A study of the vibration and lay-up optimization of rotating cross-ply laminated nanocomposite blades

Journal Pre-proofs A study of the vibration and lay-up optimization of rotating cross-ply laminated nanocomposite blades Rengjin Xiang, Zhou-Zhou Pan, Hua Ouyang, Lu-Wen Zhang PII: DOI: Reference:

S0263-8223(19)34554-4 https://doi.org/10.1016/j.compstruct.2019.111775 COST 111775

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Composite Structures

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30 November 2019 3 December 2019

Please cite this article as: Xiang, R., Pan, Z-Z., Ouyang, H., Zhang, L-W., A study of the vibration and lay-up optimization of rotating cross-ply laminated nanocomposite blades, Composite Structures (2019), doi: https:// doi.org/10.1016/j.compstruct.2019.111775

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A study of the vibration and lay-up optimization of rotating cross-ply laminated nanocomposite blades Rengjin Xiang1, §, Zhou-Zhou Pan2, §, Hua Ouyang3, Lu-Wen Zhang1, * 1 Department

of Engineering Mechanics, School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, China

2 Department

of Architecture and Civil Engineering, City University of Hong Kong, Kowloon, Hong Kong, China

3School

of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai 200240, China

§ the authors contribute equally to this work * Corresponding author: [email protected]

Abstract A comprehensive understanding of the vibration characteristics of rotating blades is critical to avoid resonance fracture. This paper presents the vibration analysis and layup optimization of rotating pre-twisted laminated functionally graded CNTs reinforced composite (FG-CNTRC) shallow conical shells. A numerical shallow conical shell model is proposed, and the displacement fields of the shell model are described via the first-order shear deformation theory. Considering the rotating speed, the governing equation of the rotating pre-twisted laminated conical shell is derived, and then solved using the meshless kp-Ritz method. We carry out the comparison studies to demonstrate the accuracy and efficiency of the present model in dealing with the vibrations of this kind of structures. Detailed parametric studies are then performed and demonstrate that the CNT configuration, pre-twisted angle, and stacking sequence play essential roles in vibration behaviors of rotating FG-CNTRC blade. With the nondimensional frequency parameter being the optimization objective function, a genetic algorithm is employed to search for the optimal layering sequence of the rotating shallow conical shells. The optimized results demonstrate that the GA serves as a useful tool for sequence optimization of rotating blade in terms of improving the vibration characteristics. Keywords: free vibration; rotating pre-twisted blade; FG-CNTRC; stacking sequence

optimization; genetic algorithm 1. Introduction Traditionally, alloys are the most widely used for fabricating blades. However, more advanced materials have been emerging mainly due to the advances of the material science and fabrication technology. For example, due to the high specific modulus and strength, carbon fiber reinforced composites have served as a better substitute for the heavy metals. More recently, carbon nanotubes have been experimentally and theoretically justified as a kind of excellent nano-reinforcement due to their extremely large modulus and high specific area [1-3]. For example, compared with the neat high-density polyethylene, the Young’s modulus was reported with an increment of 22.23% by adding the CNTs with a volume fraction of 0.44% [4]. It was also shown that the tensile strength increased from 16 to 58 MPa and elongation at break increased from 2.33% to 5.22%, after adding 1 wt.% plasma treated CNTs into the epoxy [5]. Moreover, many investigations showed that arranging the CNTs in functionally graded configurations could further improve the mechanical properties of carbon nanotube reinforced composite structures. Researchers interested in this area are recommended to refer to the earlier works by Liew et al. [3, 6-9]. Therefore, the unique properties of FG-CNTRC have made them becoming great potential applications in rotating blades. Rotating structures, such as turbomachinery blades, aerial fans, and marine propellers are widely used in civil and ocean engineering, aeronautical and aerospace industries. Many events have shown that resonance fracture of rotating structure plays a key role in causing the catastrophe of property and life [10-12]. Therefore, a comprehensive understanding and optimization of vibration behaviors of pre-twisted blades are helpful to avoid the disasters caused by resonance fractures. Rotating blades generally possess complex geometries [13]. In order to model the blade structure, especially at the predesign stage, some simplifications of the real blade may not only grasp the main mechanical behaviors but also accelerate the product design process. In previous works, the blades were modeled as beams. The beam theories can accurately predict the behaviors of slender blades, while they are failed in predicting the chordwise

bending and coupled chordwise and spanwise bending vibration modes which are surely existed in short blade structures [14, 15]. To overcome the limitations of beam models, the plate [16-19] and shell [14, 20-23] models have been adopted to model the blade structures with small aspect ratios. However, the investigations on the mechanical behaviors of rotating blades consisting of FG-CNTRC can be hardly found. To the best knowledge of the authors, they can only find one study in a very recent publication. Rout and Karmakar [24] modeled the turbomachinery blades by a twisted cantilever cylindrical shell consisting of a single FG-CNTRC layer and investigated the vibration characteristics of the rotating blades. However, the pre-set angle and the hub radius were not considered in the study. Thus, their effects on the vibration characteristics of FG-CNTRC blades are not well understood. In the real product, a blade is always made of a stacking of layers with varied ply angles instead of a single layer to improve the mechanical properties in every direction [25]. It was reported that stacking sequence was one of the most important and most frequently used design variables in composite structures [26, 27]. So there arises a question about the optimal stacking sequence of rotating pre-twisted laminated FG-CNTRC shallow conical shells. laminated composite structures are usually consisting of a stack of layers with fiber orientations being limited to a set of angles such as 0◦, ±45◦ and 90◦ [28]. As a consequence, searching for the optimal stacking sequence of a laminated composite structure is an integer programming problem. Numerous methods, including simulated annealing approach [29], particle swarm optimization algorithm [30-32], elitist ant system optimization [33], hybrid particle swarm optimization and simulated annealing method [34], genetic algorithm (GA) [27, 28, 35], and other optimization method [26] have been employed to solve the optimal stacking sequence problems of laminated composite structures. Among the algorithms, GA is mostly widely used in the optimization of the stacking sequence due to its great ability in dealing with optimization problems with discrete design variables and mixed discrete and continuous variables. Abouhamze and Shakeri [36] carried out a stacking sequence optimization to maximize the weighted sum of first natural frequency and

buckling loads of a laminated composite cylindrical panel. Using a GA in combination with the NURMS-based isogeometric analysis, Le-Manh and Lee [37] optimize the stacking sequence and predict the maximum strengths of laminated composite plates subjected to transverse loads and compress loads. Wei et al. [27] proposed a stiffness coefficient-based method to search the best stacking sequence of a composite cylindrical shell that led to the maximum buckling load. The method was based on the optimal results obtained by using the GA coupling with FEM. Chen et al. [28] conducted a stacking sequence optimization problems with the objective as the structural weight based on a two-level approximation and GA. The stacking sequence optimizations of laminated composite structures can also be found in [38-42]. The aim of this paper is to study the free vibration and search for the optimal stacking sequence of rotating pre-twisted laminated FG-CNTRC shallow conical shells. The FG-CNTRC layers with varied ply angles are assumed to be bonded perfectly. Three types of FG-CNTRC layer are considered in the present study. The geometric properties of the pre-twisted shallow conical shell are modeled based on the analytic geometry theory, and the displacement fields are described using the FSDT. The meshless kp-Ritz method is adopted for the numerical computation of the governing equations. The detailed parametric studies are conducted to mainly determine the effects of nano fillers and lay-up sequences on the vibration behavior of the nanocomposite blades. In addition, with the nondimensional frequency parameter being the optimization objective function, a genetic algorithm is presented to search for the optimal layering sequence of the rotating shallow conical shells.

2. Problem definition 2.1 Mathematical model of the pre-twisted blade As shown in Figure 1(a) and (b), an untwisted shallow conical shell structure may be obtained by truncating a cone with oval cross section. The major radius and minor radius of the base of the cone are 0 and 0 , respectively. The length and the vertex angle of the cone are denoted by a0 and φ1 , respectively. The length a, width b, and

base subtended angle φ0 are demonstrated in Figure 1(b). The reciprocal of curvatures along the  1 and  2 direction are denoted by R1 and R2 , respectively. Obviously,

R1 is infinity and R2  1 ,  2  is a function of  1 and  2 . According to [15], the

R2  1 ,  2  can be given as 3

 1  2 1 1  2 R2  1 ,  2    2  2  2  22  2  2   ,      

(1)

   tan( 1 )  a0  x  ,

(2)

where

2

b1 tan(



4  tan ( 2

2

0 2

0 2

) ,

(3)

)b

2 1

in which

   b1  b 1  1  ,  a0 

b  2a0 sin

tan 2

1 2

cos 2

1 2

(4)

0 2

+tan 2

0

.

(5)

2

The pre-twisted laminated shallow conical shell with thickness h and of twisted angle  is shown in Figure 2(a). And the rotating system consisting of a stiff hub and a shallow conical shell is illustrated in Figure 2(b). The radius of the hub is r and the pre-set angle of the blade is 1 . The rotating speed is denoted by 0 . According to[43], the twist curvature R12 can be expressed as R12 

-a . tan  

(6)

2.2 CNTR-FG layers The armchair (10, 10) Single-Walled Carbon Nanotubes (SWCNTs) are used as

fillers in the present study. The distributions of CNTs through the thickness of conical shells are denoted by UD, FG-O and FG-X. And the corresponding volume fractions

V 1  3  with respect to thickness  3 is given as   V*  1   2 3 V 1  3   2 1  h    4 3  V1*  h

 UD   *  V1  FG-O  . 

(7)

 FG-X 

Using the extended rule of mixture, we can calculate the elastic modulus, Poisson’s ratio of FG-CNTRC layer [44] as the following E1  1V 1 E11  V 2 E 2 ,

2 E2

3 G12

(8)



V1 V 2  , E21 E 2

(9)



V1 V 2  , G121 G 2

(10)

 12  V1* 121  V 2 2 ,

(11)

2 2 where the properties of isotropic matrix are expressed as E 2 ,  and G , the properties 1 of the CNT are denoted by E11 , E21 ,  12 and G121 . Here E , G , and  denote the

Young’s modulus, shear modulus, and Poisson’s ratio, respectively. The superscript 1 denotes the matrix while 2 denotes the CNT. The subscript 1 and 2 represent the parameters along the axial direction and transverse direction, respectively. 1 , 2 and 3 are efficiency parameters to account for the small-scale effect of CNT and the interfacial interaction between CNT and the polymer matrix [44].

3. Theoretical analysis 3.1. Energy formulation The FSDT indicates that the displacement at any point  1 ,  2 ,  3  of the blade can be expressed as [45],

 u ( 1 ,  2 ,  3 )   u0 ( 1 ,  2 )       v( 1 ,  2 ,  3 )    v0 ( 1 ,  2 )     w( ,  ,  )   w ( ,  )  1 2 3    0 1 2 

 1 ( 1 ,  2 )      2 ( 1 ,  2 )  ,   0  

(12)

where u0 ( 1 ,  2 ) , v0 ( 1 ,  2 ) and w0 ( 1 ,  2 ) denote the translational motions at the middle surface along the  1 ,  2 , and  3 respectively,  1 ( 1 ,  2 ) and  2 ( 1 ,  2 ) represent the functions of the rotation angle of surface normal at location of ( 1 ,  2 ) . The relationships between strain and displacement are expressed as

 12 

1 

u0 w0  1 , + + 3  1 R1  1

(13)

2 

v0 w0  2 , + + 3  2 R2  2

(14)

u0 v0 2 w0  2  1 , + 3 + 3 + +  1  2  2  1 R12

(15)

 13   1 +

w0 u0 v0 , -  1 R1 R12

(16)

 23   2 +

w0 v0 u0 . -  2 R2 R12

(17)

The total strains can be decomposed into in-plane strain ε and the transverse shear strain γ

    1  u0 w0 +      1 R1  1         2  v0 w0 + ε  ε0   κ    + 3  ,  2 R2  2      u0 v0 2 w0    1  2    +       2  1    2  1 R12 

(18)

w0 v0 u0    2   - R - R   2 2 12  γ .  w u v 0 0 0    -  1  1 R1 R12 

(19)

The linear constitutive relationship of the kth layer is expressed as

k

  1   K11    K 2 12       12    K16    0  23    13    0 

K12

K16

0

K 22 K 26 0 0

K 26 K 66 0 0

0 0 K 44 K 45

0  0  0   K 45  K 55 

k

 1     2     12  ,    23    13  

(20)

where

K11 =K11cos 4θ+2(K12 +2K 66 )sin 2θcos 2θ+K 22sin 4θ K16 =(K11 -K12 -2K 66 )sinθcos3θ+(2K 66 -K12 +K11 )sin 3θcosθ K12 =(K11 +K12 -4K 66 )sin 2θcos 2θ+K12 (sin 4θ+cos 4θ) K 26 =(K11 -K12 -2K 66 )sin 3θcosθ+(2K 66 -K12 +K11 )sinθcos3θ K 22 =K11sin 4θ+2(K12 +2K 66 )sin 2θcos 2θ+K 22 cos 4θ K 66 =K 66 (sin 4θ+cos 4θ)+(K11 +K 22 -2K12 -2K 66 )sin 2θcos 2θ ,

(21)

K 45 =(K 55 -K 44 )sinθcosθ K 55 =K 55cos 2θ+K 44sin 2θ K11 =

E1

1- 12 21

,K 22 =

E2

1- 12 21

,K12 =

 12 E1 1- 12 21

K11 =G11 , K 44 =G23 , K 55 =G13 in which K ij is the transformed stiffness matrix,  is the lamination angle. Strain energy stored in the laminated shell due to deformation can be expressed as

1 ε T SεdΩ ,  Ω 2

(22)

ε 0      κ  , γ  

(23)

V where

 A1  S  A2 0 

A2 A3 0

0  0 , A 4 

(24)

in which the components of matrixes A1 , A 2 , A 3 and A 4 are described as [45]

( A1ij , A2ij , A3ij )  

h /2

 h /2

Aij4  K 

K ij (1,  3 ,  32 )d 3 , (i, j  1, 2, 6),

h /2

 h /2

K ij d 3 , (i, j  4,5),

(25) (26)

in which K denotes the shear correction factor. The kinetic energy of laminated conical shell can be represented as

1 V3   h（u 2 +v2 +w 2） dΩ . 2

(27)

For moderate rotations, the Coriolis effects are relatively small. Thus, they are neglected in the present study. However, the effect of body forces due to the rotation is considered. Consequently, two additional terms of potential energy V1

and V2 are

introduced

w   N1   2   N01 2 0

1  w Ω 2    1

V1  

V2  -

 hΩ 0 2 2

 w  N1 2    1   dΩ ,  N02   w      2 0

sin  r  w2 dΩ ,

(28)

(29)



in which N01  N 2  0

 hΩ 0 2 2

 hΩ 0 2 2

 r+a 

2

  r+ 1 

2

,

  b 2  cos  r      2 2  ,  2    

(30)

2

(31)

N01 2  0 .

(32)

Finally, the total energy of the rotating system can be written as

 = V -V3 +V1 +V2 .

(33)

3.2. Discrete system equations To solve the differential equation, the meshless kp-Ritz method [46, 47] is used in this article. For the meshless method, a series of node x I , I=1, ... , K, are generated to represent the computational domain, and the generalized displacement at any point can be obtained by L

 u, v, w, 1 , 2     I (ς)  u I , vI , wI , 1I , 2 I eit , I 1

(34)

where  I (ζ) denoted the shape function, which can be represented as the product of kernel function and correction function

 I (ς)  C(ς; ς  ς I ) a (ς  ς I ) ,

(35)

in which the correction function is C(ς; ς  ς I )  H T (ς  ς I )M 1 (ς )H (0) ,

(36)

H T (ς  ς I )  [1, ς1  ς1I , ς 2  ς 2 I , (ς1  ς1I )(ς 2  ς 2 I ), (ς1  ς1I ) 2 , (ς 2  ς 2 I ) 2 ] ,

(37)

where

J

M (ς )   H (ς  ς I )H T (ς  ς I ) a (ς  ς I ) ,

(38)

H (0)  [1, 0, 0, 0, 0, 0, ]T .

(39)

I 1

When there are only two parameters, the kernel function  a (ς  ς I ) is expressed as

 a (ς  ς I )   a (ς1 )   a (ς 2 ) ,

(40)

where

 a (ς )   (

ς  ς I ), (  1, 2) . a

(41)

For the meshless method, it is crucial to select a proper shape function to ensure the efficient implementation of the algorithm. Considering the shape of integral region, the spline function would be an alternative with desired computational efficiency

2 2 3  3  4 zI  4 zI  4 4  z ( z I )    4 z I  4 z I2  z I3 3 3 0   Here z I 

 ς  ςI  , dI dI

1  2   1  for  zI  1  . 2  otherwise    for 0  z I 

(42)

is the support domain size of point I and can be

expressed as

d I  d max cI ,

(43)

where d max denoted the scaling factor, cI is selected to have the determinant of M nonzero. The limitation of meshless methods has been identified as being not straightforward to treat essential boundary conditions. Many solutions have been proposed to address this problem, the full transformation method [48] is employed in the present study. Generalized displacement

can be written as L

u K  uˆ (ς K )   LJK u J ,

(44)

LJK   J (ς K ) ,

(45)

J 1

then L

J . u J   LJKT u

(46)

J 1

By combining Eq. (46) with Eq. (44), the generalized displacement can be rewritten as L

u K    I (ς )u I ,

(47)

I 1

where L

 J (ς)   LJKT K (ς) .

(48)

K 1

Thus, the reconstruction shape function possesses the Kronecker delta property

 J (ς K )   JK

(49)

Substituting Eq. (34) into Eq. (33) and using the principle of minimum potential energy, the free vibration eigenvalue equation is derived as

 Η - ω Θ  u  0 . 2

(50)

 are given as follows: Frequency is denoted by ω ; Η and Θ  = Λ 1ΘΛ -T , Η = Λ -1KΛ -T，Θ

(51)

K = K b  K m  K s  K r1  K r2 ,

(52)

K bIJ   BbI A 3 BbJ dΩ , T

(53)

Ω

K mIJ   B mI A1B mJ dΩ   B mI A 2 BbJ dΩ   BbI A 2 B mJ dΩ ,

(54)

K sIJ   B sI A 4 B sJ dΩ ,

(55)

K rIJ1   B rI1 N1B rJ1 dΩ ,

(56)

K rIJ2   B rI2 B rJ2 dΩ ,

(57)

  G ΘG dΩ . Θ IJ  I J

(58)

T

T

Ω

T

Ω

Ω

T

Ω

T

Ω

T

Ω

Ω

where

 J - R 1 s BJ =   J - R  12

-

J

-

J

R12 R2

 J  1

J

 J  2

0

 J  0 0  1 B rJ1 =   J  0 0  2 

 J1   Θ=   J2   I   GI =    

 0 0 ,  0 0 

J2 J1 J1 J3 J2

I

 N01 N1 =  0  N1 2

(60)

(61)

   ,    I 

(62)

I

BrI2 =  0 0 J

(59)

 J 2  ,   J 3 

0

0

 0 ,  J  

I

0 0 , N01 2  , N02 

(63) (64)

in which h /2

 J1 , J 2 , J 3  =  h /2  (1,  ,  2 )d .

(65)

In this paper, the bending stiffness matrix K b is calculated by the stabilized nodal integration[49], while the other stiffness matrices K m , K s , K r1 , K r2 and mass matrix

 are determined by direct nodal integration[50]. Θ IJ 3.3. Genetic algorithm for lay-up optimization GA, a class of evolutionary algorithms inspired by evolutionary biology such as inheritance, mutation, selection, and crossover, are widely used in generating true or approximate solutions to optimal search problems. In order to simulate the evolution of organisms, the variables in the actual problem are used to represent genes. The variables are arranged in a certain order to form chromosomes which represent individuals in the population. In this article, the stacking sequence

 1, 2 ,, i ,, n  ( i denotes

the

layer angle) represent chromosomes. The evolution process is divided into reproduction, crossover and mutation, genes in chromosomes are encoded in binary code, and the flowchart is illustrated in Figure 4. The random generation of initial population marks the beginning of the algorithm, and the fitness of each individual is calculated by decoding the genes on the chromosome. The next generation will be selected based on the fitness of the previous generation of individuals. After the new population is determined, all chromosomes will undergo crossover and mutation operations through the defined mutation rate, crossover rate. In the process, we keep all the genes on the best chromosome until the next one comes along. By repeating the above steps, the result will tend to be closer to the optimal results. 4. Results and discussion We firstly validate the accuracy of the proposed model and the numerical method by the comparison study. Secondly, the vibration of the rotating pre-twisted laminated FG-CNTRC shallow conical shell is then studied. The material properties of the polymer matrix and CNTs [46] used in the study are:  2 =0.34, E 2 =2.5GPa, 1 1 1 1 E11 =5.6466TPa, E21 =7.08TPa, G12 =G13 =G23 =1.9455TPa, and 12 =0.175.

The

efficiency parameters of CNT are listed in Table 1[51]. For the boundary condition of the pretwisted blade, the root of the blade is clamped and the other three edges are free (CFFF):  1  0, u0  v0  w0   1   2  0. To eliminate the effect of the geometric parameter, the nondimensional frequency parameter    a 2  / E 2 is used. 4.1 Validation study To validate the convergence and accuracy of the theoretical and numerical method, several convergence and comparison studies are conducted. Firstly, we investigate the vibration of a cantilevered (CFFF) shallow conical Eglass/epoxy plate. The geometry factors of the plate are: a / h  100.0 , a / b0  1.5 , θ v =15  ,

θ 0 =30  ,   15 , and the material properties of the E-glass/epoxy are:

E1 =60.7GPa, E2 =24.8GPa, G12 =12.0GPa,  12 =0.23. The convergence results of the first frequency parameter obtained by the present model is shown in Figure 5. It is observed that the frequency parameter converges quickly to a stable value. Accounting for the balance of efficiency and accuracy, number of nodes 18×18 are used for discretizing the physical domain in the study. Secondly, we conduct a comparison study of the free vibration of a cantilevered (CFFF) shallow conical shell with the angle-plied glass/epoxy structure. The geometrical parameters and material properties are the same as those used in the convergence analysis. The results are listed in Table 2. We predict the results with highly agreement with the reference [15]. The third case is a comparison study of the vibration behaviors of a pre-twisted cantilevered (CFFF) 3-ply graphite/epoxy plate. The geometry factors are: a / b  1.0 ,

c r  1.0 , a / h  100.0 ,  15 ; stacking sequence of the plate is (β, -β, β); and the material properties are: E1 =138GPa , E2 =8.96GPa , G12 =7.1GPa , 12 =0.30 . Table 3 shows the comparisons between the present results and the values from reference [52]; a good agreement has been observed. Accordingly, it can be concluded that the present numerical model is accurate in predicting the vibration characteristics of pre-twisted laminated conical shells. 4.2. Free vibration of pre-twisted laminated FG-CNTRC shallow conical shell

In this section, the influence of geometric parameters, CNT configuration and volume fraction, and stacking circumstances on the nondimensional frequency parameters    a 2  / E 2 of rotating pre-twisted laminated FG-CNTRC shallow conical blades are investigated in details. The frequency parameters of rotating pre-twisted laminated shallow conical shells with different distribution configurations and volume fractions of CNT are presented in Table 4. The geometry factors are: a / h  100.0 , a / b  1.5 , r / a  1 ,   15 ,

1  45 , 1  15 , 0  30 , respectively, with the stacking sequence (-45˚, 45˚, -45˚, 45˚). The rotation speed is selected as 0 =300 / a (rad / s ) . It is observed that the frequency parameters increase with the enhancement of the content of CNT. For the same volume fraction of CNT, the frequency parameters of blades are the highest with CNTs in FG-X type while the lowest ones with CNTs in FG-O type, which implies that FG-X composites have the highest stiffness while FG-O composites exhibit the lowest. Therefore, distribution patterns of CNTs might be designed to improve the vibration behaviors of rotating blades. The effect of stacking sequence of FG-CNTRC layers on the nondimensional frequency parameters of pre-twisted laminated FG-CNTRC shallow conical shells is presented in Table 5. And the variation of the first-order frequency parameter with respect to β is illustrated in Figure 6. According to the results presented in Table 5 and Figure 6, it is evident that as β increasing, the first eight frequency parameters rise firstly and drop later for all the blades regardless of the CNT distribution configurations. why? The large effect of stacking sequence on the vibration of rotating blade demonstrates the importance of selecting the appropriate stacking sequence of FGCNTRC layers. The frequency parameters of the pre-twisted laminated shallow conical shell corresponding to various twisted angles are reported in Table 6. The curves of the first order frequency parameter with respect to 

are displayed in Figure 7. In the

simulation, 0 =300 / a (rad / s ) , V1*  0.17 , the stacking sequence is chosen as (-45˚, 45˚, -45˚, 45˚)), the geometry factors are

a / h  100.0,

r / a  1,

a / b  1.5

φ 0  3 0  , φ 1  1 5  , 1  45 and  varies from 0˚ to 45˚. The results show that the

fundamental frequency parameters decrease quickly as the increasing of twisted angle, while the trends of higher frequency parameters are not monolithic. The effect of the pre-set angle θ on the frequency parameters of rotating conical shell is illustrated in 1 Table 10 and Figure 10. It is observed that the frequency parameters decrease very slightly as the pre-set angle increases. In Table 7 and Table 8, the influences of the length-to-thickness ratio ( a / h =60, 80, 100, 200, 500) and aspect ratio ( a / b =1.5, 2.0, 2.5, 3.0, 3.5) are studied. The trends of the first-order frequency parameter with respect to a / h and a / b are shown in Figure 8 and Figure 9, respectively. As can be seen, the frequency parameters decrease dramatically with the increase of a / h or a / b . It can be explained by the fact that the structure becomes more flexible when it is slender and thinner. Table 11 and Figure 11 show the values and trends of frequency parameters of the pre-twisted laminated shallow conical shells with radius-to-length varying from 0 to 2 with an increment of 0.5. It is found that the ratio r / a has a little effect on the frequency parameter. The effect of number of layers on the frequency parameters is investigated and shown in Table 9. The stacking sequence is (-45˚, 45˚)k , where the layer k is considered from 2 to10 with an increment of 2. It is shown that as the number of layers increases from 2 to 4, the frequency parameters of the pre-twisted laminated shallow conical shell increased substantially, while the increasing rate slows down with further increasing the number of layers and the frequency parameters keep almost steady when the number of layers reaches 8. 4.3. Optimization of stacking sequence From the above study, we have investigated the vibration characteristics of rotating blades with stacking sequences of (-β, β, -β, β) and (-45˚,45˚)k. Considering the actual situation of industrial manufacture that the ply angles of blade layers are among the values of 0, 30, 45, 60,90, there are numerous of stacking circumstances. Then it would be essential to optimize the stacking sequence for the purpose of regulating the static and dynamic performance of the structure. This optimal problem

can be mathematically expressed by

  max. f ( X )  ,  X  1 ,  2 , , i , ,  n    i  0, 30, 45, 60,90 where n denotes the number of layers which is set as 20 in the present study. Using the GA presented in the section 3.3, the optimized stacking sequence is generated in this study. Here the parameters are set as a / h  100,

a / b  1.5, r / a  0, φ 1  1 5  ,

φ 0  3 0  ,   15 , 0 =300 / a (rad / s ) . With a randomly generated initial stacking

sequence, the evolution of the frequency parameter with respect to the generation N is monitored. To show the robustness of the proposed algorithm, three independent tests about the stacking sequence optimization are conducted, and the evolution processes of the first frequency parameters are illustrated in Figure 12 (a) and (b), for shells with CNT volume fraction of 0.12 and 0.17, respectively. The results show that the frequency parameter converges quickly to a stable value with less than 100 evolution generations. When compared with the total number (260) of possible cases in terms of stacking sequence, about 2 thousand times of calculation would be accurate enough to generate the optimal results. This demonstrates that genetic algorithm is a very efficient tool in searching for the optimal stacking sequences of laminated composite blades. The optimized sequences with related frequencies are listed in Table 12. The nondimensional fundamental frequencies of the blade with the optimal sequence are almost the same for all three test cases, revealing that the implemented genetic algorithm is robust in dealing with the sequence optimization of rotating blade problems. Therefore, the GA may serve as a useful tool for sequence optimization in the design process of rotating blade for improving its vibration characteristics. 5. Conclusion The free vibration of rotating pre-twisted laminated FG-CNTRC shallow conical shell is conducted. A shallow conical shell model is proposed to model the geometric properties and the FSDT is adopted to describe the displacement fields of the shell

structure. The governing equation of the rotating pre-twisted laminated conical shell is derived within the energy framework, and the computation of the derived equations are performed using the meshless kp-Ritz method. The effects of many factors, including CNT volume fractions, ratio of radius to length, number of layers, stacking sequence, twisted angle, aspect ratio, pre-set angle, distribution type of CNT and length-tothickness ratio on the nondimensional free vibration frequency parameters are investigated and discussed. In addition, with frequency parameters as the objective function, the optimization of the stacking sequence of layers is solved using a genetic algorithm. The optimization progress and results show that the GA is of high efficiency and accuracy in dealing with the stacking sequence optimization problem of rotating pre-twisted laminated FG-CNTRC shallow conical shells. Within the present numerical framework, the forced vibration of rotating FG-CNTRC blades subjected to external forces will be investigated in the future.

Acknowledgements The work is supported by Natural Science Foundation of Shanghai (Grant No. 19ZR1474400) and National Natural Science Foundation of China (Grant No. 11872245).

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Table 1. Efficiency parameters of the CNT[51] V1*

 1

 2

 3

0.12

0.137

1.022

0.715

0.17

0.142

1.626

1.138

0.28

0.141

1.585

1.109

Table 2. Frequency parameter    a  / E11 of a CFFF E/E shallow conical shell ( a / h  100.0 , a / b0  1.5 , θ v  15 , θ 0  15 ,   15 ) Lamination

Reference

β

Mode sequence 1

2

3

4

0.021271

0.066337

0.12163

0.13323

30˚ 0.022170

0.064450

0.12275

0.13856

Liew and Lim[15] 45˚ 0.020996

0.059658

0.11491

0.14857

60˚ 0.019264

0.055021

0.10636

0.15282

90˚ 0.016933

0.050176

0.095991

0.14031

0˚

0.020931

0.063924

0.120923

0.129254

30˚ 0.021066

0.061573

0.118102

0.140472

45˚ 0.020036

0.057735

0.111538

0.150229

60˚ 0.018543

0.053385

0.103655

0.148480

90˚ 0.016475

0.048362

0.093689

0.135827

0˚

0.021362

0.066337

0.12163

0.13323

30˚ 0.021362

0.063875

0.12016

0.13726

Liew and Lim[15] 45˚ 0.020405

0.059428

0.011332

0.14583

60˚ 0.018954

0.055037

0.10550

0.15022

90˚ 0.016933

0.050176

0.095991

0.14031

0˚

0.020932

0.063924

0.120923

0.129254

30˚ 0.020407

0.060917

0.116626

0.138502

45˚ 0.019417

0.057240

0.110001

0.147070

60˚ 0.018095

0.053069

0.102353

0.146465

90˚ 0.016475

0.048362

0.093689

0.135827

0˚

[(-β,β)4]sym

present

[(-β,β)4]unsym

present

Table 3. Frequency parameter ωa 2  / E11h3 of the pre-twisted 3-ply graphite/epoxy plate with a / b0  1.0 , c r  1.0 , b / h  100.0 ,  45 , and stacking sequence(β, -β, β) Reference β

Mode sequence number 1

2

3

4

5

6

7

8

9

10

0˚ 0.8608 5.1012 12.764 14.449 15.861 16.347 21.273 25.315 26.535 27.149 Qatu and 30˚ 0.6549 3.8350 11.708 11.967 14.012 15.750 22.133 26.077 27.522 28.092 Leissa [52] 60˚ 0.3135 1.8384 5.9160 9.4600 13.645 17.099 18.990 19.603 22.953 25.397 90˚ 0.2192 1.3058 4.2192 9.1152 9.2419 12.747 17.889 18.888 19.675 20.771 0˚ 0.9678 4.8009 13.093 14.768 15.488 15.969 21.495 23.944 25.889 26.308 present

30˚ 0.7347 3.6268 11.244 12.072 14.075 15.451 21.548 23.248 24.149 25.701 60˚ 0.3515 1.7438 5.5747 9.5004 10.714 11.497 16.900 18.607 19.635 22.899 90˚ 0.2488 1.2450 4.0001 8.1707 9.2492 9.3011 13.709 18.251 18.836 19.139

Table 4. Frequency parameter    a 2  / E 2 for pre-twisted laminated FGCNTRC shallow conical shell ( a / h  100.0, a / b  1.5, r / a  1, φ1  15 , φ 0  30 ,

  15 , 1  45 , 0 =300 / a (rad / s ), d max =3.0 , and stacking sequence (-45˚, 45˚, -45˚, 45˚)) * Types V1

Mode sequence number 1

2

3

4

5

6

7

8

0.12 0.57219 1.79022 3.01424 5.22450 5.33320 7.32495 8.01916 9.06155 UD 0.17 0.71935 2.20567 3.79073 6.39900 6.59249 9.17281 10.22670 11.14880 0.28 0.81937 2.67182 4.33978 7.85409 7.91023 10.64461 11.36446 13.56843 0.12 0.57189 1.76870 3.00086 5.14824 5.27762 7.27359 8.00887 8.94698 FG-O 0.17 0.73018 2.19222 3.79668 6.31235 6.54020 9.12971 10.23543 11.03380 0.28 0.82736 2.65311 4.38007 7.77310 7.87939 10.69090 11.56235 13.45950 0.12 0.57728 1.81471 3.03844 5.30532 5.39846 7.39842 8.06851 9.18812 FG-X 0.17 0.72700 2.23832 3.83116 6.50406 6.68357 9.28384 10.31704 11.31244 0.28 0.85105 2.73135 4.44214 8.01341 8.07541 10.88567 11.55653 13.81203

Table 5. Frequency parameter    a 2  / E 2 for pre-twisted laminated FGCNTRC shallow conical shell ( a / h  100.0, a / b  1.5, r / a  1, φ1  15 , φ 0  30 ,

  15 , 1  45 , 0 =300 / a (rad / s ), V1* =0.17, d max =3.0 , and stacking sequence (-β, β, -β, β)) β Types UD

Mode sequence number 1

2

3

4

5

6

7

8

0.95505 2.66901 3.56617 5.54096 6.01721 6.74213 7.41522 9.12173

0˚ FG-O 0.94564 2.66515 3.57348 5.50132 5.98780 6.75043 7.42167 9.12507 FG-X 0.96757 2.69206 3.59963 5.60580 6.08642 6.81788 7.49258 9.22812 UD

1.07777 2.83054 4.69731 5.68058 7.62897 10.09082 11.02347 11.35367

30˚ FG-O 1.06741 2.84360 4.66656 5.62730 7.58205 9.99865 10.90599 11.26009 FG-X 1.09187 2.85931 4.75841 5.75879 7.71457 10.23754 11.18401 11.51545 UD

0.71935 2.20567 3.79073 6.39900 6.59249 9.17281 10.22670 11.14880

45˚ FG-O 0.71862 2.18396 3.78612 6.31566 6.54432 9.14000 10.24797 11.05282 FG-X 0.72700 2.23832 3.83116 6.50406 6.68357 9.28384 10.31704 11.31244 UD

0.58001 1.88136 3.17086 5.50569 7.74745 8.14088 9.14357 9.83363

60˚ FG-O 0.57808 1.85856 3.16722 5.41755 7.69869 8.05349 9.08224 9.81476 FG-X 0.58411 1.90802 3.19930 5.57996 7.84428 8.23208 9.25956 9.96526 UD

0.34257 0.98378 1.96038 2.79418 4.37050 5.77462 6.90296 7.52005

90˚ FG-O 0.34440 0.99042 1.97299 2.81224 4.39651 5.80949 6.94523 7.56699 FG-X 0.34574 0.99291 1.97905 2.81932 4.41228 5.82786 6.96650 7.58654

Table 6. Frequency parameter    a 2  / E 2 for pre-twisted laminated FGCNTRC shallow conical shell ( a / h  100.0, a / b  1.5, r / a  1, φ1  15 , φ 0  30 ,

1  45 , 0 =300 / a (rad / s ), V1* =0.17, d max =3.0 , and stacking sequence (-45˚, 45˚, 45˚, 45˚)). Types

UD

FG-O

FG-X

Mode sequence number

 1

2

3

4

5

6

7

8

0˚

0.81308 1.94774 4.24422 5.91741 6.63861 9.21725 9.90232 10.98238

15˚

0.71935 2.20567 3.79073 6.39900 6.59249 9.17281 10.22670 11.14880

30˚

0.57002 2.46934 3.54675 6.12010 7.05669 9.95366 10.86061 11.89488

45˚

0.45503 1.95060 4.69487 5.47674 7.17914 10.83770 12.10774 13.58599

0˚

0.81915 1.91760 4.25699 5.83177 6.56191 9.17444 9.89844 10.90114

15˚

0.73018 2.19222 3.79668 6.31235 6.54020 9.12971 10.23543 11.03380

30˚

0.56440 2.44421 3.54274 6.05686 6.96813 9.93205 10.83801 11.75660

45˚

0.44952 1.92909 4.70222 5.42499 7.09354 10.74041 12.08114 13.50950

0˚

0.81970 1.98115 4.28207 6.01605 6.74126 9.32936 9.99506 11.12737

15˚

0.72700 2.23832 3.83116 6.50406 6.68357 9.28384 10.31704 11.31243

30˚

0.57777 2.50326 3.58376 6.20926 7.16583 10.06037 10.97039 12.07222

45˚

0.46197 1.97971 4.73768 5.55554 7.29165 10.98947 12.23733 13.75229

Table 7. Frequency parameter    a 2  / E 2 for pre-twisted laminated FGCNTRC composite shallow conical shell ( a / b  1.5, r / a  1, φ1  15 , φ 0  30 ,

  15 , 1  45 , V1* =0.17, 0 =300 / a (rad / s ), d max =3.0 , and stacking sequence (-45˚, 45˚, -45˚, 45˚)). Types

UD

FG-O

a/h

Mode sequence number 1

2

3

4

5

6

7

8

60

0.86278 3.26170 4.58414 9.28748 9.79196 10.84254 12.24396 16.60579

80

0.77629 2.59513 4.12711 7.65105 7.75051 10.23725 10.54619 13.54201

100

0.71935 2.20567 3.79073 6.39900 6.59248 9.17281 10.22670 11.14880

200

0.53307 1.46497 2.72440 3.44484 4.51537 6.15320 6.59047 7.82200

500

0.31552 1.02118 1.55754 1.74266 2.96379 3.15048 3.45740 4.14293

60

0.86112 3.21015 4.57764 9.20101 9.64985 10.86642 12.13029 16.33444

80

0.78047 2.57180 4.14823 7.60718 7.68551 10.26336 10.65314 13.13588

100

0.73018 2.19222 3.79668 6.31235 6.54020 9.12971 10.23543 11.03380

FG-X

200

0.53009 1.45731 2.70996 3.39654 4.49524 6.07984 6.55565 7.72455

500

0.31279 1.01891 1.53833 1.73615 2.93236 3.13213 3.42352 4.11024

60

0.87152 3.31749 4.63220 9.39753 9.92781 10.93191 12.38748 16.76109

80

0.78530 2.63487 4.16882 7.75466 7.87472 10.35859 10.64990 13.38226

100

0.72700 2.23832 3.83116 6.50406 6.68357 9.28384 10.31704 11.31244

200

0.53927 1.48256 2.75789 3.50326 4.56726 6.26091 6.66633 7.95134

500

0.31956 1.03174 1.58241 1.76361 3.00799 3.18885 3.50958 4.20029

Table 8. Frequency parameter    a 2  / E 2 for pre-twisted composite FGCNTRC shallow conical shell ( a / h  100.0, r / a  1, φ1  15 , φ 0  30 ,   15 ,

1  45 , 0 =300 / a (rad / s ), V1* =0.17, d max =3.0 , and stacking sequence (-45˚, 45˚, 45˚, 45˚)). Types a / b

UD

FG-O

Mode sequence number 1

2

3

4

5

6

7

8

1.5

0.71935 2.20567 3.79073 6.39900 6.59248 9.17281 10.22670 11.14880

2.0

0.60931 2.65076 3.29586 7.45494 8.17616 8.68706 11.16280 13.30776

2.5

0.51965 2.74375 3.27358 6.45315 7.72877 9.32808 13.75649 15.79048

3.0

0.45315 2.36964 5.45001 6.77601 10.68612 13.00148 18.30324 19.77424

3.5

0.41940 2.14446 4.47225 4.90053 6.03563 11.84978 12.75008 18.87431

1.5

0.72048 2.18823 3.80215 6.32707 6.57420 9.17971 10.24785 11.10442

2.0

0.60927 2.61518 3.31337 7.42409 8.19707 8.74073 11.06517 13.22197

2.5

0.52094 2.75872 3.21945 6.49247 7.80954 9.20064 13.88016 15.62035

3.0

0.45803 2.38950 3.86120 5.49863 6.85152 10.79713 13.21242 18.14998

FG-X

3.5

0.41036 2.10972 4.40316 4.83993 6.05942 12.00469 12.43730 19.33827

1.5

0.72986 2.24402 3.85391 6.52023 6.72554 9.33878 10.36854 11.38404

2.0

0.61483 2.69411 3.34249 7.58036 8.28184 8.85602 11.37946 13.59448

2.5

0.52469 2.78152 3.31740 6.50456 7.88810 9.48062 14.09726 15.96936

3.0

0.45995 2.40780 3.96466 5.48650 6.91669 11.12432 13.37407 18.61328

3.5

0.41381 2.13014 4.51080 4.90296 6.12376 12.14048 12.87781 19.59937

Table 9. Frequency parameter    a 2  / E 2 for pre-twisted laminated FGCNTRC shallow conical shell ( a / b  1.5, a / h  100.0, r / a  1, φ1  15 , φ 0  30 ,

  15 , 1  45 , 0 =300 / a (rad / s ), V1* =0.17, d max =3.0 , and stacking sequence (-45˚, 45˚)k). Types

UD

FG-O

Mode sequence number

No. of layers

1

2

3

4

5

6

7

8

2

0.59858 1.66846 3.07601 4.37900 5.21546 7.13552 7.72219 8.67197

4

0.71935 2.20567 3.79073 6.39900 6.59248 9.17281 10.22670 11.14880

6

0.73575 2.29786 3.88975 6.70710 6.86083 9.48347 10.39192 11.67000

8

0.73966 2.32487

10

0.74154 2.33827 3.92560 6.84924 6.97309 9.60557 10.43285 11.89375

2

0.57989 1.52168 2.90327 3.73697 4.64278 6.42660 6.79380 8.06118

4

0.72048 2.18823 3.80215 6.32707 6.57420 9.17971 10.24785 11.10442

6

0.73678 2.28908 3.89520 6.67574 6.84509 9.48336 10.43384 11.62697

3.91430 6.80419 6.93773 9.56619 10.42026 11.82147

FG-X

8

0.74178 2.32273 3.92461 6.79145 6.93609 9.58195 10.46973 11.80653

10

0.74403 2.33850 3.93800 6.84461 6.97779 9.62788 10.48495 11.89121

2

0.65081 1.83216 3.51035 5.04404 5.69136 8.04596 9.99610 10.78233

4

0.72986 2.24402 3.85391 6.52023 6.72554 9.33878 10.36854 11.38404

6

0.74028 2.31206 3.91541 6.75566 6.90813 9.54581 10.45604 11.74633

8

0.74374 2.33588 3.93559 6.83579 6.97094 9.61771 10.48169 11.87584

10

0.74526 2.34685 3.94490 6.87271 6.99991 9.65052 10.49224 11.93523

Table 10. Frequency parameter    a 2  / E 2 for pre-twisted laminated FGCNTRC shallow conical shell ( a / b  1.5, a / h  100.0, r / a  1, φ1  15 , φ 0  30 ,

  15 , 1  45 , 0 =300 / a (rad / s ), V1* =0.17, d max =3.0 , and stacking sequence (-45˚, 45˚)k). Types

UD

1

Mode sequence number 1

2

3

4

5

6

7

8

0˚

0.86710 3.26191 4.61687 9.32350 9.80880 10.91441 12.31835 16.63981

30˚

0.86662 3.26158 4.61676 9.32339 9.80865 10.91439 12.31827 16.63974

45˚

0.86613 3.26131 4.61665 9.32330 9.80852 10.91436 12.31820 16.63968

60˚

0.86565 3.26107 4.61654 9.32323 9.80843 10.91434 12.31813 16.63963

90˚

0.86516 3.26090 4.61645 9.32318 9.80836 10.91433 12.31808 16.63959

FG-O

FG-X

0˚

0.86468 3.21462 4.61040 9.24339 9.67878 10.93593 12.19580 16.38578

30˚

0.86419 3.21428 4.61029 9.24328 9.67861 10.93591 12.19571 16.38570

45˚

0.86370 3.21401 4.61018 9.24319 9.67849 10.93588 12.19564 16.38564

60˚

0.86322 3.21377 4.61008 9.24311 9.67838 10.93586 12.19558 16.38558

90˚

0.86273 3.21360 4.60998 9.24306 9.67831 10.93585 12.19553 16.38554

0˚

0.87504 3.31605 4.66320 9.44690 9.96613 11.00264 12.48121 16.91550

30˚

0.87456 3.31572 4.66309 9.44679 9.96599 11.00261 12.48112 16.91543

45˚

0.87407 3.31546 4.66299 9.44670 9.96587 11.00258 12.48105 16.91538

60˚

0.87359 3.31522 4.66288 9.44663 9.96577 11.00256 12.48099 16.91533

90˚

0.87311 3.31505 4.66279 9.44658 9.96571 11.00255 12.48094 16.91529

Table 11. Frequency parameter    a 2  / E 2 for pre-twisted laminated FGCNTRC shallow conical shell ( a / h  100.0, a / b  1.5, φ1  15 , φ 0  30 ,   15 ,

1  45 , 0 =300 / a (rad / s ), V1* =0.17, d max =2.8 , and stacking sequence (-45˚, 45˚, 45˚, 45˚)). Types

UD

FG-O

FG-X

r/a

Mode sequence number 1

2

3

4

5

6

7

8

0

0.71868 2.21002 3.80978 6.41352 6.63197 9.22476 10.27428 11.21871

0.5

0.72063 2.21077 3.81152 6.41421 6.63308 9.22681 10.27497 11.22001

1

0.72257 2.21152 3.81327 6.41490 6.63420 9.22887 10.27567 11.22132

1.5

0.72450 2.21227 3.81501 6.41559 6.63531 9.23092 10.27635 11.22262

2

0.72643 2.21302 3.81675 6.41627 6.63643 9.23297 10.27704 11.22393

0

0.71659 2.18671 3.79864 6.32569 6.57193 9.17558 10.24512 11.10201

0.5

0.71854 2.18747 3.80039 6.32638 6.57306 9.17764 10.24649 11.10321

1

0.72048 2.18823 3.80215 6.32707 6.57420 9.17971 10.24785 11.10442

1.5

0.72242 2.18898 3.80390 6.32775 6.57533 9.18178 10.24920 11.10563

2

0.72435 2.18974 3.80565 6.32843 6.57647 9.18384 10.25052 11.10684

0

0.72602 2.24254 3.85047 6.51885 6.72335 9.33475 10.36728 11.38137

0.5

0.72794 2.24328 3.85219 6.51954 6.72444 9.33677 10.36791 11.38271

1

0.72986 2.24402 3.85391 6.52023 6.72554 9.33878 10.36854 11.38404

1.5

0.73177 2.24476 3.85563 6.52091 6.72663 9.34080 10.36917 11.38537

2

0.73367 2.24549 3.85735 6.52160 6.72773 9.34281 10.36979 11.38670

Table 12. Optimization results of laminated conical shell V1*

Test No.



Stacking sequence [-30°/-30°/-30°/60°/0°/-

1

1.15305

30°/0°/0°/30°/30°/30°/30°/30°/90°/0°/0°/-30°/-30°/30°/-30°] [-30°/-30°/-30°/60°/-

0.12

2

1.15442

30°/0°/0°/30°/0°/30°/30°/30°/30°/90°/0°/0°/-30°/-30°/30°/-30°] [-30°/-30°/-30°/60°/0°/-

3

1.15334

30°/0°/0°/0°/30°/30°/30°/30°/90°/0°/0°/-30°/-30°/-30°/30°] [-30°/-30°/60°/0°/-30°/-

1

1.38724

30°/0°/0°/0°/30°/30°/30°/90°/30°/0°/0°/-30°/-30°/-30°/30°] [-30°/-30°/60°/0°/-30°/-

0.17

2

1.38668

30°/0°/0°/30°/30°/30°/30°/90°/30°/0°/0°/-30°/-30°/30°/-30°] [-30°/-/60°/0°/-30°/-

3

1.38668

30°/0°/0°/30°/30°/30°/30°/90°/30°/0°/0°/-30°/-30°/30°/-30°]

Figure 1. (a) The base of the cone; (b) The geometry of the untwisted laminated shallow conical shell.

Figure 2. (a) Pre-twisted shallow conical shell; (b) Pre-twisted shallow conical shell fixed to rotating cylindrical hub.

Figure 3. The distribution of CNTs through the conical shell (a) UD, (b) FG-O, and (c) FG-X.

Figure 4. Flowchart of the genetic algorithm

0.036

Frequency parameter 1

0.034 0.032 0.030

dmax=2.4

0.028

dmax=2.6

0.026

dmax=2.8

0.024 0.022 0.020 2

4

6

8

10

12

14

16

18

20

22

M

Figure 5. Convergence study of the first frequency parameter.

UD

FG-X 1.4

1.2

1.2

1.0

1.0

0.8

0.8

0.6

0.6

0.4

0.4

V1=0.12

1.4

V1=0.17

Frequency parameter 1

FG-O 1.4

V1=0.28

1.2 1.0 0.8 0.6 0.4 0.2

0

0.2 20 40 60 80 100

0

0.2 20 40 60 80 100

0

20 40 60 80 100

  Figure 6. Effect of the stacking sequence (-β, β, -β, β) on the first frequency parameter

   a 2  / E 2 of pre-twisted laminated FG-CNTRC shallow conical shell.

FG-O

UD  1  1  1

v =0.12

0.9

v =0.17

Frequency parameter 1

v =0.28 0.8

FG-X

1.0

0.9 0.9 0.8 0.8 0.7

0.7

0.7 0.6

0.6

0.6 0.5

0.5

0.5 0.4

0.4

0.4 0

10 20 30 40 50

0.3

0

10 20 30 40 50

0

10 20 30 40 50

Twisted angle  

Figure 7. Effect of the twisted angle on the first frequency parameter

   a 2  / E 2 of pre-twisted laminated FG-CNTRC shallow conical shell. FG-O

UD

Frequency parameter 1

1.0 0.9

 1  1  1

v =0.12 v =0.17

0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.5

0.4

0.4

0.4

0.3

0.3

0.3

0.2

0 100 200 300 400 500

FG-X

1.0 0.9

v =0.28 0.8

1.1

1.0

0.2

0.7 0.6

0 100 200 300 400 500

0.2

0 100 200 300 400 500

a/h

Figure 8. Effect of the length-to-thickness ration on the first frequency parameter

   a 2  / E 2 of pre-twisted laminated FG-CNTRC shallow conical shell.

UD

Frequency prameter 

0.8

0.7

v1=0.12 v1=0.17 v1=0.28

0.6

0.5

0.4

0.3

1.5 2.0 2.5 3.0 3.5

0.9

FG-O

0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

1.5 2.0 2.5 3.0 3.5

FG-X

1.5 2.0 2.5 3.0 3.5

a/b

Figure 9. Effect of aspect ratio on the first frequency parameter    a 2  / E 2 of pre-twisted laminated FG-CNTRC shallow conical shell. v1=0.12

Frequency prameter 1

0.580

0.578

0.576

UD FG-O FG-X

v1 =0.17 0.769 0.768

0.838 0.836 0.834

0.767

0.832

0.766

0.830

0.765 0.574

v1 =0.28

0.764

0.828 0.826 0.824

0.572

0.570

0.763

0.822

0.762

0.820

0.761 0 20 40 60 80 100

0.818 0 20 40 60 80 100

0 20 40 60 80 100

Pre-set angle  

Figure 10. Effect of the pre-set angle on the first frequency parameter

   a 2  / E 2 of pre-twisted laminated FG-CNTRC shallow conical shell.

v1=0.12

0.584

0.580

v1=0.28

0.732

UD FG-O FG-X

0.582

Frequency parameter 1

v1=0.17 0.840

0.730 0.835

0.728

0.578

0.726

0.576

0.830

0.724

0.574

0.722

0.572

0.825

0.720

0.570

0.820

0.718

0.568

0.716

0.566 0.0 0.5 1.0 1.5 2.0

0.714

0.815 0.0 0.5 1.0 1.5 2.0

0.0 0.5 1.0 1.5 2.0

r/a

Figure 11. Effect of the radius-to-length on the first frequency parameter

   a 2  / E 2 of pre-twisted laminated FG-CNTRC shallow conical shell. v1=0.12

v1=0.17 1.40

Frequency parameter 1

1.15

1.35

1.10

1.30

1.05 1.25

1.00 the first time the second time the third time

0.95 0

50

100

N

150

1.20 1.15 0

50

100

150

N

(a) (b) Figure 12. The variation of the first frequency parameter during the evolution process of stacking sequence for laminated conical shell with CNT volume fraction of (a) 0.12, and (b) 0.17.