Nonlinear modal interaction in rotating composite Timoshenko beams

Composite Structures 96 (2013) 121–134

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Nonlinear modal interaction in rotating composite Timoshenko beams H. Arvin ⇑, F. Bakhtiari-Nejad Amirkabir University of Technology, Department of Mechanical Engineering, 424 Hafez Ave., Tehran, Iran

a r t i c l e

i n f o

Article history: Available online 29 October 2012 Keywords: Rotating composite beam Timoshenko beam Internal resonance Nonlinear normal mode Stability analysis

a b s t r a c t Nonlinear free vibration analysis of rotating composite Timoshenko beams featuring internal resonance is studied in this paper. Three nonlinear coupled equations of motion for flapping, shear and axial motions, are based on the assumptions of Timoshenko theory and the nonlinear von Karman strain–displacement relationships. Due to the small magnitude of low-order flapping/shear frequencies ratio, usually a big gap exists between the aforementioned frequencies especially in isotropic beams, while this gap reduces for composite beams which exhibit more characteristics of Timoshenko beams. Different material, geometrical and operational parameters effects on the frequencies of rotating beams are investigated. For the first time the possibility of internal resonance occurrence between low-order flapping and shear modes is proved. The direct multiple scales method is implemented for construction of the flapping nonlinear normal modes. Results for the flapping nonlinear normal modes are validated via comparison with the results of the fourth-order Runge–Kutta method. Depend on the amplitude ratios besides the nearness of the frequencies of the interacting modes, four bifurcation regions are defined through the nonlinear normal mode stability analysis; (a) one stable coupled mode, (b) two stable and one unstable coupled modes, (c) three stable coupled modes and (d) one stable coupled mode. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction Rotating composite blades are implemented variously in the aerospace engineering and energy generating structures such as helicopters and wind and gas turbines due to their high stiffness to weight ratios and different designable properties of the composite materials. Hence, study of the nonlinear dynamics characteristics such as the nonlinear normal modes (NNMs) construction and the stability of NNMs of rotating blades is crucial for engineers and designers. The concept of NNMs for continuous structures was explored by Nayfeh and Nayfeh [1,2]. They implemented the multiple scales method (MSM) for study on the NNMs of one-dimensional continuous systems with or without internal resonances with quadratic and cubic inertial and geometrical nonlinearities. The stability of NNMs of one-dimensional structures was investigated by Nayfeh et al. [3] and Lacarbonara et al. [4]. Nayfeh et al. [3] applied the direct MSM for construction of NNMs of a fixed– fixed buckled beam with internal resonance. They studied on the stability and bifurcation of the NNMs in the presence of internal resonance. Lacrbonara et al. [4] addressed the construction of NNMs for weakly nonlinear one-dimensional continuous structures with quadratic and cubic geometric nonlinearities. They ⇑ Corresponding author. E-mail address: [email protected] (H. Arvin). 0263-8223/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compstruct.2012.10.015

implemented the MSM for study on the orthogonality and stability of NNMs with 1:1, 2:1 and 3:1 internal resonances. The real nonlinear dynamics of the rotating isotropic beams was explored by few prominent authors, Hodges and Dowell [5], da Silva [6] and Lacarbonara [7], while the real formulation of the rotating composite beams was examined by Hodges [8] and Nayfeh [9]. da Silva and Hodges [6] derived the equations of motion for a straight rotating blade undergoes the aerodynamic forces. They [10] investigated the stability of the straight rotating blade, formulated in their previous research [6], under the effects of aerodynamic forces. They indicated, the third-order structural geometric nonlinearity of the torsional equation possesses the dominant nonlinear effects in comparison with the corresponding bending terms. da Silva [11] derived the fully nonlinear equations of motion of the rotating isotropic blades. His study was on the equilibrium solution and the eigenpairs of the perturbed system about its equilibrium configuration. They investigated the stability of the perturbed blade which undergoes the aerodynamic forces. The concept of construction of NNMs, using the MSM, for rotating beams has been initiated by Arvin et al. [12–14]. Arvin and Bakhtiari-Nejad [12] implemented the discretized MSM to construct the NNMs of a rotating isotropic Euler–Bernoulli beam with or without internal resonances. They studied the stability of the NNMs in the presence of three-to-one and two-to-one internal resonances, respectively, between two flapping and one flapping and one axial modes.

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Arvin and Bakhtiari-Nejad [13] implemented the von Karman strain–displacement relationships besides the Hamilton’s principle for derivation of the equations of motion for a rotating composite Timoshenko beam. They employed the differential transform method to find the prestressed configuration induced by the centrifugal forces. They executed the Galerkin disceretization approach to define the linear free vibration features including the flapping natural frequencies. They applied the direct MSM for study of the number of layers as well as the rotation speed effects on three lowest flapping backbone curves. Arvin et al. [14] applied the direct MSM to the equations of motion, which they derived in their previous research [7], for construction of the flapping NNMs of the rotating isotropic beams. They investigated the effects of the rotation speed on the flapping effective nonlinearity coefficients. In this paper, nonlinear free vibration analysis of rotating composite Timoshenko beams with internal resonance is presented. Three nonlinear coupled partial differential equations of motion for flapping, shear and axial motions, are based on the assumptions of Timoshenko beam theory and the nonlinear von Karman strain– displacement relationships. A linear frequency analysis is directed to investigate the different material, geometrical and operational parameters effects on the flapping and shear frequencies of the rotating beams. Rotating composite blade working points thereby different internal resonance possibilities especially between flapping and shear modes could occur, are specified. The direct MSM is employed for construction of the flapping NNMs in the presence of internal resonance with the shear modes. The stability analysis is established on the reduced order model obtained by the MSM to define the different stability regions of the flapping NNM (which is participated in the internal resonance with a shear mode), depend on the amplitude ratios besides the nearness of frequencies of the interacting modes.

boundary conditions, respectively, for the transverse displacement, shear deformation and axial displacement, are as follows [13]:

ga55 w0s þ a55 w00s þ w0s u00s =g þ u0s w00s =g þ b11 w0s w00s =g þ b11 w00s w0s =g 00 2 þ ð3=2Þw02 s ws =g ¼ 0;  g2 a55 ws =I22 þ k2 ws þ b11 u00s =I22 þ b11 w0s w00s =ðI22 gÞ þ d11 w00s =I22  ga55 w0s =I22 ¼ 0; k2 us þ gk2 r þ gk2 x þ u00s þ w0s w00s =g þ b11 w00s ¼ 0 and

½ws x¼0 ¼ 0; ½ws x¼0 ¼ 0; ½us x¼0 ¼ 0 and ½w0s þ gws x¼1 ¼ 0; ½w0s x¼1 ¼ 0; ½u0s þ ð1=2Þw02 s =gx¼1 ¼ 0; 0

where indicates the first derivation with respect to variable x and 2

I22 ¼ I2 =ðIo h Þ; g ¼ L=h; k2 ¼ x2R =x2f ; xf ¼

pffiffiffiffiffiffiffiffiffiffiffiffiffi A11 =Io =L;

2

b11 ¼ B11 =ðA11 hÞ; d11 ¼ D11 =ðA11 h Þ; ^ s ¼ w ; ^x ¼ x=L; ^t ¼ xf t: ^s ¼ us =h; w ^ s ¼ ws =h; w a55 ¼ KA55 =A11 ; u s The ^ sign is eliminated from the non-dimensional prestressed equations of motion for the sake of simplicity. Aij ; Bij and Dij are, respectively, the extensional, bending-extensional coupling and bending stiffnesses. K; h; Io and I2 are, respectively, the shear correction factor, the beam height and the mass and mass moment of inertia of the cross-section about y direction per unit length of the beam. On the other hand, the non-dimensional updated equations of motion, as addressed in [13], are as follows:

€ þ L  u ¼ n2 ðu; uÞ þ n3 ðu; u; uÞ; Iu

where I and L are, respectively, the linear mass and stiffness operators, given by:

2. Theory

I11 ¼ I22 ¼ I33 ¼ 1 and I12 ¼ I13 ¼ I21 ¼ I23 ¼ I31 ¼ I32 ¼ 0;

2.1. Equations of motion

2 L11 ¼ @ x ½a55 @ x ðÞ þ u0s @ x ðÞ=g þ b11 w0s @ x ðÞ=g þ ð3=2Þw02 s @ x ðÞ=g ;

Schematic of the rotating laminated blade which rotates along the z axis with the constant rotation speed, xR , is presented in Fig. 1, where L; R and N are, respectively, the beam length, the rotor radius and the number of layers of the laminate. x and z axes are, respectively, in the axial and transverse directions. The von Karman stain displacement relationships besides the Hamilton’s principal was implemented in [13] to derive the equations of motion. The equations of motion were separated into the prestressed equations induced by the centrifugal forces and the updated equations of motion about the prestressed configuration. The nondimensional prestressed equations of motion and the corresponding

ð1Þ

L12 ¼ @ x ½b11 w0s @ x ðÞ=g þ ga55 ðÞ;

L13 ¼ @ x ½w0s @ x ðÞ=g;

L21 ¼ @ x ½b11 w0s @ x ðÞ=ðI22 gÞ  ga55 ðÞ=I22 ; L22 ¼ g2 a55 ðÞ=I22 þ k2 ðÞ þ d11 @ 2x ðÞ=I22 ; L23 ¼ b11 @ 2x ðÞ=I22 ; L31 ¼ @ x ½w0s @ x ðÞ=g; L32 ¼ b11 @ 2x ðÞ and L33 ¼ k2 ðÞ þ @ 2x ðÞ; u ¼ fw w ug> and > stands for the transpose of the corresponding term. n2 and n3 , are, respectively, the second- and third-order geometric stiffness nonlinearities given by:

Fig. 1. The rotating laminated blade [13].

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H. Arvin, F. Bakhtiari-Nejad / Composite Structures 96 (2013) 121–134 ð1Þ

n2 ¼ @ x ð@ x w@ x uÞ=g þ b11 @ x ð@ x w@ x wÞ=g þ ð3=2Þ@ x ð@ x w2 w0s Þ=g2 ;

in which,

ð2Þ n2 ð1Þ n3

g;

RHSB ¼ o; RHSB1 ¼ 0; RHSB2 ¼ 0; RHSB3 ¼ 1=2½@ x w2o x¼1 =g;

0:

RHSB1 ¼ 0; RHSB2 ¼ 0; RHSB3 ¼ ½@ x wo @ x w1 x¼1 =g;

b11 @ x w@ 2x w=ðI22

gÞ; 2 2 ¼ ð3=2Þ@ x w @ x w=g2 ; ¼

ð3Þ n2 ð2Þ n3

¼ ¼

@ x w@ 2x w= ð3Þ 0; n3 ¼

ð1Þ

ð2Þ

ð2Þ

ð3Þ

ð3Þ

ð3Þ

ð2Þ

On the other hand, the non-dimensional nonlinear boundary conditions are as follows [13]:

In the case of three to one internal resonance, the first-order solution, OðeÞ, hold as:

½Bo  ux¼0 ¼ o and ½B1  ux¼1 ¼ ½nB x¼1 ;

wo ¼ w1;k Ak ðT 1 ; T 2 Þeixk T o þ w1;y Ay ðT 1 ; T 2 Þeixy T o þ CC;

ð8Þ

wo ¼ w2;k Ak ðT 1 ; T 2 Þeixk T o þ w2;y Ay ðT 1 ; T 2 Þeixy T o þ CC

ð9Þ

o

ð2Þ

1

where B and B are the linear boundary operators and nB is the nonlinear part of the boundary conditions, in which Bo is the identity matrix and B1 and nB , respectively, read as:

and

B111 ¼ @ x ðÞ; B112 ¼ gðÞ; B113 ¼ 0; B121 ¼ 0; B122 ¼ @ x ðÞ; B123 ¼ 0;

uo ¼ w3;k Ak ðT 1 ; T 2 Þeixk T o þ w3;y Ay ðT 1 ; T 2 Þeixy T o þ CC

B131 ð1Þ nB

¼ w0s @ x ðÞ= ; B132 ¼ 0 ð2Þ ¼ 0; nB ¼ 0 and

g

and

B133

¼ @ x ðÞ;

ð3Þ

nB ¼ 1=2@ x w2 =g;

2.2. Multiple scales treatment The direct multiple scales method [15], is implemented to solve the nonlinear updated equations of motion. The solution is considered as the following expansions:

wðx; T o ; T 1 ; T 2 Þ ¼ ewo ðx; T o ; T 1 ; T 2 Þ þ e2 w1 ðx; T o ; T 1 ; T 2 Þ þ e3 w2 ðx; T o ; T 1 ; T 2 Þ;

ð3Þ

where, i is the imaginary unit, Ak and Ay are, respectively, the complex-valued amplitude of the interacting modes, kth and yth modes. w1;k ; w2;k and w3;k are the kth linear normal modes and xk is the kth linear frequency of the rotating beam. xy ¼ 3xk þ e2 r in which r is a detuning parameter and CC stands for the complex conjugate of the preceding terms. Substitution of the first-order solutions, Eqs. (8)–(10), in the right hand side of the second-order equations delivers Eqs. (A.1)– (A.3); Subsequently, the corresponding secular terms hold as Eqs. (B.1) and (B.2). Enforcing the secular terms to vanish, yields:

Ak ðT 1 ; T 2 Þ ¼ Ak ðT 2 Þ and Ay ðT 1 ; T 2 Þ ¼ Ay ðT 2 Þ: Thereafter, the right hand sides of the second-order equation, suggest w1 ; w1 and u1 , respectively, as:

wðx; T o ; T 1 ; T 2 Þ ¼ ewo ðx; T o ; T 1 ; T 2 Þ þ e2 w1 ðx; T o ; T 1 ; T 2 Þ þ e3 w2 ðx; T o ; T 1 ; T 2 Þ

ð4Þ

w1 ¼ h112k ðxÞAk ðT 2 Þ2 e2ixk T o þ h112y ðxÞAy ðT 2 Þ2 e2ixy T o

and

þ h112p ðxÞAk ðT 2 ÞAy ðT 2 ÞeiT o ðxk þxy Þ 2

uðx; T o ; T 1 ; T 2 Þ ¼ euo ðx; T o ; T 1 ; T 2 Þ þ e u1 ðx; T o ; T 1 ; T 2 Þ þ e3 u2 ðx; T o ; T 1 ; T 2 Þ;

þ h112n ðxÞAk ðT 2 ÞAy ðT 2 ÞeiT o ðxk xy Þ þ h11rk ðxÞAk ðT 2 ÞAk ðT 2 Þ ð5Þ

where T o ¼ t; T 1 ¼ et and T 2 ¼ e2 t, are the fast to slow time scales and e is a book-keeping device [15]. Substituting Eqs. (3)–(5), respectively, into Eqs. (1) and (2), and keeping up to the third-order terms of e, deliverers, the following perturbation equations which their solutions provide the flapping, shear and axial motions: Oðeiþ1 Þ:

I  D2o ui þ L  ui ¼ RHSðiþ1Þ ; where Do ¼ @=@ T o , ui ¼ fwi ð1Þ

RHS

i ¼ 0...2 wi

ð6Þ

ui g> and RHS hold as:

þ h11ry ðxÞAy ðT 2 ÞAy ðT 2 Þ þ CC;

þ h212p ðxÞAk ðT 2 ÞAy ðT 2 ÞeiT o ðxk þxy Þ þ h212n ðxÞAk ðT 2 ÞAy ðT 2 ÞeiT o ðxk xy Þ þ h21rk ðxÞAk ðT 2 ÞAk ðT 2 Þ þ h21ry ðxÞAy ðT 2 ÞAy ðT 2 Þ þ CC

u1 ¼ h312k ðxÞAk ðT 2 Þ2 e2ixk T o þ h312y ðxÞAy ðT 2 Þ2 e2ixy T o þ h312p ðxÞAk ðT 2 ÞAy ðT 2 ÞeiT o ðxk þxy Þ þ h312n ðxÞAk ðT 2 ÞAy ðT 2 ÞeiT o ðxk xy Þ þ h31rk ðxÞAk ðT 2 ÞAk ðT 2 Þ

ð2Þ

þ h31ry ðxÞAy ðT 2 ÞAy ðT 2 Þ þ CC;

 ð3=2Þ@ x ð@ x w2o w0s Þ=g2 þ 2Do D1 wo ; ð2Þ

RHS2 ¼ b11 @ x wo @ 2x wo =ðI22 gÞ þ 2Do D1 wo ; ¼

ð12Þ

and

¼ o;

ð2Þ RHS3

ð11Þ

w1 ¼ h212k ðxÞAk ðT 2 Þ2 e2ixk T o þ h212y ðxÞAy ðT 2 Þ2 e2ixy T o

RHS1 ¼ @ x ð@ x wo @ x uo Þ=g  b11 @ x ð@ x wo @ x wo Þ=g

@ x wo @ 2x wo =

g þ 2Do D1 uo ;

ð3Þ

RHS1 ¼ @ x ð@ x wo @ x u1 þ @ x w1 @ x uo Þ=g  b11 @ x ð@ x wo @ x w1 þ @ x w1 @ x wo Þ=g  3@ x ð@ x wo @ x w1 w0s Þ=g2  ð1=2Þ@ x ð@ x w3o Þ=g2 þ 2Do D1 w1 þ 2Do D2 wo þ D21 wo ; ð3Þ

RHS2 ¼ b11 @ x ð@ x wo @ x w1 Þ=ðI22 gÞ þ 2Do D1 w1 þ 2Do D2 wo þ D21 wo ; ð3Þ

RHS3 ¼ @ x ð@ x wo @ x w1 Þ=g þ 2Do D1 u1 þ 2Do D2 uo þ D21 uo : For the sake of simplicity ‘‘ðx; T o ; T 1 ; T 2 Þ’’ is dropped here and henceforth. Subsequently, the corresponding boundary conditions read as: o

ð10Þ

1

½B  ui x¼0 ¼ o and ½B  ui x¼1 ¼

ðiþ1Þ RHSB ;

ð7Þ

ð13Þ

where the over-line stands for the complex conjugate of the corresponding term. Substitution of Eqs. (11)–(13) into the left hand side of the second-order equations delivers the associated boundary value problems given in C, which provide the first order spatial corrections. Thereafter, the Galerkin discretization approach is implemented to determine the first-order spatial corrections as a combination of the linear normal modes of the rotating blade. Insertion of the first- and second-order solutions, Eqs. (8)–(10) and Eqs. (11)–(13), in the right hand side of the third-order equations, lead to Eqs. (A.4)–(A.6). Applying the solvability condition to Eqs. (A.4)–(A.6) yields in: 2 A0k ¼ ð1=4iÞC11 ðk; kÞAk ðT 2 Þ2 Ay ðT 2 Þeie rT o

þ ð1=4iÞC12 ðk; kÞAy ðT 2 ÞAy ðT 2 ÞAk ðT 2 Þ þ ð1=4iÞC13 ðk; kÞAk ðT 2 ÞAk ðT 2 Þ2 and

ð14Þ

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H. Arvin, F. Bakhtiari-Nejad / Composite Structures 96 (2013) 121–134

2 A0y ¼ ð1=4iÞC11 ðy; yÞAk ðT 2 Þ3 eIe rT o

þ ð1=4iÞC12 ðy; yÞAk ðT 2 ÞAy ðT 2 ÞAk ðT 2 Þ þ ð1=4iÞC13 ðy; yÞAy ðT 2 ÞAy ðT 2 Þ2 ;

ð15Þ

where C1j ðk; kÞ and C1j ðy; yÞ are the effective nonlinearity coefficients given in D. Substitution of Eqs. (14) and (15) in the right hand side of the third-order equations, offers w2 ; w2 and u2 , respectively, as: 3 3ixk T o

w2 ¼ f1kkk ðxÞAk ðT 2 Þ e

2

þ f1yyy ðxÞAy ðT 2 Þ e

a0k ¼ 1=16C11 ðk; kÞ sinðcðT 2 ÞÞak ðT 2 Þ2 ay ðT 2 Þ;

ð20Þ

b0k;k ¼ 1=16C11 ðk; kÞ cosðcðT 2 ÞÞak ðT 2 Þay ðT 2 Þ þ 1=16C12 ðk; kÞay ðT 2 Þ2 þ 1=16C13 ðk; kÞak ðT 2 Þ2 ;

ð21Þ

ixk T o

þ f1k ðxÞAk ðT 2 Þ Ak ðT 2 Þe

a0y ¼ 1=16C11 ðy; yÞ sinðcðT 2 ÞÞak ðT 2 Þ3

þ f1kyy ðxÞAy ðT 2 ÞAy ðT 2 ÞAk ðT 2 Þeixk T o 3 3ixy T o

The evolution of the amplitude and phase of the modes are obtained by substituting the polar forms of complex-valued amplitude, into Eqs. (14) and (15) and separating the real and imaginary parts as:

ixy T o

þ f1ykk ðxÞAk ðT 2 ÞAy ðT 2 ÞAk ðT 2 Þe

and

þ f1y ðxÞAy ðT 2 ÞAy ðT 2 Þ2 eixy T o þ f1kkyn ðxÞAy ðT 2 ÞAk ðT 2 Þ2 eiT o ð2xk xy Þ þ f1kkyp ðxÞAy ðT 2 ÞAk ðT 2 Þ2 eiT o ð2xk þxy Þ þ f1kyyn ðxÞAy ðT 2 Þ2 Ak ðT 2 ÞeiT o ðxk 2xy Þ þ f1kyyp ðxÞAy ðT 2 Þ2 Ak ðT 2 ÞeiT o ðxk þ2xy Þ þ CC;

ð22Þ

ð16Þ

b0y;y ¼ 1=16C11 ðy; yÞ cosðcðT 2 ÞÞak ðT 2 Þ3 =ay ðT 2 Þ þ 1=16C12 ðy; yÞak ðT 2 Þ2 þ 1=16C13 ðy; yÞay ðT 2 Þ2 ;

ð23Þ

where cðT 2 Þ ¼ 3bk ðT 2 Þ þ by ðT 2 Þ þ re T o is the relative phase difference between the interacting modes. 2

w2 ¼ f2kkk ðxÞAk ðT 2 Þ3 e3ixk T o þ f2k ðxÞAk ðT 2 Þ2 Ak ðT 2 Þeixk T o þ f2kyy ðxÞAy ðT 2 ÞAy ðT 2 ÞAk ðT 2 Þeixk T o

2.3. Stability analysis

3

þ f2yyy ðxÞAy ðT 2 Þ e3ixy T o þ f2ykk ðxÞAk ðT 2 ÞAy ðT 2 ÞAk ðT 2 Þeixy T o þ f2y ðxÞAy ðT 2 ÞAy ðT 2 Þ2 eixy T o þ f2kkyn ðxÞAy ðT 2 ÞAk ðT 2 Þ2 eiT o ð2xk xy Þ þ f2kkyp ðxÞAy ðT 2 ÞAk ðT 2 Þ2 eiT o ð2xk þxy Þ þ f2kyyn ðxÞAy ðT 2 Þ2 Ak ðT 2 ÞeiT o ðxk 2xy Þ þ f2kyyp ðxÞAy ðT 2 Þ2 Ak ðT 2 ÞeiT o ðxk þ2xy Þ þ CC

ð17Þ

and 3

2

u2 ¼ f3kkk ðxÞAk ðT 2 Þ e3ixk T o þ f3k ðxÞAk ðT 2 Þ Ak ðT 2 Þeixk T o

The stability analysis of the flapping NNMs are investigated here by implementing Eqs. (20) and (22) and the relative phase difference via the same procedure as [3,12]. Taking the derivative of the related phase difference, cðT 2 Þ, with respect to T 2 and substituting Eqs. (21) and (23) into the outcome, yields in:

c0 =a2y ¼ 1=16C11 ðy; yÞ cosðcÞc3 þ ð3=16C13 ðk; kÞ

þ f3kyy ðxÞAy ðT 2 ÞAy ðT 2 ÞAk ðT 2 Þeixk T o 3 3ixy T o

þ f3yyy ðxÞAy ðT 2 Þ e

þ 1=16C12 ðy; yÞÞc2  3=16C11 ðk; kÞ cosðcÞc ixy T o

þ f3ykk ðxÞAk ðT 2 ÞAy ðT 2 ÞAk ðT 2 Þe

 3=16C12 ðk; kÞ þ 1=16C13 ðy; yÞ þ k;

þ f3y ðxÞAy ðT 2 ÞAy ðT 2 Þ2 eixy T o þ f3kkyn ðxÞAy ðT 2 ÞAk ðT 2 Þ2 eiT o ð2xk xy Þ 2

2

þ f3kkyp ðxÞAy ðT 2 ÞAk ðT 2 Þ eiT o ð2xk þxy Þ þ f3kyyn ðxÞAy ðT 2 Þ Ak ðT 2 ÞeiT o ðxk 2xy Þ 2

þ f3kyyp ðxÞAy ðT 2 Þ Ak ðT 2 ÞeiT o ðxk þ2xy Þ þ CC:

ð18Þ

Putting Eqs. (16)–(18) into the left hand side of the third-order equations releases the associated boundary value problems. Afterwards, the Galerkin discretization approach is employed to determine the second-order spatial corrections as a combination of the linear normal modes of the rotating blade. Substituting Eqs. (8), (11) and (16) into Eq. (3) and considering the polar forms of the complex-valued amplitude as, Ak ðT 2 Þ ¼ 1=2ak ðT 2 Þeibk;k ðT 2 Þ and Ay ðT 2 Þ ¼ 1=2ay ðT 2 Þeiby;y ðT 2 Þ , and putting e ¼ 1, provides the nonlinear dynamics configuration of the kth flapping NNM as: o NL o wðx; tÞ :¼ w1;k ak ðtÞ cosðxk t þ bNL k ðtÞ þ bk Þ þ w1;y ay ðtÞ cosðby ðtÞ þ by þ xy tÞ o þ 1=2h112k ðak ðtÞÞ2 cosð2bNL k ðtÞ þ 2bk þ 2xk tÞ

 1=16C12 ðk; kÞa2y qk þ 1=3ðb0y;y þ rÞqk þ p0k ¼ 0:

o NL o þ 1=2h112p ak ðtÞay ðtÞ cosðxk t þ xy t þ bNL k ðtÞ þ bk þ by ðtÞ þ by Þ

þ 1=2h112n ak ðtÞay ðtÞ cosðxk t  x

o  bNL y ðtÞ  by Þ

þ 1=2h11rk ðak ðtÞÞ2 þ 1=2h11ry ðay ðtÞÞ 2

o NL o þ 1=4f 1kyyn ðay ðtÞÞ ak ðtÞ cosð k t  2 y t  2bNL y ðtÞ  2by þ bk ðtÞ þ bk Þ 2 NL o NL þ 1=4f 1kkyn ay ðtÞðak ðtÞÞ cosð y t þ 2 k t  by ðtÞ  by þ 2bk ðtÞ þ 2bok Þ o þ 1=4f 1kkk ðak ðtÞÞ3 cosð3bNL k ðtÞ þ 3bk þ 3 k tÞ 2 NL þ 1=4f 1ykk ðak ðtÞÞ ay ðtÞ cosðby ðtÞ þ boy þ y tÞ

x x

x x

x x o þ 1=4f 1yyy ðay ðtÞÞ3 cosð3bNL y ðtÞ þ 3by þ 3xy tÞ o NL o þ 1=4f 1kyyp ðay ðtÞÞ2 ak ðtÞ cosðxk t þ 2xy t þ 2bNL y ðtÞ þ 2by þ bk ðtÞ þ bk Þ 3 NL o þ 1=4f 1y ðay ðtÞÞ cosðby ðtÞ þ by þ xy tÞ o þ 1=4f 1kyy ðay ðtÞÞ2 ak ðtÞ cosðxk t þ bNL k ðtÞ þ bk Þ 3 NL o þ 1=4f 1k ðak ðtÞÞ cosðxk t þ bk ðtÞ þ bk Þ þ 1=4f 1kkyp ay ðtÞðak ðtÞÞ2 cosð2xk t þ xy t þ bNL y ðtÞ þ boy

o þ 2bNL k ðtÞ þ 2bk Þ:

where k ¼ r=a2y and c ¼ ak =ay . The fix-points of Eqs. (20), (22) and (24) are corresponded to the periodic solutions of the original system. According to Eq. (24) the fix-points are located on a third-order polynomial curve. On the other hand, in accordance with Eqs. (20) and (22) there are two possible conditions for the fix-points. The first one, gives ak ¼ o and ay – o which is corresponded to the uncoupled NNMs, while the second one delivers ak – o; ay – o and sinðcÞ ¼ 0 which is corresponded to the coupled NNMs. In the former case, the complex-valued amplitudes, Ak and Ay , are inserted, respectively, in the cartesian and polar forms, Ak ðT 2 Þ ¼ 1=2ðpk ðT 2 Þ  qk ðT 2 ÞÞeisT 2 and Ay ðT 2 Þ ¼ 1=2ay ðT 2 Þeiby;y ðT 2 Þ , into Eq. (14) and s is determined as s ¼ ðb0y;y þ rÞ=3 to have the autonomous outcome (see [3,12]). The separation of the resultant into the real and imaginary parts and linearization of them with respect to pk and qk , respectively, leads to:

1=16C12 ðk; kÞa2y pk  1=3ðb0y;y þ rÞpk þ q0k ¼ 0;

o þ 1=2h112y ðay ðtÞÞ2 cosð2bNL y ðtÞ þ 2by þ 2xy tÞ

NL o y t þ bk ðtÞ þ bk 2

ð24Þ

ð19Þ

ð25Þ

Substitution of Eq. (23) into Eqs. (25) delivers the eigenvalues of the pffiffi Jacobian matrix of the outcome as k1;2 ¼  J, where, J ¼ 1=9½1=16ð3C12 ðk; kÞ  C13 ðy; yÞÞa2y þ r2 . According to J the eigenvalues are always imaginary, (except when they are zero, i.e. r ¼ 1=16ð3C12 ðk; kÞ  C13 ðy; yÞÞa2y , in this case, the degeneracy of NNMs occurs, see [3]) hence, the uncoupled NNMs are marginally stable. The stability analysis in the latter case, the coupled mode, is executed with investigation of the eigenvalues of the Jacobian matrix of Eqs. (20), (22) and (24) which is constructed as:

2

@ ak a0k

6 M ¼ 4 @ ak a0y @ ak c0

@ ay a0k @ ay a0y @ ay c

0

@ c a0k

3

@ c a0y 7 5; @ c c0

ð26Þ

125

H. Arvin, F. Bakhtiari-Nejad / Composite Structures 96 (2013) 121–134 Table 1 Material properties of the rotating laminated blade [13]. E1 (GPa)

E2 ¼ E3 (GPa)

G23 (GPa)

G12 ¼ G13 (GPa)

m12 ¼ m13

145

9.6

3.4

4.1

0.3

m23

q (kg/m3)

0.5

1389

where, its elements are introduced in E. The characteristic equation of the Jacobian matrix, M, has one zero root and the others are two imaginary or two real numbers, one is positive and the other is negative. In the former case the coupled NNMs are marginally stable and in the latter case they are unstable. 3. Results and discussion Numerical analysis is addressed in this section. At first a linear free vibration analysis is managed for finding the working point of the rotating blade thereby the necessary condition for internal resonance occurrence between the flapping and shear modes is provided. Subsequently, the obtained conditions for internal resonance occurrence is collected for determination of the flapping NNMs affected by the shear modes. The bifurcation analysis of the flapping NNMs is also presented for a beam which is under the internal resonance conditions between its flapping and shear modes. 3.1. Linear frequency analysis Linear frequency analysis is performed here which motivates us for consideration of the modal interaction possibility between flapping and shear modes for the rotating composite Timoshenko beams. Usually because of the small magnitude of the low-order flapping/shear frequencies ratio especially in isotropic beams, a big gap exists between the low-order flapping and shear frequencies, while this gap diminishes for thick composite beams which represent more characteristics of the Timoshenko type beams. The differential transform method is implemented to define the prestressed configuration and the Galerkin discretization approach is implemented to determine the linear free vibration features (see

[13]). An 20-layer anti-symmetric cross-ply laminated composite beam is considered for this investigation. The material properties are given in Table 1. The length, width, thickness of each layer and the rotor radius are, respectively, 1, 0.05, 0.00485 and 0.1 m [13]. At first a linear free vibration analysis on the material type of a stationary beam is presented in Table 2 for three different materials, one composite (the material properties are already defined in Table 1) and two different isotropic materials, aluminium and AISI 12L14 steel. The aluminium Young’s and shear moduli and mass density are, respectively, 70 and 26 GPa and 2700 kg/m3. The steel Young’s and shear moduli and mass density are, respectively, 200 and 80 GPa and 7870 kg/m3. The six lowest non-dimensional flapping and the first non-dimensional shear frequencies are given in Table 2, where ‘‘f’’ and ‘‘sh’’ stand for the flapping and shear modes, respectively. The results exhibit, selection of the composite material instead of the isotropic materials induces the significant effects on the first shear frequency, i.e. in composite beams the flapping and shear frequencies are closer to each other. Indeed, in composite material usage three times of the fifth flapping frequency is close to the first shear frequency, while in isotropic beams even three times of the sixth flapping frequency is lower than the first shear frequency. On the other hand, in isotropic materials study it is evident that with selection of the aluminium as the beam material rather than steel, the three times the sixth flapping frequency is closer to the first shear frequency. For more illustration the D11 =A55 ratio is compared in the three present cases. For the composite beam which has very low shear frequency, D11 =A55 ¼ 0:0162, while the mentioned ratio is 0:0021 and 0:0020, respectively, for the aluminium and steel beams. The implication is that with increment in the D11 =A55 ratio, the first shear frequency get closer to the low-order flapping frequencies. This ratio for the present laminated composite beam is about 8 times greater than the corresponding value for the present isotropic beams which causes the outstanding features of the shear modal interaction in composite beams. In addition, in the isotropic beams (E ¼ 2Gð1 þ mÞ), D11 =A55 ratio, directly depends on the Poisson’s coefficient (for the same geometrical properties), i.e. greater Poisson’s coefficient reduces the common gap between the low-order flapping and shear modes.

Table 2 The six and one lowest non-dimensional flapping and shear frequencies for a non-rotating, xR ¼ 0 rad=s, 20-layer antisymmetric cross-ply laminated beam in comparison with the aluminium and steel beams with the same geometrical properties. Mode number

f1

f2

f3

f4

f5

f6

sh1

20-Layer laminated beam Aluminium beam AISI 12L14 steel beam

0.0940 0.0977 0.0977

0.4792 0.5864 0.5880

1.0949 1.5450 1.5534

1.7601 2.8075 2.8316

2.4419 4.2803 4.3294

3.1176 5.8901 5.9728

7.3604 19.953 20.7018

Table 3 The six, three and one lowest non-dimensional flapping, axial and shear frequencies for a non-rotating, xR ¼ 0 rad=s, antisymmetric cross-ply laminated beam. Mode number

f1

f2

f3

f4

f5

f6

a1

a2

a3

sh1

16-Layer 20-Layer Variation (%)

0.0763 0.0940 23.1905

0.4134 0.4792 15.9237

0.9835 1.0949 11.3301

1.6289 1.7601 8.0544

2.3053 2.4419 5.9221

2.9864 3.1176 4.3956

1.5705 1.5706 0.0024

4.7067 4.7067 0.0003

7.8283 7.8284 0.0009

9.1181 7.3604 19.2769

Table 4 The six, three and first lowest non-dimensional flapping, axial and shear frequencies for an 20-layer antisymmetric cross-ply laminated beam. Mode number

f1

f2

f3

f4

f5

f6

a1

a2

a3

sh1

xR ¼ 0 rad=s xR ¼ 250 rad=s

0.0940 0.1015 7.9903

0.4792 0.4873 1.7027

1.0949 1.1054 0.9642

1.7601 1.7738 0.7732

2.4419 2.4585 0.6815

3.1176 3.1374 0.6328

1.5706 1.5702 0.0235

4.7067 4.7066 0.0032

7.8284 7.8283 0.0008

7.3604 7.3607 0.0045

Variation (%)

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H. Arvin, F. Bakhtiari-Nejad / Composite Structures 96 (2013) 121–134

(b) 1.5706

(a) 0.104 0.102

1.5705

0.1

ω 1/ω f

ω 1/ω f

1.5704 0.098

1.5703 0.096 1.5702

0.094

1.5701

0.092 0

500

1000

1500

2000

2500

0

500

ω R (rpm)

(c)

1000

1500

2000

2500

ω R (rpm)

2.46

(d)

2.456

7.3608

7.3607

ω1/ω f

ω5 /ω f

2.452 7.3606

2.448 7.3605

2.444 7.3604 2.44 0

500

1000

1500

2000

0

2500

500

1000

1500

2000

2500

ω R (rpm)

ω R (rpm)

Fig. 2. Variation of the (a) first non-dimensional flapping, (b) first non-dimensional axial, (c) fifth non-dimensional flapping and (d) first non-dimensional shear frequencies of the rotating 20-layer asymmetric cross-ply laminated blade with the angular speed xR .

5

4

ω/ω f

The number of layers effects on the different mode frequencies for a non-rotating, xR ¼ 0, anti-symmetric cross-ply laminated composite blade is presented in Table 3, where ’’a’’ represents the axial modes. The results are for the six and three lowest nondimensional flapping and axial frequencies and the first nondimensional shear frequency for the aforementioned beam, with 16 and 20 layers. The variation percent of the frequencies from 16 to 20 layers is provided in the third row which is obtained as 100ðx20  x16 Þ=x16 , where superscripts 16 and 20 are, respectively, related to the frequencies of a beam with 16 and 20 layers. The results demonstrate increment in the number of layers, rises the flapping frequencies sharply, while it decreases the first shear frequency rapidly (with the same order compared to the flapping frequencies increasing). These sharp variations reduce the common wide gap between the lowest shear modes and the low-order flapping modes. From the other point of view, centrifugal forces induced by rotation has a hardening effect on the flapping modes, i.e. with increasing of the angular speed the flapping frequencies get larger. Table 4, presents the variations of the six and three lowest non-dimensional flapping and axial frequencies and the first non-dimensional shear frequency of the 20-layer cross-ply laminated composite beam for xR ¼ 0 and xR ¼ 250 rad=s (2387.3 rpm). The third row presents the variations percent induced by the rotation, in the frequencies.

3

2

f4 a1 f3

1

f2

0

0

500

1000

1500

2000

f1 2500

ω R(rpm) Fig. 3. Variation of the first, second, third and fourth non-dimensional flapping (solid-lines) and first non-dimensional axial (dashed-lines) frequencies vs. the angular speed xR of the rotating 20-layer asymmetric cross-ply laminated blade. The filled and unfilled circles, respectively, denote twice and three times of the corresponding value.

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H. Arvin, F. Bakhtiari-Nejad / Composite Structures 96 (2013) 121–134

(a) 3.145

(b)7.8

3.14 7.7

ω/ω f

ω/ω f

3.135 3.13

7.6 7.5

3.125

7.4

3.12 3.115

7.3 0

500

1000

1500

2000

2500

0

500

ωR (rpm)

1000

1500

2000

2500

ωR (rpm)

Fig. 4. (a) Variation of the sixth non-dimensional flapping (solid-lines) and twice the first non-dimensional axial (dashed-lines) frequencies of the rotating 20-layer asymmetric cross-ply laminated blade vs. the angular speed, xR . (b) Variation of the three-times of the fifth non-dimensional flapping (solid-lines), third non-dimensional axial (dashed-lines) and first non-dimensional shear (dotted-dashed lines) frequencies of the rotating 20-layer asymmetric cross-ply laminated blade vs. the angular speed, xR .

0.05 0.04 0.03 0.02

w(L)[m]

0.01 0 −0.01 −0.02 −0.03 −0.04 −0.05

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

Time (second) Fig. 5. The tip point time history of the first flapping NNM constructed by the MSM (solid-lines) in comparison with its corresponding LNM time history (dotted-lines) and the fourth-order Runge–Kutta method result (dotted-dashed-lines) for a six-degrees-of-freedom model of the rotating 20-layer asymmetric cross-ply laminated blade.

2

1.5

1

w

0.5

0

−0.5

−1

−1.5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x(m) Fig. 6. The fifth flapping NNM (solid-lines) interacted by the first shear mode in comparison with its corresponding LNM (filled-circles) of the rotating 20-layer asymmetric cross-ply laminated blade.

128

H. Arvin, F. Bakhtiari-Nejad / Composite Structures 96 (2013) 121–134 2000

1000 Stable Un−Stable

zoom

0 50

−2000

c

c

−1000

0

−3000 −50

−4000

0

1

2

λ

3 x 10

5

−5000

−6000 −12

−10

−8

−6

−4

−2

0

2

4

6

8 8

λ

x 10

Fig. 7. The stability diagram of the fifth flapping NNM interacted by the first shear mode.

2000

(a)

(b)

(c)

(d)

1000

0

c

−1000

−2000

−3000

−4000

−5000

−6000 −12

−10

−8

−6

−4

−2

λ

0

2

4

6

8

x 108

Fig. 8. The separated regions of the stability diagram of the fifth flapping NNM interacted by the first shear mode.

The flapping frequencies, especially the first one, have experienced a large increasing, while a very narrow rising is predicted for the first shear mode. Thus the rotation speed increment is the companion tool besides the material type and number of layers increasing which reduces the common gap between low-order flapping and shear mode frequencies. The above analysis indicates, in the rotating thick laminated composite beams, the common big gap between the low-order flapping and shear frequencies does not remain as big as before and descends with increasing in the number of layers as well as the rotation speed and composite material usage instead of the isotropic material implementation. For more intensive demonstration, variations of the first nondimensional flapping and axial frequencies and the fifth flapping and the first shear frequencies vs. the rotation speed for the 20-

layer laminated beam is depicted in Fig. 2. A sharp rising in the flapping modes and a gradual decreasing and increasing, respectively, for the axial and shear modes is observed. Apparently, three times of the fifth flapping mode is close to the first shear mode which indicates the possibility of 3:1 internal resonance between the mentioned modes. A frequency loci analysis will illustrate possible internal resonances for different working points. The frequency loci of the four lowest non-dimensional flapping and the first non-dimensional axial modes for the 20-layer beam are presented in Fig. 3. For glance observation of the possible type of internal resonances, the twice and three times of the mentioned frequencies are, respectively, marked by the filled and unfilled circles. This figure exhibits no possibility of internal resonances incident between the aforementioned modes for the current rotation speed range and present

129

H. Arvin, F. Bakhtiari-Nejad / Composite Structures 96 (2013) 121–134

data. On the other hand, Fig. 4 presents the possible internal resonance cases. Fig. 4a depicts the sixth non-dimensional flapping and twice the first non-dimensional axial frequencies. It is illustrated that there is no chance for the axial modes to participate in the modal interaction with the other modes between 0 and 250 rad/s range, while beyond this range a 2:1 internal resonance occurs between the mentioned modes. Fig. 4b displays the first nondimensional shear, third non-dimensional axial and three times the fifth non-dimensional flapping frequencies. Again there is no possibility of the modal interaction with participation of the axial modes, while the possibility of 3:1 internal resonance between the first shear and the fifth flapping modes is confirmed. 3.2. Nonlinear normal mode analysis The possibility of internal resonance occurrence was confirmed in the previous section. Nonlinear normal mode analysis is performed here. At first, for validation of the results, the time history of the first NNM configuration and the corresponding linear normal mode (LNM) time history at the tip point of the blade, in the case of no-internal resonances, are shown in Fig. 5 in comparison with the fourth-order Runge–Kutta method result. The fourth-order Runge– Kutta method result is for a discretized six-degrees-of-freedom system. The angular velocity of the blade is 100 rad/s, far from the internal resonance angular speed. The results represent a good agreement between the MSM and the fourth-order Runge–Kutta method result. According to Fig. 5, both methods predict the same type of nonlinearity, the softening behaviour, for the first nonlinear natural frequency in agreement with the predicted response in [13]. According to Fig. 4 three-to-one internal resonance possibility is observed between the fifth flapping and the first shear modes when the blade rotates at 208 rad/s (1986.3 rpm). Employing this data to Eq. (19) delivers the fifth flapping NNM configuration affected by the first shear mode. The deviation of the fifth flapping NNM, interacted by the first shear mode, from its corresponding LNM is presented in Fig. 6. Following the stability analysis presented in Section 2.3, the stability diagram is presented in Fig. 7. This figure is related to the fix-points of Eq. (24). The stability of the mentioned fix-points is performed by investigation on the eigenvalues of the characteristic equation of the corresponding Jacobian matrix, Eq. (26). The four distinctive regions are separated in Fig. 8. The (a) and (d) regions define one coupled stable NNM; while the (b) and (c) regions indicate respectively, two stable and one unstable coupled NNMs and three stable coupled NNMs depend on the amplitude ratios of the interacting modes, c ¼ ak =ay , and the detuning parameter, k ¼ r=a2y .

modes were constructed in the presence of internal resonance by means of the direct multiple scales method. Results for the flapping nonlinear normal modes were validated via comparing with the results of the fourth-order Runge–Kutta method for a discretized sixdegrees-of-freedom system. The stability of the flapping nonlinear normal modes was addressed in the presence of the flapping and shear modes internal resonance. Four bifurcation regions were determined through the stability analysis; (a) one stable coupled mode, (b) two stable and one unstable coupled modes, (c) three stable coupled modes and (d) one stable coupled mode, depend on the amplitude ratios of the interacting modes and the detuning parameter (the nearness of the linear natural frequencies of the interacting modes). It could be pointed out that the modal interaction between flapping and shear modes is unavoidable for special working point in low-order-flapping frequencies for rotating composite Timoshenko blades, while decrement of the blade thickness to length ratio and isotropic material usage instead of composite one, postpones the mentioned interaction to high-flapping frequency ranges. On the other hand, the flapping modal interaction with the shear modes is in lower order in rotating blades in comparison with the stationary beams. Appendix A. The second- and third-order right hand side terms The second-order right hand side terms: ð2Þ

RHS1 ¼ ð1=2Þ½ð6w0s w01;y w001;k þ 2gw03;k w001;y þ 2gw01;y w003;k þ 2gw03;y w001;k þ 6w0s w01;k w001;y þ 2b11 gw02;k w001;y þ 2b11 gw02;y w001;k þ 2b11 gw002;k w01;y þ 6w00s w01;k w01;y þ 2gw01;k w003;y þ 2b11 gw002;y w01;k ÞAk ðT 1 ;T 2 ÞAy ðT 1 ;T 2 Þeiðxk xy ÞT o þ ð2gw03;k w001;y þ 2b11 gw02;k w001;y þ 6w0s w01;k w001;y þ 2gw01;k w003;y þ 2b11 gw02;y w001;k þ 6w00s w01;k w01;y þ 2gw01;y w003;k þ 2b11 gw002;k w01;y þ 6w0s w01;y w001;k þ 2gw03;y w001;k þ 2b11 gw002;y w01;k ÞAk ðT 1 ;T 2 ÞAy ðT 1 ;T 2 Þeiðxk þxy ÞT o þ ð2b11 gw002;k w01;k þ 2b11 gw02;k w001;k þ 2gw03;k w001;k þ 2gw01;k w003;k 2

þ 3w00s w01;k 2 þ 6w0s w01;k w001;k ÞAk ðT 1 ;T 2 Þe2ixk T o þ ð6w0s w01;y w001;y þ 2b11 gw02;y w001;y þ 2b11 gw002;y w01;y þ 3w00s w01;y 2 þ 2gw03;y w001;y þ 2gw01;y w003;y ÞA2y ðT 1 ;T 2 Þe2ixy T o þ 1=2ð4b11 gw002;k w01;k þ 4b11 gw02;k w001;k þ 12w0s w01;k w001;k þ 4gw03;k w001;k þ 4gw01;k w003;k þ 6w00s w01;k 2 ÞAk ðT 1 ;T 2 ÞAk ðT 1 ;T 2 Þ þ 1=2ð6w00s w01;y 2 þ 4gw03;y w001;y þ 4b11 gw02;y w001;y þ 4b11 gw002;y w01;y þ 12w0s w01;y w001;y þ 4gw01;y w003;y ÞAy ðT 1 ;T 2 ÞAy ðT 1 ;T 2 Þ=g2 þ 2ixk w1;k D1 Ak ðT 1 ;T 2 Þeixk T o þ 2ixy w1;y D1 Ay ðT 1 ;T 2 Þeixy T o þ CC;

ðA:1Þ

ð2Þ

4. Conclusions Nonlinear free vibration analysis of rotating composite Timoshenko beams included the modal interaction between its flapping and shear modes was addresses in this paper. Three nonlinear coupled partial differential equations of motion for flapping, shear and axial motions, were based on the assumptions of Timoshenko beam theory and the nonlinear von Karman strain–displacement relationships. A linear frequency analysis was directed for detection of the possibility of modal interaction between the low-order flapping and shear modes of the rotating beams by making variation in the material type and geometrical and operational parameters such as the number of layers and the rotation speed. For the first time the possibility of the internal resonance incident between the low-order flapping and shear modes of the rotating composite beams was proved. The flapping nonlinear normal

RHS2 ¼ b11 ½ðw01;y w001;k þ w01;k w001;y ÞAk ðT 1 ;T 2 ÞAy ðT 1 ;T 2 Þeiðxk xy ÞT o þ ðw01;y w001;k þ w01;k w001;y ÞAk ðT 1 ;T 2 ÞAy ðT 1 ;T 2 Þeiðxk þxy ÞT o þ w01;k w001;k A2k ðT 1 ;T 2 Þe2ixk T o þ w01;y w001;y A2y ðT 1 ;T 2 Þe2ixy T o þ ðw01;k w001;k Ak ðT 1 ;T 2 ÞAk ðT 1 ;T 2 Þ þ w01;y w001;y Ay ðT 1 ;T 2 ÞAy ðT 1 ;T 2 ÞÞ=ðI22 gÞ þ 2ixk w2;k D1 Ak ðT 1 ;T 2 Þeixk T o þ 2ixy w2;y D1 Ay ðT 1 ;T 2 Þeixy T o þ CC ðA:2Þ and ð2Þ

RHS3 ¼ ½ðw01;k w001;y þ w01;y w001;k ÞAk ðT 1 ;T 2 ÞAy ðT 1 ;T 2 Þeiðxk xy ÞT o þ ðw01;k w001;y þ w01;y w001;k ÞAk ðT 1 ;T 2 ÞAy ðT 1 ;T 2 Þeiðxk þxy ÞT o þ w01;k w001;k A2k ðT 1 ;T 2 Þe2ixk T o þ w01;y w001;y A2y ðT 1 ;T 2 Þe2ixy T o þ ðw01;k w001;k Ak ðT 1 ;T 2 ÞAk ðT 1 ;T 2 Þ þ w01;y w001;y Ay ðT 1 ;T 2 ÞAy ðT 1 ;T 2 ÞÞ=g þ 2ixk w3;k D1 Ak ðT 1 ;T 2 Þeixk T o þ 2ixy w3;y D1 Ay ðT 1 ;T 2 Þeixy T o þ CC: ðA:3Þ

The third-order right hand side terms:

130

H. Arvin, F. Bakhtiari-Nejad / Composite Structures 96 (2013) 121–134 0

ð3Þ

0

00

 2g

0 h312n w001;y  2

g

g

g

g g g

g

g

g g

0

0 00  6w00s w01;y h112n  2 w03;y h112n ÞAk ðT 2 ÞA2y ðT 2 Þeiðxk 2xy ÞT o = 2 0 0 0 þ ð1=2Þð6w0s h112p w001;y  2 h112y w003;k  2 h312p w001;y 00 0 00 00 0 0 0  6ws w1;k h112y  2b11 h212p w1;y  2b11 w2;k h112y

g

g

g

g

g

g

0

0

0

 2b11 gh212y w01;k  6w00s w01;k h112y  2gh112p w003;y ÞAk ðT 2 ÞA2y ðT 2 Þeiðxk þ2xy ÞT o =g2 þ ð1=2Þð2b11 g

00 0 w02;k h112p  6w00s w01;y h112k

0

00

00

0

0

 2g

g

00

00

 3w01;k 2 w001;y  2b11 g

00

00

00

 2b11 gw002;y h112k  3w01;k 2 w001;y  2b11 gh212k w01;y  2gw03;y h112k  2b11 gw02;k h112n 0

00

 2gw01;y h312k  6w0s h112k w001;y  2b11 gh212n w01;k 0

00

0

0

0

0

 6w00s w01;k h112n  6w00s w01;y h112k  2b11 gh212k w001;y  6w0s h112n w001;k  2gh312n w001;k  2b11 g

0 00 w002;k h112n  6w0s w01;k h112n

g

0

 2gh312k w001;y ÞA2k ðT 2 ÞAy ðT 2 Þeiðxy þ2xk ÞT o =g2 0 0 00 þ ð1=2Þ½ð4b11 h21rk w001;k  2 h312k w001;k  4 w01;k h31rk 0 0  4 h31rk w001;k  12w0s h11rk w001;k 00 0  9w01;k 2 w001;k  4b11 h21rk w01;k  4b11 w002;k h11rk 00 00 00 0 0  4b11 w2;k h11rk  2b11 h212k w1;k  2b11 w02;k h112k

g

g

g

g

g

g

g g g 0 0 00  2b11 gh212k w001;k  2b11 gw002;k h112k  4gw03;k h11rk 0 0 0 00 00 0 00 0  2gh112k w3;k  6ws w1;k h112k  12ws w1;k h11rk 0 00 0  4gh11rk w003;k  12w0s w01;k h11rk  6w0s h112k w001;k 00 00 00 0 0 0 0 0  2gw1;k h312k  2gw3;k h112k  6ws w1;k h112k ÞA2k ðT 2 ÞAk ðT 2 Þ þ ð12w00s w01;k h11ry 0 00 00 0  4gh31ry w1;k  2b11 gw2;y h112n 0 00 0  2b11 gw002;y h112p  2b11 gw02;y h112p  4b11 gw002;k h11ry 00 00 0  4gw01;k h31ry  4gw03;k h11ry  6w00s w01;y h112n 0 00 00 00 0  2b11 gh212n w1;y  4b11 gw2;k h11ry  4b11 gh21ry w01;k 00

00

0

 12w0s w01;k h11ry  6w0s w01;y h112n  6w0s h112p w001;y 0 0 00  6w00s w01;y h112p  2 h312p w001;y  2 w01;y h312n 00 00 0 0 0  2 w1;y h312p  2 w3;y h112p  4b11 h21ry w001;k 00 00 0  2b11 h212p w01;y  2b11 h212n w01;y  2b11 w002;y h112n 0 0 0  12w0s h11ry w001;k  4 h11ry w003;k  2 h112n w003;y 00 0 0 00  2 w03;y h112n  2 h112p w003;y  6w0s h112n w001;y  6w0s w01;y h112p 0 0 0 00 00  12w1;k w1;y w1;y  2b11 h212p w1;y 0  2 h312n w001;y  6w01;y 2 w001;k ÞAk ðT 2 ÞAy ðT 2 ÞAy ðT 2 Þ

g

g

g

g

g

g

g

g

g

g

g

g

g

g

g

ð3Þ RHS2

gw03;y h00112y

g

00 3 w02;y h112y ÞAy ðT 2 Þe3ixy T o = 2 þ CC;

g

ðA:4Þ

0 00 00 0 ¼ b11 ðh112y w001;k þ w01;y h112n þ w01;k h112y þ h112n w001;y Þ

Ak ðT 2 ÞA2y ðT 2 ÞeiT o ðxk 2xy Þ =ðI22 gÞ 00

00

0

0

 b11 ðw01;y h112p þ w01;k h112y þ h112p w001;y þ h112y w001;k Þ Ak ðT 2 ÞAy ðT 2 Þ2 eiT o ðxk þ2xy Þ =ðI22 gÞ 00

00

0

0

 b11 ðw01;y h112k þ w01;k h112p þ h112k w001;y þ h112p w001;k ÞA2k ðT 2 Þ Ay ðT 2 ÞeiT o ð2xk þxy Þ =ðI22 gÞ 0

00

00

0

 b11 ðh112k w001;y þ w01;k h112n þ w01;y h112k þ h112n w001;k Þ A2k ðT 2 ÞAy ðT 2 ÞeiT o ðxy þ2xk Þ =ðI22 gÞ 00

0

00

0

0

00

 b11 ½ð2w01;k h11ry þ 2h11ry w001;k þ w01;y h112p þ h112n w001;y þ h112p w001;y þ w01;y h112n Þ Ak ðT 2 ÞAy ðT 2 ÞAy ðT 2 Þ 00

0

00

0

þ ðw01;k h112k þ h112k w001;k þ 2w01;k h11rk þ 2h11rk w001;k Þ 2

0

Ak ðT 2 ÞAk ðT 2 Þeixk T o =ðI22 gÞ þ 2ixk w2;k Ak ðT 2 Þeixk T o 00 0 0 00 00 0  b11 ½ðw01;k h112p þ h112n w001;k þ h112p w001;k þ w01;k h112n þ 2w01;y h11rk þ 2h11rk w001;y Þ

Ak ðT 2 ÞAk ðT 2 ÞAy ðT 2 Þ 00

0

00

0

þ ðw01;y h112y þ h112y w001;y þ 2w01;y h11ry þ 2h11ry w001;y Þ 2

0

Ay ðT 2 Þ Ay ðT 2 Þeixy T o =ðI22 gÞ þ 2ixy w2;y Ay ðT 2 Þeixy T o 00

0

0

00

 b11 ðw01;k h112k þ h112k w001;k ÞA3k ðT 2 Þe3ixk T o =ðI22 gÞ  b11 ðh112y w001;y þ w01;y h112y Þ 3

Ay ðT 2 Þe3ixy T o =ðI22 gÞ þ CC

g

eixk T o =g2 þ 2ixk w1;k A0k ðT 2 Þeixk T o

g

2b11 g

0

00

0 00  2b11 h212y w001;y  3w01;y 2 w001;y  6w0s w01;y h112y 0 0 0  6w00s w01;y h112y  6w0s h112y w001;y  2b11 w002;y h112y 0 00 0 00 0 00  2 h312y w1;y  2b11 h212y w1;y  2 h112y w3;y  2

0

 2gh112n w003;k  6w01;k w01;y w001;k  2gw03;k h112n  2gh112k w003;y 00 w02;y h112k  2b11

g

g

0

þ ð1=2Þð6w0s w01;y h112k  2gw01;k h312n  2b11 gh212n w001;k 00

g

0

2

0

0

 2b11 gw002;k h112k ÞA3k ðT 2 Þe3ixk T o =g2 þ ð1=2Þð2gw01;y h312y

 2gh312p w001;k  2b11 gh212k w001;y ÞAk ðT 2 ÞAy ðT 2 Þeið2xk þxy ÞT o =g2 00

g

g g

0

00

g

g

g

0 00 w002;y h112k  6w0s w01;k h112p 0

g

00 0 00  2 w03;k h112k  2 h112k w003;k  2 w01;k h312k 0 00  2b11 h212k w001;k  3w01;k 2 w001;k  6w0s w01;k h112k 00 0 00  2b11 h212k w01;k  6w0s h112k w001;k  2b11 w02;k h112k

 2b11 gw002;k h112p  2gw01;k h312p  6w0s h112k w001;y 0

g

g

0

00

00

g

g

þ 2ixy w1;y A0y ðT 2 Þeixy T o þ ð1=2Þð2gh312k w001;k  6w00s w01;k h112k

00 h212p w01;k

 2gw01;y h312k  2gw03;k h112p  2b11 gh212k w01;y 0

g

g

 2b11 g

0

g

g g

g

00 h212n w01;k ÞAk ðT 2 ÞAk ðT 2 ÞAy ðT 2 Þeixy T o = 2

 2b11 gh212p w001;k  6w01;k w01;y w001;k  2gh312k w001;y  6w0s h112p w001;k 00 w02;y h112k  2b11

g

g g

g

0

 6w00s w01;k h112p  2gw03;y h112k  6w0s w01;y h112k  2gh112k w003;y 0 h112p w003;k  2b11

g

g

 3w01;y 2 w001;k  2b11 gw002;y h112p  2b11 gh212y w001;k 00

g

g g

g

0

g

g

g

g

g

g

g

g

g

0  6w0s h112y w001;y ÞAy ðT 2 Þ2 Ay ðT 2 Þ þ ð12w01;k w01;y w001;k 0 0 0 0  6ws h112p w001;k  6w0s h112n w001;k  12w00s w01;y h11rk 00 0 00 0 00 00 0 0  2 w3;k h112p  2 h112n w3;k  12ws h11rk w1;y  4 w03;y h11rk 00 00 0  6w0s w01;k h112p  6w0s w01;k h112n  4 h31rk w001;y 00 0 0 00  4 w01;y h31rk  6w00s w01;k h112n  6w00s w01;k h112p  12w0s w01;y h11rk 0 00 0 2 00 00 0  6w1;k w1;y  4 h11rk w3;y  2b11 w2;k h112n 0 0 00  2b11 w002;k h112p  4b11 w002;y h11rk  2 w01;k h312n 00 00 00  4b11 w02;y h11rk  4b11 h21rk w01;y  2b11 w02;k h112p 00 0 00 0 0 00 0  2 w3;k h112n  2 h312p w1;k  2 w1;k h312p  2 h312n w001;k 0 0 0 00 00 00  2 h112p w3;k  2b11 h212n w1;k  2b11 w2;k h112n 00 0 0  2b11 h212p w01;k  4b11 h21rk w001;y  2b11 h212p w001;k

g

g

00 00 00  2 w03;k h112y  2 w03;y h112p  2 w01;y h312p 00 0  2b11 h212p w01;y  6w01;k w01;y w001;y  2 h312y w001;k 00 0 00 0 0 0 0 00 0  6ws w1;y h112p  6ws h112y w1;k  6ws w1;y h112p 0 00 00 00 0 0  2b11 w2;k h112y  2 w1;k h312y  2b11 w2;y h112p

g

g g

g

0

 6w0s w01;k h112y  3w01;y 2 w001;k  2gh112n w003;y  2b11 gh212y w001;k

g

g

g

g

00

g

g

0 0 h112y w003;k  2 h312y w001;k 00 0 0 h212n w1;y  6w1;k w01;y w001;y 00 00 00 0  2 w3;k h112y  2b11 w02;k h112y  6w0s w01;y h112n 0 00 0  2b11 w002;k h112y  2b11 w02;y h112n  6w0s h112n w001;y

g g

00

00 0 0  4b11 h21ry w01;y  4b11 w002;y h11ry  4b11 h21ry w001;y 0 00 00 0 00 0 0 0  6ws w1;y h112y  6ws w1;y h112y  4 w1;y h31ry 00 0 0  4 w03;y h11ry  2 h312y w001;y  12w0s h11ry w001;y 0 00 0 00 0  4 h31ry w1;y  2 w3;y h112y  2 h112y w003;y  9w01;y 2 w001;y 00 0 00  4b11 w02;y h11ry  12w00s w01;y h11ry  12w0s w01;y h11ry 00 0 0 0 00 00  2 w1;y h312y  4 h11ry w3;y  2b11 h212y w1;y

00 0 w01;k h312y  6w00s w01;k h112y

00  2b11 h212y w01;k  2 00 0  2 w1;y h312n  2b11

0

þ ð1=2Þ½ð2b11 gw02;y h112y  2b11 gw002;y h112y  2b11 gh212y w01;y

0

RHS1 ¼ ð1=2Þð2b11 gw002;y h112n  2b11 gh212n w001;y  6w0s h112y w001;k

and

ðA:5Þ

131

H. Arvin, F. Bakhtiari-Nejad / Composite Structures 96 (2013) 121–134 0

ð3Þ

00

00

0

Ak ðT 2 ÞA2y ðT 2 Þeiðxk 2xy ÞT o =g 0 00 0 00  ðh112y w001;k þ w01;k h112y þ h112p w001;y þ w01;y h112p Þ 2 iðxk þ2xy ÞT o

Ak ðT 2 ÞAy ðT 2 Þ e

=g

0

þ

0

00

00

0

g g ¼ 0;

00

0 b11 ðw02 1;y Þ =ð2I 22

gÞ ¼ 0;

00

00

For h112p ; h212p and h312p : 0

0

00

2 ½ða55 þ u0s =g þ b11 w0s =g þ 3=2w02 s =g Þh112p

00

þ ðh112k w001;k þ 2h11rk w001;k þ 2w01;k h11rk þ w01;k h112k Þ

0

0

þ ðxk þ xy Þ2 h112p þ ½w03;k w01;y þ w01;k w03;y þ 3w0s w01;y w01;k =g

A0k ðT 2 Þeixk T o 00

00

þ w0s h312p =g0 þ b11 w0s h212p =g þ ðb11 w00s =g þ ga55 Þh212p

A2k ðT 2 ÞAk ðT 2 Þeixk T o =g þ 2ixk w3;k 0

00

0

00

 ½ðh112p w001;k þ w01;k h112p þ h112n w001;k þ w01;k h112n þ 2h11rk w001;y þ 2w01;y h11rk Þ

þ b11 ðw02;k w01;y þ w02;y w01;k Þ0 =g ¼ 0; 00

Ak ðT 2 ÞAk ðT 2 ÞAy ðT 2 Þ

00

00

b11 w0s h112p =ðI22 gÞ þ d11 h212p =I22 þ b11 h312p =I22

0 0 00 00 þ ðh112y w001;y þ 2h11ry w001;y þ w01;y h112y þ 2w01;y h11ry Þ

Ay ðT 2 Þ Ay ðT 2 Þe

4h112y x2y

0

Ak ðT 2 ÞAy ðT 2 ÞAy ðT 2 Þ

ixy T o

þ

g

00

0

0

2

þ

w03;y w01;y

0 ðw0s h112y Þ0 =g þ b11 h212y þ h312y þ ð4x2y þ k2 Þh312y þ ðw02 1;y Þ =ð2gÞ ¼ 0:

 ½ðw01;y h112p þ 2h11ry w001;k þ 2w01;k h11ry þ w01;y h112n þ h112p w001;y þ h112n w001;y Þ

0

g þ ðb11 w00s =g

0

A2k ðT 2 ÞAy ðT 2 Þeiðxy þ2xk ÞT o =g

0

0 þ a55 Þh212y þ 0 3=2w0s w02 1;y =  =

þ ðb11 w00s =g  ga55 Þh112y =I22 þ ð4x2y þ k2  g2 a55 =I22 Þh212y

 ðw01;k h112n þ w01;y h112k þ h112k w001;y þ h112n w001;k Þ 00

þ

½b11 w02;y w01;y 00

Ay ðT 2 Þeið2xk þxy ÞT o =g 00

þ

00 b11 w0s h212y =

b11 w0s h112y =ðI22 gÞ þ d11 h212y =I22 þ b11 h312y =I22

0 00 00 0  ðh112p w001;k þ w01;k h112p þ w01;y h112k þ h112k w001;y ÞA2k ðT 2 Þ

00

0

0 2 0 ½ða55 þ u0s =g þ b11 w0s =g þ 3=2w02 s =g Þh112y þ ws h312y =g

0

RHS3 ¼ ðh112y w001;k þ w01;k h112y þ w01;y h112n þ h112n w001;y Þ

0

þ ðb11 w00s =ðI22 gÞ  ga55 =I22 Þh112p þ ððxk þ xy Þ2

0 ixy T o y w3;y Ay ðT 2 Þe

=g þ 2ix

 g2 a55 =I22 þ k2 Þh212p þ b11 ðw01;k w01;y Þ0 =ðI22 gÞ ¼ 0;

00 0  ðw01;k h112k þ h112k w001;k Þ 00

0

A3k ðT 2 Þe3ixk T o =g  ðw01;y h112y þ h112y w001;y ÞA3y ðT 2 Þe3ixy T o =g þ CC:

0

ðA:6Þ

00

00

ðw0s h112p Þ0 =g þ b11 h212p þ h312p þ ððxk þ xy Þ2 þ k2 Þh312p þ ðw01;y w01;k Þ0 =g ¼ 0: For h112n ; h212n and h312n : 0

0

0 2 0 ½ða55 þ u0s =g þ b11 w0s =g þ 3=2w02 s =g Þh112n þ ws h312n =g

Appendix B. The secular terms

00

ð2kÞ

þ þ

ð2kÞ

ST ð2kÞ ¼ ST Eq  ST BS ð2kÞ

ð2kÞ

ð2kÞ

ð2kÞ

ST SC ¼ ðw1;k w0s =g þ w2;k b11 þ w3;k Þx¼1 ;

where, ST BS ¼ ST SC ST BC . ð2kÞ ST BC ¼ 0 and

¼ 2ix

þ

þ

w21;y

ST

¼ 2ix

g

00

00

00

b11 w0s h112n =ðI22 gÞ þ d11 h212n =I22 þ b11 h312n =I22

0

w23;k

þ

w22;k I22 ÞD1 Ak ðT 1 ; T 2 Þ:

þ

w22;y I22 ÞD1 Ay ðT 1 ; T 2 Þ:

00

00

ðw0s h112n Þ0 =g þ b11 h212n þ h312n þ ððxk  xy Þ2 þ k2 Þh312n ðB:1Þ

Similarly ð2yÞ

g

 g2 a55 =I22 þ k2 Þh212n þ b11 ðw01;y w01;k Þ0 =ðI22 gÞ ¼ 0;

Hence, 2 k ðw1;k

þ w01;y w03;k þ w03;y w01;k w02;k w01;y Þ0 = ¼ 0;

0

ð2kÞ

ST

½3w0s w01;y w01;k = b11 ðw02;y w01;k þ

þ ðb11 w00s =ðI22 gÞ  ga55 =I22 Þh112n þ ððxk  xy Þ2

ST Eq ¼ 2ixk ðw21;k þ w23;k þ w22;k I22 ÞD1 Ak ðT 1 ; T 2 Þ

ð2kÞ

0

þ b11 w0s h212n =g þ ðb11 w00s =g þ ga55 Þh212n þ ðxk  xy Þ2 h112n 

The same procedure as [13] leads to: Second-order secular term:

þ ðw01;y w01;k Þ0 =g ¼ 0: For h11rk ; h21rk and h31rk :

2 y ðw3;y

ðB:2Þ

0

00

The third-order secular term is defined in the same manner.

0 a55 =I22 Þh11rk

g 0

þ

þ

g

g

g

2

 g a55 =I22 Þh212k þ 0

g ¼ 0;

x gÞ ¼ 0;

0 b11 ðw02 1;k Þ =ð2I22

00

00

0

ðC:1Þ

00

g ¼ 0;

0 þ ðg2 a55 =I22 þ k2 Þh21rk þ b11 ðw02 1;k Þ =ð2I 22 gÞ ¼ 0; 00

00

For h11ry ; h21ry and h31ry : 0

0

0 2 0 ½ða55 þ u0s =g þ b11 w0s =g þ 3=2w02 s =g Þh11ry þ ws h31ry =g

ðC:2Þ

00 b11 w0s h21ry =

0 a55 Þh21ry

gþ gþg þ þ b11 w02;y w01;y 0 þ 3=2w0s w02 1;y =g =g ¼ 0; 00 00 0 00 b11 ws h11ry =ðI22 gÞ þ d11 h21ry =I22 þ b11 h31ry =I22 þ ðb11 w00s =ðI22 gÞ 0 0  ga55 =I22 Þh11ry þ ðg2 a55 =I22 þ k2 Þh21ry þ b11 ðw02 1;y Þ =ð2I 22 gÞ ¼ 0; þ

0 ðw0s h112k Þ0 =g þ b11 h212k þ h312k þ ð4x2k þ k2 Þh312k þ ðw02 1;k Þ =ð2gÞ ¼ 0:

0

For h112y ; h212y and h312y :

00

gþ

0 ðw0s h11rk Þ0 =g þ b11 h21rk þ h31rk þ k2 h31rk þ ðw02 1;k Þ =ð2gÞ ¼ 0:

4h112k x2k

00 00 þ d11 h212k =I22 þ b11 h312k =I22 0 2 a55 Þh112k =I22 þ ð4 k þ k2

gÞ gg

ðb11 w00s =

w03;k w01;k 0 =

0

00 0 b11 w0s h212k = þ ðb11 w00s = þ a55 Þh212k þ 0 0 0 ½b11 w02;k w01;k þ 3=2w0s w02 1;k = þ w3;k w1;k  =

00 b11 w0s h112k =ðI22

þ

3=2w0s w02 1;k =

00

0 2 0 ½ða55 þ u0s =g þ b11 w0s =g þ 3=2w02 s =g Þh112k þ ws h312k =g

g

½b11 w02;k w01;k

b11 w0s h11rk =ðI22 gÞ þ d11 h21rk =I22 þ b11 h31rk =I22 þ ðb11 w00s =ðI22 gÞ

For h112k ; h212k and h312k :

þ

0

þ b11 w0s h21rk =g þ ðb11 w00s =g þ ga55 Þh21rk þ

Appendix C. Boundary value problems

0

0 2 0 ½ða55 þ u0s =g þ b11 w0s =g þ 3=2w02 s =g Þh11rk þ ws h31rk =g

ðb11 w00s =

00

00

½w03;y w01;y

0 ðw0s h11ry Þ0 =g þ b11 h21ry þ h31ry þ k2 h31ry þ ðw02 1;y Þ =ð2gÞ ¼ 0:

132

H. Arvin, F. Bakhtiari-Nejad / Composite Structures 96 (2013) 121–134 ðyÞ ðyÞ ðyÞ ðyÞ ðyÞ C13 ðy; yÞ ¼ ð6hðyÞ 36;y þ 12h38;y þ 6h42;y þ 4h61;y g þ 4h63;y g þ 2h64;y g

Appendix D. The effective nonlinearity coefficients

ðyÞ

ðkÞ

ðkÞ

ðkÞ

ðkÞ

ðkÞ

ðkÞ

ð3kÞ

ðyÞ

ðkÞ

ðkÞ

ðkÞ

ðkÞ

ðkÞ

þ

ðkÞ

ðkÞ

ðkÞ

ðkÞ

ðkÞ

C12 ðk; kÞ ¼ þ

ðkÞ 3h93;k

þ

ðkÞ b11 h80;k

þ

ðkÞ 3h95;k

gþ

ðkÞ h27;k

ðkÞ

gþ

þ

ðkÞ 3h96;k

ðkÞ 2b11 h57;k

ðkÞ

gþ

þ

ðkÞ h73;k

ðkÞ h66;k

gþ

gþ

gþ

ðkÞ b11 h92;k

ðkÞ b11 h79;k

ðkÞ h16;k

gþ

gþ

ðkÞ 2h100;k

ðkÞ b11 h85;k

g

g þ b11 hðkÞ 81;k g

ðkÞ

ðkÞ

ðkÞ b11 h51;k

ðkÞ b11 h87;k

ð3kÞ

g

ðkÞ

þ 2b11 h52;k g þ ST SC C 3k g2 þ b11 h91;k g þ b11 h86;k g þ 2b11 h53;k g þ

ðkÞ 2b11 h88;k

þ

ðkÞ b11 h49;k

ðkÞ b11 h105;k

gþ

gþ

ðkÞ

ðkÞ 2b11 h82;k

ðkÞ

gþ

ðkÞ b11 h55;k

gþ

gþ

gþ

ðkÞ

ðkÞ

ðkÞ 3h33;k

g ðkÞ

þ 3h34;k

ðkÞ

ðkÞ

ðkÞ

þ 6h35;k þ 6h36;k þ 6g 3;k þ 6h25;k þ 2h65;k g þ h71;k g þ 3h78;k ðkÞ

ðkÞ

ðkÞ

ðkÞ

ðkÞ

ðkÞ

ðkÞ

þ 3g 5;k þ h67;k g þ h24;k g þ h40;k g þ h70;k g þ h72;k g þ h18;k g ðkÞ

ðkÞ

ðkÞ

ðkÞ

þ 2h99;k g þ 2h94;k g þ 2h74;k g þ 2h26;k gÞ=ðxk g2 Þ and ðkÞ ðkÞ ðkÞ ð3kÞ C13 ðk;kÞ ¼ ð4b11 h39;k g þ 2ST SC C 2k g2 þ 2b11 h69;k g þ 4b11 h21;k g ðkÞ

ðkÞ

ðkÞ

ðkÞ

ðkÞ

ðkÞ

ðkÞ

ðkÞ

ðkÞ

ðkÞ

Z

ðkÞ

g

ðkÞ þ 4h7;k

g

ðkÞ þ 4h102;k

ðkÞ

g

ðkÞ þ 2h2;k

g

ðkÞ þ 4h3;k

g

ðkÞ þ 2b11 h20;k

g

ðkÞ ðkÞ ðkÞ ðkÞ ðkÞ ðkÞ ðkÞ þ 12h1;k þ 6h4;k þ 12h5;k þ 6h12;k þ 9g 1;k þ 6h101;k þ 12h14;k Þ=ð

o

Z

2

xk g Þ 0

ð3kÞ

where ST SC ¼ ðw1;k w0s =g þ w2;k b11 þ w3;k Þx¼1 , C 1k ¼ ðw01;k h112n = 0 0 0 0 0 0 g  w1;y h112k =gÞx¼1 , C 2k ¼ ð2w1;k h11rk =g  w1;k h112k =gÞx¼1 and 0 0 0 C 3k ¼ ð2w01;k h11ry =g  w01;y h112n =g  w01;y h112p =gÞx¼1 . ðyÞ ðyÞ ðyÞ ðyÞ ðyÞ C11 ðy; yÞ ¼ ð6hðyÞ 41;y þ 6h62;y þ 6h57;y þ 3g 4;y þ 2h44;y g þ 2h1;y g ðyÞ

ðyÞ

ðyÞ

ðyÞ

ðyÞ

ðyÞ

þ 2h34;y g þ 2h33;y g þ 2h59;y g þ 2h58;y g þ 2b11 h24;y g þ 2b11 h26;y g ðyÞ

ðyÞ

ðyÞ

ðyÞ

ð3yÞ

þ 2b11 h18;y g þ 2b11 h25;y g þ 2ST SC C 1y g2 þ 2b11 h20;y g þ 2b11 h19;y gÞ=ðxy g2 Þ; ðyÞ ðyÞ ðyÞ ðyÞ 2 C12 ðy; yÞ ¼ 2ðST ð3yÞ SC C 3y g þ b11 h71;y g þ b11 h55;y g þ h3;y g þ h13;y g ðyÞ

ðyÞ

ðyÞ

ðyÞ

ðyÞ

þ b11 h70;y g þ b11 h56;y g þ b11 h68;y g þ b11 h49;y g þ 2b11 h73;y g ðyÞ

ðyÞ

ðyÞ

ðyÞ

ðyÞ

ðyÞ

ðyÞ 2h9;y

ðyÞ h15;y

ðyÞ

ðyÞ

ðyÞ

ðyÞ

þ 6h80;y þ 3h81;y þ 3h82;y þ 2b11 h76;y g þ h32;y g þ 3h87;y þ 3h88;y þ

ðyÞ 2h17;y

gþ

ðyÞ

gþ

gþ

ðyÞ

ðyÞ b11 h54;y

gþ

ðyÞ

ðyÞ 2b11 h72;y

ðyÞ

ðyÞ

ðyÞ

ðyÞ h27;y

ðyÞ h10;y

ðyÞ

ðyÞ

g

ðyÞ

ðyÞ

ðyÞ

þ 2h78;y g þ 2h14;y g þ h85;y g þ h28;y g þ h83;y g þ 2h6;y g þ

ðyÞ 2h21;y ðyÞ

gþ

ðyÞ

gþ

ðyÞ

gþ

ðyÞ 3h11;y

ðyÞ

þ

ðyÞ 3g 1;y

þ 6h29;y þ 6g 2;y þ 3h2;y þ 6h5;y Þ=ðxy g2 Þ

gþ

ðyÞ 4b11 h8;y

ðyÞ

ðyÞ

ðyÞ

gþ

ðyÞ 2b11 h7;y

ðyÞ

gþ

ðyÞ 2b11 h30;y

gÞ=ðxy g2 Þ; 0

0

Z

ðkÞ

00

1

ðkÞ

00

ðkÞ

00

ðkÞ

w1;k w03;y h112n dx ¼ h16;k ;

Zo1

w1;k w0s w01;k h112n dx ¼ h17;k ;

w1;k w03;y h112p dx ¼ h18;k ;

o

Z 1 0 ðkÞ 00 ðkÞ w1;k w00s w01;y h112k dx ¼ h19;k ; w1;k w02;k h112k dx ¼ h20;k ; o o Z 1 Z 1 0 ðkÞ 0 ðkÞ w1;k w002;k h11rk dx ¼ h21;k ; w1;k w0s h112n w001;k dx ¼ h22;k ; o o Z 1 Z 1 0 ðkÞ 0 ðkÞ w1;k h21rk w001;k dx ¼ h23;k ; w1;k h112p w003;y dx ¼ h24;k ; o o Z 1 Z 1 00 ðkÞ 00 ðkÞ w1;k w0s w01;k h11ry dx ¼ h25;k ; w1;k w01;k h31ry dx ¼ h26;k ; o o Z 1 Z 1 0 ðkÞ 00 ðkÞ w1;k h312n w001;y dx ¼ h27;k ; w1;k w0s w01;y h112k dx ¼ h28;k ; o o Z 1 Z 1 0 ðkÞ 0 ðkÞ w1;k h312k w001;y dx ¼ h29;k ; w1;k h112k w003;y dx ¼ h30;k ; Zo 1 Zo 1 00 ðkÞ ðkÞ 00 w3;k w01;k h112n dx ¼ h31;k ; w1;k w02 1;k w1;y dx ¼ g 2;k ; 1

o

o

1

Z Z

1

00

1

Z

1

Z

ðkÞ

0

1

Z

ðkÞ

0

ðkÞ

w2;k h11rk w001;k dx ¼ h38;k ; 1

00

ðkÞ

w1;k w01;y h312n dx ¼ h40;k ;

0

1

0

1

00

00

ðkÞ

w2;k w01;k h11rk dx ¼ h39;k ;

o 1 o

ðkÞ

w2;k w01;k h112k dx ¼ h37;k ;

1

Z

ðkÞ

w1;k w0s h11ry w001;k dx ¼ h35;k ;

o

o

Z

ðkÞ

w1;k w00s w01;y h112n dx ¼ h33;k ;

o

w1;k w00s w01;k h11ry dx ¼ h36;k ;

o

Z

ðkÞ

w1;k w0s w01;y h112n dx ¼ h34;k ;

o

Z

00

w1;k w02;k h11rk dx ¼ h32;k ;

o

o

and

ðyÞ

w1;k w00s w01;k h112n dx ¼ h15;k ;

o

þ b11 h69;y g þ b11 h51;y g þ b11 h75;y g þ h4;y g þ h84;y g þ h31;y g ðyÞ

ðyÞ 2b11 h89;y

1

Z

ðyÞ

þ b11 h48;y g þ 2b11 h52;y g þ 2b11 h53;y g þ b11 h74;y g þ 2b11 h50;y g ðyÞ

ðyÞ

ð3yÞ

1

Zo 1

þ 4h10;k g þ 4h9;k g þ 2h11;k g þ 2h8;k g þ 2h6;k g ðkÞ þ 2h104;k

ðyÞ

o

þ 2b11 h42;k g þ 4b11 h98;k g þ 2b11 h68;k g þ 2h13;k g þ 4h103;k g ðkÞ

ðyÞ

ðkÞ

þ 2b11 h63;k g þ 4b11 h23;k g þ 2b11 h37;k g þ 4b11 h32;k g þ 4b11 h38;k g ðkÞ

ðyÞ

Z 1 0 ðkÞ 0 ðkÞ w1;k w00s w01;k h11rk dx ¼ h1;k ; w1;k h112k w003;k dx ¼ h2;k ; o o Z 1 Z 1 0 ðkÞ 00 ðkÞ w3;k h11rk w001;k dx ¼ h3;k ; w1;k w0s w01;k h112k dx ¼ h4;k ; o o Z 1 Z 1 0 ðkÞ 00 ðkÞ w1;k w0s h11rk w001;k dx ¼ h5;k ; w1;k w03;k h112k dx ¼ h6;k ; o o Z 1 Z 1 00 ðkÞ 00 ðkÞ w1;k w03;k h11rk dx ¼ h7;k ; w3;k w01;k h112k dx ¼ h8;k ; o o Z 1 Z 1 0 ðkÞ 0 ðkÞ w1;k h31rk w001;k dx ¼ h9;k ; w1;k h11rk w003;k dx ¼ h10;k ; o o Z 1 Z 1 00 ðkÞ 0 ðkÞ w1;k w01;k h312k dx ¼ h11;k ; w1;k w0s h112k w001;k dx ¼ h12;k ; o o Z 1 Z 1 0 ðkÞ 00 ðkÞ w3;k h112k w001;k dx ¼ h13;k ; w1;k w0s w01;k h11rk dx ¼ h14;k ; o o Z 1 ðkÞ 00 w1;k w02 1;k w1;k dx ¼ g 1;k ;

Z

ðkÞ 2b11 h54;k

gþ

ðyÞ

ð3yÞ

þ 6h19;k þ 6h22;k þ 6h28;k þ 3g 2;k þ 6g 4;k Þ=ðxk g2 Þ; ðkÞ 2ðh97;k

ðyÞ

where ST SC ¼ ðw1;y w0s =g þ w2;y b11 þ w3;y Þx¼1 , C 1y ¼ ðw01;k h112k =gÞx¼1 ; 0 0 0 C 2y ¼ ð2w01;y h11ry =g  w01;y h112y =gÞx¼1 and C 3y ¼ ðw01;k h112n =g 0 0 ðkÞ ðyÞ ðkÞ ðyÞ 0 0 w1;k h112p =g  2w1;y h11rk =gÞx¼1 and hi;k ; g i;k ; hi;y and g i;y are:

ðkÞ

þ 2b11 h89;k g þ 2b11 h50;k g þ 6h41;k þ 6h15;k þ 6h17;k ðkÞ

ðyÞ

ðyÞ

þ 2b11 h61;k g þ 2b11 h59;k g þ 2b11 h48;k g þ 2b11 h84;k g ðkÞ

ðyÞ

þ 4b11 h66;y g þ 4b11 h79;y g þ 2b11 h12;y g þ 4b11 h23;y g

ðkÞ

ðkÞ

ðyÞ

ðyÞ

þ 2b11 h60;k g þ 2b11 h83;k g þ 2b11 h90;k g þ 2b11 h62;k g ðkÞ

ðyÞ

þ 2b11 h22;y g þ 2ST SC C 2y g2 þ 2b11 h77;y g þ 4b11 h90;y g þ 4b11 h86;y g

ðkÞ

þ 2h30;k g þ 2ST SC C 1k g2 þ 2b11 h56;k g þ 2b11 h58;k g ðkÞ

ðyÞ

þ 12h35;y þ 4h16;y g þ 2h46;y g þ 2h40;y g þ 6h43;y þ 12h45;y þ 9g 3;y

ðkÞ

þ 2h43;k g þ 2h45;k g þ 2h31;k g þ 2h47;k g þ 2h46;k g þ 2h75;k g ðkÞ

ðyÞ

þ 4h67;y g þ 2h65;y g þ 4h47;y g þ 2h60;y g þ 4h37;y g þ 2h39;y g

ðkÞ ðkÞ ðkÞ ðkÞ C11 ðk; kÞ ¼ ð2hðkÞ 64;k g þ 2h29;k g þ 2h44;k g þ 2h77;k g þ 2h76;k g

0

ðkÞ

w1;k w0s h112k w001;y dx ¼ h41;k ;

133

H. Arvin, F. Bakhtiari-Nejad / Composite Structures 96 (2013) 121–134

Z

1

o

Z

0

1

00

1

00

ðkÞ

w1;k w03;y h112k dx ¼ h46;k ;

o

Z

ðkÞ

w1;k w03;k h112n dx ¼ h44;k ;

o

Z

ðkÞ

w1;k w002;k h112k dx ¼ h42;k ;

1

o

Z

1

o

Z

1

o

Z

1

o

Z

1

1

o

Z

1

o

Z

1

Zo 1 Z

o

Z

1

1

Zo 1 Zo 1 Z

o

Z

1

Zo 1

Z

o

0 w1;k w002;k h112n

dx ¼ dx ¼ dx ¼

ðkÞ h57;k ; ðkÞ h59;k ; ðkÞ h61;k ;

0 w2;k h112k w001;k 00 w3;k w01;k h11ry

dx ¼

dx ¼

ðkÞ h63;k ;

dx ¼

ðkÞ h65;k ;

0

ðkÞ

w1;k h112n w003;y dx ¼ h67;k ; 0

ðkÞ

0 w3;k h112n w001;y

dx ¼

00

ðkÞ h71;k ; ðkÞ

w3;k w01;y h112n dx ¼ h73;k ; 00 w3;k w01;y h112k

dx ¼

0

ðkÞ h75;k ; ðkÞ

w3;k h112k w001;y dx ¼ h77;k ; 00

ðkÞ

00

ðkÞ

1

0

ðkÞ

w1;k h21ry w001;k dx ¼ h82;k ; 0 w1;k h212n w001;k 0 w1;k w002;y h112p

dx ¼

ðkÞ h84;k ;

dx ¼

ðkÞ h86;k ;

0

ðkÞ

w1;k w002;k h11ry dx ¼ h88;k ; 00

ðkÞ

w1;k h212n w01;k dx ¼ h90;k ; 0

ðkÞ

w1;k h212n w001;y dx ¼ h92;k ; 1

00 w1;k w03;k h11ry

dx ¼

1

o

Z Z

0

ðkÞ

w1;k h112n w003;k dx ¼ h43;k ; 1

00

o 1

00

ðkÞ

w1;k w01;y h312k dx ¼ h47;k ;

o

Z

ðkÞ

w1;k w01;k h312n dx ¼ h45;k ;

1

o

Z

1

o

Z

1

o 1

Z

o

Z

1

o

Z

00 w1;k w02;k h11ry

dx ¼

ðkÞ h50;k ;

dx ¼

ðkÞ h52;k ;

00

ðkÞ

1

o

ðkÞ h94;k ;

1

o

Z

1

1

Z o1 Zo 1 o

Z

o

Z

1

00

ðkÞ

w1;k w02;k h112n dx ¼ h56;k ; 00 w2;k w01;k h112n 00 w1;k h212k w01;y 00 w2;k w01;y h112k

dx ¼ dx ¼ dx ¼

ðkÞ h58;k ; ðkÞ h60;k ; ðkÞ h62;k ;

1

Zo 1 Zo 1 Zo 1 o

Z Z

00 w1;k w01;y h312p 00

Zo 1 Zo 1 Z o1 Zo 1

o

ðkÞ

1

1

0

dk;k ¼ Z

1

Z

1

w21;k dx þ I22

0

ðkÞ

00 w3;k w01;y h112p

dx ¼

ðkÞ h72;k ;

0

ðkÞ

0

ðkÞ

w3;k h11ry w001;k dx ¼ h74;k ; w3;k h112n w001;k dx ¼ h76;k ;

Z

0

ðkÞ

Z Z

0

ðkÞ

0

ðkÞ

w1;k w002;y h112k dx ¼ h83;k ; 0 w2;k h112p w001;y 00 w1;k h212n w01;y 00

dx ¼

ðkÞ h85;k ;

dx ¼

ðkÞ h87;k ; ðkÞ

w1;k w02;y h112k dx ¼ h89;k ;

Z

1

0

ðkÞ

0

ðkÞ

w1;k w00s w01;y h112p dx ¼ h93;k ;

Z

1

o

00

0

Z

ðyÞ

0

ðyÞ

1

0

ðyÞ

w1;y w002;k h112k dx ¼ h18;y ; 1

00

ðyÞ

w1;y h212k w01;k dx ¼ h20;y ; 1

00

ðyÞ

w2;y w01;y h112y dx ¼ h22;y ; 1

00

ðyÞ

w2;y w01;k h112k dx ¼ h24;y ; 1

00

ðyÞ

w1;y w02;k h112k dx ¼ h26;y ; 1

0

ðyÞ

w1;y h312n w001;k dx ¼ h28;y ; 1

0

ðyÞ

w1;y h212y w001;y dx ¼ h30;y ; 1

0

ðyÞ

w1;y h112p w003;k dx ¼ h32;y ; 1

0

ðyÞ

w1;y h112k w003;k dx ¼ h34;y ;

ðyÞ

0

ðyÞ

w3;y h112n w001;k dx ¼ h4;y ;

o

Z

1

Z

0

ðyÞ

w1;y h31rk w001;y dx ¼ h6;y ;

o 1

0

ðyÞ

w1;y w002;y h11ry dx ¼ h8;y ;

o 1

0

ðyÞ

w1;y h112n w003;k dx ¼ h10;y ;

o

ðyÞ

00

w23;k dx;

00

1

Z

ðyÞ

0

1

w1;y w0s w01;k h112n dx ¼ h2;y ;

o

Z

ðyÞ

w1;y h11rk w003;y dx ¼ h17;y ;

o

Z

ðyÞ

w3;y w01;k h112p dx ¼ h15;y ;

o

w1;k h212p w001;y dx ¼ h91;k ;

00

1

o

Z

ðkÞ

1

ðyÞ

w3;y h112p w001;k dx ¼ h13;y ;

o

Z

Z

w1;y w0s h112p w001;k dx ¼ h11;y ;

o

Z

0

o

ðyÞ

0

1

o

w1;k w002;y h112n dx ¼ h81;k ;

o

w1;y w03;y h11rk dx ¼ h9;y ;

o

ðkÞ

00

1

o

w1;k w0s h112p w001;y dx ¼ h78;k ;

ðkÞ

w1;k h312k w001;k dx ¼ h104;k ;

 Z w22;k dx þ

w1;y w02;y h112y dx ¼ h7;y ;

o

Z

1

ðyÞ

1

o

Z

Z

w1;y w00s w01;y h11rk dx ¼ h5;y ;

o

Z

0

1

o

Z

00

w1;k w01;k h31rk dx ¼ h102;k ;

1

o

w1;y w03;k h112n dx ¼ h3;y ;

o

Z

ðkÞ

ðkÞ

1

o

Z

0

1

o

Z

w3;y h112k w001;k dx ¼ h1;y ;

o

Z

Z

ðkÞ

ðkÞ

ðkÞ

w2;k h112n w001;y dx ¼ h105;k ;

o

o

00 w1;k w02 1;y w1;k dx ¼ g 5;k ;

Z o1

Z

dx ¼

ðkÞ h66;k ;

w1;k h212k w01;k dx ¼ h68;k ;

1

o

dx ¼

ðkÞ h64;k ;

1

o

00

00

w1;k h11ry w003;k dx ¼ h100;k ;

o

w3;k w01;k h11rk dx ¼ h103;k ;

ðkÞ

w1;k h21rk w01;k dx ¼ h98;k ;

w1;k w00s w01;k h112k dx ¼ h101;k ;

o

Z 0 w1;k h312n w001;k

w3;k h112p w001;y dx ¼ h70;k ; 1

0

0

w1;k w0s h112n w001;y dx ¼ h96;k ;

1

o Z 1

ðkÞ

1

o

Z

0

1

o

Z

ðkÞ

w1;k h31ry w001;k dx ¼ h99;k ;

o

Z

0

1

o

Z

Z

ðkÞ

w1;k h312p w001;y dx ¼ h97;k ;

o

Z

00

w1;k w0s w01;y h112p dx ¼ h95;k ;

o

1

o Z 1

Z

Z 0 w1;k h212k w001;y

w2;k w01;k h11ry dx ¼ h54;k ;

o

Z

ðkÞ

w1;k w01;k w01;y w001;y dx ¼ g 3;k ;

ðkÞ g 4;k ;

w1;k w02;y h112n dx ¼ h80;k ;

Zo 1

o

0 w2;k h112k w001;y

1

Zo 1

Z

ðkÞ

w2;k w01;y h112p dx ¼ h79;k ;

Zo 1

o

00 w1;k h21ry w01;k

1

o

Z

00

w1;k h212k w001;k dx ¼ h69;k ;

Zo 1

o

dx ¼

ðkÞ h53;k ;

w1;k w01;k w01;y w001;k

Zo 1

o

0 w2;k h11ry w001;k

dx ¼

ðkÞ h51;k ;

1

o

Z

00 w1;k h212p w01;y

dx ¼

ðkÞ h49;k ;

w2;k w01;y h112n dx ¼ h55;k ;

o

Z

ðkÞ

00 w1;k w02;y h112p

1

o

Z

0

w2;k h112n w001;k dx ¼ h48;k ;

Z

o

Z

1

1

0

ðyÞ

0

ðyÞ

1

1

0

1

1

1

1

1

ðyÞ

00

ðyÞ

00

ðyÞ

00

ðyÞ

w1;y w01;k h312k dx ¼ h33;y ;

o

o

00

w1;y w03;k h112p dx ¼ h31;y ;

o

Z

ðyÞ

w1;y w0s w01;y h11rk dx ¼ h29;y ; 1

Z

0

1

o

Z

ðyÞ

w1;y w01;k h312n dx ¼ h27;y ;

o

Z

0

w2;y h112k w001;k dx ¼ h25;y ;

o

Z

ðyÞ

w2;y h11ry w001;y dx ¼ h23;y ;

o

Z

00

w1;y w01;y h31rk dx ¼ h21;y ;

o

Z

ðyÞ

w1;y h212k w001;k dx ¼ h19;y ;

o

Z

ðyÞ

00 w1;y w02 1;k w1;y dx ¼ g 1;y ;

o

Z

ðyÞ

w1;y h11ry w003;y dx ¼ h16;y ;

o

Z

0

w1;y w002;y h112y dx ¼ h12;y ;

w3;y h11rk w001;y dx ¼ h14;y ;

o

Z

1

1

ðyÞ

w1;y w01;k w01;y w001;k dx ¼ g 2;y ;

134

Z

1

1

1

1

1

1

1

1

1

o

Z

1

o

Z

1

1

1

1

1

o

0

ðyÞ

00

0

ðyÞ

1

00

ðyÞ

w1;y h21ry w01;y dx ¼ h79;y ; 1

00

ðyÞ

1

00

ðyÞ

w3;y w01;k h112n dx ¼ h83;y ;

ðyÞ

00

Z

Z

1

o

w21;y dx þ I22

Z o

1

 Z w22;y dx þ

o

1

 w23;y dx ;

 3=16C11 ðk; kÞ cosðcÞcay  3=8C12 ðk; kÞay ;

References

ðyÞ

0

ðyÞ

00

1

ðyÞ

00

ðyÞ

0

ðyÞ

0

ðyÞ

w1;y h212p w001;k dx ¼ h70;y ; 1

00

ðyÞ

0

ðyÞ

0

ðyÞ

00

ðyÞ

w2;y w01;y h11rk dx ¼ h72;y ; w1;y w002;k h112n dx ¼ h74;y ;

1

w1;y w002;y h11rk dx ¼ h76;y ;

o 1

w3;y w01;y h11rk dx ¼ h78;y ;

o 1

0

ðyÞ

w1;y w0s h11rk w001;y dx ¼ h80;y ; Z

1 o

M2;3 ¼ 1=16C11 ðy; yÞ cosðcÞc3 a3y ;

M 3;3 ¼ 1=16C11 ðy; yÞ sinðcÞc3 a2y þ 3=16C11 ðk; kÞ sinðcÞca2y :

w1;y w01;y h312y dx ¼ h64;y ;

o

o

ðyÞ

M 3;2 ¼ 1=16C11 ðy; yÞ cosðcÞc3 ay þ 1=8C13 ðy; yÞay

w1;y w002;k h112p dx ¼ h68;y ;

Z

ðyÞ

ðyÞ

0

1

Z

00

w1;y h212y w01;y dx ¼ h89;y ;

 3=16C11 ðk; kÞ cosðcÞay  3=8C13 ðk; kÞcay ;

w1;y w0s h112k w001;k dx ¼ h62;y ;

o

w1;y w0s w01;k h112p dx ¼ h81;y ;

00

ðyÞ

1

o Z 1

ðyÞ

o

1

M 3;1 ¼ 3=16C11 ðy; yÞ cosðcÞc2 ay þ 1=8C12 ðy; yÞcay

w3;y h112y w001;y dx ¼ h60;y ;

o

Z

ðyÞ

w2;y h112y w001;y dx ¼ h77;y ;

o

Z

00

Z

w1;y w03;k h112k dx ¼ h58;y ;

1

Z

ðyÞ

0

ðyÞ

w1;y h21ry w001;y dx ¼ h90;y :

o

M 2;2 ¼ 0;

w2;y w01;y h11ry dx ¼ h66;y ;

o

w1;y w02;k h112n dx ¼ h75;y ;

o

Z

ðyÞ

0

1

o

Z

00

1

0

w1;y w00s w01;k h112n dx ¼ h87;y ;

M 1;3 ¼ 1=16C11 ðk; kÞ cosðcÞc2 a3y ; M 2;1 ¼ 3=16C11 ðy; yÞ sinðcÞc2 a2y ; ðyÞ

1

1

o Z 1

ðyÞ

w1;y w00s w01;k h112p dx ¼ h88;y ;

ðyÞ

M 1;1 ¼ 1=8C11 ðk; kÞ sinðcÞca2y ; M1;2 ¼ 1=16C11 ðk; kÞ sinðcÞc2 a2y ;

00

0

Z

o

Z

o

0

w1;y h312p w001;k dx ¼ h85;y ;

ðyÞ

w2;y h112n w001;k dx ¼ h56;y ;

o

w2;y h11rk w001;y dx ¼ h73;y ;

o

Z

ðyÞ

0

o Z 1

Appendix E. The Jacobian matrix elements

w1;y w02;y h11rk dx ¼ h52;y ;

o

Z

w1;y h212p w01;k dx ¼ h71;y ;

o

Z

dx ¼

ðyÞ h63;y ;

Z

w1;y h212n w001;k dx ¼ h69;y ;

o

Z

0

dx ¼

ðyÞ h61;y ;

w3;y w01;y h11ry dx ¼ h67;y ;

o

Z

00 w1;y w03;y h11ry

1

o

Z

00 w1;y w01;y h31ry

ðyÞ

1

ðyÞ

00

1

Z

ðyÞ

w1;y h112y w003;y dx ¼ h65;y ;

o

Z

0

w1;y h312k w001;k dx ¼ h59;y ;

o

Z

o

ðyÞ

1

ðyÞ

w2;y w01;k h112n dx ¼ h54;y ;

o

00

Z

w1;y h21rk w01;y dx ¼ h50;y ;

o 1

w1;y w0s w01;k h112k dx ¼ h57;y ; 1

00

Z

ðyÞ

w1;y w02;y h11ry dx ¼ h86;y ;

o

dy;y ¼

w3;y w01;y h112y dx ¼ h46;y ;

1

Z

ðyÞ

00

00

1

and

ðyÞ

Z

w1;y h212n w01;k dx ¼ h55;y ; 1

1

o

Z

00

w1;y w01;k h312p dx ¼ h84;y ;

ðyÞ

00 w1;y w02 1;k w1;k dx ¼ g 4;y ;

Z

ðyÞ

00

00

w3;y w01;k h112k dx ¼ h44;y ;

o

1

o

ðyÞ

0

1

1

w1;y h21rk w001;y dx ¼ h53;y ;

o

Z

0

ðyÞ

w2;y w01;k h112p dx ¼ h48;y ;

o

Z

ðyÞ

o

1

o

w1;y w00s w01;y h112y dx ¼ h42;y ;

1

Z

ðyÞ

0

Z

Z

ðyÞ

1

o

o

w2;y h112p w001;k dx ¼ h51;y ;

o

Z

00

00 w1;y w02 1;y w1;y dx ¼ g 3;y ;

o

Z

Z

Z

ðyÞ

1

o

Z

Z

w1;y w02;k h112p dx ¼ h49;y ;

o

Z

0

00

Z

ðyÞ

w3;y h11ry w001;y dx ¼ h47;y ;

o

Z

o

w1;y w00s w01;y h11ry dx ¼ h45;y ; 1

ðyÞ

w1;y w03;y h112y dx ¼ h40;y ;

ðyÞ

0

0

1

ðyÞ

00

1

o

Z

Z

w1;y w0s w01;y h112y dx ¼ h43;y ;

o

Z

0

ðyÞ

w1;y w0s h11ry w001;y dx ¼ h38;y ;

o

ðyÞ

0

w1;y w0s h112y w001;y dx ¼ h36;y ;

1

w1;y w00s w01;k h112k dx ¼ h41;y ;

o

Z

0

1

o

Z

ðyÞ

1

o

Z

0

w1;y h312y w001;y dx ¼ h39;y ;

o

Z

Z

ðyÞ

w1;y h31ry w001;y dx ¼ h37;y ;

o

Z

00

w1;y w0s w01;y h11ry dx ¼ h35;y ;

o

Z

H. Arvin, F. Bakhtiari-Nejad / Composite Structures 96 (2013) 121–134

0

ðyÞ

w1;y w0s h112n w001;k dx ¼ h82;y ;

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