Quasi-3D dynamic analysis of rotating FGM beams using a modified Fourier spectral approach

Journal Pre-proof

Quasi-3D dynamic analysis of rotating FGM beams using a modified Fourier spectral approach Guoyong Jin , Yukun Chen , Shanjun Li , Tiangui Ye , Chunyu Zhang PII: DOI: Reference:

S0020-7403(19)31685-6 https://doi.org/10.1016/j.ijmecsci.2019.105087 MS 105087

To appear in:

International Journal of Mechanical Sciences

Received date: Revised date: Accepted date:

14 May 2019 16 August 2019 17 August 2019

Please cite this article as: Guoyong Jin , Yukun Chen , Shanjun Li , Tiangui Ye , Chunyu Zhang , Quasi-3D dynamic analysis of rotating FGM beams using a modified Fourier spectral approach, International Journal of Mechanical Sciences (2019), doi: https://doi.org/10.1016/j.ijmecsci.2019.105087

This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier Ltd.

Highlights    

Quasi-3D dynamic solutions are developed for rotating FGM beams. The method is appropriate for any uniform rotating FGM beams. The method is appropriate for arbitrary boundary conditions. Parameter studies are presented on vibration of rotating FGM beams.

Quasi-3D dynamic analysis of rotating FGM beams using a modified Fourier spectral approach Guoyong Jin, Yukun Chen, Shanjun Li, Tiangui Ye*, Chunyu Zhang College of Power and Energy Engineering, Harbin Engineering University, Harbin 150001, PR China

Abstract Rotating beams and blades made of functionally graded materials (FGMs) have broad application prospects in engineering. Nonclassical features, such as anisotropy material, Coriolis effect, and both the centrifugal softening and stiffening effects, make it more challenging for one-dimensional theories to accurately model the rotating FGM beams. This work presents a quasi-three-dimensional solution for the dynamic behaviors of the rotating FGM beams. The FGM beam, attaching to a rigid hub, is assumed to have a metallic core covered with two ceramic faces. The constraint between the hub and FGM beam is simulated by the penalty method, which transforms the boundary potential energy into a quantifiable form and simplifies the selection of the trial functions. The model is established by employing the Carrera unified formulation (CUF) and modified Fourier spectral approach (MFSA) in which the displacement variables of the FGM beam are constructed by the modified Fourier series expansion. The variational method is applied for the derivation of the governing equations of the rotating FGM beams, considering the centrifugal and Coriolis effects. Several examples including the isotropic and FGM beams under different rotating conditions are performed to check the effectiveness and precision of the developed formulation. Parameter studies are then implemented to investigate the influences of the rotation velocity, material parameter, presetting angle and hub-radius ratio on the three-dimensional vibration characteristics of the rotating FGM beams. Keywords: quasi-3D solution; dynamic analysis; rotating FGM beams; modified Fourier spectral approach; Carrera unified formulation 1. Introduction 

Corresponding authors. E-mail addresses: [email protected] (G. Jin), [email protected] (T. Ye).

In the wake of developments in materials and processing technology, rotating FGM beams and blades have a progressively wide application in numerous power machineries, such as wind turbine, gas-turbine and steam turbine blades. The gradual change in the material component of FGMs gives it the ability to tailor the physical performance during the design phase, and makes it immune to stress concentration found in the discontinuous interface of traditional laminated composites. As known to us, rotating beams and blades, especially the gas-turbine blades, usually endure both thermal and mechanical loads during their operational life. FGMs generally consist of the ceramic and metallic compositions. The metallic part enhances mechanical properties of the FGM, whereas the heat resistance and inoxidizability of the material are improved by the ceramic composition. All of these have given FGMs a competitive edge and led to their extensive application prospects in rotating beams and blades. The vibration characteristics of elastic structures vary distinctly when considering large overall motions. For example, the bending frequencies of the rotating beams and blades are significantly influenced by the centrifugal effect, and the coupling phenomenon between the axial and bending motions is produced by Coriolis effect. Over the last few decades, many studies have focused on dynamic problems of the rotating FGM beam structures. Song and Librescu [1] presented a refined theory for the vibration analysis of the rotating thin-walled composite beams. The authors also extended the refined theory to the modelling of rotating thin-walled FGM beams in thermal environments [2, 3]. Fazelzadeh et al. [4] explored the vibration behaviors of rotating thin-walled FGM blades under the impact of high-speed airflow. Yoo et al. [5] provided a dynamic modeling method for the free vibration analysis of rotating beams by employing a non-Cartesian variable. Subsequently, the modeling method was extended to dynamic problems of the rotating pre-twisted isotropy blades [6], rotating pre-twisted FGM beams [7] and rotating pre-twisted tapered FGM

blades [8]. Fang et al. [9] explored the dynamic characteristics and time response of the rotating FGM cantilevered beams utilizing the classical beam theory. Li et al. [10] proposed a first-order coupled model for the vibration analysis of rotating FGM beams. Ebrahimi and Mokhtari [11] developed a dynamic analysis model for the rotating porous FGM beams based on the first order shear deformation theory (FSDT). The FSDT was also applied to study the vibration characteristics of the axially functionally graded tapered beams by Rajasekaran [12]. Vibration analysis of the rotating double tapered FGM beams was presented by Ebrahimi and Dashti [13] using the classical beam theory. From the review of literature, the classical and higher-order beam theories are the most commonly used theories for the dynamic analysis of rotating FGM beams. These beam theories usually adopt additional hypotheses and simplifications on the displacement fields, and neglect some important factors, such as shear deformation, inertia effect and warping effect [14, 15]. Rotational motions further complicate the modeling of the FGM beams due to the contributions of Coriolis and centrifugal effects [16]. When beams become less slender or the cross-section is irregular, the results obtained from these beam theories may have large errors. Moreover, the precisions of classical beam theories might be strongly impacted by the complicated cross-sections and nonhomogeneous materials of the beams [17]. Therefore, the classical and refined beam theories might not be suitable for the rotating blades with advanced geometrical shape. The primary aim of this work is to establish a quasi-3D model and study the vibration performance of the rotating FGM beams. Carrera Unified Formulation (CUF) combined with Taylor expression are employed for the refined beam theories with a quasi-3D accuracy. The CUF was first proposed for the modelling of the plates and shells by Carrera et al. [18-20] and then developed for the complex structures [21-29]. The CUF is a hierarchical formulation, in which the refined

structural models can be customized for the practical problems. For rotor-dynamic problems, the CUF has been employed for rotating composite blades [26], rotating disks [30], anisotropic rotors [31], rotating cylindrical shells [32] as well as the nonlinear dynamics of rotating blades [33]. A widely used numerical approach for the dynamic analysis of rotating structures is Rayleigh– Ritz method [9, 16, 34-38]. To ensure the convergence and accuracy, this method generally requires the trial functions to meet the essential boundary conditions. In present work, the boundary restraints of the hub-FGM beam system are imposed using the penalty method [39-49], which transforms the potential boundary energy into a quantifiable form that can be considered as part of the energy functional. This technique eliminates the need of trial functions to adjust for different boundary conditions, and makes it quite easy for the solution procedure to change from one boundary condition to another one. Modified Fourier spectral approach (MFSA) is employed to address the dynamic problems of the rotating FGM beams. The MFSA was first put forward by Li [50, 51] and extended for the dynamic characteristics of many complex problems [52-59]. In MFSA, the displacement variables are usually constructed as the combination of the Fourier cosine series and a few auxiliary functions, which are implemented to avoid certain discontinuities at the boundaries and improve the convergence speed and calculation precision of the complete Fourier approach. The organization of this paper is as following. Firstly, a rotating hub-FGM beam system is descripted and a mathematical function is presented for the description of the FGMs. Secondly, the energy functional of the hub-FGM beam system is derived based on the kinetic, strain and potential energy. Then, the variational method is applied for the governing equations of the rotating FGM beam. Different types of modes, associated with the chordwise, flapwise, longitudinal and torsional modes, are investigated accordingly. Subsequently, numerical examples including the isotropic and

FGM beams under different rotating conditions are performed to examine the convergence and precision of the developed formulation. Finally, the influences of the rotating speed, material parameter, hub-radius ratio and presetting angle on the vibration characteristics of the rotating FGM beam are investigated. 2. Theoretical formulations 2.1 Model description

Z Ω

X

γ

z

ξ

h

O x

γ b

rh

Y / y /η

θ L

ξ

Fig. 1. The dimensions of a hub-FGM beam system.

Figure 1 displays the dimensions of a hub-FGM beam system, in which a uniform rectangular FGM beam is attached to a rigid hub with a presetting angle θ. The hub is assumed to be a perfectly rigid-body, and rotates around the central axis with a rotating velocity . The rotating co-ordinate system (O-xyz) is fixed at the beam. The y-axis corresponds to the axial direction of the FGM beam, and the z-axis remains in the same direction with the rotating axis. The x-axis can then be determined using the right-handed rule. Additionally, O-ξηγ is the local co-ordinate system, where η-axis coincide with the y-axis. The ξ- and γ- axes define the cross-sectional directions. The relationship between the two co-ordinate systems can be defined as

x   cos    sin  ; y   z   cos    sin 

(1)

The FGM beam is assumed to have a metallic core covered with two ceramic faces, and can be

described by [60]. n

2z  Qm h

Q   Qc  Qm 

(2)

in which Q stands for a material property of the FGM beam, such as Young’s modulus E, density  and Poisson’s ratio υ. Additionally, Qc denotes the material properties of the ceramic component, and Qm represents those of the metal component. 2.2 Unified formulation and energy expression The CUF is applied to establish the refined theories with a 3-D accuracy. The displacement fields of the FGM beam are fitted with Taylor expansions. u ( x, y, z , t )  u1  xu2  zu3 + v( x, y, z , t )  v1  xv2  zv3 +



+z N u N 1 N  2 /2   F u  1



+z N v N 1 N  2 /2   F v

w( x, y, z , t )  w1  xw2  zw3 +

(3)

 1



+z N w N 1 N  2 /2   F w  1

where N is the order of the Taylor expression, Fτ stands for the No. τ of the Taylor expression, and Xτ (X=u, v, w) are the displacement variables of the FGM beam. Additionally, Γ can be calculated by Γ=(N+1)(N+2)/2. We note that the classical beam theories can be regarded as special cases of the present formulation. In that case, the Poisson locking may occur, and it can be eliminated by applying reduced material stiffness coefficients [61]. The vector from the rotating center O to a generic point P: rp  r  u; r   x, y  rh , z 

T

(4)

in which the displacement vector u can be grouped as u  u, v, w . T

The velocity of the generic point P can be expressed as

 0  0 v p  u  Ωrp ; Ω   0 0   0 0 0 

(5)

where (' ) represents the time derivative. The kinetic energy of the rotating FGM beam is given by 1  v p T v p dxdydz 2   uT u  2uT Ωr  2uT Ωu  1      dxdydz T T T T T T 2  2u Ω Ωr  u Ω Ωu  r Ω Ωr 

Tp 

(6)

The strain energy of the FGM beam is derived from the 3-D theory of elasticity [39]:

Vs 

1 ε n Tσ n  ε p Tσ p dxdydz  2

(7)

where εn and n are the strain and stress terms pointing to the axial direction, and εp and p represent these terms in the cross-section. The FGM beam rotates with the hub, so a centrifugal stress may appear in the FGM beam.

1 2  L  y 2   rh  L  y    2 

 c   2  y  rh  dy  2  L

y

(8)

The centrifugal potential energy is given by

  u 2  v 2  w  2  1 Vc    c           dxdydz   y   y   y   2  

(9)

In this work, the fixed constraint between the hub and the beam is realized via penalty method. Three penalty factors are placed at each end of the FGM beam. Therefore, the boundary potential energy can be calculated by

Vb 

with

1  T u k 0u    uTk Lu  dA    y 0 yL   2 A

(10)

 ku 0  k0   0  0 

0 kv 0 0

0   kuL   0 , kL   0  0 kw0  

0 kvL 0

0   0  kwL 

(11)

where kX0 (X = u, v, w) indicate the penalty factors introduced at one end of the FGM beam, and kXL are the penalty factors of the another end. Table 1 lists the classical boundary conditions that are defined by three groups of penalty factor [39]. Table 1. Classical boundary conditions defined by three groups of penalty factor. B.C.

Penalty factors

Free

ku = kv = kw = 0

Simply supported

kv = 0, ku = kw = 1012

Clamped supported

ku = kv = kw = 1012

Finally, the Lagrangian energy function can be obtained by the combination of the kinetic, strain and potential energy.   Vs  Vc  Vb  Tp

(12)

2.3 Solution procedure According to MFSA, the displacement variables (see Eq. (3)) are constructed by the modified Fourier series expansion [39].

u ( y) 

M

 a

m 2

m

 m ( y )；v ( y ) 

cos m y  m ( y)    sin m y

M

 b

m 2

m

 m ( y )；w ( y) 

m0 m0

M

 c

m 2

m

 m ( y)

m  m / L

(13)

(14)

in which M denotes the truncation index, and Xim (X = a, b, c) are the undetermined coefficients. The vector that contains the undetermined coefficients can be grouped as

q  t   {aim , bim , cim }T e jt for 1  i  , 2  m  M

(15)

Substituting the Lagrangian energy function (Eq. 12) into the Euler-Lagrange equation, one

can obtain

   t , q  t  , q  t     t , q t  , q t    0  t qim qim

(16)

Then, the following matrix relationship is obtained.

M mnij qjn  G mnij qjn   K mnij  K cmnij  K mnij  q jn  0  1  i, j  , 2  m, n  M 

(17)

mnij mnij mnij mnij mnij in which M , G , K , Kc and K are the 3×3 nucleus matrixes (see Appendix A) of the

mass, Coriolis effect, general stiffness, centrifugal stiffening and centrifugal softening matrixes, respectively. Equation (1) is applied to transform the variables between the two coordinate systems, O-xyz and Oʹ-ξηγ. The volume integrals are then rewritten in the form

 f  x, y, z dxdydz   g  ,,   J  , ,   d dd

(18)

where the limits of integration for ξ, η and γ are [–b/2, b/2], [0, L] and [–h/2, h/2], respectively. Additionally, J(ξ, η, γ) is the Jacobian matrix.

 x     y J  , ,        z   

x  y  z 

x    y    z   

(19)

The complete eigenvalue problem of the rotating FGM beam are obtained as

Mq  Gq   K  K c  K Ω  q  0

(20)

where M, G, K, Kc and K stand for the mass, Coriolis effect, general stiffness, centrifugal stiffening and centrifugal softening matrixes, respectively, and they can be obtained from the circulation of the nucleus matrixes (see Appendix A).

For the sake of the computational efficiency, an alternative procedure is introduced here to reduce the degrees of freedom of the complete eigenvalue problem shown in Equation (20).

MQ  GQ  KQ  0

(21)

in which M, G and K are the reduced matrices and they are given by

M  XTMX, G  XTGX, K  XTKX

(22)

where X is the normal eigenvector matrix of the following eigenvalue problem.

MX   K  K c  K Ω  X  0

(23)

To address the new eigenvalue problem shown in Eq. (21), the following reduced order method is applied.

 I 0   Q   0 0 M  Q  K      total

I   Q   0 G  Q

(24)

in which I represents the identity matrix, and its dimension is equal to the dimension of Q. The complex eigenvalues are then obtained with the assumption of a harmonic solution. 3. Numerical results and discussion A rotating metallic beam is performed first to assess the convergence and precision of the developed formulation. Then, the numerical model is extended to the dynamic solutions of rotating FGM beams. The ceramic component of the FGM beam is Al2O3 (Ec = 380 Gpa, c = 3800 kg/m3, υ = 0.3), whereas the metal component is Al (Em = 70 Gpa, m = 2700 kg/m3, υ = 0.3). With the perfectly rigid-body assumption of the hub, the rotating beam can be simplified as a cantilever beam. Thus, if not otherwise declared, only the clamped-free boundary condition is considered in the following discussion. For simplicity, the structure parameters are rewritten in dimensionless forms.

=

rh * ,   T * , *  T * , T *  L2 c A /  Ec I xx  L

(25)

in which δ is the hub-radius ratio, ω* represents the dimensionless natural frequencies,  * indicates the dimensionless rotating speed and Ixx denotes the moment of inertia. The vibration analysis of the rotating pure metallic beam and FGM beam is presented to access the convergence of the developed formulation. Table 2 lists the dimensionless natural frequencies of a rotating pure metallic beam obtained with different truncation numbers. For this case, the following geometrical parameters are used: L / h  70

12 , b/h = 1, δ = 0,  * = 10, θ = 0. Note

that the pure metallic material is realized by setting the material parameter to zero. Table 2 reveals that the present model converges well, and not many terms of the modified Fourier series can lead to a good accuracy. The dynamic results of the proposed method are also compared with ones from Fang et al. [9] and Kim et al. [62]. It illustrates that the results are in satisfying agreement with ones predicted by other researchers. Table 3 gives the convergence of several distinguished modes of a rotating FGM beam, when b/h = 1, L/h =10, δ = 0,  * = 10, θ = 0 and p = 5. A great convergence can also be concluded, and the truncation number will be selected as 40 terms to fit the deformation of the rotating FGM beams. Moreover, several related mode shapes are reported in Fig. 2 for illustrative purposes. Table 2. Convergence of several distinguished modes in the case of a rotating pure metallic beam ( L / h  70 12 , b/h = 1, δ = 0, * = 10, θ = 0 and p = 0). Chordwise modes

Flapwise modes

Longitudinal

Torsional

Solutions M=5 M = 10 M = 15 M = 20 M = 25 M = 30 M = 35 M = 40

B1

B2

B3

B1

B2

B3

L1

T2

5.0672 5.0043 4.9835 4.9741 4.9688 4.9656 4.9634 4.9620

32.1260 31.9329 31.8909 31.8713 31.8606 31.8541 31.8498 31.8469

73.0891 72.6558 72.5495 72.4983 72.4705 72.4533 72.4422 72.4346

11.2427 11.2138 11.2044 11.2001 11.1977 11.1962 11.1952 11.1946

33.7009 33.5168 33.4767 33.4580 33.4478 33.4416 33.4375 33.4347

73.8230 73.3938 73.2885 73.2378 73.2103 73.1933 73.1823 73.1747

112.4770 112.1701 112.0717 112.0267 112.0030 111.9893 111.9806 111.9749

190.5940 190.2718 190.1753 190.1322 190.1103 190.0983 190.0911 190.0868

Ref. [9] Ref. [62]

4.9606 5.0562

31.9230 32.1247

73.2860 74.0094

11.1950 11.2420

33.5120 33.7161

74.0150 74.7448

111.3100 111.3161

-

Table 3. Convergence of several distinguished modes in the case of a rotating FGM beam (b/h = 1, L/h =10, δ = 0, * = 10, θ = 0 and p = 5). Chordwise modes

Flapwise modes

Longitudinal

Torsional

L1

T2

Solutions M=5 M = 10 M = 15 M = 20 M = 25 M = 30 M = 35 M = 40

B1

B2

B3

B1

B2

B3

3.3432 3.2960 3.2828 3.2770 3.2739 3.2720 3.2708 3.2699

26.2487 26.1793 26.1612 26.1537 26.1497 26.1473 26.1458 26.1447

53.4897 53.3397 53.3033 53.2868 53.2785 53.2733 53.2700 53.2677

10.8268 10.8028 10.7964 10.7937 10.7922 10.7914 10.7908 10.7904

29.9399 29.8507 29.8265 29.8170 29.8121 29.8092 29.8073 29.8060

58.2540 58.0459 57.9934 57.9708 57.9601 57.9537 57.9497 57.9468

(a) Second chordwise mode, f = 26.1447

41.0150 40.9428 40.9225 40.9137 40.9091 40.9064 40.9047 40.9036

65.5192 65.3764 65.3414 65.3289 65.3231 65.3206 65.3191 65.3183

(b) Second flapwise mode, f = 29.8060

(c) First longitudinal mode, f = 40.9036 (d) Second torsional mode, f = 65.3183 Fig. 2. Several distinguished mode shapes of a rotating FGM beam (b/h = 1, L/h = 10, * = 10, δ = 0, p = 5, θ = 0).

To further verify the correctness of the computational model, Table 4 displays the fundamental natural frequencies for the flapwise and chordwise motions that are provided by the proposed model and previous research data. In the work of Yoo and Shin [36] and Chung and Yoo [63], a slender pure metallic beam ( L / h  70

12 ) was considered to guarantee the effectiveness of the classical

beam theory. Table 4 shows that the fundamental natural frequencies are well matched with the

previous research data for all kinds of rotating velocities and hub-radius ratios. The relative errors between the current results and ones provided by references [36, 63] are also shown in Table 4. The maximum relative error is not greater than 1 percent for the first flapwise and chordwise modes at any rotational speed. Table 4. Comparison of the fundamental natural frequencies for the flapwise and chordwise motions. δ

0

1

5

* 2 10 50 2 10 50 2 10 50

Flapwise

Chordwise

Ref. [36]

present

Error/%

Ref. [63]

present

Error/%

4.14 11.2 51.1 4.83 16.6 79.4 6.94 29.5 145

4.1389 11.187 50.7877 4.8344 16.576 78.9379 6.9395 29.3905 143.8392

0.03 0.12 0.61 0.09 0.14 0.58 0.01 0.37 0.80

3.6196 4.9700 7.3337 4.3978 13.0482 41.2275 6.6430 27.2660 74.0031

3.6219 4.9446 7.2652 4.3991 13.0146 41.0307 6.6412 27.1815 74.1790

0.06 0.51 0.93 0.03 0.26 0.48 0.03 0.31 0.24

Dimensionless natural frequencies of a rotating pure metallic beam are compared with ones obtained from Carrera et al. [26] in Fig. 3. The following geometrical parameters are considered:

L / h  70

12 , b/h = 1, δ = 0.1 and θ = 0. The variations in the chordwise frequencies of the

rotating beam with the dimensionless rotating speed are depicted in Fig. 3a, while the dimensionless natural frequencies of the flapwise modes are displayed in Fig.3b. From those figures, we can see that the dimensionless natural frequencies of the proposed model, both in chordwise and flapwise motions, match well with those from Ref. [26]. Although the differences of the two predictions exhibit an increasing tendency with the growth in the number of the modes, they remain constant or decrease with the increasing rotating speed. The differences are possibly due to that the TE polynomial order used in the Carrera et al. [26] is smaller than the present one. Subsequently, dimensionless natural frequencies for several distinguished modes of a rotating pure metallic beam under various rotating speeds are listed in Table 5.

(a)

(b)

Fig. 3. First six dimensionless natural frequencies of a rotating pure metallic beam with varying dimensionless rotating speed: (a) chordwise modes, and (b) flapwise modes. Table 5. Dimensionless natural frequencies of a rotating pure metallic beam with various dimensionless rotating speeds ( L / h  70 12 , b/h = 1, δ = 0.1 and θ = 0). * 0 10 20 30 40 50 60 70 80

chordwise

flapwise

Torsional

Longitudinal

B1

B2

B3

B1

B2

B3

T1

L1

3.518 6.261 9.609 12.340 14.469 16.108 17.386 18.407 19.251

21.802 33.133 53.734 75.582 97.187 118.096 138.141 157.340 175.823

60.004 74.064 104.138 138.856 174.652 210.555 246.111 281.102 315.432

3.518 11.847 22.446 33.096 43.751 54.405 65.052 75.673 86.193

21.802 34.667 57.636 82.661 107.928 133.441 159.074 184.869 210.532

60.004 74.791 106.347 142.889 181.028 219.516 261.535 300.115 339.801

63.355 63.941 65.739 68.241 71.856 76.151 81.029 86.402 92.257

110.106 112.024 117.673 126.575 138.329 152.296 167.969 184.780 202.575

Next, the variations in the first six dimensionless natural frequencies of a rotating FGM beam with varying rotating speed are displayed in Fig. 4. The FGM beam has the following configurations: b/h = 2, L/h = 10, δ = 0.1, p = 2, θ = 15. According to Fig.4, all the dimensionless natural frequencies increase due to that the centrifugal stiffening effect gets bigger as the rotating speed increasing. The increase is larger for higher modes. The first six mode shapes of a rotating FGM beam at four dimensionless rotating speeds are shown in Fig. 5. In this figure, the black box represents the undeflected beam. It can be seen that increased rotating speed does not influence the mode shapes obviously. Furthermore, curve veering phenomenon first appears between the first and second mode shapes.

Fig. 4. Variation in the dimensionless natural frequencies of a rotating FGM beam with varying dimensionless rotating speed (b/h = 2, L/h = 10, δ = 0.1, p = 2, θ = 15). * = 0

* = 10

* = 20

* = 30

Fig. 5. Mode shapes of a rotating FGM beam at four dimensionless rotating speeds (b/h = 2, L/h = 10, δ = 0.1, p = 2, θ = 15).

Fig. 6(a)-(d) presents the influences of the material parameter on the dimensionless natural frequencies of a rotating FGM beam with different dimensionless rotating speeds. The width-to-height ratio and length-to-height ratio are 1 and 10. The presetting angle and hub-radius ratio are set to 0 and 0.1, respectively. Four kinds of distinguished modes including the first chordwise bending, first flapwise bending, first longitudinal and second torsional modes, are chosen to study the influence of the material parameter. From Fig. 6, the dimensionless natural frequencies decrease rapidly at the beginning of the changing range in material parameter, and then change to stabilize. Furthermore, the material parameter does not affect each mode uniformly, for example, the longitudinal and torsional modes are more sensitive to the material parameter. Dimensionless natural frequencies for the distinguished modes of the rotating FGM beam with different material parameter are listed in Table 6.

(a) First chordwise mode

(b) First flapwise mode

(c) First longitudinal mode (d) Second torsional mode Fig. 6. Dimensionless natural frequencies of several distinguished modes in the case of a rotating FGM beam (b/h = 1, L/h = 10, δ = 0.1 and θ = 0) with varying material parameter. Table 6. Dimensionless natural frequencies of a rotating FGM beam (b/h = 1, L/h = 10, δ = 0.1 and θ = 0). *

p

chordwise

flapwise

Torsional

Longitudinal

B1

B2

B3

B1

B2

B3

T1

L1

10

0 1 2 5 10

5.91 5.37 5.08 4.70 4.46

32.22 29.9 28.86 27.58 26.89

69.93 62.67 59.36 55.31 53.12

11.79 11.69 11.60 11.45 11.36

34.32 33.21 32.40 31.14 30.28

70.79 66.89 63.96 59.86 57.16

31.93 26.45 23.88 21.50 20.47

58.41 49.97 46.04 41.18 38.54

20

0 1 2 5 10

8.10 7.21 6.75 6.10 5.72

51.55 49.45 48.37 46.9 46.03

99.49 93.01 90.87 80.65 85.38

22.29 22.13 22.01 21.85 21.76

57.02 56.28 55.72 55.46 54.38

101.91 100.86 98.10 95.03 93.06

35.33 30.12 27.84 25.83 25.02

68.99 62.12 59.07 54.91 53.59

Fig. 7(a)-(d) depicts the influences of the presetting angle on the vibration characteristics of the rotating FGM beam. The geometric properties of the FGM beam are selected as: b/h = 2, L/h = 10, δ = 0.1 and  * = 10. Four material parameters (p = 0, 1, 2, 3) are considered in this analysis. For the first chordwise mode shown in Fig. 7(a), the following phenomenon can be observed. When the presetting angle is less than 90 degrees, the first chordwise natural frequencies decrease with the presetting angle. Once the presetting angle is larger than 90 degrees, the natural frequencies begin to increase. Regarding the first flapwise mode shown in Fig. 7(b), an opposite trend shows that the natural frequencies increase first and then decrease with the presetting angle. This is due to that the structural stiffness along the chordwise direction is larger than it in the flapwise direction when a small pre-setting angle is considered. With the growth of the pre-setting angle, structural stiffness

along the chordwise direction decreases and the chordwise direction one increases. Furthermore, the longitudinal and torsional frequencies are almost constant with the increase of the presetting angle (see Fig. 7 (c) and (d)).

(a) First chordwise mode

(b) First flapwise mode

(c) First longitudinal mode (d) Second torsional mode Fig. 7. Dimensionless natural frequencies of several distinguished modes in the case of a rotating FGM beam (b/h = 2, L/h = 10, δ = 0.1 and * = 10) with varying presetting angle.

Subsequently, the dynamic analysis of rotating FGM beams with different hub-radius ratios are carried out. The dimensionless rotational velocity is selected as 10, and the presetting angle is set to zero. The length-to-height ratio and width-to-height ratio were selected as 10 and 1, respectively. Four kinds of material parameter (p = 0, 1, 2, 3) are considered as well. Fig. 8(a)-(b) displays the variation of the dimensionless natural frequencies with respect to hub-radius ratio. As expected, all the dimensionless natural frequencies of the rotating FGM beam increase as the hub-radius ratio increasing.

(a) First chordwise mode

(b) First flapwise mode

(c) First longitudinal mode (d) Second torsional mode Fig. 8. Dimensionless natural frequencies of several distinguished modes in the case of a rotating FGM beam (b/h = 1, L/h = 10, * = 10 and θ = 0) with varying hub-radius ratio.

4. Conclusion This paper presented a quasi-3D solution for the dynamic analysis of rotating FGM beams by using the modified Fourier spectral approach (MFSA). Carrera unified formulation (CUF) was utilized for the higher-order configurations of the solutions. Three groups of penalty factor based on the penalty method were introduced for the fixed constraint between the hub and FGM beam. The boundary potential energy was then considered as part of the energy functional. Without any limitation of boundary conditions, the modified Fourier series expansions were chosen as trial functions and the variational method was applied for the governing equations of the rotating FGM beams. The isotropic and FGM beams were considered as numerical examples to examine the convergence and precision of the developed formulation. Several parameter studies were carried out

and some conclusions were concluded as follows. As the rotating velocity and or hub-radius ratio increased, the centrifugal effect played a leading role and all the dimensionless natural frequencies of the rotating FGM beams increased. When the material parameter was considered, the dimensionless natural frequencies decreased with the growth of the material parameter. The investigation on the effect of presetting angle revealed that the chordwise and flapwise modes were more sensitive to the presetting angle. Taking the first chordwise mode as an example, one can observe that the chordwise natural frequencies decreased with the presetting angle increasing from 0 to 90 degrees. Once the presetting angle was larger than 90 degrees, the dimensionless natural frequencies began to increase. It should be noted that, although only the rotating FGM beams with rectangular cross-sections were considered in this study, the computational models could also be used to solve the dynamic problems of rotating FGM beam-type structures with other cross-sections and boundary conditions. Acknowledgement This work was supported by the National Natural Science Foundation of China (Nos. 51822902, 51709066 and 51775125), Heilongjiang Provincial Natural Science Foundation (No. QC2018050), the Fundamental Research Funds for the Central Universities of China (No. HEUCF180305) and Ph.D. Student Research and Innovation Fund of the Fundamental Research Funds for the Central Universities (3072019GIP0303). Appendix A. Expressions of the nucleus matrixes The elements of the nuclear stiffness matrix:

L

K

mnij xx

L

L

  C11 Fi , x Fj , x dA  m  n dy   C44 Fi Fj dA  m  n dy   C66 Fi , z F j , z dA  m  n dy A



 ku 0  m

0

n

y 0

A

y 0

 kuL  m

0

yL

n

A

  F F dA

yL

i

0

j

A

L

(A.1)

L

K xymnij   C12 Fi , x Fj dA  m  n dy   C44 Fi Fj , x dA  m  n dy A

0

A

0

L

L

K xzmnij   C13 Fi , x Fj , z dA  m  n dy   C66 Fi , z Fj , x dA  m  n dy A

0

A

0

L

K

mnij yx

L

  C12 Fi Fj , x dA  m  n dy   C44 Fi , x Fj dA  m  n dy A

0

A

0

L

L

L

K yymnij   C22 Fi Fj dA  m  n dy   C44 Fi , x Fj , x dA  m  n dy   C55 Fi , z F j , z dA  m  n dy A



＋ kv 0  m

0

y 0

A

n

y 0

0

 kvL  m

yL

n

yL

A

  F F dA i

0

(A.2)

j

A

L

L

K yzmnij   C23 Fi Fj , z dA  m  n dy   C55 Fi , z Fj dA  m  n dy A

0

A

0

L

K

mnij zx

L

  C13 Fi , z Fj , x dA  m  n dy   C66 Fi , x Fj , z dA  m  n dy A

0

A

0

L

L

K zymnij   C23 Fi , z Fj dA  m  n dy   C55 Fi Fj , z dA  m  n dy A

0

A

0

L

L

L

(A.3)

K zzmnij   C33 Fi , z Fj , z dA  m  n dy   C55 Fi Fj dA  m  n dy   C66 Fi , x F j , x dA  m  n dy A



 kw0  m

0

y 0

n

A

y 0

 kwL  m

0

yL

n

yL

  F F dA i

A

0

j

A

in which Fi,x, Fi,z and Ψ′n respectively denote the first derivative of Fi and Ψn. Nucleus matrixes of the mass, Coriolis effect, centrifugal stiffening and centrifugal softening matrixes can be expressed as

M mnij

G mnij

  Fi Fj mn A   0   0 

0

L

 0    2  Fi Fj mn A   0 

 Fi Fj

A

 m n

0 2  Fi Fj L

   0   Fi Fj A  m  n L  0

0 0

A

L

mn

L

0  0  0 

(A.4)

(A.5)

K cmnij

K mnij

  c Fi Fj m n A   0   0 

0

L

 2  Fi Fj  m n A   0   0 

 c Fi Fj

A

   0     c Fi Fj A  m  n L  0

m n

L

0 0

L

 2  Fi Fj

A

 m n

L

0

0  0  0 

(A.6)

(A.7)

where   A indicates a surface integral over the beam section, and   L represents a line integral along the axial direction. References [1] Song O, Librescu L. Structural modeling and free vibration analysis of rotating composite thin-walled beams. J Am Helicopter Soc 1997;42:358-69. [2] Oh SY, Librescu L, Song O. Thermoelastic modeling and vibration of functionally graded thin-walled rotating blades. AIAA J 2003;41:2051-61. [3] Librescu L, Oh SY, Song O, Kang HS. Dynamics of advanced rotating blades made of functionally graded materials and operating in a high-temperature field. J Eng Math 2008;61:1-16. [4] Fazelzadeh SA, Malekzadeh P, Zahedinejad P, Hosseini M. Vibration analysis of functionally graded thin-walled rotating blades under high temperature supersonic flow using the differential quadrature method. J Sound Vib 2007;306:333-48. [5] Yoo HH, Ryan RR, Scott RA. Dynamics of flexible beams undergoing overall motions. J Sound Vib 1995;181:261-78. [6] Hong HY, Park JH, Park J. Vibration analysis of rotating pre-twisted blades. J Sound Vib 2001;240:891-908. [7] Ramesh MNV, Rao NM. Free vibration analysis of pre-twisted rotating FGM beams. Int J

Mech Mater Des 2013;9:367-83. [8] Oh Y, Yoo HH. Vibration analysis of rotating pretwisted tapered blades made of functionally graded materials. Int J Mech Sci 2016;119:68-79. [9] Fang JS, Zhou D, Dong Y. Three-dimensional vibration of rotating functionally graded beams. J Vib Control 2018;24:3292-306. [10] Li L, Zhang DG, Zhu WD. Free vibration analysis of a rotating hub-functionally graded material beam system with the dynamic stiffening effect. J Sound Vib 2014;333:1526-41. [11] Ebrahimi F, Mokhtari M. Transverse vibration analysis of rotating porous beam with functionally graded microstructure using the differential transform method. J Braz Soc Meth 2015;37:1-10. [12] Rajasekaran S. Free vibration of centrifugally stiffened axially functionally graded tapered Timoshenko beams using differential transformation and quadrature methods. Appl Math Model 2013;37:4440-63. [13] Ebrahimi F, Dashti S. Free vibration analysis of a rotating non-uniform functionally graded beam. Steel Compos Struct 2015;19:1279-98. [14] Rafiee M, Nitzsche F, Labrosse M. Dynamics, vibration and control of rotating composite beams and blades: A critical review. Thin Wall Struct 2017;119:795-819. [15] Xue Y, Jin G, Ding H, Chen M. Free vibration analysis of in-plane functionally graded plates using a refined plate theory and isogeometric approach. Compos Struct 2018;192:193-205. [16] Bediz B, Romero LA, Ozdoganlar OB. Three dimensional dynamics of rotating structures under mixed boundary conditions. J Sound Vib 2015;358:176-91. [17] Volovoi VV, Hodges DH, Cesnik CES, Popescu B. Assessment of beam modeling methods for rotor blade applications. Math Comput Model 2001;33:1099-112.

[18] Carrera E. A class of two-dimensional theories for anisotropic multilayered plates analysis. Compos Struct 1995:49-87. [19] Carrera E. Theories and finite elements for multilayered, anisotropic, composite plates and shells. Arch Comput Method E 2002;9:87-140. [20] Carrera E. Theories and finite elements for multilayered plates and shells: a unified compact formulation with numerical assessment and benchmarking. Arch Comput Method E 2003;10:215-96. [21] Carrera E, Petrolo M, Nali P. Unified formulation applied to free vibrations finite element analysis of beams with arbitrary section. Shock Vib 2011;18:485–502. [22] Carrera E, Petrolo M, Varello A. Advanced beam formulations for free vibration analysis of conventional and joined wings. J Aerospace Eng 2012;25:282-93. [23] Carrera E, Pagani A, Petrolo M, Zappino E. Recent developments on refined theories for beams with applications. Mech Eng Rev 2015;2:14-00298-14-. [24] Carrera E, Zappino E. Carrera unified formulation for free-vibration analysis of aircraft structures. AIAA J 2016;54:1-13. [25] Min D, Pagani A, Carrera E. Free vibration analysis of simply supported beams with solid and thin-walled cross-sections using higher-order theories based on displacement variables. Thin Wall Struct 2016;98:478-95. [26] Carrera E, Filippi M, Zappino E. Free vibration analysis of rotating composite blades via Carrera Unified Formulation. Compos Struct 2013;106:317-25. [27] Canales FG, Mantari JL. A boundary-discontinuous-displacement based Fourier analysis of thick laminated beams via a robust 1D-CUF model. Int J Solids Struct 2017;118:109-18. [28] Canales FG, Mantari JL. Free vibration of thick isotropic and laminated beams with arbitrary

boundary conditions via unified formulation and Ritz method. Appl Math Model 2018;61:693-708. [29] Canales FG, Mantari JL. Vibrational behavior of isotropic plate structures in contact with a bounded fluid via unified formulation. Chin J Aeronaut 2019;32:921-37. [30] Carrera E, Entezari A, Filippi M, Kouchakzadeh MA. 3D thermoelastic analysis of rotating disks having arbitrary profile based on a variable kinematic 1D finite element method. J Therm Stresses 2016;39:1572-87. [31] Carrera E, Filippi M. Variable kinematic one-dimensional finite elements for the analysis of rotors made of composite materials. J Eng Gas Turbines Power-Trans ASME 2014;136:11. [32] Carrera E, Filippi M. Vibration analysis of thin/thick, composites/metallic spinning cylindrical shells by refined beam models. J Vib Acoust-Trans ASME 2015;137:9. [33] Filippi M, Pagani A, Carrera E. Accurate nonlinear dynamics and mode aberration of rotating blades. J Appl Mech-Trans ASME 2018;85:7. [34] Zirkelback NL, Ginsberg JH. Ritz series analysis of rotating shaft system: Validation, convergence, mode functions, and unbalance response. J Vib Acoust 2002;124:492-501. [35] Ramesh, M. NV, Rao, Mohan N. Free vibration analysis of pre-twisted rotating FGM beams. Int J Mech Mater Des 2013;9:367-83. [36] Yoo HH, Shin SH. Vibration analysis of rotating cantilever beams. J Sound Vib 1998;212:807-28. [37] Cao DX, Liu BY, Yao MH, Zhang W. Free vibration analysis of a pre-twisted sandwich blade with thermal barrier coatings layers. Sci China Technol Sc 2017;60:1-15. [38] Hashemi SH, Farhadi S, Carra S. Free vibration analysis of rotating thick plates. J Sound Vib 2009;323:366-84.

[39] Chen YK, Ye TG, Jin GY. Quasi-3D solutions for the vibration of solid and hollow slender structures with general boundary conditions. Comput Struct 2019;211:14-26. [40] Qin Z, Yang Z, Jean ZU, Chu F. Free vibration analysis of rotating cylindrical shells coupled with moderately thick annular plates. Int J Mech Sci 2018;142-143. [41] Su Z, Jin G, Ye T. Electro-mechanical vibration characteristics of functionally graded piezoelectric plates with general boundary conditions. Int J Mech Sci 2018;138:42-53. [42] Chen M, Jin G, Ma X, Zhang Y, Ye T, Liu Z. Vibration analysis for sector cylindrical shells with bi-directional functionally graded materials and elastically restrained edges. Compos Part B-Eng 2018;153:346-63. [43] Chen M, Jin G, Zhang Y, Niu F, Liu Z. Three-dimensional vibration analysis of beams with axial functionally graded materials and variable thickness. Compos Struct 2019;207:304-22. [44] Li HC, Pang FZ, Wang XR, Du Y, Chen HL. Free vibration analysis of uniform and stepped combined paraboloidal, cylindrical and spherical shells with arbitrary boundary conditions. Int J Mech Sci 2018;145:64-82. [45] Li HC, Pang FZ, Miao XH, Gao SY, Liu F. A semi analytical method for free vibration analysis of composite laminated cylindrical and spherical shells with complex boundary conditions. Thin Wall Struct 2019;136:200-20. [46] Chen YK, Jin GY, Zhang CY, Ye TG, Xue YQ. Thermal vibration of FGM beams with general boundary conditions using a higher-order shear deformation theory. Compos Part B-Eng 2018;153:376-86. [47] Jin G, Ye T, Wang X, Miao X. A unified solution for the vibration analysis of FGM doubly-curved shells of revolution with arbitrary boundary conditions. Compos Part B-Eng 2016;89:230-52.

[48] Ye T, Jin G, Zhang Y. Vibrations of composite laminated doubly-curved shells of revolution with elastic restraints including shear deformation, rotary inertia and initial curvature. Compos Struct 2015;133:202-25. [49] Chen M, Jin G, Ye T, Zhang Y. An isogeometric finite element method for the in-plane vibration analysis of orthotropic quadrilateral plates with general boundary restraints. Int J Mech Sci 2017;133:846-62. [50] Li W. Free vibrations of beams with general boundary conditions. J Sound Vib 2000;237:709-25. [51] Zhang X, Li W. Vibrations of rectangular plates with arbitrary non-uniform elastic edge restraints. J Sound Vib 2009;326:221-34. [52] Yang C, Jin G, Zhang Y, Liu Z. A unified three-dimensional method for vibration analysis of the frequency-dependent sandwich shallow shells with general boundary conditions. Appl Math Model 2019;66:59-76. [53] Zhou K, Su JP, Hua HX. Aero-thermo-elastic flutter analysis of supersonic moderately thick orthotropic plates with general boundary conditions. Int J Mech Sci 2018;141:46-57. [54] Shi SX, Su Z, Jin GY, Liu ZG. Vibro-acoustic modeling and analysis of a coupled acoustic system comprising a partially opened cavity coupled with a flexible plate. Mech Syst Signal Proc 2018;98:324-43. [55] Wang XZ, Xu EH, Chen Z, Jiang CB, Fang YM. A semi-analytical solution for free vibration of thick orthotropic annular sector plates with general boundary conditions, internal radial line and circumferential arc supports. Ocean Eng 2018;163:679-90. [56] Jin G, Ma X, Wang W, Liu Z. An energy-based formulation for vibro-acoustic analysis of submerged submarine hull structures. Ocean Eng 2018;164:402-13.

[57] Jin G, Ma X, Shi S, Ye T, Liu Z. A modified Fourier series solution for vibration analysis of truncated conical shells with general boundary conditions. Appl Acoust 2014;85:82-96. [58] Zhao J, Wang Q, Deng X, Choe K, Zhong R, Shuai C. Free vibrations of functionally graded porous rectangular plate with uniform elastic boundary conditions. Compos Part B-Eng 2019;168:106-20. [59] Qin Z, Chu F, Zu J. Free vibrations of cylindrical shells with arbitrary boundary conditions: A comparison study. Int J Mech Sci 2017;133:91-9. [60] Piovan MT, Sampaio R. A study on the dynamics of rotating beams with functionally graded properties. J Sound Vib 2009;327:134-43. [61] Carrera E, Giunta G, Petrolo M. Beam Structures: Classical and Advanced Theories. Wiley; 2011. [62] Kim H, Hong HY, Chung J. Dynamic model for free vibration and response analysis of rotating beams. J Sound Vib 2013;332:5917-28. [63] Chung J, Yoo HH. Dynamic analysis of a rotating cantilever beam by using the finite element method. J Sound Vib 2002;249:147-64.

Graphical abstracts