Research Article Yaolun Wang, Chaofan Yang, Yongxin Zhang ...

Research Article

Yaolun Wang, Chaofan Yang, Yongxin Zhang, Shipeng Dong, and Liang Li*

Dynamics of a rotating hollow FGM beam in thetemperature field

https://doi.org/10.1515/rams-2021-0055received June 25, 2021; accepted July 10, 2021

Abstract: Dynamic responses and vibration characteris-tics of a rotating functionally graded material (FGM)beam with a hollow circular cross-section in the tempera-ture field are investigated in this paper. The materialproperties of the FGM beam are assumed to be tempera-ture-dependent and vary along the thickness directionof the beam. By considering the rigid-flexible couplingeffect, the geometrically nonlinear dynamic equationsof a hub–FGM beam system are derived by employingthe assumed modes method and Lagrange’s equations.With the high-order coupling dynamic model, the effectof temperature variations under two different laws ofmotion is discussed, and the free vibration of the systemis studied based on the first-order approximate couplingmodel. This research can provide ideas for the design ofspace thermal protection mechanisms.

Keywords: dynamics, rotating FGM beam, hollow circularcross-section, temperature field

1 Introduction

Flexible beam structures are widely used in the field ofaerospace. These components often work in an extremelyhigh- or low-temperature environment for a long time.The components made of the isotropic materials and tra-ditional composites gradually cannot meet the require-ments of actual working conditions, and so advancedmaterials such as functionally graded materials (FGM)gradually replace them. With both good mechanicalstrength and thermal insulation properties, FGMs have

been widely used in aerospace and other fields, andresearch on mechanical behaviors of FGM has drawnthe attention of many scholars.

Hui [1] studied the effects of shear loads on thevibration and buckling of a typical antisymmetric cross-laminated cylindrical thin plate under combined loads.Aminbaghai et al. [2] investigated the effect of torsionalwarping of the FGM beam on elastic-static behavior, andthey used the transfer matrix method to obtain the finiteelement equations. Paul and Das [3] studied the vibrationbehavior of prestressed FGM beams and discussed theeffect of different FGMs on frequencies. Li et al. [4], Liand Zhang [5] and Dong et al. [6] studied the dynamics ofrotating FGM beams based on the rigid-flexible couplingdynamic theory and discussed the frequency veering andmode interactions of the flexible beam. Oh and Yoo [7]proposed a dynamic model, which can be used to analyzethe frequencies of rotating FG blades. Yang and He [8]combined the re-modified couple stress theory and therefined zigzag theory to model the functionally graded(FG) sandwich microplates. In their investigation, thevibration and buckling analysis of two types of FG micro-plates are discussed. Lee and Hwang [9] studied thegeometrical nonlinear transient behavior of carbon nano-tube/fiber/polymer composite (CNTFPC) spherical shellscontaining a central cutout, and the results indicated thatan appropriate CNT ratio and curvature are important forimproving the nonlinear dynamic properties. Parida andMohanty [10] used the finite element method to developthe dynamic equations of a skew FG plate based on thehigh order shear deformation theory (HOSDT). Shen et al.[11] studied the dynamic properties of piezoelectric coupledlaminated fiber-reinforced cylindrical shells consideringthe transverse shear effect. The results show that thismethod is more effective than the finite element method.Zhou et al. [12] developed the three-dimensional dynamicmodel of the rotating FG cantilever beam based on theTimoshenko beam theory. In their work, the nonlinearcoupling deformation term, which describes the stif-fening effect of the rotating cantilever FG beam, is con-sidered. Shahmohammadi et al. [13] investigated the freevibration of the traditional, sandwich and laminated

Yaolun Wang, Chaofan Yang, Yongxin Zhang, Shipeng Dong: Schoolof Science, Nanjing University of Science and Technology, Nanjing,People’s Republic of China

* Corresponding author: Liang Li, School of Science, NanjingUniversity of Science and Technology, Nanjing, People’s Republic ofChina, e-mail: [email protected]

Reviews on Advanced Materials Science 2021; 60: 643–662

Open Access. © 2021 Yaolun Wang et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution 4.0International License.

https://doi.org/10.1515/rams-2021-0055

mailto:[email protected]

shells containing the FG material by the isogeometricB3-spline finite strip method (IG-SFSM). Dynamics of theflexible beam structure that are made of FGM are oftenaffected by many factors, such as its shape, the type ofgraded materials, the external forces, etc. The thermaleffect of temperature is a relatively common factor. Inrecent years, more and more researchers have focusedon the thermodynamics of FGM beams. Attia et al. [14]investigated the vibration of the temperature-dependentFG plate by employing various four-variable refined platetheories. Bouchafa et al. [15] presented a new refinedhyperbolic shear deformation theory (RHSDT) to analyzethe thermoelastic bending problem of the FG sandwichplates. The influence of temperature on the stability andvibration characteristics of the prestressed sandwich beamcovered with FG sheets is investigated by Chen et al. [16].Zaman et al. [17] investigated the bending characteristicsof a curved FGM piezoelectric cantilever actuator in theelectric and temperature fields. The simulations indicatedthat the thermal load has a significant influence on theelectroelastic field of the curved actuator. Based on theclassical small deflection plate theory, Xing et al. [18]derived the governing equations of the FG plates withthe temperature field. Based on Reddy’s high-order sheardeformation theory, Bisheh et al. [19] studied the influ-ence of parameters such as temperature change and FGdistribution pattern on the natural frequency of large-amplitude vibration of FG-GRC laminated plates.

Dehrouyeh-Semnani [20] compared the thermal vibra-tion of FGM beams on simplified boundary conditions withthat on original boundary conditions and investigatedtwo different temperature types. Jiang et al. [21] studiedthe vibration behavior of composite beams in a thermalenvironment, but they only discussed the flap-wise vibra-tion and neglected the chord-wise vibration. Varmazyariand Shokrollahi [22] studied the elastic–plastic deforma-tion of rotating FG cylinders used in the strain gradienttheory. Based on the first-order shear deformation theory,Kashkoli et al. [23] investigated the time-dependent thermo-elastic creep problem of the FG thick-walled cylinder andgave the theoretical solution for this problem. CombiningDonnell’s shell theory, von Kármán nonlinearity terms,the circumferential condition in an average sense, andthree-state solution form of deflection, and applying theGalerkin procedure, Nam et al. [24] studied the nonlineardeflection torsional buckling problem of the FG carbonnanotube orthogonally reinforced composite cylindricalshells. Li et al. [25] investigated the effect of muggy envir-onment on natural frequencies and critical speed of com-posite thin-walled beams. Based on the higher-ordershear deformation beam theory, Shabanlou et al. [26]

studied the vibration characteristics of FGM beams ina thermal environment. Ghadiri and Shafiei [27] studiedthe dynamics of Timoshenko microbeams made of FGMunder four different temperature distributions in a thermalenvironment. Asadi and Beheshti [28] gave a comprehen-sive parameter analysis on the dynamics of FG compositebeams in a thermal environment. Azadi [29] investigatedthe dynamic analysis of FG beams in a thermal environ-ment. Azimi et al. [30] studied the thermomechanicalvibrations of rotating axially FG beams. Ghadiri and Jafari[31] proposed an analytical method to study the vibra-tions of FG beams with a tip mass in a thermal environ-ment. Bich et al. [32] analyzed nonlinear vibration anddynamic buckling of FG annular shells in thermal envir-onments. Li et al. [33] studied the influence of tempera-ture on vibrations and buckling behaviors of compositebeams, and it was assumed that the temperature variesalong the thickness at a constant value. Shahrjerdi andYavari [34] investigated the free vibration analysis ofFG nanocomposite beams under a thermal environment.Khosravi et al. [35] studied the influence of temperatureon the vibration behavior of rotating composite beamsbased on the Timoshenko beam theory, where the beamwas reinforced by employing carbon nanotubes; so it maycause instability and is more sensitive to temperature.

In the above studies, the cross-sections of the FGMbeams are usually rectangle, tapered, or trapezoid, whilethe study on the FGM beam with circular cross-section isquite rare. There are many studies on the nonstatic cir-cular shell and circular disks that rotate around theirlongitudinal axis. Peng and Li [36] analyzed the thermo-elastic problem of a rotating FG hollow circular disk andproposed a new analytical method, which can be used tostudy steady thermal stresses. Dai and Dai [37] used asemi-analytical approach to investigate the displacementand stress fields of the rotating FGM hollow disk in tem-perature fields. Huang et al. [38] studied the free vibra-tion of a rotating axially FG beam rotating around itslongitudinal axis and discussed the effect of axially dis-tributed FGMs on frequencies, critical speeds, and modeshapes. It should be noted that the rigid-flexible coupleddynamics of rotating FG beams with a hollow circular cross-section in the temperature field has not been reported in theopen literature.

In this paper, the dynamic responses and free vibra-tions of rotating FG beams with a hollow circular cross-section in the temperature field are studied. The materialproperties are assumed to be temperature-dependent andvary along the thickness direction of the beam. By employ-ing the assumed modes method and Lagrange’s equations,the governing equations of motion of an FG beam with

644 Yaolun Wang et al.

hollow circular cross-section attached to a rotating rigidhub are derived in Section 2. The validation of the presentdynamic model is shown in Section 3.1. The dynamicresponses of the rotating beam driven by an externaltorque are shown in Section 3.2. The dynamic responsesof the beam with the prescribed law of motion are shownin Section 3.3. Free vibrations of the beam rotatingat constant angular velocities are discussed briefly inSection 3.4. Some conclusions based on the simulationresults are given in Section 4.

2 Dynamic model of the system

Figure 1 shows the schematic of the hub–beam system inwhich the flexible beam is attached to the rigid body andthe displacement field of an arbitrary point on the beamaxis. An inertial coordinate system OXYZ and a floatingcoordinate system oxyz are defined in Figure 1, respec-tively. θ is the rotating angle of the hub.

Figure 2 shows the geometry of the FG beam with ahollow circular cross-section. The length and density ofthe beam are L and ρ r( ), respectively; the modulus ofelasticity is E r( ); the outer radius of the beam is rb; andthe inner radius is ra.

The material properties are assumed to vary alongthe thickness direction of the FG beam with a power-law distribution:

P r P P r rr r

P ,N

c ma

b am( ) ( )

⎛⎝⎜

⎞⎠⎟= −

−

−

+ (1)

where P r( ) can be replaced by ρ r( ), E r( ), the coefficient ofheat conduction K r( ), and the coefficient of thermalexpansion α rT( ); Pc and Pm are the material parameter ofceramics and metal, respectively.

It is assumed that the temperature varies along thethickness direction of the beam, and the one-dimensionalform of the steady-state heat conduction is

r rrK r T r

r1 d

dd

d0.( )

( )⎡⎣⎢

⎤⎦⎥− = (2)

The thermal Dirichlet boundary conditions are writtenas

T r T T r T, ,a m b c( ) ( )= = (3)

thus,

T r T T TC rK r

r1 d ,r

r

mc m

a

( )( )

∫= +

−

(4)

where

CrK r

r1 d .r

r

a

b

( )∫= (5)

The dependency of Young’s modulus and the expan-sion coefficient varies with the temperature T as follows:

P P P T P T P T P T1 ,0 1 1 22

33( )= / + + + +

−

(6)

where P0, P 1−, P1, P2, and P3 are relevant coefficients of

temperature.According to Figure 1, the position vector rp can be

expressed asFigure 1: Schematic of the hub–beam system and the displacementfield of an arbitrary point on the beam axis.

Figure 2: Geometry of the FG beam with a hollow circular cross-section.

Dynamics of rotating hollow FGM beam 645

a x u y u z ur i j k,x y zp ( ) ( ) ( )= + + + + + + (7)

where

u w w w w w u w u w, , ,x cy cz dy dz y z1 2 3= + + + + = = (8)

in which w1 is the axial displacement, w 1 2cyx

0( )∫= − /

w ξ ξd22( )∂ /∂ and w w ξ1 2cz

x

0 32( ) ( )∫= − / ∂ /∂ ξd are the axial

shrinkage of the beam caused by the transverse displace-ment and flapwise displacement, respectively, and wdy =

yw 2− ′ and w zwdz 3= − ′ are the axial displacements causedby the rotation of the cross-section.

According to equation (7), the velocity vector can beobtained as

u y u θ a x u θ u ur i j k ,x y x y zp [ ( ) ] [( ) ]= − + + + + + + (9)

thus, the kinetic energy of the system can be writtenas

E J θ ρ r u y u θ

a x u θ u u V

12

12

d .V

x y

x y z

kinetic oh2 2

2 2

{ ( )[[ ( ) ]

[( ) ] ]}

∫= + − +

+ + + + +

(10)

Neglecting the energy caused by shear deformation,the potential energy of the system can be expressedas

E σ ε V E r T ε V12

d 12

, d .V

x x

V

xpotential2( ) [ ( ) ]∫ ∫= = (11)

The normal strain of any point can be written as

ε wx

y wx

z wx

.x1

22

2

23

2=

∂

∂

−

∂

∂

−

∂

∂

(12)

According to Figure 2, y r αsin= and z r αcos= .Thus, the potential energy can be written as

Employing theassumedmodesmethod toapproximatevariables, the axial deformationw1, transverse deforma-tionw2, and the flapwise deformation w3 can be expressed,respectively, as follows:

w x tw x tw x t

Φ qΦ qΦ q ,

x

y

z

1 1

2 2

3 3

( ) ( )

( ) ( )

( ) ( )

⎧⎨⎪⎩⎪

=

=

=

(14)

whereΦx(x),Φy(x), andΦz(x) are modal function vectorsrelated to longitudinal vibration, transverse bendingvibration, and flapwise bending vibration of the FGMbeams, respectively. Thus, we can obtain

w xq H q12

,cy 2T

1 2( )= − (15)

w xq H q12

,cz 3T

2 3( )= − (16)

where xH1( ) and xH2( ) are coupled shape functions andcan be expressed as

x ξ ξ ξH Φ′ Φ d ,x

y y′

1

0

T( ) ( ) ( )∫= (17)

x ξ ξ ξH Φ′ Φ d .x

z z′

2

0

T( ) ( ) ( )∫= (18)

Then, we can obtain

u

r α r αuu

Φ q q H q q H q

Φ q Φ qΦ qΦ q

12

12

sin cos

.

x x

y z

y y

z z

′ ′

1 2T

1 2 3T

2 3

2 3

2

3

⎧

⎨⎪⎪⎪

⎩⎪⎪⎪

= − −

− −

=

=

(19)

Taking the first time derivative of components inequation (19), yields

ur α r α

uu

Φ q q H q q H qΦ q Φ q

Φ qΦ q

sin cos

.

x x

y z

y y

z z

′ ′1 2

T1 2 3

T2 3

2 3

2

3

⎧

⎨⎪⎪

⎩⎪⎪

= − −

− −

=

=

(20)

Let θq q q q1T

2T

3T T

= [ ] be the generalized coordinatevector, then the virtual work by temperature can bewritten as

E

E r T wx

y wx

z wx

V

rE r T wx

r α wx

r α wx

r α x π rE r T wx

r x

π r E r T wx

r x π r E r T wx

r x

12

, d

12

, sin cos d d d , d d

2, d d

2, d d .

VL π

r

r L

r

r

L

r

r L

r

r

potential

12

22

23

2

2

0 0

21

22

2

23

2

2

0

12

0

32

22

2

0

32

32

2

a

b

a

b

a

b

a

b

( )

( ) ( )

( ) ( )

⎜ ⎟

⎡⎣⎢⎢

⎛⎝⎜

⎞⎠⎟

⎤⎦⎥⎥

⎡⎣⎢⎢

⎛⎝⎜

⎞⎠⎟

⎤⎦⎥⎥

⎡⎣⎢

⎛⎝

⎞⎠

⎤⎦⎥

⎡⎣⎢⎢

⎛⎝⎜

⎞⎠⎟

⎤⎦⎥⎥

⎡⎣⎢⎢

⎛⎝⎜

⎞⎠⎟

⎤⎦⎥⎥

∫

∫∫∫ ∫∫

∫∫ ∫∫

=

∂

∂

−

∂

∂

−

∂

∂

=

∂

∂

−

∂

∂

−

∂

∂

=

∂

∂

+

∂

∂

+

∂

∂

(13)


δW E r T α r T T δε V δQ q, , Δ d ,T

V

T x TT[ ( )( ( ) ) ]∫= − = (21)

where T T r TΔ 0( )= − , T0 is the reference temperature,

and

Q Q Q Q0 ,TT

T1T

T2T

T3T T

= [ ] (22)

E r T α r T T r T VQ Φ, , d ,T

V

T x′

1 0T

[ ( ) ( )( ( ) ) ]∫= − − (23)

E r T α r T T r T VQ Φ, , d ,T

V

T y′

2 0T

[ ( ) ( )( ( ) ) ]∫= − − (24)

E r T α r T T r T VQ Φ, , d .T

V

T z′

3 0T

[ ( ) ( )( ( ) ) ]∫= − − (25)

By employing Lagrange’s equations of the secondkind,

tE E E

q q qQ Qd

d.τ T

kinetic kinetic potential⎛⎝⎜

⎞⎠⎟

∂

∂

−

∂

∂

= −

∂

∂

+ + (26)

The rigid-flexible coupling dynamic equations of thesystem can be obtained as

M θ Q τM M MM M M MM M M MM M M M

qqq

QQQ

000

QQQ

0

,

θ

T

T

T

11 12 13 14

21 22 23 24

31 32 33 34

41 42 43 44

1T

2T

3T

1

2

3

1

2

3

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥= + + (27)

where

M M ,22 1= (29)

π rρ r r xM M M H q q H2 d d ,L

r

r

33 2 4

0

1 2 2T

1

a

b

[ ( )( )]∫∫= + + ⋅ (30)

π rρ r r xM M M H q q H2 d dL

r

r

44 3 5

0

2 3 3T

2

a

b

[ ( )( )]∫∫= + + ⋅ (31)

M M q M ,12 21T

2T

6T

= = − (32)

π rρ r r x

π rρ r r x

M M S q M q Φ q H

q H q Φ q H q Φ

2 d d

d d ,

L

r

r

y

L

r

r

y y

13 31T

2 1T

6

0

2T T

2T

1

0

2T

1 2 3T

2 3

a

b

a

b

[ ( )( )]

[ ( )( )]

∫∫

∫∫

= = + + ⋅

− ⋅ + ⋅

(33)

π rρ r r xM M q Φ q H2 d d ,L

r

r

y14 41T

0

2T T

3T

2

a

b

[ ( )( )]∫∫= = ⋅ (34)

π rρ r r xM M Φ q H2 d d ,L

r

r

x23 32T

0

T2T

1

a

b

[ ( )( )]∫∫= = − ⋅ (35)

π rρ r r xM M Φ q H2 d d ,L

r

r

x24 42T

0

T3T

2

a

b

[ ( )( )]∫∫= = − ⋅ (36)

J J

π rρ r r x

π rρ r r x π rρ r r x

M S q q M q q M M C q q M C q

q H q q H q q H q q H q

q Φ q H q q Φ q H q q H q q H q

2

2d d

2 d d d d ,

L

r

r

L

r

r

x x

L

r

r

11 oh cb 1 1 1T

1 1 2T

2 4 1 2 3T

5 2 3

0

2T

1 2 2T

1 2 3T

2 3 3T

2 3

0

1T T

2T

1 2 1T T

3T

2 3

0

2T

1 2 3T

2 3

a

b

a

b

a

b

( ) ( )

[ ( )( )]

[ ( )( )] [ ( )( )]

∫∫

∫∫ ∫∫

= + + + + + − + −

+ ⋅ + ⋅

− ⋅ + ⋅ + ⋅

(28)

π rρ r r xM M H q q H2 d d ,L

r

r

34 43T

0

1 2 3T

2

a

b

[ ( )( )]∫∫= = ⋅ (37)


Q θ

πθ rρ r r x

πθ rρ r r x

πθ rρ r r x

πθ rρ r r x

π rρ r r x

S q q M q q M M C q q M C q

q H q Φ q q H q Φ q

q Φ q H q q Φ q H q



q Φ q H q q Φ q H q

2

2 d d

4 d d

2 d d

2 d d

2 d d ,

θL

r

r

x x

L

r

r

x x

L

r

r

L

r

r

L

r

r

y y

1 1 1T

1 1 2T

2 4 1 2 3T

5 2 3

0

2T

1 2 1 3T

2 3 1

0

1T T

2T

1 2 1T T

3T

2 3

0

2T

1 2 2T

1 2 3T

2 3 3T

2 3

0

3T

2 3 2T

1 2 2T

1 2 3T

2 3

0

2T T

2T

1 2 2T T

3T

2 3

a

b

a

b

a

b

a

b

a

b

[ ( ) ( ) ]

[ ( )( )]

[ ( )( )]

[ ( )( )]

[ ( )( )]

[ ( )( )]

∫∫

∫∫

∫∫

∫∫

∫∫

= − + + + − + −

+ ⋅ + ⋅

+ ⋅ + ⋅

− ⋅ + ⋅

− ⋅ + ⋅

− ⋅ + ⋅

(38)

θ π rρ r r x

θ π rρ r r x

Q K q S M q Φ q H q Φ q H q

M q Φ q H q Φ q H q

d d

2 2 d d ,

L

r

r

x x

L

r

r

x x

1 1 12

1T

1 1

0

T2T

1 2T

3T

2 3

6 2

0

T2T

1 2T

3T

2 3

a

b

a

b

{ [ ( )( )]

[ ( )( )]

⎫⎬⎪

⎭⎪∫∫

∫∫

= − + + − ⋅ + ⋅

+ + ⋅ + ⋅

(39)

θ

π rρ r r x θ

πθ rρ r r x

π rρ r r x

Q K q M M C q

H q q H q H q Φ q H q q H q M q

Φ q H q Φ q H q H q Φ q

H q q H q H q q H q

2 d d 2

4 d d

2 d d ,

L

r

r

x

L

r

r

y y y

L

r

r

2 2 22

2 4 1 2

0

1 2 2T

1 2 1 2 1 1 2 3T

2 3 6T

1

0

T2T

1 2T

3T

2 3 1 2 2

0

1 2 2T

1 2 1 2 3T

2 3

a

b

a

b

a

b

{( )

[ ( )( )]

[ ( )( )]

[ ( )( )]

⎫⎬⎪

⎭⎪∫∫

∫∫

∫∫

= − + + −

+ ⋅ − ⋅ + ⋅ −

+ ⋅ + ⋅ − ⋅

− ⋅ + ⋅

(40)

θ

π rρ r r x

πθ rρ r r x

π rρ r r x

Q K q M C q

H q q H q H q Φ q H q q H q

H q Φ q

H q q H q H q q H q

2 d d

4 d d

2 d d ,

L

r

r

x

L

r

r

y

L

r

r

3 3 32

5 2 3

0

2 3 3T

2 3 2 3 1 2 3 2T

1 2

0

2 3 2

0

2 3 2T

1 2 2 3 3T

2 3

a

b

a

b

a

b

{( )

[ ( )( )]

[ ( )( )]

[ ( )( )]

⎫⎬⎪

⎭⎪∫∫

∫∫

∫∫

= − + −

+ ⋅ − ⋅ + ⋅

− ⋅

− ⋅ + ⋅

(41)


in which the constant matrixes are expressed as

J π ρ r r a x r r x2 d d ,cb

L

r

r

0

2 3

a

b

{ ( )[ ( ) ]}∫∫= + + (42)

π rρ r a x r xS Φ2 d d ,L

r

r

x1

0 a

b

[ ( )( ) ]∫∫= + (43)

π rρ r a x r xS Φ2 d d ,L

r

r

y2

0 a

b

[ ( )( ) ]∫∫= + (44)

π rρ r a x r xC H2 d d ,L

r

r

1

0

1

a

b

[ ( )( ) ]∫∫= + (45)

π rρ r a x r xC H2 d d ,L

r

r

2

0

2

a

b

[ ( )( ) ]∫∫= + (46)

π rρ r r xM Φ Φ2 d d ,L

r

r

x x1

0

T

a

b

[ ( ) ]∫∫= (47)


r

r

y y2

0

T

a

b

[ ( ) ]∫∫= (48)


r

r

z z3

0

T

a

b

[ ( ) ]∫∫= (49)

Φ′π r ρ r r xM Φ d d ,L

r

r

y y′

4

0

3 T

a

b

[ ( ) ]∫∫= (50)

Φ′π r ρ r r xM Φ d d ,L

r

r

z z′

5

0

3 T

a

b

[ ( ) ]∫∫= (51)


r

r

x y6

0

T

a

b

[ ( ) ]∫∫= (52)

Φ′π rE r T r xK Φ2 , d d ,L

r

r

x x′

1

0

T

a

b

[ ( ) ]∫∫= (53)

Φ″π rE r T r xK Φ″, d d ,L

r

r

y y2

0

T

a

b

[ ( ) ]∫∫= (54)

Φ″π rE r T r xK Φ″, d d .L

r

r

z z3

0

T

a

b

[ ( ) ]∫∫= (55)

Equation (27) provides a three-dimensional HOCdynamic model of a rotating FGM beam with a hollowcircular cross-section. Based on the HOC dynamic model,the complex rigid-flexible coupling dynamics of the systemcan be studied. It can be seen that some of the terms inequations (28, 30, 31, 33–41) are either single under-lined or double underlined, or both. These underlinedterms are generated by the second-order coupling termswcy and wcz. If the double underlines are ignored, the HOCdynamic model can be reduced to the traditional FOACdynamic model.

3 Dynamics and vibrationcharacteristics

3.1 Comparison studies

In order to ensure the validity of the present model, nat-ural frequencies of a nonrotating beam are comparedwith the results obtained by employing ABAQUS soft-ware. Table 1 shows the first four frequencies of a non-rotatinghomogeneous beam indifferent temperaturefields.It can be seen that the results obtained by the presentmethod are well consistent with those obtained by employ-ing ABAQUS. Table 3 shows the first six frequencies of anonrotating FGM beam (N 1= ). In ABAQUS software, the

Table 1: The first four frequencies of a nonrotating homogeneous beam in different temperature fields

Mode =T 100 K =T 300 K =T 500 K

Present ABAQUS ( )RΔ % * Present ABAQUS ( )RΔ % Present ABAQUS ( )RΔ %

1 1.4951 1.4951 0.0000 1.4584 1.4584 0.0000 1.4295 1.4295 0.00002 9.3693 9.3687 0.0064 9.1395 9.1389 0.0066 8.9587 8.9581 0.00673 26.234 26.230 0.0152 25.590 25.586 0.0156 25.084 25.080 0.01594 51.405 51.390 0.0292 50.145 50.130 0.0299 49.152 49.138 0.0285

* ( )= / ×RΔ present − ABAQUS present 100%.


material properties of FGM cannot be set directly with acontinuous variation law, so multiple layers of the FGMbeam along the radial direction are shown in Figure 3.Table 2 shows the temperature-dependent coefficients ofceramic and metal which are used in this paper. As seeninTable 3, the results calculated by the presentmethod alsoagreewellwith those calculatedbyABAQUS. Therefore, theaccuracy of the present FGM beam model is guaranteed.

3.2 Dynamics of the rotating beam driven byan external torque

Neglecting the radius of the hub by choosing a 0 m= , thegeometric parameters of the FGM beam are L 5 m= ,

r 0.005 ma = and r 0.01 mb = . It is assumed that the FGMbeam is made up of the ceramic Si3N4 and the metalSUS304 with thermal conductivity “Ye et al.” is not citedas an author of ref. 8. Please indicate any changes thatare required here. and K 16.32 W mkm

1= ⋅

− , respectively.An external torque is applied on the hub to drive the

FGM beam rotating around the hub and is defined as

τ t τ tt

t t

t t

sin 2π 0

0 ,s

s

s

0( )

⎧⎨⎪⎩⎪

⎛⎝⎜

⎞⎠⎟=

≤ ≤

≥

(56)

where t 2 ss = is the cycle time.In this paper, the dynamics of the FGM beam in the

flapwise direction are neglected because the rotation haslittle effect on the responses of the beam. Figures 4 and 5show comparisons of the responses of the system fromthe FOAC model and the HOC model when τ 10 N m0 = ⋅ ,where the effect of temperature is neglected. The value ofthe gradient index is N 1= . The results show that therotating angle and angular velocity of the rigid hub,and the tip transverse deformations and velocities ofthe FGM beam from the two models are basically consis-tent, except for the tip axial deformations and the tipaxial velocities. As shown in Figure 4(a), the rigid hubrotating angle increases to 0.13 rad when t 2 s= , andthen stops and swings. In Figure 4(b), the rotatingangular velocity increases first and then decreases; laterit also stops and swings. Figure 5(a) shows that the tipmaximum transverse deformation of the FGM beam isabout 0.035m. As shown in Figure 5(c) and (d), the axialdeformations and velocities of the beam from the two

Table 3: The first six frequencies of a nonrotating FGM beam ( =N 1)

1st 2nd 3rd 4th 5th 6th

Present 0.97017 6.0799 17.0235 33.3580 55.1405 82.3656ABAQUS 0.97026 6.0800 17.022 33.349 55.115 82.306

( )RΔ % −0.0093 −0.0016 0.0088 0.0270 0.0462 0.0724

Figure 3: Multiple layers of an FGM beam along the thickness inABAQUS.

Table 2: Temperature-dependent coefficients for ceramic and metal

Materials Properties P0 P−1 P1 P2 P3

SUS304 ( )E Pam 201.04 × 109 0 3.08 × 10−4 −6.53 × 10−7 0

( )⋅α 1 KT m,−1 1.23 × 10−5 0 8.09 × 10−4 0 0

( )⋅ρ kg mm−3 8,166 0 0 0 0

Si3N4 ( )E Pac 348.43 × 10−9 0 −3.1 × 10−4 2.16 × 10−7 −8.95 × 10−11

( )⋅α 1 KT c,−1 5.87 × 10−6 0 −9.1 × 10−4 0 0

( )⋅ρ kg mc−3 2,370 0 0 0 0


Figure 4: Dynamic responses of the hub–beam system ( = ⋅τ 10 N m0 ): (a) the rotating angle of the hub and (b) the angular velocity ofthe hub.

Figure 5: Dynamic responses of the FG beam ( = ⋅τ 10 N m0 ): (a) the tip transverse deformation, (b) the tip transverse velocity, (c) the tiplongitudinal deformation, and (d) the tip longitudinal velocity.


models are not matched. It can be found that the resultscalculated by the HOC model are more stable than thoseby the FOAC model without the high-order small quan-tities associated with wcy. The high-order small quantitieshave little effect on the transverse deformation, and dueto the axial deformation of the flexible beam being asmall amount, the effect of the high-order small quanti-ties on the axial deformation is highlighted. It can also beseen that during the cycle of motion, the axial deforma-tion reaches the maximum at t 1 s= , but it is very smallcompared to the transverse deformation.

Figure 6 shows the comparisons of the tip transversedeformations and rotating angular velocities from theHOCmodel and the FOACmodel. The tip maximum trans-verse deformation can reach approximately 0.175 m, theangular velocity of the rigid hub increases first and thendecreases, and the maximum value is about 0.65 rad·s−1.After the driving torque is removed, the beam stopsrotating and swings in the final position. When theFOAC model is used to solve the problem, it is foundthat sharp divergence occurs after 0.33 s, and both thetip deformation of the FGM beam and the angular velocityof the rigid body increase to infinity, which is inconsis-tent with the actual situation. For the FGM beam with alow rotating speed, the FOAC model can meet the preciserequirement and has better computational efficiency thanthe HOC model. However, for the FGM beam with rela-tively high rotating speed, the HOC model should be usedto replace the FOAC model due to the advantage inaccuracy.

To study the effect of thermal effect on the responsesof the FGM beam, three different temperature differencesbetween inner and outer surfaces of the beam are selected

as follows: (1)T T0 K, 0 Kc m= = ; (2)T T100 K, 0 Kc m= = ;and (3)T T200 K, 0 Kc m= = . The reference temperatureis T 0 K0 = .

Figure 7 shows the influence of temperature differ-ences on the dynamics of the system. It can be found thatthe change in the temperature difference has almost noeffect on the rotating angle. As the temperature differenceincreases, the rotating angular velocity also increases.The tip deformation of the FGM beam decreases slightlywhen the temperature difference increases. The reason isthat the temperature is distributed along the radial direc-tion of the FGM beam and is symmetrical on the whole.Therefore, the thermal loads caused by the temperatureare also symmetrical; as a result, the thermal loads canceleach other out, but the axial loads caused by the tem-perature difference still exist. In addition, the tensile stiff-ness of the FGM beam also becomes smaller, and as thetemperature difference increases, the axial deformationof the beam also increases. It can be found that both thetransverse and axial velocities of the FGM beam increasewith the increase of the temperature difference, and theinfluence on the axial velocity is more obvious. It can alsobe found that when the driving torque is removed, boththe transverse deformation and velocity of the FGM beamoscillate at the equilibrium position.

Then, three different cases of constant temperaturefield are selected as follows: (1)T T 0 Kc m= = ; (2)T Tc m= =

100 K; and (3) T T 200 Kc m= = . As shown in Figure 8(a),it can be found that the transverse deformation of theFGM beam becomes smaller with the increase of theambient temperature, and the degree of deformation issmaller than that in Figure 7(c). Figure 8(c) and (d) showsthe changes in the axial deformation and velocity of the

Figure 6: Dynamic responses of the FG beam ( = ⋅τ 50 N m0 ): (a) the tip transverse deformation and (b) the angular velocity.


Figure 7: Influence of the temperature difference on the dynamics of the system: (a) rotating angle, (b) the angular velocity, (c) thetransverse deformation, (d) the transverse velocity, (e) the axial deformation, and (f) the axial velocity.


FGM beam with the ambient temperature, respectively. Itcan be observed that both the deformation and velocityincrease with temperature. Compared with Figure 7(e)and (f), the degree of deformation and velocity havealso increased. The reason is that as the temperatureincreases, the energy of the FGM beam increases andthe vibration accelerates during rotation. Because of thesymmetrical structure, the transverse deformation of theFGM beam is less affected by temperature than the axialdeformation.

3.3 Dynamics of system with prescribed lawof motion

The parameters of the FGM beam are still the same asthose in Section 3.2, the radius of the rigid body is still

zero, and the large overall motion law of the system isassumed to be

θ ω t t ω t t t tω t t

sin 2π 2π, 0.

s s s

s

0 0

0

( )⎧⎨⎩=

/ − / / ≤ ≤

≥

(57)

Figure 9 shows the effect of the functionally gradientindices on the tip deformation of the FGM beam. It can beseen from Figure 9 that as the gradient index increases,the tip transverse and axial deformation of the FGM beamare both slower to reach the maximum amplitude, andboth the maximum amplitudes also increase. When therotating angular velocity becomes uniform, the FGM beamswings at the equilibrium position and the amplitude ofthe swing also increases. The reason is that as the gra-dient index increases, the ceramic component proportionof the FGM beam decreases while the metal componentproportion increases. As a result, the FGM beam becomesmore flexible and is easier to deform.

Figure 8: Influence of temperature variation on the dynamic response of the FG beam: (a) the tip transverse deformation, (b) the tiptransverse velocity, (c) the tip axial deformation, and (d) the tip axial velocity.


As in the previous section, we set three different tem-perature differences. The temperature on the metal side isalways set to be 0 K. The temperature differences are setas T 0 K, 100 K, 200 Kcm = . Figure 10 shows the effectof temperature differences on the transverse deformationand velocity of the FGM beam. It can be found that thetransverse deformation decreases with the increase ofthe temperature difference, while the transverse velocityincreases inapparently. The larger the temperature differ-ence, the more significant the reduction of deformation.Figure 11 shows the phase diagram of the transverse

deformation and velocity of the FGM beam under differenttemperature differences. It can be observed that thevibration times increase rapidly with the increase of thetemperature difference. Figure 12 shows the effect of tem-perature differences on the axial deformation and velo-city of the FGM beam. As shown in Figure 12, it can befound that the effect of the temperature difference on thetip axial deformation and velocity of the FGM beam isobvious, as the temperature difference increases, theaxial deformation, and velocity both increase. Figure 13shows the phase diagram of the axial deformation of the

Figure 9: Influence of the gradient index on the tip deformations of the FG beam: (a) the tip transverse deformation and (b) the tip axialdeformation.

Figure 10: The tip dynamic responses of the FGM beam under different temperature differences: (a) the transverse deformation and (b) thetransverse velocity.


Figure 11: Phase diagram of the transverse deformation of the FGM beam under different temperature differences: (a) =T 0 Kcm ,(b) =T 100 Kcm , and (c) =T 200 Kcm .

Figure 12: The tip dynamic responses of the FGM beam under different temperature differences: (a) the axial deformation and (b) the axialvelocity.


Figure 13: Phase diagram of the axial deformation of the FGM beam under different temperature differences: (a) =T 0 Kcm , (b) =T 100 Kcm ,and (c) =T 200 Kcm .

Figure 14: Influence of temperature variation on the dynamic responses of the FGM beam: (a) tip transverse deformation and (b) tiptransverse velocity.


FGM beam under different temperature differences. Asshown in the figure, the vibration times increases andthe vibration becomes more pronounced as the tempera-ture difference increases.

Figure 14 shows the tip transverse deformation andvelocity of the FGM beam in three different temperaturefields:T T 0 K, 100 K, 200 Kc m= = . It can be found thatas the ambient temperature increases, the tip transversedeformation of the beam decrease, and the velocityincreases, but the increase of the velocity is not obvious.

3.4 Numerical simulation procedure

In this section, the free vibration of the FGM beam in thetemperature field will be analyzed. The rotating speed is

set as θ 100 rad s 1= ⋅

− . Figure 15 shows variations of thefirst three frequencies of the FGM beam with the ambienttemperature. It can be observed that as the temperatureincreases, the frequencies become smaller. The reason isthat as the temperature increases, the stiffness of thebeam decreases. Figure 16 shows the first four normalizedbending and tensile mode shapes of the FGM beam when

T 300 K= and θ 100 rad s 1= ⋅

− .Figure 17 shows the changes in the first three frequen-

cies of the rotating FGM beam under different temperaturedifferences between the inner and outer surfaces. And the

inner temperature isT 300 Kcm = , the rotating speed is θ =

200 rad s 1⋅

− . It can be found that the natural frequenciesdecrease slightly with the increase of the temperaturedifference.

Figure 15: Effects of temperature on the first three damped frequencies of the FG beam when = ⋅θ 100 rad s−1: (a) first damped frequency,(b) second damped frequency, and (c) third damped frequency.


Figure 16: The first four mode shapes of the FG beam when =T 300 K and = ⋅θ 100 rad s−1: (a) first modal bending mode, (b) first modalstretching mode, (c) second modal bending mode, (d) second modal stretching mode, (e) third modal bending mode, (f) third modalstretching mode, (g) fourth modal bending mode, and (f) fourth modal stretching mode.


4 Discussion

The dynamic modeling and vibration analysis of therotating FGM beam with a hollow circular cross-sectionin the temperature field are investigated in this paper.The influences of the temperature difference betweenthe inner and outer surfaces of the beam, the ambienttemperature, and the functional gradient index on thedynamic responses of the rotating FGM beam under twolaws of motion are discussed. Finally, the free vibration ofthe rotating FGM beam is discussed. The main conclu-sions are as follows:• For the flexible FGM beam system with low rotatingspeed, the results obtained from the FOAC model arealmost the same as those obtained from the HOC model.But for the system with a high rotating speed, the

convergence performance of the numerical results fromthe HOC model is better than the FOAC model.

• Thermal loads will induce oscillation when the rotatingFGM beam is in the temperature field. For rotating FGMbeams with a radial temperature gradient, the influ-ences of the temperature on the transverse deformationand velocity of the beam are small, while those on theaxial deformation and velocity of the beam are large.

• The variation of the functional gradient index has asignificant influence on the dynamic responses of theFGM beam, so it is possible to select the proper func-tional gradient index for the structure to meet therequirements in engineering applications such as sup-pression of the unwanted vibration.

• Since only the modulus of elasticity and the coefficientof thermal expansion of the FGMbeamare assumed to be

Figure 17: Influence of temperature difference on the first three damped frequencies of the beam when = ⋅θ 200 rad s−1 and =T 300 Kcm :(a) first damped frequency, (b) second damped frequency, and (c) third damped frequency.


temperature-dependent, it is found that the temperaturehas little influence on thenatural frequencies of the beambased on the present model.

Acknowledgements: Grateful acknoledgement is made toProf. Dingguo Zhang who gave us considerable help bymeans of suggestion, comments and criticism. In addition,we deeply appreciate the contribution to this paper made invarious ways by Mr Yongbin Guo.

Funding information: This paper is funded by the NationalUndergraduate Training Program for Innovation andEntrepreneurship (Item number: 202010288129Z), andgrants from the National Natural Science Foundationof China (Project No. 12072159).

Author contributions: Yaolun Wang: writing – originaldraft, writing – review and editing, methodology, formalanalysis, project administration; Chaofan Yang: writing –original draft, formal analysis; Yongxin Zhang: visuali-zation, investigation; ShipengDong: Validation, writing –original draft; Liang Li: writing – review and editing,methodology, supervision.

Conflict of interest: No conflict of interest.

References

[1] Hui, D. Effects of shear loads on vibration and buckling ofantisymmetric cross-ply cylindrical panels. InternationalJournal of Non-Linear Mechanics, Vol. 23, No. 3, 1988,pp. 177–187.

[2] Aminbaghai, M., J. Murin, V. Kutiš, J. Hrabovský, and H. A.Mang. Torsional warping elastostatic analysis of FGM beamswith longitudinally varying material properties. EngineeringStructures, Vol. 200, No. C, 2019, id. 109694.

[3] Paul, A. and D. Das. Free vibration analysis of pre-stressedFGM Timoshenko beams under large transverse deflectionby a variational method. Engineering Science andTechnology, an International Journal, Vol. 19, No. 2, 2016, pp.1003–1017.

[4] Li, L., D. G. Zhang, and W. D. Zhu. Free vibration analysis of arotating hub-functionally graded material beam system withthe dynamic stiffening effect. Journal of Sound and Vibration,Vol. 333, No. 5, 2014, pp. 1526–1541.

[5] Li, L. and D. G. Zhang. Dynamic analysis of rotating axially FGtapered beams based on a new rigid–flexible coupled dynamicmodel using the B-spline method. Composite Structures,Vol. 124, 2015, pp. 357–367.

[6] Dong, S. P., L. Li, and D. G. Zhang. Vibration analysis ofrotating functionally graded tapered beams with hollow

circular cross-section. Aerospace Science and Technology, Vol.95, 2019, id. 105476.

[7] Oh, Y. and H. H. Yoo. Vibration analysis of rotating pre-twistedtapered blades made of functionally graded materials.International Journal of Mechanical Sciences, Vol. 119, 2016,pp. 68–79.

[8] Yang, Z. H. and D. He. Vibration and buckling of functionallygraded sandwich micro-plates based on a new size-dependentmodel. International Journal of Applied Mechanics, Vol. 11,No. 1, 2019, id. 34.

[9] Lee, S.-Y. and J.-G. Hwang. Finite element nonlinear transientmodelling of carbon nanotubes reinforced fiber/polymercomposite spherical shells with a cutout. NanotechnologyReviews, Vol. 8, No. 1, 2019, pp. 444–451.

[10] Parida, S. and S. C. Mohanty. Free vibration analysis of func-tionally graded skew plate in thermal environment usinghigher order theory. International Journal of AppliedMechanics, Vol. 10, No. 1, 2018, id. 1850007.

[11] Shen, H.-S., Y. Xiang, and Y. Fan. Large amplitude vibration ofdoubly curved FG-GRC laminated panels in thermal environ-ments. Nanotechnology Reviews, Vol. 8, No. 1, 2019,pp. 467–483.

[12] Zhou, D., J. S. Fang, H. W. Wang, and X. P. Zhang. Three-dimensional dynamics analysis of rotating functionally gra-dient beams based on Timoshenko beam theory. InternationalJournal of Applied Mechanics, Vol. 11, No. 4, 2019, id. 23.

[13] Shahmohammadi, A. M., M. Azhari, and M. M. Saadatpour.Free vibration analysis of sandwich FGM shells using isogeo-metric B-spline finite strip method. Steel and CompositeStructures, Vol. 34, No. 3, 2020, pp. 361–376.

[14] Attia, A., A. Tounsi, E. A. Bedia Adda, and S. R. Mahmoud. Freevibration analysis of functionally graded plates with tem-perature-dependent properties using various four variablerefined plate theories. Steel and Composite Structures, Vol. 18,No. 1, 2015, pp. 187–212.

[15] Bouchafa, A., B. M. Bouiadjra, M. Houari Ahmed Sid, andA. Tounsi. Thermal stresses and deflections of functionallygraded sandwich plates using a new refined hyperbolic sheardeformation theory. Steel and Composite Structures, Vol. 18,No. 6, 2015, pp. 1493–1515.

[16] Chen, C. S., F. H. Liu, and W. R. Chen. Vibration and stability ofinitially stressed sandwich plates with FGM face sheets inthermal environments. Steel and Composite Structures,Vol. 23, No. 3, 2017, pp. 251–261.

[17] Zaman, M., Z. Yan, and L. Y. Jiang. Thermal effect on thebending behavior of curved functionally graded piezoelectricactuators. International Journal of Applied Mechanics, Vol. 2,No. 4, 2010, pp. 787–805.

[18] Xing, Y. F., Z. K. Wang, and T. F. Xu. Closed-form analyticalsolutions for free vibration of rectangular functionally gradedthin plates in thermal environment. International Journal ofApplied Mechanics, Vol. 10, No. 3, 2018, id. 1850025.

[19] Bisheh, H., N. Wu, and D. Hui. Polarization effects on wavepropagation characteristics of piezoelectric coupled laminatedfiber-reinforced composite cylindrical shells. InternationalJournal of Mechanical Sciences, Vol. 161–162, No. C, 2019,id. 105028.

[20] Dehrouyeh-Semnani, A. M. On boundary conditions for ther-mally loaded FG beams. International Journal of EngineeringScience, Vol. 119, 2017, pp. 109–127.


[21] Jiang, B. K., J. Xu, and Y. H. Li. Flapwise vibration analysis of arotating composite beam under hygrothermal environment.Composite Structures, Vol. 117, 2014, pp. 201–211.

[22] Varmazyari, S. and H. Shokrollahi. Analytical solution forstrain gradient plasticity of rotating functionally graded thickcylinders. International Journal of Applied Mechanics, Vol. 12,No. 7, 2020, pp. 201–211.

[23] Kashkoli, M. D., K. N. Tahan, and M. Z. Nejad. Time-dependentthermomechanical creep behavior of FGM thick hollow cylind-rical shells under non-uniform internal pressure. InternationalJournal of Applied Mechanics, Vol. 9, No. 6, 2017, id. 26.

[24] Nam, V. H., N. T. Trung, N. T. Phuong, V. M. Duc, and V. T. Hung.Nonlinear torsional buckling of functionally graded carbonnanotube orthogonally reinforced composite cylindrical shellsin thermal environment. International Journal of AppliedMechanics, Vol. 12, No. 7, 2020, id. 27.

[25] Li, X., Y. H. Li, and Y. Qin. Free vibration characteristics of aspinning composite thin-walled beam under hygrothermalenvironment. International Journal of Mechanical Sciences,Vol. 119, 2016, pp. 253–265.

[26] Shabanlou, Gh, S. A. A. Hosseini, and M. Zamanian. Vibrationanalysis of FG spinning beam using higher-order sheardeformation beam theory in thermal environment. AppliedMathematical Modelling, Vol. 56, 2018, pp. 325–341.

[27] Ghadiri, M. and N. Shafiei. Vibration analysis of rotating func-tionally graded Timoshenko microbeam based on modifiedcouple stress theory under different temperature distributions.Acta Astronautica, Vol. 121, 2016, pp. 221–240.

[28] Asadi, H. and A. R. Beheshti. On the nonlinear dynamicresponses of FG-CNTRC beams exposed to aerothermal loadsusing third-order piston theory. Acta Mechanica, Vol. 229,No. 6, 2018, pp. 2413–2430.

[29] Azadi, M. Free and forced vibration analysis of FG beam con-sidering temperature dependency of material properties.Journal of Mechanical Science and Technology, Vol. 25, No. 1,2011, pp. 69–80.

[30] Azimi, M., S. S. Mirjavadi, N. Shafiei, and A. M. S. Hamouda.Thermo-mechanical vibration of rotating axially functionallygraded nonlocal Timoshenko beam. Applied Physics A,Vol. 123, No. 1, 2017, id. 104.

[31] Ghadiri, M. and A. Jafari. Thermo-mechanical analysis of FGnanobeam with attached tip mass: an exact solution.Applied Physics A, Vol. 122, No. 12, 2016, id. 1017.

[32] Bich, D. H., D. G. Ninh, B. H. Kien, and D. Hui. Nonlineardynamical analyses of eccentrically stiffened functionallygraded toroidal shell segments surrounded by elastic foun-dation in thermal environment. Composites Part B:Engineering, Vol. 95, 2016, pp. 355–373.

[33] Li, J., Y. C. Bao, and P. Hu. A dynamic stiffness method foranalysis of thermal effect on vibration and buckling of alaminated composite beam. Archive of Applied Mechanics,Vol. 87, No. 8, 2017, pp. 1295–1315.

[34] Shahrjerdi, A. and S. Yavari. Free vibration analysis of func-tionally graded graphene-reinforced nanocomposite beamswith temperature-dependent properties. Journal of theBrazilian Society of Mechanical Sciences and Engineering,Vol. 40, No. 1, 2018, id. 25.

[35] Khosravi, S., H. Arvin, and Y. Kiani. Vibration analysis ofrotating composite beams reinforced with carbon nanotubesin thermal environment. International Journal of MechanicalSciences, Vol. 164, No. C, 2019, id. 105187.

[36] Peng, X. L. and X. F. Li. Thermal stress in rotating functionallygraded hollow circular disks. Composite Structures, Vol. 92,No. 8, 2010, pp. 1896–1904.

[37] Dai, T. and H. L. Dai. Thermo-elastic analysis of a functionallygraded rotating hollow circular disk with variable thicknessand angular speed. Applied Mathematical Modelling, Vol. 40,No. 17–18, 2016, pp. 7689–7707.

[38] Huang, Y. X., T. S. Wang, Y. Zhao, and P. P. Wang. Effect ofaxially functionally graded material on whirling frequenciesand critical speeds of a spinning Timoshenko beam.Composite Structures, Vol. 192, 2018, pp. 355–367.


Research Article Yaolun Wang, Chaofan Yang, Yongxin Zhang ...

Documents