, the nonlinear dynamic responses for different temperature variations are presented. The jump phenomenon occurs in these curves. It is known that the dynamic response curves of the rotating composite shell bend to the left when they have softening property and to the right when they have hardening one. It is known that the nonlinear amplitude-frequency response curve of the rotating composite shell bends to the left and right when it has the softening and hardening characteristics, respectively. The resonance amplitudes increase significantly and the resonance region moves to the left with the increase in the temperature change. The results show that the energy increment induced by the temperature change has a notable effect on the dynamic responses. Besides, for the UT type, more peaks appear on the response curve by increasing the temperature change.  Figure 7 illustrates the nonlinear dynamic responses for different porosity volume fractions. Here, α = 0 corresponds to the rotating composite shell without porosity. One can find that the porosity volume fraction has a notable effect on the nonlinear vibration behavior of the rotating composite thin shell. Examining Figs. 7(a) and 7(b) shows that the peaks with respect to the unstable solution decrease with the increase in the porosity. This is because that addition will lead to a decrease in the shell stiffness, consequently weakening the nonlinear coupling induced   by the large deformation of the rotating composite shell. Additionally, a trend is observed that the resonance region moves towards the direction of the larger dimensionless nonlinear frequency with the porosity volume fraction increasing. The nonlinear dynamic responses for different power-law indexes are presented in Fig. 8. As the dimensionless nonlinear frequency increases, the amplitude peaks of the driving mode w 1,4,c /h increase as the power-law index increases. At the same time, the partial stable solution on the first resonance peak is converted to an unstable solution for all the three temperature types. Furthermore, it is revealed from these figures that the resonance regions of the driven mode w 1,4,c move to the lower dimensionless nonlinear frequency by increasing the power-law index.
As shown in Fig. 9, the nonlinear dynamic responses for different dimensionless rotating speeds are investigated. The results demonstrate that the nonlinear dynamic response curves are sensitive to the rotating speed. A very significant difference is that the nonlinear hardening/softening spring behaviors attenuate as the dimensionless rotating speed increases. That is because the increase in the rotating speed weakens the coupled effect between the modes. Moreover, as the rotating speed increases, the peak values of the nonlinear amplitude-frequency response curves gradually decrease, and the resonance regions move to the direction of the larger dimensionless nonlinear frequency.

Conclusions
The multiple internal resonances of composite cylindrical shells with porosities induced by rotation with varying temperature fields are investigated. It is shown that the nonlinear vibration response exhibits both hardening and softening characteristics, and four coupled peaks and an energy transfer phenomenon are detected due to the multiple internal resonances induced by the rotation effect. In addition, the locations of the BPC and NS points vary with the temperature fields, but the dynamic response curves under the NLT and LT fields have little difference. Besides, the energy increment induced by the temperature variation has notable influence on the nonlinear dynamic responses. For the UT type, more peaks appear on the response curves as the temperature change increases. Furthermore, the resonance region moves towards the direction of the larger dimensionless nonlinear frequency by increasing the porosity volume fraction. As the dimensionless nonlinear frequency increases, the amplitude peaks of the driving mode w 1,4,c /h increase with the increase in the power-law index. Moreover, the nonlinear hardening/softening spring behaviors attenuate as the dimensionless rotating speed increases.
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