Vibration characteristics of a rotating composite laminated cylindrical shell in subsonic air flow and hygrothermal environment

International Journal of Mechanical Sciences 150 (2019) 356–368

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International Journal of Mechanical Sciences journal homepage: www.elsevier.com/locate/ijmecsci

Vibration characteristics of a rotating composite laminated cylindrical shell in subsonic air flow and hygrothermal environment X. Li, Y.H. Li∗, T.F. Xie School of Mechanics & Engineering, Southwest Jiaotong University, Chengdu 610031, PR China

a r t i c l e

i n f o

Keywords: Subsonic air flow Hygrothermal environment Rotating composite laminated cylindrical shell Critical rotating velocity Initial hoop tension

a b s t r a c t This paper focuses on vibration characteristics of a rotating composite laminated cylindrical shell subjected to both subsonic air flow and hygrothermal effects. Based on Love’s nonlinear shell theory, and introducing hygrothermal strains into the constitutive relation of single layer material, the dynamic equations of the shell considering rotation, subsonic air flow and hygrothermal effects are obtained by Hamilton’s principle. The frequency parameters of the equations are derived by means of Galerkin’s method. Some numerical results are performed to conduct detailed parametric studies on vibration characteristics of the shell. In particular, combined effects of subsonic air flow and hygrothermal environment on natural frequencies of forward and backward travelling waves and critical rotating velocity of the shell are discussed, and the influence of initial hoop tension on those frequencies is also carried out. From the results it is shown that rotating angular velocity, subsonic air flow velocity and hygrothermal effects show the significant influence on vibration characteristics of the shell.

1. Introduction Composite structures rotating about their longitudinal axis have gained wide applications in many engineering fields, such as rotor system, high velocity centrifugal separator, gas turbine engines and so on [1–7], due to their high strength, lightweight and high temperature resistance. In practice, these types of structures are generally simplified as rotating composite cylindrical shells [8–10]. Dynamic characteristics of rotating composite laminated cylindrical shells has attracted much popularity over the years [11,12]. Lam and Loy [13] studied natural frequencies of forward and backward travelling waves of rotating composite cylindrical shells by adopting Donnell’s, Flügge’s, Love’s and Sanders’ shell theories, respectively. Specifically, they obtained that Donnell’s shell theory is accurate merely when length-to-radius ratio and circumferential wave number of the shells are small, while Love’s shell theory is more reasonable for the shells with large length-to-radius ratio compared with other three shell theories. In Refs. [14,15], they further examined free vibration characteristics of rotating laminated cylindrical shells considering Coriolis force and the initial hoop tension, and discussed the effects of different boundary conditions on frequencies of the shells. Taking account of the first order shear deformation theory, free vibration of a rotating functionally graded (FG) cylindrical shell was researched by Malekzadeh and Heydarpour [16]. However, traveling wave vibrations generated by rotation were not concerned in their study. Civalek [17] carried out free

∗

vibration of rotating composite cylindrical shells by means of discrete singular convolution technique [18], and analyzed the effects of different rotating velocity, boundary conditions, and geometric parameters on frequencies parameters of the shells in detail. Employing Rayleigh– Ritz method, traveling wave vibration of rotating laminated cylindrical shells with arbitrary edges was conducted in Ref. [19], the influence of various boundaries on frequencies of the shells was studied. Wang et al. [20] investigated the large-amplitude traveling wave vibrations of rotating composite cylindrical shells under transverse harmonic excitation. Effects of rotating angular velocity and system parameters on nonlinear dynamic response of the shells were discussed. Composite shell structures are generally exposed to hygrothermal environment. Kollár [21] presented a three-dimensional elasticity solution of composite cylinders with axially varying loads considering hygrothermal effects. The strains due to hygrothermal loads were incorporated into the stress-strain relation of composite cylinders. Considering the non-linear prebuckling deformations and initial geometric imperfections of cylindrical shell structure, Shen [22] adopted a singular perturbation technique to analyze the influence of hygrothermal environment on bucking and post-bucking of laminated cylindrical shells under both axial loads and external pressures. In Ref. [23], he further studied hygrothermal analysis of composite cylindrical shells with the consideration of higher order shear deformation theory. Employing modified first order shear deformation theory including Green–Lagrange nonlinear strain, nonlinear transient response and free vibration of composite laminated shell under hygrothermal environment were carried out in Refs.

Corresponding author. E-mail address: [email protected] (Y.H. Li).

https://doi.org/10.1016/j.ijmecsci.2018.10.024 Received 29 June 2018; Received in revised form 1 October 2018; Accepted 11 October 2018 Available online 22 October 2018 0020-7403/© 2018 Elsevier Ltd. All rights reserved.

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International Journal of Mechanical Sciences 150 (2019) 356–368

[24,25], respectively. Shen and Yang [26] investigated hygrothermal effects on nonlinear free vibration of shear deformable fiber-reinforced composite cylindrical shells. Of special interest in their conclusion was that nonlinear to linear frequency ratios of the shell seemed to be few affected by temperature and moisture concentration. Based on higherorder shear deformation theory, Mahapatra et al. [27–31] dealt with nonlinear vibration analysis of composite laminated curved shell panels of different geometries (cylindrical, hyperboloid, elliptical, spherical and flat panels) under hygrothermal environment, and discussed the influence of hygrothermal effects and system parameters on frequency parameters and nonlinear responses. Using a micromechanical approach, they further studied nonlinear bending behavior and free vibration of the panels [32–37]. For experimental validation of composite shell structures, Biswal et al. [38,39] conducted experimental and numerical studies on vibration characteristic of composite laminated cylindrical shallow shells in hygrothermal environment using B&K FFT analyzer. Sahoo et al. [40] studied static, free vibration and transient behavior of composite laminated curved shallow panel by means of an experimental approach. Sharma et al. [41,42] carried out experimental validation on vibro-acoustic analysis of un-baffled composite curved shell panels. In addition, cylindrical shell structures considering aeroelastic effects are extensively used in aeronautics and aerospace fields [43–45]. There is a good deal of literatures related to dynamics of cylindrical shell structures in the air flow. According to three-dimensional elasticity theory, transverse vibration of isotropic cylindrical shells with internal compressible fluid was researched by Chen and Ding [46], in which an analytical solution of frequency equation of the fluid-filled shell was proposed. In Ref. [47], nonlinear flutter of simply supported circular cylindrical shells in supersonic air flow was investigated with the consideration of viscous structural damping and linear piston theory. Karagiozis et al. [48] conducted an experimental research on nonlinear vibration and stability of cylindrical thin shells under axial flow. Experimental data illustrated that the shell surrounded by axial flow occurred unstable phenomena due to divergence. Prado et al. [49] studied parametric instability and nonlinear vibrations of orthotropic cylindrical shells coupled with fluid under combined harmonic axial and lateral loads. The influence of material orthotropy on natural frequencies, critical loads, nonlinear buckling and frequency–amplitude relations the shells was examined systematacially. For composite cylindrical shell with aeroelastic effects, Li and Yao [50] analyzed 1/3 subharmonic resonance of composite cylindrical shells subjected to subsonic air flow and radial parametric excitation. Effects of subsonic air flow velocity on amplitude–frequency curves of 1/3 subharmonic resonance of the shells were discussed specially. As mentioned above, abundant researches associated with rotating composite laminated cylindrical shells have been published in the previous studies. However, to our best knowledge, there are no works in literatures on combined effects of subsonic air flow and hygrothermal environment on traveling wave vibration of rotating composite laminated cylindrical shells. This paper investigates combined influence of subsonic air flow and hygrothermal effects on natural frequencies of forward and backward travelling waves and critical rotating velocity of the shells. The effects of rotating angular velocity, subsonic air flow velocity, temperature, moisture and fiber orientation angle on vibration characteristics of the shell is treated systematically.

Fig. 1. Rotating composite laminated cylindrical shell configuration in subsonic air flow.

(3) Each layer is made of homogeneous, orthotropic and linearly elastic material. (4) The subsonic air flow around the shell is assumed to be nonviscous, irrotational and incompressible. 2.1. Constitutive relation Fig. 1 shows a rotating composite laminated cylindrical shell in subsonic air flow. The length, radius and thickness of the shell are denoted by L, R and h respectively. The cylindrical coordinate system (x, 𝜃, z) is built on the middle surface of the shell, and (u, v, w) is the displacement vector. The shell rotates about its longitudinal x-axis at a constant angular velocity Ω. Furthermore, the subsonic air flow with flow velocity U∞ along x-axis direction of the shell is considered in Fig. 1. In accordance with Love’s nonlinear shell theory [52], the strains take the form of 𝜀𝑥 = 𝜀1 + 𝑧𝜅1 , 𝜀𝜃 = 𝜀2 + 𝑧𝜅2 , 𝜀𝑥𝜃 = 𝛾 + 𝑧𝜒

(1)

where 𝜀1 , 𝜀2 and 𝛾 are the middle surface strains, while 𝜅 1 , 𝜅 2 and 𝜒 the middle surface curvatures. They are expressed as 𝜀1 =

( ) ( ) ( ) 𝜕𝑢 1 𝜕𝑤 2 1 𝜕𝑣 1 𝜕𝑤 2 1 𝜕𝑢 𝜕𝑣 + , 𝜀2 = ,𝛾 = +𝑤 + + 𝜕𝑥 2 𝜕𝑥 𝑅 𝜕𝜃 2 𝑅𝜕𝜃 𝑅 𝜕𝜃 𝜕𝑥 𝜕𝑤 𝜕𝑤 , + 𝜕𝑥 𝑅𝜕𝜃

𝜅1 = −

𝜕2 𝑤 1 , 𝜅2 = 𝜕 𝑥2 𝑅2

(

) ( ) 𝜕𝑣 𝜕 2 𝑤 2 𝜕𝑣 𝜕2 𝑤 ,𝜒 = − . − 𝜕𝜃 𝑅 𝜕𝑥 𝜕 𝑥𝜕 𝜃 𝜕𝜃2

(2)

(3)

Taking account of the hygrothermal effect, the stress-strain relation of single layer composite cylindrical shell is ( ) ̄ 𝛆 − 𝛆𝑇 − 𝛆𝐻 (4) 𝛔=𝐐 where 𝝈 = (𝜎 x 𝜎 𝜃 𝜏 x𝜃 )t and 𝜺 = (𝜀x 𝜀𝜃 𝜀x𝜃 )t denote stress and strain vec̄ tors, respectively. The superscript t is transposition of the vectors, 𝐐 the stiffness matrix of single layer material [53]. 𝛆𝑇 = (𝜀𝑇𝑥 𝜀𝑇𝜃 𝜀𝑇𝑥𝜃 )𝑡 and 𝐻 𝐻 𝑡 𝛆𝐻 = (𝜀𝐻 𝑥 𝜀𝜃 𝜀𝑥𝜃 ) are the thermal and the humid strain vectors, which are given by

2. Dynamic model

𝛆𝑇 = 𝐓𝑡 𝛂12 Δ𝑇

(5)

Following assumptions are given to build the dynamic model of a rotating composite laminated cylindrical shell in subsonic air flow and hygrothermal environment:

𝛆𝐻 = 𝐓𝑡 𝛃12 Δ𝐶

(6)

where 𝜶 12 = (𝛼 1 𝛼 2 0)t and 𝜷 12 = (𝛽 1 𝛽 2 0)t . 𝛼 1 and 𝛼 2 are the thermal expansion coefficients in material principal orientation, while 𝛽 1 and 𝛽 2 the hygroscopic expansion coefficients. ΔT is the temperature variation, and ΔC the moisture concentration. Tt is the transformation matrix [53].

(1) The deformation of the cross-section satisfies the Kirchhoff–Love assumption [51]. (2) Effects of shear deformation and rotary inertia are neglected. 357

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International Journal of Mechanical Sciences 150 (2019) 356–368

2.2. Potential energy

𝐵𝑥𝑇 𝐻 =

The strain energy of the shell is expressed as ℎ 2

𝐿

1 𝑈= 2 ∫− ℎ ∫0 ∫𝑜

2𝜋

𝛔𝑡 𝛆𝑅d𝑥d𝜃d𝑧

𝐵𝜃𝑇 𝐻 = (7) 𝑇𝐻 𝐵𝑥𝜃 =

2

If we set H = (Hx H𝜃 Hx𝜃 )t = 𝜺T + 𝜺H , and introducing Eq. (4) into Eq. (7) yields w 𝐿 2𝜋 ∫0 ∫𝑜

𝑈ℎ =

(8)

1 2

∫

ℎ 2 − ℎ2

𝐿

2𝜋

∫0 ∫𝑜

ℎ∕2

(

) 𝑄̄ 12 𝐻𝑥 +𝑄̄ 22 𝐻𝜃 +𝑄̄ 26 𝐻𝑥𝜃 𝑧d𝑧

(

) 𝑄̄ 16 𝐻𝑥 +𝑄̄ 26 𝐻𝜃 +𝑄̄ 66 𝐻𝑥𝜃 𝑧d𝑧

∫−ℎ∕2 ℎ∕2

∫−ℎ∕2

{( )]2 ) [ ( 𝐿 2𝜋 1 1 𝜕𝑢 2 1 𝜕𝑣 𝑁𝜃0 + +𝑤 2 ∫0 ∫𝑜 𝑅 𝜕𝜃 𝑅 𝜕𝜃 [ ( )]2 } 𝜕𝑤 1 + 𝑅d𝑥d𝜃 − +𝑣 𝑅 𝜕𝜃

𝑁𝜃0 = 𝜌𝑡 Ω2 𝑅2

(17)

(18)

(19)

where 𝜌t is the density per unit area written as

(

𝑄11 𝜀2𝑥 + 𝑄22 𝜀2𝜃 + 𝑄66 𝜀2𝑥𝜃 + 2𝑄12 𝜀𝑥 𝜀𝜃 ) +2𝑄̄ 16 𝜀𝑥 𝜀𝑥𝜃 +2𝑄̄ 26 𝜀𝜃 𝜀𝑥𝜃 𝑅d𝑥d𝜃d𝑧

) 𝑄̄ 11 𝐻𝑥 +𝑄̄ 12 𝐻𝜃 +𝑄̄ 16 𝐻𝑥𝜃 𝑧d𝑧

where 𝑁𝜃0 is the initial hoop tension due to the centrifugal force defined as

where the first six terms in Eq. (8) are strain energy caused by initial strains, the remaining terms are coupled strain energy generated by hygrothermal- elastic coupling effects. The corresponding strain energy in Eq. (8) can be written as respectively 𝑈𝐷 =

(

The strain energy generated by the centrifugal force is given by [54]

[ 𝑄̄ 11 𝜀2𝑥 +𝑄̄ 22 𝜀2𝜃 +𝑄̄ 66 𝜀2𝑥𝜃 +2𝑄̄ 12 𝜀𝑥 𝜀𝜃 +2𝑄̄ 16 𝜀𝑥 𝜀𝑥𝜃 +2𝑄̄ 26 𝜀𝜃 𝜀𝑥𝜃 𝑈 = 12 ∫ ( ) ( ) −𝜀𝑥 𝑄̄ 11 𝐻𝑥 + 𝑄̄ 12 𝐻𝜃 + 𝑄̄ 16 𝐻𝑥𝜃 − 𝜀𝜃 𝑄̄ 12 𝐻𝑥 + 𝑄̄ 22 𝐻𝜃 + 𝑄̄ 26 𝐻𝑥𝜃 ] ( ) −𝜀𝑥𝜃 𝑄̄ 16 𝐻𝑥 + 𝑄̄ 26 𝐻𝜃 + 𝑄̄ 66 𝐻𝑥𝜃 𝑅d𝑥d𝜃d𝑧 ℎ 2 − ℎ2

ℎ∕2

∫−ℎ∕2

𝜌𝑡 =

(9)

ℎ 2

∫− ℎ

𝜌(𝑧)d𝑧

(20)

2

in which 𝜌 represent the density of the single layer material. Consequently, the total potential energy of a rotating composite laminated cylindrical shell in hygrothermal environment is

ℎ ) 𝐿 2𝜋 [ ( 𝑈𝑇 𝐻 = − 12 ∫ 2ℎ ∫0 ∫𝑜 𝜀𝑥 𝑄̄ 11 𝐻𝑥 + 𝑄̄ 12 𝐻𝜃 + 𝑄̄ 16 𝐻𝑥𝜃 −2 ( ) ( )] +𝜀𝜃 𝑄̄ 12 𝐻𝑥 +𝑄̄ 22 𝐻𝜃 +𝑄̄ 26 𝐻𝑥𝜃 +𝜀𝑥𝜃 𝑄̄ 16 𝐻𝑥 +𝑄̄ 26 𝐻𝜃 +𝑄̄ 66 𝐻𝑥𝜃 𝑅d𝑥d𝜃d𝑧

𝑈 = 𝑈 𝐷 +𝑈 𝑇 𝐻 +𝑈 ℎ

(10)

(21)

2.3. Kinetic energy

Substituting Eq. (1) intoEq. (9) has 𝑈𝐷 = 𝑈𝑒 +𝑈𝑏 +𝑈𝑐

The displacement vector of an arbitrary point of the shell is

(11)

𝐫 = 𝑢𝐢 + 𝑣𝐣 + 𝑤𝐤

where Ue , Ub and Uc represent strain energy of extension, bending and extension-bending coupling respectively, which are expressed as 𝑈𝑒 =

𝐿

1 2 ∫0 ∫𝑜

2𝜋

(

) 𝐴11 𝜀21 +𝐴22 𝜀22 +𝐴66 𝛾 2 +2𝐴12 𝜀1 𝜀2 + 2𝐴16 𝜀1 𝛾+2𝐴26 𝜀2 𝛾 𝑅d𝑥d𝜃 (12)

𝑈𝑏 =

𝐿 2𝜋 ( ∫0 ∫𝑜 𝐷11 𝜅12 + 𝐷22 𝜅22 + 𝐷66 𝜒 2 + 2𝐷12 𝜅1 𝜅2 + 2𝐷16 𝜅1 𝜒 ) +2𝐷26 𝜅2 𝜒 𝑅d𝑥d𝜃 1 2

(22)

in which i, j, k are the unit vectors along the x-, 𝜃- and z-axes, and the velocity vector of arbitrary point is expressed as 𝐯 = 𝐫̇ + 𝛀 × 𝐫

(23)

in which 𝛀 = −Ωi. Substituting Eqs. (22) and (23), the velocity vector can be further known that

(13)

𝐯 = 𝑢̇ 𝐢 + (𝑣̇ + Ω𝑤)𝐣 + (𝑤̇ − Ω𝑣)𝐤

(24)

Employing Eq. (24), the kinetic energy of the shell in arbitrary time

) ( ) ( 𝐿 2𝜋 [ 𝑈𝑐 = ∫0 ∫𝑜 𝐵11 𝜀1 𝜅1 + 𝐵22 𝜀2 𝜅2 + 𝐵66 𝛾𝜒 + 𝐵12 𝜀1 𝜅2 + 𝜀2 𝜅1 + 𝐵16 𝜀1 𝜒 )] ( +𝜅1 𝛾 + 𝐵26 𝜀2 𝜒 + 𝜅2 𝛾 𝑅d𝑥d𝜃

is 𝑇 =

(14)

𝐿

1 2 ∫0 ∫𝑜

2𝜋

[ ] 𝜌𝑡 𝑢̇ 2 + (𝑣̇ + Ω𝑤)2 + (𝑤̇ − Ω𝑣)2 𝑅d𝑥d𝜃

(25)

where ( ) 𝐴𝑖𝑗 , 𝐵𝑖𝑗 , 𝐷𝑖𝑗 =

ℎ∕2

∫−ℎ∕2

( ) 𝑄̄ 𝑖𝑗 1, 𝑧, 𝑧2 d𝑧 (𝑖, 𝑗 = 1, 2, 6)

Substituting Eqs. (1), (5) and (6) intoEq. (10) yields 𝐿 2𝜋 [ 𝑈𝑇 𝐻 = − 12 ∫0 ∫𝑜 𝐴𝑇𝑥 𝐻 𝜀1 + 𝐴𝑇𝜃 𝐻 𝜀2 + 𝐴𝑇𝑥𝜃𝐻 𝛾 + 𝐵𝑥𝑇 𝐻 𝜅1 + 𝐵𝜃𝑇 𝐻 𝜅2 ] 𝑇 𝐻 𝜒 𝑅d𝑥d𝜃 +𝐵𝑥𝜃

2.4. External work (15) The aerodynamic pressure of composite laminated cylindrical shell surrounded by the subsonic air flow are given by [50] ( 2 ) 2 𝜕 w 𝜕2 w 2 𝜕 w 𝑃 = 𝜌∞ 𝑓𝑚𝑛 + 2𝑈∞ + 𝑈∞ (26) 2 2 𝜕 𝑥𝜕 𝑡 𝜕𝑡 𝜕𝑥

(16)

where 𝜌∞ is the density of the air flow, and fmn is defined as [55] ( ) 𝑅𝐾𝑛 𝜆𝑚 𝐿 𝑓𝑚𝑛 = (27) ( ) ( ) 𝑛𝐾𝑛 𝜆𝑚 𝐿 − 𝜆𝑚 𝐾𝑛+1 𝜆𝑚 𝐿

where𝐴𝑇𝑥 𝐻 , 𝐴𝑇𝜃 𝐻 and 𝐴𝑇𝑥𝜃𝐻 are extensional stiffness terms caused by 𝑇 𝐻 extensionhygrothermal expansion deformation, 𝐵𝑥𝑇 𝐻 , 𝐵𝜃𝑇 𝐻 and𝐵𝑥𝜃 bending coupling ones, which are given by 𝐴𝑇𝑥 𝐻 = 𝐴𝑇𝜃 𝐻 = 𝐴𝑇𝑥𝜃𝐻 =

ℎ∕2

∫−ℎ∕2 ℎ∕2

∫−ℎ∕2 ℎ∕2

∫−ℎ∕2

( ) 𝑄̄ 11 𝐻𝑥 +𝑄̄ 12 𝐻𝜃 +𝑄̄ 16 𝐻𝑥𝜃 d𝑧 ( ) 𝑄̄ 12 𝐻𝑥 +𝑄̄ 22 𝐻𝜃 +𝑄̄ 26 𝐻𝑥𝜃 d𝑧

in which K is the Bessel functions of the second kind, m axial half-wave number, n circumferential wave number, and 𝜆m L the corresponding eigenvalues related to the boundary conditions [56]. The work performed by aerodynamic pressure is written as

( ) 𝑄̄ 16 𝐻𝑥 +𝑄̄ 26 𝐻𝜃 +𝑄̄ 66 𝐻𝑥𝜃 d𝑧

𝑊 =− 358

𝐿

∫0 ∫𝑜

2𝜋

𝑃 𝑤𝑅d𝑥d𝜃

(28)

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International Journal of Mechanical Sciences 150 (2019) 356–368

Table 1 Elastic modulus variation of graphite/epoxy laminate [61].

2.5. Governing equations The governing equations of a rotating composite laminated cylindrical shell subjected to both subsonic air flow and hygrothermal environment are derived by Hamilton’s variational principle 𝛿

𝑡2

∫𝑡1

(𝑇 − 𝑈 )d 𝑡 +

𝑡2

∫𝑡1

𝛿𝑊 d𝑡 = 0

(29)

Substituting Eqs. (21), (25) and (28) into Eq. (29) and ignoring nonlinear terms, the governing equations of the shell can be written as ⎡𝐿11 ⎢𝐿 ⎢ 21 ⎣𝐿31

𝐿12 𝐿22 𝐿32

𝐿13 ⎤⎡ 𝑢 ⎤ 𝐿23 ⎥⎢ 𝑣 ⎥ = 0 ⎥⎢ ⎥ 𝐿33 ⎦⎣𝑤⎦

𝑚=1 𝑛=1 ∞ ∑ ∞ ∑

𝑤=

𝑚=1 𝑛=1 ∞ ∑ ∞ ∑

(31)

𝐌∗ 𝐪̈ + 𝐂∗ 𝐪̇ + 𝐊∗ 𝐪 = 𝟎

(32)

(34)

] Csi Ω , Cri U∞

=

(𝑞𝑖∗ , 𝑝∗𝑖 )𝑡 ,

j=

√

(41)

4.1. Validation

(36) 𝐾𝑜𝑖 𝐾𝑖 + 𝐾𝑟𝑖 U2∞ + 𝐾𝑠𝑖 Ω2

(40)

Numerical results of a rotating composite laminated cylindrical shell subjected to both hygrothermal environment and subsonic air flow are performed in this section. The material properties of graphite/epoxy laminate in different hygrothermal environments is given in Table 1, where thermal expansion coefficients 𝛼 1 = −0.3 × 10−6 / °C and 𝛼 2 = 28.1 × 10−6 / °C, hygroscopic expansion coefficients 𝛽 1 = 0 and 𝛽 2 = 0.44 × 10−6 , Poisson’s ratio 𝜈 21 = 0.3, and material density 𝜌 = 1600 kg/m3 . In addition, ΔT = T-T0 in Section 2.1 denote temperature variation, and T0 is defined as initial temperature. In this paper, we set T0 = 25 °C as a reference value. A symmetric (𝜃 0 /-𝜃 0 )s lay up of laminates is considered in the present work, 𝜃 0 is the fiber orientation angle defined as the angle from x-axis to the fiber orientation. Unless otherwise mentioned, the fiber orientation angle 𝜃 0 = 45° in the following.

]

In this section, the validity of the present theory is verified. Ignoring hygrothermal effects and aerodynamic pressure, and considering a three-layered (0°/90°/0°) simply supported rotating composite laminated√ cylindrical shell, the dimensionless frequency parameter 𝜔∗ = 𝜔𝑅 𝜌∕𝐸2 of the shell are displayed in Table 2 and compared with the results obtained by Lam and Loy [13], in which m = 1, L/R = 10, h/R = 0.002. The material properties are E2 = 7.6 GPa, E1 = 2.5E2 , G12 = 4.1 GPa, 𝜈 12 = 0.26, 𝜌 = 1643 kg/m3 . It is shown from

(37)

Based on Theory of Ordinary Differential Equations [59], the solutions of Eq. (34) is supposed as the following forms

where𝐪∗

150 130 6.75 4.5

4. Numerical results and discussion

whereq = (qi ,pi )t . The mass matrix M∗ , damping matrix C∗ and stiffness matrix K∗ are introduced below [ ] 𝑀𝑖 0 𝐌∗ = , (35) 0 𝑀𝑖

𝐪 = 𝐪∗ ej𝜆𝑡

125 130 7.0 4.75

in which the real part 𝜔k (k = 1, 6) are three coupled roots for each combination of m and n for the shell, which represents positive and negative natural frequencies for u, v, w respectively, and 𝛾 k are the imaginary part of the solution. Based on Refs. [6,60], the positive root of 𝜔k represent backward travelling wave, while the negative root is forward travelling wave. Special attention is given to the frequencies of forward and backward travelling waves related to w, which is because the frequencies for u and v are much higher than the ones for w. In addition, the absolute value of frequency of backward travelling wave is larger than the one of forward travelling wave. When the frequency 𝜔k is equal to zero, the lowest value of corresponding velocity is defined as critical velocity. In this analysis, critical velocity includes critical rotating velocity and critical flow velocity, which indicate that the system exists instability phenomena.

where Mi is the mass coefficients of the shell for the i-th mode (m, n), Cri U∞ and Csi Ω damping coefficients due to the subsonic air flow and gyroscopic effects respectively, Ki , Koi system stiffness coefficients, 𝐾𝑟𝑖 U2∞ and Ksi Ω2 stiffness coefficients generated by the subsonic air flow and rotating effects respectively, which are listed in the Appendix B. For brevity, Eq. (33) can be further written as

[ 𝐾𝑖 + 𝐾𝑟𝑖 U2∞ + 𝐾𝑠𝑖 Ω2 −𝐾𝑜𝑖

100 130 7.5 5.0

2,•••,

where 𝜆m L is the root of frequency equation cos𝜆m L•cosh𝜆m L = 1. Substituting Eq. (31) into Eq. (30) and multiplying mode shapes of Eq. (31) respectively, integrating over the surface and employing the Galerkin’s method, one leads to [ ] 𝑀𝑖 𝑞̈𝑖 + 2𝐶𝑟𝑖 U∞ 𝑞̇ 𝑖 + 2𝐶𝑠𝑖 Ω𝑝̇ 𝑖 + 𝐾𝑖 + 𝐾𝑟𝑖 U2∞ + 𝐾𝑠𝑖 Ω2 𝑞𝑖 + 𝐾𝑜𝑖 𝑝𝑖 = 0 [ ] (33) 𝑀𝑖 𝑝̈𝑖 + 2𝐶𝑟𝑖 U∞ 𝑝̇ 𝑖 − 2𝐶𝑠𝑖 Ω𝑞̇ 𝑖 + 𝐾𝑖 + 𝐾𝑟𝑖 U2∞ + 𝐾𝑠𝑖 Ω2 𝑝𝑖 − 𝐾𝑜𝑖 𝑞𝑖 = 0

𝐊∗ =

130 8.5 6.0

It can be noted that the presence of the damping matrix C∗ will result in complex values of natural frequencies. Solving Eq. (40), we obtain

[ ] 𝑞𝑖 (𝑡) cos (𝑛𝜃) − 𝑝𝑖 (𝑡) sin (𝑛𝜃) 𝜑𝑤𝑚 (𝑥)

Cri U∞ −Csi Ω

130 8.5 6.0

𝜆𝑘 = 𝜔𝑘 ± j𝛾 𝑘

[ ] 𝑏𝑖 𝑞𝑖 (𝑡) sin (𝑛𝜃) + 𝑝𝑖 (𝑡) cos (𝑛𝜃) 𝜑𝑣𝑚 (𝑥)

( ) ( ) 𝜑𝑣𝑚 (𝑥) = sinh 𝜆𝑚 𝑥 − sin 𝜆𝑚 𝑥 ( )( ( ) ( )) sinh (𝜆 𝐿)−sin (𝜆 𝐿) cosh 𝜆𝑚 𝑥 − cos 𝜆𝑚 𝑥 (𝑚 = 1, 2, ⋯ , ∞) − cosh 𝜆𝑚 𝐿 −cos 𝜆𝑚 𝐿 ( 𝑚 ) ( 𝑚 )

[

1.50

130 8.5 6.0

| − 𝜆2 𝐌∗ + j𝜆𝐂∗ + 𝐊∗ | = 0

where qi (t), pi (t) are generalized displacement, ai , bi are mode shape ratios [51], i = 1, 2,•••, mn. The axial modal functions 𝜑jm (x)(j = u, v, w) are defined as the beam functions and satisfy boundary conditions. In the present study, 𝜑um (x) = d𝜑vm (x)/dx = d𝜑wm (x)/dx. 𝜑vm (x) are written as

𝐂∗ = 2

1.25

130 8.75 6.0

Then, Eq. (39) has untrivial solution if

[ ] 𝑎𝑖 𝑞𝑖 (𝑡) cos (𝑛𝜃) − 𝑝𝑖 (𝑡) sin (𝑛𝜃) 𝜑𝑢𝑚 (𝑥)

𝑚=1 𝑛=1

130 9.0 6.0 T ( °C) 75 130 8.0 5.5

Substituting Eq. (38) into Eq. (34) and eliminating the time variable t, we obtain ( 2 ∗ ) (39) −𝜆 𝐌 + j𝜆𝐂∗ + 𝐊∗ 𝐪∗ = 𝟎

In this section, Galerkin’s method is employed to obtain the vibration characteristics of the shell. The displacement functions of the shell are assumed as following [57,58]

𝑣=

130 130 9.5 9.25 6.0 6.0 (b)Temperature, 25 50 130 130 9.5 8.5 6.0 6.0

(30)

3. Solution method

𝑢=

E1 E2 G12

E1 E2 G12

in which Lij are differential operators as shown in the Appendix A.

∞ ∑ ∞ ∑

Modulus(a) Moisture concentration, ΔC (%) (GPa) 0.00 0.25 0.50 0.75 1.00

(38) −1. 359

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Table 2 Dimensionless frequency parameter for a (0°/90°/0°) simply supported rotating composite laminated cylindrical shell (m = 1, L/R = 10, h/R = 0.002).

Table 4 Natural frequencies (Hz) of composite cylindrical shell (m = 1, L/R = 5, h/R = 0.01, T = 75 °C, ΔC = 0.5%).

Ω n (rev/s)

𝜔̂ ∗𝑓 Present

Ref. [13]

𝜔̂ ∗𝑏 Present

Ref. [13]

n

Present

FEM

0.1

0.083847 0.029942 0.015317 0.012472 0.015793 0.082811 0.029532 0.014839 0.012417 0.015752 0.081139 0.028014 0.015205 0.014948 0.019714

0.083624 0.029778 0.015036 0.012085 0.015181 0.082770 0.029122 0.014783 0.012351 0.015747 0.081060 0.027961 0.015177 0.014810 0.019409

0.084637 0.030717 0.015393 0.012516 0.015829 0.085773 0.031083 0.016463 0.013593 0.016895 0.086844 0.032759 0.018725 0.017786 0.021773

0.084193 0.030245 0.015387 0.012360 0.015406 0.085046 0.030987 0.016185 0.013451 0.016646 0.086749 0.032625 0.018684 0.017561 0.021657

1 2 3 4

290.4724 127.4789 62.5831 37.1766

287.1647 124.9874 60.3862 35.3394

0.4

1.0

1 2 3 4 5 1 2 3 4 5 1 2 3 4 5

h/R = 0.01, T = 75 °C, ΔC = 0.5%. It indicates that there is a reasonable agreement between the present solutions and the FEM results. Moreover, vibration modes of composite cylindrical shell for various circumferential wave numbers (n = 1, 2, 3, 4) in hygrothermal environment corresponding to Table 4 are shown in Fig. 2. In the following, the dimensionless quantities are introduced for convenience 𝜔𝑓 ∗ = 𝜔𝑓 ∗ 𝜔0 , 𝜔𝑏 ∗ = 𝜔𝑏 ∗ 𝜔0

Table 3 Comparison of natural frequency (Hz) for a clamped-free isotropic cylindrical shell (E = 2.1 × 1011 N/m2 , 𝜈 = 0.28, 𝜌 = 7.8 × 103 kg/m3 , L = 502 mm, h = 1.63 mm, R = 63.5 mm). n

m=1

3 4 5 6

Ref. [62] 760.0 1451.0 2336.0 3429.0

(42)

in which 𝜔f and 𝜔b are dimensionless natural frequencies of forward √ and backward traveling waves respectively, and𝜔0 = 𝑅 𝜌∕𝐸1 . ∗

∗

4.2. Combined effects of rotating angular velocity and subsonic air flow velocity on vibration characteristics

m=2 Present 768.5 1463.7 2353.2 3460.3

Ref. [62] 886.0 1503.0 2384.0 3476.0

In this section, the influence of rotating angular velocity and subsonic air flow velocity on vibration characteristics are investigated. Fig. 3 presents combined effects of rotating angular velocity Ω and subsonic air flow velocity U∞ on dimensionless natural frequencies for modes (1, 1) and (1, 2) in the condition that L/R = 10, h/R = 0.01, T = 50 °C, ΔC = 1.0%. As shown in the figure, with the increase of Ω, a bifurcation of dimensionless natural frequencies is found for different U∞ , where the higher branches corresponds to dimensionless natural frequencies of backward traveling wave 𝜔b ∗ , and lower ones to dimensionless natural frequencies of forward traveling wave 𝜔f ∗ , they are represented by solid and dot lines respectively. It is viewed from Fig. 3(a) that, 𝜔b ∗ increases monotonically with Ω for mode (1, 1), while 𝜔f ∗ decreases continuously with Ω until the dot line intersects the abscissa, and then increases with Ω again. When 𝜔f ∗ = 0, the value of corresponding abscissa is the critical rotating velocity Ωc , which reveals that possible instability phenomena occur. That makes sense physically, Coriolis force due to the rotation (corresponding to the term Csi Ω in Eq. (33)) induce the asymmetric influence on bending stiffness of forward and backward traveling waves [63], which results in different trends of frequency curves of both traveling waves. For mode (1, 2), Fig. 3(b) shows that the trend of the curve of 𝜔b ∗ with Ω is similar to that for mode (1, 1). However, 𝜔f ∗ displays a slight decline in lower rotating angular velocity range and then ascends with Ω. It is obvious from Fig. 3 that subsonic air flow velocity U∞ induces the reduction of dimensionless natural frequencies of both travelling waves for every mode before the systems became unstable, and the regions bounded by 𝜔f ∗ and 𝜔b ∗ is narrowed with the increase of U∞ . This is expected that air flow velocity weakens the stiffness of the systems by means of the term 𝐾𝑟𝑖 U2∞ in Eq. (33). It is also noted that the critical rotating velocity Ωc descends with U∞ for mode (1, 1). Specifically, the variation of Ωc with U∞ for different modes (m, 1) is plotted in Fig. 4. It is clearly seen that U∞ impair Ωc for different m, which is concluded that the damping term Cri U∞ in Eq. (33) caused by the subsonic air flow can affect the system stability and even destabilize the system. In order to further discuss vibration characteristics of composite laminated cylindrical shell with fluid structure interaction, we set rotating angular velocity to be zero. Fig. 5 shows that effects of subsonic air flow velocity U∞ on dimensionless natural frequencies for four different modes. It is seen that the frequencies diminish with air flow velocity

Present 894.3 1521.8 2401.4 3498.2

Table 2 that, due to the gyroscopic effect, the natural frequencies of rotating composite laminated cylindrical shells are replaced by forward and backward traveling waves. 𝜔̂ ∗𝑓 and 𝜔̂ ∗𝑏 in Table 2 denote dimensionless natural frequencies of forward and backward traveling waves respectively. It is evident that the present results are very close to the ones of literature. To further show the accuracy level of the present solution, the experimental results in Ref. [62] are employed. Table 3 shows a comparison of natural frequency of a clamped-free isotropic cylindrical shell obtained by the experimental results in Ref. [62] and the present theory. It indicates that natural frequency derived by the present theory is in good agreement with the one given by experiment. Furthermore, in order to exhibit the accuracy and reliability of the present solutions considered hygrothermal effects, a finite element simulation model is implemented using Patran &Nastran software. In this case, the rotating angular velocity is equal to zero. The following steps are followed to obtain natural frequencies and mode shapes of composite laminated cylindrical shell in hygrothermal environment: (1) The cylindrical shell model is developed in Patran software, and the quadrilateral shell elements with four nodes (Nastran element type CQUAD4) are used to discretize the model. (2) The support condition in the present analysis is u = v = w = 𝜕 w/𝜕 x = 0 at x = 0 and L. (3) Hygrothermal stress analysis is performed using Nastran software. (4) The results in (3) are imposed on the cylindrical shell model in the form of initial stress. (5) The modal analysis is performed using Nastran software, and natural frequencies and mode shapes are obtained. Table 4 presents a comparison of the natural frequencies of composite laminated cylindrical shell obtained by the present theory and Finite Element Method (FEM) results under the condition that m = 1, L/R = 5, 360

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Fig. 2. Vibration modes of composite cylindrical shell for different circumferential wave numbers (m = 1, L/R = 5, h/R = 0.01, T = 75 °C, ΔC = 0.5%): (a) n = 1; (b) n = 2; (c) n = 3; (d) n = 4; (a) overall vibration modes; circumferential vibration modes, (b) overall vibration modes; circumferential vibration modes, (c) overall vibration modes; circumferential vibration modes, (d) overall vibration modes; circumferential vibration modes.

361

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International Journal of Mechanical Sciences 150 (2019) 356–368

Fig. 3. Combined effects of rotating angular velocity and subsonic air flow velocity on vibration characteristics (L/R = 10, h/R = 0.01, T = 50 °C, ΔC = 1.0%): (a) mode (1, 1); (b) mode (1, 2).

variation of dimensionless natural frequencies with rotating angular velocity Ω for different air flow velocities. From Fig. 6(a), it can be seen that neglecting initial hoop tension, dimensionless natural frequencies of both travelling waves vary linearly with Ω for mode (1, 1), and they have a tiny decrease compared with those frequencies in Fig. 3(a) considering initial hoop tension. Fig. 6(b) reveals that critical rotating velocity exist for mode (1, 2) without considering initial hoop tension, and 𝜔f ∗ and 𝜔b ∗ converge to one value for each air flow velocity after instability phenomena occurring. Results in Figs. 3(b) and 5(b) illustrates initial hoop tension would have dominant contribution in vanishing of critical rotating velocity for higher order circumferential mode (1, 2). In Figs. 7 and 8, effects of circumferential wave number n on dimensionless natural frequencies for different rotating angular velocities and air flow velocities are displayed respectively, where m = 1. It is noticeable that dimensionless natural frequencies of forward and backward travelling waves all initially descend and then ascend as n increase, and the frequencies of forward travelling wave are lower than those of backward travelling wave for certain mode (1, n), as shown in Eq. (41) of Section 3. It is also found that the frequencies of both travelling waves are more sensitive to circumferential wave number in the higher rotating angular velocities and air flow velocities ranges than in the lower ones.

Fig. 4. Effects of subsonic air flow velocity on critical rotating velocity for different axial half-wave number (n = 1, L/R = 5, h/R = 0.01, T = 50 °C, ΔC = 1.0%).

4.3. Combined effects of hygrothermal environment and rotating angular velocity on vibration characteristics The effects of hygrothermal environment on dimensionless natural frequencies of forward and backward travelling waves are discussed in this subsection. Results in Fig. 9 which displays combined effect of hygrothermal environment and rotating angular velocity on the frequencies, illustrate various combinations of temperature and moisture affect obviously the frequencies of the shell. It is also shown that the rotating composite cylindrical shell with higher temperature and moisture concentration exhibits lower critical rotating velocity for mode (1, 1). Physically, the influence of hygrothermal environment on vibration characteristics of the shell is superposition of two contributions composed of hygrothermal expansion deformation and material property variation.

Fig. 5. Effects of subsonic air flow velocity on vibration characteristics for different modes (L/R = 10, h/R = 0.01, T = 50 °C, ΔC = 1.0%, Ω = 0 rad/s).

U∞ , and eventually vanish. The value of flow velocity which results in a zeroing of the corresponding frequency is critical flow velocity Ucr , which predicts that the system occurs a divergent instability [55]. In addition, it is observed from Fig. 3(b) that there is no critical rotating velocity for mode (1, 2). This phenomenon is attributed to the initial hoop tension 𝑁𝜃0 generated by the centrifugal force from Section 2.2. Ignoring the strain energy of centrifugal force Uh in Eq. (18), Fig. 6 depict

4.4. Effects of temperature on vibration characteristics In this section, effects of temperature on vibration characteristics of the shell are examined, in which ΔC = 0.0%. Figs. 10 and 11 plot the influence of thermal expansion deformation and variation of material property generated by temperature on dimensionless natural frequen362

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International Journal of Mechanical Sciences 150 (2019) 356–368

Fig. 6. Variation of dimensionless natural frequencies with rotating angular velocity for different air flow velocities without considering initial hoop tension (L/R = 10, h/R = 0.01, T = 50 °C, ΔC = 1.0%): (a) mode (1, 1); (b) mode (1, 2).

Fig. 7. Effects of circumferential wave number on vibration characteristics for different rotating angular velocities (m = 1, L/R = 5, h/R = 0.01, T = 50 °C, ΔC = 1.0%, U∞ = 100 m/s).

Fig. 8. Effects of circumferential wave number on vibration characteristics for different air flow velocities (m = 1, L/R = 5, h/R = 0.01, T = 50 °C, ΔC = 1.0%, Ω = 200 rad/s).

cies of both travelling waves, respectively. It is seen from Fig. 10 that, only considering thermal expansion deformation, temperature leads to decrease of dimensionless natural frequencies of both travelling waves for modes (1, 3) and (1, 4). However, as shown in Fig. 11, the frequen-

cies seem to be negligibly affected by temperature merely considering material property variation. It can be concluded that thermal expansion deformation has more prominent effects on the frequencies than material property variation in the case of same temperature change.

Fig. 9. Combined effects of hygrothermal environment and rotating angular velocity on vibration characteristics (L/R = 10, h/R = 0.01, U∞ = 100 m/s): (a) mode (1, 1); (b) mode (1, 2). 363

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Fig. 10. Effects of thermal expansion deformation on vibration characteristics (m = 1, L/R = 10, h/R = 0.01, Ω = 200 rad/s, U∞ = 100 m/s, ΔC = 0.0%): (a) n = 3; (b) n = 4.

Fig. 11. Effects of material property variation generated by temperature on vibration characteristics (m = 1, L/R = 10, h/R = 0.01, Ω = 200 rad/s, U∞ = 100 m/s, ΔC = 0.0%): (a) n = 3; (b) n = 4.

Fig. 12. Effects of material property variation generated by moisture on vibration characteristics (m = 1, n = 2, L/R = 10, h/R = 0.01, Ω = 200 rad/s, U∞ = 100 m/s, T = 25 °C).

4.5. Effects of moisture on vibration characteristics

from 0.0% to 1.0%, and keep a constant when moisture concentration exceeds 1.0%, which is because elastic modulus E2 in Table 1 is a constant when ΔC ≥ 1.0%. Considering solely hygroscopic expansion deformation, Table 5 shows a tiny variation of the frequencies with moisture concentration. It is found from Fig. 12 and Table 5 that the influence of moisture on vibration characteristics is mainly attributed to change of material property generated by moisture instead of hygroscopic expansion deformation. Furthermore, results in Figs. 9–12 indicate that,

In order to illustrate effects of moisture concentration on vibration characteristics of the shell, we set T = 25 °C. The influence of material property variation generated by moisture and hygroscopic expansion deformation on dimensionless natural frequencies of both travelling waves are described in Fig. 12 and Table 5 for mode (1, 2), respectively. It is shown from Fig. 12 that, taking account of only material property variation, the frequencies descend with moisture concentration ΔC being 364

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Fig. 13. Effects of (a) temperature and (b) moisture on critical rotating velocities (m = 1, n = 2, L/R = 10, h/R = 0.01, U∞ = 100 m/s).

with fiber orientation angle 𝜃 0 for different hygrothermal environment, where 𝜃 0 means (𝜃 0 /-𝜃 0 )s lay up of laminates. It is clearly shown that the frequencies of both travelling waves firstly increase and then decrease with fiber orientation angle for modes (1, 1) and (1, 2), and reach their peak at 𝜃 0 = 45°. However, for modes (1, 3) and (1, 4), the frequencies ascend continuously with fiber orientation angle and reach their peak at 𝜃 0 = 90°. From a physical point of view, lamination schemes of composites can affect the bending stiffness of the system, which results in the change of the frequencies. In addition, effects of fiber orientation angles on the frequencies of lower order circumferential modes are more dramatical than on the frequencies of the higher ones.

Table 5 Effects of hygroscopic expansion deformation on vibration characteristics (m = 1, n = 2, L/R = 10, h/R = 0.01, Ω = 200 rad/s, U∞ = 100 m/s, T = 25 °C). ΔC

𝜔b ∗

𝜔f ∗

0.0% 0.5% 1.0% 1.5%

0.21138123 0.21138122 0.21138120 0.21138118

0.08179864 0.08179862 0.08179861 0.08179859

Table 6 Variation of critical rotating velocities with moisture concentration ΔC (m = 1, n = 2, L/R = 10, h/R = 0.01, U∞ = 100 m/s). ΔC

critical speeds Ωc

0.0% 0.5% 1.0% 1.5%

258.58586 258.58583 258.58581 258.58581

5. Conclusion Free vibration characteristics of a rotating composite laminated cylindrical shell under both subsonic air flow and hygrothermal effects are studied in this paper. The effects of rotating angular velocity, subsonic air flow velocity, temperature, moisture and fiber orientation angle on vibration characteristics of the shell are treated systematically, both analytically and numerically. Of great interest in this study is combined influence of hygrothermal environment and subsonic air flow on traveling wave vibration of the shell. The results of this work are drawn as follows: (1) Coriolis force caused by the rotation induces natural frequencies of forward and backward traveling waves. Subsonic air flow velocity and hygrothermal effects both weaken dimensionless natural frequencies of forward and back travelling waves and critical rotating velocity of the shell. (2) The frequencies descend with moisture concentration ΔC being from 0.0% to 1.0%, and keep a constant when moisture concentration exceeds 1.0%. In addition, compared with temperature, moisture has less prominent influence on the frequencies and critical rotating velocity. (3) Initial hoop tension is dominant factor leading to the disappearance of critical rotating velocity for higher order circumferential modes. (4) The frequencies of both travelling waves initially ascend and then descend with fiber orientation angle for lower order circumferential modes. However, they ascend continuously for higher ones, as fiber orientation angle increase.

temperature has larger influence on vibration characteristics than moisture. 4.6. Effects of temperature and moisture on critical rotating velocities In this section, effects of temperature and moisture on critical rotating velocities Ωc are depicted in Fig. 13 for mode (1, 2) respectively, which ignoring initial hoop tension caused by rotation. It is observed from Fig. 13(a) that critical rotating velocities descend with the temperature. However, Fig. 13(b) shows that critical rotating velocities seem to be little affected by moisture. To further exhibit effects of moisture on critical rotating velocities, Table 6 displays the variation of critical rotating velocities with moisture concentration ΔC. It is seen that critical rotating velocities have a tiny decrease when ΔC locates between 0.0% and 1.0%, and then keep a constant when ΔC ≥ 1.0%, which is similar with the trend of the frequency with moisture. 4.7. Effects of fiber orientation angles on vibration characteristics

Acknowledgments

In this subsection, the influence of fiber orientation angles on vibration characteristics is studied. Results in Fig. 14 demonstrate variation of dimensionless natural frequencies of forward and back travelling waves

This research work was supported by the National Natural Science Foundation of China (Grant Nos. 11872319, 51878559, 11702230, 11602208 and 51674216). 365

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Fig. 14. Variation of dimensionless natural frequencies with fiber orientation angles (m = 1, L/R = 10, h/R = 0.01, Ω = 200 rad/s, U∞ = 100 m/s): (a) T = 75 °C, ΔC = 0.5%; (b) T = 125 °C, ΔC = 1.0%.

Appendix A 𝐿31 = −𝐿13

(A.7)

𝐿32 = −𝐿23

(A.8)

4 4 4 2(𝐷12 +2𝐷66 ) 𝜕 4 4𝐷 𝐴 𝐷 𝐿33 = − 𝑅222 − 𝐷11 𝜕𝜕𝑥4 − 𝑅224 𝜕𝜕𝜃4 − − 𝑅16 𝜕 𝑥𝜕3 𝜕𝜃 2 𝑅2 𝑇 𝐻 ) 𝜕 𝑥2 𝜕 𝜃( ( ) 2 ( 𝑇𝐻 ) 4 2 𝐴 2𝐴 4𝐷 4𝐵 2𝐵 2𝐵 𝜕 𝜕2 − 𝑅326 𝜕 𝑥𝜕 + 𝑅12 −𝐴𝑇𝑥 𝐻 𝜕𝜕𝑥2 + 𝑅322 − 𝑅𝜃 2 𝜕𝜕𝜃2 + 𝑅226 − 𝑅𝑥𝜃 𝜕 𝑥𝜕 𝜃3 𝜃 ( ) ( ) 2 2 2 2 2 2 𝜕 +𝜌𝑡 Ω2 𝜕𝜕𝜃2 − 𝜕𝜕𝑡2 − 𝜌∞ 𝑓𝑚𝑛 𝜕𝜕𝑡2 +2𝑈∞ 𝜕𝜕𝑥𝜕 𝑡 +𝑈∞ 𝜕 𝑥2

(A.9)

The differential operators Lij in Eq. (30) are expressed as 𝐿11 = 𝐴11

𝐴 𝜕2 2𝐴16 𝜕 2 𝜕2 𝜕2 𝜕2 + 66 + − 𝜌𝑡 + 𝜌 𝑡 Ω2 2 2 2 2 𝑅 𝜕 𝑥𝜕 𝜃 𝜕𝑥 𝑅 𝜕𝜃 𝜕𝜃 𝜕 𝑡2

(A.1)

( ) 2 ( ) ( ) 2 2𝐵 𝐴26 𝐵26 𝜕 2 𝐴12 +𝐴66 𝐵12 +2𝐵66 𝜕 𝜕 𝐿12 = 𝐴16 + 16 + + 3 + + 2 2 2 2 𝑅 𝑅 𝜕 𝑥𝜕 𝜃 𝜕𝑥 𝑅 𝑅 𝜕𝜃 𝑅 (A.2)

𝐿13 =

Appendix B

3𝐵 𝐵 𝜕3 𝐴12 𝜕 𝐴26 𝜕 𝜕 3 𝐵 +2𝐵 𝜕3 𝜕3 + 2 − 16 2 − 26 −𝐵11 3 − 12 2 66 2 𝑅 𝜕𝑥 𝑅 𝜕𝜃 𝑅 𝜕 𝑥 𝜕𝜃 𝑅3 𝜕 𝜃 3 𝜕𝑥 𝑅 𝜕 𝑥𝜕 𝜃 (A.3)

𝐿21 = 𝐿12 ( ) 2 ( 2𝐷 3𝐵 𝐴 𝐷 𝐿22 = 𝐴66 + 𝑅266 + 𝑅66 𝜕𝜕𝑥2 + 𝑅222 + 𝑅224 + ( ) 2 ( ) 2 2 2𝐴 6𝐵 4𝐷 𝜕 + 𝑅26 + 𝑅226 + 𝑅326 𝜕 𝑥𝜕 + 𝜌𝑡 Ω2 𝜕𝜕𝜃2 − 𝜕𝜕𝑡2 𝜃

The coefficients in Eq. (33)Equation Chapter (Next) Section 1 are given as ( 𝑀 𝑖 = 𝜌𝑡 𝜋 𝑎 𝑖 𝑎 𝑖

(A.4) 2𝐵22 𝑅3

)

𝜕2 𝜕𝜃2

𝐿

∫0

𝜑2 𝑢𝑚 (𝑥)d𝑥+𝑏𝑖 𝑏𝑖

( ) + 𝜌𝑡 +𝜌∞ 𝑓𝑚𝑛 𝜋 (A.5) 𝐶𝑟𝑖 = 𝜌∞ 𝑓𝑚𝑛 𝜋

𝐿

∫0

∫0

∫0

𝜑2 𝑤𝑚 (𝑥)d𝑥

𝐿

∫0

𝜑𝑣𝑚 (𝑥) ⋅ 𝜑𝑤𝑚 (𝑥)d𝑥

𝐾𝑖 = 𝜋𝐾 𝐾𝑟𝑖 = 𝜌∞ 𝑓𝑚𝑛 𝜋

(A.6) 366

𝐿

∫0

) 𝜑2 𝑣𝑚 (𝑥)d𝑥

𝜑′𝑤𝑚 (𝑥) ⋅ 𝜑𝑤𝑚 (𝑥)d𝑥

( ) 𝐶𝑠𝑖 = −𝜌𝑡 𝜋 𝑏𝑖 + 𝑏𝑖

( ) ( ) ( ) 3 ( ) 3 𝐴 2𝐵 2𝐷 𝐴 𝐵 𝐷 𝐵 𝜕 𝜕 𝐿23 = 𝑅26 + 𝑅226 𝜕𝑥 + 𝑅222 + 𝑅223 𝜕𝜃 − 𝐵16 + 𝑅16 𝜕𝜕𝑥3 − 𝑅224 + 𝑅223 𝜕𝜕𝜃3 ( ) 3 ( ) 3 ( ) 𝐷 +4𝐷 𝐵 +2𝐵 3𝐵 4𝐷 𝜕 𝜕 − 12 𝑅2 66 + 12 𝑅 66 𝜕 𝑥𝜕2 𝜕𝜃 − 𝑅226 + 𝑅326 𝜕 𝑥𝜕 + 2𝜌𝑡 Ω2 𝜕𝜃 − Ω 𝜕𝑡𝜕 𝜃2

𝐿

𝐿

𝜑′′𝑤𝑚 (𝑥) ⋅ 𝜑𝑤𝑚 (𝑥)d𝑥

X. Li et al.

International Journal of Mechanical Sciences 150 (2019) 356–368

𝐾𝑠𝑖 = 𝜌𝑡 𝜋𝐾𝑟 𝐾𝑜𝑖 = 𝜋𝐾𝑜

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(B.1)

where ( ) 𝐿 𝐿 𝐴 𝐾 = 𝑎𝑖 𝑎𝑖 −𝐴11 ∫0 𝜑′′𝑢𝑚 (𝑥) ⋅ 𝜑𝑢𝑚 (𝑥)d𝑥 + 𝑅662 𝑛2 ∫0 𝜑2 𝑢𝑚 (𝑥)d𝑥 ( ) 𝐿 𝐴 +𝐴 𝐵 +2𝐵 −𝑛𝑎𝑖 𝑏𝑖 12 𝑅 66 + 12 𝑅2 66 ∫0 𝜑′𝑣𝑚 (𝑥) ⋅ 𝜑𝑢𝑚 (𝑥)d𝑥 [( ) ] 𝐿 𝐿 𝐵12 +2𝐵66 2 𝐴12 −𝑎𝑖 + 𝑅2 𝑛 ∫0 𝜑′𝑤𝑚 (𝑥) ⋅ 𝜑𝑢𝑚 (𝑥)d𝑥 − 𝐵11 ∫0 𝜑′′′𝑤𝑚 (𝑥) ⋅ 𝜑𝑢𝑚 (𝑥)d𝑥 𝑅 ( ) 𝐿 𝐴 +𝐴 𝐵 +2𝐵 +𝑛𝑏𝑖 𝑎𝑖 12 𝑅 66 + 12 𝑅2 66 ∫0 𝜑′𝑢𝑚 (𝑥) ⋅ 𝜑𝑣𝑚 (𝑥)d𝑥 [( ) ( ) 𝐿 𝐿 2𝐷 3𝐵 𝐴22 𝐷22 2𝐵22 +𝑏𝑖 𝑏𝑖 𝑅2 + 𝑅4 + 𝑅3 𝑛2 ∫0 𝜑2 𝑣𝑚 (𝑥)d𝑥 − 𝐴66 + 𝑅266 + 𝑅66 ∫0 𝜑′′𝑣𝑚 (𝑥) [( ) ( ) ] ] 𝐿 𝐴 𝐵 𝐷 𝐵 ⋅𝜑𝑣𝑚 (𝑥)d𝑥 + 2𝑏𝑖 𝑅222 + 𝑅223 𝑛 + 𝑅224 + 𝑅223 𝑛3 ∫0 𝜑𝑤𝑚 (𝑥) ⋅ 𝜑𝑣𝑚 (𝑥)d𝑥 ( ) 𝐿 𝐷 +4𝐷 𝐵 +2𝐵 −2𝑏𝑖 12 𝑅2 66 + 12 𝑅 66 𝑛 ∫0 𝜑′′𝑤𝑚 (𝑥) ⋅ 𝜑𝑣𝑚 (𝑥)d𝑥 [( ) ] 𝐿 𝐿 𝐵12 +2𝐵66 2 𝐴12 +𝑎 𝑖 + 𝑅2 𝑛 ∫0 𝜑′𝑢𝑚 (𝑥) ⋅ 𝜑𝑤𝑚 (𝑥)d𝑥 − 𝐵11 ∫0 𝜑′′′𝑢𝑚 (𝑥) ⋅ 𝜑𝑤𝑚 (𝑥)d𝑥 𝑅 ( ) 𝐴𝑇 𝐻 𝐿 𝐿 𝐴 𝐷 2𝐵 + 𝑅222 + 𝑅224 𝑛4 + 𝑅322 𝑛2 − 𝑅𝜃 2 𝑛2 ∫0 𝜑2 𝑤𝑚 (𝑥)d𝑥+𝐷11 ∫0 𝜑′′′′𝑤𝑚 (𝑥) ⋅ 𝜑𝑤𝑚 (𝑥)d𝑥 [ 2 𝐷 +2𝐷 ] 𝐿 ( 12 66 ) 2 2𝐵12 − 𝑛 + 𝑅 − 𝐴𝑇𝑥 𝐻 ∫0 𝜑′′𝑤𝑚 (𝑥) ⋅ 𝜑𝑤𝑚 (𝑥)d𝑥 𝑅2

(B.2) [ ] 𝐿 𝐿 𝐿 𝐾𝑟 = 𝑛2 𝑎𝑖 𝑎𝑖 ∫0 𝜑2 𝑢𝑚 (𝑥)d𝑥 + 𝑏𝑖 𝑏𝑖 ∫0 𝜑2 𝑣𝑚 (𝑥)d𝑥 + ∫0 𝜑2 𝑤𝑚 (𝑥)d𝑥 𝐿

+4𝑛𝑏𝑖 ∫0 𝜑𝑣 (𝑥) ⋅ 𝜑𝑤 (𝑥)d𝑥

2𝐴

(B.3)

𝐿

𝐾𝑜 = 𝑛𝑎𝑖 𝑎𝑖 𝑅16 ∫0 𝜑′ 𝑢𝑚 (𝑥) ⋅ 𝜑𝑢𝑚 (𝑥)d𝑥 ( )( ) 𝐿 𝐿 2𝐵 +𝑎𝑖 𝑏𝑖 𝐴16 + 𝑅16 ∫0 𝜑′′ 𝑢𝑚 (𝑥) ⋅ 𝜑𝑣𝑚 (𝑥)d𝑥 − ∫0 𝜑′′ 𝑣𝑚 (𝑥) ⋅ 𝜑𝑢𝑚 (𝑥)d𝑥 ( ) 𝐿 𝐿 3𝐵 +𝑛𝑎𝑖 𝑅16 ∫0 𝜑′′ 𝑢𝑚 (𝑥) ⋅ 𝜑𝑤𝑚 (𝑥)d𝑥 − ∫0 𝜑′′ 𝑤𝑚 (𝑥) ⋅ 𝜑𝑢𝑚 (𝑥)d𝑥 ( ) 𝐿 2𝐴 6𝐵 4𝐷 +𝑛𝑏𝑖 𝑏𝑖 𝑅26 + 𝑅226 + 𝑅326 ∫0 𝜑′ 𝑣𝑚 (𝑥) ⋅ 𝜑𝑣𝑚 (𝑥)d𝑥 [( ) ( )]( 𝐿 𝐿 𝐴26 2𝐵26 3 𝐵 2 𝐷 ∫0 𝜑′ 𝑤𝑚 (𝑥) ⋅ 𝜑𝑣𝑚 (𝑥)d𝑥 + ∫0 𝜑′ 𝑣𝑚 (𝑥) +𝑏𝑖 + 𝑅2 + 𝑛2 𝑅226 + 𝑅326 𝑅 )( ) ) ( 𝐿 𝐿 2𝐷 ⋅𝜑𝑤𝑚 (𝑥)d𝑥 −𝑏𝑖 𝐵16 + 𝑅16 ∫0 𝜑′′′ 𝑤𝑚 (𝑥)⋅𝜑𝑣𝑚 (𝑥)d𝑥+ ∫0 𝜑′′′ 𝑣𝑚 (𝑥)⋅𝜑𝑤𝑚 (𝑥)d𝑥 ( ) 2𝐴𝑇 𝐻 𝐿 𝐿 4𝐷 4𝐷 4𝐵 − 𝑅16 𝑛 ∫0 𝜑′′′ 𝑤𝑚 (𝑥) ⋅ 𝜑𝑤𝑚 (𝑥)d𝑥 + 𝑅326 𝑛3 + 𝑅226 𝑛 − 𝑅𝑥𝜃 𝑛 ∫0 𝜑′ 𝑤𝑚 (𝑥) ⋅𝜑𝑤𝑚 (𝑥)d𝑥

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