Investigation on Free Vibration of Rotating Cylindrical Shells with Variable Thickness

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Introduction
Te rotating cylindrical shell (RCS) is widely used in the industrial feld, such as high-speed rotating centrifugal separator and rotor system of aeroengine. Te RCS is increasingly demanding complex working condition in the engineering equipment; meanwhile, the wall thickness of the RCS shows the trend of more and more thin to meet the increasing of performance and efciency, which lead to the structural vibration becoming more and more prominent. In addition, the RCS will have special traveling wave vibration phenomenon due to the infuence of centrifugal force, Coriolis force, and the initial hoop tension compared with the static cylindrical shell; thus, it is of great signifcance to study the vibration characteristics of the RCS.
Tere are various solutions for the free vibration analysis of the RCS; for example, some scholars used Galerkin's method to analyze the free vibration characteristics of the RCS [1][2][3][4][5], and the diferential quadrature method was also used to study the traveling wave properties of the free vibration for the RCS [6][7][8][9][10][11]. In addition, Ritz method was relatively simple in calculation process and could maintain high accuracy, so it was widely used in the research of vibration characteristics of RCS. For instance, Yan et al. [12] solved an accurate solution for the vibration-damping characteristics of a constrained stand-of-layer damping cylindrical shell and obtained the loss factor and the frequency based on the Hamilton principle. Song et al. [13] presented an improved Rayleigh-Ritz method to investigate the vibration and damping behavior of thin short cylindrical shell. Nguyen et al. [14] established sandwich structures and derived equations for the free vibration of rotating shells by the frst-order, shear-deformation shell theory. Abbaspour and Hosseini [15] employed the Ritz method with Chebyshev polynomials to perform the free vibration analysis of the graphene platelets reinforced piezoelectric cylindrical microshells with various boundary conditions. Qin et al. [16][17][18] took Chebyshev polynomials as the admissible functions, employed the Rayleigh-Ritz method to derive the motion equations for the rotating shell-plate combination, investigated the vibration of an RCS coupled with an annular plate based on the Sanders shell theory and Mindlin plate theory, and calculated the strain energy of the shell and plate. Ten, they studied the free vibration of rotating functionally graded carbon nanotube reinforced composite cylindrical shells with arbitrary boundary conditions. Some scholars utilized Ritz method and Gram-Schmidt polynomials to form approximate functions and analyzed the free vibration of RCS [19]. Furthermore, some scholars studied the nonlinear vibration of cylindrical shells of composite materials [20][21][22][23][24].
In addition, the abovementioned studies did not consider the infuence of the thickness variation of the RCS. However, sometimes, the RCS needs to be designed with variable thickness in practice; that is, the thickness varies along the axial direction, and this can further reduce the weight. Terefore, some scholars analyzed the traveling wave vibration characteristics of rotating cylindrical shells with variable thickness (VTRCS). Quoc et al. [25] used Galerkin method to study the vibration characteristics of VTRCS under thermal environment, but only one thickness variation form was considered in this study.
Based on the above discussion, three diferent thickness variations are considered in this study, and Chebyshev-Ritz method is used to investigate the free vibration traveling wave behavior of VTRCS. Furthermore, the infuences of parameters such as rotational speed, thickness variation parameters, and aspect ratio of cylindrical shells on the traveling wave behavior of free vibration for the VTRCS are discussed, which has signifcance for the lightweight design of rotating cylindrical shell structures.
High speed rotary drum has important application in centrifuge, which is abstracted to the rotating cylindrical shell (RCS) to reduce its mass and improve its performance and efciency. Te scientifc and efective method is to design the variable thickness of RCS along the axial direction. In addition, the research can be applied to the aircraft engine disk or other mechanical structures and industrial felds, such as submarine, ship, drilling oil and gas feld development, and factory workshop. Terefore, the research in this work has important application value in the felds of national defense and military.

Theoretical Modeling
Firstly, a theoretical model is established to study the traveling wave vibration characteristics of VTRCS. Te cylindrical shell structure rotates around its central axis with a rotational speed Ω, its length is L, and its average radius is R. Te (x, θ, z) is an orthogonal coordinate system established on the surface of the cylindrical shell, and u, v, and w are the dispositions of any point on the cylindrical shell along the three directions of x, θ, and z, respectively, as shown in Figure 1. It is assumed that the thickness h(x) of the rotating cylindrical shell varies linearly along its axis, as shown in Figure 2, which can be divided into three variation forms, denoted as V1, V2, and V3, respectively.
Te coordinates of the upper and lower surfaces of the cylindrical shell in the Z direction of the coordinate axis will change under the three diferent thickness changes. Te specifc expressions are as follows.
Te thickness change form of V1 is expressed as follows: where h 1 (x) and h 2 (x), respectively, represent the coordinates of the upper and lower surfaces of the cylindrical shell in the Z direction of the coordinate axis; h 0 represents the initial thickness of the cylindrical shell, that is, the thickness at x � 0; k h represents the thickness variation parameter.
Te thickness change form of V2 is expressed as follows: Te thickness change form of V3 is expressed as follows:

Solution of Kinetic and Potential Energy.
Te Chebyshev-Ritz method is used to study the VTRCS in order to solve the natural frequency of the model. Firstly, the kinetic energy equation and potential energy equation of the cylindrical shell need to be given. Te velocity vector of any point on the VTRCS can be expressed as follows: where k is the unit vectors along z direction and r represents the displacement vector of the VTRCS at any point on the coordinate system (x, θ, z), which can be described as follows: where i, j, and k are, respectively, unit vectors along x, θ, and z directions. Te kinetic energy of the VTRCS can be calculated as follows: Ten, the kinetic energy equation of VTRCS is obtained by substituting equation (4) into equation (6), which is denoted as follows: According to Sanders shell theory, the strain at any point on the rotating cylindrical shell is written as follows: xθ are the strain components of the middle surface and ε (1) xx , ε (1) θθ , and c (1) xθ are the curvature components of the middle surface. Tey can be expressed by the following formula [26]: x, u

Shock and Vibration 3
Te stress-strain relationship of the VTRCS can be expressed as follows: where Te strain energy of the VTRCS caused by the deformation can be described as follows: Te strain energy of the VTRCS caused by centrifugal force [27] is expressed as follows: Terefore, the total potential energy equation of the VTRCS with variable cross section can be expressed as follows: Te total potential energy solution of the VTRCS lays a foundation for studying its natural frequency and mode shape.

Solution of Natural Frequency and Mode Shape.
Te natural frequency and mode shape of the VTRCS are solved in this section, and the solution process is given on the basis of the kinetic energy equation and potential energy equation obtained by the Chebyshev-Ritz method in the previous section.
Firstly, the displacement feld of the VTRCS can be expressed as follows: where w is the natural frequency and n is the toroidal wave number of the traveling wave modal. U(x), V(x), and W(x) are modal functions, which are approximately expanded by the product of Chebyshev polynomials and their corresponding boundary functions, and the specifc formula is expressed as follows: where a i , b j , and c k are unknown coefcients and P i (ξ), P j (ξ), and P k (ξ) are the frst kind Chebyshev expressions, which can be expressed in trigonometric terms as follows: Chebyshev polynomials can achieve fast convergence speed and maintain high accuracy at a lower computational cost, but they are defned on the interval [−1, 1] and have orthogonality only on this interval, so coordinate transformation is needed, , and f w (ξ) represent boundary functions along the ξ direction, which need to satisfy the corresponding geometric boundary conditions of the VTRCS, and the specifc expressions are shown in Table 1.
Secondly, the displacement feld equation (14) of the VTRCS is substituted into the kinetic energy equation (7), which can be further expressed as follows: Similarly, equation (14) is substituted into equation (13), and the potential energy equation of the VTRCS can be expressed as follows: where A 11 , A 12 , and A 66 represent the tensile stifness and D 11 , D 12 , and D 66 represent the bending stifness. Te specifc expressions are denoted as follows: Gdz, Ten, according to equations (17) and (18), the energy expression of the VTRCS can be obtained as follows: Te most likely approximate value is determined by minimizing the energy expression with respect to the unknown coefcient according to Rayleigh's principle, so the derivative of the energy expression with respect to the unknown coefcient is performed as follows: Ten, equation (21) can be further transformed into an eigenvalue matrix, which is written as follows: where K is the stifness matrix, M1 and M2 are the mass matrices, w represents the natural frequency, and P is the eigenvector composed of unknown coefcients, namely, the mode shape of the VTRCS, which is expressed as follows: ..., a i , b 1 , ..., b j , c 1 , ..., c k T .
Te natural frequencies and corresponding mode shapes can be obtained by solving equation (22).

Numerical Results and Discussion
In this section, the calculation results are compared with those in the existing literature, and their convergence is studied to verify the accuracy of the established model. Ten, the free vibration traveling wave behavior of the VTRCS is analyzed through parameter research. Unless otherwise Shock and Vibration mentioned, the length of the VTRCS L � 1 m, the initial thickness h 0 � 0.02 m, the thickness variation parameter k h � 0.5, the density ρ � 1072 kg/m 3 , the elastic modulus E � 172 GPa, the shear modulus G � 4.2 GPa, and the Poisson's ratio v � 0.31, the average radius R � 0.2, the speed ω � 25 r/s, the dimensionless frequency parameter w * � wR ��� ρ/E, and the dimensionless speed Ω * � ΩR ��� ρ/E.

Comparison and Convergence
Research. Two cases are studied in this section to verify the accuracy and convergence of the proposed method.

Case One.
Te dimensionless frequency parameters (DFPs) of the backward wave and the forward wave for the RCS with uniform thickness under fxed (C-C) boundary conditions at both ends are given, respectively, and compared with the results in literature [27]. Also, the convergence research is conducted, and the calculation results under diferent number of truncation terms are listed, as shown in Tables 2 and 3. In this example, the aspect ratio of the RCS L/R � 10, the thick-aspect ratio h/R � 0.05, the Poisson's ratio v � 0.3, the dimensionless frequency parameter w * � wR ��������� � ρ(1 − μ 2 )/E, and the dimensionless speed Ω * � ΩR ��������� � ρ(1 − μ 2 )/E. It can be seen from Tables 2 and 3 that the results are basically consistent with the results in literature [27]. Te study shows that the calculation results converge to a certain value with the increase of the number of truncated terms nmax, which have converged to a sufciently accurate value when the number of truncated terms is 11, so nmax is 11 in the following calculation.

Case Two.
Te second case gives the natural frequency (Hz) of the RCS with uniform thickness under simply supported (S-S) boundary conditions at both ends as shown in Table 4, which is compared with the results in literature [28]. Te cylindrical shell length L � 0.256 m, average radius R � 0.16 m, thickness h � 0.0025 m, elastic modulus E � 110 GPa, Poisson's ratio v � 0.31, density ρ � 4480 kg/m 3 , axial half wave number m � 1, and rotational speed Ω � 20000 r/min. It is observed in Table 4 that the results obtained by the proposed model are basically consistent with those in literature [28], and the maximum error is less than 1%.
In short, the above two examples verify the correctness and convergence of the proposed method.

Parameter Study.
Te efects of diferent thickness variations, rotational speed, thickness variation parameters, aspect ratio, and thickness-to-diameter ratio on the traveling wave behavior of free vibration are discussed in order to obtain the vibration characteristics of the VTRCS under diferent thickness variations, as shown in Figures 3-7.
Te variation of the DFPs w * b and w * f with the circumferential wave number n of the VTRCS is shown in Figure 3. In the fgure, V0: k h � 0, that is, the thickness of the RCS is uniform and remains unchanged in the x-axis direction. Figure 3 shows that the DFPs w * b and w * f for the RCS under thickness change rise with the increase of the wave number n in the boundary conditions of the S-S or C-C. Te DFPs of the VTRCS are the maximum under V0 thickness changes form, and it is the minimum under V1 thickness changes form. Te DFPs of the VTRCS under V2 thickness variation form are higher than those under the other two thickness variations when V0 thickness variation form is not considered, which are close to those under V0 thickness variation form. It is also obvious from Figure 3 that the DFPs of the VTRCS under the four thickness changes frstly disperse from the same initial value and then converge to the same value with the increase of circumferential wave number n.
Te variation of the DFPs w * b and w * f with the thickness parameter k h for the RCS under the three thickness forms when the rotational speed is, respectively, 0 r/s, 25 r/s, and 50 r/s, is shown in Figure 4.
It is seen from Figure 4 that the DFPs for the VTRCS decrease gradually with the increase of k h under diferent rotational speed and thickness, and it is minimum in the condition of the V2 thickness change. In addition, it can also be found that the rotational speed has almost no infuence on the variation trend of the k h .
Te variation of the DFP w * as the changing of Ω is shown in Figure 5 to further study the infuence of rotational speed on the traveling wave for the VTRCS. Te BW represents the backward wave, and FW represents the forward wave. Figure 5 shows that the DFPs of backward wave rise with the increase of the rotational speed for the VTRCS, and the DFPs of forward wave reduce gradually with the increase of rotational speed. Te DFP value of backward wave and forward wave under diferent rotational speed is the biggest under V0 thickness change form. Te DFP value increases, and their diference becomes smaller under V2 thickness variation form when the boundary conditions change from S-S to C-C. In addition, it can also be found that the difference between the DFPs under diferent thicknesses becomes signifcantly larger when the circumferential wave number changes from 1 to 2.
Finally, the infuence of geometric parameters on the DFPs of VTRCS is studied in Figures 6 and 7. Figure 6 gives the variation of the DFP for the VTRCS with the changing of the aspect ratio L/R, and Figure 7 shows the changing of the aspect ratio R/h 0 . Figure 6 manifests that the DFPs of VTRCS decrease gradually with the increase of length-to-diameter ratio L/R. Particularly, the DFPs decrease rapidly when the length-todiameter ratio is less than 3, and it decreases slowly when the length-to-diameter ratio is more than 3. It can be seen from Figure 7 that the backward wave w * b and the forward wave w * f both decrease frst and then increase with the increase of the thickness-to-diameter ratio for the DFP for the VTRCS. It can also be found from Figure 7 that the diference between the DFPs under the four thickness variation forms becomes smaller as the boundary conditions change from S-         S to C-C. Specifcally, the infuence of thickness variation forms on the DFPs is little when the thickness-to-diameter ratio is higher than 25.

Conclusion
(1) Te vibration characteristics of the VTRCS are investigated based on the Sanders shell theory via the Chebyshev-Ritz method. Te correctness and convergence of the modeling method are verifed through comparison and convergence research, and it is proved that the proposed model can efectively predict the traveling wave behavior of the free vibration for the VTRCS. (2) Te traveling wave DFP of the VTRCS rises with the increase of the wave number n under three thickness variation forms. Te traveling wave DFP is the largest under V0, and it is higher than the other two kinds of thickness under V2 when V0 is not considered, which is close to the DFP in the V0. In addition, the DFP of the VTRCS frst discretizes from the same initial value and then converges to the same value with the increase of circumferential wave number n under the four thickness variations. (3) Te traveling wave DFP of the VTRCS gradually decreases with the increase of the thickness change parameter kh at diferent speeds and diferent thicknesses. Among them, the traveling wave DFP change is the smallest under V2 and is signifcantly smaller than that of the other two thickness changes. Te rotational speed has little efect on the variation trend of traveling wave frequency with thickness. (4) Te traveling wave DFP of VTRCS decreases gradually with the increase of aspect ratio L/R and decreases frst and then increases with the increase of aspect ratio h 0 /R. Te diference between the DFPs becomes smaller under the four thickness variations as the boundary conditions change from S-S to C-C.
Te variation of thickness has little efect on the traveling wave frequency when the thickness-todiameter ratio is higher than a certain value.

Data Availability
Te data used to support the fndings of this study are currently under embargo while the research fndings are commercialized. Requests for data, [6/12 months] after publication of this article, will be considered by the corresponding author.