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Jun 11, 2011 - Abstract This paper treats nonlinear vibration of pipes conveying fluid in the supercritical regime. If the flow speed is larger than the critical ...
Nonlinear Dyn (2012) 67:1505–1514 DOI 10.1007/s11071-011-0084-5

O R I G I N A L PA P E R

Internal resonance of pipes conveying fluid in the supercritical regime Yan-Lei Zhang · Li-Qun Chen

Received: 22 December 2010 / Accepted: 11 May 2011 / Published online: 11 June 2011 © Springer Science+Business Media B.V. 2011

Abstract This paper treats nonlinear vibration of pipes conveying fluid in the supercritical regime. If the flow speed is larger than the critical value, the straight equilibrium configuration becomes unstable and bifurcates into two possible curved equilibrium configurations. The paper focuses on the nonlinear vibration around each bifurcated equilibrium. The disturbance equation is derived from the governing equation, a nonlinear integro-partial-differential equation, via a coordinate transform. The Galerkin method is applied to truncate the disturbance equation into a two-degree-of-freedom gyroscopic systems with weak nonlinear perturbations. The internal resonance may occur under the certain condition of the supercritical flow speed for the suitable ratio of mass per unit length of pipe and that of fluid. The method of multiple scales is applied to obtain the relationship between the amplitudes in the two resonant modes. The time histories predicted by the analytical method are compared with Y.-L. Zhang · L.-Q. Chen () Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai 200072, China e-mail: [email protected] L.-Q. Chen Department of Mechanics, Shanghai University, Shanghai 200444, China L.-Q. Chen Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai 200072, China

the numerical ones and the comparisons validate the analytical results when the nonlinear terms are small. Keywords Nonlinearity · Internal resonance · Supercritical motion · Method of multiple scales · Pipes conveying fluid

1 Introduction Pipes conveying fluid can be found in boilers, nuclear reactors, heat exchangers, and steam generators. Understanding nonlinear vibrations of the pipes is significant for the design of the devices. Vibrations of pipes conveying fluid have been studied extensively for a long time as comprehensively reviewed by Ibrahim [1]. The monographs by Païdoussis [2] and [3] presented a perspective on the whole field of pipes conveying fluid. The early investigation on pipes conveying fluid is within the linear model, which was initiated by Païdoussis and Issid [4] in 1974. They applied Bolotin’s method to present the stability diagrams of parametric instabilities, for pipes with pined, clamped or free ends. More and more effort has been devoted to the study of the nonlinear dynamics of the pipes conveying fluid, staring with Holmes’ [5] benchmark paper in 1977. They examined in detail the symmetric saddlenode bifurcation that occurs at the critical speed via the method of center manifold reduction.

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Nonlinear models can predict that, if the flow speed is larger than the critical value that can be determined by the linear models [4], the straight pipe equilibrium configuration becomes unstable and bifurcates into two possible curved equilibria. Yoshizawa et al. [6] determined the critical velocity of the fluid and a stable equilibrium solution with finite deflection of an inextensible clamped-hinged pipe, which could slide axially at its hinged downstream end. Nikoli´c and Rajkovi´c [7] studies stationary bifurcations in several nonlinear models of fluid-conveying clampedclamped pipes via the method of Lyapunov–Schmidt reduction and singularity theory. Plaut [8] found that fluid-conveying pipes are conservative in the subcritical regime and nonconservative in the supercritical regime. Modarres-Sadeghi and Païdoussis [9] found that an extensible fluid-conveying pipe supported at both ends is stable at its original equilibrium configuration up to the flow velocity at which it loses stability via static divergence via a supercritical pitchfork bifurcation. Ghayesh and Païdoussis [10] observed 3-D flutter, divergence, quasiperiodic and chaotic motions, with the increasing flow velocity, in three-dimensional motions of a cantilever tube conveying fluid and having additionally supported by an intermediate spring array. Ghayesh et al. [11] observe that the system loses the stability at the flow speed large enough, bifurcates into 3-D periodic, quasiperiodic motions and chaos in three-dimensional dynamics of a fluid-conveying cantilevered pipe fitted with an end-mass and additional intra-span spring-support. Internal resonance is an exclusive phenomenon in nonlinear vibration. Researchers found internal resonance in vibration of pipes conveying fluid. For fluidconveying pipes, Xu and Yang [12] studies internal resonances under external sinusoidal excitation at certain flow velocity via the method of multiple scales. Panda and Kar [13, 14] applied the method of multiple scales to the governing nonlinear integro-partialdifferential equations of a pipe conveying pulsating fluid with combination, principal parametric and internal resonances. However, the available works on internal resonances are all in the in the subcritical regime. There are no investigations on internal resonances of pipes conveying fluid in the supercritical regime. To address the lack of research in this aspect, the present work treats internal resonance in free nonlinear vibration of pipes conveying fluid in the supercritical regime.

Y.-L. Zhang, L.-O. Chen

2 Formulations 2.1 The governing equation of pipes conveying fluid Consider a uniform pipe hinged at both ends conveying fluid, as illustrated schematically in Fig. 1. It is assumed that the motion is planar, the uniform crosssection remains plane during the motion and the pipe behaves like an Euler–Bernoulli beam in bending vibration. The fluid is assumed to be incompressible and steady with mean velocity U . The material of the pipe is elastic. The effect of internal and external dampings are neglected here. The gravity effect is also neglected. The governing equation of transverse motion of the pipe including the nonlinearity due to midline stretching is a nonlinear partial-integro-differential equation [2, 5]: EI

 ∂ 2y ∂ 4y  2 + MU − T ∂x 4 ∂x 2   2  L EA ∂ y − (y  )2 dx 2L 0 ∂x 2 + (M + m)

∂ 2y ∂ 2y =0 + 2MU ∂x∂t ∂t 2

(1)

and the boundary conditions of a simply supported pipe y(0, t) = y(L, t) =

∂ 2y ∂ 2y (0, t) = 2 (L, t) = 0 2 ∂x ∂x

(2)

where x is the longitudinal coordinate, y is the transverse deflection, U is the fluid velocity, T is the externally imposed axial tension, m and M are, respectively, the mass per unit length of pipe and fluid materials, A is the cross sectional area of the pipe, L is the length, EI is the flexural stiffness of the pipe. Incorporation of the following dimensionless quantities

Fig. 1 A horizontal pipe conveying fluid

Internal resonance of pipes conveying fluid in the supercritical regime



x y , x= , L L  1 M 2 LU, u= EI

t↔

η=

 Mr =

M M +m

EI M +m

1 2

2.3 The disturbance equation in the supercritical regime

t , L2 (3)



1 2

γ=

,

AL2 , I

T L2 T¯ = EI

into (1) and (2) leads to the dimensionless governing equation of transverse motion

η,tt + 2Mr uη,xt + u2 − T¯ η,xx + η,xxxx  1 γ2 η,xx = (η,x )2 dx (4) 2 0 with the boundary conditions (5)

η,xx (0, t) = η,xx (1, t) = 0

where the comma-subscript notation denotes the partial differentiation with respect to the dimensionless time.

Equilibrium solutions η(x) ˆ satisfy    γ 2 1 ˆ 2   ηˆ + u2 − T¯ + η dx ηˆ = 0 2 0

γ2

− T¯ + =

 γ2 2

ηˆ k±  



ηˆ (0) = ηˆ (1) = 0

2 

0



 2 ηˆ k± dx η,xx + η,xxxx

1

0 1 0

1

+ η,xx



2 η,x dx − 2η,xx

1

0



ηˆ k± η dx

2 η,x dx

(9)

for small but finite-amplitude vibration about a specified non-trivial equilibrium. Substitution of (8) into (9) for u > u(k) leads to a continuous gyroscopic system with nonlinear disturbances Mη,tt + Gη,t + Kη = N (η)

M = I, (6) (7)

where the prime indicates the partial differentiation with respect to x and the superscript indicates the sense of the equilibrium displacement. The trivial configuration ηˆ 0 = 0 is always an equilibrium solution. In addition, the pairs of non-trivial equilibrium solutions [15] 2  2 ηˆ k± (x) = ± u − u2(k) sin(kπx), kπγ k = 1, 2, 3, . . .



(10)

where the mass, gyroscopic, and stiffness operators are defined as

2.2 The equilibria of pipes conveying fluid



The substitution η(x, t) ↔ ηk± (x) + η(x, t) in (4) yields the disturbance equation about the non-trivial equilibria in the supercritical regime  1 η,tt + 2Mr uη,xt + u2 η,xx + γ 2 ηˆ k± ηˆ k± η dx

0

η(0, t) = η(1, t) = 0,

η(0) ˆ = η(1) ˆ = 0,

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G = 2Mr u

∂ ∂x

(11)

 K = (2kπ)2 u2 − u2k sin(kπx) + (kπ)2



1

sin(kπx) dx 0

∂2 ∂4 + ∂x 2 ∂x 4

(12)

and the disturbing term is    2 2 N (η) = ±kπγ u − uk 2η,xx 

1

− sin(kπx) 0



2 η,x

1

sin(kπx)η dx 0

γ2 η,xx dx + 2

 0

1

2 η,x dx

(13)

(8)

bifurcate from  the straight configuration and exist for u > u(k) = T¯ + (kπ)2 , where u(k) is termed the critical speed for linear mode k. For sub-critical speeds, only the straight configuration exists. In the supercritical range, u(1) < u < u(2) , there exists the three solutions ηˆ 0 , ηˆ 1− and ηˆ 1+ .

3 Discretization and analyses 3.1 Discretization The nonlinear partial-integro-differential (10) can be discretized by the Galerkin technique based on the stationary beam eigenfunctions. Suppose η(x, t) can be

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Y.-L. Zhang, L.-O. Chen

η(x, t) =

n 

φr (x)qr (t)

(14)

r=1

where φr (x) are eigenfunctions for the free undamped vibrations of a beam satisfying the pined-pined bound√ ary conditions (5), namely φr (x) = 2 sin(rπx), and qr (t) the rth generalized coordinate. In the following, only the first two terms in superposition (14) is retained so that the discretized system is of two degrees of freedom. In the case of k = 1, substituting (14) (with r = 2) into (10), multiplying the resulting equation by weighted function φr (x) and integrating the product from 0 to 1 yield q¨1 − λq˙2 + αq1



 = εα11 3q12 + 4q22 + ε 2 α12 q1 q12 + 4q22

 q¨2 + λq˙1 + βq2 = εβ11 q2 q1 + ε 2 β12 q2 q12 + 4q22 (15) where

 16 λ = Mr u, α = 2π 2 u2 − u21 , 3  1 γ2 α12 = − π 4 , α11 = ∓ √ π 3 γ u2 − u21 , 2 (16) 2 √ 3  4 2 β11 = ∓4 2π γ u2 − u1 , β = 12π , β12 = −2γ 2 π 4 The dot represents the differentiation with respect to nondimensional time t. To express the smallness of the amplitude of motion q r , it is scaled as q r ↔ εq r where the small parameter ε is a book keeping device in the subsequent multi-scale analysis. Equation (15) defines a two-degree-of-freedom gyroscopic system with small nonlinear terms. For the whole supercritical regime, the Galerkin discretization is √

 2 sin(kπ) 16 2 2 q¨1 − Mr uq˙2 + (2kπ) u − uk 3 π − k2π √ √   2 sin(kπ) 2 2 sin(kπ) × q1 + q 2 π − k2π (−4 + k 2 )π − k 2 π 4 q1 + π 4 q1  = ±kπγ u2 − u2k −2π 2 q1

√ √  2 sin(kπ) 2 2 sin(kπ) + q2 × q1 π − k2π (−4 + k 2 )π √   2 sin(kπ) 2 2 2 π − q + 4q 1 2 π − k2π 

approximated by



 γ2 4 2 π q1 q1 + 4q22 2

(17)



 16 2 2 2 2 sin(kπ) q¨2 + Mr uq˙1 + (2kπ) u − uk 3 (−4 + k 2 )π √   √ 2 sin(kπ) 2 2 sin(kπ) + q2 × q1 π − k2π (−4 + k 2 )π − 4k 2 π 4 q2 + 16π 4 q2  √  2 sin(kπ) = ±kπγ u2 − u2k −8π 2 q2 q1 π − k2π √ √  2 2 sin(kπ) 2 2 2 sin(kπ) − π + q2 (−4 + k 2 )π (−4 + k 2 )π  



× q12 + 4q22 − 2γ 2 π 4 q2 q12 + 4q22

(18)

It is interesting to notice that (15) can be derived form (17) and (18) by letting k → 1. In the supercritical regime, the pipes conveying fluid vibrate about non-trivial equilibria whose numbers and shape depend on the flow speed. Here, only the speed range u(1) < u < u(2) is investigated. 3.2 The multi-scale analysis Nayfeh and Mook [16] proposed the method of multiple scales to analyze the internal resonance in the two-degree-of-freedom gyroscopic system with disturbances. Here internal resonance will be investigated for pipes conveying fluid in the supercritical range, u(1) < u < u(2) via the method of multi-scales. Assume an approximate expansion of the solution of (15) in the form q1 = q11 (T0 , T1 ) + εq12 (T0 , T1 ) + · · · q2 = q21 (T0 , T1 ) + εq22 (T0 , T1 ) + · · ·

(19)

where T0 = t and T1 = εt are, respectively, the fast and slow time scales. Denote D0 = ∂/∂T0 and D1 = ∂/∂T1 . Substituting (19) into (15) and equating coefficients of like powers of ε, in resulting equation, one

Internal resonance of pipes conveying fluid in the supercritical regime

obtains at order ε 0 D02 q11 − λD0 q21 + αq11 = 0 D02 q21 + λD0 q11 + βq21 = 0

(20)

+ 4α11 Λ21 A21 exp(i2ω1 T0 )

D02 q12 − λD0 q22 + αq12

D02 q22 + λD0 q12 + βq22

(21)

= −2D1 D0 q21 − λD1 q11 + β11 q11 q21

q11 = A1 (T1 ) exp(iω1 T0 ) + A2 (T1 ) exp(iω2 T0 ) + cc (22)

where the ωn2 (n = 1, 2) are the two real roots of its frequency equation

 ω4 − ω2 α + β + λ2 + αβ = 0 (23) and substitution of q11 and q21 into (20) yields α

=0

βΛn + iλωn − Λn ωn2 = 0

iλωn i(α − ωn2 ) = − λωn β − ωn2

= (λΛ1 − 2iω1 )A1 exp(iω1 T0 ) − (2iω2 − λΛ2 )A2 exp(iω2 T0 )

+ β11 Λ1 A21 exp(i2ω1 T0 ) + β11 Λ2 A22 exp(i2ω2 T0 ) + β11 (Λ1 + Λ2 )A1 A2 

× exp i(ω1 + ω2 )T0

 

+ β11 Λ¯ 1 + Λ2 A¯ 1 A2 exp i(ω2 − ω1 )T0 (28)

Then in the case ω2 = 2ω1 , there is possible internal resonant, which will be focused upon in the following section.

(25)

4 Internal resonance in the supercritical regime

√    2

2 ω1,2 = α + β + λ2 ∓ α − β + λ2 + 4βλ2 2 (26)

D02 q12 − λD0 q22 + αq12

= −(2iω1 Λ1 + λ)A1 exp(iω1 T0 )

(24)

It should be remarked that ωn can be solved from (23) as

which are distinct for the parameter given in (16). Substitution of (22) into (21) yields

(27)

D02 q22 + λD0 q12 + βq22

+ β11 Λ¯ 1 A1 A¯ 1 + β11 Λ¯ 2 A2 A¯ 2 + cc

Thus Λn = −

+ 4α11 Λ¯ 2 Λ2 A2 A¯ 2 + cc

− (2iω2 Λ2 + λ)A2 exp(iω2 T0 )

× exp(iω2 T0 ) + cc

− iλΛn ωn − ωn2

+ 4α11 Λ22 A22 exp(i2ω2 T0 )

 + 8α11 Λ1 Λ2 A1 A2 exp i(ω1 + ω2 )T0 

+ 8α11 Λ¯ 1 Λ2 A¯ 1 A2 exp i(ω2 − ω1 )T0 + 4α11 Λ¯ 1 Λ1 A1 A¯ 1

The solution of linear gyroscopic system (20) can be expressed as

q21 = Λ1 A1 (T1 ) exp(iω1 T0 ) + Λ2 A2 (T1 )

+ 3α11 A21 exp(i2ω1 T0 ) + 3α11 A22 exp(i2ω2 T0 ) 

+ 6α11 A1 A2 exp i(ω1 + ω2 )T0 

+ 6α11 A¯ 1 A2 exp i(ω2 − ω1 )T0 + 3α11 A1 A¯ 1 + 3α11 A2 A¯ 2

and at order ε 1

2  2 = −2D1 D0 q11 + λD1 q21 + α11 3q11 + 4q21

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4.1 Possibility of internal resonance Under certain conditions, internal resonances may occur if the two natural frequencies are commensurable or nearly commensurable. For example, consider the pipes conveying fluid with T¯ = −5 and Mr = 0.447. Its natural frequencies depend on the fluid speed as shown in Fig. 2. When u ≈ 5.02655 in Fig. 2(b), the natural frequency ω1 = 18.5197 and ω2 = 37.0424 so that ω2 ≈ 2ω1 . When u ≈ 3.48717 in Fig. 2(c), the natural frequency ω1 = 11.6145 and ω2 = 35.3134 so that ω2 ≈ 3ω1 . However, 3 : 1 internal resonance will not occur for the secular terms of (27) and (28). For other system parameters with T¯ = −5 and Mr = 0.78, if u ≈ 3.89557 in Fig. 3(c), the natural frequency ω1 =

1510 Fig. 2 The first two natural frequencies changing with the fluid speed

Fig. 3 The first two natural frequencies changing with the fluid speed

Y.-L. Zhang, L.-O. Chen

Internal resonance of pipes conveying fluid in the supercritical regime

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12.7031 and ω2 = 38.3874 so that ω2 ≈ 3ω1 . Nevertheless internal resonance may not occur as shown in Fig. 3.

or

4.2 Solvability condition

which is the solvability condition.

A detuning parameter σ is introduced to quantify the deviation of ω2 from 2ω1 . Then ω2 is described by

4.3 Steady-state response

ω2 = 2ω1 + εσ

(29)

R1n = −Λ¯ n R2n

(36)

Substituting (33) and (34) into (36) and rearranging the resulting equation yield

and 2ω1 T0 and (ω2 − ω1 )T0 are, respectively, expressed as

D1 A1 = Γ1 A¯ 1 A2 exp(iσ T1 )

2ω1 T0 = ω2 T0 −σ T1 ,

where

(ω2 −ω1 )T0 = ω1 T0 +σ T1 (30)

To establish the solvability condition of (27), (28), assume a particular solution in the form q12 = P11 exp(iω1 T0 ) + P12 exp(iω2 T0 ) q22 = P21 exp(iω1 T0 ) + P22 exp(iω2 T0 )

(31)

where R11 = −(2iω1 − λΛ1 )A1 + 6α11 A¯ 1 A2 exp(iσ T1 ) R12 = −(2iω2 − λΛ2 )A2

(33)

+ 3α11 A21 exp(−iσ T1 )

+ 4α11 Λ21 A21 exp(−iσ T1 )

6α11 + 8α11 Λ¯ 1 Λ2 + β11 Λ¯ 1 (Λ¯ 1 + Λ2 ) Λ¯ 1 (2iω1 Λ1 + λ) + (2iω1 − λΛ1 )

= Re(Γ1 ) + i Im(Γ1 ) Γ2 =

(37)

3α11 + 4α11 Λ21

(38) + β11 Λ¯ 2 Λ1

Λ¯ 2 (2iω2 Λ2 + λ) + (2iω2 − λΛ2 ) (39)

Substituting (25) into (38) and (40), one obtains Re(Γ1 ) = 0

Im(Γ1 ) = − β(8α11 + β11 )λ2 ω1 ω2

 − (8α11 + β11 )λ2 ω13 ω2 + 6α11 β 2 β − ω22 (40)  

− 12α11 β + β11 λ2 ω12 β − ω22



 + 6α11 ω14 β − ω22 / 2ω1 β β + λ2   − 2βω12 + ω14 β − ω22

Re(Γ2 ) = 0

R21 = −(2iω1 Λ1 + λ)A1

 + β11 Λ¯ 1 + Λ2 A¯ 1 A2 exp(iσ T1 )

Γ1 =

= Re(Γ2 ) + i Im(Γ2 )

Substituting (31) into (27), (28), using (30), and equating the coefficients of exp(iω1 T0 ) and exp(iω2 T0 ) on both hands of the resulting equations, one obtains

 α − ωn2 P1n + (−iωn λ)P2n = R1n (32) 

(iωn λ)P1n + β − ωn2 P2n = R2n

+ 8α11 Λ¯ 1 Λ2 A¯ 1 A2 exp(iσ T1 )

D1 A2 = Γ2 A21 exp(−iσ T1 )

(34)

R22 = −(2iω2 Λ2 + λ)A2 + β11 Λ1 A21 exp(−iσ T1 ) Due to (23), the coefficient determinant of linear algebraic (32) respect to P1n and P2n vanishes. Therefore, the existence of solutions to (32) implies    R1n −iωn λ    (35)  R2n β − ω2  = 0 n

2

2 

Im(Γ2 ) = − β − ω22 3α11 β − ω12 β − ω22

 (41) − 4α11 λ2 ω12 β − ω22



 + β11 λ2 ω1 ω2 β − ω12 /2ω2 β β + λ2 2  − 2βω22 + ω24 β − ω12 β − ω22 Therefore Γn (n = 1, 2) are pure imaginary numbers. Express the solution to (37) in the polar form: 1 An = an exp(iθn ) 2

(42)

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Y.-L. Zhang, L.-O. Chen

Substituting (42) into (37), separating the resulting equations into real and imaginary parts and eliminating θ1 and θ2 yield 1 D1 a1 = − a1 a2 Im(Γ1 ) sin(θ ) 2 1 2 D1 a2 = a1 Im(Γ2 ) sin(θ ) (43) 2 1 a2 D1 θ = a2 σ + a12 Im(Γ2 ) cos(θ ) − a22 Im(Γ1 ) cos(θ ) 2 where θ = θ2 − 2θ1 + σ T1 . For the steady-state response, the amplitude an (n = 1, 2) and the new phase angle γ in (43) should be constants. Setting D1 an = 0 and D1 θ = 0, one derives the non-trivial periodic solution with amplitudes given by 2[Im(Γ1 )a22 ∓ σ a2 ] (44) a1 = Im(Γ2 ) Equation (44) reveals that the amplitude a1 and a2 are associated with the detuning parameterσ in the internal resonance. Choose the fluid-pipe mass ratio Mr = 0.447, initial tension parameter T¯ = −5, and the nondimensional parameter γ = 4. It is found that for nondimensional mean flow velocity u = 5.02655, the natural frequency of the second mode is approximately equal to two times that of the first mode (ω2 ≈ 2ω1 + εσ ) indicating the presence of 2 : 1 internal resonance. For different detuning parameters, the relations between the two amplitudes are shown in Fig. 4 (εσ = 0.01, 0.5, 1.0) by the solid lines. It should be pointed out that only the negative sign in (44) is adopted because the numerical simulations indicate the instability of (44) with the positive sign. The corresponding numerical results by solving (43) are also shown in Fig. 4 by the dots. Both the analytical and numerical results demonstrate that the amplitude of the first mode increases with the amplitude of the second mode. The analytical and numerical results have good agreement in the internal resonance.

5 Numerical results To explore the validation of the multi-scale analysis, the analytical results are compared with the numerical solution to (15) for different values of the small

Fig. 4 Comparison of the perturbation and numerical solutions for the amplitude with different detuning parameters εσ = 0.01, 0.5, 1.0

parameter. The time histories of two generalized coordinates can be analytically predicted by the (19) with (22), (42) and (44). For exceptionally small ε, the analytical results are in prefect agreement with the numerical ones, as shown in Fig. 5 in which ε = 0.01 and initial conditions are q1 (0) = 0.0891, q2 (0) = 0, q˙1 (0) = 0, q˙2 (0) = 0. For reasonably small ε, the analytical results are in good agreement with the numerical ones, as shown in Fig. 6 in which ε = 0.1 and initial conditions are q1 (0) = 0.0896, q2 (0) = 0, q˙1 (0) = 0, q˙2 (0) = 0. For rather large ε, the frequency predicted by the analytical method is still in good agreement with the numerical one while the amplitude has certain differences, as shown in Fig. 7 in which ε = 0.5 and initial conditions are q1 (0) = 0.0937, q2 (0) = 0, q˙1 (0) = 0, q˙2 (0) = 0. For very large ε, the analytical results are totally different from the numerical results, as shown in Fig. 8 in which ε = 0.5 and initial conditions are q1 (0) = 0.1047, q2 (0) = 0, q˙1 (0) = 0, q˙2 (0) = 0. The calculations confirm that the analytical results are valid for small ε.

6 Conclusions Previous work in the pipe conveying fluid focused on nonlinear vibration about the trivial equilibria. However, if the flow speed is larger than the critical value, the straight pipe equilibrium configuration becomes unstable and bifurcates into two possible curved equilibria. This paper is devoted to analyzing the internal resonance of the pipe conveying fluid in the supercritical regime. The disturbance equations about the non-

Internal resonance of pipes conveying fluid in the supercritical regime Fig. 5 Comparisons between the analytical and numerical results for exceptionally small ε

Fig. 6 Comparisons between the analytical and numerical results for reasonably small ε

Fig. 7 Comparisons between the analytical and numerical results for rather large ε

Fig. 8 Comparisons between the analytical and numerical results for very large ε

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trivial equilibria are derived. It is found that two-toone internet resonance may occur under the certain conditions. The method of multiple scales is applied to establish the relationship between the amplitude of steady-state response in the two modes in the internet resonance. The amplitude in the first mode increase with the amplitude in the second mode. For explore the validation of the multi-scale analysis, dynamic behaviors of the system are presented in the form of time histories by the numerical method. The analytical results are numerically confirmed in case of weak nonlinearities. Acknowledgements This work was supported by the National Outstanding Young Scientists Foundation of China (No. 10725209), the National Natural Science Foundation of China (No. 10902064), Shanghai Subject Chief Scientist Project (No. 09XD1401700), Shanghai Leading Talent Program, Shanghai Leading Academic Discipline Project (No. S30106), Innovation Foundation for Graduates of Shanghai University Project (No. SHUCX111011) and the program for Changjiang Scholars and Innovative Research Team in University (No. IRT0844).

References 1. Ibrahim, R.A.: Overview of mechanics of pipes conveying fluids—Part I: Fundamental studies. J. Press. Vessel Technol. 132(3), 034001 (2010) 2. Païdoussis, M.P.: Fluid-Structure Interactions: Slender Structures and Axial Flow, vol. 1. Academic Press, London (1998) 3. Païdoussis, M.P.: Fluid-Structure Interactions: Slender Structures and Axial Flow, vol. 2. Academic Press, London (2004)

Y.-L. Zhang, L.-O. Chen 4. Païdoussis, M.P., Issid, N.T.: Dynamic stability of pipes conveying fluid. J. Sound Vib. 33(3), 267–294 (1974) 5. Holmes, P.J.: Bifurcations to divergence and flutter in flowinduced oscillations: a finite dimensional analysis. J. Sound Vib. 53, 471–503 (1977) 6. Yoshizawa, M., Nao, H., Hasegawa, E., Tsujioka, Y.: Buckling and postbuckling behavior of a flexible pipe conveying fluid. Bull. JSME 28(240), 1218–1225 (1985) 7. Nikoli´c, M., Rajkovi´c, M.: Bifurcations in nonlinear models of fluid-conveying pipes supported at both ends. J. Fluids Struct. 22(2), 173–195 (2006) 8. Plaut, R.H.: Postbuckling and vibration of end-supported elastica pipes conveying fluid and columns under follower loads. J. Sound Vib. 289(1–2), 264–277 (2006) 9. Modarres-Sadeghi, Y., Païdoussis, M.P.: Nonlinear dynamics of extensible fluid-conveying pipes, supported at both ends. J. Fluids Struct. 25(3), 535–543 (2009) 10. Ghayesh, M.H., Païdoussis, M.P.: Three-dimensional dynamics of a cantilevered pipe conveying fluid, additionally supported by an intermediate spring array. Int. J. NonLinear Mech. 45(5), 507–524 (2010) 11. Ghayesh, M.H., Païdoussis, M.P., Modarres-Sadeghi, Y.: Three-dimensional dynamics of a fluid-conveying cantilevered pipe fitted with an additional spring-support and an end-mass. J. Sound Vib. 330, 2869–2899 (2011) 12. Xu, J., Yang, Q.B.: Flow-induced internal resonances and mode exchange in horizontal cantilevered pipe conveying fluid. Appl. Math. Mech. 27, 819–832 (2006) 13. Panda, L.N., Kar, R.C.: Nonlinear dynamics of a pipe conveying pulsating fluid with parametric and internal resonances. Nonlinear Dyn. 49, 9–30 (2007) 14. Panda, L.N., Kar, R.C.: Nonlinear dynamics of a pipe conveying pulsating fluid with combination, principal parametric and internal resonances. J. Sound Vib. 309, 375–406 (2008) 15. Wickert, J.A.: Non-linear vibration of a traveling tensioned beam. Int. J. Non-Linear Mech. 27, 503–517 (1992) 16. Nayfeh, A.H., Mook, D.T.: Nonlinear Oscillations. Wiley, New York (1979)