Nonlinear galloping of internally resonant iced transmission lines considering eccentricity

Journal of Sound and Vibration 331 (2012) 3599–3616

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Nonlinear galloping of internally resonant iced transmission lines considering eccentricity Zhimiao Yan a,b, Zhitao Yan a,b,n, Zhengliang Li a,b, Ting Tan a a

College of Civil Engineering, Chongqing University, Chongqing 400045, China Key Laboratory of Construction and New Technology of Mountainous Region Cities (Chinese Ministry of Education), Chongqing University, Chongqing 400045, China

b

a r t i c l e i n f o

abstract

Article history: Received 27 April 2011 Received in revised form 9 November 2011 Accepted 9 March 2012 Handling Editor: L.N. Virgin Available online 12 April 2012

Based on the curved-beam theory, a nonlinear galloping model considering three displacement (normal, bi-normal and tangential) components and twist is formulated. According to the property of transmission line, one reduced (normal and bi-normal) galloping model, with regard of bending, rotation and eccentricity of cross section, is obtained. Moreover, the initial rotation angle is also introduced in galloping and aerodynamic models. Additionally, based on the reduced model, the bifurcation and stability of the two cases (1:1 resonance and 2:1 resonance) are analyzed. The results turn out that the importance of ice eccentricity needs to be highlighted. Finally, multiple stabilities are found through the analyses of bifurcation and stability and proved by the reduced model and Reduced Amplitude Modulation Equations (RAME) numerically integrated in time history. & 2012 Elsevier Ltd. All rights reserved.

1. Introduction The galloping of the iced transmission line is the self-exciting vibration with high amplitude and low frequency at the mean wind speed. As a result of the asymmetrical cross section caused by ice, aerodynamic coefficients of transmission line change, thus, at certain wind speed, the transmission line becomes unstable and then the galloping appears. The earliest galloping models of transmission line are mainly based on the analytical theory of the low-sag cable which was developed by Irvine [1]: Den Hartog [2] and Parkinson [3] have developed the single-degree-of-freedom (vertical) galloping model; Jones [4] and Luongo [5] then have considered the interaction of vertical and horizontal effects and proposed two-degree-of-freedom galloping model; moreover, Blevins [6] and Yu [7] have found that the rotation plays great effect on the initiation of galloping under certain conditions, then Yu [8,9] has developed three-degree-of-freedom (vertical, horizontal and rotational) galloping model with regard of ice eccentricity. However, all these models cannot take the bending effect into consideration. Recently, based on the curved-beam theory, Luongo [10–12] has proposed a new galloping model which can consider both bending and rotation. In these three papers, the importance of considering the bending effects with curved-beam theory has been proved. Yet, it did not take the eccentricity of ice into account. In this paper, the curved beam model [13] is introduced to analyze the iced transmission line. In order to describe the curved beam and its motion, a system of converted curvilinear coordinates which are moved along with the beam material is adopted. Also, three-dimensional Lagrange strain tensor is used to analyze the deformation of curved beam [14]. Then, according to the Hamilton principle, a nonlinear galloping model taking three displacement (normal, bi-normal and tangential) components, twist and eccentricity of ice into account is developed. Moreover, according to the property of the transmission line and Galerkin

n

Corresponding author at: College of Civil Engineering, Chongqing University, Chongqing 400045, China. E-mail address: [email protected] (Z. Yan).

0022-460X/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jsv.2012.03.011

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Z. Yan et al. / Journal of Sound and Vibration 331 (2012) 3599–3616

Nomenclature A A1,A2

area of cross section unknown complex normal and bi-normal amplitudes depending on the slow time scales, respectively ai amplitude varying as the time t b1,b2,b3 aerodynamic lift force, drag force and aerodynamic moment, respectively ba aerodynamic force on the section ci structural damping coefficients cd ½g, cl ½g aerodynamic drag and lift coefficients, respectively d sag at the mid-span EA axial stiffness EI1,EI2 bending stiffness about a1 and a2 axes, respectively e0 mean bi-nominal aerodynamic force ex,ey eccentricity of cross section about a1 and a2 axes, respectively e0y initial eccentricity along Y-axis f 1 ½s, f 2 ½s, f 3 ½s normal, bi-normal and rotational cable model shapes, respectively GJ torsional stiffness h0 static rotational aerodynamic moment L horizontal distance between adjacent towers l cable length M1,M2 bending forces about a1 and a2 axes, respectively M3 torsional forces at time t ¼0

q1 ½t, q2 ½t, q3 ½t normal, bi-normal and rotational amplitudes varying as the time t, respectively r radius of the transmission line S1,S2 first moments of area about the a1 and a2 axes, respectively T tensional forces at time t ¼0 T0 fast time scale governing the dynamic phenomenon T 1 ,T 2 slow time scales on which the amplitudes and phases of are modulated by nonlinearity u1,u2,u3 normal, bi-normal and tangential displacements at time t, respectively V relative velocity of the wind with respect to section k curvature along the axis a2 at time t ¼0 g attack angle y3 rotational displacements at time t ai phase varying as the time t r density of transmission line rair density of the air o1 , o2 normal and bi-normal circular frequencies of the iced transmission line, respectively c1 , c2 curvature rates of beam about the a1 and a2 axes, respectively j rotation caused by static aerodynamic force j0 initial rotation caused by the eccentricity of ice e incremental axial strain Z torsional curvature rate

method, a simplified two-degree-of-freedom (normal and bi-normal) model with consideration of torsion, bending and eccentricity has been obtained. Furthermore, in order to better evaluate the actual galloping of iced transmission line, the initial rotation angle is introduced into both the galloping and aerodynamic models. Finally, bifurcation and stability of two special cases (1:1 resonance and 2:1 resonance) are analyzed with Multiple Scale Perturbation Method (MSM). From the analyses, the importance of considering the eccentricity of ice is highlighted and one special phenomenon (multiple stabilities) is found. Also, multiple stabilities is proved by the reduced model and RAME numerically integrated in time history. 2. Nonlinear cable model based on curved beam theory In order to better describe the curved beam and its motions, the choice of coordinates is made as following: directions 1, 2 and 3 denote the normal, bi-normal and tangential directions of transmission line, respectively (see Fig. 1). Also, the positive twists of all angles are assumed to be counterclockwise. Moreover, it is assumed that the transmission line is uniformly iced and the mean wind ðU ¼ UYÞ blows horizontally. There are four different configurations. (a, b) The initial configurations C10 (the eccentricity is not considered) and C20 (the eccentricity is considered) are under the action of the gravity (including the ice). The configuration C10 belongs to the vertical (X, Z) plane. In this configuration, the relations of coordinates are a110 ¼ X and a120 ¼ Y. In the configuration C20 , the first moments of area S1 and S2 are introduced to consider iced eccentricity. Because of eccentricity of iced transmission line, the configuration C20 rotates the initial angle j0 from shape C10 . (c) The reference configuration C is assumed as the body at the time t¼0, in which static aerodynamic forces act on transmission line. Due to static aerodynamic forces, the iced transmission line swings the angle j from the shape C20 . (d) The actual configuration C is assumed as the body at time t 4 0, in which the loads acting on the transmission line include both static and dynamic aerodynamic forces. Because of the dynamic aerodynamic force, the dynamic translation and rotation are u and y, respectively. 2.1. Mechanical model Following Refs. [13,15], the Lagrange strain–displacement relationship can be calculated with the consideration of low tension as below:

eL ¼ u3;3 ku1 , eN ¼ 1=2ðu1;3 þ ku3 Þ2 þ1=2u22;3 þ1=2ðu3;3 ku1 Þ2

Z. Yan et al. / Journal of Sound and Vibration 331 (2012) 3599–3616

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Fig. 1. (a) Cable configurations and corresponding coordinates; (b) aerodynamic forces on transversal section and corresponding coordinates of _ configurations ðV ¼ UuÞ.

cL1 ¼ u2;33 þ ky3 , cN1 ¼ y3 ðu1;33 þ2ku3;3 k2 u1 Þ þu2;3 ðu3;33 2ku1;3 k2 u3 Þ cL2 ¼ u1;33 þ ku3;3 þ keL , cN2 ¼ y3 u2;33 ðu1;3 þ ku3 Þðu3;33 2ku1;3 k2 u3 Þ

ZL ¼ y3;3 þ ku2;3 , ZN ¼ ky3 ðu1;3 þ ku3 Þ þ u2;3 ðu1;33 þ ku3;3 Þ

(1)

where the subscript ‘‘,’’ denotes derivation; e, c1 , c2 and Z are the incremental axial strain and curvature rates of the neutral axis; the right superscripts L and N represent the linear and nonlinear components, respectively; u1, u2, u3 and y3 are nominal, bi-nominal, tangential and rotational displacements at time t, respectively; k denotes the curvature along the axis a2 at time t ¼0. In these two papers, Zhu established the Lagrange strain–displacement relationship with the assumption that the curvature is relatively large. However, according to Ref. [16], the curvature k and tangential displacement u3 are very small in the transmission line system. Therefore, terms on the high order of k and u3 in the linear equations turn into the nonlinear equations. Also, terms on the quadratic order of k and u3 or high in Eq. (1) are neglected, leading to

eL ¼ u3;3 ku1 , eN ¼ 1=2u21;3 þ1=2u22;3 cL1 ¼ u2;33 , cN1 ¼ ky3 þ y3 u1;33 þu2;3 ðu3;33 2ku1;3 Þ cL2 ¼ u1;33 , cN2 ¼ y3 u2;33 u1;3 ðu3;33 2ku1;3 Þ

ZL ¼ y3;3 , ZN ¼ ku2;3 þ ky3 u1;3 þ u2;3 u1;33

(2)

Based on the Hamilton principle and Ref. [1], the equations of motion can be extended by ignoring the shear resistance of curved beam as following: Z tZ l N N f½ðEAede þ EI1 c1 dc1 þ EI2 c2 dc2 þGJ ZdZÞ þðT deN þ M 1 dc1 þ M 2 dc2 þM 3 dZN Þ 0

0

þ rAðu€ 1 du1 þ u€ 2 du2 þ u€ 3 du3 Þ þ rðJ y€ 3 dy3 þ u€ 1 S1 dy3 þ y€ 3 S1 du1 þ u€ 2 S2 dy3 þ y€ 3 S2 du2 Þ ½ðb c u_ Þdu þ ðb c u_ Þdu c u_ du þ ðb c y_ Þdy g ds dt ¼ 0 1

L

N

c1 ¼ cL1 þ cN1 ;

1

1

1

cL2 þ cN2 ;

2

2

L

2

2

3

3

3

3

4 3

3

(3)

N

c2 ¼ Z ¼ Z þ Z ; S1 ¼ Aey ; S2 ¼ Aex ; S1 and S2 are the first moments of areas where e ¼ e þ e ; about the a1 and a2 axes, respectively; T ¼ rAgL2 =ð8dÞ; M 1 ¼ 0; M 2 ¼ EI2 k; M 3 ¼ ðsl=2ÞrAgðey cos½j þ ex sin½jÞ; T, M1, M2 and M3 represent the tensional, bending and torsional forces at time t ¼0, respectively; ‘‘.’’ denotes derivation of time t; ex and ey represent eccentricity of cross section about the a1 and a2 axes, respectively; r is the density of transmission line; A is the area of cross section; EA, EI1 , EI2 , and GJ are the axial, bending and torsional stiffness, respectively; b1, b2 and b3 are aerodynamic forces; c1, c2, c3 and c4 are structural damping coefficients; l denotes the cable length. L represents the horizontal distance between adjacent towers; d is sag at the mid-span. Substituting Eq. (2) into Eq. (3), the equations of motion can be obtained in the four directions as following: Normal direction: EAfkðu3;3 ku1 þ 1=2u21;3 þ 1=2u22;3 Þ þ ½ðu3;3 ku1 þ 1=2u21;3 þ 1=2u22;3 Þu1;3 0 g þTu1;33 rAu€ 1 rS1 y€ 3 þðb1 c1 u_ 1 Þ ¼ 0

(4)

Bi-normal direction: EA½ðu3;3 ku1 þ 1=2u21;3 þ 1=2u22;3 Þu2;3 0 þ Tu2;33 rAu€ 2 rS2 y€ 3 þ ðb2 c2 u_ 2 Þ ¼ 0

(5)

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Tangential direction: EAðu3;3 ku1 þ1=2u21;3 þ1=2u22;3 Þ0 rAu€ 3 c3 u_ 3 ¼ 0

(6)

GJ y003 þ ðGJ þ EI1 Þku002 EI1 ky3 M 1 ðu1;33 þ kÞM 2 u2;33 M 3 ku1;3 rJ y€ 3 rS1 u€ 1 rS2 u€ 2 þ b3 c4 y_ 3 ¼ 0

(7)

Torsional direction:

where the boundary conditions are u1 ¼ u2 ¼ u3 ¼ y3 ¼ 0;

at s ¼ 0,l

(8)

where ‘‘0 ’’ denotes differentiation with respect to s. Due to the fact that normal-to-tangential and normal-to-torsional squared frequency ratios are small [11], the components, such as rAu€ 3 , c3 u_ 3 , rJ y€ 3 , b3 and c4 y_ 3 , can be assumed to be equal to zero. Furthermore, u€ 1 and u€ 2 in Eq. (7) are assumed as o21 u1 and o22 u2 , respectively. Thus, Eqs. (6) and (7) can be simplified as Z Z s s l ðku1 1=2u21;3 1=2u22;3 Þ ds þ ½ðku1 ðx,tÞ1=2u01 ðx,tÞ2 1=2u02 ðx,tÞ2 Þ dx (9) u1 ðs,tÞ ¼  l 0 0 GJy003 þðGJ þ EI1 Þku002 EI1 ky3 M1 ðu1;33 þ kÞM2 u2;33 M3 ku1;3 rS1 o21 u1 rS2 o22 u2 ¼ 0

(10)

Additionally, according to the actual transmission line, S1 and S2 are very small compared with A. Thereby, rS1 y€ 3 in Eq. (4) and rS2 y€ 3 in Eq. (5) can be ignored. Besides, substituting Eq. (9) into Eqs. (4) and (5), leads to " Z #0 ) ( Z l u0 k l EA ðku1 1=2u21;3 1=2u22;3 Þ ds þ 1 ðku1 1=2u21;3 1=2u22;3 Þ ds (11) þTu001 rAu€ 1 þ ðb1 c1 u_ 1 Þ ¼ 0 l 0 l 0

EA

" Z #0 u02 l ðku1 1=2u21;3 1=2u22;3 Þ ds þTu002 rAu€ 2 þ ðb2 c2 u_ 2 Þ ¼ 0 l 0

(12)

From the above simplification, the equations of the motion can be expressed as Eq. (11) (normal direction), Eq. (12) (binormal direction) and Eq. (10) (torsional direction). Eqs. (11) and (12), which are the same as those in Refs. [11,16], show that the normal and bi-normal dynamics of the transmission line are governed by the classical equations of the perfectly flexible model. Ref. [11] and this paper consider effect of the asymmetric iced transmission line to aerodynamic forces. Besides this, S1 and S2 are also introduced into Eq. (10) to consider the eccentricity of cross area because of asymmetrically distributed ice in this paper. 2.2. Aerodynamic model Following Refs. [10,11], the aerodynamic forces can be obtained under the following assumptions: (1) the quasi-steady theory [17] is adopted; (2) the curvature of the transmission line is neglected; (3) loads are evaluated taking into account the twist angle, but neglecting the flexural rotations; (4) the ice is uniformly distributed along the transmission line; (5) the aerodynamic couples are neglected. Based on the these assumptions, the aerodynamic forces can be calculated as ba ¼ 1=2rair Vrðcd ½gV þ cl ½ga3  VÞ

(13)

where V denotes relative velocity of the wind with respect to section; ba represents aerodynamic force on the section; the attack angle g ¼ j0 jy þ ðu_ 1 =UÞcos½jðu_ 2 =UÞsin½j; the relative velocity V ¼ Uð1ðu_ 1 =UÞsin½jðu_ 2 =UÞcos½jÞ; the initial rotation angle j0 , the rotation angle j caused by the static aerodynamic force, and the aerodynamic coefficients cd ½g and cl ½g are reported in Appendix A. 2.3. Discrete model An asymptotic model for Eqs. (10)–(12) is discreted via the Galerkin method as below: u1 ½s,t ¼ f 1 ½sq1 ½t; u2 ½s,t ¼ f 2 ½sq2 ½t; y3 ½s,t ¼ f 3 ½sq3 ½t

(14)

where f 1 ½s, f 2 ½s and f 3 ½s are the normal, bi-normal and rotational cable model shapes, respectively; q1 ½t, q2 ½t and q3 ½t describe the normal, bi-normal and rotational amplitudes varying as the time t, respectively. Substituting Eq. (14) into Eq. (10), leads to q3 ¼ b1 q1 þ b2 q2

(15)

where the coefficients b1 and b2 are defined in Appendix B. Then, substituting Eqs. (14) and (15) into Eqs. (11) and (12), the discrete model can be gained as follows: q001 þ o21 q1 ðm5 q21 þ m15 q31 þ m4 q2 þ m6 q1 q2 þm16 q21 q2 þ m7 q22 þm17 q1 q22 þ m18 q32 þ m1 q01 þm8 q1 q01 þm19 q21 q01 þ m9 q2 q01 þm20 q1 q2 q01 þm21 q22 q01 þ m10 q01 2þ m22 q1 q01 2 þ m23 q2 q01 2 þm24 q01 3 þ m2 q02 þ m11 q1 q02 þ m25 q21 q02 þ m12 q2 q02 þ m26 q1 q2 q02 þ m27 q22 q02 þ m13 q01 q02 þ m28 q1 q01 q02 þ m29 q2 q01 q02 þ m30 q01 2q02 þ m14 q02 2 þ m31 q1 q02 2 þ m32 q2 q02 2 þm33 q01 q02 2þ m34 q02 3Þ ¼ 0

Z. Yan et al. / Journal of Sound and Vibration 331 (2012) 3599–3616

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q002 þ o22 q2 ðn3 q1 þ n5 q21 þn15 q31 þ n6 q1 q2 þ n16 q21 q2 þ n7 q22 þ n17 q1 q22 þ n18 q32 þn1 q01 þ n8 q1 q01 þn19 q21 q01 þn9 q2 q01 þ n20 q1 q2 q01 þ n21 q22 q01 þ n10 q01 2þ n22 q1 q01 2 þn23 q2 q01 2 þn24 q01 3 þ n2 q02 þ n11 q1 q02 þ n25 q21 q02 þ n12 q2 q02 þ n26 q1 q2 q02 þ n27 q22 q02 þ n13 q01 q02 þ n28 q1 q01 q02 þ n29 q2 q01 q02 þn30 q01 2q02 þn14 q02 2þ n31 q1 q02 2 þn32 q2 q02 2 þn33 q01 q02 2 þn34 q02 3Þ ¼ 0

(16)

where ‘‘ ’’ denotes the differentiation with respect to t; o1 and o2 are the normal and bi-normal circular frequencies of the iced transmission line; the coefficients mi and ni are reported in Appendix B. 0

3. Perturbation analysis Periodic solutions to Eq. (16) are found for weakly nonlinear response, thus the Multiple Scale Perturbation Method (MSM) can be used to calculate Eq. (18) [16]. A second order approximation is sought, so that new time variables are introduced, such as T n ¼ en t,

n ¼ 0; 1,2

(17)

and the solutions are expanded in power series of e: qi ðt, eÞ ¼

3 X

ej qi,j ðT 0 ,T 1 ,T 2 Þ;

i ¼ 1; 2

(18)

j¼1

where T0 is the fast time scale governing the dynamic phenomenon; T1 and T2 denote slow time scales on which the amplitudes and phases of qi are modulated by nonlinearity. Moreover, the coefficients (m1, m2, m4, n1, n2 and n3) of Eq. (16) are assumed to be small as the same order of e as follows: mi ¼ emmi ;

ni ¼ enni

(19)

where Oðmmi Þ ¼ Oðnni Þ ¼ 1,i ¼ 14. Substitute Eqs. (17)–(19) into Eq. (16) and collect terms with powers of e, as follows: e order: D20 q1;0 þ o21 q1;0 ¼ 0,

D20 q2;0 þ o22 q2;0 ¼ 0

(20)

2

e order: D20 q1;1 þ o21 q1;1 ¼ ðD0 D1 q1;0 Þ þ m5 q21;0 þmm4 q2;0 þ m6 q1;0 q2;0 þ m7 q22;0 þ mm1 D0 q1;0 þ m8 q1;0 D0 q1;0 þ m9 q2;0 D0 q1;0 þ m10 ðD0 q1;0 Þ2 þmm2 D0 q2;0 þm11 q1;0 D0 q2;0 þ m12 q2;0 D0 q2;0 þ m13 D0 q1;0 D0 q2;0 þm14 ðD0 q2;0 Þ2 D20 q2;1 þ o22 q2;1 ¼ ðD0 D1 q2;0 Þ þnn3 q1;0 þ n5 q21;0 þ n6 q1;0 q2;0 þ n7 q22;0 þnn1 D0 q1;0 þn8 q1;0 D0 q1;0 þ n9 q2;0 D0 q1;0 þn10 ðD0 q1;0 Þ2 þ nn2 D0 q2;0 þn11 q1;0 D0 q2;0 þ n12 q2;0 D0 q2;0 þn13 D0 q1;0 D0 q2;0 þ n14 ðD0 q2;0 Þ2

(21)

3

e order: D20 q1;2 þ o21 q1;2 ¼ ðD0 D1 q1;1 ÞD0 D2 q1;0 þ m15 q31;0 þ 2m5 q1;0 q1;1 þ m16 q21;0 q2;0 þ m6 q1;1 q2;0 þ m17 q1;0 q22;0 þ m18 q32;0 þ mm4 q2;1 þm6 q1;0 q2;1 þ 2m7 q2;0 q2;1 þ m19 q21;0 D0 q1;0 þ m8 q1;1 D0 q1;0 þm20 q1;0 q2;0 D0 q1;0 þ m21 q22;0 D0 q1;0 þ m9 q2;1 D0 q1;0 þ m22 q1;0 ðD0 q1;0 Þ2 þm23 q2;0 ðD0 q1;0 Þ2 þ m24 ðD0 q1;0 Þ3 þmm1 v1;1 þ m8 q1;0 D0 q1;1 þ m9 q2;0 D0 q1;1 þ2m10 D0 q1;0 D0 q1;1 þ m25 q21;0 D0 q2;0 þ m11 q1;1 D0 q2;0 þ m26 q1;0 q2;0 D0 q2;0 þ m27 q22;0 D0 q2;0 þ m12 q2;1 D0 q2;0 þ m28 q1;0 D0 q1;0 D0 q2;0 þm29 q2;0 D0 q1;0 D0 q2;0 þ m30 ðD0 q1;0 Þ2 D0 q2;0 þ m13 D0 q1;1 D0 q2;0 þ m31 q1;0 ðD0 q2;0 Þ2 þ m32 q2;0 ðD0 q2;0 Þ2 þm33 D0 q1;0 ðD0 q2;0 Þ2 þ m34 ðD0 q2;0 Þ3 þ mm2 D0 q2;1 þ m11 q1;0 D0 q2;1 þ m12 q2;0 D0 q2;1 þ m13 D0 q1;0 D0 q2;1 þ2m14 D0 q2;0 D0 q2;1 D20 q2;2 þ o22 q2;2 ¼ ðD0 D1 q2;1 ÞD0 D2 q2;0 þn15 q31;0 þ nn3 q1;1 þ2n5 q1;0 q1;1 þn16 q21;0 q2;0 þn6 q1;1 q2;0 þ n17 q1;0 q22;0 þ n18 q32;0 þ n6 q1;0 q2;1 þ 2n7 q2;0 q2;1 þn19 q21;0 D0 q1;0 þ n8 q1;1 D0 q1;0 þn20 q1;0 q2;0 D0 q1;0 þn21 q22;0 D0 q1;0 þ n9 q2;1 D0 q1;0 þ n22 q1;0 ðD0 q1;0 Þ2 þn23 q2;0 ðD0 q1;0 Þ2 þ n24 ðD0 q1;0 Þ3 þ nn1 D0 q1;1 þ n8 q1;0 D0 q1;1 þ n9 q2;0 D0 q1;1 þ 2n10 D0 q1;0 D0 q1;1 þn25 q21;0 D0 q2;0 þ n11 q1;1 D0 q2;0 þ n26 q1;0 q2;0 D0 q2;0 þ n27 q22;0 D0 q2;0 þ n12 q2;1 D0 q2;0 þn28 q1;0 D0 q1;0 D0 q2;0 þ n29 q2;0 D0 q1;0 D0 q2;0 þ n30 ðD0 q1;0 Þ2 D0 q2;0 þn13 D0 q1;1 D0 q2;0 þ n31 q1;0 ðD0 q2;0 Þ2 þ n32 q2;0 ðD0 q2;0 Þ2 þ n33 D0 q1;0 ðD0 q2;0 Þ2 þn34 ðD0 q2;0 Þ2 þ nn2 D0 q2;1 þ n11 q1;0 D0 q2;1 þn12 q2;0 D0 q2;1 þ n13 D0 q1;0 D0 q2;1 þ2n14 D0 q2;0 D0 q2;1

(22)

where Dn ¼ @=@T n ,n ¼ 0; 1,2. Eq. (20) admits the solution: qj,0 ¼ eiT 0 oj Aj ½T 1 ,T 2  þ c:c:;

j ¼ 1; 2 (23) pffiffiffiffiffiffiffi where i ¼ 1; A1 and A2 are unknown complex normal and bi-normal amplitudes depending on the slow time scales, respectively; c.c. stands for the complex conjugate.

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3.1. 1:1 Resonant case In this case, the ratio of the circular frequencies o1 and o2 is close to 1:1. Specifically, the difference o1 o2 is at the same order of e. Thus, it is assumed:

o2 ¼ o1 þ es1

(24)

According to Ref. [18], the solution (23) is substituted into systems (21) and (22) and the secular producing terms due to the resonance conditions stated by Eq. (24) are eliminated. Then, Amplitude Modulation Equations (AME) of 1:1 resonant case can be obtained as A_ 1 ¼ A1 ðpp1;1 þipp1;2 Þ þeies1 t A2 ðpp2;1 þ ipp2;2 Þ þ A21 A1 ðpp3;1 þ ipp3;2 Þ þ eies1 t A1 A2 A1 ðpp4;1 þipp4;2 Þ þe2ies1 t A22 A1 ðpp5;1 þ ipp5;2 Þ þ eies1 t A21 A2 ðpp6;1 þ ipp6;2 Þ þ A1 A2 A2 ðpp7;1 þipp7;2 Þ þ eies1 t A22 A2 ðpp8;1 þipp8;2 Þ A_ 2 ¼ eies1 t A1 ðqq1;1 þiqq1;2 Þ þ A2 ðqq2;1 þiqq2;2 Þ þ eies1 t A21 A1 ðqq3;1 þ iqq3;2 Þ þ A1 A2 A1 ðqq4;1 þiqq4;2 Þ þ eies1 t1 A22 A1 ðqq5;1 þ iqq5;2 Þ þe2ies1 t A21 A2 ðqq6;1 þ iqq6;2 Þ þ eies1 t A1 A2 A2 ðqq7;1 þiqq7;2 Þ þA22 A2 ðqq8;1 þ iqq8;2 Þ (25) pffiffiffiffiffiffiffi where i ¼ 1; A1 and A2 are unknown complex normal and bi-normal amplitudes; Aj denotes the complex conjugate of Aj ; the coefficients ppi,j and qqi,j are shown in Appendix C. Then, the amplitudes Aj are adopted by the polar form, namely Aj ¼ 12aj ½texp½ðiaj ½tÞ;

j ¼ 1; 2

(26)

where aj and aj are the amplitude and the phase varying as the time t. Moreover, in order to calculate Reduced Amplitude Modulation Equations (RAME) [18], it is assumed that

c1 ½t ¼ a2 ½ta1 ½t þ es1 t

(27)

Thus, substituting Eqs. (26) and (27) into AME (25), the RAME of 1:1 resonant case can be obtained: a_ 1 ¼ pp1;1 a1 þ 1=4pp3;1 a31 þ ðcos½c1 pp2;1 sin½c1 pp2;2 Þa2 þ 1=4ðcos½c1 pp4;1 sin½c1 pp4;2 þ cos½c1 pp6;1 þ sin½c1 pp6;2 Þa21 a2 þ 1=4ðcos½2c1 pp5;1 sin½2c1 pp5;2 þ pp7;1 Þa1 a22 þ1=4ðcos½c1 pp8;1 sin½c1 pp8;2 Þa32 a_ 2 ¼ ðcos½c1 qq1;1 þ sin½c1 qq1;2 Þa1 þ 1=4ðcos½c1 qq3;1 þ sin½c1 qq3;2 Þa31 þ qq2;1 a2 þ 1=4ðqq4;1 þ cos½2c1 qq6;1 þsin½2c1 qq6;2 Þa21 a2 þ1=4ðcos½c1 qq5;1 sin½c1 qq5;2 þcos½c1 qq7;1 þsin½c1 qq7;2 Þa1 a22 þ1=4qq8;1 a32 _ ¼ ðsin½c qq þcos½c qq Þa2 þ 1=4ðsin½c qq þcos½c qq Þa4 þ ðes pp þ qq Þa a þ1=4ðpp þ qq a1 a2 c 1 1;1 1;2 1 3;1 3;2 1 1;2 2;2 1 2 3;2 4;2 1 1 1 1 1 sin½2c1 qq6;1 þ cos½2c1 qq6;2 Þa31 a2 þ ðsin½c1 pp2;1 cos½c1 pp2;2 Þa22 þ 1=4ðsin½c1 pp4;1 cos½c1 pp4;2 þ sin½c1 pp6;1 cos½c1 pp6;2 þsin½c1 qq5;1 þ cos½c1 qq5;2 sin½c1 qq7;1 þ cos½c1 qq7;2 Þa21 a22 þ1=4ðsin½2c1 pp5;1 cos½2c1 pp5;2 pp7;2 þqq8;2 Þa1 a32 þ 1=4ðsin½c1 pp8;1 cos½c1 pp8;2 Þa42

(28)

3.2. 2:1 resonant case In this case the relationship of the circular frequencies can be expressed as

o2 ¼ 12o1 þ es2

(29)

Substitute the solution (23) into systems (21) and (22) and eliminate the secular producing terms due to the resonance conditions stated by Eq. (29). Then, AME of 2:1 resonant case can be gained as A_ 1 ¼ A1 ðpp9;1 þ ipp9;2 Þ þ e2ies2 t A22 ðpp10;1 þ ipp10;2 Þ þ A21 A1 ðpp11;1 þ ipp11;2 Þ þ A1 A2 A2 ðpp12;1 þ ipp12;2 Þ A_ 2 ¼ A2 ðqq9;1 þiqq9;2 Þ þe2ies2 t A1 A2 ðqq10;1 þiqq10;2 Þ þ A1 A2 A1 ðqq11;1 þ iqq11;2 Þ þ A22 A2 ðqq12;1 þiqq12;2 Þ

(30)

where the coefficients ppi,j and qqi,j are reported in Appendix C. Furthermore, it is assumed that

c2 ½t ¼ a1 ½t2ða2 ½t þ es2 tÞ

(31)

Then, substituting Eqs. (26) and (31) into Eq. (30), the RAME of 2:1 resonant case can be calculated as follows: a_ 1 ¼ pp9;1 a1 þ 1=4pp11;1 a31 þ 1=2ðcos½c2 pp10;1 þsin½c2 pp10;2 Þa22 þ 1=4pp12;1 a1 a22 a_ 2 ¼ qq9;1 a2 þ 1=2ðcos½c2 qq10;1 sin½c2 qq10;2 Þa1 a2 þ 1=4qq11;1 a21 a2 þ 1=4qq12;1 a32 2 3 _ ¼ ð2es þpp 2qq Þa a þ ðsin½c qq a1 a2 c 2 9;2 9;2 1 2 10;1 cos½c2 qq10;2 Þa1 a2 þ 1=4ðpp11;2 2qq11;2 Þa1 a2 þ 1=2ðsin½c2 pp10;1 2 2

þ cos½c2 pp10;2 Þa32 þ1=4ðpp12;2 2qq12;2 Þa1 a32

(32)

Z. Yan et al. / Journal of Sound and Vibration 331 (2012) 3599–3616

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4. Analyses of bifurcation and stability In the case of 1:1 resonance, the following branches of fixed points can be found from the analysis of RAME (28): Branch I, a1 ¼ a2 ¼ 0, 8c1 ; Branch II, a1 ¼ a1 ðUÞ,a2 ¼ a2 ðUÞ, c1 ¼ c1 ðUÞ. The stability of this case can be evaluated by the eigenvalues of the Jacobian matrix of RAME (28). In the case of 2:1 resonance, evaluating RAME (32), the branches of fixed points can be obtained as following: Branch I, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a1 ¼ a2 ¼ 0, 8c2 ; Branch II, a1 ¼ a1 ðUÞ,a2 ¼ a2 ðUÞ, c2 ¼ c2 ðUÞ; Branch III, a1 ¼ 4pp9;1 =pp11;1 ,a2 ¼ 0, 8c2 . The stability of Branch II can be evaluated by the eigenvalues of the Jacobian matrix of RAME (32) as that in the case of 1:1 resonance. However, the stability of Branch III cannot be assessed by the Jacobian matrix of RAME (32), since the polar form of RAME (32) becomes singular. Thus, a mixed polar-Cartesian form is adopted [19] to overcome this problem. It is assumed that A1 ¼ 1=2a1 ½texp½ðiz1 ½tÞ; A2 ¼ 1=2a1 ½tðuu2 ½t þivv2 ½tÞexp½ðiz2 ½tÞ

(33)

where z2 ½t ¼ 12z1 ½tes2 t. Then substituting Eq. (33) into Eq. (30), the mixed polar-Cartesian form can be calculated as: a_ 1 ¼ pp9;1 a1 þ1=4pp11;1 a31 þ1=2pp10;1 a21 uu22 þ 1=4pp12;1 a31 uu22 pp10;2 a21 uu2 vv2 1=2pp10;1 a21 vv22 þ 1=2pp12;1 a31 vv22 _ 2 ¼ ðpp9;1 þ qq9;1 Þuu2 þ 1=2qq10;1 a1 uu2 þ 1=4ðpp11;1 þqq11;1 Þa21 uu2 1=2pp10;1 a1 uu32 þ 1=4ðpp12;1 þ qq12;1 Þa21 uu32 uu þ 1=2ð2es2 þ pp9;2 2qq9;2 Þvv2 þ1=2qq10;2 a1 vv2 þ 1=8ðpp11;2 2qq11;2 Þa21 vv2 þ 5=4pp10;2 a1 uu22 vv2 þ1=8ðpp12;2 2qq12;2 Þa21 uu22 vv2 þ pp10;1 a1 uu2 vv22 þ 1=4ðpp12;1 þqq12;1 Þa21 uu2 vv22 1=4pp10;2 a1 vv32 þ 1=8ðpp12;2 2qq12;2 Þa21 vv32 _ 2 ¼ 1=2ð2es2 pp9;2 þ 2qq9;2 Þuu2 þ1=2qq10;2 a1 uu2 þ 1=8ðpp11;2 þ 2qq11;2 Þa21 uu2 vv 1=4pp10;2 a1 uu32 þ1=8ðpp12;2 þ 2qq12;2 Þaa21 uu32 þ ðpp9;1 þ qq9;1 Þvv2 1=2qq10;1 a1 vv2 þ1=4ðpp11;1 þ qq11;1 Þaa21 vv2 pp10;1 a1 uu22 vv2 þ1=4ðpp12;1 þqq12;1 Þa21 uu22 vv2 þ 5=4pp10;2 a1 uu2 vv22 þ1=8ðpp12;2 þ 2qq12;2 Þa21 uu2 vv22 þ 1=2pp10;1 a1 vv32 þ1=4ðpp12;1 þ qq12;1 Þa21 vv32

(34)

In this condition, the stability of Branch III can be evaluated by the Jacobian matrix of mixed polar-Cartesian form (34). 5. Results and discussion 5.1. Parameters In order to analyze the above-mentioned model of iced transmission line, a U shaped iced cross section shown in Fig. 2 is chosen. The model type of this transmission line is 4XLGJ400=50, thus the parameters are calculated as the following: axial stiffness EA ¼ 31:3  106 N, torsional stiffness GJ ¼ 393 N m2 , bending stiffness EI ¼ 1965 Nm2 , mass per unit length (including ice) rA ¼ 1:82 kg=m, the initial eccentricity of the cross section e0y ¼ 0:00326 m and the initial rotation j0 ¼ 0:395. Moreover, according to Ref. [9], the damping ratio coefficients x1 and x2 are assumed to be 0.45 percent. In the case of 1:1 resonance, it is assumed that the distance between horizontal supports L and sag at the mid-span d, shown in Fig. 1, are 250 m and 1.3 m, respectively. Thus, the circular frequencies of normal displacement ðo1 Þ and bi-normal displacement ðo2 Þ are calculated as 3.11 rad/s and 3.05 rad/s at the configuration C20 , respectively. In the case of 2:1 resonance, L and d are assumed to be 250 m and 5.6 m, then o1 and o2 are 2.96 rad/s and 1.47 rad/s, respectively. Furthermore, the scale parameter e is assumed as 0.1 in these two cases. With regard to aerodynamic properties, the aerodynamic coefficients of this kind of cross section are obtained from the experiments recently completed in Chinese Aerodynamic Research and Development Center. In this paper, g ¼ 0 is assumed at the configure C20 , shown in Fig. 2. Thus, the drag and the lift coefficients are fitted, shown in Fig. 2, as follows: cd ½g ¼ 4:5712g3 þ 1:3518g2 1:7591g þ 0:9874 cl ½g ¼ 8:483g3 þ 3:3187g2 1:7491g0:3046

(35)

where the range of g is [ 0.215, 0.395]; g is assumed to be j. 5.2. Calculations of bifurcation and stability _ ¼ 0 into Eq. (28), a and a varying as the wind speed U can In the 1:1 resonant case, substituting a_ 1 ¼ 0, a_ 2 ¼ 0 and c 1 2 1 be obtained under the assumption of e0y ¼ 0 (without considering the eccentricity of cross section), shown in Fig. 3. When U o 3:4 m=s, it is at the Branch I, namely, a1 ¼ a2 ¼ 0. When U Z 3:4 m=s, it is at the Branch II. Moreover, as the wind speed increases, the values of a1 and a2 increase, too. Furthermore, according to Jacobian matrix of RAME (28), it can be found that Branch II is stable.

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Fig. 2. U-shaped conductor: (a) sectional dimension and initial aerodynamic shape C20 ðg ¼ 0Þ; (b) drag and lift aerodynamic coefficients.

Fig. 3. Bifurcation and stability of 1:1 resonant case: (a) without considering the effect of eccentricity; (b) considering the effect of eccentricity.

Fig. 4. Bifurcation and stability of 2:1 resonant case: (a) without considering the effect of eccentricity; (b) considering the effect of eccentricity.

When the eccentricity of ice is considered, in other words, e0y ¼ 0:00326 m, a1 ½U and a2 ½U are calculated, shown in Fig. 3. When U o3:3 m=s, a1 ¼ a2 ¼ 0. When U Z 3:3 m=s, it enters into Branch II. As the wind speed increases, the values of a1 and a2 first increase, then decline. Moreover, according to Jacobian matrix of RAME (28), it can be found that Branch II is stable except the wind speed range [19.9 m/s, 21.7 m/s]. Besides, when U Z28:1 m=s, a1 ¼ a2 ¼ 0. _ ¼ 0 into Eq. (32), a ½U and a ½U are calculated under the In the case of 2:1 resonance, substituting a_ 1 ¼ 0, a_ 2 ¼ 0 and c 1 2 2 assumption of e0y ¼ 0 (without considering the eccentricity of ice), shown in Fig. 4. When U o3:7 m=s, a1 ¼ a2 ¼ 0. When U Z 3:7 m=s, it gets into the Branch III, namely, a1 4 0,a2 ¼ 0. Moreover, as the wind speed increases, the value of a1 increases, too. Furthermore, according to Jacobian matrix of RAME (34), the stable wind speed range of Branch III is [3.7 m/s, 5.5 m/s]. When U Z 4:8 m=s, it goes into the Branch II. Moreover, as the wind speed increases, the values of a1 and a2 increase, too. Furthermore, according to Jacobian matrix of RAME (32), the stable wind speed range of Branch II is [4.8 m/s, 10.3 m/s].

Z. Yan et al. / Journal of Sound and Vibration 331 (2012) 3599–3616

3607

Fig. 5. Multiple stabilities: (a) bifurcation and stability; (b) RAME (32) numerically integrated in time history (U ¼5.1 m/s).

Fig. 6. Reduced form (16) numerically integrated in time history (U ¼5.1 m/s): (a) nominal vibration; (b) bi-nominal vibration.

When the eccentricity of ice is considered, a1 ½U and a2 ½U are obtained, shown in Fig. 4. When U o 3:7 m=s, a1 ¼ a2 ¼ 0. When U Z 3:7 m=s, it enters into Branch III. As the wind speed increases, the value of a1 first increases, then declines. Moreover, according to Jacobian matrix of RAME (34), Branch III are found to be stable. Besides, when U Z28:3 m=s, a1 ¼ a2 ¼ 0. From the above analyses, it can be easily found that the effect of eccentricity of ice is very obvious in both 1:1 resonant and 2:1 resonant cases. When the eccentricity of ice is not considered, the values of a1 and a2 in Branch II and Branch III increase as the wind speed increase. However, with the consideration of the eccentricity, the values of a1 and a2 in Branch II and Branch III first increase then decline as the wind speed increases. 5.3. Multiple stabilities In the 2:1 resonant case without considering the eccentricity of ice, both Branch II and Branch III are found stable in the wind speed range of [4.8 m, 5.5 m], shown in Fig. 5. In order to analyze this special phenomenon, RAME (32) is calculated by the numerical integration (Runge–Kutta Method with the initial values a1 ð0Þ ¼ 0:1,a2 ð0Þ ¼ 0:1, c2 ð0Þ ¼ 0:4) in time history (U ¼5.1 m/s), shown in Fig. 5. From Fig. 5, it can be found: a1 ¼ 3:59 m, a2 ¼ 0 (Branch III) at the time range of [1000 s, 5000 s]; a1 ¼ 3:21 m, a2 ¼ 0:64 m (Branch II) at the time t 47000 s. Furthermore, calculating Eq. (16) with numerical integration (Runge–Kutta Method with the initial values q1 ð0Þ ¼ 0:05,q2 ð0Þ ¼ 0:05,q01 ð0Þ ¼ 0:01,q02 ð0Þ ¼ 0:01) in time history, u1 and u2 varying as the time t are obtained, shown in Fig. 6. It can also be found: the value of u1 is 0.36 m at the time range of [500 s, 2500 s]; the values of u1 and u2 are 0.32 m and 0.065 m at the time t 4 3000 s, respectively. From the numerical integration of Eqs. (32) and (16), it can be found that both Branch II and Branch III exist. Moreover, as the time t increases, Branch III first appears, then Branch II emerges. 6. Conclusion In this paper, one nonlinear galloping model of iced transmission line which contains four motions (normal, bi-normal, tangential and torsional directions) has been formulated, with regard of bending stiffness of the transmission line and eccentricity of ice. According to the property of the transmission line and Galerkin method, a simplified model which is expressed as two-degree-of-freedom (normal and bi-normal) equations with consideration of the effects of torsion,

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bending and eccentricity has been obtained. Moreover, the initial rotation angle which is caused by the eccentricity of ice is introduced into the galloping and aerodynamic models. Additionally, a mixed polar-Cartesian form of RAME is chosen to evaluate the stability of bifurcation when the polar form of RAME is not suitable. The 1:1 resonant and 2:1 resonant cases are analyzed with the reduced model. Moreover, the aerodynamic coefficients are fitted as cubic equations according to the experimental data. The results turn out that the eccentricity of ice has great effect on the galloping of iced transmission line: with the consideration of eccentricity, the values of a1 and a2 in Branch II and Branch III first increase then decline as the wind speed increases; while without considering eccentricity, the values of a1 and a2 in Branch II and Branch III increase as the wind speed increase. Specially, in the 2:1 resonant case without considering eccentricity, it is found that multiple stabilities exist. Through the analysis of numerical integration of RAME and reduced model in time history, it is proved that this phenomenon actually appears as the wind speed is in the range of multiple stabilities, which is obtained by anlyses of bifurcation and stability of RAME.

Acknowledgement This work was supported by the Natural Science Funding (China, 51178489) and the Fundamental Research Funds for the Central Universities (No. CDJZR11200015). Appendix A. The initial rotation angle u0 , the rotation angle j under static aerodynamic forces, and aerodynamic coefficients c d ½g and c l ½g In order to obtain wind force from the experimental data, the initial ice shape is assumed to be symmetrical to (Ya120 ) axis in the shape C10 , shown in Fig. 1. Due to eccentricity of ice, the actual shape C20 of the iced transmission line only under the action of gravity (including the ice) rotates the angle j0 and becomes to be symmetrical to a220 axis, shown in Fig. 1. The initial rotation angle j0 can be calculated as follows: R L=2 RL ðsL=2Þðe1x þ e1y ÞrAg ds f ½s ds  L j0 ¼ 0 3 (A.1) L GJ where e1x ¼ e0y sin½j0 ; e1y ¼ e0y cos½j0 ; e0y is the initial eccentricity along Y-axis; g is the acceleration of gravity. Furthermore, the reference configuration C , which is loaded by the static aerodynamic forces, turns the angle j from the shape C20 . The angle j can be obtained as follows: RL RL f ½s ds e0 L=2 h0 ðsL=2Þ ds  (A.2)  j¼ 0 3 GJ rAg L where the static rotational aerodynamic moment h0 ¼ 12 r 2 U 2 rair aa3;0 ½c; the mean bi-normal aerodynamic force e0 ¼ 12 rU 2 rair ðaa2;0 cos½c þ aa1;0 sin½cÞ; rair is the density of the air. Finally, according to Ref. [10], the aerodynamic coefficients cd ½g and cl ½g can be extended as cl ½g ¼ aa1;0 þ aa1;1 w þ1=2aa1;2 w2 þ 1=6aa1;3 w3 cd ½g ¼ aa2;0 þ aa2;1 w þ 1=2aa2;2 w2 þ 1=6aa2;3 w3 0

(A.3) 00

000

where w ¼ y þ ðu_ 1 =UÞcos ½jðu_ 2 =UÞsin ½j; aa1;0 ¼ cl ½g; aa1;1 ¼ ðcl ½gÞ ; aa1;2 ¼ ðcl ½gÞ ; aa1;3 ¼ ðcl ½gÞ ; aa2;0 ¼ cd ½g; aa2;1 ¼ ðcd ½gÞ0 ; aa2;2 ¼ ðcd ½gÞ00 ; aa2;3 ¼ ðcd ½gÞ000 ; ‘‘’’’ denotes differentiation with respect to g; cd ½g and cl ½g are the aerodynamic coefficients at the reference configuration C . In order to fit cd ½g and cl ½g, g ¼ 0 is assumed from the experimental data at j0 . Thus, cd ½g ¼ cd ½j and cl ½g ¼ cl ½j. Appendix B. Coefficients of Galerkin discredited equations The coefficients of Eqs. (15) and (16) are RL ð rS f dsÞo21 þ kM 3 ðf 1 ½0 þ f 1 ½LÞ þ M1 ððf 1 Þ0 ½0 þ ðf 1 Þ0 ½L þ kLÞ b1 ¼  0 1 1 RL 00 2 0 ðEI1 k f 3 ½sGJf 3 Þ ds

b2 ¼

m1 ¼ t1

Z 0

L

RL ð 0 rS2 f 2 dsÞo22 þ ðEI1 k þGJ kM 2 Þððf 2 Þ0 ½0ðf 2 Þ0 ½LÞ RL 00 2 0 ðEI1 k f 3 GJf 3 Þ ds

 ð2rAx1 o1 f 1 þ kðcos½jsin½jaa1;0 f 1 aa2;0 f 1 sin½j2 aa2;0 f 1 cos½j2 f 1 aa1;1 þ cos½jsin½jf 1 aa2;1 ÞÞ ds

m2 ¼ t1

Z 0

L

 kðaa1;0 f 2 þ cos½j2 aa1;0 f 2 cos½jsin½jaa2;0 f 2 þ cos½jsin½jf 2 aa1;1 sin½j2 f 2 aa2;1 Þ ds

Z. Yan et al. / Journal of Sound and Vibration 331 (2012) 3599–3616

Z

m3 ¼ t1

L

b1 kUðcos½jf 3 aa1;1 sin½jf 3 aa2;1 Þ ds

0

Z

m4 ¼ t1

L

0

m5 ¼ t1

Z

L 0

@1 kb2 Uðcos½jf 2 aa1;2 þ sin½jf 2 aa2;2 Þ 3 3 2 1 Z

L

m7 ¼ t1

Z

L

EAk

Z

L

Z

L

R  1 L ds þ 2EAk 0 f 1 ds f 01 Ads 2L

2

2 0 ðf 2 Þ

2L

ds

þ

! 1 2 2 2 kb2 Uðcos½jf 3 aa1;2 þsin½jf 3 aa2;2 Þ ds 2

b2 kðcos½jsin½jf 1 f 3 aa1;1 þcos½j2 f 1 f 3 aa1;2 þ f 1 f 3 aa2;1 þsin½j2 f 1 f 3 aa2;1 cos½jsin½jf 1 f 3 aa2;2 Þ ds

0

Z

m10 ¼ t1

0 2 0 ðf 1 Þ

kb1 ðcos½jsin½jf 1 f 3 aa1;1 þcos½j2 f 1 f 3 aa1;2 þ f 1 f 3 aa2;1 þsin½j2 f 1 f 3 aa2;1 cos½jsin½jf 1 f 3 aa2;2 Þ ds

0

m9 ¼ t1

RL

2

RL

0

EAk

kb1 b2 Uðcos½jf 3 aa1;2 þ sin½jf 3 aa2;2 Þ ds

0

m8 ¼ t1

kb2 Uðcos½jf 3 aa1;1 sin½jf 3 aa2;1 Þ ds

0

m6 ¼ t1

3609

L

0

k 2 2 2 2 2 ð2 sin½ja2;0 f 1 þ2 cos½j2 sin½jf 1 aa1;1 cos½j3 f 1 aa1;2 2 cos½jf 1 aa2;1 2 cos½jsin½j2 f 1 aa2;1 2U 2

þ cos½j2 sin½jf 1 aa2;2 Þ ds m11 ¼ t1

Z

L

0

m12 ¼ t1

Z

L

0

kb1 ðf 2 f 3 aa1;1 cos½j2 f 2 f 3 aa1;1 cos½jsin½jf 2 f 3 aa1;2 þcos½jsin½jf 2 f 3 aa2;1 þ sin½j2 f 2 f 3 aa2;2 Þ ds kb2 ðf 2 f 3 aa1;1 cos½j2 f 2 f 3 aa1;1 cos½jsin½jf 2 f 3 aa1;2 þcos½jsin½jf 2 f 3 aa2;1 þ sin½j2 f 2 f 3 aa2;2 Þ ds

m13 ¼ t1

Z

L

0

k sin½jaa1;0 f 1 f 2 þ cos½jaa2;0 f 1 f 2 þcos½jf 1 f 2 aa1;1 þcos½j3 f 1 f 2 aa1;1 U

cos½jsin½j2 f 1 f 2 aa1;1 þ cos½j2 sin½jf 1 f 2 aa1;2 þ sin½jf 1 f 2 aa2;1 cos½j2 sin½jf 1 f 2 aa2;1  þ sin½j3 f 1 f 2 aa2;1 cos½jsin½j2 f 1 f 2 aa2;2 ds m14 ¼ t1

Z

L 0

k 2 2 2 ð2 cos½jaa1;0 f 2 2 sin½jf 2 aa1;1 2 cos½j2 sin½jf 2 aa1;1 ½j 2U 2

2

2

cos½jsin½j2 f 2 aa1;2 þ 2 cos½jsin½j2 f 2 aa2;1 þ sin½j3 f 2 aa2;2 Þ ds m15 ¼ t1

Z

L 0

EAð 1 3 3 3 kb Uðcos½jf 3 aa1;3 sin½jf 3 aa2;3 Þ þ 6 1

m16 ¼ t1

Z

L 0

m17 ¼ t1

Z 0

L

Z 0

Z

L

0

m20 ¼ t1

Z

L 0

m21 ¼ t1

Z 0

L

0 2 0 ðf 1 Þ

2L

dsÞf 001

! ds

1 3 3 2 kU b1 b2 ðcos½jf 3 aa1;3 sin½jf 3 aa2;3 Þ ds 2

! RL EAð 0 ðf 02 Þ2 dsÞf 001 1 2 3 3 kU b1 b2 ðcos½jf 3 aa1;3 sin½jf 3 aa2;3 Þ þ ds 2 2L

m18 ¼ t1

m19 ¼ t1

RL

L

1 3 3 3 kU b2 ðcos½jf 3 aa1;3 sin½jf 3 aa2;3 Þ ds 6

1 2 2 2 2 2 2 kb ðcos½jsin½jf 1 f 3 aa1;2 cos½j2 f 1 f 3 aa1;3 f 1 f 3 aa2;2 sin½j2 f 1 f 3 aa2;2 þ cos½jsin½jf 1 f 3 aa2;3 Þ ds 2 1 2

2

2

2

2

kb1 b2 ðcos½jsin½jf 1 f 3 aa1;2 cos½j2 f 1 f 3 aa1;3 f 1 f 3 aa2;2 sin½j2 f 1 f 3 aa2;2 þ cos½jsin½jf 1 f 3 aa2;3 Þ ds 1 2 2 2 2 2 2 kb ðcos½jsin½jf 1 f 3 aa1;2 cos½j2 f 1 f 3 aa1;3 f 1 f 3 aa2;2 sin½j2 f 1 f 3 aa2;2 þ cos½jsin½jf 1 f 3 aa2;3 Þ ds 2 2

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Z. Yan et al. / Journal of Sound and Vibration 331 (2012) 3599–3616

m22 ¼ t1

Z

L

k b ð2 cos½j2 sin½jf 21 f 3 aa1;2 þ cos½j3 f 21 f 3 aa1;3 2 sin½jf 21 f 3 aa2;1 2U 1

0

2

2

2

þ 2 cos½jf 1 f 3 aa2;2 þ 2 cos½jsin½j2 f 1 f 3 aa2;2 cos½j2 sin½jf 1 f 3 aa2;3 Þ ds m23 ¼ t1

Z

L

k b ð2 cos½j2 sin½jf 21 f 3 aa1;2 þ cos½j3 f 21 f 3 aa1;3 2 sin½jf 21 f 3 aa2;1 2U 2

0

2

2

2

þ 2 cos½jf 1 f 3 aa2;2 þ 2 cos½jsin½j2 f 1 f 3 aa2;2 ½jcos½j2 sin½jf 1 f 3 aa2;3 Þ ds m24 ¼ t1

Z

L

k

3

6U 2

0

3

3

3

ð3 cos½j3 sin½jf 1 aa1;2 cos½j4 f 1 aa1;3 þ 6 cos½jsin½jf 1 aa2;1 3 cos½j2 f 1 aa2;2 3

3

3 cos½j2 sin½j2 f 1 aa2;2 þ cos½j3 sin½jf 1 aa2;3 Þ ds m25 ¼ t1

Z

L 0

m26 ¼ t1

Z

L

0

m27 ¼ t1

1 2 2 2 2 2 2 kb ðf f aa1;2 þ cos½j2 f 2 f 3 aa1;2 þ cos½jsin½jf 2 f 3 aa1;3 cos½jsin½jf 2 f 3 aa2;2 sin½j2 f 2 f 3 aa2;3 Þ ds 2 1 2 3 2

2

2

2

2

kb1 b2 ðf 2 f 3 aa1;2 þ cos½j2 f 2 f 3 aa1;2 ½j þ cos½jsin½jf 2 f 3 aa1;3 cos½jsin½jf 2 f 3 aa2;2 sin½j2 f 2 f 3 aa2;3 Þ ds

Z

L 0

1 2 2 2 2 2 2 kb ðf f aa1;2 þ cos½j2 f 2 f 3 aa1;2 þ cos½jsin½jf 2 f 3 aa1;3 cos½jsin½jf 2 f 3 aa2;2 sin½j2 f 2 f 3 aa2;3 Þ ds 2 2 2 3 Z

m28 ¼ t1

L

0 2

k b ðsin½jf 1 f 2 f 3 aa1;1 cos½jf 1 f 2 f 3 aa1;2 cos½j3 f 1 f 2 f 3 aa1;2 ½j U 1

þ cos½jsin½j f 1 f 2 f 3 a1;2 cos½j2 sin½jf 1 f 2 f 3 aa1;3 cos½jf 1 f 2 f 3 aa2;1 sin½jf 1 f 2 f 3 aa2;2 þ cos½j2 sin½jf 1 f 2 f 3 aa2;2 sin½j3 f 1 f 2 f 3 aa2;2 þcos½jsin½j2 f 1 f 2 f 3 aa2;3 Þ ds m29 ¼ t1

Z 0

L

k b ðsin½jf 1 f 2 f 3 aa1;1 cos½jf 1 f 2 f 3 aa1;2 U 2

cos½j3 f 1 f 2 f 3 aa1;2 þcos½jsin½j2 f 1 f 2 f 3 aa1;2 cos½j2 sin½jf 1 f 2 f 3 aa1;3 cos½jf 1 f 2 f 3 aa2;1 sin½jf 1 f 2 f 3 aa2;2 þ cos½j2 sin½jf 1 f 2 f 3 aa2;2 sin½j3 f 1 f 2 f 3 aa2;2 þ cos½jsin½j2 f 1 f 2 f 3 aa2;3 ½jÞ ds Z

m30 ¼ t1

L 0

k 2 2 2 ð2 cos½jsin½jf 1 f 2 aa1;1 þ cos½j2 f 1 f 2 aa1;2 þcos½j4 f 1 f 2 aa1;2 2Uz

2

2

2

2

2

2 cos½j2 sin½j2 f 1 f 2 aa1;2 þ cos½j3 sin½jf 1 f 2 aa1;3 þ 2 cos½j2 f 1 f 2 aa2;1 2 sin½j2 f 1 f 2 aa2;1 þ 2 cos½jsin½jf 1 f 2 aa2;2 cos½j3 sin½j

2 f 1 f 2 aa2;2 ½ 2

j þ 2 cos½jsin½j

2 3 f 1 f 2 aa2;2

2

cos½j sin½j2 f 1 f 2 aa2;3 Þ ds m31 ¼ t1

Z

L 0

k b ð2 cos½jf 22 f 3 aa1;1 þ 2 sin½jf 22 f 3 aa1;2 þ 2 cos½j2 sin½jf 22 f 3 aa1;2 2U 1 2

2

2

þ cos½jsin½j2 f 2 f 3 aa1;3 2 cos½jsin½j2 f 2 f 3 aa2;2 sin½j3 f 2 f 3 aa2;3 Þ ds m32 ¼ t1

Z

L 0

k b ð2 cos½jf 22 f 3 aa1;1 þ 2 sin½jf 22 f 3 aa1;2 þ 2 cos½j2 sin½jf 22 f 3 aa1;2 2U 2 2

2

2

þ cos½jsin½j2 f 2 f 3 aa1;3 2 cos½jsin½j2 f 2 f 3 aa2;2 sin½j3 f 2 f 3 aa2;3 Þ ds m33 ¼ t1

Z

L

k 2U 2

0

2

2

2

ð2 cos½j2 f 1 f 2 aa1;1 þ 2 sin½j2 f 1 f 2 aa1;1 2 cos½jsin½jf 1 f 2 aa1;2 2

2

2

2 cos½j3 sin½jf 1 f 2 aa1;2 þ cos½jsin½j3 f 1 f 2 aa1;2 cos½j2 sin½j2 f 1 f 2 aa1;3 2

2

2

2

2

2 cos½jsin½jf 1 f 2 aa2;1 sin½j2 f 1 f 2 aa2;2 þ 2 cos½j2 sin½j2 f 1 f 2 aa2;2 sin½j4 f 1 f 2 aa2;2 þcos½jsin½j3 f 1 f 2 aa2;3 Þ ds m34 ¼ t1

Z

L 0

k 6U 2

3

3

3

ð6 cos½jsin½jf 2 aa1;1 þ 3 sin½j2 f 2 aa1;2 þ 3 cos½j2 sin½j2 f 2 aa1;2 3

3

3

þ cos½jsin½j3 f 2 aa1;3 3 cos½jsin½j3 f 2 aa2;2 sin½j4 f 2 aa2;3 Þ ds n1 ¼ t2

Z 0

L

kðaa1;0 f 1 sin½j2 aa1;0 f 1 cos½jsin½jaa2;0 f 1 þ cos½jsin½jf 1 aa1;1 þcos½j2 f 1 aa2;1 Þ ds

Z. Yan et al. / Journal of Sound and Vibration 331 (2012) 3599–3616

n2 ¼ t2

Z

L

0

ð2rAx2 o2 f 2 þkðcos½jsin½jaa1;0 f 2 aa2;0 f 2 cos½j2 aa2;0 f 2 sin½j2 f 2 aa1;1 cos½jsin½jf 2 aa2;1 ÞÞ ds Z

n3 ¼ t2

L

kU b1 ðsin½jf 3 aa1;1 cos½jf 3 aa2;1 Þ ds

0

Z

n4 ¼ t2

L 0

Z

n5 ¼ t2

L 0

n6 ¼ t2

Z

L 0

L

0

n9 ¼ t2

Z

L

0

1 2 2 2 kU b1 ðsin½jf 3 aa1;2 þ cos½jf 3 aa2;2 Þ ds 2 2

Z

L 0

Z

kU b2 ðsin½jf 3 aa1;1 cos½jf 3 aa2;1 Þ ds

2

kU b1 b2 ðsin½jf 3 aa1;2 þ cos½jf 3 aa2;2 Þ

n7 ¼ t2

n8 ¼ t2

3611

EAkð

RL 0

f 1 dsÞf 002 L

! ds

1 2 2 2 kU b2 ðsin½jf 3 aa1;2 þ cos½jf 3 aa2;2 Þ ds 2

b1 kðf 1 f 3 aa1;1 þ sin½j2 f 1 f 3 aa1;1 cos½jsin½jf 1 f 3 aa1;2 þ cos½jsin½jf 1 f 3 aa2;1 cos½j2 f 1 f 3 aa2;2 Þ ds kb2 ðf 1 f 3 aa1;1 þ sin½j2 f 1 f 3 aa1;1 cos½jsin½jf 1 f 3 aa1;2 þ cos½jsin½jf 1 f 3 aa2;1 cos½j2 f 1 f 3 aa2;2 Þ ds

n10 ¼ t2

Z

L 0

k 2 2 2 2 ð2 sin½jaa1;0 f 1 2 cos½jf 1 aa1;1 2 cos½jsin½j2 f 1 aa1;1 þ cos½j2 sin½jf 1 aa1;2 2U 2

2

2 cos½j2 sin½jf 1 aa2;1 þcos½j3 f 1 aa2;2 Þ ds n11 ¼ t2

Z

L 0

n12 ¼ t2

Z

L 0

n13 ¼ t2

kb1 ðcos½jsin½jf 2 f 3 aa1;1 þsin½j2 f 2 f 3 aa1;2 þ f 2 f 3 aa2;1 þcos½j2 f 2 f 3 aa2;1 þ cos½jsin½jf 2 f 3 aa2;2 Þ ds kb2 ðcos½jsin½jf 2 f 3 aa1;1 þsin½j2 f 2 f 3 aa1;2 þ f 2 f 3 aa2;1 þcos½j2 f 2 f 3 aa2;1 þ cos½jsin½jf 2 f 3 aa2;2 Þ ds

Z 0

L

k ðcos½jaa1;0 f 1 f 2 þ sin½jaa2;0 f 1 f 2 þ sin½jf 1 f 2 aa1;1 cos½j2 sin½jf 1 f 2 aa1;1 þ sin½j3 f 1 f 2 aa1;1 U

cos½jsin½j2 f 1 f 2 aa1;2 cos½jf 1 f 2 aa2;1 cos½j3 f 1 f 2 aa2;1 þ cos½jsin½j2 f 1 f 2 aa2;1 cos½j2 sin½jf 1 f 2 aa2;2 Þ ds n14 ¼ t2

Z

L 0

k 2 2 2 2 ð2 cos½jaa2;0 f 2 þ 2 cos½jsin½j2 f 2 aa1;1 þsin½j3 f 2 aa1;2 þ2 sin½jf 2 aa2;1 2U 2

2

þ cos½j2 sin½jf 2 aa2;1 þ cos½jsin½j2 f 2 aa2;2 Þ ds n15 ¼ t2

Z

L 0

n16 ¼ t2

Z

L 0

! RL EAð 0 ðf 01 Þ2 dsÞf 002 1 3 3 2 kU b1 b2 ðsin½jf 3 aa1;3 cos½jf 3 aa2;3 Þ þ ds 2 2L

n17 ¼ t2

Z

L

1 3 3 2 kU b1 b2 ðsin½jf 3 aa1;3 cos½jf 3 aa2;3 Þ ds 2

L

1 2 3 3 kU b1 b2 ðsin½jf 3 aa1;3 cos½jf 3 aa2;3 Þ ds 2

0

n17 ¼ t2

Z 0

n19 ¼ t2

Z

L 0

n20 ¼ t2

Z

L 0

n21 ¼ t2

Z 0

1 3 3 3 kU b1 ðsin½jf 3 aa1;3 cos½jf 3 aa2;3 Þ ds 6

1 2 2 2 2 2 2 kb ðf 1 f 3 aa1;2 sin½j2 f 1 f 3 aa1;2 þcos½jsin½jf 1 f 3 aa1;3 cos½jsin½jf 1 f 3 aa2;2 þ cos½j2 f 1 f 3 aa2;3 Þ ds 2 1 2

2

2

2

2

kb1 b2 ðf 1 f 3 aa1;2 sin½j2 f 1 f 3 aa1;2 þ cos½jsin½jf 1 f 3 aa1;3 cos½jsin½jf 1 f 3 aa2;2 þ cos½j2 f 1 f 3 aa2;3 Þ ds L

k 2 b ðf 1 f 23 aa1;2 sin½j2 f 1 f 23 aa1;2 þ cos½jsin½jf 1 f 23 aa1;3 cos½jsin½jf 1 f 23 aa2;2 þcos½j2 f 1 f 23 aa2;3 Þ ds 2 2

3612

Z. Yan et al. / Journal of Sound and Vibration 331 (2012) 3599–3616

n22 ¼ t2

Z

L

k b ð2 sin½jf 21 f 3 aa1;1 þ 2 cos½jf 21 f 3 aa1;2 þ 2 cos½jsin½j2 f 21 f 3 aa1;2 2U 1

0

2

2

2

cos½j2 sin½jf 1 f 3 aa1;3 þ 2 cos½j2 sin½jf 1 f 3 aa2;2 cos½j3 f 1 f 3 aa2;3 Þ ds n23 ¼ t2

Z

L

k b ð2 sin½jf 21 f 3 aa1;1 þ 2 cos½jf 21 f 3 aa1;2 þ 2 cos½jsin½j2 f 21 f 3 aa1;2 2U 2

0

2

2

2

cos½j2 sin½jf 1 f 3 aa1;3 þ 2 cos½j2 sin½jf 1 f 3 aa2;2 cos½j3 f 1 f 3 aa2;3 Þ ds n24 ¼ t2

Z

L

k

3

6U 2

0

3

3

ð6 cos½jsin½jf 1 aa1;1 3 cos½j2 f 1 aa1;2 3r cos½j2 sin½j2 f 1 aa1;2 3

3

3

þcos½j3 sin½jf 1 aa1;3 3 cos½j3 sin½jf 1 aa2;2 þ cos½j4 f 1 aa2;3 Þ ds n25 ¼ t2

Z

L 0

n26 ¼ t2

Z

L

0

n27 ¼ t2

Z

L 0

1 2 2 2 2 2 2 kb ðcos½jsin½jf 2 f 3 aa1;2 sin½j2 f 2 f 3 aa1;3 f 2 f 3 aa2;2 cos½j2 f 2 f 3 aa2;2 cos½jsin½jf 2 f 3 aa2;3 Þ ds 2 1 2

2

2

2

2

kb1 b2 ðcos½jsin½jf 2 f 3 aa1;2 sin½j2 f 2 f 3 aa1;3 f 2 f 3 aa2;2 cos½j2 f 2 f 3 aa2;2 cos½jsin½jf 2 f 3 aa2;3 Þ ds 1 2 2 2 2 2 2 kb ðcos½jsin½jf 2 f 3 aa1;2 sin½j2 f 2 f 3 aa1;3 f 2 f 3 aa2;2 cos½j2 f 2 f 3 aa2;2 cos½jsin½jf 2 f 3 aa2;3 Þ ds 2 2 Z

n28 ¼ t2

L 0

k b ðcos½jf 1 f 2 f 3 aa1;1 sin½jf 1 f 2 f 3 aa1;2 þ cos½j2 sin½jf 1 f 2 f 3 aa1;2 U 1

sin½j3 f 1 f 2 f 3 aa1;2 þ cos½jsin½j2 f 1 f 2 f 3 aa1;3 sin½jf 1 f 2 f 3 aa2;1 þ cos½jf 1 f 2 f 3 aa2;2 þ cos½j3 f 1 f 2 f 3 aa2;2 cos½jsin½j2 f 1 f 2 f 3 aa2;2 þcos½j2 sin½jf 1 f 2 f 3 aa2;3 Þ ds Z

n29 ¼ t2

L 0

k b ðcos½jf 1 f 2 f 3 aa1;1 sin½jf 1 f 2 f 3 aa1;2 þ cos½j2 sin½jf 1 f 2 f 3 aa1;2 U 2

sin½j3 f 1 f 2 f 3 aa1;2 þ cos½jsin½j2 f 1 f 2 f 3 aa1;3 sin½jf 1 f 2 f 3 aa2;1 þ cos½jf 1 f 2 f 3 aa2;2 þ cos½j3 f 1 f 2 f 3 aa2;2 cos½jsin½j2 f 1 f 2 f 3 aa2;2 þcos½j2 sin½jf 1 f 2 f 3 aa2;3 Þ ds n30 ¼ t2

Z

L

 0

k

2

2U 2

2

2

ð2 cos½j2 f 1 f 2 aa1;1 þ 2 sin½j2 f 1 f 2 aa1;1 2 cos½jsin½jf 1 f 2 aa1;2 2

2

2

þ cos½j3 sin½jf 1 f 2 aa1;2 2 cos½jsin½j3 f 1 f 2 aa1;2 þ cos½j2 sin½j2 f 1 f 2 aa1;3 2

2

2

2 cos½jsin½jf 1 f 2 aa2;1 þcos½j2 f 1 f 2 aa2;2 þ cos½j4 f 1 f 2 aa2;2 2 2 f 1 f 2 aa2;2 þ cos½

2 cos½j2 sin½j n31 ¼ t2

Z

L

 0

j3 sin½j

2 f 1 f 2 aa2;3 Þ

ds

k b ð2 cos½jsin½j2 f 22 f 3 aa1;2 þ sin½j3 f 22 f 3 aa1;3 þ2 cos½jf 22 f 3 aa2;1 2U 1 2

2

2

þ 2 sin½jf 2 f 3 aa2;2 þ 2 cos½j2 sin½jf 2 f 3 aa2;2 þ cos½jsin½j2 f 2 f 3 aa2;3 Þ ds n32 ¼ t2

Z

L

 0

k b ð2 cos½jsin½j2 f 22 f 3 aa1;2 þ sin½j3 f 22 f 3 aa1;3 þ2 cos½jf 22 f 3 aa2;1 2U 2 2

2

2

þ 2 sin½jf 2 f 3 aa2;2 þ 2 cos½j2 sin½jf 2 f 3 aa2;2 þ cos½jsin½j2 f 2 f 3 aa2;3 Þ ds n33 ¼ t2

Z

L 0

k 2U 2

2

2

2

2

ð2 cos½jsin½jf 1 f 2 aa1;1 sin½j2 f 1 f 2 aa1;2 þ 2 cos½j2 sin½j2 f 1 f 2 aa1;2 sin½j4 f 1 f 2 aa1;2 2

2

2

2

þ cos½jsin½j3 f 1 f 2 aa1;3 þ 2 cos½j2 f 1 f 2 aa2;1 2 sin½j2 f 1 f 2 aa2;1 þ 2 cos½jsin½jf 1 f 2 aa2;2 2

2

2

þ 2 cos½j3 sin½jf 1 f 2 aa2;2 cos½jsin½j3 f 1 f 2 aa2;2 þ cos½j2 sin½j2 f 1 f 2 aa2;3 Þ ds n34 ¼ t2

Z

L 0

k

3

6U 2

3

3

ð3 cos½jsin½j3 f 2 aa1;2 sin½j4 f 2 aa1;3 6 cos½jsin½jf 2 aa2;1 3

3

3

3 sin½j2 f 2 aa2;2 3 cos½j2 sin½j2 f 2 aa2;2 cos½jsin½j3 f 2 aa2;3 Þ ds sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z L o1 ¼ t1 EAk2 f 1 ds þTððf 1 Þ0 ½0ðf 1 Þ0 ½LÞÞm3 0

Z. Yan et al. / Journal of Sound and Vibration 331 (2012) 3599–3616

o2 ¼ where t1 ¼ ð

RL 0

RL

rAf 1 ½s dsÞ1 ; t2 ¼ ð

0

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi t1 Tððf 2 Þ0 ½0ðf 2 Þ0 ½LÞn4

rAf 2 ½s dsÞ1 ; k ¼ rU rair =2.

Appendix C. Coefficients of the amplitude equations of 1:1 resonant case and 2:1 resonant case The coefficients appearing in the amplitude equations (25) of 1:1 resonant case are pp1;1 ¼ 60t3 eðemm4 nn1 o1 þ 4mm1 o31 emm2 nn3 o2 Þ pp1;2 ¼ 60t3 e2 ðmm4 nn3 þ o1 ðmm21 o1 þmm2 nn1 o2 ÞÞ pp2;1 ¼ 60t3 eo1 ðmm4 enn2 þ emm1 mm4 4o1 o2 Þ pp2;2 ¼ 60t3 eo1 ðemm1 mm2 o1 þ 4mm4 o1 þ emm2 nn2 o2 Þ pp3;1 ¼ 80t3 e2 o1 ð3m5 m8 2m11 n5 m6 n8 þ 3m8 m10 o21 þ3m19 o21 2m13 n8 o21 þ2m11 n10 o21 þ 9m24 o41 þ m9 ð7n5 þ5n10 o21 ÞÞ pp3;2 ¼ 80t3 e2 ð10m25 þ 10m5 m10 o21 þ m6 ð5n5 þ 7n10 o21 Þ þ o21 ðm28 þ 9m15 2m13 n5 m9 n8 þ 2m11 n8 þ4m210 o21 þ 3m22 o21 þ 2m13 n10 o21 ÞÞ pp4;1 ¼ 160t3 e2 ð2m9 n6 o1 m11 n6 o1 þ m6 ðm8 2n9 þn11 Þo1 2m9 m10 o31 2m10 m11 o31 þ2m8 m13 o31 m13 n9 o31 m13 n11 o31 þ m9 n13 o31 þm11 n13 o31 þ 3m12 n5 o2 þ 3m10 m11 o21 o2 þ 3m25 o21 o2 þ 3m12 n10 o21 o2 þ 3m30 o41 o2 þ m5 ð4m9 o1 þm11 ð2o1 þ 3o2 ÞÞÞ pp4;2 ¼ 160t3 e2 ðm6 n6 þ 2m8 m9 o21 þ m6 m10 o21 m8 m11 o21 þ3m16 o21 m13 n6 o21 þ m9 n9 o21 þ m11 n9 o21 2m9 n11 o21 þ m11 n11 o21 þ 2m6 n13 o21 þ 2m10 m13 o41 þ 3m23 o41 þ m13 n13 o41 þ m5 ð5m6 þ 4m13 o21 Þ þ 6m7 ðn5 þ n10 o21 ÞÞ pp5;1 ¼ 80t3 e2 ð2m5 m12 o1 þ m9 n7 o1 2m11 n7 o1 m7 ðm8 þ 6n9 6n11 Þo1 4m10 m12 o31 þm8 m14 o31 3m21 o31 2m13 n12 o31 m9 n14 o31 þ2m11 n14 o31 þ 3m12 n6 o2 þ 3m11 m13 o21 o2 þ3m26 o21 o2 þ3m12 n13 o21 o2 þ3m33 o31 o22 þm6 ð3m9 o1 n12 o1 þ 3m11 ðo1 þ o2 ÞÞÞ pp5;2 ¼ 80t3 e2 ð3m26 þ 6m7 n6 4m7 m10 o21 þm8 m12 o21 þ 3m17 o21 2m13 n7 o21 m9 n12 o21 þ 2m11 n12 o21 þ 6m7 n13 o21 þ 4m10 m14 o41 þ 2m13 n14 o41 2m5 ðm7 m14 o21 Þ þ m6 ðn7 þ ð3m13 þ n14 Þo21 Þ þ 3m9 m11 o1 o2 3m211 o1 o2 þ3m12 n9 o1 o2 3m12 n11 o1 o2 þ 3m29 o31 o2 3m31 o21 o22 Þ pp6;1 ¼ 80t3 e2 ð2m12 n5 o1 þ 3m9 n6 o1 2m7 n8 o1 þ m6 ð2m8 þ 3n9 3n11 Þo1 þ 2m9 m10 o31 þ 3m8 m13 o31 þ 3m20 o31 þ 2m12 n10 o31 þ 3m9 n13 o31 þ m12 n5 o2 m10 m11 o21 o2 2m8 m13 o21 o2 3m25 o21 o2 4m14 n8 o21 o2 m12 n10 o21 o2 þ3m30 o41 o2 þ m5 ð4m9 o1 þ m11 ð6o1 þ o2 ÞÞÞ pp6;2 ¼ 80t3 e2 ð3m6 n6 m8 m9 o21 þm6 m10 o21 þ 3m8 m11 o21 þ3m16 o21 þ2m12 n8 o21 3m9 n9 o21 þ3m9 n11 o21 þ3m6 n13 o21 2m7 ðn5 n10 o21 Þ3m23 o41 þ m5 ð5m6 þ 2m13 o1 ð3o1 o2 ÞÞ m8 m11 o1 o2 4m14 n5 o1 o2 m12 n8 o1 o2 þ 2m10 m13 o31 o2 þ3m28 o31 o2 þ4m14 n10 o31 o2 Þ pp7;1 ¼ 160t3 e2 ðm12 n6 o1 þ 3m9 n7 o1 þm7 ð3m8 þ 2n9 4n11 Þo1 þ m9 m13 o31 þ3m8 m14 o31 þ 3m21 o31 þ m12 n13 o31 þ3m9 n14 o31 þ2m12 n6 o2 m9 m13 o21 o2 2m14 n9 o21 o2 2m14 n11 o21 o2 þ m12 n13 o21 o2 þ3m33 o31 o22 þ 2m6 m11 ðo1 þ o2 ÞÞ pp7;2 ¼ 160t3 e2 ðm26 þ 2m7 n6 þm29 o21 þm9 m11 o21 þ 3m17 o21 þ m12 n9 o21 þ m12 n11 o21 þ 6m5 ðm7 þ m14 o21 Þ þ m6 ð3n7 þ o1 ð3n14 o1 þ m13 ð2o1 o2 ÞÞÞþ m12 n9 o21 þ 4m7 n13 o21 2m9 m11 o1 o2 þ m211 o1 o2 2m14 n6 o1 o2 2m12 n9 o1 o2 þ m12 n11 o1 o2 þm213 o31 o2 þ2m14 n13 o31 o2 þ 3m31 o21 o22 Þ pp8;1 ¼ 80t3 e2 ðm6 m12 o1 2m12 n7 o1 þ 2m9 m14 o31 þ2m12 n14 o31 þ 7m12 n7 o2 2m12 m13 o21 o2 þ 5m11 m14 o21 o2 þ 3m27 o21 o2 4m14 n12 o21 o2 þ 5m12 n14 o21 o2 þ 9m34 o21 o32 þ m7 ð2m9 o1 2n12 o1 þ 7m11 o2 ÞÞ

3613

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pp8;2 ¼ 80t3 e2 ðm6 ð5m7 þ 7m14 o21 Þ þ 2m7 ð5n7 þ 7n14 o21 m13 o1 o2 Þ þ o1 ð2m9 m12 o1 þ 9m18 o1 þ 2m12 n12 o1 m11 m12 o2 4m14 n7 o2 m12 n12 o2 þ2m13 m14 o21 o2 þ 4m14 n14 o21 o2 þ 3m32 o1 o22 ÞÞ qq1;1 ¼ 60t4 eðemm1 nn3 o1 þ 4nn1 o31 enn2 nn3 o2 Þ qq1;2 ¼ 60t3 eo1 ðemm1 nn1 o1 þ4nn3 o1 þ enn1 nn2 o2 Þ qq2;1 ¼ 60t3 eo1 ðemm4 nn1 þ emm2 nn3 þ 4nn2 o1 o2 Þ qq2;2 ¼ 60t3 e2 ðmm4 nn3 þ o1 ðmm2 nn1 o1 þnn22 o2 ÞÞ qq3;1 ¼ 80t3 e2 o1 ð5m5 n8 n6 n8 þ 7n5 n9 2n5 n11 þ 7m10 n8 o21 þ 5n9 n10 o21 þ 2n10 n11 o21 2n8 n13 o21 þ 3n19 o21 þ 9n24 o41 2m8 ðn5 þ 2n10 o21 ÞÞ qq3;2 ¼ 80t3 e2 ð2m5 ð5n5 2n10 o21 Þ þ n5 ð5n6 þ2ð7m10 n13 Þo21 Þ þ o21 ðm8 n8 þ 7n6 n10 n8 ðn9 2n11 Þ þ 9n15 þ 4m10 n10 o21 þ2n10 n13 o21 þ 3n22 o21 ÞÞ qq4;1 ¼ 160t3 e2 ð4m9 n5 o1 2m11 n5 o1 m6 n8 o1 2m13 n8 o31 þ 2m9 n10 o31 þ 2m11 n10 o31 3m5 n11 o2 3n5 n12 o2 3m10 n11 o21 o2 3n10 n12 o21 o2 3n25 o21 o2 3n30 o41 o2 Þ qq4;2 ¼ 160t3 e2 ð3m5 n6 þn26 þ 6n5 n7 þ 4m13 n5 o21 þ 3m10 n6 o21 þ 2m9 n8 o21 m11 n8 o21 þ n29 o21 þ6n7 n10 o21 n9 n11 o21 þ n211 o21 þ n6 n13 o21 þ3n16 o21 þ 2m13 n10 o41 þ n213 o41 þ3n23 o41 þ 2m6 ðn5 n10 o21 ÞÞ qq5;1 ¼ 80t3 e2 ð3m9 n6 o1 þ 3m11 n6 o1 m7 n8 o1 5n7 n9 o1 þ 4n7 n11 o1 n6 n12 o1 þ m14 n8 o31 2n12 n13 o31 n9 n14 o31 þ 2n11 n14 o31 3n21 o31 2m12 o1 ðn5 þ 2n10 o21 Þ þ 3m6 n11 o2 þ 3n6 n12 o2 þ 3m13 n11 o21 o2 þ3n12 n13 o21 o2 þ 3n26 o21 o2 þ3n33 o31 o22 Þ qq5;2 ¼ 80t3 e2 ð2m7 n5 3m6 n6 5n6 n7 2m14 n5 o21 3m13 n6 o21 m12 n8 o21 þ4m7 n10 o21 þn9 n12 o21 2n11 n12 o21 4n7 n13 o21 n6 n14 o21 3n17 o21 4m14 n10 o41 2n13 n14 o41 3m9 n11 o1 o2 þ 3m11 n11 o1 o2 3n9 n12 o1 o2 þ 3n11 n12 o1 o2 3n29 o31 o2 þ3n31 o21 o22 Þ qq6;1 ¼ 80t3 e2 ð6m9 n5 o1 6m11 n5 o1 m8 n6 o1 þ3m6 n8 o1 2n7 n8 o1 2m5 n9 o1 þ6n6 n9 o1 3n6 n11 o1 2n5 n12 o1 þ 3m13 n8 o31 þ 2m10 n9 o31 þ2n10 n12 o31 þ3n9 n13 o31 þ 3n20 o31 þ m5 n11 o2 þ n5 n12 o2 m10 n11 o21 o2 n10 n12 o21 o2 2m8 n13 o21 o2 4n8 n14 o21 o2 3n25 o21 o2 þ 3n30 o41 o2 Þ qq6;2 ¼ 80t3 e2 ð6m6 n5 þ3n26 2n5 n7 þ6m13 n5 o21 þ m10 n6 o21 3m9 n8 o21 þ 3m11 n8 o21 þ 2m8 n9 o21 3n29 o21 þ 2n7 n10 o21 þ3n9 n11 o21 þ 2n8 n12 o21 þ 3n6 n13 o21 þ 3n16 o21 3n23 o41 m8 n11 o1 o2 n8 n12 o1 o2 4n5 n14 o1 o2 þ2m10 n13 o31 o2 þ4n10 n14 o31 o2 þ3n28 o31 o2 m5 ðn6 þ2n13 o1 o2 ÞÞ qq7;1 ¼ 160t3 e2 ð3m7 n8 o1 m6 n9 o1 þ 5n7 n9 o1 4n7 n11 o1 n6 n12 o1 þ 3m14 n8 o31 þ m13 n9 o31 þn12 n13 o31 þ3n9 n14 o31 þ 3n21 o31 þ 2m6 n11 o2 þ 2n6 n12 o2 þm13 n11 o21 o2 þ n12 n13 o21 o2 2n9 n14 o21 o2 2n11 n14 o21 o2 þ 3n33 o31 o22 þm9 o1 ðn6 n13 o1 o2 Þm11 o1 ð2n6 þ n13 o1 o2 ÞÞ qq7;2 ¼ 160t3 e2 ð6m7 n5 þ5n6 n7 þ 6m14 n5 o21 þ 2m13 n6 o21 þ m9 n9 o21 þm11 n9 o21 þ n9 n12 o21 þ n11 n12 o21 þ4n7 n13 o21 þ 3n6 n14 o21 þ3n17 o21 2m9 n11 o1 o2 þ m11 n11 o1 o2 2n9 n12 o1 o2 þn11 n12 o1 o2 2n6 n14 o1 o2 þ m13 n13 o31 o2 þ 2n13 n14 o31 o2 þ 3n31 o21 o22 þm6 ðn6 n13 o1 o2 ÞÞ qq8;1 ¼ 80t3 e2 ð4n7 n12 o1 2m14 n9 o31 2n12 n14 o31 7n7 n12 o2 5m14 n11 o21 o2 n12 n14 o21 o2 3n27 o21 o2 9n34 o21 o32 þm7 ð2n9 o1 7n11 o2 Þ þ m12 o1 ðn6 þ 2n13 o1 o2 ÞÞ qq8;2 ¼ 80t3 e2 ð10n27 þ2n7 n14 o1 ð7o1 2o2 Þ þ m7 ð5n6 2n13 o1 o2 Þ þ o1 ð2n212 o1 þ 9n18 o1 n212 o2 þ4n214 o21 o2 þ 3n32 o1 o22 þ m12 ð2n9 o1 n11 o2 Þ þ m14 o1 ð7n6 þ2n13 o1 o2 ÞÞÞ The coefficients appearing in the amplitude equations (30) of 2:1 resonant case are pp9;1 ¼ 80t3 eo1 ð4emm4 nn1 þ4emm2 nn3 3mm1 o21 Þ pp9;2 ¼ 20t3 e2 ð16mm4 nn3 ð3mm21 þ16mm2 nn1 Þo21 Þ

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pp10;1 ¼ 5t3 eð32em6 mm2 o1 þ 32em9 mm4 o1 64emm2 n7 o1 32emm4 n12 o1 12em7 ðmm1 4nn2 Þo1 þ 3em14 mm1 o31 þ 16emm2 n14 o31 þ 64em11 mm4 o2 þ 48m12 o21 o2 16em13 mm2 o21 o2 þ 48em14 nn2 o1 o22 Þ pp10;2 ¼ 10t3 eð32em6 mm4 8emm4 ð4n7 þ o1 ðn14 o1 þ 2m13 o2 ÞÞ þ o1 ð24m7 o1 þ 3em12 mm1 o1 8ðem9 mm2 o1 þ 3m14 o1 o22 þmm2 ð2en12 o1 þ 2em11 o2 ÞÞÞÞ pp11;1 ¼ 8t3 e2 o1 ð15m5 m8 8m11 n5 4m6 n8 þ 15m8 m10 o21 þ 15m19 o21 8m13 n8 o21 þ 8m11 n10 o21 þ 45m24 o41 þ 4m9 ð31n5 þ 29n10 o21 ÞÞ pp11;2 ¼ 8t3 e2 ð50m25 þ50m5 m10 o21 þ 4m6 ð29n5 þ 31n10 o21 Þ þ o21 ð5m28 þ 45m15 8m13 n5 4m9 n8 þ 8m11 n8 þ 20m210 o21 þ15m22 o21 þ8m13 n10 o21 ÞÞ pp12;1 ¼ 2t3 e2 ð90m12 n6 o1 þ 960m9 n7 o1 þ 60m7 ð4m8 þ 2n9 3n11 Þo1 þ112m9 m13 o31 þ 60m8 m14 o31 þ240m21 o31 þ45m12 n13 o31 þ240m9 n14 o31 128m6 m11 ðo1 2o2 Þ þ 60m12 n6 o2 224m9 m13 o21 o2 180m14 n9 o21 o2 90m14 n11 o21 o2 30m12 n13 o21 o2 þ 240m14 n9 o1 o22 120m14 n11 o1 o22 þ240m33 o31 o22 Þ pp12;2 ¼ 2t3 e2 ð64m26 þ120m7 n6 þ 64m29 o21 þ 112m9 m11 o21 þ 240m17 o21 þ90m12 n9 o21 þ 45m12 n11 o21 þ 180m7 n13 o21 þ 120m5 ð4m7 þ m14 o21 Þ256m9 m11 o1 o2 þ32m211 o1 o2 þ16m6 ð60n7 þ o1 ð15n14 o1 þ 2m13 ð4o1 7o2 ÞÞÞ180m14 n6 o1 o2 60m12 n9 o1 o2 30m12 n11 o1 o2 þ 32m213 o31 o2 þ90m14 n13 o31 o2 þ 240m14 n6 o22 þ 240m31 o21 o22 þ 120m14 n13 o21 o22 Þ qq9;1 ¼ 80t4 eo1 ð2emm4 nn1 þ2emm2 nn3 þ 3nn2 o1 o2 Þ qq9;2 ¼ 40t4 e2 ð8mm4 nn3 þ o1 ð2mm2 nn1 o1 þ 3nn22 o2 ÞÞ qq10;1 ¼ 20t4 eð8emm2 o1 ð2n5 þ n10 o21 Þ þ 3emm1 o1 ðn6 þn13 o1 o2 Þ2ð4emm4 n8 o1 þ 4em6 nn1 o1 16en7 nn1 o1 3en6 nn2 o1 þ8em9 nn3 o1 4em11 nn3 o1 8en12 nn3 o1 þ6n9 o31 þ 2em13 nn1 o31 6en6 nn2 o2 þ 8en12 nn3 o2 6n11 o21 o2 16en14 nn1 o21 o2 ÞÞ qq10;2 ¼ 20t4 eð16em6 nn3 32en7 nn3 þ12n6 o21 þ4emm2 n8 o21 þ3emm1 n9 o21 8em9 nn1 o21 þ 4em11 nn1 o21 þ 16en12 nn1 o21 þ6en9 nn2 o21 þ8em13 nn3 o21 þ16emm4 ð2n5 þ n10 o21 Þ þ 3emm1 n11 o1 o2 16en12 nn1 o1 o2 þ12en9 nn2 o1 o2 32en14 nn3 o1 o2 þ12n13 o31 o2 Þ qq11;1 ¼ 8t4 e2 ð8m6 n8 o1 15n6 n11 o1 þ16m13 n8 o31 15n9 n13 o31 þ 16m11 o1 ðn5 n10 o21 Þ 16m9 ð8n5 o1 þ 7n10 o31 Þ þ 60m5 n11 o2 þ30n6 n11 o2 þ240n5 n12 o2 þ 60m10 n11 o21 o2 þ 240n10 n12 o21 o2 þ 30n9 n13 o21 o2 þ 60n25 o21 o2 þ 60n30 o41 o2 Þ qq11;2 ¼ 2t4 e2 ð240m5 n6 þ 60n26 þ 1920n5 n7 þ256m13 n5 o21 þ 240m10 n6 o21 þ128m9 n8 o21 16m11 n8 o21 þ60n29 o21 þ 1920n7 n10 o21 þ 45n211 o21 þ 240n16 o21 þ 224m13 n10 o41 þ 45n213 o41 þ240n23 o41 þ128m6 ðn5 n10 o21 Þ þ 120n9 n11 o1 o2 60n211 o1 o2 120n6 n13 o1 o2 60n213 o31 o2 Þ qq12;1 ¼ 5t4 e2 ð128n7 n12 o1 þ 3m14 n9 o31 þ 16n12 n14 o31 þ 448n7 n12 o2 þ 27m14 n11 o21 o2 þ16n12 n14 o21 o2 þ 48n27 o21 o2 þ144n34 o21 o32 12m7 ðn9 o1 7n11 o2 Þ þ 6m12 o1 ðn6 n13 o1 o2 ÞÞ qq12;2 ¼ 5t4 e2 ð640n27 þ 32n7 n14 o1 ð7o1 4o2 Þ þ 12m7 ð9n6 n13 o1 o2 Þ þ o1 ð6m12 ðn9 o1 þn11 o2 Þ þ 3m14 o1 ð7n6 þn13 o1 o2 Þ þ16ð2n212 ðo1 o2 Þ þ o1 ð9n18 þ2n214 o1 o2 þ 3n32 o22 ÞÞÞÞ where t3 ¼ ð480o31 Þ1 and t4 ¼ ð480o21 o2 Þ1 . References [1] [2] [3] [4] [5] [6] [7] [8]

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