Published: 31 March 2014

Dynamic stiffness method for free vibration analysis of variable diameter pipe conveying fluid

Liu Yongshou1
Zhang Zijun2
Li Baohui3
Gao Hangshan4
1, 2, 3, 4Department of Engineering Mechanics, Northwestern Polytechnical University, Xi’an, P. R. China
Corresponding Author:
Zhang Zijun
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Abstract

The governing equation of flow-induced vibration is deduced in terms of Hamilton’s principle for a variable diameter pipe conveying axial steady flow. Frobenius method is applied to analyze the governing equation. After the recursion formulas of coefficients are obtained, dynamic stiffness method is proposed for free vibration analysis of the variable diameter pipe conveying fluid. In the example, the natural frequencies of uniform pipes conveying fluid are computed and comparisons are made to validate the dynamic stiffness method. Then, the natural frequencies and modal shapes are obtained for the variable diameter pipe conveying fluid with different section variation coefficients and fluid velocities.

1. Introduction

Fluid-conveying pipe is widely used in nuclear industry, petroleum industry, and many other industries. The failure caused by the interaction between fluid and pipe wall occurs very frequently in practice. So flow-induced vibration of pipe conveying fluid attracts more and more interests of researchers. Ashley and Haviland [1] begun the research of bending vibrations of pipe line containing flowing fluid. Paidoussis [2] has studied the dynamic characteristics of pipe conveying fluid for many years, the great achievement he accomplished in nonlinear vibration of pipes set the foundation of pipe dynamics. Recently, flow-induced vibration of pipe conveying fluid is researched more exhaustively. Doare et al. [3] studied the dissipation effect on the local and global stability of pipe conveying fluid and proposed a numerical method to analyze the stability of finite-length system. Jung, Chung [4] investigated the stability of semi-circular pipe conveying harmonically oscillating fluid. They considered the extensibility and nonlinearity of the fluid-conveying pipe with different boundary conditions through Hamilton principle. Rinaldi, Prabhakar [5] studied micro-scale resonators containing internal flow, modeled as micro-fabricated pipe conveying fluid, and investigated the effects of flow velocity on damping, stability, and frequency shift. Huang, Liu [6] studied the natural frequencies of fluid conveying pipe with different boundary conditions through Garlerkin approach. Ragulskis, Fedaravicius [7] have developed a harmonic balance method in the FEM analysis of the fluid in pipe with taking the interactions between the vibrating pipe and fluid flow into consideration.

Uniform pipe was always adopted in the flow-induced vibration analysis of pipe conveying fluid. However, fluid-conveying pipes with variable section are widespread in nuclear engineering, such as pipe-expanding and nozzle etc. Hannoyer and Paidoussis [8, 9] have studied the dynamics of a slender tapered beam with internal and external flow in theory and experiment. Based on the researches of Hannoyer and Paidoussis [8, 9], we studied the dynamic characteristics of variable diameter pipes conveying fluid through stiffness matrix method. Natural frequencies and modal shapes of the variable diameter pipe were calculated in this paper and contrasts were made between the results in this paper and previous researches, which prove that the method employed in this paper is reasonable.

2. Motion equation of variable diameter pipe conveying fluid

Here, we use Hamilton’s principle to build the dynamic model of variable diameter pipe conveying fluid. Because that the fluid-conveying pipe is an open system. The Hamilton’s principle for open system is used here, i.e.:

1
δt1t2Ldt+t1t2δWdt-t1t2mfVxr˙e +Vxetδredt=0,

where, L is the system’s Lagrange function, that is L=Tk-Tp, where Tk, Tp stand for the kinetic energy and potential energy of the system respectively. δW is the work done by non-conservative force, mf is the fluid mass per unit length, re is the position vector of pipe exit. et is the unit vector in tangent direction of the deformed pipeline, as shown in Fig. 1.

In the coordinate shown in Fig. 1, the absolute fluid velocity can be written as follow:

2
Va(x)=u˙+Vxu'e1+w˙+Vxw'e2.

In Eq. (2), V(x) stands for the velocity of the fluid relative to the pipe. e1 and e2 are unit vectors in x- and y-directions. ' and ˙ stand for the partial derivatives with respect to x (coordinate) and t (time). According to the law of conservation of fluid mass, as shown in Eq. (3), the velocity of the fluid is change along the length of the conical pipe. So we use V(x) and Va(x) here:

3
VxAfx=C,

where C is a constant.

Fig. 1The displacement sketch of simply supported pipe

The displacement sketch of simply supported pipe

Fig. 2The sketch of variable diameter pipe

The sketch of variable diameter pipe

The sketch of a variable diameter pipe is shown in Fig. 2. The fluid flows from left end to right end. The outer and inner diameters of left end of the pipe are Di and dis respectively. The non-uniformity of the pipe is characterized by the diameter variation coefficient γ. The outer and inner diameters of the section at coordinate x can be expressed as D(x)=1-γxLDi and d(x)=1-γxLdi respectively, where L is length of the pipe. Because that the area of the fluid is Af(x)=14πd2(x), the velocity of the fluid along the length of the pipe can be expressed as:

4
Vx=C1-γxL2di2.

Given that the flow velocity at the entrance of the pipe is V0, then Eq. (4) can be transformed into following form:

5
Vx=11-γxL2V0.

Now, we use Hamilton’s principle for open system to deduce the motion equation of variable diameter pipe conveying fluid. The kinetic energy of fluid in pipe can be written in the following form:

6
Tkf=120LρfAfxVa2xdx=120LρfAfxw˙2+V2x+2w˙w'Vx+2Vxu˙dx.

Here, the axial inextensible assumption was used, i.e. ux2+wx2=1. And high order infinitesimals have been ignored in Eq. (6). ρf is the density of fluid in pipe, Af(x) is area of the fluid section located at x. mf(x)dx=ρfAf(x)dx is mass of the fluid element analyzed.

The kinetic energy of the pipe is as follow:

7
Tkp=120LρpApxw˙2dx=120Lmp(x)w˙2dx,

where, ρp is density of the pipe material, Ap(x) is area of the pipe cross section at x. mp(x)dx=ρpAp(x)dx is mass of the pipe element.

The potential energy of the pipe is as follow:

8
Tpp=12E0LIxw2dx.

In Eq. (8), Ix is the inertial moment of the pipe section. Obviously, Ix also changes along xaxis. Only considering the work done by fluid pressure, the work done by non-conservative force is zero [2], that is:

9
δW=0.

Substituting Eq. (6)-(9) into Eq. (1), the following equation can be obtained:

10
δt1t2120L[mf(w˙2+V2+2w˙w'V+2Vu˙)+mpw˙2-EIxw2]dxdt-t1t2mfVr˙e+Vetδredt=0.

Here, we take the simply supported pipe (as shown in Fig. 1) for example to deduce the lateral vibration equation of the variable diameter pipe conveying fluid.

The boundary conditions of simply supported pipe are:

11
w0=w1=0, EI2wx2x=0=EI2wx2x=1=0.

At time t1 and t2:

12
δut1=δut2=0,δwt1=δwt2=0.

Combine Eq. (11) with Eq. (12), the following equation is obtained:

13
t1t20LE(Ixw)+(mp+mf)w¨+(mfVw˙'+mf'Vw˙+mfV'w˙)+m'fw'V2+mfwV2+2mfVV'w'δwdxdt=0.

The detail manipulation of the third term in Eq. (10) is as follow:

14
δt1t20LmfVw˙w'dxdt=t1t20L(mfVw'δw˙+mfVw˙δw')dxdt=t1t20L-mfVw˙'-Vmfw˙' δwdxdt=-t1t20LmfVw˙'+mf'Vw˙+mfV'w˙δwdxdt.

Considering the assumption of axial inextensible, we obtain that ue=-120Lw'2dx, where the subscript ‘e’ means “exit”, and the following equation can be gained:

15
t1t2mfVr˙e+Vetδredt=t1t2mfV2δuedt=-t1t20LmfV2w'δw'dxdt=t1t20Lmfw'V2'δwdxdt=t1t20Lm'fw'V2+mfwV2+2mfVV'w'δwdxdt.

Now, we can get the motion equation of variable diameter pipe conveying fluid as follow:

16
EIxw''+mp+mfw¨+mfVw˙'+mfVw˙+m'fV'w˙+m'fw'V2+mfwV2+2mfVV'w'=0.

3. Free vibration analysis of variable diameter pipe

3.1. Frobenius method

To harmonic vibration, we have:

17
wx,t=Wxeiωt.

Substituting Eq. (17) into Eq. (16), we can get the following ordinary differential equation:

18
EIxd4Wdx4+2EdIxdxd3Wdx3+Ed2Ixdx2+mfV2d2Wdx2+mfViω+mf'V2+2mfVV'dWdx+mf'V+mfV'iω-mp+mfω2W=0,

where:

19
mfx=ρfAfx=ρfπ1-γxL2di24=1+α1xL+α2x2L2ρfAf0,
mpx=ρpApx=ρfπ1-γxL2Di2-di24=1+α1xL+α2x2L2ρpAp0,

and

20
Ix=πDx4-dx464=π1-γxL4Di4-di464=1+β1xL+β2x2L2+β3x3L3+β4x4L4I0.

For the purpose of analyzing Eq. (18) conveniently, the following dimensionless parameters are introduced here:

21
ξ=xL, η=WL,   Ω=ρfAf0+ρpAp0EI0ωL2,
v0=ρfAf0EI0LV0, β=ρfAf(0)ρfAf(0)+ρpAp(0).

The dimensionless form of Eq. (18) is as follow:

22
1+β1ξ+β2ξ2+β3ξ3+β4ξ4d4ηdξ4+2β1+2β2ξ+3β3ξ2+4β4ξ3d3ηdξ3+2β2+6β3ξ+12β4ξ2+1+α1ξ+α2ξ2v021-γξ4d2ηdξ2+α1+2α2ξv021-γξ4+1+α1ξ+α2ξ2γv021-γξ5+iΩv0141-γξ2βdηdξ+iΩβα1+2α2ξ11-γξ2+1+α1ξ+α2ξ2γv01-γξ3-1+α1ξ+α2ξ2Ω2η=0.

Given that the contraction angle of the conical pipe is very small, that is, (γξ) is sufficiently small so that we can use Tylor series here as:

23
11-γξ=n=0γξn1+γξ+γξ2+γξ3+γξ4+γξ5+γξ6,
1(1-γξ)21+2γξ+3γξ2+4γξ3+5γξ4+6γξ5+7γξ6,
1(1-γξ)31+3γξ+6γξ2+10γξ3+15γξ4+21γξ5+28γξ6,
1(1-γξ)41+4γξ+10γξ2+20γξ3+35γξ4+56γξ5+84γξ6,
1(1-γξ)51+5γξ+15γξ2+35γξ3+70γξ4+126γξ5+210γξ6.

Here, we apply Frobenius method to analyze Eq. (22). Given that the solution of Eq. (22) takes the following form:

24
ηξ=n=0anξn+k, a00.

In which an is the coefficient of the polynomial. The derivatives of η(ξ) are:

25
dηdξ=n=0ann+kξn+k-1, d2ηdξ2=n=0ann+kn+k-1ξn+k-2,
d3ηdξ3=n=0ann+kn+k-1n+k-2ξn+k-3,
d4ηdξ4=n=0an(n+k)(n+k-1)(n+k-2)(n+k-3)ξn+k-4.

When n=0, substituting Eq. (24) and Eq. (25) into Eq. (22), because that the lowest power of ξ equals to zero, so the following equation can be obtained:

26
a0k(k-1)(k-2)(k-3)=0.

Because that a00, so we obtain:

27
k(k-1)(k-2)(k-3)=0.

Eq. (27) is namely indicial equation and it plays an important role in the analysis [10]. Obviously, Eq. (27) have four roots, i.e. kj= 0, 1, 2, 3 (j= 1, 2, 3, 4) respectively. To each k, only one coefficient an(k) is related. Substituting Eq. (24) and Eq. (25) into Eq. (22), then all the coefficients can be obtained:

28a
V0=0,
28b
a1k=-β1k-1k+1a0k,
28c
a2k=-β1kk+2a1k-β2kk-1+ζ1k+2k+1a0k,
28d
a3k=-β1k+1k+3a2k-β2kk+1+ζ1k+3k+2a1k
-β3k+1kk-1+α1ζ1k+4ζ1γk-1+ζ1+ζ2k+3k+2k+1a0k,
28e
a4k=-β1k+2k+4a3k-β2k+2k+1+ζ1k+4k+3a2k
-β3k+2k+1k+α1ζ1k+1+4γζ1k+ζ1+ζ2k+4k+3k+2a1(k)-β4k+2k+1kk-1+4α1γ+10γ2+α2ζ1kk-1+4α1γ+2α2+5γ+α1ζ1+2γζ2k+α1ζ3+ζ4-ζ5(k+4)(k+3)(k+2)(k+1)a0(k),
28f
an+5k=-β1k+n+3k+n+5an+4k-β2k+n+3k+n+2+ζ1k+n+5k+n+4an+3k
-β3k+n+3k+n+2k+n+1+α1ζ1k+n+2+4γζ1k+n+1+ζ1+ζ2k+n+5k+n+4k+n+3an+2k-β4k+n+3k+n+2k+n+1k+n+4α1γζ1+10γ2ζ1+α2k+n+1k+n+4α1γ+2α2+5γ+α1ζ1+2γζ2k+n+1+α1ζ3+ζ4-ζ5k+n+5k+n+4k+n+3k+n+2an+1k-ζ32α2+2α1γ+ζ43γ+α1-ζ5α1k+n+5k+n+4k+n+3k+n+2ank,

where:

29
ζ1=v02, ζ2=iΩv0β, ζ3=iΩβ, ζ4=iΩβγ, ζ5=Ω2.

Here, the high orders of γ have been neglected. After getting these coefficients, the solution of Eq. (22) can be written in the following form:

30
ηξ=A1f1ξ+A2f2ξ+A3f3ξ+A4f4ξ,

where Aj is constant. The expression of fj(ξ) is as follow:

31
f1(ξ)=a0(k1)+a1(k1)ξ+a2(k1)ξ2+a3(k1)ξ3+,
f2(ξ)=a0(k2)ξ+a1(k2)ξ2+a2(k2)ξ3+a3(k2)ξ4+,
f3ξ=a0(k3)ξ2+a1(k3)ξ3+a2(k3)ξ4+a3(k3)ξ5+,
f4(ξ)=a0(k4)ξ3+a1(k4)ξ4+a2(k4)ξ5+a3(k4)ξ6+.

Fig. 3The sketch of node force

The sketch of node force

3.2. Dynamic stiffness method

Here, we use dynamic stiffness method to analyze vibration of variable diameter pipe conveying fluid. As shown in Fig. 3, the node displacement of the pipe is:

32
δe=PA,

where:

33
δe=η1η1'η1η2', A=A1A2A3A4,
P=a0k1000a1k1a0k200f11f21f31f41f1'1f2'1f3'1f4'1.

As shown in Fig. 3, the shear force and bending moment can be expressed as:

34
M=EIx2wx2,
Q=-Mx=-EIx3wx3-EIxx2wx2.

Introducing the dimensionless form of shear force and bending moment as follows:

35
M-=MLEIi=1+β1ξ+β2ξ2+β3ξ3+β4ξ4d2ηdξ2,
Q-=QL2EIi=-1+β1ξ+β2ξ2+β3ξ3+β4ξ4d3ηdξ3
-β1+2β2ξ+3β3ξ2+4β4ξ3d2ηdξ2.

The directions of M, Q are shown in Fig. 3, then it is easy to get the vector of node force:

36
F=[Q-1M-1Q-2M-2]T,

where Q-1=-Q-(0), M-1=-M-(0), Q-2=Q-(1), M-2=M-(1).

The relation between vector of node force and vector A can now be expressed as follow:

37
F=BA,

where:

38
B=f1'''0+2β1f1''0f2'''0+2β1f2''0f3'''0+2β1f3''0f4'''0+2β1f4''0-f1''0-f2''0-f3''0-f4''0σ1f1'''1-σ2f1''1σ1f2'''1-σ2f2''1σ1f3'''1-σ2f3''1σ1f4'''1-σ2f4''1f1''1f2''1f3''1f4''1.

In which σ1=1+β1+β2+β3+β4, σ2=β1+2β2+3β3+4β4, and what should be noted here is that, ' stands for the derivative with respect to ξ, which is different from that in Section 2.

According to Eq. (32), we can get the coefficient vector as follow:

39
A=P-1δe.

Now, substituting Eq. (39) into Eq. (37), the following result will be gained:

40
F=BP-1δe=Sδe.

Here, the matrix S is named as the dynamic stiffness matrix of the system. Once the dynamic stiffness matrix is obtained, the natural frequencies and modal shapes can be easily obtained.

Here, we take the simply supported pipe for an example to illuminate how to compute natural frequencies of the pipe through dynamic stiffness method. The boundary conditions of simply supported pipe can be written as:

41
ξ=0:η1=0, M-1=0,
ξ=1:η2=0, M-2=0.

Rewriting Eq. (41) in the matrix form, we have:

42
Q-10Q-20=S11S12S13S14S21S22S23S24S31S32S33S34S41S42S43S440η1'0η2'.

The following characteristic equation can be obtained from Eq. (42):

43
S22S24S42S44η1'η2'=0.

It is easy to find from Eq. (43) that the following equation must be satisfied:

44
h(ω)=detS22S24S42S44=0.

The natural frequencies can be obtained from Eq. (44). In the same way, we can get the characteristic equation for cantilevered pipe and clamped-pinned pipe conveying fluid as follows:

Cantilevered pipe (left clamped and right free):

45
h(ω)=detS33S34S43S44=0.

Clamped-pinned pipe (left clamped and right pinned):

46
h(ω)=detS11S12S14S21S22S24S41S42S44=0.

4. Examples

4.1. Uniform pipe conveying fluid

In order to validate the proposed method, we calculated the natural frequencies of a uniform pipe conveying fluid. The uniform pipe is actually the case of variable diameter pipe with section variation coefficient γ=0. The pipe and fluid parameters are chosen the same as previous reference, that is, Yang’s modulus is E=210 GPa, the outer and inner diameters at origin of the pipe are Di=324 mm and di=292 mm respectively, and the pipe span is L=32 m, the density of pipe material is ρp=8200 kg/m3, the density of fluid is ρf=908.2 kg/m3 [11]. The first five orders of natural frequencies are computed under three different velocities, i.e. V=0, V=15 m/s, V=25 m/s respectively. And comparisons are presented between the results obtained by the proposed method and that published in the research of Housner [11]. The computation results of natural frequencies and error are listed in Table 1.

Table 1Natural frequencies of uniform simply supported pipe conveying fluid under different fluid velocities (rad/s)

Fluid velocity
Natural frequency
ω1
ω2
ω3
ω4
ω5
V=0
This paper
4.373
17.493
39.359
69.971
109.330
Reference
4.3732
17.4928
39.3587
error (%)
0.004
0.001
0.0007
V=15 m/s
This paper
4.287
17.417
39.286
69.900
109.259
Reference
4.2870
17.4171
39.2858
error (%)
0
0.0006
0.0005
V=25 m/s
This paper
4.131
17.282
39.156
69.772
109.133
References
4.1293
17.2816
39.1559
error (%)
0.04
0.002
0.0003

It can be found from Table 1 that a well agreement is obtained between the results in this paper and that in reference.

Fig. 4Curves of lg(h)-frequency under different fluid velocities: a) V=0, b) V=15 m/s, c) V=25 m/s

Curves of lg(h)-frequency under different fluid velocities: a) V=0, b) V=15 m/s, c) V=25 m/s

a)

Curves of lg(h)-frequency under different fluid velocities: a) V=0, b) V=15 m/s, c) V=25 m/s

b)

Curves of lg(h)-frequency under different fluid velocities: a) V=0, b) V=15 m/s, c) V=25 m/s

c)

Fig. 4 shows the curve of lgh(ω) versus frequency ω under different fluid velocities, and the natural frequencies have been marked in the figure. It should be noted here that when the slope of the curve is negative infinite, the real part and the imaginary part of h(ω) are both zeros. In another word, the frequencies related to the negative cuspidal points are the natural frequencies of the fluid-conveying pipe.

4.2. Variable diameter pipe conveying fluid

To simply supported variable diameter pipes conveying fluid, we choose three different diameter variation coefficients, that is 0.1, 0.2, 0.3, respectively. Given that the Yang’s modulus of the pipe material is E=70 GPa, the outer and inner diameters of pipe at the origin of the pipe are Di=80 mm and di=72 mm respectively. The pipe length is L=15 m, the density of pipe material is ρp=2800 kg/m3, and the density of fluid is V0=25 m/s. The natural frequencies are computed under the following three different fluid velocities: V=0,V=15 m/s,V=25 m/s. The results are listed in Table 2.

Table 2Natural frequencies of variable diameter pipe conveying fluid with different diameter variation coefficients under different fluid velocities

γ and fluid velocities
Natural frequencies
ω1
ω2
ω3
ω4
ω5
γ=0.1
V0=0
3.54
14.14
31.82
56.56
88.36
V0=10 m/s
3.10
13.80
31.52
56.26
88.08
V0=25 m/s
11.92
29.84
54.68
86.54
γ=0.2
V0=0
3.34
13.48
30.32
53.88
84.18
V0=10 m/s
2.88
13.12
29.98
53.56
83.88
V0=25 m/s
11.10
28.20
51.88
82.84
γ=0.3
V0=0
3.16
12.92
29.04
51.62
80.58
V0=10 m/s
2.66
12.52
28.68
51.28
80.42
V0=25 m/s
10.30
26.76
49.44
77.98

From Table 2, we can find that to the same diameter variation coefficient γ, the natural frequencies decrease as the fluid velocity increases, and that the larger γ we choose, the smaller the natural frequencies we get. The results obtained above agree with that in the researches of Hannoyer and Paidoussis [8].

Additionally, we notice that when the fluid velocity reaches 25 m/s, the first order natural frequency has already vanished. So we can conclude that such velocity has already exceeded the critical velocity of the pipe. Actually, the critical velocities of variable diameter pipe conveying fluid with different diameter variation coefficients are γ= 0.1, Vcr=19.96 m/s;γ= 0.2, Vcr=18.82 m/s; γ= 0.3, Vcr=17.77 m/s, respectively, as shown in Fig. 5.

Fig. 5Curves of first order natural frequency versus fluid velocity

Curves of first order natural frequency versus fluid velocity

Fig. 6Buckling of the variable diameter pipe conveying fluid

Buckling of the variable diameter pipe conveying fluid

When the fluid velocity reaches the critical value, the simply supported variable diameter pipe would yield buckling, as a form of static instability, as shown in Fig. 6. Further, the instability would result in the failure of the pipe.

It is obvious that the larger the diameter variation coefficient is, the smaller the critical velocity we get, which agree with the result in the researches of Hannoyer and Paidoussis [8].

Fig. 7 shows the curves of lgh(ω) versus frequency ω for variable diameter pipe conveying fluid with different diameter variation coefficients under velocity V0=0 and V0=25 m/s.

Fig. 7Curves of log(h)-frequency with different diameter variation ratio under different fluid velocities: a) γ=0.1, V0=0; b) γ=0.1, V0=25 m/s; c) γ=0.2, V0=0; d) γ=0.2, V0=25 m/s; e) γ=0.3, V0=0; f) γ=0.3, V0=25 m/s

Curves of log(h)-frequency with different diameter variation ratio under different fluid velocities: a) γ=0.1, V0=0; b) γ=0.1, V0=25 m/s; c) γ=0.2, V0=0;  d) γ=0.2, V0=25 m/s; e) γ=0.3, V0=0; f) γ=0.3, V0=25 m/s

a)

Curves of log(h)-frequency with different diameter variation ratio under different fluid velocities: a) γ=0.1, V0=0; b) γ=0.1, V0=25 m/s; c) γ=0.2, V0=0;  d) γ=0.2, V0=25 m/s; e) γ=0.3, V0=0; f) γ=0.3, V0=25 m/s

b)

Curves of log(h)-frequency with different diameter variation ratio under different fluid velocities: a) γ=0.1, V0=0; b) γ=0.1, V0=25 m/s; c) γ=0.2, V0=0;  d) γ=0.2, V0=25 m/s; e) γ=0.3, V0=0; f) γ=0.3, V0=25 m/s

c)

Curves of log(h)-frequency with different diameter variation ratio under different fluid velocities: a) γ=0.1, V0=0; b) γ=0.1, V0=25 m/s; c) γ=0.2, V0=0;  d) γ=0.2, V0=25 m/s; e) γ=0.3, V0=0; f) γ=0.3, V0=25 m/s

d)

Curves of log(h)-frequency with different diameter variation ratio under different fluid velocities: a) γ=0.1, V0=0; b) γ=0.1, V0=25 m/s; c) γ=0.2, V0=0;  d) γ=0.2, V0=25 m/s; e) γ=0.3, V0=0; f) γ=0.3, V0=25 m/s

e)

Curves of log(h)-frequency with different diameter variation ratio under different fluid velocities: a) γ=0.1, V0=0; b) γ=0.1, V0=25 m/s; c) γ=0.2, V0=0;  d) γ=0.2, V0=25 m/s; e) γ=0.3, V0=0; f) γ=0.3, V0=25 m/s

f)

After we obtain the natural frequencies of the variable diameter pipe, the modal shapes can be obtained correspondingly. The modal shapes of variable diameter pipe conveying fluid are shown in Fig. 8 and Table 3.

Table 3Dimensionless amplitudes of the modal shape under different flow velocities

Fluid velocity
Amplitudes
1st order
2nd order
3rd order
4th order
V0=0
0.22
0.29
0.21
0.12
V0=10 m/s
0.25
0.31
0.23
0.15
V0=25 m/s
0.30
0.35
0.26
0.18

Table 3 shows that the amplitude of the modal shape increases with increasing flow velocity. This illuminate that the increasing fluid velocity can weaken the pipe stiffness.

Fig. 8The first four orders modal shapes of variable diameter pipe conveying fluid with different diameter variation coefficients under different fluid velocities

The first four orders modal shapes of variable diameter pipe conveying fluid  with different diameter variation coefficients under different fluid velocities

4.3. Variable diameter pipe with damping

Damping is ignored in above calculations. But in practice, damping cannot be neglected. Here we just consider the viscous damping, then the motion equation governing the vibration of the pipe is:

47
EIxw''+mp+mfw¨+mfVw˙'+mfVw˙+m'fV'w˙+cw˙+m'fw'V2+mfwV2+2mfVV'w'=0,

where the term (cw˙) is the viscous damping force, which may be introduced by immersing the pipe in liquid. Suppose that c= 0.05, then we can calculate the natural frequencies of the variable diameter pipe. A contrast is illustrated in Fig. 9 between damping pipe and non-damping pipe (c= 0).

Fig. 9First order natural frequencies of damping pipe and non-damping pipe with variable fluid velocity

First order natural frequencies of damping pipe and non-damping pipe with variable fluid velocity

From Fig. 9, we conclude that damping would decrease the natural frequencies and critical flow velocity of the system. This has been illustrated by Paidoussis [2] in the research for a uniform fluid-conveying pipe. Moreover, damping delays the decreasing speed of natural frequency versus flow velocity. That is, as seen in Fig. 9, there is just a small difference between the two critical flow velocities, but a relatively larger difference between the natural frequencies when flow velocity is zero.

5. Conclusion

In this paper, the dynamic model was built by open system’s Hamilton’s principle on the basis of Euler-Bernoulli beam model and axial inextensible assumption. The motion equation of variable diameter pipe conveying fluid is more complex than that of uniform pipe. We employed Frobenius method to analyze the motion equation and proposed the dynamic stiffness method for free vibration analysis of variable diameter pipe conveying fluid. By using the presented method, the natural frequencies of a uniform pipe conveying fluid under different fluid velocities were obtained and the comparisons between our results and that in previous researches were made. A well agreement validates our method. The natural frequencies and modal shapes were obtained for variable diameter pipe conveying fluid with different diameter variation coefficients under different fluid velocities. We find that the natural frequencies and critical velocities of variable diameter pipe both decreased with increasing diameter variation coefficient.

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About this article

Received
07 December 2013
Accepted
24 January 2014
Published
31 March 2014
Keywords
variable diameter pipe conveying fluid
dynamic stiffness method
natural frequencies
modal shapes
Acknowledgements

This work was supported by Northwestern Polytechnical University Basic Research Fund (Grant No. JC201114) and Aeronautical Science Foundation of China (2011ZA53014).