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The dynamic stability and nonlinear vibration analysis of stiffened functionally graded cylindrical shells
Accepted Manuscript
The dynamic stability and nonlinear vibration analysis of stiffened functionally graded cylindrical shells G.G. Sheng , X. Wang PII: DOI: Reference:
S0307-904X(17)30752-7 10.1016/j.apm.2017.12.021 APM 12102
To appear in:
Applied Mathematical Modelling
Received date: Revised date: Accepted date:
2 June 2017 1 November 2017 7 December 2017
Please cite this article as: G.G. Sheng , X. Wang , The dynamic stability and nonlinear vibration analysis of stiffened functionally graded cylindrical shells, Applied Mathematical Modelling (2017), doi: 10.1016/j.apm.2017.12.021
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Highlights
The dynamic stability and nonlinear vibrations of stiffened FG shells are investigated. A reduction nonlinear model of stiffened FG shells is presented. The effects of key parameters on the dynamic stability and nonlinear
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vibrations are analyzed.
A series of comparison are performed and the investigations demonstrate
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good reliability.
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The dynamic stability and nonlinear vibration analysis of stiffened functionally graded cylindrical shells G.G. Shenga,*[email protected], X.Wangb a
School of Civil Engineering, Changsha University of Science and Technology, Changsha, Hunan 410114, People’s Republic of China
School of Naval Architecture, Ocean and Civil Engineering (State Key Laboratory of Ocean
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b
Engineering), Shanghai Jiaotong University, Shanghai 200240, People’s Republic of China *
Corresponding author.
Abstract
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A theoretical model is developed to study the dynamic stability and nonlinear vibrations of the stiffened functionally graded (FG) cylindrical shell in thermal environment. Von Kármán nonlinear theory, first-order shear deformation theory, smearing stiffener approach and Bolotin method are used to model stiffened FG
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cylindrical shells. Galerkin method and modal analysis technique is utilized to obtain the discrete nonlinear ordinary differential equations. Based on the static condensation
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method, a reduction model is presented. The effects of thermal environment, stiffeners number, material characteristics on the dynamic stability, transient responses and
Keywords
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primary resonance responses are examined.
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Stiffened; Functionally graded cylindrical shell; Nonlinear vibration; Dynamic stability
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1 Introduction
FG shells are being used as structural components in modern industries such as
aerospace, mechanical, chemical industry and nuclear engineering [1-5]. The main applications of FG shells have been in high temperature environments. To enhance the stiffness and stability of FG shells, we can use stiffeners. The axial and ring stiffeners in FG shells are very good stiffening elements to raise the stiffness and stability without great mass increase. The knowledge of their characteristics is
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necessary, and the natural frequencies, responses, static and dynamic stability of stiffened FG shells are of special interest for their applications. The stiffened shells have been investigated for many years. When the number of stiffeners is small, and the stiffeners are not closely and evenly spaced, the discrete stiffener theory is used to develop the characteristics of stiffened shells [6]. For the non-uniform ring stiffener distribution, Jafari and Bagheri [7] studied the free
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vibration of stiffened shells based on the Ritz method and the discrete stiffener theory. Using Reissner-Naghdi shell theory and the discrete element method, Qu et al.[8] reported the dynamic characteristics of the stiffened shell. When stiffeners are closely and uniformly-spaced, their discrete nature can be neglected and the characteristics of
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stiffened shells are similar with that of orthotropic shells. Based on this idea, the smearing method is proposed [9, 10]. There are a large number of literatures on the smearing method for stiffened structures. Using the Donnel shell theory, Li and Qiao [11] presented the nonlinear free vibration and primary resonance analysis for
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stiffened cylindrical shells, and proposed an improved smearing stiffener approach. Using the smearing method, an investigation was presented for stiffened FG shells in
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thermal environment [12]. The effects of rotatory inertia and shear deformation were considered. According to the smeared stiffener approach and shear deformation theory,
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Sun et al.[13] analyzed the buckling behaviors of FG stiffened cylindrical shells under the thermal load. The dynamic stability of FG structures was also studied by many
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researchers. In case of a harmonic parametric excitation, the motion amplitude of the structures will define the instability regions through the Mathieu equation. For FG
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cylindrical shells, the dynamic stability was studied using Bolotin method by Ng et al.[1]. They found that material distribution (volume fraction exponent) has significant influence on the positions and sizes of instability regions. The free vibration and dynamic stability were presented by Yang and Shen [14] for FG shells under the harmonic parametric excitation and thermal load, and Galerkin technique and Bolotin method were used to analyze the dynamic characteristics of shells. Using the shear deformation theory and Galerkin method, the dynamic stability of FG cylindrical shells was studied by Torki et al. [15]. The effect of the material 3
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characteristics on the critical mode number and the flutter load was addressed in their paper. Using linear Donnell type equations, Galerkin method and Bolotin method, Sofiyev and Kuruoglu [16] analyzed the dynamic stability of FG conical shells subjected to time dependent loads. Sahmani et al.[17] studied the dynamic stability of FG cylindrical shells based on the shear deformation theory, Hamilton’s principle and Bolotin method. Asadi and Wang [18] investigated the dynamic stability for a FG
Donnell theory and the shear deformation theory.
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reinforced composite cylindrical shell. The dynamic model was established using
The structures can be vibrated with large amplitude. The linear theory is not sufficient to analyze the large amplitude vibration. To design a stable and reliable
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structure, the nonlinear vibration theory should be employed. A lot of literatures on the nonlinear vibration are concerned for plates and shells [19–24]. Using the smearing stiffener method and von Kármán nonlinear theory, Dung and Hoa [25], Ninh and Bich [26] studied the nonlinear characteristics for stiffened FG shells in
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thermal environment. Using the stress function, Duc and Thang [27] developed a theoretical model to analyze the nonlinear dynamic response for stiffened FG shells.
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Similarly, Duc et al.[28-30] discussed the nonlinear thermal dynamic behavior of stiffened FG shells, and the changes in geometric shapes were considered after the
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thermal deformation process. Furthermore, using Galerkin method, Kumar et al. [31] reported the dynamic stability of FG plates considering von Kármán geometric
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nonlinearity. Employing Donnell nonlinear shell theory and Bolotin methods, Dey and Ramachandra [32] discussed the dynamic stability for the composite cylindrical
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shell.
As stated above literatures, only a few studies are concerned on the dynamic
stability and nonlinear characteristics of stiffened FG shells. In this paper, the authors attempt to solve this subject. Hamilton’s principle, the smeared technique, the shear deformation and von Kármán nonlinear theory are used to obtain the theoretical model of stiffened FG shells. This study is an extension of our earlier works [33-35] on the dynamic characteristics of stiffened and non-stiffened FG cylindrical shells. A reduced model is given based on the static condensation method, and it can capture 4
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the main dynamic stability and nonlinear vibration characteristics of stiffened shells. The dynamic stability and natural frequencies have been compared and verified with that of previous studies for non-stiffened FG cylindrical shells and isotropic stiffened cylindrical shells. 2 Theoretical formulations
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Now consider FG cylindrical shells reinforced by closely, uniformly-spaced eccentric ring and axial stiffeners. The geometry and coordinate system are shown in Fig.1. The FG cylindrical shell has length L ¸ thickness h , and mean radius R . The ring and axial stiffeners have rectangular cross section with width ( br , ba ) and
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thickness ( hr , ha ). The functionally graded material is Type B (inner surface: ceramic, outer surface: metal). The properties of the stiffener material are the same as that of the surface material. That means inner stiffeners are assumed to be made of full ceramic material, and outer stiffeners are made of full metal. The stiffened FG shell is
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a complete whole, with stiffeners and shell.
The FG material property Feff is varied according to the expressions (a power
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law) along the thickness
zh/2 ) (0 ) h
(1)
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Feff ( z) FO A( z) FI (1 A( z)) , A( z ) (
where denotes the volume fraction exponent of FG material, FI denotes the
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material property of the inner surface, FO denotes the material property of the outer
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surface. Here, mass density , elastic modulus E , thermal expansion , Poisson’s ratio , and thermal conductivity k vary according to Eq. 1. In this paper, the temperature variation is assumed to occur in radial direction
only. The temperature field is defined by the following equation [36] d dT (k eff ( z ) ) 0 dz dz h h T ( ) Tc , T ( ) Tm 2 2
(2) (3)
where the thermal conductivity k eff satisfies the Eq. (1). The solution to Eq. (2) can 5
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be obtained
T ( z ) Tc
Tcm h 2 h 2
dz
z
h 2
dz k e f (f z )
(4)
k e f (f z )
h h where Tcm T ( ) T ( ) (temperature difference). 2 2 The stress-strain relations are of the form
where
σ x
x xz z
T
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σ C( z)[ ε α( z)T ( z)]
(5)
ε x
(stresses),
x
xz z
T
(strains) , T ( z ) T ( z ) T0 and T0 Tm (room temperature). α(z ) and C(z ) is the
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thermal expansion coefficients and elasticity matrix of FG materials, respectively:
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0 0 0 Q11 ( z ) Q12 ( z ) Q ( z ) Q ( z ) 0 0 0 22 21 C( z ) 0 0 Q66 ( z ) 0 0 , 0 0 Q55 ( z ) 0 0 0 0 0 0 Q44 ( z ) α( z ) xxe ( z ) e ( z) 0 0 0
Eeff
1 eff
2
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Q11 ( z )
ED
where
Eeff
1 eff 2
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Q22 ( z )
, Q12 ( z )
T
(6)
eff Eeff E , Q21 ( z ) eff eff2 , 2 1 eff 1 eff
, Q44 ( z ) Q55 ( z ) Q66 ( z )
Eeff 2(1 eff )
(7)
FG material properties are uniform distribution in plane, and the thermal expansion
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coefficients ( xxe ( z ) , e ( z ) ) are equal in axial and circumferential directions ( x , ) ( xxe ( z ) e ( z ) eff ( z ) ). Using Eq. (1), the effective Young’s modulus E eff , effective thermal expansion coefficients eff and effective Poisson’s ratio eff can be obtain for a given volume fraction exponent . The arbitrary point strains ε can be expressed using mid-surface strains ( x , , x , xz and z ) and curvatures( x , and x ): 6
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x x z x , ( z ) / (1 z / R) , x ( x z x ) / (1 z / R) , xz xz , z z / (1 z / R)
(8)
According to the smearing method [9, 10], an equivalent non-stiffened shell model can be obtained. Substituting Eq. 8 into the stress-strain relations in Eq. 5, the following constitutive equations can be derived by integrating over the entire
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thickness of the shell:
s s T T N x ( A11 A11 ) x A12 ( B11 B11 ) x B12 N x N a s s T T N A21 x ( A22 A22 ) B21 x ( B22 B22 ) N N r N A66 x B66 x N x T x
(9)
(10)
(11)
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( A55 A55 s ) xz Qxz G s Q z ( A44 A44 ) z
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s s T T M x ( B11 B11 ) x B12 ( D11 D11 ) x D12 M x M a s s T T M B21 x ( B22 B22 ) D21 x ( D22 D22 ) M M r M B66 x D66 x M x T x
where N x , N , N x , M x , M , M x , Q x and Q are the usual stress
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resultants. G is the shear correction factor ( G 5 / 6 [36]), and the coefficients Aij ,
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Dij and Bij appearing in Eqs.(9-11) are defined in terms of elasticity coefficients
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Qij :
h 2 h 2
( Aij , Bij , Dij )= Qij (1, z, z ) dz 2
h 2 h 2
h 2 h 2
( i, j 1,2,6) , A55 Q55 dz , A44 Q44 dz
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Q11 Q11 ( z ) , Q12 Q12 ( z ) / (1 z / R) , Q21 Q21 ( z ) , Q22 Q22 ( z ) / (1 z / R) ,
Q66 Q66 ( z ) / (1 z / R) , Q55 Q55 ( z ) , Q44 Q44 ( z ) / (1 z / R)
(12)
N xT , N T , N x T , M xT , M T and M x T are thermal stress resultants of the shell: N xT T N N T x
M xT Q11 ( z ) xxe ( z ) Q12 ( z ) e ( z ) h T M 2h Q12 ( z ) xxe ( z ) Q22 ( z ) e ( z ) T ( z ) 1 z dz 2 M x T 0
(13)
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Aij s , Bij s and Dij s denote stiffener stiffnesses, and they are given in terms of the geometric and material parameters of stiffeners (subscripts a : axial stiffeners; subscripts r : ring stiffeners)
A11s Ea Aa / da , B11s Ea Aa ea / da , D11s Ea ( I a Aa ea 2 ) / da ,
A44 s
Ea A Er A a , A55 s r 2(1 a ) d a 2(1 r ) d r
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A22 s Er Ar / dr , B22 s Er Ar er / dr , D22 s Er ( I r Ar er 2 ) / dr , (14)
E and denote the Young’s modulus and Poisson’s ratio of stiffeners, respectively. The stiffeners are of eccentricity
e
(positive: outside, negative: inside),
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cross-sectional area A , intertia moment I , ring stiffener spacing d r ( dr L / Nr ), axial stiffener spacing d a ( da 2 R / Na ), ring stiffener number N r and axial stiffener number N a . The thermal stress resultants in Eqs. (9) and (10) for stiffeners
M aT Ea a Aa / d a M r T Er r Ar / d r
Ea a Aa ea / d a T (hs ) Er r Ar er / d r
(15)
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N aT T Nr
M
are given by
where outside stiffeners hs h / 2 and inside stiffeners hs h / 2 .
u 1 w 2 1 v 1 1 w 2 ( ) , ( w) ( ) , x 2 x R 2 R
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x
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The strains in Eqs. (9-11) have the form
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x
x
w 1 w v 1 u 1 w w , xz x , z , x R x R R x
x 1 1 x , , x . x R x R
(16)
where u , v and w are the midsurface displacement components along the x , and z direction; x and are the rotation functions. For dynamic stability, the stiffened FG cylindrical shell is under a periodically pulsating load (negative in compression) in axial direction:
Na N0 Nd cos Pt
(17) 8
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where P denotes the frequency of parameter excitation. Considering the rotary inertia and transverse shear, the equations of motion under the parameter excitation are obtained using the first-order shear deformation theory and Hamilton’s principle [35, 36]:
u :
( N x N xT N aT ) 1 N x x R
v :
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( I 0 I 0r I 0a )u ( I1 I1r I1a )x 1 ( N N T N r T ) N x 1 2v Q z ( N0 N d cos Pt ) 2 R x R x
( I 0 I 0r I 0a )v ( I1 I1r I1a ) 1 Q z Qxz 1 ( N N T N r T ) R x R
( N0 N d cos Pt )
2w (u, v, w, x , ) q( x, , t ) ( I 0 I 0 r I 0 a ) w x 2
( M x M xT M aT ) 1 M x Qxz ( I1 I1r I1a )u ( I 2 I 2 r I 2 a )x x R
M
x :
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w :
(18)
PT
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1 ( M M T M r T ) M x : Q z ( I1 I1r I1a )v ( I 2 I 2r I 2a ) R x
where q( x, , t ) is the force per unit area acting radial direction, (u, v, w, x , ) is
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the nonlinear term from the von Kármán nonlinear strains:
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(u, v, w, x , )
w [( N x N xT N aT ) ] x x
1 w 1 w 1 w [( N N T N r T ) ] ( N x ) ( N x ) 2 R R x R x
(19)
The mass inertia terms of the shell and the stiffeners are defined by
( I 0 , I1 , I 2 )
h /2
h /2
eff ( z )(1, z, z 2 )dz
( I 0r , I1r , I 2r ) ( r Ar / dr , r Ar er / dr , r ( I r Ar er 2 ) / dr )
(20) (21)
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( I 0a , I1a , I 2a ) ( a Aa / da , a Aa ea / da , a ( I a Aa ea 2 ) / da )
(22)
r and a are mass densities of the ring and axial stiffeners, respectively. Utilizing the strain-displacement relations in Eq. (16), the nonlinear equations are obtained in terms of the generalized displacements ( u, v, w, x , ) by substituting Eqs.
L11u L12v L13w L14x L15 wx L16 w w L17 w
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(9-11), (13), (15) and (17) into Eq. (18)
( I 0 I 0r I 0a )u ( I1 I1r I1a )x
(23)
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2 L21u [ L22 ( N0 N d cos Pt ) 2 ]v L23w L24x L25 wx L26 w w L27 w x
( I 0 I 0r I 0a )v ( I1 I1r I1a ) L31u L32v [ L33 ( N 0 N d cos Pt )
（24）
2 ]w L34x L35 x 2
(25)
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wx L36 w w L37 w (u, v, w, x , ) q( x, , t ) ( I 0 I 0r I 0a )w
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L41u L42v L43w L44x L45 wx L46 w w L47 w
( I1 I1r I1a )u ( I 2 I 2r I 2a )x
(26)
( I1 I1r I1a )v ( I 2 I 2r I 2a )
(27)
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L51u L52v L53w L54x L55 wx L56 w w L57 w
where Lij are linear operators (see Appendix A).
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The following representation of the linear ( u, v, x , ) and nonlinear ( w )
displacement fields is used for simply supported boundary conditions at x 0 and x L [35, 37] M
N
u u mn (t ) cos m x cos n , m 1 n 1 M
N
M
N
v v mn (t ) sin m x sin n , m 1 n 1 M
N
x xmn (t ) cos m x cos n , mn (t ) sin m x sin n , m 1 n 1
m 1 n 1
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M
N
2
M
w (t ) w wmn (t ) sin m x cos n mn sin m x 2R m 1 n 1 m 1
where m
(28)
m , n represents the number of circumferential waves and m represents L
the number of axial half-waves. The radial distributed force q( x, , t ) in Eq. (25) can also be expanded as N
m 1 n 1
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M
q( x, , t ) qmn (t )sin m x cos n
(29)
Substituting Eqs. (28) and (29) into Eqs. (23-27), the continuous system model is , discretized by multiterm Galerkin s technique, and we obtain the coupled nonlinear
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ordinary differential equations:
C11mnumn (t ) C12mnvmn (t ) C13mnxmn (t ) C14mn mn (t ) C15mn wmn (t ) ijkl Cumn wij (t ) wkl (t ) ijkl
C
ijklpq umn
wij (t )wkl (t )wpq (t )
ijklpq
(30)
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( I 0 I 0 r I 0 a )umn (t ) ( I1 I1r I1a )xmn (t )
mn mn mn mn C21 umn (t ) [C221 ( N0 Nd cos Pt )C222 ]vmn (t ) C23 xmn (t ) C24mn mn (t ) C25mn wmn (t )
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ijkl ijklpq Cvmn wij (t )wkl (t ) Cvmn wij (t )wkl (t )wpq (t ) ijkl
ijklpq
(31)
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( I 0 I 0 r I 0 a )vmn (t ) ( I1 I1r I1a ) mn (t )
AC
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mn mn mn C41 umn (t ) C42 vmn (t ) C43 xmn (t ) C44mn mn (t ) C45mn wmn (t )
Cijklxmn wij (t )wkl (t ) Cijklpq xmn wij (t ) wkl (t ) w pq (t ) ijkl
ijklpq
( I1 I1r I1a )umn (t ) ( I 2 I 2r I 2a )xmn (t )
(32)
C51mnumn (t ) C52mnvmn (t ) C53mnxmn (t ) C54mn mn (t ) C55mn wmn (t ) ijkl ijklpq C mn wij (t ) wkl (t ) C mn wij (t ) wkl (t ) w pq (t ) ijkl
ijklpq
( I1 I1r I1a )vmn (t ) ( I 2 I 2r I 2a ) mn (t )
(33)
( I 0 I 0r I 0a )wmn (t ) C31mnumn (t ) C32mnvmn (t ) mn mn mn mn [C331 ( N0 Nd cos Pt )C332 ( N xT NaT )C333 ( N T NrT )C334 ]wmn (t )
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C
ijkl 4 mn
ijkl
ijkl
ijkl 5 mn
wij (t ) kl (t ) C
wij (t ) xkl (t ) C
ijkl
ijklpq 6 mn
ijkl
ijkl
wij (t )wkl (t )wpq (t ) qmn (t )
(34)
ijklpq
where u mn (t ) , vmn (t ) , wmn (t ) , xmn (t ) and mn (t ) is the time-dependent variables, and the coefficients can be determined by operators Lij and the nonlinear term
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(u, v, w, x , ) . According to the static condensation method [36], Eqs. (30-33) can be written as
C12mn
C13mn
mn mn C221 C222 No
mn C23
C42mn
mn C43
C52mn
C53mn
C14mn umn C15mn mn mn C24 vmn C25 w mn xmn C45mn mn C44 C54mn mn C55mn
(35)
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C11mn mn C21 C41mn mn C51
Solving Eq. (35) with respect to the time-dependent variables ( umn (t ) , vmn (t )
xmn (t ) , mn (t ) ), and then substituting the results into the right of Eqs. (30-33), these equations can be expressed as
C13mn
mn mn C221 C222 No
C42mn C52mn
C14mn umn C15mn C1 mn mn C24 vmn C25 C2 w mn wmn C44mn xmn C45mn C3 mn mn C4 C54 mn C55
M
C12mn
mn C23
C43mn
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C11mn mn C21 C41mn mn C51
C53mn
(36)
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C i ( i 1,2,3,4 ) can be obtained by the radial direction, in-plane and rotary inertias, including the effects of axial and ring stiffeners. Repeating the static condensation
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procedure to eliminate umn , vmn , xmn and mn , solving Eq. (36) and then
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substituting the results into Eq. (34), Eqs. (30-34) are transformed into a reduced equation with respect to wmn (t ) :
M mn ( I 0 , I1 , I 2 , I 0r , I1r , I 2r , I 0a , I1a , I 2a )wmn (t )
[ Kmno ( No , N xT , N T , NrT , NaT ) cos Pt Kmnd ( Nd )]wmn (t ) ijkl ijklpq C mn 2 wij (t ) wkl (t ) Cmn 3 wij (t )wkl (t ) wpq (t ) qmn (t ) ijkl
where m 1,
, M , n 1,
(37)
ijklpq
, N . The generalised mass M mn depends on the shell
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and stiffeners inertia. The generalized linear stiffness K mno is related to the axial load (the static component of the pulsating load) and the thermal stress, and K mnd is ijkl associated with the oscillating component of the parameter excitation. C mn 2 and ijklpq are the nonlinear stiffness components. C mn 3
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These discretized equations in (37) can also be expressed by a vectorial equation Mq (K 0 K d cos Pt )q K NL 2q K NL3q F
(38)
For purposes of numerical simulations, Eq.(38) is reduced to its first order form,
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q(t) associated with the state vector X , leading to q (t)
0 E 0 X X 1 1 M (K 0 K d cos Pt ) 0 M [F(t) (K NL 2 K NL3 )q(t)]
(39)
The numerical solutions of the nonlinear differential equations are carried out using the Runge–Kutta method. Once the solution of Eq. (39) is obtained, the dynamic
M
responses can be computed using Eq. (28).
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Neglecting the quadratic ( K NL 2 ), cubic ( K NL3 ) nonlinear stiffness and the
PT
generalised force F , Eq. (38) is reduced to a Mathieu–Hill type equation Mq (K 0 K d cos Pt )q 0
(40)
3. Free vibration
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For free vibrations of the stiffened FG cylindrical shell, setting the oscillating
AC
component ( N d ) of the parameter excitation to zero, and substituting q(t ) qeit into Eq. (40), we have (K 0 2M)q 0
(41)
The natural frequencies can be obtain from the eigenvalue problem in Eq. (41). 4. Dynamic stability analysis Using Bolotin method [38], the unstable regions of Mathieu-Hill Eq. (40) are separated by the periodic solution:
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q
a
k 1,3...
k
sin
kPt kPt b k cos 2 2
(42)
where a k and b k are arbitrary time-independent vectors. Substituting Eq. (42) into Eq. (40), we obtain:
1 1 1 (K 0 P 2M K d )a1 K d a3 0 4 2 2 k2 2 1 P M)ak K d (ak 2 ak 2 ) 0 k 3 4 2
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(K 0
(43) (44)
(45)
k2 2 1 (K 0 P M)b k K d (b k 2 b k 2 ) 0 k 3 4 2
(46)
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1 1 1 (K 0 P 2M K d )b1 K d b3 0 4 2 2
A good approximation can be obtained when k 1 , therefore, the critical excitation frequency P can be computed from the following eigenvalue problems:
M
1 1 K 0 K d P 2M 0 2 4
(48)
ED
1 1 K 0 K d P 2M 0 2 4
(47)
For a given axial load (oscillating component N d ), Eqs. (47) and (48) delivers two
PT
critical excitation frequencies P (see Fig. 2). In Fig. 2, the unstable region size can be
CE
determined by the angle i , and the branches emanate at N d 0 (i.e., K d 0 ) from the Pi ( Pi 2i , i 1, 2,3... ). AB is arbitrary line segment that is parallel to the
AC
horizontal axis and is in the unstable region. The left of A and the right of B are in stable region. 5 Primary resonance Eq. (37) represents a coupled nonlinear ordinary differential equation. We can obtain the single mode approximation by neglecting parameter excitation and modal interaction:
M mn ( I 0 , I1 , I 2 , I 0r , I1r , I 2r , I 0a , I1a , I 2a )wmn (t ) 14
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Kmno ( No , N xT , N T , Nr T , NaT )wmn (t ) 2 3 Cmn wmn 2 (t ) Cmn wmn3 (t ) qmn (t )
(49)
Here the external periodic excitation is considered qmn (t ) Fmn cos t
(50)
is the external excitation frequency in radial direction. An approximate solution of
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Eq. (49) can be obtained using the multiple scales method. Eq. (49) is rewritten in the following form:
wmn (t ) mn 2 wmn (t ) 2mn wmn (t )2 23mn wmn (t )3 2 mn cos t
2 M mn , is perturbation parameter, mn 2 Kmno M mn , 2mn Cmn
3 3mn Cmn 2 M mn ,
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where
(51)
mn Fmn 2 M mn .
For the primary resonance of stiffened FG cylindrical shells, the detuning parameter ( ) is introduced in order to express the nearness of the external excitation
M
frequency ( ) to the natural frequency ( mn ):
mn 2
ED
(52)
The frequency-response curves of stiffened FG cylindrical shells are then easily obtained for the desired mode ( m , n ) [39], and it can be expressed as a relation
CE
PT
between the detuning parameter ( ) and coefficients of Eq. (51):
Λmn a 2
mn , 2mn a
AC
where a is the vibration amplitude, Λmn
(53)
1 8 mn
[3 3mn
10 2 mn 3 mn
2
2
] . For a single
nonlinear mode motion, the hardening or softening nonlinearity can be determined by the Λmn . 6 Results and discussion 6.1 Verification Example 1
For the free vibration of stiffened cylindrical shells, the natural 15
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frequencies can be computed using Eq. (41).
Fig. 3 contains the plot of the
fundamental frequency versus circumferential wave number ( n ) for an isotropic cylindrical shell with eight internal axial stiffeners, and these results are compared with the experimental results of Schnell and Heinrichsbauer [40]. The geometric and material properties of the stiffened cylindrical shell adopted here are R 194.49 mm,
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h 0.4638 mm, L 986.79 mm, E 200.0 GPa, 0.3 and 7998.97 kg/m3 ,
and the internal axial stiffener cross-sectional area As 2.5218 105 mm2. As can be seen from comparisons (for different axial wave numbers m ), the agreement with the experimental results of Schnell and Heinrichsbauer is good. However, the validation case also presents some important deviation for a few points in Fig. 3. The maximum
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deviation occurred in mode ( m 1 , n 5 ) and the deviation data is larger than 10％. This is because there are some differences between the theoretical model in this paper and the experimental model of Schnell and Heinrichsbauer [40], inclouding boundary
condensation method [36].
M
conditions etc.. In addition, in-plane and rotary inertias are simplified in the static
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Example 2 Numerical results of the dynamic stability, based on Eqs. (47) and (48), have been computed and plotted for non-stiffened FG cylindrical shells, in order to establish reasonable comparisons. The geometric and material constants used are
PT
m n 1 , N0 0.5Ncr , L / R 1 , R / h 100 , c 2370 kg/m3, e 8900 kg/m3,
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c 0.24 , e 0.31. The elastic moduli are given by
AC
Ec 348.43 109 (1 3.070 10 4 T 2.160 10 7 T 2 8.946 10 11T 3 ) ,
Em 223.95 109 (1 2.794 10 4 T 3.998 10 9 T 2 ) ,
where T is assumed to be 300K. For the FG Type A (inner surface: metal rich, outer surface: ceramic rich) and Type B (inner surface: ceramic rich, outer surface: metal rich), the points of origin P1 and unstable region size 1 are presented in Fig. 4 and Fig. 5 ( P1 2 1 and the nondimensionalized coefficient 2 R
I0
A11
).
16
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The comparison shows excellent agreement with those results obtained by Ng et al. [1], and the deviation data is less than 0.3％. 6.2 Dynamic behavior of stiffened FG cylindrical shells Here some numerical results are presented for internal stiffened FG cylindrical shells. The functionally graded material is Type B (inner surface: Zirconia, outer
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surface: Aluminum). The internal stiffeners are assumed to be made of ceramic material (Zirconia). The material properties for Aluminum and Zirconia are listed below [36]. Aluminum
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Em 70 GPa, m 0.3 , m 2707 kg/m3, km 204 W/mK, m 23 10 6 0C Zirconia
Ec 151 GPa, c 0.3 , c 3000 kg/m3, kc 2.09 W/mK, c 10 10 6 0C
The temperature distribution through the thickness can be calculated by solving
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Eq. 2 (see Eq. 4). The applied temperature fields for various values of the volume fraction exponent are shown in Fig. 6. The metal surface (outer) is exposed to
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300K (zero thermal stress state) and the ceramic surface (inner) is exposed to 600K, i.e., the temperature difference Tcm between the inner surface and the outer surface is
PT
assumed to be 300K in Fig. 6.
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Fig. 7 clearly shows the effect of the temperature difference Tcm on the dynamic stability region based on Eqs. (47) and (48). Here the static axial load (see Eq. 17) is
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N0 0.5Ncr . N cr is the axial buckling load of non-stiffened FG cylindrical shell, and it can be obtained from the Eq. (47) or Eq. (48) by set K d 0 and M 0 . As the temperature difference ( Tcm ) increases, the points of origin Pi ( i 1 ) move from the right to the left (3.23→1.46→0.69), and the dynamic unstable region ( 1 ) increases. The effect of larger temperature difference ( Tcm ) is to reduce the stiffness and the natural frequencies of stiffened FG cylindrical shells, and the stability
17
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of the system will be reduced rapidly. Based on Eqs. (47) and (48), Fig. 8 shows the effect of stiffener numbers ( N s , N r ) on the dynamic stability region of FG cylindrical shells. Here the static axial load is
N0 0.5Ncr . As the stiffener number increases, the points of origin Pi ( i 1 ) move from the left to the right (3.23→4.40→4.86), and the dynamic unstable region
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i ( i 1 ) decreases. The effect of more stiffeners is to increase the stiffness and the natural frequencies of FG cylindrical shells, and enhance the stability of the system.
Neglecting the modal interaction terms and using the single mode method [see Eqs. (37) and (39)], Figs. 9 and 10 show the plots of the linear and nonlinear response
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(time histories, phase plots) in the dynamic unstable region and the stable region, respectively. Here the static and dynamic loads ( parameter excitation, see Eq. 17) used are N0 0.5Ncr , Nd 0.8N0 . The external excitation amplitude and excitation frequency used in the calculus are Fmn 0.0012h 2 mmn 2 ,
0.8mn
M
(see Eq. 50). For the numerical results of dynamic stability in this example, the point
ED
of origin is P1 =10.06, and coordinates are A(9.55, 0.8) and B(10.55, 0.8), respectively (see Fig. 2). The parameter values in the unstable region are taken to be
PT
(10.00, 0.8) (between A and B). The linear and nonlinear responses in the unstable region are compared in Fig.9. Fig.9(a,b) corresponds to the linear parameter vibration,
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and Fig.9(c,d) corresponds to the nonlinear parameter vibration. From the time histories and phase plots, it is observed that the linear parameter vibration is
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unbounded and the nonlinear parameter vibration is bounded in the unstable regions. Similarly, the parameter values in the stable region are taken to be (10.568, 0.8), (10.569, 0.8), (10.570, 0.8) and (10.580, 0.8) (see Fig. 2, the right of B point), and the linear and nonlinear responses are compared in Fig. 10. It can be observed that the linear response is getting closer to nonlinear response as the parameter excitation frequency P (10.568→ 10.569→10.570→10.580) increases and departs from the unstable region. For different numbers of stiffeners, the nonlinear dynamic responses are 18
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compared in Fig.11 ( N s Nr 6 , N s Nr 8 and N s Nr 10 ). It is clearly observed from Fig.11 that the amplitude of the nonlinear dynamic response decreases as the number of stiffeners increases. The reason of this change in behavior can be explained by the fact that the system stiffness increases with the increase of the number of stiffeners.
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Fig. 12 shows the effect of volume fraction exponent on the nonlinear dynamic response of the FG cylindrical shell with ring and axial stiffeners. As the volume fraction exponent (0.0→1.0→5.0→20.0) increases, the amplitude of the nonlinear dynamic response decreases. This is expected because ceramic content increases with the increasing of volume fraction exponent , and the ceramic is
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relatively more stiffness than the metal. When 5 ( near the material properties of fully ceramic), the nonlinear responses of stiffened FG shells are almost the same for different volume fraction exponents .
To investigate the effect of stiffener number on primary resonance responses,
M
Ns Nr 4 , Ns Nr 6 , Ns Nr 8
and
N s Nr 12
are
considered
ED
respectively. In Fig.13, plots of primary resonance responses are presented based on Eqs. (51-53). Fig.13 indicates that the hardening nonlinearity becomes small as the
PT
number of ring and axial stiffeners increases. The reason is similar with Fig.11. The system stiffness increases with the increase of the stiffener number, and the
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nonlinearity decreases. 7 Conclusions
AC
In this paper, the theoretical method is developed to investigate the nonlinear
vibration and dynamic stability of stiffened FG cylindrical shells. The natural frequencies of stiffened cylindrical shells are compared with the previous experimental results, and numerical results of the dynamic stability are also consistent with that of the published article for non-stiffened FG cylindrical shells. Some conclusions are summarized as follows: (1)
As
the
temperature
difference
(thermal load)
increases,
the
dynamic unstable region of stiffened FG cylindrical shells increases rapidly. The 19
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stiffeners enhance the dynamic stability, and the more the stiffeners, the better the stability of FG cylindrical shells. (2) The linear parameter vibration is unbounded and the nonlinear parameter vibration is bounded in the unstable regions, but the linear response is getting closer to nonlinear response as the parameter excitation frequency increases and departs from the unstable region.
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(3) The stiffener number and the volume fraction exponent of the material significantly impact on the nonlinear dynamic responses of FG cylindrical shells. The system stiffness increases with the increase of the stiffener number, and the nonlinearity decreases.
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The new features of the nonlinear vibration and dynamic stability should be useful for the design and application of stiffened FG cylindrical shells in high-temperature and other environments. Acknowledgements
Appendix A.
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of China under No. 13JJ4053.
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The authors thank the supports of the Hunan Provincial Natural Science Foundation
A12 A66 2 A 2 1 2 , , L13 12 , A L 66 12 2 2 2 R x R x x R
L14 B11
B12 B66 2 2 B66 2 , L 15 R x x 2 R 2 2
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PT
L11 A11
,
L16 L11 ,
A A12 2 E A 2 L12 2 , L21 66 , L22 A66 2 22 552 , 2 2 R x R x R R
L23
B66 B12 2 E55 A22 E55 2 B22 2 , , , L L B 24 25 66 R x R x 2 R 2 2 R2
AC
L17
L26 L21 ,
L33 E44
L27
A66 2 E A22 2 A A , L31 21 , L32 55 , 222 2 3 2 2 R x R x R R R
E B22 A 2 E55 2 B21 2 222 , L34 E 44 , L35 55 , 2 2 x R x R R 2 x R R 20
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L36
A21 A , L37 223 , 2 R x 2R
L41 B11
B12 B66 2 2 B66 2 , , L 42 R x x 2 R 2 2
D12 D66 2 B12 2 D66 2 , L44 D11 2 2 , E44 , L45 L43 ( E 44 ) R x R x x R 2
L46 L41 , L47
L56 L51
, L57
B66 2 B22 2 . R x 2 R 2 2
References
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D12 D66 2 B22 E55 R D22 2 2 , , , L L E D 54 55 55 66 R x R 2 2 x 2 R2
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L53
B B12 2 E L42 2 B 2 , L51 66 , L52 B66 2 22 55 , 2 2 R x R R x R
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of ring-stiffened functionally graded shell, J. Therm. Stresses 30 (2007) 1249–1267.
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cylindrical shells, Nonlinear Dyn. 87 (2017) 1095–1109. [36] J.N. Reddy, Mechanics of Laminated Composite Plates and Shells: theory and
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longitudinally-stiffened thin-walled, circular cylindrical shells, NASA, TT
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CE
PT
ED
M
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F-8856, 1964.
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Fig. 1: Geometry and coordinate system of an FG cylindrical shell with ring
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CE
PT
ED
M
and axial stiffeners.
Fig. 2: Illustrative plot of the unstable region in the frequency.
26
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800
Frequency(Hz)
m=4
Present Exp.[40]
m=3
600
m=2 400 m=1
2
3
4
5
6
7
8
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200
9 10 11 12
Circumferential wave number (n)
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Fig. 3: The natural frequencies of stiffened cylindrical shells obtained by the present
[40]. 11.0 Present Ng et al.[1]
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Point of origin P1
10.8 10.6
10.2 0
2 4 6 8 10 Volume fraction exponent
PT
10.0
(a)
ED
10.4
Unstable region size 1
work and the experimental results obtained by Schnell and Heinrichsbauer
0.095 0.090 0.085 0.080 0.075 0.070 0.065 0.060
Present Ng et al.[1] (b)
0
2 4 6 8 10 Volume fraction exponent
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Fig. 4: Comparison of the unstable regions for the FG Type A cylindrical shell: (a)
AC
point of origin Pi ( i 1 ), (b) unstable region size i ( i 1 ).
27
10.8 10.6 Present Ng et al.[1]
10.4 10.2 10.0
(a)
0 2 4 6 8 10 Volume fraction exponent
0.100 0.095 0.090 0.085 0.080 0.075 0.070 0.065 0.060
Present Ng et al.[1]
(b)
0
2 4 6 8 10 Volume fraction exponent
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Point of origin P1
11.0
Unstable region size 1
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Fig. 5: Comparison of the unstable regions for the FG Type B cylindrical shell: (a)
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point of origin Pi ( i 1 ), (b) unstable region size i ( i 1 ).
= 0.0 = 1.0. = 2.0
0.25
-0.25
M
0.00
ED
Thickness coordinate, z/h
0.50
350
400 450 500 Temperature (K)
550
600
PT
-0.50 300
AC
CE
Fig. 6: Variation of the temperature through the shell thickness.
28
Nd / N0
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1.2
Tcm=300K
1.0
Tcm=550K Tcm=600K
0.8 0.6 0.4
0.0
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0.2 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 Nondimensional frequency parameter ( P)
Fig. 7: Variation of the unstable region with the temperature difference ( Tcm ) between
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the inner surface and the outer surface for the FG cylindrical shell with ring and axial stiffeners ( m 1 , n 4 , R 1.0 , L 2.0 , h 0.02 , ba br 0.04R , ha hr 0.08R , N s Nr 4 , 1.0 ).
M
1.2 1.0
Nd / N0
ED
0.8
N s= N r= 4 N s= N r= 8 Ns= Nr= 16
0.6
PT
0.4 0.2
CE
0.0
3.3 3.6 3.9 4.2 4.5 4.8 Nondimensional frequency parameter ( P)
AC
Fig. 8: Effect of the stiffener number on the unstable region for the FG cylindrical shell with ring and axial stiffeners ( m 1 , n 4 , R 1.0 , L 2.0 , h 0.02 , ba br 0.04R , ha hr 0.08R , Tcm 300 K, 1.0 ).
29
20
0.002
linear, P = 10.00, Nd / N0 = 0.8
15 10
0.001
0 -5
-0.001
linear, P = 10.00, Nd / N0 = 0.8
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wmn (m)
5
0.000
CR IP T
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-10
-0.002
-15
(a) 0.010
0.015 t (s)
0.0002
ED
CE
AC
0.28
t (s)
(b)
-0.002
0.000
wmn
0.002
0.9 nonlinear, P = 10.00 Nd / N0 = 0.8
0.6
0.0 -0.3 -0.6
(c)
0.27
-20 -0.004
0.3
PT
wmn (m)
0.0000
-0.0002 0.26
0.025
nonlinear, P = 10.00 Nd / N0 = 0.8
0.0001
-0.0001
0.020
M
-0.003 0.005
0.29
0.30
(d) -0.0001
0.0000
0.0001
wmn
Fig. 9: Comparison of the linear and nonlinear response in the unstable region, (a) linear time histories, (b) linear phase plot; (c) nonlinear time histories, (d) nonlinear phase plot ( m 1 , n 4 , R 1.0 , L 2.0 ,
h 0.02 ,
ba br 0.04R , ha hr 0.08R , N s Nr 8 , Tcm 300 K, 1.0 ).
30
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linear parametric vibration nonlinear parametric vibration
0.004
linear parametric vibration nonlinear parametric vibration
0.003 0.002
0.002
wmn (m)
wmn (m)
0.001 0.000
0.000
-0.001
-0.002
-0.002 P = 10.568
(a)
0.165
0.170
0.175 t (s)
0.180
0.185
-4
1.2x10
linear parametric vibration nonlinear parametric vibration
0.002
-5
8.0x10
0.001
-5
w mn(m)
wmn (m)
0.170
0.175 t (s)
0.180
0.185
linear parametric vibration nonlinear parametric vibration
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4.0x10
0.000
P = 10.569
(b)
-0.003 0.165
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-0.004
0.0 -5
-4.0x10
-0.001
-5
P = 10.570 0.175 t (s)
0.180
0.185
-8.0x10
P = 10.580
(d)
0.17
M
-0.002 ( c ) 0.165 0.170
0.18 t (s)
0.19
Fig. 10: Comparison of the linear and nonlinear response in the stable region, ( a)
L 2.0 ,
h 0.02 ,
ba br 0.04R ,
ha hr 0.08R ,
PT
R 1.0 ,
ED
P 10.568 , (b) P 10.569 , ( c) P 10.570 , (d) P 10.580 ( m 1 , n 4 ,
AC
CE
N s Nr 8 , Tcm 300 K, 1.0 ).
31
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-7
3.0x10
0.0 -7
wmn (m)
-3.0x10
-7
-6.0x10
-7
-9.0x10
-6
-1.2x10
-6
Ns = Nr = 6
Ns = Nr = 8
Ns = Nr = 10
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-1.5x10
-6
-1.8x10
0.007 0.008 0.009 0.010 0.011 0.012 t (s)
Fig. 11: Effect of the number of ring and axial stiffeners on the response of the FG n6 ,
R 1.0 ,
L 2.0 ,
h 0.02 ,
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( m 1 ,
cylindrical shell
ba br 0.04R , ha hr 0.08R , N0 0.2 Ncr , Nd 0.8N0 , P 40 rad/s, 20 rad/s, Tcm 300 K, 1.0 ). -7
-7
-4.0x10
-7
-6.0x10
-7
-8.0x10
-7
ED
-2.0x10
CE
PT
wmn (m)
0.0
M
2.0x10
-1.0x10
-6
-1.2x10
-6
=0. =5.
=1. =20.
0.007 0.008 0.009 0.010 0.011 0.012 t (s)
AC
Fig. 12: Effect of the volume fraction exponent on the response of the FG cylindrical shell with ring and axial stiffeners ( m 1 , n 6 , R 1.0 , L 2.0 ,
h 0.02 , ba br 0.04R ,
ha hr 0.08R ,
N0 0.2 Ncr , Nd 0.8N0 , P 40 rad/s, 20 rad/s,
Ns Nr 8 ,
Tcm 300 K).
32
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0.7
Ns = Nr = 4 Ns = Nr = 6 Ns = Nr = 8 Ns = Nr = 12
0.6
a/h
0.5 0.4 0.3 0.2 0.1
0.98
1.00
1.02 /mn
1.04
1.06
1.08
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0.96
Fig.13: Influence of the stiffener number on the primary resonance responses of FG shells ( m 1 , n 6 ,
R 1.0 , L 2.0 , h 0.02 , ba br 0.04R ,
AC
CE
PT
ED
M
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ha hr 0.08R , N0 0.2 Ncr , 1.0 , Tcm 300 K).
33
The dynamic stability and nonlinear vibration analysis of stiffened functionally graded cylindrical shells G.G. Sheng , X. Wang PII: DOI: Reference:
S0307-904X(17)30752-7 10.1016/j.apm.2017.12.021 APM 12102
To appear in:
Applied Mathematical Modelling
Received date: Revised date: Accepted date:
2 June 2017 1 November 2017 7 December 2017
Please cite this article as: G.G. Sheng , X. Wang , The dynamic stability and nonlinear vibration analysis of stiffened functionally graded cylindrical shells, Applied Mathematical Modelling (2017), doi: 10.1016/j.apm.2017.12.021
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Highlights
The dynamic stability and nonlinear vibrations of stiffened FG shells are investigated. A reduction nonlinear model of stiffened FG shells is presented. The effects of key parameters on the dynamic stability and nonlinear
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vibrations are analyzed.
A series of comparison are performed and the investigations demonstrate
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good reliability.
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The dynamic stability and nonlinear vibration analysis of stiffened functionally graded cylindrical shells G.G. Shenga,*[email protected], X.Wangb a
School of Civil Engineering, Changsha University of Science and Technology, Changsha, Hunan 410114, People’s Republic of China
School of Naval Architecture, Ocean and Civil Engineering (State Key Laboratory of Ocean
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b
Engineering), Shanghai Jiaotong University, Shanghai 200240, People’s Republic of China *
Corresponding author.
Abstract
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A theoretical model is developed to study the dynamic stability and nonlinear vibrations of the stiffened functionally graded (FG) cylindrical shell in thermal environment. Von Kármán nonlinear theory, first-order shear deformation theory, smearing stiffener approach and Bolotin method are used to model stiffened FG
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cylindrical shells. Galerkin method and modal analysis technique is utilized to obtain the discrete nonlinear ordinary differential equations. Based on the static condensation
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method, a reduction model is presented. The effects of thermal environment, stiffeners number, material characteristics on the dynamic stability, transient responses and
Keywords
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primary resonance responses are examined.
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Stiffened; Functionally graded cylindrical shell; Nonlinear vibration; Dynamic stability
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1 Introduction
FG shells are being used as structural components in modern industries such as
aerospace, mechanical, chemical industry and nuclear engineering [1-5]. The main applications of FG shells have been in high temperature environments. To enhance the stiffness and stability of FG shells, we can use stiffeners. The axial and ring stiffeners in FG shells are very good stiffening elements to raise the stiffness and stability without great mass increase. The knowledge of their characteristics is
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necessary, and the natural frequencies, responses, static and dynamic stability of stiffened FG shells are of special interest for their applications. The stiffened shells have been investigated for many years. When the number of stiffeners is small, and the stiffeners are not closely and evenly spaced, the discrete stiffener theory is used to develop the characteristics of stiffened shells [6]. For the non-uniform ring stiffener distribution, Jafari and Bagheri [7] studied the free
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vibration of stiffened shells based on the Ritz method and the discrete stiffener theory. Using Reissner-Naghdi shell theory and the discrete element method, Qu et al.[8] reported the dynamic characteristics of the stiffened shell. When stiffeners are closely and uniformly-spaced, their discrete nature can be neglected and the characteristics of
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stiffened shells are similar with that of orthotropic shells. Based on this idea, the smearing method is proposed [9, 10]. There are a large number of literatures on the smearing method for stiffened structures. Using the Donnel shell theory, Li and Qiao [11] presented the nonlinear free vibration and primary resonance analysis for
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stiffened cylindrical shells, and proposed an improved smearing stiffener approach. Using the smearing method, an investigation was presented for stiffened FG shells in
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thermal environment [12]. The effects of rotatory inertia and shear deformation were considered. According to the smeared stiffener approach and shear deformation theory,
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Sun et al.[13] analyzed the buckling behaviors of FG stiffened cylindrical shells under the thermal load. The dynamic stability of FG structures was also studied by many
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researchers. In case of a harmonic parametric excitation, the motion amplitude of the structures will define the instability regions through the Mathieu equation. For FG
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cylindrical shells, the dynamic stability was studied using Bolotin method by Ng et al.[1]. They found that material distribution (volume fraction exponent) has significant influence on the positions and sizes of instability regions. The free vibration and dynamic stability were presented by Yang and Shen [14] for FG shells under the harmonic parametric excitation and thermal load, and Galerkin technique and Bolotin method were used to analyze the dynamic characteristics of shells. Using the shear deformation theory and Galerkin method, the dynamic stability of FG cylindrical shells was studied by Torki et al. [15]. The effect of the material 3
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characteristics on the critical mode number and the flutter load was addressed in their paper. Using linear Donnell type equations, Galerkin method and Bolotin method, Sofiyev and Kuruoglu [16] analyzed the dynamic stability of FG conical shells subjected to time dependent loads. Sahmani et al.[17] studied the dynamic stability of FG cylindrical shells based on the shear deformation theory, Hamilton’s principle and Bolotin method. Asadi and Wang [18] investigated the dynamic stability for a FG
Donnell theory and the shear deformation theory.
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reinforced composite cylindrical shell. The dynamic model was established using
The structures can be vibrated with large amplitude. The linear theory is not sufficient to analyze the large amplitude vibration. To design a stable and reliable
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structure, the nonlinear vibration theory should be employed. A lot of literatures on the nonlinear vibration are concerned for plates and shells [19–24]. Using the smearing stiffener method and von Kármán nonlinear theory, Dung and Hoa [25], Ninh and Bich [26] studied the nonlinear characteristics for stiffened FG shells in
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thermal environment. Using the stress function, Duc and Thang [27] developed a theoretical model to analyze the nonlinear dynamic response for stiffened FG shells.
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Similarly, Duc et al.[28-30] discussed the nonlinear thermal dynamic behavior of stiffened FG shells, and the changes in geometric shapes were considered after the
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thermal deformation process. Furthermore, using Galerkin method, Kumar et al. [31] reported the dynamic stability of FG plates considering von Kármán geometric
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nonlinearity. Employing Donnell nonlinear shell theory and Bolotin methods, Dey and Ramachandra [32] discussed the dynamic stability for the composite cylindrical
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shell.
As stated above literatures, only a few studies are concerned on the dynamic
stability and nonlinear characteristics of stiffened FG shells. In this paper, the authors attempt to solve this subject. Hamilton’s principle, the smeared technique, the shear deformation and von Kármán nonlinear theory are used to obtain the theoretical model of stiffened FG shells. This study is an extension of our earlier works [33-35] on the dynamic characteristics of stiffened and non-stiffened FG cylindrical shells. A reduced model is given based on the static condensation method, and it can capture 4
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the main dynamic stability and nonlinear vibration characteristics of stiffened shells. The dynamic stability and natural frequencies have been compared and verified with that of previous studies for non-stiffened FG cylindrical shells and isotropic stiffened cylindrical shells. 2 Theoretical formulations
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Now consider FG cylindrical shells reinforced by closely, uniformly-spaced eccentric ring and axial stiffeners. The geometry and coordinate system are shown in Fig.1. The FG cylindrical shell has length L ¸ thickness h , and mean radius R . The ring and axial stiffeners have rectangular cross section with width ( br , ba ) and
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thickness ( hr , ha ). The functionally graded material is Type B (inner surface: ceramic, outer surface: metal). The properties of the stiffener material are the same as that of the surface material. That means inner stiffeners are assumed to be made of full ceramic material, and outer stiffeners are made of full metal. The stiffened FG shell is
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a complete whole, with stiffeners and shell.
The FG material property Feff is varied according to the expressions (a power
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law) along the thickness
zh/2 ) (0 ) h
(1)
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Feff ( z) FO A( z) FI (1 A( z)) , A( z ) (
where denotes the volume fraction exponent of FG material, FI denotes the
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material property of the inner surface, FO denotes the material property of the outer
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surface. Here, mass density , elastic modulus E , thermal expansion , Poisson’s ratio , and thermal conductivity k vary according to Eq. 1. In this paper, the temperature variation is assumed to occur in radial direction
only. The temperature field is defined by the following equation [36] d dT (k eff ( z ) ) 0 dz dz h h T ( ) Tc , T ( ) Tm 2 2
(2) (3)
where the thermal conductivity k eff satisfies the Eq. (1). The solution to Eq. (2) can 5
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be obtained
T ( z ) Tc
Tcm h 2 h 2
dz
z
h 2
dz k e f (f z )
(4)
k e f (f z )
h h where Tcm T ( ) T ( ) (temperature difference). 2 2 The stress-strain relations are of the form
where
σ x
x xz z
T
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σ C( z)[ ε α( z)T ( z)]
(5)
ε x
(stresses),
x
xz z
T
(strains) , T ( z ) T ( z ) T0 and T0 Tm (room temperature). α(z ) and C(z ) is the
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thermal expansion coefficients and elasticity matrix of FG materials, respectively:
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0 0 0 Q11 ( z ) Q12 ( z ) Q ( z ) Q ( z ) 0 0 0 22 21 C( z ) 0 0 Q66 ( z ) 0 0 , 0 0 Q55 ( z ) 0 0 0 0 0 0 Q44 ( z ) α( z ) xxe ( z ) e ( z) 0 0 0
Eeff
1 eff
2
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Q11 ( z )
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where
Eeff
1 eff 2
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Q22 ( z )
, Q12 ( z )
T
(6)
eff Eeff E , Q21 ( z ) eff eff2 , 2 1 eff 1 eff
, Q44 ( z ) Q55 ( z ) Q66 ( z )
Eeff 2(1 eff )
(7)
FG material properties are uniform distribution in plane, and the thermal expansion
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coefficients ( xxe ( z ) , e ( z ) ) are equal in axial and circumferential directions ( x , ) ( xxe ( z ) e ( z ) eff ( z ) ). Using Eq. (1), the effective Young’s modulus E eff , effective thermal expansion coefficients eff and effective Poisson’s ratio eff can be obtain for a given volume fraction exponent . The arbitrary point strains ε can be expressed using mid-surface strains ( x , , x , xz and z ) and curvatures( x , and x ): 6
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x x z x , ( z ) / (1 z / R) , x ( x z x ) / (1 z / R) , xz xz , z z / (1 z / R)
(8)
According to the smearing method [9, 10], an equivalent non-stiffened shell model can be obtained. Substituting Eq. 8 into the stress-strain relations in Eq. 5, the following constitutive equations can be derived by integrating over the entire
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thickness of the shell:
s s T T N x ( A11 A11 ) x A12 ( B11 B11 ) x B12 N x N a s s T T N A21 x ( A22 A22 ) B21 x ( B22 B22 ) N N r N A66 x B66 x N x T x
(9)
(10)
(11)
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( A55 A55 s ) xz Qxz G s Q z ( A44 A44 ) z
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s s T T M x ( B11 B11 ) x B12 ( D11 D11 ) x D12 M x M a s s T T M B21 x ( B22 B22 ) D21 x ( D22 D22 ) M M r M B66 x D66 x M x T x
where N x , N , N x , M x , M , M x , Q x and Q are the usual stress
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resultants. G is the shear correction factor ( G 5 / 6 [36]), and the coefficients Aij ,
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Dij and Bij appearing in Eqs.(9-11) are defined in terms of elasticity coefficients
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Qij :
h 2 h 2
( Aij , Bij , Dij )= Qij (1, z, z ) dz 2
h 2 h 2
h 2 h 2
( i, j 1,2,6) , A55 Q55 dz , A44 Q44 dz
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Q11 Q11 ( z ) , Q12 Q12 ( z ) / (1 z / R) , Q21 Q21 ( z ) , Q22 Q22 ( z ) / (1 z / R) ,
Q66 Q66 ( z ) / (1 z / R) , Q55 Q55 ( z ) , Q44 Q44 ( z ) / (1 z / R)
(12)
N xT , N T , N x T , M xT , M T and M x T are thermal stress resultants of the shell: N xT T N N T x
M xT Q11 ( z ) xxe ( z ) Q12 ( z ) e ( z ) h T M 2h Q12 ( z ) xxe ( z ) Q22 ( z ) e ( z ) T ( z ) 1 z dz 2 M x T 0
(13)
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Aij s , Bij s and Dij s denote stiffener stiffnesses, and they are given in terms of the geometric and material parameters of stiffeners (subscripts a : axial stiffeners; subscripts r : ring stiffeners)
A11s Ea Aa / da , B11s Ea Aa ea / da , D11s Ea ( I a Aa ea 2 ) / da ,
A44 s
Ea A Er A a , A55 s r 2(1 a ) d a 2(1 r ) d r
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A22 s Er Ar / dr , B22 s Er Ar er / dr , D22 s Er ( I r Ar er 2 ) / dr , (14)
E and denote the Young’s modulus and Poisson’s ratio of stiffeners, respectively. The stiffeners are of eccentricity
e
(positive: outside, negative: inside),
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cross-sectional area A , intertia moment I , ring stiffener spacing d r ( dr L / Nr ), axial stiffener spacing d a ( da 2 R / Na ), ring stiffener number N r and axial stiffener number N a . The thermal stress resultants in Eqs. (9) and (10) for stiffeners
M aT Ea a Aa / d a M r T Er r Ar / d r
Ea a Aa ea / d a T (hs ) Er r Ar er / d r
(15)
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N aT T Nr
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are given by
where outside stiffeners hs h / 2 and inside stiffeners hs h / 2 .
u 1 w 2 1 v 1 1 w 2 ( ) , ( w) ( ) , x 2 x R 2 R
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x
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The strains in Eqs. (9-11) have the form
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x
x
w 1 w v 1 u 1 w w , xz x , z , x R x R R x
x 1 1 x , , x . x R x R
(16)
where u , v and w are the midsurface displacement components along the x , and z direction; x and are the rotation functions. For dynamic stability, the stiffened FG cylindrical shell is under a periodically pulsating load (negative in compression) in axial direction:
Na N0 Nd cos Pt
(17) 8
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where P denotes the frequency of parameter excitation. Considering the rotary inertia and transverse shear, the equations of motion under the parameter excitation are obtained using the first-order shear deformation theory and Hamilton’s principle [35, 36]:
u :
( N x N xT N aT ) 1 N x x R
v :
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( I 0 I 0r I 0a )u ( I1 I1r I1a )x 1 ( N N T N r T ) N x 1 2v Q z ( N0 N d cos Pt ) 2 R x R x
( I 0 I 0r I 0a )v ( I1 I1r I1a ) 1 Q z Qxz 1 ( N N T N r T ) R x R
( N0 N d cos Pt )
2w (u, v, w, x , ) q( x, , t ) ( I 0 I 0 r I 0 a ) w x 2
( M x M xT M aT ) 1 M x Qxz ( I1 I1r I1a )u ( I 2 I 2 r I 2 a )x x R
M
x :
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w :
(18)
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1 ( M M T M r T ) M x : Q z ( I1 I1r I1a )v ( I 2 I 2r I 2a ) R x
where q( x, , t ) is the force per unit area acting radial direction, (u, v, w, x , ) is
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the nonlinear term from the von Kármán nonlinear strains:
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(u, v, w, x , )
w [( N x N xT N aT ) ] x x
1 w 1 w 1 w [( N N T N r T ) ] ( N x ) ( N x ) 2 R R x R x
(19)
The mass inertia terms of the shell and the stiffeners are defined by
( I 0 , I1 , I 2 )
h /2
h /2
eff ( z )(1, z, z 2 )dz
( I 0r , I1r , I 2r ) ( r Ar / dr , r Ar er / dr , r ( I r Ar er 2 ) / dr )
(20) (21)
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( I 0a , I1a , I 2a ) ( a Aa / da , a Aa ea / da , a ( I a Aa ea 2 ) / da )
(22)
r and a are mass densities of the ring and axial stiffeners, respectively. Utilizing the strain-displacement relations in Eq. (16), the nonlinear equations are obtained in terms of the generalized displacements ( u, v, w, x , ) by substituting Eqs.
L11u L12v L13w L14x L15 wx L16 w w L17 w
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(9-11), (13), (15) and (17) into Eq. (18)
( I 0 I 0r I 0a )u ( I1 I1r I1a )x
(23)
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2 L21u [ L22 ( N0 N d cos Pt ) 2 ]v L23w L24x L25 wx L26 w w L27 w x
( I 0 I 0r I 0a )v ( I1 I1r I1a ) L31u L32v [ L33 ( N 0 N d cos Pt )
（24）
2 ]w L34x L35 x 2
(25)
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wx L36 w w L37 w (u, v, w, x , ) q( x, , t ) ( I 0 I 0r I 0a )w
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L41u L42v L43w L44x L45 wx L46 w w L47 w
( I1 I1r I1a )u ( I 2 I 2r I 2a )x
(26)
( I1 I1r I1a )v ( I 2 I 2r I 2a )
(27)
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L51u L52v L53w L54x L55 wx L56 w w L57 w
where Lij are linear operators (see Appendix A).
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The following representation of the linear ( u, v, x , ) and nonlinear ( w )
displacement fields is used for simply supported boundary conditions at x 0 and x L [35, 37] M
N
u u mn (t ) cos m x cos n , m 1 n 1 M
N
M
N
v v mn (t ) sin m x sin n , m 1 n 1 M
N
x xmn (t ) cos m x cos n , mn (t ) sin m x sin n , m 1 n 1
m 1 n 1
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M
N
2
M
w (t ) w wmn (t ) sin m x cos n mn sin m x 2R m 1 n 1 m 1
where m
(28)
m , n represents the number of circumferential waves and m represents L
the number of axial half-waves. The radial distributed force q( x, , t ) in Eq. (25) can also be expanded as N
m 1 n 1
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M
q( x, , t ) qmn (t )sin m x cos n
(29)
Substituting Eqs. (28) and (29) into Eqs. (23-27), the continuous system model is , discretized by multiterm Galerkin s technique, and we obtain the coupled nonlinear
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ordinary differential equations:
C11mnumn (t ) C12mnvmn (t ) C13mnxmn (t ) C14mn mn (t ) C15mn wmn (t ) ijkl Cumn wij (t ) wkl (t ) ijkl
C
ijklpq umn
wij (t )wkl (t )wpq (t )
ijklpq
(30)
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( I 0 I 0 r I 0 a )umn (t ) ( I1 I1r I1a )xmn (t )
mn mn mn mn C21 umn (t ) [C221 ( N0 Nd cos Pt )C222 ]vmn (t ) C23 xmn (t ) C24mn mn (t ) C25mn wmn (t )
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ijkl ijklpq Cvmn wij (t )wkl (t ) Cvmn wij (t )wkl (t )wpq (t ) ijkl
ijklpq
(31)
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( I 0 I 0 r I 0 a )vmn (t ) ( I1 I1r I1a ) mn (t )
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mn mn mn C41 umn (t ) C42 vmn (t ) C43 xmn (t ) C44mn mn (t ) C45mn wmn (t )
Cijklxmn wij (t )wkl (t ) Cijklpq xmn wij (t ) wkl (t ) w pq (t ) ijkl
ijklpq
( I1 I1r I1a )umn (t ) ( I 2 I 2r I 2a )xmn (t )
(32)
C51mnumn (t ) C52mnvmn (t ) C53mnxmn (t ) C54mn mn (t ) C55mn wmn (t ) ijkl ijklpq C mn wij (t ) wkl (t ) C mn wij (t ) wkl (t ) w pq (t ) ijkl
ijklpq
( I1 I1r I1a )vmn (t ) ( I 2 I 2r I 2a ) mn (t )
(33)
( I 0 I 0r I 0a )wmn (t ) C31mnumn (t ) C32mnvmn (t ) mn mn mn mn [C331 ( N0 Nd cos Pt )C332 ( N xT NaT )C333 ( N T NrT )C334 ]wmn (t )
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C
ijkl 4 mn
ijkl
ijkl
ijkl 5 mn
wij (t ) kl (t ) C
wij (t ) xkl (t ) C
ijkl
ijklpq 6 mn
ijkl
ijkl
wij (t )wkl (t )wpq (t ) qmn (t )
(34)
ijklpq
where u mn (t ) , vmn (t ) , wmn (t ) , xmn (t ) and mn (t ) is the time-dependent variables, and the coefficients can be determined by operators Lij and the nonlinear term
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(u, v, w, x , ) . According to the static condensation method [36], Eqs. (30-33) can be written as
C12mn
C13mn
mn mn C221 C222 No
mn C23
C42mn
mn C43
C52mn
C53mn
C14mn umn C15mn mn mn C24 vmn C25 w mn xmn C45mn mn C44 C54mn mn C55mn
(35)
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C11mn mn C21 C41mn mn C51
Solving Eq. (35) with respect to the time-dependent variables ( umn (t ) , vmn (t )
xmn (t ) , mn (t ) ), and then substituting the results into the right of Eqs. (30-33), these equations can be expressed as
C13mn
mn mn C221 C222 No
C42mn C52mn
C14mn umn C15mn C1 mn mn C24 vmn C25 C2 w mn wmn C44mn xmn C45mn C3 mn mn C4 C54 mn C55
M
C12mn
mn C23
C43mn
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C11mn mn C21 C41mn mn C51
C53mn
(36)
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C i ( i 1,2,3,4 ) can be obtained by the radial direction, in-plane and rotary inertias, including the effects of axial and ring stiffeners. Repeating the static condensation
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procedure to eliminate umn , vmn , xmn and mn , solving Eq. (36) and then
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substituting the results into Eq. (34), Eqs. (30-34) are transformed into a reduced equation with respect to wmn (t ) :
M mn ( I 0 , I1 , I 2 , I 0r , I1r , I 2r , I 0a , I1a , I 2a )wmn (t )
[ Kmno ( No , N xT , N T , NrT , NaT ) cos Pt Kmnd ( Nd )]wmn (t ) ijkl ijklpq C mn 2 wij (t ) wkl (t ) Cmn 3 wij (t )wkl (t ) wpq (t ) qmn (t ) ijkl
where m 1,
, M , n 1,
(37)
ijklpq
, N . The generalised mass M mn depends on the shell
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and stiffeners inertia. The generalized linear stiffness K mno is related to the axial load (the static component of the pulsating load) and the thermal stress, and K mnd is ijkl associated with the oscillating component of the parameter excitation. C mn 2 and ijklpq are the nonlinear stiffness components. C mn 3
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These discretized equations in (37) can also be expressed by a vectorial equation Mq (K 0 K d cos Pt )q K NL 2q K NL3q F
(38)
For purposes of numerical simulations, Eq.(38) is reduced to its first order form,
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q(t) associated with the state vector X , leading to q (t)
0 E 0 X X 1 1 M (K 0 K d cos Pt ) 0 M [F(t) (K NL 2 K NL3 )q(t)]
(39)
The numerical solutions of the nonlinear differential equations are carried out using the Runge–Kutta method. Once the solution of Eq. (39) is obtained, the dynamic
M
responses can be computed using Eq. (28).
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Neglecting the quadratic ( K NL 2 ), cubic ( K NL3 ) nonlinear stiffness and the
PT
generalised force F , Eq. (38) is reduced to a Mathieu–Hill type equation Mq (K 0 K d cos Pt )q 0
(40)
3. Free vibration
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For free vibrations of the stiffened FG cylindrical shell, setting the oscillating
AC
component ( N d ) of the parameter excitation to zero, and substituting q(t ) qeit into Eq. (40), we have (K 0 2M)q 0
(41)
The natural frequencies can be obtain from the eigenvalue problem in Eq. (41). 4. Dynamic stability analysis Using Bolotin method [38], the unstable regions of Mathieu-Hill Eq. (40) are separated by the periodic solution:
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q
a
k 1,3...
k
sin
kPt kPt b k cos 2 2
(42)
where a k and b k are arbitrary time-independent vectors. Substituting Eq. (42) into Eq. (40), we obtain:
1 1 1 (K 0 P 2M K d )a1 K d a3 0 4 2 2 k2 2 1 P M)ak K d (ak 2 ak 2 ) 0 k 3 4 2
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(K 0
(43) (44)
(45)
k2 2 1 (K 0 P M)b k K d (b k 2 b k 2 ) 0 k 3 4 2
(46)
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1 1 1 (K 0 P 2M K d )b1 K d b3 0 4 2 2
A good approximation can be obtained when k 1 , therefore, the critical excitation frequency P can be computed from the following eigenvalue problems:
M
1 1 K 0 K d P 2M 0 2 4
(48)
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1 1 K 0 K d P 2M 0 2 4
(47)
For a given axial load (oscillating component N d ), Eqs. (47) and (48) delivers two
PT
critical excitation frequencies P (see Fig. 2). In Fig. 2, the unstable region size can be
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determined by the angle i , and the branches emanate at N d 0 (i.e., K d 0 ) from the Pi ( Pi 2i , i 1, 2,3... ). AB is arbitrary line segment that is parallel to the
AC
horizontal axis and is in the unstable region. The left of A and the right of B are in stable region. 5 Primary resonance Eq. (37) represents a coupled nonlinear ordinary differential equation. We can obtain the single mode approximation by neglecting parameter excitation and modal interaction:
M mn ( I 0 , I1 , I 2 , I 0r , I1r , I 2r , I 0a , I1a , I 2a )wmn (t ) 14
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Kmno ( No , N xT , N T , Nr T , NaT )wmn (t ) 2 3 Cmn wmn 2 (t ) Cmn wmn3 (t ) qmn (t )
(49)
Here the external periodic excitation is considered qmn (t ) Fmn cos t
(50)
is the external excitation frequency in radial direction. An approximate solution of
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Eq. (49) can be obtained using the multiple scales method. Eq. (49) is rewritten in the following form:
wmn (t ) mn 2 wmn (t ) 2mn wmn (t )2 23mn wmn (t )3 2 mn cos t
2 M mn , is perturbation parameter, mn 2 Kmno M mn , 2mn Cmn
3 3mn Cmn 2 M mn ,
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where
(51)
mn Fmn 2 M mn .
For the primary resonance of stiffened FG cylindrical shells, the detuning parameter ( ) is introduced in order to express the nearness of the external excitation
M
frequency ( ) to the natural frequency ( mn ):
mn 2
ED
(52)
The frequency-response curves of stiffened FG cylindrical shells are then easily obtained for the desired mode ( m , n ) [39], and it can be expressed as a relation
CE
PT
between the detuning parameter ( ) and coefficients of Eq. (51):
Λmn a 2
mn , 2mn a
AC
where a is the vibration amplitude, Λmn
(53)
1 8 mn
[3 3mn
10 2 mn 3 mn
2
2
] . For a single
nonlinear mode motion, the hardening or softening nonlinearity can be determined by the Λmn . 6 Results and discussion 6.1 Verification Example 1
For the free vibration of stiffened cylindrical shells, the natural 15
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frequencies can be computed using Eq. (41).
Fig. 3 contains the plot of the
fundamental frequency versus circumferential wave number ( n ) for an isotropic cylindrical shell with eight internal axial stiffeners, and these results are compared with the experimental results of Schnell and Heinrichsbauer [40]. The geometric and material properties of the stiffened cylindrical shell adopted here are R 194.49 mm,
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h 0.4638 mm, L 986.79 mm, E 200.0 GPa, 0.3 and 7998.97 kg/m3 ,
and the internal axial stiffener cross-sectional area As 2.5218 105 mm2. As can be seen from comparisons (for different axial wave numbers m ), the agreement with the experimental results of Schnell and Heinrichsbauer is good. However, the validation case also presents some important deviation for a few points in Fig. 3. The maximum
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deviation occurred in mode ( m 1 , n 5 ) and the deviation data is larger than 10％. This is because there are some differences between the theoretical model in this paper and the experimental model of Schnell and Heinrichsbauer [40], inclouding boundary
condensation method [36].
M
conditions etc.. In addition, in-plane and rotary inertias are simplified in the static
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Example 2 Numerical results of the dynamic stability, based on Eqs. (47) and (48), have been computed and plotted for non-stiffened FG cylindrical shells, in order to establish reasonable comparisons. The geometric and material constants used are
PT
m n 1 , N0 0.5Ncr , L / R 1 , R / h 100 , c 2370 kg/m3, e 8900 kg/m3,
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c 0.24 , e 0.31. The elastic moduli are given by
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Ec 348.43 109 (1 3.070 10 4 T 2.160 10 7 T 2 8.946 10 11T 3 ) ,
Em 223.95 109 (1 2.794 10 4 T 3.998 10 9 T 2 ) ,
where T is assumed to be 300K. For the FG Type A (inner surface: metal rich, outer surface: ceramic rich) and Type B (inner surface: ceramic rich, outer surface: metal rich), the points of origin P1 and unstable region size 1 are presented in Fig. 4 and Fig. 5 ( P1 2 1 and the nondimensionalized coefficient 2 R
I0
A11
).
16
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The comparison shows excellent agreement with those results obtained by Ng et al. [1], and the deviation data is less than 0.3％. 6.2 Dynamic behavior of stiffened FG cylindrical shells Here some numerical results are presented for internal stiffened FG cylindrical shells. The functionally graded material is Type B (inner surface: Zirconia, outer
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surface: Aluminum). The internal stiffeners are assumed to be made of ceramic material (Zirconia). The material properties for Aluminum and Zirconia are listed below [36]. Aluminum
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Em 70 GPa, m 0.3 , m 2707 kg/m3, km 204 W/mK, m 23 10 6 0C Zirconia
Ec 151 GPa, c 0.3 , c 3000 kg/m3, kc 2.09 W/mK, c 10 10 6 0C
The temperature distribution through the thickness can be calculated by solving
M
Eq. 2 (see Eq. 4). The applied temperature fields for various values of the volume fraction exponent are shown in Fig. 6. The metal surface (outer) is exposed to
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300K (zero thermal stress state) and the ceramic surface (inner) is exposed to 600K, i.e., the temperature difference Tcm between the inner surface and the outer surface is
PT
assumed to be 300K in Fig. 6.
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Fig. 7 clearly shows the effect of the temperature difference Tcm on the dynamic stability region based on Eqs. (47) and (48). Here the static axial load (see Eq. 17) is
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N0 0.5Ncr . N cr is the axial buckling load of non-stiffened FG cylindrical shell, and it can be obtained from the Eq. (47) or Eq. (48) by set K d 0 and M 0 . As the temperature difference ( Tcm ) increases, the points of origin Pi ( i 1 ) move from the right to the left (3.23→1.46→0.69), and the dynamic unstable region ( 1 ) increases. The effect of larger temperature difference ( Tcm ) is to reduce the stiffness and the natural frequencies of stiffened FG cylindrical shells, and the stability
17
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of the system will be reduced rapidly. Based on Eqs. (47) and (48), Fig. 8 shows the effect of stiffener numbers ( N s , N r ) on the dynamic stability region of FG cylindrical shells. Here the static axial load is
N0 0.5Ncr . As the stiffener number increases, the points of origin Pi ( i 1 ) move from the left to the right (3.23→4.40→4.86), and the dynamic unstable region
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i ( i 1 ) decreases. The effect of more stiffeners is to increase the stiffness and the natural frequencies of FG cylindrical shells, and enhance the stability of the system.
Neglecting the modal interaction terms and using the single mode method [see Eqs. (37) and (39)], Figs. 9 and 10 show the plots of the linear and nonlinear response
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(time histories, phase plots) in the dynamic unstable region and the stable region, respectively. Here the static and dynamic loads ( parameter excitation, see Eq. 17) used are N0 0.5Ncr , Nd 0.8N0 . The external excitation amplitude and excitation frequency used in the calculus are Fmn 0.0012h 2 mmn 2 ,
0.8mn
M
(see Eq. 50). For the numerical results of dynamic stability in this example, the point
ED
of origin is P1 =10.06, and coordinates are A(9.55, 0.8) and B(10.55, 0.8), respectively (see Fig. 2). The parameter values in the unstable region are taken to be
PT
(10.00, 0.8) (between A and B). The linear and nonlinear responses in the unstable region are compared in Fig.9. Fig.9(a,b) corresponds to the linear parameter vibration,
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and Fig.9(c,d) corresponds to the nonlinear parameter vibration. From the time histories and phase plots, it is observed that the linear parameter vibration is
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unbounded and the nonlinear parameter vibration is bounded in the unstable regions. Similarly, the parameter values in the stable region are taken to be (10.568, 0.8), (10.569, 0.8), (10.570, 0.8) and (10.580, 0.8) (see Fig. 2, the right of B point), and the linear and nonlinear responses are compared in Fig. 10. It can be observed that the linear response is getting closer to nonlinear response as the parameter excitation frequency P (10.568→ 10.569→10.570→10.580) increases and departs from the unstable region. For different numbers of stiffeners, the nonlinear dynamic responses are 18
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compared in Fig.11 ( N s Nr 6 , N s Nr 8 and N s Nr 10 ). It is clearly observed from Fig.11 that the amplitude of the nonlinear dynamic response decreases as the number of stiffeners increases. The reason of this change in behavior can be explained by the fact that the system stiffness increases with the increase of the number of stiffeners.
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Fig. 12 shows the effect of volume fraction exponent on the nonlinear dynamic response of the FG cylindrical shell with ring and axial stiffeners. As the volume fraction exponent (0.0→1.0→5.0→20.0) increases, the amplitude of the nonlinear dynamic response decreases. This is expected because ceramic content increases with the increasing of volume fraction exponent , and the ceramic is
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relatively more stiffness than the metal. When 5 ( near the material properties of fully ceramic), the nonlinear responses of stiffened FG shells are almost the same for different volume fraction exponents .
To investigate the effect of stiffener number on primary resonance responses,
M
Ns Nr 4 , Ns Nr 6 , Ns Nr 8
and
N s Nr 12
are
considered
ED
respectively. In Fig.13, plots of primary resonance responses are presented based on Eqs. (51-53). Fig.13 indicates that the hardening nonlinearity becomes small as the
PT
number of ring and axial stiffeners increases. The reason is similar with Fig.11. The system stiffness increases with the increase of the stiffener number, and the
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nonlinearity decreases. 7 Conclusions
AC
In this paper, the theoretical method is developed to investigate the nonlinear
vibration and dynamic stability of stiffened FG cylindrical shells. The natural frequencies of stiffened cylindrical shells are compared with the previous experimental results, and numerical results of the dynamic stability are also consistent with that of the published article for non-stiffened FG cylindrical shells. Some conclusions are summarized as follows: (1)
As
the
temperature
difference
(thermal load)
increases,
the
dynamic unstable region of stiffened FG cylindrical shells increases rapidly. The 19
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stiffeners enhance the dynamic stability, and the more the stiffeners, the better the stability of FG cylindrical shells. (2) The linear parameter vibration is unbounded and the nonlinear parameter vibration is bounded in the unstable regions, but the linear response is getting closer to nonlinear response as the parameter excitation frequency increases and departs from the unstable region.
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(3) The stiffener number and the volume fraction exponent of the material significantly impact on the nonlinear dynamic responses of FG cylindrical shells. The system stiffness increases with the increase of the stiffener number, and the nonlinearity decreases.
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The new features of the nonlinear vibration and dynamic stability should be useful for the design and application of stiffened FG cylindrical shells in high-temperature and other environments. Acknowledgements
Appendix A.
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of China under No. 13JJ4053.
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The authors thank the supports of the Hunan Provincial Natural Science Foundation
A12 A66 2 A 2 1 2 , , L13 12 , A L 66 12 2 2 2 R x R x x R
L14 B11
B12 B66 2 2 B66 2 , L 15 R x x 2 R 2 2
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PT
L11 A11
,
L16 L11 ,
A A12 2 E A 2 L12 2 , L21 66 , L22 A66 2 22 552 , 2 2 R x R x R R
L23
B66 B12 2 E55 A22 E55 2 B22 2 , , , L L B 24 25 66 R x R x 2 R 2 2 R2
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L17
L26 L21 ,
L33 E44
L27
A66 2 E A22 2 A A , L31 21 , L32 55 , 222 2 3 2 2 R x R x R R R
E B22 A 2 E55 2 B21 2 222 , L34 E 44 , L35 55 , 2 2 x R x R R 2 x R R 20
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L36
A21 A , L37 223 , 2 R x 2R
L41 B11
B12 B66 2 2 B66 2 , , L 42 R x x 2 R 2 2
D12 D66 2 B12 2 D66 2 , L44 D11 2 2 , E44 , L45 L43 ( E 44 ) R x R x x R 2
L46 L41 , L47
L56 L51
, L57
B66 2 B22 2 . R x 2 R 2 2
References
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D12 D66 2 B22 E55 R D22 2 2 , , , L L E D 54 55 55 66 R x R 2 2 x 2 R2
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L53
B B12 2 E L42 2 B 2 , L51 66 , L52 B66 2 22 55 , 2 2 R x R R x R
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of ring-stiffened functionally graded shell, J. Therm. Stresses 30 (2007) 1249–1267.
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cylindrical shells, Nonlinear Dyn. 87 (2017) 1095–1109. [36] J.N. Reddy, Mechanics of Laminated Composite Plates and Shells: theory and
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[37] M. Rougui, F. Moussaoui, R. Benamar, Geometrically non-linear free and forced vibrations of simply supported circular cylindrical shells: A semi-analytical approach, Int. J. Non-Linear Mech. 42 (2007) 1102–1115. [38] V.V. Bolotin, The Dynamic Stability of Elastic Systems, Holden-Day, San Francisco, 1964. [39] A.H. Nayfeh, D.T. Mook, Non-linear Oscillation, Wiley, NewYork, 1979. [40] W. Schnell, F.J. Heinrichsbaue, The determination of free vibrations of 24
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longitudinally-stiffened thin-walled, circular cylindrical shells, NASA, TT
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CE
PT
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M
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F-8856, 1964.
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Fig. 1: Geometry and coordinate system of an FG cylindrical shell with ring
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CE
PT
ED
M
and axial stiffeners.
Fig. 2: Illustrative plot of the unstable region in the frequency.
26
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800
Frequency(Hz)
m=4
Present Exp.[40]
m=3
600
m=2 400 m=1
2
3
4
5
6
7
8
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200
9 10 11 12
Circumferential wave number (n)
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Fig. 3: The natural frequencies of stiffened cylindrical shells obtained by the present
[40]. 11.0 Present Ng et al.[1]
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Point of origin P1
10.8 10.6
10.2 0
2 4 6 8 10 Volume fraction exponent
PT
10.0
(a)
ED
10.4
Unstable region size 1
work and the experimental results obtained by Schnell and Heinrichsbauer
0.095 0.090 0.085 0.080 0.075 0.070 0.065 0.060
Present Ng et al.[1] (b)
0
2 4 6 8 10 Volume fraction exponent
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Fig. 4: Comparison of the unstable regions for the FG Type A cylindrical shell: (a)
AC
point of origin Pi ( i 1 ), (b) unstable region size i ( i 1 ).
27
10.8 10.6 Present Ng et al.[1]
10.4 10.2 10.0
(a)
0 2 4 6 8 10 Volume fraction exponent
0.100 0.095 0.090 0.085 0.080 0.075 0.070 0.065 0.060
Present Ng et al.[1]
(b)
0
2 4 6 8 10 Volume fraction exponent
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Point of origin P1
11.0
Unstable region size 1
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Fig. 5: Comparison of the unstable regions for the FG Type B cylindrical shell: (a)
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point of origin Pi ( i 1 ), (b) unstable region size i ( i 1 ).
= 0.0 = 1.0. = 2.0
0.25
-0.25
M
0.00
ED
Thickness coordinate, z/h
0.50
350
400 450 500 Temperature (K)
550
600
PT
-0.50 300
AC
CE
Fig. 6: Variation of the temperature through the shell thickness.
28
Nd / N0
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1.2
Tcm=300K
1.0
Tcm=550K Tcm=600K
0.8 0.6 0.4
0.0
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0.2 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 Nondimensional frequency parameter ( P)
Fig. 7: Variation of the unstable region with the temperature difference ( Tcm ) between
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the inner surface and the outer surface for the FG cylindrical shell with ring and axial stiffeners ( m 1 , n 4 , R 1.0 , L 2.0 , h 0.02 , ba br 0.04R , ha hr 0.08R , N s Nr 4 , 1.0 ).
M
1.2 1.0
Nd / N0
ED
0.8
N s= N r= 4 N s= N r= 8 Ns= Nr= 16
0.6
PT
0.4 0.2
CE
0.0
3.3 3.6 3.9 4.2 4.5 4.8 Nondimensional frequency parameter ( P)
AC
Fig. 8: Effect of the stiffener number on the unstable region for the FG cylindrical shell with ring and axial stiffeners ( m 1 , n 4 , R 1.0 , L 2.0 , h 0.02 , ba br 0.04R , ha hr 0.08R , Tcm 300 K, 1.0 ).
29
20
0.002
linear, P = 10.00, Nd / N0 = 0.8
15 10
0.001
0 -5
-0.001
linear, P = 10.00, Nd / N0 = 0.8
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wmn (m)
5
0.000
CR IP T
ACCEPTED MANUSCRIPT
-10
-0.002
-15
(a) 0.010
0.015 t (s)
0.0002
ED
CE
AC
0.28
t (s)
(b)
-0.002
0.000
wmn
0.002
0.9 nonlinear, P = 10.00 Nd / N0 = 0.8
0.6
0.0 -0.3 -0.6
(c)
0.27
-20 -0.004
0.3
PT
wmn (m)
0.0000
-0.0002 0.26
0.025
nonlinear, P = 10.00 Nd / N0 = 0.8
0.0001
-0.0001
0.020
M
-0.003 0.005
0.29
0.30
(d) -0.0001
0.0000
0.0001
wmn
Fig. 9: Comparison of the linear and nonlinear response in the unstable region, (a) linear time histories, (b) linear phase plot; (c) nonlinear time histories, (d) nonlinear phase plot ( m 1 , n 4 , R 1.0 , L 2.0 ,
h 0.02 ,
ba br 0.04R , ha hr 0.08R , N s Nr 8 , Tcm 300 K, 1.0 ).
30
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linear parametric vibration nonlinear parametric vibration
0.004
linear parametric vibration nonlinear parametric vibration
0.003 0.002
0.002
wmn (m)
wmn (m)
0.001 0.000
0.000
-0.001
-0.002
-0.002 P = 10.568
(a)
0.165
0.170
0.175 t (s)
0.180
0.185
-4
1.2x10
linear parametric vibration nonlinear parametric vibration
0.002
-5
8.0x10
0.001
-5
w mn(m)
wmn (m)
0.170
0.175 t (s)
0.180
0.185
linear parametric vibration nonlinear parametric vibration
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4.0x10
0.000
P = 10.569
(b)
-0.003 0.165
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-0.004
0.0 -5
-4.0x10
-0.001
-5
P = 10.570 0.175 t (s)
0.180
0.185
-8.0x10
P = 10.580
(d)
0.17
M
-0.002 ( c ) 0.165 0.170
0.18 t (s)
0.19
Fig. 10: Comparison of the linear and nonlinear response in the stable region, ( a)
L 2.0 ,
h 0.02 ,
ba br 0.04R ,
ha hr 0.08R ,
PT
R 1.0 ,
ED
P 10.568 , (b) P 10.569 , ( c) P 10.570 , (d) P 10.580 ( m 1 , n 4 ,
AC
CE
N s Nr 8 , Tcm 300 K, 1.0 ).
31
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-7
3.0x10
0.0 -7
wmn (m)
-3.0x10
-7
-6.0x10
-7
-9.0x10
-6
-1.2x10
-6
Ns = Nr = 6
Ns = Nr = 8
Ns = Nr = 10
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-1.5x10
-6
-1.8x10
0.007 0.008 0.009 0.010 0.011 0.012 t (s)
Fig. 11: Effect of the number of ring and axial stiffeners on the response of the FG n6 ,
R 1.0 ,
L 2.0 ,
h 0.02 ,
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( m 1 ,
cylindrical shell
ba br 0.04R , ha hr 0.08R , N0 0.2 Ncr , Nd 0.8N0 , P 40 rad/s, 20 rad/s, Tcm 300 K, 1.0 ). -7
-7
-4.0x10
-7
-6.0x10
-7
-8.0x10
-7
ED
-2.0x10
CE
PT
wmn (m)
0.0
M
2.0x10
-1.0x10
-6
-1.2x10
-6
=0. =5.
=1. =20.
0.007 0.008 0.009 0.010 0.011 0.012 t (s)
AC
Fig. 12: Effect of the volume fraction exponent on the response of the FG cylindrical shell with ring and axial stiffeners ( m 1 , n 6 , R 1.0 , L 2.0 ,
h 0.02 , ba br 0.04R ,
ha hr 0.08R ,
N0 0.2 Ncr , Nd 0.8N0 , P 40 rad/s, 20 rad/s,
Ns Nr 8 ,
Tcm 300 K).
32
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0.7
Ns = Nr = 4 Ns = Nr = 6 Ns = Nr = 8 Ns = Nr = 12
0.6
a/h
0.5 0.4 0.3 0.2 0.1
0.98
1.00
1.02 /mn
1.04
1.06
1.08
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0.96
Fig.13: Influence of the stiffener number on the primary resonance responses of FG shells ( m 1 , n 6 ,
R 1.0 , L 2.0 , h 0.02 , ba br 0.04R ,
AC
CE
PT
ED
M
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ha hr 0.08R , N0 0.2 Ncr , 1.0 , Tcm 300 K).
33