Nonlinear vibration analysis of laminated composite angle-ply cylindrical and conical shells

Composite Structures 255 (2021) 112867

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Nonlinear vibration analysis of laminated composite angle-ply cylindrical and conical shells Shahin Mohammadrezazadeh ⇑, Ali Asghar Jafari K. N. Toosi University of Technology, Tehran, Iran

A R T I C L E

I N F O

Keywords: Nonlinear free vibration Laminated composite angle‐ply Cylindrical shells Conical shells Multiple scales method Modal analysis

A B S T R A C T This paper presents the combination of multiple scale method and modal analysis in order to investigate nonlinear vibration of laminated composite angle‐ply cylindrical and conical shells. The shells are modeled considering the shear deformation and rotary inertia while the geometrical nonlinearity is modeled using von Karman approach. Hamilton principle is used for obtaining the basic equations of the system. These equations are converted to nonlinear ordinary differential equations depending on time variable using Ritz method. The results of this study are validated against the results of open literature and good agreement is observed. The effects of several parameters including the layers' angle, the number of the layers, semi‐vertex angle, length, radius and also each layer's thickness on nonlinear frequency ratio, fundamental linear frequency and nonlinear frequency are illustrated in details.

1. Introduction Structural mechanics is a field that numerous researches have been done [1–7]. Cylindrical and conical shells are from mechanical structures which have a lot of applications in engineering. Vibration phenomenon can cause resonance condition [8] in mechanical structures such as cylindrical and conical shells. In addition, it could cause fatigue and fracture of mechanical systems [8]. Therefore, vibration analysis is important in order to improve the behavior of structures such as cylindrical and conical shells. Several researchers have studied vibration of cylindrical [9–19] and conical shells [20–27]. For large amplitudes, the nonlinear model of the system can better describe the structure's vibration behavior. In this case, geometrical nonlinearity takes place in system which is due to the nonlinear relation of strains and displacement [28]. There are some researches which study nonlinear vibration of cylindrical and conical shells. Yong‐gang et al. [29] have investigated the nonlinear vibration of thin shallow conical shells under peripheral moment and transverse loads. Shen [30] has adopted a higher order shear deformation shell theory and perturbation method to evaluate large amplitude vibration of a shear deformable functionally graded cylindrical shell in an elastic medium and in thermal environment. Bich and Nguyen [31] have investigated nonlinear vibration of functionally graded cylindrical shells under axial and transverse mechani-

⇑ Corresponding author. E-mail address: [email protected] (S. Mohammadrezazadeh).

https://doi.org/10.1016/j.compstruct.2020.112867 Received 18 March 2020; Revised 11 August 2020; Accepted 18 August 2020 Available online 26 August 2020 0263-8223/© 2020 Elsevier Ltd. All rights reserved.

cal loads via Galerkin and Runge–Kutta methods. Jafari et al. [32] have presented nonlinear vibration responses of functionally graded cylindrical shells with piezoelectric layers. Sofiyev [33] has investigated nonlinear vibration of orthotropic heterogeneous conical shells resting on elastic foundations using homotopy perturbation method. Sofiyev [34] has investigated nonlinear free vibration of functionally graded orthotropic cylindrical shells considering shear stresses based on the shear deformation theory. Shen et al. [35] have studied nonlinear vibration of graphene‐reinforced composite laminated cylindrical shells which are in thermal environments. Hasrati et al. [36] have developed a new numerical approach to study nonlinear free and forced vibrations of cylindrical shells based on first‐order shear deformation theory. Ansari et al. [37] have investigated nonlinear vibration of functionally graded carbon nanotube reinforced composite conical shells through variational differential quadrature method. Li et al. [38] have studied nonlinear vibration of thin composite cylindrical shells with arbitrary boundary conditions based on nonlinear theory of Donnell for thin shells. Li et al. [39] have utilized Donnell’s nonlinear shell theory to present responses for nonlinear forced vibration of non‐continuous elastic‐supported thin laminated composite cylindrical shells which are under periodic radial point loading. Aris and Ahmadi [40] have studied nonlinear vibration and resonance of functionally graded conical shells subjected to harmonic excitation. Bakhtiari et al. [41] have adopted Donnell, Sanders and Nemeth shell theories

S. Mohammadrezazadeh, A.A. Jafari

Composite Structures 255 (2021) 112867

layers, semi‐vertex angle, length, radius and thickness of each layer on the vibration characteristics are studied with details.

to study nonlinear free vibration of conical shells. Kamaloo et al. [42] have studied nonlinear free vibration of composite laminated cylindrical shells using Galerkin method. As mentioned there are several studies about the nonlinear vibration of cylindrical shells; but it seems that the works about nonlinear vibration of conical shells are fewer. According to the literature review, it seems that no study has been done about nonlinear free vibration of angle‐ply laminated composite cylindrical and conical shells using a combination of multiple scales method and modal analysis. The use of multiple scales method and modal analysis leads to approximate results which are more reliable than numerical results. Therefore, in this paper, nonlinear free vibration of angle‐ply laminated composite cylindrical and conical shells with simply supported boundary conditions via a combination of multiple scales method and modal analysis is handled. There are several theories for modeling of the structures. Reddy [43] has explained first‐order shear deformation theory as well as second and higher order equivalent single layer theories for plates. The higher order theories utilize higher order polynomials for modeling of the displacement components in thickness direction of a laminate [43]. In addition, the higher order theories include additional unknowns for which it is frequently difficult to find physical concept [43]. The accuracy of third‐order theories is a slight better that first‐ order shear deformation theory, but third‐order theories require more computational effort [43]. It should be mentioned that first‐order shear deformation theory with transverse extensibility is the best equivalent single layer theory in terms of economy, accuracy and simplicity for plates [43]. Therefore, in this study, a theory is used for modeling of the shells' kinematics which is similar to first order shear deformation theory of plates [43] and considers shear deformation and rotary inertia. Also, geometric nonlinearity is modeled based on von Karman theory [43]. Partial differential equations of the motion can be derived using Hamilton principle. These equations are converted to ordinary differential equations via Ritz method. It is obvious that there is not any analytical solution for a wide range of structural problems; therefore numerical approaches are used in order to obtain responses for many structural problems. Discrete singular convolution (DSC) method, differential quadrature method (DQM), meshless method and finite element method (FEM) are from numerical methods which are used in mechanical structures problems. There are several researches in literature which have used numerical approaches. Civalek [1] has investigated nonlinear dynamic of doubly curved shallow shells on elastic foundation for two different loading types through a couple of harmonic differential quadrature (HDQ) and finite differences (FD) methods. Civalek and Acar [2] have utilized DSC in order to study bending of Mindlin plates which are on elastic foundations. Akgoz and Civalek [3] have presented nonlinear free vibration of thin laminated plates on non‐linear elastic foundations through von Karman nonlinear theory and DSC method. Liang et al. [6] have studied dynamic responses of functionally graded piezoelectric cylindrical panels using DQM and state space approach. Regardless of the advantages of the numerical methods, analytical solutions seem more attractive for parametric studies and considering physics of the problems [44–45]. Besides, analytical methods can be used as a reference framework for validation and verification of numerical results [44–45]. One instance for application of semi‐analytical methods is that Jing [46] has used a semi‐analytical optimal solution in order to investigate buckling of simply supported rectangular orthotropic plates subjected to axial compression. Multiple scales method can be classified among approximate analytical methods [47]. Therefore, in this study, the fundamental linear frequency, amplitude, nonlinear frequency and nonlinear frequency ratio are extracted by combination of multiple scales method and modal analysis. The results of literature are used for validation of this study's results. The effect of several parameters such as the angle of the layers, the number of the

2. Problem formulation 2.1. Modeling of cylindrical and conical shells In Fig. 1 cylindrical and conical shells with reference coordinate system are shown. The coordinate system is attached to the middle surface of the small edge of the conical shell while it can be attached to middle surface of a random edge of the cylindrical shell. It should be noted that x, θ and z axes are along the generator, circumferential and thickness directions of the shells, respectively. The terms L, hT and h denote the length, the whole thickness and thickness of each layer of the shells. In addition, α denotes the semi‐vertex angle of the conical shell. It should be mentioned that R1 and R2 respectively refer to the small and large edge radiuses of the conical shell while the radius of the cylindrical shell is constant and is denoted by R. It should be denoted that the radiuses are considered for points on the middle surface of the shells. The radius of an arbitrary point on the middle surface of the conical shell is obtained as: RðxÞ ¼ R1 þ xsinα. The shells are simply supported at both ends and consist of N orthotropic layers which are stacked on each other. The lamination scheme of the shells is angle‐ply and the angle of each layer in anticlockwise sense with x axis of reference coordinate is introduced with φ. Fig. 2 shows the laminated scheme of the shells. In the next, the relations for the conical shells are extracted. It is obvious that by substituting α ¼ 0, the relations for cylindrical shells are obtained. It should be mentioned that the relations are obtained by considering shear deformation and rotary inertia. The relations between displacements of a point on the middle surface and displacements of any point on the shell along x, θ and z directions are obtained as [49]: u ¼ u0 þ zψ x ; v ¼ v0 þ zψ θ ; w ¼ w0

ð1Þ

While u(u0 ), v(v0 ) and w(w0 ) are displacements of any point (a point on the middle surface) along x, θ and z directions, respectively. In addition Ψx and ψθ are respectively used for total rotations accurded about θ and x directions. The relations of strains with displacements and rotations obtain as following [49]: ɛx ¼ ɛ0x þ zkx ; ɛ θ ¼ ɛ 0θ þ zkθ ; ɛ xθ ¼ ɛ 0xθ þ zkxθ ; 1 0 þ ψ x ; ɛ θz ¼ RðxÞ ɛxz ¼ @w @x

@w0 @θ

0 cosα  vRðxÞ þ ψθ

ð2Þ

While considering nonlinear von Karman theory leads to the membrane strains (ɛ0x ; ɛ0θ ; ɛ0xθ ) and curvatures (kx ; kθ ; kxθ ) as following [43,49]:
0 2 0 ɛ0x ¼ @u þ 12 @w @x @x
@w0 2 u0 sinα 1 @v0 1 0 cosα ɛ0θ ¼ RðxÞ þ þ wRðxÞ þ 2RðxÞ 2 RðxÞ @θ @θ ð3Þ v0 sinα 1 @u0 1 @w0 @w0 0 x ɛ0xθ ¼ @v þ  þ ; kx ¼ @ψ @x @x RðxÞ RðxÞ @θ RðxÞ @x @θ
@ψ θ  1 1 @ψ x θ θ sinα kθ ¼ RðxÞ þ ψ x sinα ; kxθ ¼ @ψ þ RðxÞ  ψRðxÞ @θ @x @θ Eq. (3) reveals that there are nonlinear relations between strains and displacement which demonstrate that the source of nonlinearity is geometrical in this study. The relation of stresses and strains for any layer is in the following type [43]: 2 3ðkÞ   9ðkÞ 9 8 8 Q11 Q12 0 0 Q16 > > > σx > > ɛx > 6 7 > >    > > > > > > > > 6Q > 0 0 Q26 7 > > > σθ > > ɛθ > 6 12 Q22 7 > = = < < 6 7   6 7 ð4Þ σ θz ¼6 0 ɛθz 0 Q Q 0 7 44 45 > > > > > > > 6 7 >   > > > > > > > σ xz > > ɛxz > 6 0 > 0 Q45 Q55 0 7 > > > > 4 5 > ; ; : :    σ xθ ɛ xθ Q16 Q26 0 0 Q66

2

Composite Structures 255 (2021) 112867

S. Mohammadrezazadeh, A.A. Jafari

Fig. 1. The schematic of cylindrical and conical shells and coordinate system [48].

9 2 8 A11 Nx > > > > > > > > 6 > > N A θ > 12 > > 6 > > = 6 6 A16 xθ 6 ¼6 > > 6 B11 > Mx > > > > > 6 > > > 4 B12 > Mθ > > > > ; : M xθ B16 

Qθ Qx



 ¼ Ks

A12

A16

B11

B12

A22

A26

B12

B22

A26

A66

B16

B26

B12

B16

D11

D12

B22

B26

D12

D22

B26

B66

D16

D26

A44

A45

A45

A55



ɛ θz



ð8Þ

ɛ xz

ðkÞ

Aij ¼ ∑k¼1 Qij ðzkþ1  zk Þ Fig. 2. Schematic of the lamination scheme for considered cylindrical and conical shells [48].

ð7Þ



While [43]: 
 ðkÞ R N Aij ; Bij ; Dij ¼ ∑k¼1 Qij z ð1; z; z2 Þdz N

9 38 B16 > ɛ0x > > > > > > ɛ0θ > > B26 7 > > 7> > > > 7> B66 7< ɛ 0xθ = 7 7 D16 7> > > kx > > > > 7> > > kθ > D26 5> > > > > ; : D66 kxθ

i; j ¼ 1; 2; 6 i; j ¼ 4; 5

ð9Þ

2.2. Hamilton principle Hamilton principle [49] is used for obtaining partial differential equations of systems. For the nonlinear free vibration of the cylindrical and conical shells it can be written in the following type [49]: Z t2 ðδT  δU ɛ Þdt ¼ 0 ð10Þ

While superscript k denotes each layer's number and is omitted for brevity in the rest of the paper. Transformed reduced stiffness coeffi

cients (Qij ) are in relation with reduced stiffnesses (Qij ) and the angle of each layer with x axis of the coordinate system in anticlockwise sense (φ) [43]:

t1

While t indicates time and T and U ɛ refer respectively to kinetic and strain energies. The kinetic energy is in relation with the density ~ in the following type [49]: of each layer ρk and Velocity V Z Z Z n o N ρk ~V ~ RðxÞdxdθdz; V ~ ¼ u_^i þ v_^j þ w_ ^k V: ð11Þ T¼ ∑ z θ x k¼1 2



Q11 ¼ Q11 cos4 φ þ Q22 sin4 φ þ 2ðQ12 þ 2Q66 Þsin2 φcos2 φ 

Q12 ¼ Q12 ðsin4 φ þ cos4 φÞ þ ðQ22 þ Q11  4Q66 Þsin2 φcos2 φ 

Q22 ¼ Q11 sin4 φ þ Q22 cos4 φ þ 2ð2Q66 þ Q12 Þsin2 φcos2 φ 

Q16 ¼ ðQ12 þ 2Q66  Q22 Þcosφsin3 φ þ ðQ11  2Q66  Q12 Þcos3 φsinφ

Some simplifications lead to the variational form of the kinetic energy: R R δT ¼ θ x J 1 ðu_ 0 δu_ 0 þ v_ 0 δ_v0 þ w_ 0 δw_ 0 ÞRðxÞdxdθ R R þ θ x J 2 ðψ_ x δu_ 0 þ ψ_ θ δ_v0 þ u_ 0 δψ_ x þ v_ 0 δψ_ θ ÞRðxÞdxdθ ð12Þ R R þ θ x J 3 ðψ_ x δψ_ x þ ψ_ θ δψ_ θ ÞRðxÞdxdθ



Q26 ¼ þðQ12 þ 2Q66  Q22 Þcos3 φsinφ þ ðQ11  2Q66  Q12 Þcosφsin3 φ 

Q66 ¼ ðQ22 þ Q11  2Q66  2Q12 Þcos2 φsin2 φ þ Q66 ðcos4 φ þ sin4 φÞ 

Q44 ¼ Q55 sin2 φ þ Q44 cos2 φ 

Q45 ¼ ðQ55  Q44 Þsinφcosφ 

While [50]:

Q55 ¼ Q44 sin2 φ þ Q55 cos2 φ

N

ð5Þ

ðJ 1 ; J 2 ; J 3 Þ ¼ ∑ k¼1

While [43]: υ12 E2 Q11 ¼ 1υE121 υ21 ; Q12 ¼ 1υ ; Q22 ¼ 1υE122 υ21 12 υ21

Q66 ¼ G12 ; Q44 ¼ G23 ; Q55 ¼ G13

Z

zkþ1


 ρðkÞ 1; z; z2 dz

ð13Þ

zk

In addition, the variational form of the strain energy is obtained in the following form [49]: Z Z  N x δɛ 0x þ N θ δɛ 0θ þ N xθ δɛ0xθ þ M x δkx þ M θ δkθ δU ɛ ¼ RðxÞdxdθ þM xθ δkxθ þ Qx δɛxz þ Qθ δɛθz θ x

ð6Þ

It should be noted that E 1 , E 2 and υij denote the Young's moduli in x and θ directions and poisson's ratio [43]. In addition, G12 , G23 and G13 are shear moduli in x  θ, θ  z and x  z planes [43]. By employing some mathematical efforts, the in‐plane forces, moments and shear forces can be extracted in the following types [43]:

ð14Þ Using some mathematical efforts, variational form of the strain energy can be rewritten as following: 3

S. Mohammadrezazadeh, A.A. Jafari

0

0 0 @δw0 0 N x RðxÞ @δu þ N x RðxÞ @w þ N θ @δv þ N θ δu0 sinα @x @x @x @θ B N θ @w0 @δw0 0 0 þN θ δw0 cosα þ RðxÞ @θ @θ þ N xθ RðxÞ @δv þ N xθ @δu @x @θ Z Z B B B @δψ x @w0 @δw0 @w0 @δw0 δU ɛ ¼ B N xθ δv0 sinα þ N xθ @θ @x þ N xθ @x @θ þ M x RðxÞ @x θ x B B @δψ @δψ @δψ @ þM θ @θ θ þ M θ δψ x sinα þ M xθ RðxÞ @x θ þ M xθ @θ x  M xθ δψ θ sinα

Composite Structures 255 (2021) 112867

1

The equation obtained from multiple scales method seems to be complicated [53]. This conclusion is true; but the advantages of this method justify its usage [53]. According to Eqs. (20) and (22), the following relation is established:
 €y ¼ ɛD20 y1 þ ɛ 2 2D0 D1 y1 þ D20 y2
 þ ɛ 3 2D0 D2 y1 þ D21 y1 þ 2D0 D1 y2 þ D20 y2 ð23Þ

C C C C Cdxdθ C C A

0 0 þQx RðxÞ @δw þ Qx RðxÞδψ x þ Qθ @δw  Qθ δv0 cosα þ Qθ RðxÞδψ θ @x @θ

ð15Þ

Eqs. (2), (3), (7) and (8) should be substituted into Eq. (15). After this substitution, substituting Eqs. (12) and (15) into Eq. (10) leads to variational relation of the problem.

Substituting Eqs. (20), (21) and (23) into Eq. (18) and removing the terms with ɛ to power of greater than 3 lead to the following formulation:
M ɛðD20 y1 þ 2ɛD0 D1 y1 þ 2ɛ 2 D0 D2 y1 þ ɛ 2 D21 y1 Þ þ ɛ 2 ðD20 y2 þ 2ɛD0 D1 y2 Þ 
 ð24Þ þɛ 3 D20 y3 þ K ɛy1 þ ɛ 2 y2 þ ɛ3 y3 þ ɛ 3 Bw31 ¼ 0

2.3. Ritz method The Ritz method [43,51] uses variational form of Hamilton principle for obtaining the ordinary differential equations of the systems. The approximate function of the Ritz method is only necessary to satisfy geometrical boundary conditions of mechanical structures [43]. The geometrical boundary conditions for cylindrical and conical shells with simply supported ends are as following [49–50]: 8 > < v0 ðx; θ; zÞ ¼ 0 w0 ðx; θ; zÞ ¼ 0 ; > : ψ θ ðx; θ; zÞ ¼ 0

x ¼ 0; L

Arranging of Eq. (24) with respect to the coefficients of different powers of ɛ leads to the following equations:

ð16Þ

ð27Þ

ð28Þ

ð30Þ

While As is complex parameter related to variables T 1 and T 2 . In addition, tcc denotes the complex conjugate of the term before it. In order to solve Eq. (26), the modal analysis leads to the following relation [49]: y2 ¼ Yfη2 g

ð31Þ

ð18Þ

Substituting Eq. (31) into Eq. (26) and some simplifications lead to the following formulation: D20 fη2 g þ ω2i fη2 g ¼ f2D0 D1 η1 g ð32Þ

ð19Þ

The s row, which contains the fundamental linear frequency (ωL ), is written as following: D20 η2s þ ω2L η2s ¼ 2D0 D1 η1s

3. Problem solution

yðtÞ ¼ ɛy1 ðT 0 ; T 1 ; T 2 Þ þ ɛ2 y2 ðT 0 ; T 1 ; T 2 Þ

ð20Þ

wt ðtÞ ¼ ɛw1 ðT 0 ; T 1 ; T 2 Þ þ ɛ 2 w2 ðT 0 ; T 1 ; T 2 Þ

ð21Þ

ð33Þ

Substituting Eq. (30) into Eq. (33), the following relation is extracted:

In this section, multiple scales method is combined with modal analysis in order to obtain an approximate solution for the defined nonlinear vibration problem. In the multiple scales method, the vectors y and wt , which are functions of the variable t, are written as functions of the independent variables T 0 , T 1 and T 2 [52]:

D20 η2s þ ω2L η2s ¼ 2iωL D1 As expðiωL T 0 Þ þ tcc

ð34Þ

In order to have bounded response, it is necessary that the coefficient of expðiωL T 0 Þ becomes zero; otherwise, the term T 0 expðiωL T 0 Þ appears which leads to infinite response. Therefore: 2iωL D1 As ¼ 0 ! D1 As ¼ 0 ! As ðT 2 Þ

ð35Þ

Thus: η2s ¼ 0

While [52–53]: T 0 ¼ t; T 1 ¼ ɛt; T 2 ¼ ɛ 2 t


 ɛ3 ! M 2D0 D2 y1 þ D21 y1 þ 2D0 D1 y2 þ D20 y3 þ Ky3 þ Bw31 ¼ 0

η1s ¼ As ðT 1 ; T 2 ÞexpðiωL T 0 Þ þ tcc

While the vector y is defined as: y ¼ fut ; vt ; wt ; ψxt ; ψθt gT

ð26Þ

While Y and fη1 g denote modal matrix and the vector of the modal coordinates, respectively [49]. Substituting Eq. (28) into Eq. (25) leads to the following equation:

2  2 D0 η1 þ ωi fη1 g ¼ 0; ω2i ¼ Y1 M1 KY ð29Þ 2 It is obvious from Eq. (29) that the components of matrix ωi are related to components of mass and stiffness matrices and decrease and increase with increase of the components of mass and stiffness matrices, respectively. The response of the s row, which contains the fundamental linear frequency ωL , is in the following type:

In Eq. (17), m and n refer respectively to longitudinal and circumferential wave numbers. In addition, M T demonstrates the upper bounding value for longitudinal wave number. It should be noted that the approximate function of Eq. (17) is written based on reference [49]. In this paper, initial excitations are assumed such that the approximate function of Eq. (17) is not written in series forms for θ. Substituting Eq. (17) into Eq. (10) and employing some simplifications lead to the ordinary differential equation of the system in the following form: ¼0


 ɛ2 ! M 2D0 D1 y1 þ D20 y2 þ Ky2 ¼ 0

y1 ¼ Yfη1 g

ð17Þ

M€ y þ Ky þ

ð25Þ

In order to solve Eq. (25), the vector y1 is written in the following form through modal analysis [49]:

The following approximate function satisfies geometric boundary conditions which are shown in Eq. (16); so it can be introduced for use in Ritz method. 9 8 MT 9 8 8 9 ÞcosðnθÞ > ∑m¼1 cosðmπx > u0 > ut ðtÞ > L > > > > > > > > > > > > > > MT > > > > > > > > mπx > > > > > > sinð ÞsinðnθÞ ∑ v ðtÞ v > > > > > > 0 t m¼1 L = =  < < = < MT ; θ ¼ 1; 2; ::: w0 ¼ ϕf : wt ðtÞ ; ϕf ¼ ∑m¼1 sinðmπx ÞcosðnθÞ L > > > > > > > > > > > > > > MT > ψ ðtÞ > >ψ > mπx > > > > > > x> xt ∑m¼1 cosð L ÞcosðnθÞ > > > > > > > > > > > ; ; > : : > ; : MT ψθ ψ θt ðtÞ ∑m¼1 sinðmπx ÞsinðnθÞ L

Bw3t

ɛ ! MD20 y1 þ Ky1 ¼ 0

ð36Þ

In order to obtain the response of Eq. (27), using modal analysis leads to the following relation [49]:

ð22Þ

4

Composite Structures 255 (2021) 112867

S. Mohammadrezazadeh, A.A. Jafari

y3 ¼ Yfη3 g

ð37Þ

Eqs. (51) and (52) show that nonlinear frequency and nonlinear frequency ratio are directly related to the power 2 of the amplitude (A); therefore the increase of amplitude leads to increase of nonlinear frequency and nonlinear frequency ratio. In addition, Eq. (51) reveals that for small values of amplitude, nonlinear frequency relates approximately only to fundamental linear frequency. On the other hand, for great values of the amplitude, nonlinear frequency value increases with increase of Gs and decrease of ωL . Furthermore, according to Eq. (52), it can be concluded that the nonlinear frequency ratio increases with increase of Gs and decrease of fundamental linear frequency ωL .

Substitution Eq. (37) into Eq. (27) and doing some mathematical efforts lead to the following formulation:

2  2

 ^ 3 D0 η3 þ ωi fη3 g ¼ 2fD0 D2 η1 g  D21 η1  2fD0 D1 η2 g  Bw ð38Þ 1 While: ^ ¼ Y1 M1 B B

ð39Þ

Considering the s row of Eq. (38), which contains the fundamental linear frequency (ωL ) of the shells, leads to the following formulation: D20 η3s þ ω2s η3s ¼ 2D0 D2 η1s  P1 η31s

ð40Þ

4. Results and discussions

While: m

^ jÞðYð2m þ j; sÞÞ3 P1 ¼ ∑ Bðs;

In this section, the results of the nonlinear free vibration of laminated composite angle‐ply cylindrical and conical shells are obtained with the help of supplementary source code. Before obtaining the results, it is necessary to validate the method used in this paper. For this purpose, Table 1 compares the frequency parameter pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (ωR ð1  υ212 Þρ=E1 ) results of this study with the results of literature for different values of the circumferential wave number n. The results of Table 1 are for an isotropic cylindrical shell with m ¼ 1, L=R ¼ 20, h=R ¼ 0:01 and υ12 ¼ 0:3. According to Table 1, one can conclude that there is excellent agreement between the results of this study and literature. It is necessary to investigate the convergence of the used method. In this way, Table 2 presents the effect of the upper bounding value for longitudinal wave number (M T ) on the frequency parameter pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (ωR2 ð1  υ212 Þρ=E1 ) responses of an isotropic conical shell with  h=R2 ¼ 0:01, Lsinα=R2 ¼ 0:25, α ¼ 30 and υ12 ¼ 0:3 for different values of circumferential wave number n and compares the results with literature. The results of this table indicate that the rate of convergence is appropriate for considered constants and M T ¼ 3 leads to the results which have good convergence. Also, adaptation of the results with the determined results of literature is good. In order to further validate this study's method for conical shells, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Table 3 compares the frequency parameter (ωR2 ð1  υ212 Þρ=E1 ) results for an isotropic conical shell with literature for different values of circumferential wave number n. The results are obtained for isotro  pic conical shells with semi‐vertex angles of α ¼ 30 and α ¼ 45 while M T ¼ 3 and other constants are considered like constants of Table 2. Table 3 reveals that there is good adaptation between the results of this study and determined results of literature. After validation of the method of this study, it is time to obtain the results for nonlinear vibration of composite laminated angle‐ply cylindrical and conical shells. For this purpose, shells with the lamination scheme of ½60 =  60 =60 =  60 s and constants of L ¼ 0:9 m and h ¼ 1 mm are considered while n ¼ 3. The shells are constructed from carbon fiber‐reinforced Polymeric (CFRP) [43] material with constants

ð41Þ

j¼1

Substituting Eq. (30) into Eq. (40) and making some mathematical simplifications lead to the following equation: 

D20 η3s þ ω2L η3s ¼ ð2iωL D2 As  3P1 A2s As ÞexpðiωL T 0 Þ  P1 A3s expð3iωL T 0 Þ þ tcc

ð42Þ

In order to obtain a bounded response from Eq. (42), the coefficient of expðiωs T 0 Þ must be considered zero: 

ð2iωL D2 As  3P1 A2s As Þ ¼ 0

ð43Þ

In order to solve Eq. (43), As is considered to be a complex term as following: As ¼

1 Q expðiγ s Þ 2 s

ð44Þ

While Qs and γ s demonstrate real terms dependent to T 2 and i is the imaginary number. The T 2 derivative of As is obtained as: D2 As ¼

1 i D2 Qs expðiγ s Þ þ Qs D2 γ s expðiγ s Þ 2 2

ð45Þ

Substituting relations (44) and (45) into relation (43) leads to responses for Qs and γ s : γs ¼

3 P1 Q2s T 2 þ Q2 ; Qs ¼ constant;Q2 ¼ constant 8ωs

ð46Þ

Therefore the time response is obtained as: ^ s cosððωL þ 3 P1 Q ^ s ¼ ɛQs ^ 2 Þt þ Q2 Þ; Q ηs ¼ ɛη1 ¼ Q s 8ωL

ð47Þ

Given the Eq. (47), the amplitude and nonlinear frequency are extracted as: ^ s jmaxðXð2m þ 1 : 3m; sÞÞj A¼Q

ð48Þ

3P1 ^ 2 Q 8ωL s

ð49Þ

ωNL ¼ ωL þ

Table 1 Comparison of the results of frequency parameter for a cylindrical shell with literature.

The following relation is considered: Gs ¼

3P1 8ðmaxðXð2m þ 1 : 3m; sÞÞÞ2

ð50Þ

According to Eqs. (48)–(50), nonlinear frequency and nonlinear to linear frequency ratio (nonlinear frequency ratio) are respectively obtained from Eqs. (51) and (52): ωNL ¼ ωL þ

Gs 2 A ωL

ωNL Gs ¼ 1 þ 2 A2 ωL ωL

ð51Þ ð52Þ

5

n

Present

Ref. [54]

1 2 3 4 5 6 7 8 9 10

0.016098 0.009387 0.022104 0.042084 0.067978 0.099665 0.137116 0.180316 0.229252 0.283912

0.016102 0.009387 0.022108 0.042096 0.068008 0.099730 0.0137239 0.180527 0.229594 0.284435

S. Mohammadrezazadeh, A.A. Jafari

Composite Structures 255 (2021) 112867

Table 2 The effect of M T on the convergence of the frequency parameter results of an isotropic conical shell.

Table 3 Comparison of the frequency parameter results of this study with literature for isotropic conical shells. α ¼ 30

n

2 3 4 5 6 7 8 9





α ¼ 45

Present

Ref. [55]

Ref. [56]

Present

Ref. [55]

Ref. [56]

0.8316 0.7331 0.6340 0.5513 0.4937 0.4647 0.4639 0.4884

0.7910 0.7284 0.6352 0.5531 0.4949 0.4653 0.4645 0.4892

0.8420 0.7376 0.6362 0.5528 0.4950 0.4661 0.4660 0.4916

0.7606 0.7175 0.6711 0.6300 0.6013 0.5896 0.5969 0.6232

0.6879 0.6973 0.6664 0.6304 0.6032 0.5918 0.5992 0.6257

0.7655 0.7212 0.6739 0.6323 0.6035 0.5921 0.6001 0.6273

of: ρ ¼ 1824 kg=m3 , E 11 ¼ 138:6 GPa, E 22 ¼ 8:27 GPa; G13 ¼ 4:96 GPa, G23 ¼ 4:96 GPa, G12 ¼ 4:12 GPa and υ12 ¼ 0:26[57]. It should be mentioned that the cylindrical shell's radius is R ¼ 1 m while the large edge radius and semi‐vertex angle of the conical shell  are R2 ¼ 1 m and α ¼ 30 , respectively. These constants are used in this study unless otherwise mentioned. In order to investigate the effect of M T on the convergence of the nonlinear results of the considered cylindrical and conical shells, Table 4 shows the values of nonlinear frequency ratio (ωNL =ωL ) and nonlinear frequency (ωNL ) for different values of M T . This table indicates that the obtained results for M T ¼ 3 have suitable convergence.

Table 4 The influence of M T on the convergence of nonlinear frequency ratio and nonlinear frequency of considered cylindrical and conical shells. MT

1 2 3 4

Cylindrical shells

Conical shells

ωNL =ωL

ωNL

ωNL =ωL

ωNL

1.0270 1.0270 1.0270 1.0270

2895.0282 2895.0282 2895.0282 2895.0282

1.0796 1.0936 1.0936 1.0946

2761.7560 2607.3304 2598.4372 2582.5228

Fig. 3. The effect of the angle of the layers on the diagrams of the amplitude versus the nonlinear frequency ratio for, a: cylindrical shells, b: conical shells. 6

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Table 5 The effect of the angle of the layers on some characteristics of cylindrical and conical shells. φ ðdegreeÞ

Shells' type

ωL

Gs =ωL

Gs

Cylindrical shells

15 30 60 75

1795.3550 2382.6993 2818.7956 1996.0964

1.1266107068637 × 10 9.013680980757 × 108 5.372100725085 × 108 6.092418369959 × 108

Conical shells

15 30 60 75

1793.1737 2387.4759 2375.9744 1730.1572

1.2384991932511 3.4162425244513 1.3214147596455 1.6969347195804

ωNL =ωL jðφ¼75 Þ > ωNL =ωL jðφ¼60 Þ . On the other hand, according to Fig. 3(b), one can conclude that for a constant value of amplitude for considered conical shells, ωNL =ωL jðφ¼30 Þ > ωNL =ωL jðφ¼75 Þ > ωNL =ωL jðφ¼15 Þ > ωNL =ωL jðφ¼60 Þ . These results are due to this fact that according to Eq. (52), the nonlinear frequency ratio increases with increase of the parameter Gs =ω2L , and Table 5 shows that for cylindrical     shells, Gs =ω2L φ¼15 > Gs =ω2L φ¼30 > Gs =ω2L φ¼75 > Gs =ω2L φ¼60 , and for     2 2 2  > Gs =ω  > Gs =ω . conical shells, Gs =ω2   > Gs =ω L φ¼75

L φ¼15

109 109 109 109

Gs =ω2L 5

6.275141826 × 10 3.782970449 × 105 1.90581422 × 105 3.052166462 × 105

349.5210 158.7683 67.6109 152.9068

6.906744083 × 105 1.4309013381 × 106 5.561569812 × 105 9.807980157 × 105

385.1687 599.3364 234.0753 566.8838

cylindrical shells, when the amplitude is small, ωNL jφ¼60 > ωNL jφ¼30 > ωNL jφ¼75 > ωNL jφ¼15 . This is because of the fact that according to Eq. (51), for small amplitudes, the relation between linear and nonlinear frequencies is approximately linear, and according to Table 5 for the considered cylindrical shells, ωL jφ¼60 > ωL jφ¼30 > ωL jφ¼75 > ωL jφ¼15 . On the other hand, Table 5 shows that for the considered conical shells, ωL jφ¼30 > ωL jφ¼60 > ωL jφ¼15 > ωL jφ¼75 which leads to the result of Fig. 4(b) for small specified values of amplitude (ωNL jφ¼30 > ωNL jφ¼60 > ωNL jφ¼15 > ωNL jφ¼75 ). Besides, Fig. 4(a) demonstrates that for specified great values of A for cylindrical shells, ωNL jφ¼15 > ωNL jφ¼30 > ωNL jφ¼75 > ωNL jφ¼60 ; on the other hand, Fig. 4 (b) shows that for a constant value of amplitude, for conical shells, ωNL jφ¼30 > ωNL jφ¼75 > ωNL jφ¼15 > ωNL jφ¼60 ; because for great values of A, the relation between nonlinear frequency and Gs =ωL is direct, which is proven in Eq. (51), and as shown in Table5, for cylindrical shells, Gs =ωL jφ¼15 > Gs =ωL jφ¼30 > Gs =ωL jφ¼75 > Gs =ωL jφ¼60 , and for conical shells, Gs =ωL jφ¼30 > Gs =ωL jφ¼75 > Gs =ωL jφ¼15 > Gs =ωL jφ¼60 . Figs. 5(a) and (b) show the effect of the number of the layers on the curves of amplitude versus nonlinear frequency ratio for cylindrical and conical shells, respectively. These figures depict that as the number of layers increases, the nonlinear frequency ratio becomes smaller for a constant value of A; this is because of this fact that nonlinear frequency ratio is related to 1=ω2L and also Gs , and Table 6 shows that as the number of the layers for cylindrical and conical shells increases, the fundamental linear frequency increases and Gs decreases. Figs. 6(a) and (b) depict the variation of the nonlinear frequency with amplitude and the number of the layers. One can conclude from

In addition, it should be mentioned that initial excitations are assumed such that the use of M T ¼ 3 is appropriate. Therefore, in the next of this study, M T ¼ 3 unless other value for M T is mentioned. Figs. 3(a) and (b) show the effect of the layers' angle (φ) on the curves of A versus ωNL =ωL for cylindrical and conical shells, respectively. It is obvious from these figures that for a specified value of the layers' angle, the increase of amplitude A leads to the increase of nonlinear frequency ratio which can be proven according to Eq. (52). In addition, Fig. 3(a) reveals that for a constant value of A for considered cylindrical shells, ωNL =ωL jðφ¼15 Þ > ωNL =ωL jðφ¼30 Þ >

L φ¼30

× × × ×

9

L φ¼60

The effect of φ on the fundamental linear frequency ωL and Gs can 

be described as: According to Eq. (5), Qij is related to φ. Eqs. (7)–(9) indicate that in‐plane forces and moments which affected the compo

nents of K and B matrices are related to Qij . ω2L which is one of the components of ω2i is related to the components of K matrix. In addition, according to Eqs. (39), (41) and (50), Gs is related to matrix B. Therefore the value of φ has effect on the values of ωL and Gs . Figs. 4(a) and (b) depict the curves of amplitude versus nonlinear frequency for different values of φ. Fig. 4 (a) shows that for considered

Fig. 4. The curves of amplitude versus nonlinear frequency for different values of the layers' angle for, a: cylindrical shells, b: conical shells. 7

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Composite Structures 255 (2021) 112867

Fig. 5. The effect of the number of the layers on the curves of amplitude- nonlinear frequency ratio for, a: cylindrical shells, b: conical shells.

Table 6 The influence of the number of the layers on several characteristics of laminated composite angle-ply cylindrical and conical shells. 

Lamination scheme (θ ¼ 60 )

Cylindrical shell ωL ðrad=sÞ

[θ/−θ]s [θ/−θ/θ/−θ]s [θ/−θ/θ/−θ/θ/−θ]s [θ/−θ/θ/−θ/θ/−θ/θ/−θ]s

2811.3649 2818.7956 2831.0853 2848.0953

Conical shell Gs 5.373831936968 5.372100725085 5.369224345008 5.365216124797

× × × ×

8

10 108 108 108

ωL ðrad=sÞ

Gs

2354.2722 2375.9744 2411.2829 2459.0339

1.3215895509335 1.3214147596455 1.3210556449605 1.3204374304414

× × × ×

109 109 109 109

Fig. 6. Diagrams of amplitude versus nonlinear frequency for different values of the number of the layers for, a: cylindrical shells, b: conical shells.

Gs . According to Table 6, as the number of the layers increases, ωL increases and Gs decreases, leading to the results of Figs. 6(a) and (b). Table 7 shows the values of nonlinear frequency ratio and nonlinear frequency for different values of amplitude and semi‐vertex angle of the conical shells. According to Table 7, one can conclude that for each considered amplitude, the increase of the semi‐vertex angle leads to increase of nonlinear frequency ratio; because according to Table 8, the increase of semi‐vertex angle leads to decrease of fundamental lin-

Figs. 6(a) and (b) that for small values of amplitude, the value of nonlinear frequency increases with increase of the number of the layers. On the other hand, for great values of amplitude, the increase of the number of the layers leads to smaller values for the nonlinear frequency. These conclusions are obtained due to these facts: According to Eq. (51), for small values of A, ωNL has almost linear relation with ωL , while for great values of A, ωNL and ωL have almost inverse relation. In addition, for great values of amplitude, ωNL is also related to

8

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Table 7 Nonlinear frequency ratio and nonlinear frequency of conical shells for different values of amplitude and semi-vertex angle. Parameter

A ðmÞ

α 15°

30°

45°

60°

75°

ωNL =ωL

0.007 0.008 0.009 1

1.0059 1.0077 1.0097 1.0120

1.0115 1.0150 1.0190 1.0234

1.0228 1.0298 1.0377 1.0465

1.0499 1.0652 1.0825 1.1019

1.1429 1.1866 1.2362 1.2916

ωNL ðrad=sÞ

0.007 0.008 0.009 1

2571.0692 2575.6743 2580.8934 2586.7265

2403.2261 2411.5685 2421.0231 2431.5901

2326.0664 2341.9376 2359.9249 2380.0284

2086.4612 2116.8356 2151.2600 2189.7342

1449.0493 1504.4999 1567.3439 1637.5813

and conical shells. This table demonstrates that Gs =ω2L decreases and then increases with increase of the length which is the reason of the behavior of nonlinear frequency ratio in Figs. 7(a) and (b). Figs. 8(a) and (b) depict the variation of nonlinear frequency with length for small values of amplitude for cylindrical and conical shells, respectively. According to these figures, nonlinear frequency value decreases with increase of the length. The reason of the mentioned result can be presented as: According to Eq. (51), for small values of the amplitude, the nonlinear frequency relates directly to fundamental linear frequency. According to Table 9, fundamental linear frequency gets smaller values with increase of the length; so, the nonlinear frequency decreases with increase of the length value. Fig. 9(a) shows the curves of amplitude versus the nonlinear frequency ratio for different values of the cylindrical shell's radius (R). Fig. 9(b) depicts the diagrams of amplitude versus nonlinear frequency ratio for various values of large edge radius (R2 ) of the conical shell. According to these figures, for a constant value of A, the increase of radiuses (R and R2 ) leads to decrease of nonlinear frequency ratio. The cause of the mentioned behavior can be described as: According to Eq. (52), nonlinear frequency ratio has direct and inverse relationship with Gs and ω2L , respectively; according to the results of Table 10, it is concluded that the increase of the radiuses (R and R2 ) leads to decrease of Gs and increase of ωL , which reduces the nonlinear frequency ratio. Figs. 10(a) and (b) demonstrate the curves of A‐ωNL which are obtained for different values of R(cylindrical shells) and R2 (conical shells), respectively. According to these figures, for small values of the amplitude, the increase of the radiuses (R and R2 ) leads to larger

Table 8 The effect of the semi-vertex angle on the fundamental linear frequency and Gs of the conical shells. α

ωL ðrad=sÞ

Gs

15° 30° 45° 60° 75°

2556.0259 2375.9744 2274.2205 1987.2382 1267.9106

7.847153099082 × 108 1.3214147596455 × 109 2.4063039100693 × 109 4.0240773377045 × 109 4.6870938946235 × 109

ear frequency and increase of Gs , which leads to the greater values for nonlinear frequency ratio according to Eq. (52). Besides, Table 7 depicts that the increase of semi‐vertex angle leads to decrease of the values of nonlinear frequency for considered values of A; because the considered values for A are small and in this condition, the nonlinear frequency and the fundamental linear frequency are almost directly related to each other according to Eq. (51). In addition, Table 7 demonstrates that for a constant value of semi‐vertex angle, as the amplitude increases, the nonlinear frequency ratio and nonlinear frequency get greater values. Figs. 7(a) and (b) show the diagrams of nonlinear frequency ratio versus length for different values of amplitude for cylindrical and conical shells, respectively. According to these figures, the nonlinear frequency ratio decreases and then increases with increase of the length. The reason of this behavior can be expressed as: Table 9 shows the values of ωL , Gs and Gs =ω2L for different values of the length of cylindrical

Fig. 7. Diagrams of nonlinear frequency ratio versus length for different values of amplitude for, a: cylindrical shells, b: conical shells. 9

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Composite Structures 255 (2021) 112867

Table 9 The effect of the length on some characteristics of the cylindrical and conical shells. L ðmÞ

Shell's type

ωL ðrad=sÞ

Gs =ω2L

Gs 10

490.8936 174.1768 88.1015 60.1876 52.7165 56.1091 67.6109 565.8483 544.1878 181.5693 97.1791 114.6102 160.4409 234.0753

Cylindrical shell

0.3 0.4 0.5 0.6 0.7 0.8 0.9

5191.5950 5189.4773 5089.5141 4729.0336 4106.0496 3420.2751 2818.7956

1.32308875055209 × 10 4.6906982660544 × 109 2.2821073310406 × 109 1.3460204206740 × 109 8.887814040770 × 108 6.563804324513 × 108 5.372100725085 × 108

Conical shell

0.3 0.4 0.5 0.6 0.7 0.8 0.9

4881.6049 4849.1949 4743.2712 4247.2220 3478.3730 2832.5888 2375.9744

1.34842021040847 × 1010 1.27964076120480 × 1010 4.0850599445418 × 109 1.7530042711136 × 109 1.3866776485188 × 109 1.2873069066752 × 109 1.3214147596455 × 109

Fig. 8. The curves of nonlinear frequency against length for different amplitude values of, a: cylindrical shells, b: conical shells.

Fig. 9. Diagrams of amplitude versus nonlinear frequency ratio for different values of, a: the radius of cylindrical shell, b: the large edge radius of conical shell. 10

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Table 10 The effect of the radius of cylindrical shell and large edge radius of conical shell on the fundamental linear frequency and Gs ωL ðrad=sÞ

Variable

Table 11 The effect of the thickness of each layer of cylindrical and conical shells on the fundamental linear frequency and Gs

Gs

Shell's type

h ðmmÞ

ωL ðrad=sÞ

Gs

R ðmÞ (Cylindrical shell)

1 1.2 1.4

2818.7956 3129.1343 3198.2067

5.372100725085 × 10 3.631375208651 × 108 2.896302005442 × 108

Cylindrical shells

1 1.5 2

2818.7956 2831.0853 2848.0953

5.372100725085 × 108 5.369224345009 × 108 5.365216124797 × 108

R2 ðmÞ (Conical shell)

1 1.2 1.4

2375.9744 2603.4184 2789.6567

1.3214147596455 × 109 6.339242596677 × 108 4.037342802746 × 108

Conical shells

1 1.5 2

2375.9744 2411.2829 2459.0339

1.3214147596455 × 109 1.3210556449607 × 109 1.3204374304414 × 109

8

increasement. The explanation of this result is: According to Eq. (51), for large values of A, nonlinear frequency decreases with increase of fundamental linear frequency and decrease of Gs ; therefore, according to the results obtained for ωL and Gs in Table 10, the behavior of the nonlinear frequency in Fig. 10(a) and (b) is logical for large values of amplitude.

values for nonlinear frequency; because at small amplitudes, according to Eq. (51), the relation of nonlinear frequency with fundamental linear frequency is approximately linear, and as shown in Table 10, the increase of the radiuses (R and R2 ) leads to the increase of the fundamental linear frequency. On the other hand, for larger values of A, nonlinear frequency gets smaller values with the radiuses (R and R2 )

Fig. 10. The curves of amplitude versus nonlinear frequency for different values of, a: the radius of cylindrical shell, b: the large edge radius of conical shell.

Fig. 11. The curves of amplitude against nonlinear frequency ratio for different values of each layer's thickness for, a: cylindrical shells, b: conical shells. 11

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Composite Structures 255 (2021) 112867

Fig. 12. The influence of the layer's thickness on the curves of amplitude versus nonlinear frequency for, a: cylindrical shells, b: conical shells.

Figs. 11(a) and (b) show the effect of each layer's thickness on the nonlinear frequency ratio of cylindrical and conical shells, respectively. These figures reveal that for a specified value of A, nonlinear frequency gets smaller values with increase of the thickness. This is due to this fact that according to Table 11, as h becomes greater, the values of fundamental linear frequency and Gs become greater and smaller, respectively. Fig. 12(a) shows the curves of amplitude against nonlinear frequency for different values of each layer's thickness of the cylindrical shells. Fig. 12(b) depicts the diagrams of amplitude versus nonlinear frequency for different values of the thickness of each layer of the conical shells. These figures reveal that for small values of amplitude, the increase of thickness leads to greater values for nonlinear frequency, while for great values of amplitude, the nonlinear frequency decreases with increase of the thickness value. These behaviors' reasons can be explained as: Table 11 shows that fundamental linear frequency and Gs get higher and smaller values with increase of the thickness of each layer, respectively. Eq. (51) shows that when amplitude is small, the nonlinear frequency is approximately related in a linear form to the fundamental linear frequency; therefore, in this condition, the increase of thickness leads to greater values for nonlinear frequency. On the other hand, for large values of amplitude, Eq. (51) indicates approximately an inverse relation of nonlinear frequency with fundamental linear frequency, as well as direct relation of nonlinear frequency with Gs ; therefore, for large values of amplitude, the increase of thickness leads to smaller values for nonlinear frequency.

semi‐vertex angle, length, radius and thickness of each layer on the nonlinear vibration characteristics are discussed in details. The main results of this study can be classified as following:

5. Conclusion

Shahin Mohammadrezazadeh: Software, Validation, Formal analysis, Investigation, Resources, Writing ‐ original draft, Writing ‐ review & editing. Ali Asghar Jafari: Supervision.

1. The results obtained for different values of the upper bounding value for longitudinal wave number reveal that the convergence of the results of this study is appropriate for determined constants of cylindrical and conical shells. 2. The increase of amplitude leads to the increase of nonlinear frequency ratio as well as nonlinear frequency. 3. The increase of the number of the layers, radiuses (radius of cylindrical shell and large edge radius of conical shell) or thickness of each layer leads to smaller values for nonlinear frequency ratio and greater values for fundamental linear frequency. 4. The increase of the semi‐vertex angle leads to greater and smaller values for nonlinear frequency ratio and fundamental linear frequency, respectively. 5. For small values of amplitude, the increase of the number of the layers, radiuses or thickness of each layer leads to higher values for nonlinear frequency, while for high values of amplitude, the mentioned variables' increasement leads to smaller values for nonlinear frequency. 6. For small values of amplitude, the increase of semi‐vertex angle or length leads to decrease of nonlinear frequency. 7. Fundamental linear frequency decreases as the length increases. CRediT authorship contribution statement

In this paper, nonlinear free vibration of laminated composite angle‐ply cylindrical and conical shells have been analyzed and discussed using the combination of multiple scales method and modal analysis. The equations of the shells are obtained considering shear deformation and rotary inertia based on von Karman nonlinear theory. The partial differential equations of the system are derived by means of Hamilton principle. The ordinary differential equations of the cylindrical and conical shells are extracted using Ritz method. Then the results for fundamental linear frequency, amplitude, nonlinear frequency and nonlinear frequency ratio are obtained using multiple scales method with the help of modal analysis. The effects of several parameters including the angle of the layers, the number of the layers,

Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgements This research did not receive any specific grant from funding agencies in the public, commercial, or not‐for‐profit sectors. 12

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[26] Sofiyev AH. The buckling and vibration analysis of coating-FGM-substrate conical shells under hydrostatic pressure with mixed boundary conditions. Compos Struct 2019;209:686–93. [27] Permoon MR, Shakouri M, Haddadpour H. Free vibration analysis of sandwich conical shells with fractional viscoelastic core. Compos Struct 2019;214:62–72. [28] Emam SA. A Theoretical and Experimental Study of Nonlinear Dynamics of Buckled Beams PhD Thesis. Virginia: Virginia Polytechnic Institute; 2002. [29] Yong-gang Z, Xin-zhi W, Kai-yuan Y. Nonlinear vibration of thin shallow conic shells under combined action of peripheral moment and transverse loads. Appl Math Mech 2003;24(12):1381–9. [30] Shen H-S. Nonlinear vibration of shear deformable FGM cylindrical shells surrounded by an elastic medium. Compos Struct 2012;94(3):1144–54. [31] Bich DH, Xuan Nguyen N. Nonlinear vibration of functionally graded circular cylindrical shells based on improved Donnell equations. J Sound Vib 2012;331 (25):5488–501. [32] Jafari AA, Khalili SMR, Tavakolian M. Nonlinear vibration of functionally graded cylindrical shells embedded with a piezoelectric layer. Thin-Walled Structures 2014;79:8–15. [33] Sofiyev AH. The combined influences of heterogeneity and elastic foundations on the nonlinear vibration of orthotropic truncated conical shells. Compos B Eng 2014;61:324–39. [34] Sofiyev AH. Nonlinear free vibration of shear deformable orthotropic functionally graded cylindrical shells. Compos Struct 2016;142:35–44. [35] Shen H-S, Xiang Y, Fan Y. Nonlinear vibration of functionally graded graphenereinforced composite laminated cylindrical shells in thermal environments. Compos Struct 2017;182:447–56. [36] Hasrati E, Ansari R, Torabi J. A novel numerical solution strategy for solving nonlinear free and forced vibration problems of cylindrical shells. Appl Math Model 2018;53:653–72. [37] Ansari R, Hasrati E, Torabi J. Nonlinear vibration response of higher-order shear deformable FG-CNTRC conical shells. Compos Struct 2019;222:110906. https:// doi.org/10.1016/j.compstruct.2019.110906. [38] Li C, Zhong B, Shen Z, Zhang J. Modeling and nonlinear vibration characteristics analysis of symmetrically 3-layer composite thin circular cylindrical shells with arbitrary boundary conditions. Thin-Walled Structures 2019;142:311–21. [39] Li C, Li P, Zhong B, Wen B. Geometrically nonlinear vibration of laminated composite cylindrical thin shells with non-continuous elastic boundary conditions. Nonlinear Dyn 2019;95(3):1903–21. [40] Aris H, Ahmadi H. Nonlinear vibration analysis of FGM truncated conical shells subjected to harmonic excitation in thermal environment. Mech Res Commun 2020;104:103499. https://doi.org/10.1016/j.mechrescom.2020.103499. [41] Bakhtiari M, Lakis AA, Kerboua Y. Nonlinear vibration of truncated conical shells: Donnell, Sanders and Nemeth theories. International Journal of Nonlinear Sciences and Numerical Simulation 2020; 21(1): 83-97. [42] Kamaloo A, Jabbari M, Yarmohammad Tooski M, Javadi M. Nonlinear free vibration analysis of delaminated composite circular cylindrical shells. J Vib Control 2020;26(19-20):1697–707. [43] Reddy JN. Mechanics of laminated composite plates and shells: theory and analysis. Second Edition. CRC Press; 2003. [44] Pirbodaghi T, Ahmadian MT, Fesanghary M. On the homotopy analysis method for non-linear vibration of beams. Mech Res Commun 2009;36(2):143–8. [45] Motallebi AA, Poorjamshidian M, Sheikhi J. Vibration analysis of a nonlinear beam under axial force by homotopy analysis method. J Solid Mech 2014;6(3):289–98. [46] Jing Z. Semi-analytical optimal solution for maximum buckling load of simply supported orthotropic plates. Int J Mech Sci 2020;187:105930. https://doi.org/ 10.1016/j.ijmecsci.2020.105930. [47] Elliott AJ, Cammarano A, Neild SA, Hill TL, Wagg DJ. Comparing the direct normal form and multiple scales methods through frequency detuning. Nonlinear Dyn 2018;94(4):2919–35. [48] Shakouri M, Kouchakzadeh MA. Analytical solution for vibration of generally laminated conical and cylindrical shells. Int J Mech Sci 2017;131-132:414–25. [49] Rao SS. Vibration of Continuous Systems. INC: JOHN WILEY & SONS; 2007. [50] Qatu MS. In: Vibration of Laminated Shells and Plates. Elsevier; 2004. p. 109–79. https://doi.org/10.1016/B978-008044271-6/50006-5. [51] Reddy JN. Energy principles and variational methods in applied mechanics. John Wiley & Sons; 2002. [52] Nayfeh AH, Mook DT, editors. Nonlinear Oscillations. Wiley; 1995. [53] Nayfeh AH. Introduction to Perturbation Techniques. JOHN WILEY & SONS, INC., 1993. [54] Loy CT, Lam KY, Reddy JN. Vibration of functionally graded cylindrical shells. Int J Mech Sci 1999;41(3):309–24. [55] Irie T, Yamada G, Tanaka K. Natural frequencies of truncated conical shells. J Sound Vib 1984;92(3):447–53. [56] Lam KY, Hua Li. On free vibration of a rotating truncated circular orthotropic conical shell. Compos B Eng 1999;30(2):135–44. [57] Pradhan SC, Reddy JN. Vibration control of composite shells using embedded actuating layers. Smart Mater. Struct. 2004;13(5):1245–57.

Appendix A. Supplementary data Supplementary data to this article can be found online at https://doi.org/10.1016/j.compstruct.2020.112867. References [1] Civalek Ö. Geometrically nonlinear dynamic analysis of doubly curved isotropic shells resting on elastic foundation by a combination of harmonic differential quadrature-finite difference methods. Int J Press Vessels Pip 2005;82(6):470–9. [2] Civalek Ö, Acar MH. Discrete singular convolution method for the analysis of Mindlin plates on elastic foundations. Int J Press Vessels Pip 2007;84(9):527–35. [3] Akgoz B, Civalek O. Nonlinear vibration analysis of laminated plates resting on nonlinear two-parameters elastic foundations. Steel Compos Struct 2011;11 (5):403–21. [4] Demir Ç, Civalek Ö. A new nonlocal FEM via Hermitian cubic shape functions for thermal vibration of nano beams surrounded by an elastic matrix. Compos Struct 2017;168:872–84. [5] Akgöz B, Civalek Ö. A size-dependent beam model for stability of axially loaded carbon nanotubes surrounded by Pasternak elastic foundation. Compos Struct 2017;176:1028–38. [6] Liang Xu, Deng Yu, Cao Z, Jiang X, Wang T, Ruan Y, Zha X. Three-dimensional dynamics of functionally graded piezoelectric cylindrical panels by a semianalytical approach. Compos Struct 2019;226:111176. https://doi.org/10.1016/ j.compstruct.2019.111176. [7] Jing Z, Sun Q, Liang Ke, Chen J. Closed-form critical buckling load of simply supported orthotropic plates and verification. Int J Str Stab Dyn 2019;19 (12):1950157. https://doi.org/10.1142/S0219455419501578. [8] Rao SS. Mechanical vibrations. fifth edition. Prentice Hall; 2011. [9] ZHANG XM, LIU GR, LAM KY. Vibration analysis of thin cylindrical shells using wave propagation approach. J Sound Vib 2001;239(3):397–403. [10] Zhang L, Xiang Y, Wei GW. Local adaptive differential quadrature for free vibration analysis of cylindrical shells with various boundary conditions. Int J Mech Sci 2006;48(10):1126–38. [11] Sobhani Aragh B, Yas MH. Static and free vibration analyses of continuously graded fiber-reinforced cylindrical shells using generalized power-law distribution. Acta Mech 2010;215(1-4):155–73. [12] Sofiyev AH, Kuruoglu N. Buckling and vibration of shear deformable functionally graded orthotropic cylindrical shells under external pressures. Thin-Walled Structures 2014;78:121–30. [13] Tadi Beni Y, Mehralian F, Razavi H. Free vibration analysis of size-dependent shear deformable functionally graded cylindrical shell on the basis of modified couple stress theory. Compos Struct 2015;120:65–78. [14] 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. [15] Nurul Izyan MD, Aziz ZA, Viswanathan KK. Free vibration of anti-symmetric angleply layered circular cylindrical shells filled with quiescent fluid under first order shear deformation theory. Compos Struct 2018;193:189–97. [16] Lin H, Cao D, Shao C. An admissible function for vibration and flutter studies of FG cylindrical shells with arbitrary edge conditions using characteristic orthogonal polynomials. Compos Struct 2018;185:748–63. [17] Zhao J, Choe K, Zhang Y, Wang A, Lin C, Wang Q. A closed form solution for free vibration of orthotropic circular cylindrical shells with general boundary conditions. Compos B Eng 2019;159:447–60. [18] Sofiyev AH, Hui D. On the vibration and stability of FGM cylindrical shells under external pressures with mixed boundary conditions by using FOSDT. Thin-Walled Structures 2019;134:419–27. [19] Baghlani A, Khayat M, Dehghan SM. Free vibration analysis of FGM cylindrical shells surrounded by Pasternak elastic foundation in thermal environment considering fluid-structure interaction. Appl Math Model 2020;78:550–75. [20] Wu C-P, Lee C-Y. Differential quadrature solution for the free vibration analysis of laminated conical shells with variable stiffness. Int J Mech Sci 2001;43 (8):1853–69. [21] Civalek O. Free vibration analysis of composite conical shells using the discrete singular convolution algorithm. Steel and Composite Structures 2006;6 (4):353–66. [22] Sofiyev AH, Kuruoglu N, Halilov HM. The vibration and stability of nonhomogeneous orthotropic conical shells with clamped edges subjected to uniform external pressures. Appl Math Model 2010;34(7):1807–22. [23] Viswanathan KK, Javed S, Prabakar K, Aziz ZA, Abu Bakar I. Free vibration of antisymmetric angle-ply laminated conical shells. Compos Struct 2015;122:488–95. [24] Sofiyev AH, Osmancelebioglu E. The free vibration of sandwich truncated conical shells containing functionally graded layers within the shear deformation theory. Compos B Eng 2017;120:197–211. [25] Sofiyev AH, Kuruoglu N. Determination of the excitation frequencies of laminated orthotropic non-homogeneous conical shells. Compos B Eng 2018;132:151–60.

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