Journal of Mechanical Science and Technology 25 (5) (2011) 1235~1246 www.springerlink.com/content/1738-494x

DOI 10.1007/s12206-011-0313-3

C0-continuous isoparametric Timoshenko beam element for rotating† Yong-Woo Kim1,* and Jaeho Jeong2 1 Department of Mechanical Engineering, Sunchon National University, Chonnam, 540-742, Korea Department of Mechanical Engineering, Graduate School, Sunchon National University, Chonnam 540-742, Korea

2

(Manuscript Received March 5, 2010; Revised December 27, 2010; Accepted February 7, 2011) ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

Abstract This paper presents a C0-continuous isoparametric finite element for free vibration analysis of a rotating, tapered Timoshenko beam mounted on the periphery of a rotating rigid hub, at a setting angle with the plane of rotation. The finite element has three nodes and two degrees of freedom per node and employs modified shape functions for rotational displacement associated with the shear strain energy to avoid shear locking. To obtain a finite element equation of the generalized eigenvalue problem, Hamilton's principle is applied to kinetic and potential energy expressions of a rotating Timoshenko beam with non-zero setting angle. The numerical solutions for various situations including variations of rotational speed, taper ratio, slenderness ratio, hub radius and setting angle are compared with other numerical results available in the literature whenever possible. The results show that the new 3-noded isoparametric element yields accurate results when compared to other numerical ones. Keywords: Rotating Timoshenko beam; Isoparametric element; Shear locking; Finite element ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

1. Introduction Rotating beams are often used to model blades of wind turbines, steam and gas turbines, helicopter rotors and aircraft propellers. The accurate prediction of rotating frequencies using finite element methods is therefore an active area of research. The free vibration of rotating Euler-Bernoulli beams [1-8] and/or Timoshenko beams [8-20] has been studied by a number of investigators using different methods. Some researchers have used the Frobenius method [1-3, 9] or power series [10, 11] for the solution of differential equations. Some have employed the dynamic stiffness method [4, 12], dynamic discretization technique [8], extended Galerkin’s method [13], Rayleigh’s principle [14] or finite element methods [5-7, 1520] to solve for the natural frequencies of rotating beams. This paper presents C0-continuous finite element for rotating Timoshenko beam. So, formulating methods for rotating Timoshenko beams are surveyed briefly as follows: Wang et al. [13] presented an extended Galerkin’s method for rotating cantilever beam vibration using Legendre polynomials and showed that their method is effective in solving rotating beam problems. Downs [8] calculated the natural frequencies of linearly tapered non-rotating beams, based on both EulerBernoulli and Timoshenko beam theories, by using a dynamic †

This paper was recommended for publication in revised form by Associate Editor Seokhyun Kim Corresponding author. Tel.: +82 61 750 3537, Fax.: +82 61 750 3530 E-mail address: [email protected] © KSME & Springer 2011 *

discretization technique. Putter and Manor [16] calculated natural frequencies and mode shapes of a radial beam mounted on a rotating disc at a 90˚ setting angle by means of the finite element method. They employed the fifth degree polynomial as displacement function and considered the effects of shearing force, rotary inertia and varying centrifugal force. Du et al. [10] presented a convergent power series solution for the exact natural frequencies and modal shape of rotating uniform Timoshenko beams. Yokoyama [17] developed a finite element procedure for determining the free vibration characteristics of rotating uniform Timoshenko beams. The effects of hub radius, setting angle, shear deformation and rotary inertia on the natural frequencies of the rotating beams have been examined. Mulmule et al. [18] reported the flexural vibrations of tapered rotating Timoshenko beam including the effects of setting angle and hub radius. Bazoune and Khulief [19] presented a finite beam element for the dynamic and modal analysis of rotating tapered Timoshenko beam. They extended their work of Ref. [19] to include the effect of hub radius and in-plane vibration [20]. Most investigators [15-20] have employed C1-continuous displacement functions for finite element analysis of the flexural vibrations of rotating Timoshenko beams. In this paper, by using independent polynomial displacement fields for deflection and section rotation, a simple quadratic 3-noded isoparametric C0-continuous beam element having two degrees of freedom per node for tapered rotating Timoshenko beam is suggested. Since a quadratic beam ele-

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Fig. 2. A beam linearly tapered in two planes.

Fig. 1. Configuration of a tapered beam mounted on a rotating hub and coordinate systems.

ment with independent polynomial displacement fields of the same degree introduces numerical error called ‘shear locking’, field-consistent displacement functions for shear deformation are employed to avoid the locking phenomenon [22, 23]. To develop the finite element procedure, the strain energy and kinetic energy expressions defining the free vibration characteristics of a rotating tapered Timoshenko beam are used. The kinetic energy expression includes the energy due to the centrifugal force effects, which is usually treated as the potential energy due to the centrifugal force [16-20]. The kinetic energy expression also gives an extra term which is neglected by some researchers. It can be significant for higher rotational speed [12, 21]. To show this fact, both of the natural frequencies with and without the extra term are calculated to compare with other numerical solutions. Various numerical tests are performed by varying setting angle, hub radius, rotational speed, taper ratio and slenderness ratio of Timoshenko beam. The numerical results show higher accuracy compared with other numerical solutions.

cross-sectional area of the beam. The beam undergoes vibration in a plane fixed in a local system rotating with the beam. For ψ = 0o the motion of the beam is perpendicular to the plane of rotation and for ψ = 90o the motion of the beam is purely in the plane of rotation. For a beam linearly tapered in two planes ( x - y plane and x - z plane) as shown in Fig. 2, the taper is assumed to be such that 1⎞ ⎛ 1⎞ ⎛ A( x) = A0 ⎜1 − α y ⎟ ⎜1 − α z ⎟ L⎠ ⎝ L⎠ ⎝

(1)

3

x⎞ ⎛ x⎞ ⎛ I ( x) = I 0 ⎜ 1 − α y ⎟ ⎜ 1 − α z ⎟ L L ⎝ ⎠ ⎝ ⎠

(2)

where A( x) and I ( x) are cross-sectional area and second moment of area about the z axis at a distance x away from x = 0 , respectively, and A0 and I 0 are the area and the second moment of area at x = 0 , respectively. α y and α z are the taper ratios in the x - y plane and the x - z plane, respectively. They are defined as α y = L Ly and α z = L Lz , where Ly and Lz are the untruncated lengths of the beam in the x - y plane and the x - z plane, respectively. The taper ratios must be such that 0 ≤ α y < 1 and 0 ≤ α z < 1 because otherwise the beam tapers to zero between its ends.

2. Energy expressions and equations of motion Fig. 1 shows a typical rotating tapered cantilever beam model of length L with its left-hand end at a distance R from the axis of rotation. The beam is assumed to be rotating at a constant angular speed Ω . In this figure, the ( XYZ ) axes represent a global orthogonal coordinate system with the origin at the center of mass of the hub such that the Y -axis corresponds to the spin axis. The coordinate system ( xyz ) represents a body coordinate system that is rigidly attached to the hub. The hub is considered as rigid. The coordinate system ( xyz ) is obtained by shifting the origin of the ( XYZ ) axes by R at first and then by rotating it about the X -axis by an angle ψ called the setting angle. The X - and x -axes are collinear and coincident with the undeformed beam centerline, while the y - and z -axes lie along the principal axes of the

2.1 Strain energy expression Using the coordinate system and notation of Fig. 1, the centrifugal force at a distance x is L

T ( x) = ∫ ρ A( x)Ω 2 ( R + x)dx , x

(3)

where ρ is mass density. The strain ε 0 ( x) due to the action of the centrifugal force T ( x) is given by

ε 0 ( x) =

du0 ( x) T ( x) = , dx EA( x)

(4)

where u0 ( x) is the axial displacement due to the centrifugal

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2.2 Kinetic energy expression The velocity at a point P in Fig. 3 can be obtained as follows. G G G G K VP = Ω × (r + u ) + u ,

(8)

G G where Ω is the angular velocity vector of the hub; r is a position vector from the center of hub to the location of P ; G G u is the elastic deformation vector; and u is the time deG rivative of u . When the rigid hub rotates at a constant angular speed Ω , these vectors are expressed, using the body coordinates, as follows: G G G Ω = Ω cosψ j − Ω sinψ k G G G r = ( R + x) i + η j G G G u = (u0 − ηφ ) i + w j G G G u = −ηφ i + w j

Fig. 3. Configurations of a beam before and after deformation; PQ is displaced to P′Q′ after bending deformation.

force and E is Young’s modulus. By introducing a flexural displacement w( x, t ) of the beam neutral axis in the y -direction for an element of length dx at a distance x from the origin and by following Timoshenko’s beam theory, the shearing strain ( γ ) induced under flexural displacement ( w ) and section rotation ( φ ) in the element is given by ∂w −φ . ∂x

γ=

(5)

As a result of the combined axial and flexural deformation of the element dx , the following strain energy expression is provided by the authors of Refs. [11], [12] and [21]: 1 L 1 L EI ( x )(φ ′) 2 dx + ∫ T ( x)( w′) 2 dx ∫ 0 2 2 0 L 1 + ∫ κ GA( x )( w′ − φ ) 2 dx + C1 , 2 0

(6)

where G is shear modulus, κ is shear correction factor, and C1 is constant and is given by

(10) (11) (12)

G G G where i , j and k are unit vectors in x -, y - and z directions, respectively, and an over dot denotes differentiaEqs. (9)-(12) into Eq. tion with respect to time t . By inserting G (8), the following velocity vector VP is obtained: G G G G VP = Vx i + Vy j + Vz k

(13)

where Vx = −ηφ + Ω( w + η )sinψ Vy = w − Ω sinψ ⋅ ( R + x + u0 − ηφ )

(14) (15)

Vz = −Ω cosψ ⋅ ( R + x + u0 − ηφ ) .

(16)

Thus the kinetic energy of the rotating Timoshenko beam is expressed as [11, 12, 21] 1 L ρ (Vx 2 + Vy 2 + Vz 2 ) dAdx 2 ∫0 ∫A 1 L 1 L = ∫ ρ I ( x) Ω 2φ 2 + (φ) 2 dx + ∫ ρ A( x) w 2 dx 0 2 2 0

K=

{

+

U=

(9)

1 2 2 Ω sin ψ 2

}

∫

L

0

ρA( x) w2 dx

L

 −Ω sinψ ∫ ρ A( x)( R + x + u0 ) wdx 0

L −Ω sinψ ∫ ρ I ( x)φdx + C2 + C3 ,

(17)

0

where C2 and C3 are constants and they are given by

L

C1 = ∫

L

{∫ ρ A( x)Ω 2 ( R + x) dx}2 x

2 EA( x)

0

dx .

(7)

In Eq. (6), ( • ) means the partial differentiation of the variable • with respect to x . '

2

L x ⎧ 1 T ( x) ⎫ C2 = Ω 2 ∫ ρ A( x) ⎨ R + x + ∫ dx ⎬ dx 0 0 2 EA ( x) ⎭ ⎩ L

C3 = Ω 2 sin 2 ψ ∫ ρ I ( x)dx . 0

(18) (19)

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2.3 Equations of motion The equations of motion are obtained by applying Hamilton’s principle:

δ∫

t2

t1

Ldt = 0 ,

(20)

where the Lagrangian is given by L = U − K . Using variational principles, the differential equations of motion for a rotating, non-uniform Timoshenko beam are derived as follows:

ρ I ( x) ⋅ φ − ρ I ( x) ⋅ Ω 2φ ′ − { EI ( x) ⋅ φ ′} − κ GA( x) ⋅ ( w′ − φ ) = 0 ,

(21)

and ′

 − {T ( x ) ⋅ w′} − {κ GA( x ) ⋅ ( w′ − φ )} ρ A( x) ⋅ w − ρ A( x) ⋅ Ω 2 sin 2 ψ ⋅ w = 0 .

(22)

The term ρ I ( x) ⋅ Ω 2φ , which appears in Eq. (21), can be important when the constant rotational speed ( Ω ) is high. The term is usually omitted, whereas it is considered by the authors of Refs. [11, 12, 21]. If the rotary inertia terms including the term ρ I ( x) ⋅ Ω 2φ are neglected, Eq. (21) is reduced to ′

κ GA( x) ( w′ − φ ) = − {EI ( x) ⋅ φ ′} .

(23)

Inserting the above equation into Eq. (22) gives ′

{

′

 − (T ( x) w′ ) + ( EI ( x) ⋅ φ ′ ) ρ A( x) w

′

}

− ρ A( x)Ω 2 sin 2 ψ ⋅ w = 0 .

(24)

Additionally, if the beam is so slender that shear deformation can be neglected, i.e., γ = w′ − φ = 0 or φ = w′ , then Eq. (23) is automatically satisfied and Eq. (24) is reduced to

Fig. 4. Numbering system of element number, node number and local node number.

3. Finite element formulation 3.1 Elemental equation Fig. 4 shows the numbering system of element number, node number and local node number for 3-noded element. The element numbers i = 1, 2, 3, ", N are written in parentheses and they are allocated in ascending order from the fixed end to the free end as shown in Fig. 4. The three node numbers of the element (i ) are 2i − 1, 2i, 2i + 1 . The direction of x -axis is coincident with the direction of natural coordinate ξ (−1 ≤ ξ ≤ 1) , and the local node numbers are allocated from left ( ξ = −1 ) to right ( ξ = 1 ) as shown in Fig. 4. The relationship between the body coordinate x ( i ) and the natural coordinate ξ in (i ) th element can be expressed, by using isoparametric 3-noded element, as 3

x ( i ) = ∑ N k (ξ ) xk( i )

(26)

k =1

′

″

 − (T ( x) w′ ) + { EI ( x) ⋅ w′′} ρ A( x) w − ρ A( x)Ω sin ψ ⋅ w = 0 , 2

2

(25)

which is the equation of motion for a rotating, non-uniform Euler-Bernoulli beam. The above discussion shows that the equation of motion for Timoshenko beam becomes that of Euler-Bernoulli beam by neglecting the terms of rotary inertia and shear in the equations for the Timoshenko beam. Thus, the solutions of the Timoshenko beam merge to the solutions of the fourth order differential Eq. (25) as the slenderness ratio increases.

where xk( i ) denotes the x -coordinate of the k th local node in element (i ) and the shape functions N k (k = 1, 2,3) are given by N1 (ξ ) = 0.5ξ (ξ − 1) ,

N 2 (ξ ) = 1 − ξ 2 ,

(27)

N 3 (ξ ) = 0.5ξ (ξ + 1) .

The area and the second moment of area in an element (i ) are expressed, using Eqs. (1), (2) and (26), as

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Y.-W. Kim and J. Jeong / Journal of Mechanical Science and Technology 25 (5) (2011) 1235~1246 3 ⎛ ⎞ A( i ) (ξ ) = A0 ⎜1 − α y ∑ N k xk( i ) L ⎟ × k =1 ⎝ ⎠ 3 ⎛ ⎞ (i ) ⎜1 − α z ∑ N k xk L ⎟ k =1 ⎝ ⎠

x3( i )

−Ω sinψ ∫ ( i ) ρ A( x) w ⋅ ( R + x + u0 ) dx x1

(i)

(28)

3

3 ⎛ ⎞ I ( i ) (ξ ) = I 0 ⎜1 − α y ∑ N k xk( i ) L ⎟ × k =1 ⎝ ⎠

3 ⎛ ⎞ (i ) ⎜1 − α z ∑ N k xk L ⎟ . k =1 ⎝ ⎠

x3 −Ω sinψ ∫ ( i ) ρ I ( x)φdx + C2(i ) + C3( i ) .

(36)

x1

The constants in Eqs. (35) and (36) are: (29)

The centrifugal force induced by an infinitesimal length ds , at a distance s away from the hub surface, shown in Fig. 4, is given by

L

x3( i )

{∫ ρ A( x)Ω 2 ( R + x)dx}2

(i ) 1

= ∫ (i )

(i ) 2

x3( i ) x ⎧ 1 T ( x) ⎫ = Ω2 ∫ ( i ) ρ A( x) ⎨ R + x + ∫ dx ⎬ dx , 0 x1 EA 2 ( x) ⎭ ⎩

C

x

2 EA( x)

x1

dx ,

(37) 2

C

x3( i )

C3( i ) = Ω 2 sin 2 ψ ∫ ( i ) ρ I ( x) dx .

(39)

x1

dT = ρ A( s )Ω 2 ( R + s )ds ,

(30)

where s -axis is collinear and coincident with the x -axis. Thus, the centrifugal force acting at a distance x is given by L

T ( x) = ∫ ρ A( s )Ω 2 ( R + s )ds .

(31)

x

By using the above equation, the centrifugal force acting on the section at ξ in the element (i ) , is expressed as T ( i ) (ξ ) = Ω 2T( i ) ,

(32)

where

ξ

−1

3

ρ A( i ) (ξ ) ( R + ∑ N k xk( i ) )J ( i ) dξ .

(33)

k =1

In Eq. (33), N is the element number located at the free end as shown in Fig. 4 and J ( i ) is the Jacobian determinant, 3

J (i ) =

dx = dξ

d (∑ N k xk( i ) ) k =1

dξ

3

= ∑ N k ,ξ xk( i ) .

Applying Hamilton’s principle to the Lagrangian L(i ) yields the following equation: t2

x3( i )

t1

x1( i )

∫ [ −∫

ρ I ( x)(δφ )φdx + ∫

(34)

k =1

x1( i )

x3( i )

x3( i )

x1

x1

x3( i )

(i) 3

− ∫ ( i ) T ( x)(δ w, x ) w, x dx − ∫ ( i ) κ GA( x){δ ( w, x −φ )}( w, x −φ )dx x1

x1

x3( i )

+ ∫ ( i ) ρ A( x)Ω 2 (sin 2 ψ ) w(δ w)dx ] dt = 0 . x1

w( x, t ) = [ N w (ξ )]{u (i ) (t )} ,

φ ( x , t ) = [ N φ (ξ )]{ u

K

1 x3( i ) 1 x3( i ) EI ( x)(φ ′) 2 dx + ∫ ( i ) T ( x)( w′) 2 dx (i ) ∫ 2 x1 2 x1 1 x3( i ) (35) + ∫ ( i ) κ GA( x)( w, x −φ ) 2 dx + C1( i ) , 2 x1 1 x3( i ) 1 x3( i ) = ∫ ( i ) ρ I ( x) Ω 2φ 2 + (φ) 2 dx + ∫ ( i ) ρ A( x ) w 2 dx x 2 1 2 x1 (i ) x3 1 + Ω 2 sin 2 ψ ∫ ( i ) ρ A( x) w2 dx xi 2

{

}

(40)

(i )

(41)

( t )}

,

(42)

where [ N w ] = [ N1 0 N 2 0 N 3 0] ,

(43)

[ Nφ ] = [0 N1 0 N 2 0 N 3 ] , {u (t )} = [ w

If the Lagrangian of element (i ) is denoted by L( i ) = U (i ) − K ( i ) , the strain energy U ( i ) and the kinetic energy K ( i ) of the element (i ) are expressed as

(i )

ρ I ( x)Ω 2 (δφ )dx

 − ∫ ( i ) EI ( x)(δφ , x )φ , x dx − ∫ ( i ) ρ A( x)(δ w) wdx

(i )

U (i ) =

x3( i )

The transverse displacement w( x, t ) and the bending rotation φ ( x, t ) at an arbitrary point within the element (i ) are expressed in terms of the element nodal displacement vector {u ( i ) (t )} as

N 3 ⎧ +1 ⎫ T( i ) = ∑ ⎨ ∫ ρ A( j ) (ξ ) ( R + ∑ N k xk( j ) ) J ( j ) dξ ⎬J ( i ) dξ −1 j =i ⎩ k =1 ⎭

−∫

(38)

T

(i ) 1

φ

(i ) 1

(i ) 2

w

φ

(i ) 2

(44) (i ) 3

w

φ ]. (i ) 3

(45)

In Eq. (45), wk( i ) and φk( i ) ( k = 1, 2,3 ) are nodal displacements of the element (i ) . To avoid shear locking, modified shape functions N k (k = 1, 2,3) for the rotational displacement associated with the shear strain energy are employed: w, x −φ = (

d[ Nw ] − [ Nφ ] ){u ( i ) } , dx

(46)

where [ Nφ ] = [0 N1 0 N 2 0 N 3 ] .

(47)

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Y.-W. Kim and J. Jeong / Journal of Mechanical Science and Technology 25 (5) (2011) 1235~1246

{u} = {u }eiωt ,

The modified shape functions are given by [22] N1 (ξ ) = 0.5(1 3 − ξ ) , N 2 (ξ ) = 2 3 ,

(48)

N 3 (ξ ) = 0.5(1 3 + ξ ) .

( [M

(i ) t

one obtains the generalized eigenvalue problem for rotating Timoshenko beam:

[ (

By using the displacements in Eqs. (41), (42) and (46), Eq. (40) can be written as

−ω

[ K b ] + [ K s ] + Ω 2 ( [ K c ] − sin 2 ψ [ M t ] − [ M r ] ) 2

(

[M t ] + [M r ]

)

) ]{u } = {0} ,

(59)

where {u } is the eigenvector and ω is the natural frequencies of free vibration.

] + [ M r( i ) ] ){u( i ) } + ( [ K b(i ) ] + [ K s(i ) ]

+Ω 2 ( [ K c( i ) ] − sin 2 ψ [ M t( i ) ] − [ M r( i ) ] ) ){u (i ) } = {0}.

(58)

(49)

4. Numerical results and discussion In Eq. (49), the elemental mass matrices, the elemental stiffness matrices, and the elemental acceleration vector are given as: [ M t( i ) ] = ∫

+1

[ M r( i ) ] = ∫

+1

−1

−1

[K ] = ∫ (i ) b

[ K s(i ) ] =

+1

−1

∫

+1

−1

ρ A( i ) (ξ ) [ N w ]T [ N w ] J ( i ) dξ ,

(50)

ρ I (i ) (ξ ) [ Nφ ]T [ Nφ ]J (i ) dξ ,

(51)

T

⎛ [N′ ] ⎞ ⎛ [N′ ] ⎞ EI (ξ ) ⎜ (φi ) ⎟ ⎜ (φi ) ⎟ J ( i ) d ξ , ⎝ J ⎠ ⎝ J ⎠ (i )

T

⎛ [ N w′ ] ⎞ ⎛ [N′ ] ⎞ − [ Nφ ] ⎟ ⎜ (wi ) − [ Nφ ] ⎟ J (i ) dξ (i ) J ⎝ ⎠ ⎝ J ⎠

κGA(i ) (ξ ) ⎜

, (53)

T

⎛ [N′ ] ⎞ ⎛ [N′ ] ⎞ T( i ) (ξ ) ⎜ (wi ) ⎟ ⎜ (wi ) ⎟ J (i ) dξ , ⎝ J ⎠ ⎝ J ⎠ (i ) T ( i ) ( i ) ( i ) ( i ) ( i ) ( i ) 1 φ1 w 2 φ2 w 3 φ3 ] , {u } = [ w [ K c( i ) ] = ∫

(52)

+1

−1

(54) (55)

where [ N w′ ] =

d [ Nφ ] d[ Nw ] , [ Nφ′ ] = . dξ dξ

(56)

It should be noted that the term ρ I ( x) ⋅ Ω 2φ , which appears in Eq. (21), corresponds to the term −Ω 2 [ M r( i ) ]{u (i ) } in Eq. (49). 3.2 The generalized eigenvalue problem The global equation of motion of the rotating Timoshenko beam can be written as

( (

[M t ] + [M r ]

){u}

[ K b ] + [ K s ] + Ω 2 ( [ K c ] − sin 2 ψ [ M t ] − [ M r ] ) ){u} = {0} ,

(57) where {u} is the vector of all nodal variables of the beam. The matrices [ M t ] and [ M r ] are the global translational mass and rotational mass matrices, respectively. The matrices [ K b ] , [ K s ] and [ K c ] are the global bending stiffness matrix, shear stiffness matrix and centrifugal stiffness matrix, respectively. On assuming the solution of Eq. (57) in the form

Several numerical examples are now considered and the predictions of the new element are compared with the published literature. To do this, the following parameters are used:

µ=

ρ A0ω 2 L4

λ=

ρ A0Ω 2 L4

EI 0 EI 0

= (non-dimensional natural frequency), = (non-dimensional rotation speed),

S = L r = (non-dimensional slenderness ratio), and r = r L = 1 S = (non-dimensional radius gyration), where r is radius of gyration defined as r = I 0 A0 .

4.1 Convergence test For a uniform beam, convergence tests are carried out. Table 1 shows that convergence for the first mode for λ = 0 ~ 4 was achieved using 20 uniform elements to get the desired accuracy (tolerance of ≤ 0.00001). In Table 1, the present_1 denotes the frequencies that are calculated with the extra term ρ I Ω 2φ . Throughout this paper, 20 uniform elements for each numerical test are used. 4.2 Variation of slenderness ratio in a uniform Timoshenko beam Tables 2 and 3 show a comparison of the fundamental natural frequencies of rotating uniform cantilever Timoshenko beam obtained by using the present element with the results from Refs. [12] and [14]. The results of Ref. [12] were obtained by using dynamic stiffness formulation, while those of [14] were obtained by using the transfer matrix method by the authors of Ref. [12]. Table 2 illustrates the effect of the non-dimensional rotation speed ( λ ) on the fundamental natural frequency of the Timoshenko beam, where present_1 denotes the frequencies calculated with the extra term ρ I Ω 2φ and present_2 denotes the frequencies calculated without the term. These designations will be used throughout this paper. As expected, when λ = 0 , complete agreement between the four sets of results in Table 2 is evident. The discrepancy between the present_1 and the results of Ref. [14] increases with increasing the non-

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Table 1. Convergence of the fundamental non-dimensional natural frequencies of a rotating cantilever Timoshenko beam for various values of the non-dimensional rotation speed when R = 0 , S = 30 , E κ G = 3.059 and ψ = 0o .

λ =0

Table 2. Fundamental non-dimensional natural frequency of a rotating cantilever Timoshenko beam for various values of the non-dimensional rotation speed when R = 0, S = 30, E κ G = 3.059 and ψ = 0o . Fundamental natural frequency

λ

Present_1

(Present_2)

Ref. [12]

Ref. [14]

0 1 2 3 4 5

3.47984 3.64452 4.09710 4.75157 5.53140 6.38584

(3.47984) (3.64520) (4.09949) (4.75610) (5.53814) (6.39471)

3.4798 3.6445 4.0971 4.7516 5.5314 6.3858

3.4798 3.6452 4.0994 4.7558 5.5375 6.3934

Table 3. Fundamental non-dimensional natural frequency of a rotating cantilever Timoshenko beam for various values of S , when λ =0 and λ = 5 with R = 0, E κ G = 3.059 and ψ = 0o .

λ =0

S

λ =1 20 30 40 50 80

λ=2

100 150 200 300 1000

λ =3

λ=4

dimensional rotation speed, but present_2 agrees with the results of Ref. [14]. The reason for this is that Ref. [14] omitted the term ρ I Ω 2φ whereas Ref. [12] considered it. Table 3 shows the effect of slenderness ratio ( S ) on the

Present_1 (Present_2) 3.43643 (3.43643) 3.47984 (3.47984) 3.49550 (3.49550) 3.50284 (3.50284) 3.51084 (3.51084) 3.51270 (3.51270) 3.51454 (3.51454) 3.51519 (3.51519) 3.51565 (3.51565) 3.51598 (3.51598)

λ =5

Ref. [12]

Ref. [14]

3.4364

3.4364

3.4798

3.4798

3.4955

3.4954

3.5028

3.5028

3.5108

3.5108

3.5127

3.5126

3.5145

3.5144

3.5152

3.5152

3.5156

3.5155

3.51602* (Euler-Bernoulli Beam)

Present_1 (Present_2) 6.31259 (6.33174) 6.38584 (6.39471) 6.41312 (6.41818) 6.42604 (6.42931) 6.44029 (6.44157) 6.44361 (6.44443) 6.44690 (6.44727) 6.44806 (6.44826) 6.44888 (6.44897) 6.44949 (6.44949)

Ref. [12]

Ref. [14]

6.3126

6.3241

6.3858

6.3934

6.4131

6.4179

6.4260

6.4294

6.4403

6.4418

6.4436

6.4446

6.4469

6.4476

6.4481

6.4485

6.4489

6.4493

6.44954* (Euler-Bernoulli Beam)

* Reference [5].

fundamental frequency of the beam when the non-dimensional rotation speed is set to 0 and 5, respectively. Here, again, the results of the present_1 match exactly with those of Refs. [12] and [14] for the case when λ = 0 . Clearly, the present_1 and the Ref. [12] do not match with those from Ref. [14] for the case when λ = 5 because of the above reason. With increasing S , the difference between present_1 and present_2 diminishes. But the results of Ref. [14] are much closer to the results of present_2 than those of present _1 for the same reason. In Table 4, the illustrative examples of Refs. [12] and [10] are used for comparison with the present results, where the authors obtained the solutions by using dynamic stiffness method and power series method, respectively. Since r = 0 implies zero thickness physically, the frequencies for an extremely thin beam with r = 0.001 are calculated to compare it with the frequencies of Euler-Bernoulli beam [5] and the

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Y.-W. Kim and J. Jeong / Journal of Mechanical Science and Technology 25 (5) (2011) 1235~1246

Table 4. Fundamental non-dimensional natural frequency of a rotating cantilever Timoshenko beam for various values of r (= 1 S ) and λ , when R = 0, κ = 2 3, E G = 8 3 and ψ = 0o .

λ=0

λ=4

r

Present_1

-

-

3.51602*

-

3.5160†

-

3.516‡

3.51597 (3.51597)

0 0.001 0.01

Present_2

Present_1

Ref.

Present_2

0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

Ref.

Present_2

λ = 12 Present_1

Ref.

Present_2

Mode No.

Ref.

5.58500*

-

9.25684*

-

-

5.5850†

-

9.2568†

-

13.170†

-

5.585‡

-

9.257‡

-

13.170‡

-

5.58494

-

9.25672

-

13.16992

-

-

(5.58495)

-

(9.25673)

-

(13.16995)

-

9.2447†

13.14819

13.148†

3.51194 3.5119†

5.57908 5.5791† 9.24471

(3.51194) 3.512‡

(5.57971) 5.580‡ (9.24607)

†

0.02

λ =8 Present_1

3.49980 3.4998

(3.49980) 3.500‡

†

5.56158 5.5616 (5.56405)

9.20959

5.564 (9.21495)

3.47990 3.4799†

5.53322 5.5332† 9.15493

(3.47990) 3.480‡

(5.53867) 5.539‡ (9.16669)

3.45267 3.4527†

5.49511 5.4951† 9.08544

‡

(3.45267) 3.453

‡

(5.50455) 5.505

(9.10567)

3.41872 3.4187†

5.44866 5.4487† 9.00598

(3.41872) 3.419‡

(5.46294) 5.463‡ (9.03645)

3.37873 3.3787†

5.39542 5.3954† 8.92085

(3.37873) 3.379‡

(5.41523) 5.415‡ (8.96305)

3.33347 3.3335†

5.33699 5.3370† 8.83330

(3.33347) 3.333‡

(5.36284) 5.363‡ (8.88859)

3.28370 3.2837†

5.27486 5.2749† 8.74555

(3.28370) 3.248‡

(5.30716) 5.307‡ (8.81525)

3.23022 3.2302†

5.21041 5.2104† 8.65881

(3.23022) 3.230‡ 3.17377 3.1738† ‡

(3.17377) 3.174

(5.24943)

5.249 (8.74437)

5.14482 5.1448† 8.57351 ‡

(5.19075) 5.191

(8.67666)

13.17015*

9.246‡ (13.15018) 13.150‡ †

9.2096

13.08700

Table 5. Non-dimensional natural frequencies of a rotating uniform slender beam with cantilever end condition for various values of λ , when R = 0, r = 1 1000, κ = 0.85, E G = 13 5, ν = 0.3 and ψ = 0o .

1

2

†

13.087

9.215‡ (13.09482) 13.095‡ 9.1549†

12.99792

12.998†

9.167‡ (13.01500) 13.015‡ 9.0854† ‡

9.106

9.0060†

12.89339

12.893†

3

‡

(12.92271) 12.923 12.78297

12.783†

9.036‡ (12.82721) 12.827‡ 8.9208†

12.67236

12.672†

8.963‡ (12.73411) 12.734‡ 8.8333†

12.56400

4

12.564†

8.889‡ (12.64606) 12.646‡ 8.7456†

12.45809

12.458†

8.815‡ (12.56383) 12.564‡ 8.6588†

12.35323

5

12.353†

8.744‡ (12.48715) 12.487 8.5735† ‡

8.677

12.24673

12.247† ‡

(12.41516) 12.415

* Ref. [5]; † Ref. [12]; ‡ Ref. [10]

frequency corresponding to r = 0 [10, 12]. It is worthwhile to pay attention to the fact that these frequencies are nearly the same. Table 4 shows a complete agreement between present_1 and the results of Ref. [12], whereas there is a discrepancy between present_1 and the results of Ref. [10], except the case when λ = 0 . It also shows that the discrepancy increases with increasing values of λ . The reason for this discrepancy is the omission of the term ρ I Ω 2φ by the authors of Ref. [10]. However, there is also a complete agreement between the present_2 and the results from Ref. [10] because both sets of results are obtained by omitting the term ρ I Ω 2φ . In Tables 3 and 4, the values with superscript * are taken from Ref. [5], and they are the frequencies of the EulerBernoulli beam. The comparison of the present_2 with them shows that the values of present_2 become closer to the frequency of Euler-Bernoulli beam from the lower value than the frequency of Euler-Bernoulli beam as the slenderness ratio increases. This fact is reported partially by several authors. [12-14, 16, 17] For instance, Banerjee [12] has shown that the natural frequencies of rotating Timoshenko beam approach those of rotating Euler-Bernoulli beam as r decreases and coincide with those of rotating Euler-Bernoulli beam eventually when r = 0 . This phenomenon or trend can be utilized as one of the criteria to estimate if a numerical solution of a slender beam is converged or to compare the accuracy of numerical solutions of slender beams under the assumption that

6

7

8

9

10

λ

Present_1

(Present_2)

Ref. [5]

Ref. [1, 8]

Ref. [19]

0 1 3 5 7 10 0 1 3 5 7 10 0 1 3 5 7 10 0 1 3 5 7 10 0 1 3 5 7 10 0 1 3 5 7 10 0 1 3 5 7 10 0 1 3 5 7 10 0 1 3 5 7 10 0 1 3 5 7 10

3.51598 3.68161 4.79724 6.44949 8.29955 11.2022 22.0332 22.1797 23.3190 25.4448 28.3327 33.6389 61.6906 61.8351 62.9783 65.1984 68.3793 74.6425 120.890 121.039 122.223 124.554 127.960 134.871 199.869 200.021 201.232 203.631 207.169 214.468 298.688 298.843 300.073 302.517 306.141 313.685 417.503 417.659 418.903 421.378 425.061 432.770 556.579 556.736 557.990 560.488 564.212 572.036 716.332 716.490 717.751 720.266 724.021 731.928 897.374 897.532 898.799 901.326 905.102 913.069

(3.51598) (3.68161) (4.79724) (6.44949) (8.29956) (11.2022) (22.0332) (22.1797) (23.3190) (25.4448) (28.3327) (33.6389) (61.6906) (61.8351) (62.9783) (65.1984) (68.3793) (74.6425) (120.890) (121.039) (122.223) (124.554) (127.960) (134.871) (199.869) (200.021) (201.232) (203.631) (207.169) (214.468) (298.688) (298.843) (300.073) (302.517) (306.141) (313.685) (417.503) (417.659) (418.903) (421.378) (425.061) (432.770) (556.579) (556.736) (557.990) (560.488) (564.212) (572.036) (716.332) (716.490) (717.751) (720.266) (724.021) (731.928) (897.374) (897.532) (898.799) (901.326) (905.102) (913.069)

3.51602 3.68165 4.79728 6.44954 8.29964 11.2023 22.0345 22.1810 23.3203 25.4461 28.3341 33.6404 61.6972 61.8418 62.9850 65.2050 68.3860 74.6493 -

3.51602† 3.6817‡ 4.7973‡ 6.4495‡ 8.2996‡ 11.2023‡ 22.0345† 22.1810‡ 23.3203‡ 25.4461‡ 28.3341‡ 33.6404‡ 61.6972† 61.8418‡ 62.9850‡ 65.2050‡ 68.3860‡ 74.6493‡ 120.902† 121.051‡ 122.236‡ 124.566‡ 127.972‡ 134.884‡ 199.860† 200.012‡ 201.223‡ 203.622‡ 207.161‡ 214.461‡ 298.566† 416.991†

3.51602 3.68165 4.79728 6.44955 8.29967 11.2024 22.0348 22.1814 23.3206 25.4464 28.3345 33.6410 61.7049 61.8494 62.9925 65.2124 68.3931 74.6566 120.959 121.108 122.292 124.662 128.026 134.936 200.110 200.262 201.472 203.868 207.403 214.697 299.369 299.522 300.750 303.188 306.805 314.334 419.135 419.291 420.529 422.994 426.662 434.342 560.027 560.183 561.428 563.909 567.608 575.380 722.815 723.008 724.255 726.743 730.457 738.281 908.338 908.494 909.741 912.228 915.945 923.788

-

555.165† -

† Ref. [8]; ‡ Ref. [1].

numerical solutions are converged. In Table 5, the solutions from Refs. [1], [5], [8] and [19] are based on Euler-Bernoulli beam theory, whereas the present results are based on Timoshenko beam theory. Thus, it is expected that the results of present_2 will be slightly smaller than the other results if each set of the results in Table 5 is converged. Comparing the results of present_2 with the other

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Y.-W. Kim and J. Jeong / Journal of Mechanical Science and Technology 25 (5) (2011) 1235~1246

Table 6. Non-dimensional natural frequencies ( µi , i = 1, 2, "10 ) of a rotating cantilever Timoshenko beam for various values of λ when R = 0, r = 0.08, κ = 0.85, E G = 13 5, ν = 0.3 and ψ = 0o . Mode No.

1

2

3

4

5

6

7

8

9

10

λ

Present_1

(Present_2)

Ref. [8]

Ref. [19]

0 1 3 5 7 10 0 1 3 5 7 10 0 1 3 5 7 10 0 1 3 5 7 10 0 1 3 5 7 10 0 1 3 5 7 10 0 1 3 5 7 10 0 1 3 5 7 10 0 1 3 5 7 10 0 1 3 5 7 10

3.32405 3.48538 4.56208 6.13713 7.88940 10.6387 16.2891 16.4528 17.7033 19.9505 22.8704 27.9664 36.7087 36.9140 38.5004 41.4187 45.3010 52.1958 58.2833 58.5389 60.5156 64.1481 68.9368 77.0927 80.2265 80.5142 82.7191 86.6364 91.3380 97.0801 94.4620 94.5889 95.4935 96.9236 98.7480 103.130 106.859 107.080 108.795 111.835 115.547 122.329 114.752 114.956 116.580 119.832 124.542 131.976 133.319 133.628 135.880 139.253 143.008 151.360 139.796 140.018 141.902 146.293 153.166 163.575

(3.32405) (3.48887) (4.58501) (6.18152) (7.95325) (10.7299) (16.2891) (16.4561) (17.7319) (20.0253) (23.0077) (28.2258) (36.7087) (36.9159) (38.5176) (41.4658) (45.3931) (52.3876) (58.2833) (58.5405) (60.5301) (64.1895) (69.0230) (77.3019) (80.2265) (80.5157) (82.7326) (86.6793) (91.4421) (97.3912) (94.4620) (94.5924) (95.5260) (97.0171) (98.9263) (103.432) (106.859) (107.083) (108.814) (111.889) (115.652) (122.503) (114.752) (114.959) (116.599) (119.880) (124.634) (132.206) (133.319) (133.629) (135.895) (139.302) (143.100) (151.479) (139.796) (140.020) (141.916) (146.321) (153.220) (163.751)

3.32405 16.2890 36.7078 58.2788 80.2127 94.4520 106.836 114.732 -

3.32485 3.48952 4.58641 6.18416 7.95761 10.7377 16.2862 16.4534 17.7310 20.0216 23.0142 28.2394 36.7960 37.0035 38.6801 41.5617 45.4960 52.5039 58.7155 58.9728 60.9631 65.6224 69.4524 77.7166 81.4035 81.6906 83.8921 87.8000 92.4923 98.3338 95.5754 95.6941 96.5552 97.9783 99.8844 104.512 109.131 109.356 111.090 114.161 117.992 125.058 117.552 117.763 119.445 122.775 127.475 134.985 138.255 138.532 140.537 143.733 147.790 156.790 145.115 145.385 147.645 152.462 159.296 169.423

solutions, Table 5 shows that µ present _ 2 ≤ µ[1,8] < µ[19] for lower modes and µ[1,8] < µ present _ 2 < µ[19] for higher modes. But the difference between µ present _ 2 and µ[1,8] is smaller than the difference between µ present _ 2 and µ[19] for higher modes. Table 5 shows that µ present _ 2 ≈ µ[1,8] < µ[19] in average sense. This fact implies that the results of µ[19] may be slightly overestimated. Here, µ["] denotes the nondimensional frequency from the Ref. in ["] . This designa-

Table 7. Non-dimensional natural frequencies of rotating cantilever slender beams when ψ = 00 , r = 1 1000 , ν = 0.3 and κ = 0.85 . ψ = 00 R

λ

0 2 4 0 6 8 10 0 2 4 1 6 8 10

1st mode

2nd mode

Present_1

Ref.

Ref.

Ref.

Present_1

Ref.

Ref.

Ref.

(Present_2)

[5]

[17]

[18]

(Present_2)

[5]

[17]

[18]

3.51598 (3.51598) 4.13728 (4.13729) 5.58495 (5.58496) 7.36030 (7.36031) 9.25674 (9.25675) 11.2022 (11.2022) 3.51598 (3.51598) 4.83365 (4.83365) 7.47498 (7.47499) 10.4438 (10.4438) 13.5072 (13.5072) 16.6061 (16.6061)

3.5160

3.516

3.515

4.1373

4.137

4.137

5.5850

5.585

-

7.3604

7.360

-

9.2568

9.257

-

11.2023 11.203 11.203 3.5160

3.516

3.515

4.8337

4.834

4.834

7.4750

7.475

-

10.4439 10.444

-

13.5074 13.509

-

16.6064 16.609 16.609

22.0332 (22.0332) 22.6136 (22.6136) 24.2720 (23.2720) 26.8077 (26.8078) 29.9940 (29.9940) 33.6389 (33.6389) 22.0332 (22.0332) 23.3647 (23.3647) 26.9559 (26.9559) 32.0258 (32.0258) 37.9522 (37.9522) 44.3664 (44.3664)

22.0345 22.036 22.031 22.6149 22.617 22.617 24.2733 24.275

-

26.8091 26.811

-

29.9954 29.998

-

33.6404 33.643 33.643 22.0345 22.036 22.031 23.3660 23.368 23.368 26.9573 26.959

-

32.0272 32.030

-

37.9538 37.959

-

44.3682 44.378 44.378

tion will be used throughout this paper. All the solutions in Table 6 are based on Timoshenko beam theory. For λ = 0 , Table 6 shows that µ[8] ≤ µ present _ 2 < µ[19] but the difference between µ present _ 2 and µ[8] is small compared to the difference between µ present _ 2 and µ[19] . This fact will be used in the later section to discuss the example of a linearly tapered beam in two planes. 4.3 Variation of setting angle and hub radius in uniform Timoshenko beam Tables 7 and 8 compare the present results for a slender beam with those from Refs. [5], [16], [17] or [18] for different setting angles. Ref. [5] is based on Euler-Bernoulli beam theory, whereas Refs. [16-18] are based on Timoshenko beam theory. The present solutions in Tables 7 and 8 are calculated for a slender beam with r = 1 1000 . The authors in Refs. [17] and [18] obtained their solutions by disregarding the effects of rotary inertia and shear deformation. Thus, it is expected that the frequencies of present_2 are less than the frequencies of the Euler-Bernoulli beam in Ref. [5]. It is also expected that the results from Refs. [17] and [18] will be coincident with those from Ref. [5] if the results from Ref. [17] and [18] are converged ones. As expected, it shows that the values of the present_2 are slightly less than the values in Ref. [5]. However, the results for λ = 10 from Refs. [17] and [18] in Table 7 are greater than the results of the Euler-Bernoulli beam from Ref. [5]. This fact implies that the natural frequencies from Refs.

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Y.-W. Kim and J. Jeong / Journal of Mechanical Science and Technology 25 (5) (2011) 1235~1246

Table 8. Non-dimensional natural frequencies of rotating cantilever slender beams when ψ = 900 , r = 1 1000 , ν = 0.3 and κ = 0.85 . ψ = 900 R λ

0 2 5 0 10 20 50 0 2 5 1 10 20 50

1st mode

2nd mode

Present_1

Ref.

Ref.

Ref.

Present_1

Ref.

Ref.

Ref.

(Present_2)

[16]

[17]

[18]

(Present_2)

[16]

[17]

[18]

-

3.516

3.515

3.6118 3.622

3.622

4.0739 4.074

-

5.0490 5.050

5.050

6,7757 6.794

-

10.4806 10.899

-

3.51598 (3.51598) 3.62176 (3.62176) 4.07381 (4.07383) 5.04871 (5.04874) 6.77436 (6.77446) 10.4367 (10.4370) 3.51598 (3.51598) 4.40048 (4.40048) 7.41139 (7.41139) 13.2576 (13.2576) 25.2860 (25.2860) 61.5831 (61.5832)

-

3.516

3.515

4.4005 4.401

4.400

7.4115 7.412

-

13.2580 13.261 13.260 25.2881 25.318

-

61.6408 61.881

-

22.0332 (22.0332) 22.5250 (22.5250) 24.9487 (24.9487) 32.1182 (32.1182) 51.3506 (51.3507) 116.1744 (116.1748) 22.0332 (22.0332) 23.2790 (23.2790) 28.92247 (28.92248) 43.2248 (43.2248) 76.5880 (76.5881) 181.8230 (181.8233)

-

22.036 22.031

22.5263

22.528 22.528

24.9500

24.952

32.1197

32.123 32.123

51.3531

51.372

-

116.1996 116.417

-

Table 9. Effect of setting angle ( ψ ) on natural frequency of rotating cantilever Timoshenko beams with different slenderness ratios ( R / L = 3, λ = 10, ν = 0.3(or E G = 13 5) and κ = 0.85 ) . Mode No. ψ

0o 1

90o

-

-

22.036 22.031

23.2803

23.282 23.282

28.9238

28.926

43.2267

43.237 43.237

76.5942

76.659

-

181.9361 182.386

-

0o 2

45o 90o 0o

3

45o 90o

-

[17] and [18] have been slightly overestimated or the results have been not converged sufficiently yet. In Table 8, the results from Refs. [17] and [18] are the solutions neglecting the effects of rotary inertia and shear deformation, whereas the present results and the results from Ref. [16] are the solutions considering shearing forces and rotary inertia. The authors of Ref. [16] discretized the uniform beam with five equal elements that are based on a fifth degree polynomial being used as displacement function. If the accuracy of each set of solutions is sufficiently accurate, it is expected that the results of Refs. [17] and [18] are greater than those of the present_2 and Ref. [16]. Table 8 shows that µ present _ 2 ≤ µ[16] < µ[17] ≅ µ[18] for high rotation speeds λ = 10 , 20 and 50 . This indicates that the results from Refs. [17] and [18] have been slightly overestimated and the results of present_2 are very accurate along with the implication obtained from Table 7. Table 9 compares the present solutions with other numerical results for Timoshenko beams that are not slender beams. Wang et al. [13] calculated the first four frequencies by using Galerkin’s method that employs Legendre polynomials with nine terms, which yields very accurate solutions. Table 9 shows ( µ present _ 2 ≈ µ[13] ) < ( µ[17] ≈ µ[18] ) . This implies that the results of present_2 are accurate and that µ[17] and µ[18] might be slightly overestimated.

45o

0o 4

45o 90o

S = 10 Present_1 Present_2 22.9376 (23.0362) 21.8735 (21.9726) 20.7525 (20.8528) 44.7814 (45.4290) 44.5502 (45.1950) 44.3147 (44.9566) 66.2880 (66.8625) 66.2004 (66.7712) 66.1105 (66.6774) 71.9680 (72.3157) 71.7927 (72.1490) 71.6209 (71.9858)

Ref. [13]

S = 20 Ref. [17]

Ref. [18]

23.037 23.050 23.050 21.974 21.987

-

20.850 20.867 20.867 45.428 45.598 45.598 45.194 45.359

-

44.955 45.115 45.115 66.854 67.716

-

66.763 67.619

-

66.668 67.520

-

72.313 73.076

-

72.146 72.914

-

71.982 72.756

-

Present_1 Present_2 23.4906 (23.5098) 22.4112 (22.4311) 21.2770 (21.2979) 55.9846 (56.0614) 55.5750 (55.6522) 55.1623 (55.2398) 96.9164 (96.9926) 96.6964 (96.7726) 96.4757 (96.5521) 143.730 (143.795) 143.588 (143.654) 143.447 (143.512)

Ref. [13]

Ref. [17]

Ref. [18]

23.514 23.524

-

22.436 22.446

-

21.302 21.313

-

56.072 56.105

-

55.662 55.696

-

55.250 55.284

-

97.011 97.188

-

96.792 96.968

-

96.570 96.747

-

143.815 144.490

-

143.673 144.349

-

143.531 144.208

-

Table 10. Non-dimensional natural frequencies of rotating cantilever slender beams with taper ratio α y = 0.5 and α z = 0.0 when ψ = 00 , r = 1 1000 , R = 0 and κ = 0.85 .

1st mode Present_1 (Present_2) 4.43677 2 (4.43677) 5.87872 4 (5.87872) 7.65508 6 (7.65508) 9.55387 8 (9.55388) 11.50143 10 (11.50144)

λ

2nd mode

Ref. [5]

Ref. [18]

4.43680

4.4368

5.87876

5.8788

7.65514

7.6551

9.55396

9.5540

11.50154 11.5015

Present_1 (Present_2) 18.93603 (18.93604) 20.68455 (20.68456) 23.30864 (23.30865) 26.54297 (26.54229) 30.18199 (30.18202)

Ref. [5]

Ref. [18]

18.93663 18.9367 20.68516 20.6852 23.30927 23.3049 26.54366 26.5437 30.18274 30.1827

4.4 Linearly tapered beam in one plane Table 10 shows that µ present _ 2 ≤ µ[5] ≈ µ[18] . It should be noted that the results of present_2, based on Timoshenko beam theory, are smaller than the results of the EulerBernoulli beam from Ref. [5]. However, the results from Ref. [18] agree with those of the Euler-Bernoulli beam from Ref. [5] because they obtained their solutions by disregarding the rotary inertia and shear deformation.

Y.-W. Kim and J. Jeong / Journal of Mechanical Science and Technology 25 (5) (2011) 1235~1246

Table 11. Non-dimensional natural frequencies of rotating cantilever Timoshenko beam with taper ratio α y = α z = 0.6 when R = 0, r = 0.08, κ = 0.85, E G = 13 5, ν = 0.3 and ψ = 0o . Mode No.

1

2

3

4

5

6

7

8

9

10

λ

0 1 3 5 7 10 0 1 3 5 7 10 0 1 3 5 7 10 0 1 3 5 7 10 0 1 3 5 7 10 0 1 3 5 7 10 0 1 3 5 7 10 0 1 3 5 7 10 0 1 3 5 7 10 0 1 3 5 7 10

Present_1 4.74979 4.87995 5.80245 7.25598 8.93858 11.6235 15.9106 16.0457 17.0867 18.9960 21.5359 26.0942 32.7702 32.9188 34.0827 36.2932 39.3641 45.1625 52.6372 52.8092 54.1629 56.7605 60.4185 67.4390 74.3047 74.5042 76.0769 79.1061 83.3922 91.6532 96.8637 97.0921 98.8931 102.360 107.246 116.424 119.536 119.778 121.642 124.782 127.119 128.508 128.570 128.601 128.885 129.871 133.377 143.382 143.885 144.172 146.440 150.811 156.772 160.966 159.597 159.646 159.960 160.392 161.096 169.624

(Present_2) (4.74979) (4.88161) (5.81463) (7.28190) (8.97808) (11.6824) (15.9106) (16.0471) (17.0988) (19.0263) (21.5883) (26.1828) (32.7702) (32.9199) (34.0925) (36.3193) (39.4126) (45.2537) (52.6372) (52.8101) (54.1712) (56.7829) (60.4615) (67.5247) (74.3047) (74.5050) (76.0837) (79.1250) (83.4295) (91.7318) (96.8637) (97.0928) (98.8990) (102.377) (107.282) (116.521) (119.536) (119.779) (121.650) (124.824) (127.269) (128.833) (128.570) (128.605) (128.913) (129.932) (133.429) (143.456) (143.885) (144.173) (146.444) (150.824) (156.812) (161.229) (159.597) (159.649) (159.984) (160.460) (161.216) (169.689)

Ref. [8] 4.74979 15.9107 32.7692 52.6316 74.2587 96.8403 119.904 128.763 -

-

Ref. [19] 4.75611 4.79747 5.11557 5.69567 6.46308 7.83303 15.9325 15.9839 16.3890 17.1675 18.2666 20.3867 32.8667 32.9303 33.4340 34.4126 35.8162 38.5856 53.0132 53.0917 53.7131 54.9229 56.6633 60.1120 75.3768 75.4699 77.2069 77.6429 79.7107 83.8124 99.2476 99.3544 100.200 101.846 104.207 108.849 123.671 123.773 124.553 125.923 127.433 129.015 130.768 130.796 131.045 131.680 133.060 137.222 151.629 151.765 152.835 154.902 157.793 162.529 163.888 163.904 164.026 164.274 164.697 166.495

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more accurate than those from Ref. [19]. The comparison in Table 1-11 shows good agreement with analytical or numerical solutions of the rotating uniform or tapered beams. Although there are slight differences between present and other numerical solutions, the present solutions follow the general trend with high accuracy, that is, frequencies based on Timoshenko beam theory are less than those based on Euler-Bernoulli beam theory.

5. Conclusions A C0-continuous isoparametric 3-noded finite element is presented for the modal analysis of the free vibration of a tapered Timoshenko beam mounted on the periphery of a rotating rigid hub, at a setting angle with the plane of rotation. Starting from the expression of flexural deformation energy and kinetic energy, we derived the governing differential equations of motion and finite element equation that includes an extra term, which is neglected by some researchers. The finite element formulation uses the independent polynomial displacement fields for deflection and rotation of section, and employs a field-consistent displacement functions for the shear strain to avoid shear locking. The results obtained from the present formulation are compared with published results. The present numerical results display high accuracy compared with other numerical solutions.

Nomenclature-----------------------------------------------------------------------A0 : The cross-sectional area at fixed end I0 : The second moment of area at fixed end E : Young’s modulus G : Shear modulus L : Length of beam r = I 0 A0 : Radius of gyration r =r L : Non-dimensional radius gyration R : Hub radius S=L r : Non-dimensional slenderness ratio

κ λ µ ν ρ ω ψ Ω

: Shear correction factor : Non-dimensional rotation speed : Non-dimensional natural frequency : Poisson’s ratio : Mass density : Natural frequency : Setting angle : Rotational speed

References 4.5 Linearly tapered beam in two planes All the solutions in Table 11 are based on Timoshenko beam theory. Table 11 shows that the results of present_2 for λ = 0 are closer to the results from Ref. [8] than the results from Ref. [19]. Because of this fact along with the results in Table 6, it can be considered that the present solutions are

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Yong-Woo Kim is currently Professor at Department of Mechanical Engineering at Sunchon National University in Sunchon, Korea. Dr. Kim’s research interests include structural analysis, machine design and tire mechanics.