Vibration analysis of rotating functionally graded tapered beams with hollow circular cross-section

Aerospace Science and Technology 95 (2019) 105476

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Vibration analysis of rotating functionally graded tapered beams with hollow circular cross-section Shipeng Dong, Liang Li ∗ , Dingguo Zhang School of Science, Nanjing University of Science and Technology, Nanjing 210094, PR China

a r t i c l e

i n f o

Article history: Received 6 June 2019 Received in revised form 1 July 2019 Accepted 10 October 2019 Available online 14 October 2019 Keywords: Rotating FG beam Hollow circular cross-section Dynamic modeling Coupling Frequency veering Mode shape interaction

a b s t r a c t The dynamic modeling and free vibrations of rotating functionally graded (FG) tapered cantilever beams with hollow circular cross-section are studied in this paper. To capture the additional dynamic stiffening terms, the axial shrinkage of the beam caused by the transverse displacement is considered. The dynamic equations of the system governing stretching motion, flapwise bending motion, and chordwise bending motion are derived via employing assumed modes method and Lagrange’s equations. Based on the first order approximate coupling (FOAC) dynamic model, natural frequencies and mode shapes of the beam system are calculated by solving eigenvalue problem of the deduced dimensionless vibration equations. Influences of the angular speed, the hub radius, the slenderness ratio, the ratio of hollow radius to the root radius, the taper ratio, and the functional gradient index on natural frequencies are studied. Frequency veering and mode shape interaction are discussed when the bending-stretching mode coupling effect of the beam is considered. © 2019 Elsevier Masson SAS. All rights reserved.

1. Introduction Dynamics of flexible attachments of multibody systems with large overall motions are very complicated due to the coupling effect between the rigid motion and elastic deformation of the flexible body. A rotating hub-beam system can be used to do modeling and dynamic analysis of space manipulators or slender rotor blades, which are usually made of advanced composite materials. For such a rigid-flexible coupled system, there are also complicated coupling effect between different vibrations modes of the flexible beam moving in the three-dimensional space, and the vibration characteristics of the flexible beam structures should be well examined. Natural frequencies of the flapwise bending vibration are the same as those of the chordwise bending vibration for static cantilever beams. However, they are found to be quite different for rotating cantilever beams due to the change of angular speed. In recent years, there are many studies about flapwise bending vibration and chordwise bending vibration of rotating beams made of homogeneous materials. Yoo and Shin [1] studied the three-dimensional vibration of rotating cantilever beams, and found that the coupling effect of transverse bending vibration and axial stretching vibration should not be neglected when the angular speed is quite big. Shen et al. [2] used Absolute Nodal Coordinate Formulation to study the vibration induced by absorbed heat of the beam. Jafari-Talookolaei et al. [3] investigated the in-plane and out-of-plane vibrations of composite beams. Banerjee and Jackson [4] investigated vibration problems of rotating tapered beams, both the rigidity effect of centrifugal force and rotatory inertia were taken into account. Tang et al. [5] used the Wittrick–Williams algorithm to study the vibrations of rotating tapered beams, where they neglected the shear deformation and considered the rotary inertia of the beam. Xi et al. [6] studied transverse vibrations of a standing and hanging Rayleigh beam-column, found that rotary inertia could not be neglected when the system is in high frequency vibration. Kaya [7] studied the flapwise bending vibration characteristics of rotating tapered beams, and differential transform method (DTM) was employed to obtain the natural frequencies. Huang et al. [8] studied flapwise vibrations of the beam which rotates at high speed, what’s more, the coupling effect of lagged bending and longitudinal vibration was also studied. Zhao et al. [9] studied vibration problems of rotating three-dimensional beams, and found that axial deformation has scare impact on flapwise vibration while has large impact on chordwise vibration. Functionally graded materials (FGM) are well received in many engineering fields because they can integrate the excellent properties of two or more materials. Beam structures with large overall motions such as turbine blades, helicopter rotor blades, and manipulators can

*

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

https://doi.org/10.1016/j.ast.2019.105476 1270-9638/© 2019 Elsevier Masson SAS. All rights reserved.

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S. Dong et al. / Aerospace Science and Technology 95 (2019) 105476

be designed to be made of FGM so that they can possess excellent dynamic characteristics and work well under some extreme conditions. With the vigorous development of aerospace, robotics and other engineering fields, the dynamics of composite beam structures made of FGM has drawn more and more attention by researchers. Ramesh et al. [10] used Rayleigh-Ritz method to investigate transverse bending vibrations of rotating FG beams with lumped mass. Wattanasakulpong et al. [11] studied linear and nonlinear vibrations of FG beams which have porosities. Rajasekaran [12] studied the dynamic behavior of functionally graded beams, the material properties of the FG beam were hypothetical to vary in two directions, and the beam has single or multiple cracks. Su et al. [13] proposed a valid formulation to investigate the vibration of multiple stepped FG beams, the variational method and the first-order shear deformation theory was employed to establish the model. Shenas et al. [14] studied vibration characteristics of FG carbon nanotube reinforced composite beams, the effect of variations of pre-twisted angle was investigated. In order to study vibration behavior of rotating FG blades, Oh and Yoo [15] showed an advanced dynamic model to conduct research, they discussed the effect of the pretwist angle and found the proposed dynamic model is better than finite element model. Viet et al. [16] proposed a new analytical model to study the relevant problems of composite beams composed of FGM and shape memory alloy with a lump tip load. Shafiei et al. [17] studied the buckling behavior of FG tapered beams. Kumar et al. [18] first used Mori Tanaka Method to assess the material properties, then used DTM to investigated flapwise vibrations of rotating FG uniform beams. Pradhan et al. [19] studied free vibrations of FG beams, where different higher order shear deformation theories were taken into account, and discussed vibrations in different boundary conditions. Lee et al. [20] proposed a transfer matrix method which can be used to study the vibration problems of rotating FG beams, and the coupling effect of axial deformation and transverse displacement were also taken into account. Tang et al. [21] studied the buckling behavior of double-directionally porous beam. Fang et al. [22] investigated free vibrations of rotating FG microbeams, discussed effects of material properties and size-dependency. In addition, Kien [23] used the finite element method to investigate the response of axially FG tapered cantilever beams, where the shear deformation was considered. Li et al. [24] investigated vibration characteristics of rotating FG beams and discussed the influence of functional gradient index on the structure. Li and Zhang [25] also used the B-spline method to investigate the response and vibration problems of axially FG tapered beams. In Refs. [24] and [25], the coupling effect of longitudinal displacement and transverse displacement of FG beams was included. Since the FGM beam was modeled in a planar coordinate system, the flapwise bending vibration of the beam was ignored. Rajasekaran [26] used DTM in conjunction with minimum order differential quadrature element method to study the vibration problems of rotating FG beams in different boundary conditions. Mazanoglu et al. [27] first used Rayleigh-Ritz method to investigate vibrations of rotating axially FG tapered beams, both flapwise bending and chordwise bending vibrations were studied, and the effect of centrifugal rigidity was also considered. Huang et al. [28] studies free vibrations of rotating axially FG Timoshenko beams, they discussed the influence of gradient materials in different boundary conditions and found that the superiority of the FG beam cannot be observed under clamped-clamped, clamped-pinned, and pinned-pinned boundary conditions. Gao et al. [29] used the asymptotic development method (ADM) to study the vibration of axially FG beams, found the method is more convenient to analyze axially FG beams. Xie et al. [30] studied the dynamic response of an axially functionally graded (AFG) beam based on the classical beam theory and Timoshenko beam theory, where the coupling effect was taken into account. Chen et al. [31] first used isogeometric analysis method to investigate vibration characteristics of three-dimensional axially FG beams. In Refs. [32–34], axially FG beams are also studied. Li and Wang [35] first investigated transverse vibration of rotating tapered cantilever beams with hollow circular cross-section based on Rayleigh beam theory. They used DTM to obtain the bending natural frequencies of the beam. However, they neglected the axial deformation of the beam, which leads to a missing of the coupling effect between the transverse and axial vibration. To the authors’ knowledge, research on dynamics of rotating FG tapered beam with hollow circular cross-section has not been reported in literatures. In this paper, vibration characteristics of a rotating FG tapered beam with hollow circular cross-section attached to a rigid hub are investigated, where rotary inertia is included. The material characteristics of the FG beam are assumed to change along the thickness direction. The three sets of coupled dynamic equations of the system, governing stretching motion, flapwise bending motion, and chordwise bending motion, respectively, are derived via employing the assumed modes method and Lagrange’s equations in section 2. The natural frequencies of flapwise and chordwise bending vibration are shown in section 3.1 and section 3.2, respectively. With the bendingstretching (B-S) coupling effect considered, frequency veering and mode shape interaction of the system are observed and discussed in section 3.3. 2. Derivation of differential equations of motion 2.1. Physical model Fig. 1 shows the schematic of a flexible rotating FG tapered beam attached to a rigid hub. A body-fixed coordinate system OXYZ is employed as a reference for the rigid hub that rotates about the vertical Z axis. To describe the displacement field of any point P 0 on the beam in space, a floating coordinate system oxyz, with its origin o locates on the connection point between the beam and the rigid hub, is also defined. θ is the rotating angle of the hub. r 0 is a position vector of the origin o in the floating coordinate system oxyz with respect to the point O in the inertial coordinate system OXYZ. x p is the position vector of P 0 , which is on the axis of the beam before deformation. The point P 0 moves to point P after deformation, and u p is a displacement vector of the point P 0 . w 1 , w 2 and w 3 are the axial stretching displacement, the chordwise bending displacement, and flapwise bending displacement of the FGM beam, respectively. Fig. 2 shows the geometry of the FG tapered beam with hollow circular cross-section. The length of the beam is L, the density of the beam is ρ (n), the modulus of elasticity of the beam is E (n), the outer diameter of the beam varies linearly along the x axis, and the inner diameter is kept constant along the x axis. The root radius is R 1 , the end radius is R 2 , and the wall thickness e (x) changes with x. R (x) is the radius of the middle line of the wall thickness of any section, and d is the hollow diameter. The average radius R (x) and the wall thickness e (x) can be expressed as

R (x) = and

R1 2

 1 + (λ − 1)

x L

 +

d 4

(1)

S. Dong et al. / Aerospace Science and Technology 95 (2019) 105476

3

Fig. 1. Description of the configuration of the FG tapered beam with hollow circular cross-section attached to a rigid hub.

Fig. 2. Geometry of the FG tapered beam with hollow circular cross-section.

e (x) = 2R (x) − d,

(2)

respectively; λ = R 2 / R 1 is the taper ratio of cross-section. The effective material properties of the FG beam are assumed to vary along the thickness with a power law relation:

N  n 1 E (n) = ( E c − E m ) + + Em e



ρ (n) = (ρc − ρm )

n e

(3)

2

+

1

N + ρm

2

(4)

where n ∈ [−e /2, e /2], E c and E m are the elastic modulus of ceramics and metals, respectively; ρc and ρm are the density of ceramics and metals, respectively; N is the functional gradient index. When n = −e /2, E m , ρm are material parameters of inter surface; when n = e /2, E c , ρc are material parameters of outer surface. 2.2. Description of the deformation field According to Fig. 1, the position vector r p can be expressed as

r p = A θ (r 0 + x p + u p )

(5)

where

⎡

− sin θ cos θ

cos θ A θ = ⎣ sin θ 0



r0 = a



0

0 0

T

,

⎤

0 0⎦ 1



xp = x

0 0

T

u p = w 1 + w c y + w cz + w dy + w dz w 2 w 3

T

A θ is a coordinate transformation matrix; w cy and w cz are the second order coupling terms of the axial shrinkage caused by the transverse displacements associate with chordwise direction and flapwise direction, respectively; w dy and w dz are axial displacements caused by the rotation of the cross section associate with chordwise direction and flapwise direction, respectively. The velocity vector of r p is

˙ θ (r 0 + x p + u p ) + A θ u˙ p r˙ p = A

(6)

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S. Dong et al. / Aerospace Science and Technology 95 (2019) 105476

2.3. Kinetic energy of the system The kinetic energy of the hub-beam system can be expressed as follows:

T=

1 2

J oh θ˙ 2 +

1



ρ (n)˙r Tp r˙ p dV

2

(7)

V

˙2

1 2

1

T2 =

1 2

where T 1 = J oh θ is the kinetic energy of the hub; T 2 = expression is given as

ρ (n)˙r Tp r˙ p dV is the kinetic energy of the flexible beam, and its expansion

V

e L 2 2π

ρ (n)( R + n)˙r Tp r˙ p dα dndx

2 0 −e 0 2

1

=

L

2





˙ 22 + ( w ˙1+w ˙ cy + w ˙ cz )2 mc θ˙ 2 (a + x + w 1 + w c y + w cz )2 + w 22 + w

0

 

˙1+w ˙ cy + w ˙ cz ) + w ˙ 23 dx ˙ 2 (a + x + w 1 + w c y + w cz ) − w 2 ( w + 2θ˙ w +

1

L









ρc I m γN θ˙ 2 w 22,x + w˙ 22,x dx

2 0

+

1

L

ρc I m γN θ˙ 2 w 23,x + w˙ 23,x dx

2

(8)

0

where mc is effective mass per unit length of the FG beam, and can be written as follows: e 2 2π

mc =

ρ (n)( R + n)dα dn

− 2e

0

e 2

2 π R =

ρ (n)

− 2e

R +n R

dsdn

0

e

N   2 2 π R n 1 R +n (ρc − ρm ) = + + ρm dsdn e

− 2e

2

R

0

= ρc S α N

(9)

in which S = S (x) = 2π R (x)e (x) = π R 21 {[1 + (λ − 1)x/ L ]2 − β 2 } is the area of any cross-section.

αN =

1 + β1 N N +1

 + (1 − β1 )β3

I m = π R (x)3 e (x)

γN =

1 + β1 N N +1

β1 =

e (x) R (x)

ρm , ρc

N +2



+ 3(1 − β1 )β3 

+ (1 − β1 )β33 β3 =

=2−

1

1 N +4

−

−

1 N +2

2( N + 3)

4β 1 + (λ − 1) xL + β

,

(10)

2( N + 1)

−

3



1



1 2( N + 1)

+

3 4( N + 2)



1

+ β1 β32 + 3(1 − β1 )β32 4

−

1 8( N + 1)



1 N +3

−

1 N +2

+

1



(11)

4( N + 1) (12) (13) (14)

where β = d/(2R 1 ) is the ratio of hollow radius to the root radius. Since the last two terms in Eq. (8) are associated with the rotary inertia effect of the beam, the FG beam follows the Rayleigh beam theory. One can obtain the kinetic energy of an Euler–Bernoulli (EB) FG beam by setting ρ I m = 0 in Eqs. (8).

S. Dong et al. / Aerospace Science and Technology 95 (2019) 105476

5

The kinetic energy of the system can be rewritten as

1

T=

2

J oh θ˙ 2 +

1

L





ρc S (x)αN θ˙ 2 (a + x + w 1 + w c y + w cz )2 + w 22 + ( w˙ 1 + w˙ c y + w cz )2 + w˙ 22

2 0

  1 ˙1+w ˙ cy + w ˙ cz ) + w ˙ 23 dx + ˙ 2 (a + x + w 1 + w c y + w cz ) − w 2 ( w + 2θ˙ w

L

2





ρc I m γN θ˙ 2 w 22,x + w˙ 22,x dx

0

L

1

+

2





ρc I m γN θ˙ 2 w 23,x + w˙ 23,x dx

(15)

0

2.4. Potential energy of the system Neglecting torsional effect and deformation energy caused by shear deformation, the potential energy can be expressed as

1

U=



σx εx dV =

2

1



E (n)εx2 dV

2

V

(16)

V

where σx , εx are normal stress and normal strain, respectively. The normal strain of any point can be written as

εx =

∂ w1 ∂2 w2 ∂2 w3 −y −z 2 ∂x ∂x ∂ x2

(17)

According to Fig. 2 (b), one can get

dy ds dz ds

  = − sin α = − sin   s = cos α = cos

s

(18a)

R

(18b)

R

thus, the potential energy of the rotating FG tapered cantilever beam with hollow circular cross-section can be obtained as

1

U=

e L 2 2π

σx εx ( R + n)dα dndx

2 0

1

=

− 2e

0

e 2

L 2 π R

2 0 −e 2

+

1

1

0 −e 2

0 e 2

E (n)

2

L

2



L 2 π R 0 −e 2

=

e 2 2   2 2   L 2 2 π R dz(s) 1 R +n ∂ w1 R + n ∂ w2 − y (s) − n E (n) E (n) dsdndx + dsdndx ∂x R 2 ds R ∂ x2

∂2 w3 ∂ x2

0

2   2 d y (s) R +n − z (s) + n dsdndx ds

R

0

E˜ ( w 1,x )2 dx +

0

1

L

2

˜I z ( w 2,xx )2 dx + 1

L

2

0

˜I y ( w 3,xx )2 dx

(19)

0

where E˜ is compressional stiffness, ˜I y and ˜I z are bending stiffness, and can be written as follows:

e/2 2 π R

E˜ =

E (n)

−e/2 0

e/2 2 π R

˜I z =

R +n R

dsdn = E c S α N1

2   dz R +n E (n) − y (s) − n dsdn = E c I m γ N1 ds

−e/2 0

˜I y =

e/2 2 π R

−e/2 0

(20)

  E (n) − z(s) + n

dy ds

R

2

R +n R

dsdn = E c I m γ N1

(21)

(22)

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S. Dong et al. / Aerospace Science and Technology 95 (2019) 105476

in which

αN1 = γN1 =

β2 =

1 + β2 N N +1

 + (1 − β2 )β3

1 N +2



1 + β2 N

−

1 2( N + 1)

1

1

 (23)



1



1

1

1

+ β2 β32 + 3(1 − β2 )β32 + 3(1 − β2 )β3 − − + N +2 2( N + 1) 4 N +3 N +2 4( N + 1)   1 3 3 1 + (1 − β2 )β33 − + − N +4 2( N + 3) 4( N + 2) 8( N + 1)



N +1

Em

(24) (25)

Ec

2.5. Assumed modes discretization Employing the assumed modes method to approximate variables, w 1 , w 2 , and w 3 can be expressed as

⎧ ⎨ w 1 = Φ x (x)q1 (t ) w 2 = Φ y (x)q2 (t ) ⎩ w 3 = Φ z (x)q3 (t )

(26)

where Φ x (x) ∈ R 1× K , Φ y (x) ∈ R 1× K and Φ z (x) ∈ R 1× K are modal function vectors related to longitudinal vibration, transverse bending vibration and flapwise bending vibration of FGM beams, respectively; q 1 (t ) ∈ R K ×1 , q2 (t ) ∈ R K ×1 and q3 (t ) ∈ R K ×1 are generalized coordinate vectors related to longitudinal vibration, chordwise bending vibration and flapwise bending vibration of the FGM beam, respectively. The vectors in Eq. (26) are given as follows:



 Φ x (x) = φx1 (x), φx2 (x), . . . , φxK (x)

T q1 (t ) = q1,1 (t ), q1,2 (t ), . . . , q1, K (t ) 

 Φ y (x) = φ y1 (x), φ y2 (x), . . . , φ y K (x)

T q2 (t ) = q2,1 (t ), q2,2 (t ), . . . , q2, K (t ) 

 Φ z (x) = φz1 (x), φz2 (x), . . . , φzK (x)

T q3 (t ) = q3,1 (t ), q3,2 (t ), . . . , q3, K (t ) where

  π x , φxi (x) = sin (2i − 1) · · 2

L

i = 1, 2, . . . , K

φ yi (x) = (cos βi x − chβi x) + γi (sin βi x − shβi x),

i = 1, 2, . . . , K

φzi (x) = (cos βi x − chβi x) + γi (sin βi x − shβi x),

i = 1, 2, . . . , K

in which

β1 L = 1.875,

β2 L = 4.694

βi L = (i − 0.5) · π , i ≥ 3 cos βi L + chβi L γi = − sin βi L + shβi L The second order coupling terms are expressed as

1 w c y = − qT2 H 1 (x)q2 2 1 T w cz = − q3 H 2 (x)q3 2

(27) (28)

where H 1 (x) and H 2 (x) are the coupling shape functions, and can be expressed as:

x H 1 (x) =

Φ yT (ζ )Φ y (ζ )dζ

(29)

Φ zT (ζ )Φ z (ζ )dζ

(30)

0

x H 2 (x) = 0

Φ  is the derivative of Φ with respect to ζ .

S. Dong et al. / Aerospace Science and Technology 95 (2019) 105476

7

The discretized kinetic energy of the system is

T=

1 2

L

1

J oh θ˙ 2 +

2

  1 ρc S (x)αN θ˙ 2 (a + x)2 + qT1 Φ Tx Φ x q1 + qT2 H 1 q2 qT2 H 1 q2 4

0

1

+ qT3 H 2 q3 qT3 H 2 q3 + 2(a + x)Φ x q1 − (a + x)qT2 H 1 q2 − (a + x)qT3 H 2 q3 + qT2 Φ Ty Φ y q2 4   1 1 − qT1 Φ Tx qT2 H 1 q2 − qT1 Φ Tx qT3 H 2 q3 + qT2 H 1 q2 qT3 H 2 q3 + 2θ˙ −qT2 Φ Ty Φ x q˙ 1 − qT2 H 1 q2 Φ y q˙ 2 2 2  1 T T T T T T T T T + q2 Φ y q2 H 1 q˙ 2 + q2 Φ y q3 H 2 q3 + (a + x)Φ y q˙ 2 + q1 Φ x Φ y q˙ 2 − q3 H 2 q3 Φ y q˙ 2 2



q˙ T2 Φ Ty Φ y q˙ T2

+

+ q˙ T1 Φ Tx Φ x q˙ T1

+ q˙ T3 Φ Tz Φ z q˙ 3

+ q˙ T2 H 1 q2 qT2 H 1 q˙ 2 

− 2q˙ T1 Φ Tx qT2 H 1 q˙ 2 − 2q˙ T1 Φ Tx qT3 H 2 q˙ 3 + 2q˙ T2 H 1 q2 qT3 H 2 q˙ 3 1

+

L

2















+ q˙ T3 H 2 q3 qT3 H 2 q˙ 3

dx

ρc I m γN θ˙ 2 qT2 Φ yT Φ y q2 + q˙ T2 Φ yT Φ y q˙ 2 dx

0

1

+

L

2

ρc I m γN θ˙ 2 qT3 Φ zT Φ z q3 + q˙ T3 Φ zT Φ z q˙ 3 dx

(31)

0

The discretized potential energy of the beam is

U=

1

L

2

E c S (x)α N1 qT1 Φ xT Φ x q1 dx +

1

L

2

0

E c I m γ N1 qT2 Φ y T Φ y q2 dx +

0

1

L

2

E c I m γ N1 qT3 Φ z T Φ z q3 dx

(32)

0

2.6. Equations of motion



Let q = qT1 second kind:

d



∂T dt ∂ q˙

qT2

 −

T

qT3

be the generalized coordinate vector, then substituting Eqs. (31) and (32) into Lagrange’s equations of the

∂T ∂U =− + Fq ∂q ∂q

(33)

yields the governing equations of the system:

M q¨ = Q

(34)

The mass matrix M and force matrix Q can be written as follows

⎡

M 11 M = ⎣ M 21 M 31

M 12 M 22 M 32

⎤

M 13 M 23 ⎦ , M 33

⎤

⎡

Q Q =⎣ Q Q

1 2

⎦

(35)

3

where

M 11 = M 1

(36)

L M 12 = M T21 = −

ρc S (x)αN Φ Tx qT2 H 1 dx

(37)

L = − ρc S (x)α N Φ Tx qT3 H 2 dx

(38)

0

M 13 =

M T31

0

L

ρc S (x)αN H 1 q2 qT2 H 1 dx

M 22 = M 2 +

(39)

0

L M 23 =

M T32

ρc S (x)αN H 1 q2 qT3 H 2 dx

= 0

(40)

8

S. Dong et al. / Aerospace Science and Technology 95 (2019) 105476

L

ρc S (x)αN H 2 q3 qT3 H 2 dx

M 33 = M 3 + M 4 +

(41)

0

Q

1

= − K 1 q1 + θ˙ 2 ( S x + M 1 q1 ) + 2θ˙ M xy q˙ 2 L +





1





ρc S (x)αN Φ Tx q˙ T2 H 1 q˙ 2 + Φ Tx q˙ T3 H 2 q˙ 3 − θ˙ 2 Φ Tx qT2 H 1 q2 + Φ Tx qT3 H 2 q3 dx + θ¨ M xy q2 2

(42)

0

Q

2

T = − K 2 q2 + θ˙ 2 ( M 2 − C 1 )q2 − 2θ˙ M xy q˙ 1

+ θ˙ 2



L

ρc S (x)αN

1 2

H 1 q2 qT2 H 1 q2 − H 1 q2 Φ x q1 +

1 2

 H 1 q2 qT3 H 2 q3 dx

0

L

− 2θ˙





ρc S (x)αN H 1 q2 Φ y q˙ 2 − Φ Ty qT2 H 1 q˙ 2 − Φ Ty qT3 H 2 q3 dx 0

L





ρc S (x)αN H 1 q2 q˙ T2 H 1 q˙ 2 + H 1 q˙ 2 qT3 H 2 q˙ 3 + H 1 q2 q˙ T3 H 2 q˙ 3 dx

− 0

 − θ¨ S y + M Txy q1 +



L

ρc S (x)αN H 1 q2 Φ y q2 −

1 2

Φ Ty qT2 H 1 q2

−

1 2

  Φ Ty qT3 H 2 q3

dx

(43)

0

Q

˙2

˙2



L

3 = − K 3 q3 − θ C 2 q3 + M 4 q3 + θ

ρc S (x)αN

1 2

 H 2 q3 qT3 H 2 q3 − H 2 q2 Φ x q1 dx

0

+ θ˙

L





ρc S (x)αN qT2 Φ Ty H 2 q˙ 3 − 2H 2 q3 Φ y q˙ 2 − H 2 q˙ 3 Φ y q2 dx 0

L





ρc S (x)αN 2H 2 q3 q˙ T3 H 2 q˙ 3 + 2H 2 q˙ 3 qT2 H 1 q2 + 2H 2 q3 q˙ T2 H 1 q˙ 2 dx

− 0

− θ¨

L

ρc S (x)αN H 2 q3 Φ y q2 dx

(44)

0

in which the constant matrices can be written as follows

L

ρc S (x)αN (a + x)2 dx

(45)

ρc S (x)αN (a + x)Φ x dx

(46)

ρc S (x)αN (a + x)Φ y dx

(47)

J ob = 0

L Sx = 0

L Sy = 0

L

ρc S (x)αN Φ Tx Φ x dx

M1 =

(48)

0

L M2 =

L

ρc S (x)α 0

T N Φ y Φ y dx + 0

ρc I m γN Φ yT Φ y dx

(49)

S. Dong et al. / Aerospace Science and Technology 95 (2019) 105476

9

L

ρc S (x)αN Φ Tx Φ y dx

M xy =

(50)

0

L M3 =

ρc S (x)αN Φ Tz Φ z dx

(51)

ρc I m γN Φ zT Φ z dx

(52)

0

L M4 = 0

L C1 =

ρc S (x)αN (a + x) H 1 dx

(53)

ρc S (x)αN (a + x) H 2 dx

(54)

E c S (x)α N1 Φ xT Φ x dx

(55)

E c I m γ N1 Φ y T Φ y dx

(56)

E c I m γ N1 Φ z T Φ z dx

(57)

0

L C2 = 0

L K1 = 0

L K2 = 0

L K3 = 0

Eqs. (35)–(57) provide a high order coupled (HOC) dynamic model of the rotating FG tapered beam with hollow circular cross-section. Based on the HOC dynamic model, the complicated rigid-flexible coupling dynamic behaviors of the system can be studied. The underlined terms in each equation result from the second order coupling terms w cy (x, t ) and w cz (x, t ). If the double-underlined terms are neglected, the HOC dynamic model can be degraded into the first order approximate coupling (FOAC) dynamic model. With the linearized vibration equations based on the FOAC dynamic model, the vibration of the rotating FG tapered beam with hollow circular cross-section can be investigated. In the following study of this paper, we will focus on free vibration characteristics of the three dimensional rotating FG tapered beam with hollow circular cross-section. 3. Simulation results and discussions 3.1. Flapwise bending vibration analysis The rotating angular speed is assumed to be a constant, θ˙ = Ω , hence θ¨ = 0. Note that there is no mode coupling effect between the flapwise bending vibration and other vibration modes of the beam, the flapwise bending vibration equation is

M 33 q¨ 3 + K 33 q3 = 0

(58)

K 33 = K 3 + θ˙ 2 C 2 − θ˙ 2 M 4

(59)

where

The following dimensionless variables are introduced:



ς = t /T , where T =



ρc S 1 L 4 Ec I1

ζ = x/ L ,

κ3 = q3 / L ,

, S 1 = π R 21 , I 1 =

π R 41 4

,

δ = a/ L ,

˙ γ = T θ,

r=

I1 S 1 L2

(60)

γ is the dimensionless angular speed, δ is the ratios of hub radius to beam length and r is the

inverse of slenderness ratio. Thus, dimensionless expression of equation (58) can be expressed as

¯ κ¨ 3 + K¯ κ 3 = 0 M where

(61)

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S. Dong et al. / Aerospace Science and Technology 95 (2019) 105476

Table 1 The first three natural frequencies of non-rotating solid tapered EB beams with different taper ratios of cross-section;

ω1 λ=1 λ = 0.75 λ = 0.5 λ = 0.25

ω2

γ = 0, δ = 0, β = 0, N = 0, and r = 0. ω3

K =4

K = 10

Ref. [5]

K =4

K = 10

Ref. [5]

K =4

K = 10

Ref. [5]

3.5160 3.95672 4.6252 5.8296

3.5160 3.9567 4.6252 5.8231

3.5160 3.9567 4.6252 5.8231

22.0345 20.8099 19.5678 18.6818

22.0345 20.807 19.5477 18.4804

22.0345 20.807 19.5476 18.480

61.6972 55.3444 48.8923 43.8353

61.6972 55.3305 48.5793 41.3309

61.6972 55.3304 47.5789 41.321

Table 2 The first three flapwise bending natural frequencies of rotating solid tapered EB beams with different dimensionless angular speeds and taper ratios of cross-section; δ = 0, β = 0, N = 0, and r = 0.

γ 0

5

10

λ=1

ω ω1 ω2 ω3 ω1 ω2 ω3 ω1 ω2 ω3

¯ 33 = M

1

λ = 0.75

λ = 0.5

λ = 0.25

Present

Ref. [5]

Present

Ref. [5]

Present

Ref. [5]

Present

Ref. [5]

3.5160 22.0345 61.6972 6.4496 25.4461 65.2051 11.2025 33.6409 74.6507

3.5160 22.0345 61.6972 6.4495 25.4661 65.2050 11.2023 33.6404 74.6493

3.9567 20.8070 55.3305 6.7729 24.066 58.6364 11.4856 31.8896 67.5318

3.9567 20.807 55.3304 6.7729 24.0660 58.6364 11.4856 31.8895 67.5316

4.6252 19.5477 48.5792 7.2901 22.6360 51.6921 11.9415 30.0299 60.0403

4.6252 19.5476 48.5789 7.2901 22.6360 51.6918 11.9415 30.0299 60.0399

5.8231 18.4804 41.3309 8.2621 21.3842 44.2787 12.7912 28.3020 52.1119

5.8231 18.480 41.321 8.2620 21.384 44.269 12.791 28.301 52.100



1 + (λ − 1)ζ

2

 − β 2 α¯ N Φ Tz (ζ )Φ z (ζ )dζ

0

1

r2

+



1 + (λ − 1)ζ + β

2

3



1 + (λ − 1)ζ − β γ¯N Φ zT (ζ )Φ z (ζ )dζ

(62)

0

¯ 33 = K

1

1



1 + (λ − 1)ζ + β

2

3



1 + (λ − 1) − β γ¯N1 Φ z T (ζ )Φ z (ζ )dζ

0

1 +γ

2



1 + (λ − 1)ζ

2

 − β 2 α¯ N (δ + ζ ) H 2 (ζ )dζ

0

−

r2γ 2

1



1 + (λ − 1)ζ + β

2

3



1 + (λ − 1) − β γ¯N Φ zT (ζ )Φ z (ζ )dζ

(63)

0

To solve Eq. (60), one can define

κ 3 = e jωt Θ where j is an imaginary number, rewritten as

¯ 33 Θ = K¯ 33 Θ ω2 M

(64)

ω is the dimensionless natural frequency, and Θ is the corresponding eigenvector. Eq. (61) can be (65)

Table 1 shows the first three natural frequencies of non-rotating solid tapered EB beams with different taper ratios when δ = 0, β = 0, N = 0, and r = 0. By selecting the above values of the dimensionless parameters, the present model can be degraded into the model for traditional solid homogeneous beams. To investigate the accuracy of the present results, four and ten trial functions are used and compared with those from literature, respectively. As it can be seen from Table 1, for a uniform beam (λ = 1), four assumed modes could meet the requirement of accuracy, while for non-uniform beams, more assumed modes are needed as the taper of the flexible beam increases. And results calculated by ten assumed modes from the present model are much more consistent with those from Ref. [5], where the integral equation method was developed to analyze flapwise bending vibration of a rotating tapered Rayleigh beam. Thus, in order to ensure the simulation accuracy, ten assumed modes for each vibration mode are used in the following study. Furthermore, as the taper ratio decreases, the first frequency increases, while the second and the third decrease. Table 2 and Table 3 show the first three flapwise bending natural frequencies of rotating solid tapered beams with different dimensionless angular speeds and taper ratios of cross-section when δ = 0, β = 0, N = 0 based on the EB beam theory (r = 0) and Rayleigh beam theory (r = 1/50), respectively. As expected, because dynamic stiffening effect exists, the first three natural frequencies all increase with the dimensionless angular speed. Both results from the present EB beam model and Rayleigh beam model agree well with those from Ref. [5]. It can also be observed that, due to the consideration of the rotary inertia effect of the beam, the natural frequencies calculated by the Rayleigh beam model are slightly smaller than those by the EB beam model.

S. Dong et al. / Aerospace Science and Technology 95 (2019) 105476

11

Table 3 The first three flapwise bending natural frequencies of rotating solid tapered Rayleigh beams with different dimensionless angular speeds and taper ratios of cross-section; δ = 0, β = 0, N = 0, and r = 1/50.

γ 0

5

10

λ=1

ω ω1 ω2 ω3 ω1 ω2 ω3 ω1 ω2 ω3

λ = 0.75

λ = 0.5

λ = 0.25

Present

Ref. [5]

Present

Ref. [5]

Present

Ref. [5]

Present

Ref. [5]

3.5128 21.8929 60.7650 6.4407 25.2734 64.2071 11.1872 33.3945 73.4711

3.5128 21.8929 60.7649 6.4407 25.2734 64.2069 11.1870 33.3940 73.4711

3.9537 20. 7125 54.7233 6.7654 23.9509 57.9849 11.4725 31.7270 66.7601

3.9537 20.7125 54.7231 6.7654 23.9509 57.9848 11.4724 31.7269 66.7601

4.6221 19.4885 48.2208 7.2835 22.564 51.3061 11.9295 29.9292 59.5811

4.6221 19.4885 48.2204 7.2835 22.5640 51.3057 11.9295 29.9292 59.5805

5.8197 18.4455 41.1503 8.2554 21.3418 44.0827 12.7788 28.2424 51.8755

5.8196 18.4449 41.1401 8.2554 21.3411 44.0719 12.7788 28.2417 51.8625

Table 4 The first three flapwise natural frequencies of rotating FG beams for different gradient indices; δ = 0, β = 0.3, λ = 0.5, γ = 5, and r = 1/50. N

First

Second

Third

0.5 1 2 3 4 5 10 20 50 100 500 1000

6.8401 6.5632 6.3344 6.2288 6.1662 6.1243 6.0265 5.9674 5.9272 5.9127 5.9008 5.8992

19.4953 17.9339 16.6210 16.0141 15.6564 15.4183 14.8698 14.5454 14.3281 14.2510 14.1874 14.1793

42.6968 38.3699 34.6486 32.8997 31.8605 31.1650 29.5526 28.5925 27.9486 27.7173 27.5279 27.5038

Table 5 Flapwise natural frequencies with different ratios of hollow radius to the root radius and angular speeds; λ = 0.75, δ = 0, N = 1, and r = 1/50.

γ

ω

β = 0.1

β = 0.2

β = 0.3

β = 0.4

β = 0.5

β = 0.6

0

ω1 ω2 ω3 ω1 ω2 ω3 ω1 ω2 ω3

2.7469 14.3990 38.0838 6.0975 18.7700 42.6499 10.9774 27.9952 53.9400

2.7419 14.3314 37.8646 6.0950 18.7029 42.4301 10.9753 27.9195 53.7097

2.7890 14.4816 38.1727 6.1189 18.7845 42.6535 10.9938 27.9091 53.7693

2.8929 14.8362 38.9415 6.1728 19.0016 43.2556 11.0356 27.9445 54.0502

3.0745 15.4140 40.1502 6.2701 19.3622 44.2053 11.1109 28.0056 54.4874

3.4028 16.3264 41.8950 6.4549 19.9346 45.5549 11.2525 28.0747 55.0171

5

10

For a rotating FG tapered beam with hollow circular cross-section, in order to neglect the torsional effect, wall thickness should not be too small. The FG beam is assumed to be made of ceramics Si3 N4 and metal SUS304, the elastic modulus of Si3 N4 is 348.43 × 109 Pa, the density of Si3 N4 is 2370 kg/m3 , the elastic modulus of SUS304 is 201.04 × 109 Pa and the density of SUS304 is 8166 kg/m3 , thus, β1 = 8166/2370, β2 = 20104/34843. Table 4 shows flapwise bending natural frequencies of rotating FG tapered beams with different gradient indices when δ = 0, β = 0.3, λ = 0.5, γ = 5, and r = 1/50. As it is shown in Table 4, as the gradient index increases, the natural frequencies decrease, which means that the beam becomes more flexible. And growth of frequencies tend to be gentle when the gradient index is very large, the reason is that when N → +∞, the FG beam degenerate into homogeneous material beam. Table 5 shows flapwise bending natural frequencies of rotating FG tapered beams with different ratios of hollow radius to the root radius and angular speeds; λ = 0.75, δ = 0, N = 1, and r = 1/50. As it is shown in Table 5, the first three flapwise bending natural frequencies of the FG tapered beam decrease firstly, then increase with the increase of hollow diameter, the reason is that when β is smaller, variations of mass of the beam is slightly slower than stiffness of the beam, and as β becomes bigger, the beam is lighter and stiffer Fig. 3 shows flapwise bending natural frequency variations of rotating FG tapered beams with δ = 0, 0.5, and 1 when λ = 0.75, β = 0.3, N = 10, and r = 1/50. As it is shown in Fig. 3, when the ratio of hub radius to beam length increases, the first three natural frequencies will increase, and the differences of the results caused by the parameter δ increase with the dimensionless angular speed. Fig. 4 shows flapwise bending natural frequency variations of rotating FG beams with r = 1/10, 1/30, and 1/50 when λ = 0.5, δ = 0, β = 0.3, and N = 10. It is found that the second and third natural frequencies with r = 1/30 are close to those when r = 1/50, while they are far away from those when r = 1/10, the result shows that as the variation of the slenderness ratio, the effect of shear deformation should not be neglected, the reason is that the normal of the beam cross section is not coincident with the beam axis. Fig. 5 shows flapwise bending natural frequency variations of rotating FG beams with λ = 1, 0.75, and 0.5 when δ = 0, β = 0.4, N = 1, and r = 1/30. The impacts of the taper ratio on the first natural frequencies are consistent with the results obtained from Table 1. Fig. 6 shows flapwise bending natural frequency variations of rotating FG beams with N = 2, 5, and10 when δ = 0, β = 0.3, λ = 0.5, and r = 1/30. It can be seen from Fig. 6 that the differences of the results caused by gradient indices decrease with the increase of dimensionless angular speed. From Fig. 4, Fig. 5, and Fig. 6, one can see that the variations of the taper ratio, the slenderness ratio and functional gradient index have small impacts on

12

S. Dong et al. / Aerospace Science and Technology 95 (2019) 105476

Fig. 3. Variations of flapwise bending natural frequencies with δ = 0, 0.5, and 1; λ = 0.75, β = 0.3, N = 10, and r = 1/50.

Fig. 4. Variations of flapwise bending natural frequencies with r = 1/10, 1/30, and 1/50; λ = 0.5, δ = 0, β = 0.3, and N = 10.

Fig. 5. Variations of flapwise bending natural frequencies with λ = 1, 0.75, and 0.5; δ = 0, β = 0.4, N = 1, and r = 1/30.

the first natural frequencies, while they have obviously impacts on the second and third natural frequencies, and all frequencies increase with the dimensionless angular speed.

S. Dong et al. / Aerospace Science and Technology 95 (2019) 105476

13

Fig. 6. Variations of flapwise bending natural frequencies with N = 2, 5, and 10; δ = 0, β = 0.3, λ = 0.5, and r = 1/30.

3.2. Chordwise bending vibration analysis without the mode coupling effect Ignoring the mode coupling effect between the chordwise bending vibration and stretching vibration of the beam, the chordwise bending vibration equation of the system is

M 22 q¨ 2 + K 22 q2 = 0

(66)

K 22 = K 2 + θ˙ 2 C 1 − θ˙ 2 M 2

(67)

where

κ 2 = q2 / L, the dimensionless expression of equation (66) can be expressed as

Define an additional dimensionless variable

¯ 22 κ¨ 2 + K¯ 22 κ 2 = 0 M

(68)

where

¯ 22 = M

1



1 + (λ − 1)ζ

2

 − β 2 α¯ N Φ Ty (ζ )Φ y (ζ )dζ

0

1

r2

+



1 + (λ − 1)ζ + β

2

3



1 + (λ − 1)ζ − β γ¯N Φ yT (ζ )Φ y (ζ )dζ

(69)

0

¯ 22 = K

1

1



1 + (λ − 1)ζ + β

2

3



1 + (λ − 1) − β γ¯N1 Φ y T (ζ )Φ y (ζ )dζ

0

1 −γ

2



2

 − β 2 α¯ N Φ Ty (ζ )Φ y (ζ )dζ



2

 − β 2 α¯ N (δ + ζ ) H 1 (ζ )dζ

1 + (λ − 1)ζ

0

1 +γ2

1 + (λ − 1)ζ

0

−

r2γ 2

1

2



1 + (λ − 1)ζ + β

3



1 + (λ − 1) − β γ¯N Φ yT (ζ )Φ y (ζ )dζ

(70)

0

Table 6 shows comparisons of the first three natural frequencies of chordwise bending and flapwise bending of FG tapered beams with

δ = 0, β = 0, N = 0, and r = 0. One can know that the first three chordwise bending natural frequencies increase with the dimensionless angular speed, and the first natural frequencies of chordwise bending vibration increase much slower than those of flapwise bending vibration, while the second and the third frequencies of chordwise bending vibration increase slightly slower than those of flapwise bending vibration. As the beam becomes more tapered, the first natural frequencies increase, while the second and the third decrease. Table 7 shows the chordwise bending natural frequencies of rotating FG tapered beams with different gradient indices; λ = 0.5, δ = 0, β = 0.3, γ = 5, and r = 1/100. As it is shown in Table 7, when the gradient index increases, natural frequencies of the FG beam decrease,

14

S. Dong et al. / Aerospace Science and Technology 95 (2019) 105476

Table 6 Comparisons of the first three natural frequencies of chordwise bending and flapwise bending with λ = 1, 0.75, and0.5; δ = 0, β = 0, N = 0, and r = 0.

γ 0

5

10

ω ω1 ω2 ω3 ω1 ω2 ω3 ω1 ω2 ω3

λ=1

λ = 0.75

λ = 0 .5

flapwise

chordwise

flapwise

chordwise

flapwise

chordwise

3.5160 22.0345 61.6972 6.4496 25.4461 65.2051 11.2025 33.6409 74.6507

3.5160 22.0345 61.6972 4.0747 24.9520 65.0194 5.0638 32.1408 74.0623

3.9567 20.8070 55.3305 6.7729 24.0660 58.6364 11.4856 31.8896 67.5318

3.9567 20.8070 55.3305 4.5687 23.5409 58.4228 5.6493 30.2811 68.7873

4.6252 19.5477 48.5792 7.2901 22.6360 51.6921 11.9415 30.0299 60.0403

4.6252 19.5477 48.5792 5.3053 22.0769 51.4497 6.52681 28.3160 59.2016

Table 7 The first three chordwise bending natural frequencies with different gradient indices; λ = 0.5, δ = 0, β = 0.3, γ = 5, and r = 1/100. N

First

Second

Third

0.5 1 2 3 4 5 10 20 50 100 500 1000

4.6764 4.2605 3.8985 3.7243 3.6187 3.5469 3.3753 3.2689 3.1949 3.1681 3.1458 3.1429

18.8976 17.2734 15.8997 15.2616 14.8845 14.6330 14.0522 13.7075 13.4761 13.3939 13.3261 13.3175

42.6650 38.2774 34.5026 32.7276 31.6725 30.9662 29.3276 28.3513 27.6943 27.4606 27.2678 27.2433

Table 8 Comparisons of the first three natural frequencies of chordwise bending and flapwise bending with β = 0.2, 0.4, and0.6; λ = 0.75, δ = 0, N = 1, and r = 1/50.

γ 0

5

10

ω ω1 ω2 ω3 ω1 ω2 ω3 ω1 ω2 ω3

β = 0.2

β = 0.4

β = 0.6

flapwise

chordwise

flapwise

chordwise

flapwise

chordwise

2.7419 14.3314 37.8646 6.0950 18.7029 42.4301 10.9753 27.9195 53.7097

2.7419 14.3314 37.8646 3.4899 18.0278 42.1402 4.5351 26.0829 52.7894

2.8929 14.8362 38.9415 6.1728 19.0016 43.2556 11.0356 27.9445 54.0502

2.8929 14.8362 38.9415 3.6251 18.3388 42.9726 4.6821 26.1134 53.1401

3.4028 16.3264 41.8950 6.4549 19.9346 45.5549 11.2525 28.0747 55.0171

3.4028 16.3264 41.8950 4.0891 19.3059 45.2883 5.1790 26.2587 54.1300

it also confirms that the flexibility of the beam becomes larger when the gradient index increases, and frequencies increase gently when the gradient index is very large. Table 8 shows comparisons of the first three natural frequencies of chordwise bending and flapwise bending of FG tapered beams for different ratio of the hollow radius to the root radius when λ = 0.75, δ = 0, N = 1, and r = 1/50. It can be seen that the first three natural frequencies increase with the increase of the hollow diameter. Fig. 7 shows chordwise bending natural frequency variations of FG tapered beams with different angular speeds with λ = 0.5, 0.75, and 1 when δ = 0, β = 0.25, N = 5, and r = 1/100. It is found from Fig. 7 that the taper ratio variation of the beam has little impact on the first dimensionless natural frequencies, but has observable impact on the second and third frequencies. As mentioned above, the first frequencies increase while the second and third natural frequencies decrease with the taper ratio, respectively. Fig. 8 shows chordwise bending natural frequency variations of FG tapered beams with different angular speeds with N = 2, 5, and 10 when δ = 0, β = 0.25, λ = 0.5, and r = 1/100. One can see from Fig. 8 that the first three dimensionless natural frequencies decrease with the increase of the gradient index. Compared with the first natural frequency, the gradient index has more obviously impact on the second and the third natural frequencies. Fig. 9 shows chordwise bending natural frequency variations of FG tapered beams with different angular speeds with different ratios of hub radius to beam length δ = 0, 0.5, and 1 when λ = 0.5, β = 0.3, N = 5, and r = 1/100. As it is shown in Fig. 9, when the ratio of hub radius to beam length increases, the first three natural frequencies also increase, and the differences of the results caused by the parameter δ increase rapidly with the dimensionless angular speed. Fig. 10 shows chordwise bending natural frequency variations of FG tapered beams with different angular speeds with r = 1/100, 1/50, and 1/30 when λ = 0.75, δ = 0, β = 0.3, and N = 5. It is found from Fig. 10 that the impact of the slenderness ratio on chordwise bending natural frequencies of the beam is slight (see the third dimensionless natural frequency loci), the first two dimensionless natural frequencies have low sensitive to this parameter. The difference between the third frequencies of the beams with small slenderness ratios (r = 1/100 and 1/50) is quite small.

S. Dong et al. / Aerospace Science and Technology 95 (2019) 105476

15

Fig. 7. Variations of chordwise bending natural frequencies with λ = 0.5, 0.75, and 1; δ = 0, β = 0.25, N = 5, and r = 1/100.

Fig. 8. Variations of chordwise bending natural frequencies with N = 2, 5, and 10; δ = 0, β = 0.25, λ = 0.5, and r = 1/100.

Fig. 9. Variations of chordwise bending natural frequencies with δ = 0, 0.5, and 1; λ = 0.5, β = 0.3, N = 5, and r = 1/100.

3.3. Analysis of frequency veering and mode shape interaction with the mode coupling effect Following the basic definitions in the above two sections, the transverse and longitudinal vibration mode coupled free vibration equations of flexible beams can be written as:

16

S. Dong et al. / Aerospace Science and Technology 95 (2019) 105476

Fig. 10. Variations of chordwise bending natural frequencies with r = 1/100, 1/50, and 1/30; λ = 0.75, δ = 0, β = 0.3, and N = 5.



M 11 0

0 M 22



q¨ 1 q¨ 2



 +

0 G 12 G 21 0



q˙ 1 q˙ 2



 +

K 11 0 0 K 22



q1 q2



  =

0 0

(71)

where

K 11 = K 1 − θ˙ 2 M 1 G 12 =

(72)

= −2θ˙ M xy

− G T21

(73)

κ 1 = q1 / L, the dimensionless equation can be obtained as        ¯ 11 0 0 G¯ 12 κ˙ 1 K κ1 0 + ¯ + = ¯ 22 0 κ˙ 2 κ2 G 21 0 0 K

Define another dimensionless variable



¯ 11 M 0

0 ¯ 22 M



κ¨ 1 κ¨ 2





(74)

where T G¯ 12 = − G¯ 21 = −2γ

1



1 + (λ − 1)ζ

2

 − β 2 α¯ N Φ Tx (ζ )Φ y (ζ )dζ

(75)

0

¯ 11 = M

1



1 + (λ − 1)ζ

2

 − β 2 α¯ N Φ Tx (ζ )Φ x (ζ )dζ

(76)

0

¯ 11 = 1 K

1



1 + (λ − 1)ζ

r2

2



T

1



− β α¯ N1 Φ x (ζ )Φ x (ζ )dζ − γ 2

0

2



1 + (λ − 1)ζ

2

 − β 2 α¯ N Φ Tx (ζ )Φ x (ζ )dζ

(77)

0

In order to solve Equation (74), the state space is introduced, and the following expression is obtained:

¯ z˙ + K¯ z = 0 M where

⎡

¯ 11 M ⎢ 0 ¯ =⎢ M ⎣ 0 0

(78)

0 ¯ 22 M 0 0

⎤

0 0 0 0⎥ ⎥ I 0⎦ 0 I

⎡

0 ⎢ G¯ 21 ⎢ ¯ K =⎣ −I 0

G¯ 12 0 0 −I

¯ 11 K 0 0 0

⎤

0 ¯ 22 ⎥ K ⎥ 0 ⎦ 0

⎡

⎤

κ˙ 1 ⎢ κ˙ 2 ⎥ ⎥ ⎢ z=⎣ κ1 ⎦ κ2

(79)

The eigenvalue problem associated with equation (74) is

¯ + K¯ ) Z = 0 ( jω M

(80)

where ω is the dimensionless natural frequency, and Z is the corresponding complex eigenvector. One can see from Table 9 and Table 10 that, when the rotating angular speed increases, the bending-stretching coupling effect becomes obvious, and the effect of the coupling on the first frequency is larger than the second. As the inverse of slenderness ratio increases, the effect of coupling increases. Thus, in order to get more accurate results, it is necessary to take the coupling effect into account when the angular speed is high. For the beam with a relatively smaller slenderness ratio, it needs to consider the coupling effect although the rotating angular speed is small. Furthermore, when the inverse of slenderness ratio is increased to 1/30, the difference of the first natural frequency caused by the bending-stretching coupling increases first and then decreases.

S. Dong et al. / Aerospace Science and Technology 95 (2019) 105476

17

Table 9 The bending-stretching coupling effect on the first natural frequency of rotating FG tapered beams; λ = 0.75, β = 0.3, and N = 1.

δ 0

1

γ 10 20 30 50 10 20 30 50

r = 1/100

r = 1/50

r = 1/30

Without B-S

With B-S

Difference (%)

Without B-S

With B-S

Difference (%)

Without B-S

With B-S

Difference (%)

4.5927 6.2974 7.6940 10.2221 13.3129 25.6957 38.1533 63.1740

4.5249 5.9359 6.7595 7.3194 13.1149 24.1728 33.2033 43.2137

1.48 5.7 12.1 28.4 1.49 5.93 13 31.6

4.5808 6.2679 7.6422 10.1174 13.3033 25.6772 38.1259 63.1287

4.3205 5.0215 4.8033 2.6596 12.5184 20.0609 22.3584 14.3557

5.68 19.9 37.1 73.7 5.9 21.9 41.4 77.3

4.5526 6.1974 7.5183 9.8656 13.2804 25.6333 38.0609 63.0213

3.8928 3.5320 1.9741 3.8872 11.1878 13.4178 8.6546 20.3834

14.5 43 73.7 60.6 15.8 47.7 77.3 67.7

Table 10 The bending-stretching coupling effect on the second natural frequency of rotating FG tapered beams; λ = 0.75, β = 0.3, and N = 1.

δ 0

1

γ 10 20 30 50 10 20 30 50

r = 1/100

r = 1/50

r = 1/30

Without B-S

With B-S

Difference (%)

Without B-S

With B-S

Difference (%)

Without B-S

With B-S

Difference (%)

26.1750 45.4610 65.7577 107.079 37.8449 70.6364 104.206 171.966

26.1139 45.0596 64.4524 100.97 37.7534 69.9308 101.405 145.292

0.23 0.88 1.98 5.7 0.24 1.0 2.69 15.5

26.0734 45.2743 65.4808 106.617 37.6992 70.3441 103.759 171.205

25.8230 43.5803 59.8276 81.7661 37.2910 64.5971 80.4358 109.372

0.96 3.74 8.63 23.3 1.08 8.17 22.5 36.1

25.8356 44.8354 64.8282 105.524 37.2577 69.6551 102.701 169.393

25.0957 39.7952 49.5641 52.7556 35.4309 51.4478 65.5822 65.2191

2.86 11.2 23.5 50 4.9 26.1 36.1 61.5

Fig. 11. Chordwise bending natural frequency variations (a) without and (b) with the B-S coupling effect.

Fig. 11 shows chordwise bending natural frequency variations when λ = 0.75, β = 0.3, N = 1, r = 1/100, and δ = 0. The fifth frequency is expressed by S1, the ninth frequency is expressed by S2, because they correspond to the first and the second order frequencies of stretching vibration when γ = 0, respectively; the others are expressed by B1-B8. One can see from Fig. 11 (a) that if the B-S coupling effect is ignored, the natural frequencies of transverse bending of cantilever beams increase with angular speed, the longitudinal vibration frequencies decrease with the angular speed, and frequency veering will not occur between the two homologous modes; as shown in Fig. 11(b), if the B-S coupling effect is considered, frequencies of each mode emerge bigger differences with the increase of angular speed, frequency veering occurs between the homologous modes and also occurs between the nonhomologous modes, such as B4 and S1, B5 and B6, B6 and B7, and B7 and S2. According to the data from Table 9 and Table 10, the hub radius and slenderness ratio have large impact on frequency veering and mode shape interaction of the beam. In order to observe in detail, the sharpest two veering areas are enlarged in Fig. 12. As it is shown in Fig. 12, as the dimensionless angular speed varies, modes of B4, S1, B6, B7, and S2 also change. γ = 18.7 is a turn angular speed related to B4 and S1, the B4 mode transforms from transverse bending vibration dominance to longitudinal stretching vibration dominance, while the S1 mode is contrary; γ = 36 is a turn angular speed related B4 and B3, the B4 mode transforms from longitudinal stretching vibration dominance to transverse bending vibration dominance, while the B3 is contrary. Similarly, γ = 19 is a turn angular speed related to B7 and S2, the B7 mode transforms from transverse bending vibration dominance to longitudinal stretching vibration dominance, while the S2 mode is contrary; γ = 32 is a turn angular speed related to B7 and B6, the B7 mode transforms from longitudinal stretching vibration dominance to transverse bending vibration dominance, while the B6 is contrary. Fig. 13 shows the bending and stretching mode shape vibration along B3, B4, and S1; Fig. 14 shows the bending and stretching mode shape vibration along B5, B6, B7, and S2; In order to observe easily, all the amplitudes are normalized.

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Fig. 12. (a) The enlarged drawing of the first frequency veering region; (b) The enlarged drawing of the second frequency veering region.

Fig. 13. (a) chordwise bending mode shape variations in B4 when γ = 15, S1 when γ = 25, respectively; (b) chordwise bending mode shape variations in B3 when B4 when γ = 40, respectively; (c) stretching mode shape variations in S1 when γ = 15, B4 when γ = 25, B3 when γ = 40, respectively.

γ = 25,

S. Dong et al. / Aerospace Science and Technology 95 (2019) 105476

19

Fig. 14. (a) chordwise bending mode shape variations in B7 when γ = 15, S2 when γ = 25, respectively; (b) chordwise bending mode shape variations in B6 when γ = 25, B7 when γ = 40, respectively; (c) chordwise bending mode shape variations in B5 when γ = 40, B6 when γ = 50, respectively; (d) stretching mode shape variations in S2 when γ = 15, B7 when γ = 25, B6 when γ = 40, B5 when γ = 50, respectively.

4. Conclusions In this paper, vibration characteristics of rotating FG tapered beams with hollow circular cross-section are investigated based on the rigid-flexible coupled modeling method. Free vibrations of both flapwise bending and chordwise bending are studied by using the linearized dynamic model. Results show that variations of the hub radius, the slenderness ratio, the ratio of hollow radius to the root radius, the taper ratio, and the functional gradient index have different degrees of impacts on natural frequencies of the rotating composite beams. Due to the included dynamic stiffening effect, influence of these parameter factors on beam natural frequencies of each mode increase with the rotating angular speed of the hub. For flapwise bending vibration, the longitudinal vibration has little impact on natural frequencies of the beam, while it has considerable impact on chordwise bending vibration. The B-S coupling effect cannot be neglected when the angular speed is large. With such mode coupling effect, frequency veering and mode shape interaction can be observed. Research in this paper are helpful to understand the dynamic characteristics of hollow circular cross-section beams which are frequently used in practical engineering fields and may provide theoretical guidance for vibration prediction of such flexible structures. Declaration of competing interest There is no competing interest. Acknowledgements This research is funded by the grants from the National Natural Science Foundation of China (Project Nos. 11502113 and 11772158), the grant from the Natural Science Foundation of Jiangsu Province (Project No. BK20170820) and the Fundamental Research Funds for the Central Universities (Project No. 30917011103).

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