- Email: [email protected]
In-plane vibration analyses of curved pipes conveying fluid using the generalized differential quadrature rule
Available online at www.sciencedirect.com
Computers and Structures 86 (2008) 133–139 www.elsevier.com/locate/compstruc
In-plane vibration analyses of curved pipes conveying fluid using the generalized differential quadrature rule Wang Lin *, Ni Qiao Department of Mechanics, Huazhong University of Science and Technology, Wuhan 430074, Hubei, People’s Republic of China Received 24 November 2006; accepted 2 May 2007 Available online 18 June 2007
Abstract In-plane vibrations of curved pipes conveying fluid are investigated by using the generalized differential quadrature rule (GDQR) proposed. The ‘‘modified inextensible’’ theory for the curved pipes is considered, and the steady-state combined force is taken into account. Several examples of curved pipes conveying fluid with different boundaries are presented to illustrate the validity of the GDQR. The obtained natural frequencies compare quite well with those predicted by the finite element method. Based on the GDQR, the effect of some key parameters on the natural frequencies of the pipe system is further discussed. Compared with other methods, GDQR is more convenient to deal with the boundary conditions of the pipe and gives acceptable precision in the numerical results. This method can be useful for further study of the nonlinear dynamics of pipes conveying fluid. Ó 2007 Elsevier Ltd. All rights reserved. Keywords: Curved pipes conveying fluid; Generalized differential quadrature rule; Natural frequency; In-plane vibration; Natural frequency; DQM
1. Introduction Vibration and stability of curved pipes conveying fluid has been the subject of increasing attention in current engineering practice encountered often in hydroelectric and nuclear power plants, suction and pressure pipes, and fuel feeding lines in aerospace. Refs. [1–3] have presented a perspective review of the available research on this dynamical problem. If the curved pipe has a constant cross-section, one obtains a governing equation with constant coefficients. Based on this governing equation, one can study the dynamic behaviour of curved pipe conveying fluid. Hence, many methods, such as the finite element method (FEM), analytical method, Galerkin method, multiple scales method, transfer matrix method and differential quadrature method (DQM), have been applied to investigate this dynamic model during the previous decades. Chen [4,5] and Aithal and Steven Gipson [6] have studied curved pipes conveying fluid by an analytical method, where natural fre*
Corresponding author. E-mail address: wanglinfl[email protected] (W. Lin).
0045-7949/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruc.2007.05.011
quencies and critical flow velocities were obtained; however, the equations of motion they used neglect the steady-state combined force due to the flow in curved pipes. Ko and Bert [7] solved the nonlinear governing equation of a curved pipe conveying fluid by the method of multiple scales in conjunction with the Bubnov–Galerkin method, but his process proposed is very complex. Misra et al. [8,9] investigated the vibrations of curved pipes by finite element method in 1988, developed three models: (i) inextensible theory, (ii) extensible theory and (iii) a modified inextensible theory. For pipes with both ends supported, the results predicted by extensible theory are remarkably close to those of the modified inextensible one. Hence, the extensible (or the modified inextensible) theory, in which the real effects neglected by inextensible theory are taken into account, is the correct one. Next, Huang et al. [10] presented a transfer matrix method for solving vibration and stability of curved pipes conveying fluid. Quite recently, Ni and Huang [11], Wang et al. [12–14] employed the conventional DQM to study the dynamic vibrations of curved pipes, the boundary conditions were imposed by using the approximated double d technique. All these methods mentioned in the foregoing
134
W. Lin, N. Qiao / Computers and Structures 86 (2008) 133–139
have proved to be effective in analysis of certain curved pipes conveying fluid, but each method has its own disadvantages. For instance, the finite element method requires a great deal of degrees of freedom as the number of discrete nodes becomes relatively large; the convergence of DQM with d-point technique requires suitable computing parameters. The GDQR has been proposed recently as a general numerical method to solve high-order differential equations and applied to various problems [15–18]. As pointed out [15], the GDQR is a generalization of the conventional differential quadrature method since it can be applied to any finite order differential equations in a strict form. The GDQR employs the same number of independent variables as that of the conditions at a point, while the conventional DQM uses only one function value at one point. Thus the GDQR applications in mechanics include fourth-order differential equations of Bernoulli–Euler beams [16,19], sixth-order differential equations for structural arches [20], eighth-order differential equations for cylindrical roofs [21], and fourth-order equations for static and free vibrational analyses of rectangular and circular plates [22,23]. In this paper, the GDQR is presented to study the inplane vibrations of modified-inextensible curved pipes conveying fluid, which is governed by a sixth-order differential equation [8] and constrained by three boundary conditions at each end. The vibration analysis of curved pipe conveying fluid is much more complex than that of the circular rings or arches, because the curved pipe is conveying internal fluid and displays much richer dynamics. The study will present ample examples for the clamped–clamped, clamped–hinged and hinged–hinged pipes to illustrate the validity of the GDQR. The numerical results are compared with the FEM solutions developed by Misra et al. [8], and it will be seen that the results are quite satisfactory. Moreover, the effect of some other key parameters on the natural frequency of the system is discussed subsequently. 2. Governing equation
w
u
R θ0
V
Fig. 1. A curved pipe conveying fluid.
where prime and dot denote differentiation with respect to f and t, respectively; P0 is the steady-state combined force, which depends on the dimensionless fluid flow velocity, in addition to the gravity loading and the orientation of the pipe. If both ends of the pipe are supported and the gravity effect is neglected, then P0 ¼ u2 [8]. In Eq. (1), several nondimensional parameters are defined as follows: g ¼ w=L; b ¼ mf =ðmf þ mp Þ; t ¼ EI=ðmf þ mp Þ 1=2 t=L2 ;
u ¼ ðmf =EIÞ1=2 LV ; ð2Þ
f ¼ s=L; ba ¼ ma =ðmf þ mp Þ; a ¼ m a =ðmf þ mp Þ; h0 ¼ L=R; b 1=2 D ¼ cL2 = EIðmf þ mp Þ ; D ¼ Dðc=cÞ
in which s is the curvilinear co-ordinate along the deformed centerline; ma and c are the added mass per unit length and the coefficient of viscous damping due to the surrounding fluid, associated with the transverse motion; and c play similar roles in longitudinal motion. It is more convenient to use the GDQR for analyzing the equations of motion in another form by defining the following dimensionless quantities: b ¼ mf =ðmf þ mp Þ; 1=2
ð1Þ
V
O
n ¼ w=R;
Consider curved pipes conveying fluid as shown in Fig. 1. The system consists of a uniformly curved pipe of radius R, length L, mass per unit length mp, effective flexural rigidity EI, and conveying fluid of mass per unit length mf, flowing with a constant velocity V, and the opening angle is h0. Denote the displacements as w and u corresponding to the tangential and radial displacements, respectively. If the modified inextensible theory associated with Ref. [8] are employed, in which the steady-state combined force is taken into account, the non-dimensional equation of motion of the curved pipe can be expressed as [8]: vi u2 giv þ 2h20 g00 þ h40 g g þ 2h20 giv þ h40 g00 þ a Þ€ þ 2b1=2 g00 h20 ð1 þ b g u g_ 000 þ h20 g_ 0 þ ð1 þ ba Þ€ 00 2 2 2 2 0 0 00 00 00 þ Dg_ h0 D g_ þ P g þ h0 g þ h0 P g þ h20 g ¼ 0
θ
v ¼ ðmf =EIÞ
1=2
RV ;
2
s ¼ ½EI=ðmf þ mp Þ t=R ; h ¼ s=Rh0 ; ba ¼ ma =ðmf þ mp Þ; a ¼ m a =ðmf þ mp Þ; h0 ¼ L=R; b 1=2
D1 ¼ cR2 =½EIðmf þ mp Þ
;
ð3Þ
D1 ¼ D1 ðc=cÞ
Hence, Eq. (1) can be rewritten as 1 o6 n 1 o4 n 1 o2 n þ ð2 þ v2 Þ 4 4 þ ð1 þ 2v2 Þ 2 2 þ v2 n 6 6 h0 oh h0 oh h0 oh 1
þ 2b2 v
1 1 o4 n 1 o2 n 1 o4 n 2v þ þ 2b ð1 þ b Þ a h30 osoh3 h20 osoh h20 os2 oh2
2 3 a Þ o n þ 1 D1 o n D1 on ð1 þ b os2 h20 oh2 os os ! 4 2 1 on 1 on þ P0 4 4 þ 2 2 2 þ n ¼ 0 h0 oh h0 oh
ð4Þ
W. Lin, N. Qiao / Computers and Structures 86 (2008) 133–139
The boundary conditions which will be considered in the following are expressed by: A: for clamped–clamped pipes, 9 nðh; sÞ ¼ 0 > = /ðh; sÞ ¼ 0 at h ¼ 0 and at h ¼ 1; > ; o/ðh; sÞ=oh ¼ 0
N dr wðxi Þ X ðrÞ ¼ Eij wj dxr j¼1
ð5Þ
ð6Þ
C: for clamped–hinged pipes, 9 nðh; sÞ ¼ 0 > = /ðh; sÞ ¼ 0 at h ¼ 0 and > ; o/ðh; sÞ=oh ¼ 0 9 nðh; sÞ ¼ 0 > = at h ¼ 1 /ðh; sÞ ¼ 0 > ; 3 3 o nðh; sÞ=oh ¼ 0
ð7Þ
For h 2 [0, 1], the dimensionless domain h is divided into N points hi (i = 1, 2, . . ., N) using the cosine type sampling points proposed in Refs. [16,17]. At boundary point h1, one has three boundary conditions as described in Eqs. ð1Þ (5), (6) or (7). Thus, three independent variables n1, n1 ð2Þ and n1 , are used in the GDQR. At boundary point hN, one also has three boundary conditions and three indepenð1Þ ð2Þ dent variables nN, nN and nN . It can be seen that, the governing equation is to be implemented at the non-boundary points, and only their function values are the independent variables. Thus, one has n1 = nN = 3, n2 = n3 = = nN1 = 1. From the theory mentioned in the foregoing, the PN total number of independent variables is M ¼ i¼1 ni ¼ N þ 4. Then Eq. (8) can be written as nðrÞ ðxi Þ ¼
For completeness of the present work, the GDQR is briefed first. To solving high-order differential equation with given conditions at any point, the GDQR is formulated directly as [14,15] nj 1 N X M X dr wðxi Þ X ðrÞ ðlÞ ðrÞ ¼ E w ¼ Eik U k ijl j dxr j¼1 l¼0 k¼1
ði ¼ 1; 2; . . . ; N Þ ð8Þ
in which N is the number of sampling points, ni (i = 1, 2, . . ., N) the number of equations at point which the function has to satisfy, wið0Þ ¼ wðxi Þ ¼ wi function vaðrÞ ðkÞ lue, wðkÞ i ¼ w ðxi Þ (k = 1, 2, . . ., ni 1) its derivatives, E ik ðrÞ (the convenient expression of Eijl Þ the weightingPcoefficient N of the rth-order derivative at point xi, and M ¼ i¼1 ni . The total independent variables Uk can be expressed in series as
ðrÞ
ði ¼ 1; 2; . . . ; N Þ
Eik U k
ð11Þ
where
ðrÞ
þ ð1 þ ba Þ
T
ð0Þ
ð1Þ
ðn 1Þ
; . . . ; wN ; wN ; . . . ; wN N
N þ4 X 1 ð2Þ € a €ni þ P0 ni ¼ 0 E n 1þb 2 ij j j¼1 h0
ði ¼ 2; 3; . . . ; N 1Þ
g ð9Þ
As it is known that the DQM only chooses function values as independent variables, while in the GDQR, the independent variables are chosen as function values and their derivatives of possible lowest order wherever necessary. It can be seen that, if ni = 1 (i = 1, 2, . . ., N) are applied in Eq. (8), the GDQR becomes the DQM as
ð12Þ
It should be noted that the weighting coefficients Eik in Eq. (11) have been explicitly obtained and used in papers and will be directly applied in this study. Based on the GDQR, applying Eq. (11) to governing Eq. (4) leads to " N þ4 X 1 ð6Þ 1 1 ð4Þ ð2Þ E þ 4 ð2 þ v2 ÞEij þ 2 ð1 þ 2v2 ÞEij 6 ij h0 h0 j¼1 h0 # 1 0 ð4Þ 1 0 ð2Þ þ 4 P Eij þ 2 2P Eij nj þ v2 ni h0 h0 ! N þ4 N þ4 X X 1 ð3Þ 1 ð1Þ _ 1 ð2Þ _ 1=2 n þ 2b v E þ E þ D E n D1 n_ i j 1 ij ij 3 2 ij j h h h 0 j¼1 j¼1 0 0
fU g ¼ fU 1 ; U 2 ; . . . ; U k ; . . . ; U m g ðn 1Þ
N þ4 X
fU 1 ; U 2 ; . . . ; U k ; . . . ; U N þ4 g n o ð1Þ ð2Þ ð1Þ ð2Þ ¼ n1 ; n1 ; n1 ; n2 ; n3 ; . . . ; nN 1 ; nN ; nN ; nN
3. The generalized differential quadrature rule
ð1Þ
ð10Þ
k¼1
where / = u/R denotes the dimensionless radial displacement.
ð0Þ
ði ¼ 1; 2; . . . ; N Þ
4. Apply the GDQR to the curved pipes conveying fluid
B: for hinged–hinged pipes, 9 nðh; sÞ ¼ 0 > = at h ¼ 0 and at h ¼ 1; /ðh; sÞ ¼ 0 > ; 3 3 o nðh; sÞ=oh ¼ 0
¼ fw1 ; w1 ; . . . ; w1 1
135
ð13Þ
Similarly, the GDQR’s analogues of the boundary conditions Eqs. (5)–(7) are written according to Eq. (11) as follows: A: for clamped–clamped pipes, ni ¼ 0;
nið1Þ ¼ 0;
nið2Þ ¼ 0 ði ¼ 1 or N Þ
ð14Þ
136
W. Lin, N. Qiao / Computers and Structures 86 (2008) 133–139
B: for clamped–hinged pipes, ni ¼ 0;
nið1Þ
ni ¼ 0;
nið1Þ ¼ 0;
¼ 0;
nð2Þ i N þ4 X
¼ 0 ði ¼ 1Þ ð3Þ
Eij nj ¼ 0 ði ¼ N Þ
ð15aÞ ð15bÞ
j¼1
C: for hinged–hinged pipes, ni ¼ 0;
nið1Þ ¼ 0;
N þ4 X
ð3Þ
Eij nj ¼ 0 ði ¼ 1 or N Þ
ð16Þ
j¼1
where the subscript b denotes elements associated with the boundary points while d the remainder, such as n o ð1Þ ð2Þ ð1Þ ð2Þ fU b g ¼ n1 ; n1 ; n1 ; nN ; nN ; nN ; ð18Þ
Obviously, all the sub-matrixes in Eq. (17) can be determined by Eq. (13) and its corresponding equations of boundary conditions. Eq. (17) forms a dynamical motion equation and can be solved by Newmark method. It should be noted that, as the centerline of the curved pipe is inextensible, the tangential displacement is related with the radial displacement. Thus, if the status of tangential displacement is calculated by solving Eq. (17), the radial one can be obtained by the following relationship: /i ¼
N þ4 X
ð1Þ
Eik nk
ði ¼ 1; 2; . . . ; N Þ
ð19Þ
k¼1
For a self-excited vibration, the solution of Eq. (17) is written in the following form: fU g ¼ fU g expðXsÞ
ð20Þ
where
T T T fU g ¼ fU b g fU d g
ðX2 ½M þ X½G þ ½KÞfU d g ¼ f0g
ð21Þ
fU g is an undetermined function of amplitude, X is a dimensionless frequency related to the circular frequency of motion x by rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mf þ mp 2 X¼R x ð22Þ EI
ð23Þ
To obtain a non-trivial solution of the above equation, it is required that the determinant of the coefficient matrix vanish, namely, detðX2 ½M þ X½G þ ½KÞ ¼ 0
One has (N 2) equations from Eq. (13), and 6 equations from the equations of boundary conditions (Eqs. (14), (15) or (16)). Therefore, there are a total of N 2 + 6 = N + 4 equations corresponding to (N + 4) independent variables. By rearranging Eq. (13) and the equations of boundary conditions, an assembled form is given as follows: ( _ ) Ub ½K bb ½K bd ½0 ½0 fU b g þ
½K db ½K dd ½Gdb ½Gdd fU d g U_ d ( € ) Ub ½0 ½0 þ ð17Þ
¼0 €d ½M db ½M dd U
fU d g ¼ fn2 ; n3 ; . . . ; nN 2 ; nN 1 g
Substituting Eq. (20) into Eq. (17) and after having eliminated fU b g, one can obtain a homogeneous equation, which is corresponding to a generalized eigenvalue problem:
ð24Þ
where [M], [G] and [K] denote the structural mass matrix, damping matrix and stiffness matrix, respectively. The matrix elements of [G] and [K] are associated with some parameters of the system, such as dimensionless flow velocity and mass ratio. Therefore, one can compute the eigenvalue numerically from Eq. (24) and obtain the natural frequency of the curved pipe for different boundary conditions. It ought to be mentioned that, Eq. (24) forms a generalized eigenvalue equation and can be reduced to standard eigenvalue equation. For that purpose, we let
T
fUg ¼ ½fU d g fU d g
ð25Þ
and one obtains ½AfUg ¼ XfUg in which " 1 ½M ½G ½A ¼ ½I
ð26Þ
1
½M ½K ½0
# ð27Þ
5. Numerical results In this section, the first purpose of the numerical calculations is to check the validity of the GDQR applying to the curved pipe systems. As mentioned in the foregoing, the frequency equation is obtained by setting the determinant of coefficients in Eq. (23) equal to zero. It is a functional relation between the frequency and system parameters, such as the total angle, mass ratio and flow velocity. Therefore, if the total angle and mass ratio are chosen to be constant values, the natural frequencies of the curved pipes can be solved from Eq. (24). Consider the in-plane vibrations of a semi-circular pipe conveying a ¼ D1 ¼ D1 ¼ 0, which has been studied fluid with ba ¼ b by Misra et al. [8]. Some results are shown in Figs. 2–4. The natural frequencies obtained are for the three cases of support conditions specified in Eqs. (14), (15) and (16). In these three figures, seventeen sampling points (N = 17) were used to obtain convergence in the calculating scheme. It is shown that the natural frequencies vary slightly as the dimensionless flow velocity increases. It can be clearly observed that the lowest four natural frequencies are reasonably in agreement with the FEM results obtained by Misra et al. [8]. However, the GDQR natural frequencies
30
30
25
25 Dimensionless frequency
Dimensionless frequency
W. Lin, N. Qiao / Computers and Structures 86 (2008) 133–139
20
15
10
5
0
20
15
10
5
0
0.5
1
1.5
2
2.5
0
3
0
0.5
Dimensionless flow velocity Fig. 2. Natural frequencies of a clamped–clamped curved pipe as a function of dimensionless flow velocity for b = 0.5, h0 = p, N = 17 (the real lines denote the GDQR solutions and the dashed denote the FEM solutions).
30
30
25
25
20
15
10
1 1.5 2 2.5 Dimensionless flow velocity
3
Fig. 4. Natural frequencies of a clamped–hinged curved pipe as a function of dimensionless flow velocity for b = 0.5, h0 = p, N = 17 (the real lines denote the GDQR solutions and the dashed denote the FEM solutions).
Dimensionless frequency
Dimensionless frequency
137
20
15
10
5 5
0 0
0
0.5
1
1.5
2
2.5
3
Dimensionless flow velocity Fig. 3. Natural frequencies of a hinged–hinged curved pipe as a function of dimensionless flow velocity for b = 0.5, h0 = p, N = 17 (the real lines denote the GDQR solutions and the dashed denote the FEM solutions).
are always a little less than the FEM solutions, especially for the hinged–hinged support condition. Fig. 5 shows the convergence, with increasing number of sampling points, of the lowest four natural frequencies associated with v = 2. It may be seen that convergence is very fast. For a small number of sampling points (7 or so), the results are not reliable at all. However, for a rather large number of sampling points (e.g., N > 13), the obtained natural frequencies are fairly close to those
8
10
12
14
16
18
Sampling number Fig. 5. Natural frequencies of a clamped–clamped curved pipe as a function of sampling points for b = 0.5, h0 = p, v = 2.0.
obtained utilizing the FEM developed in Ref. [8]. Obviously, a much higher computational cost will be required with a very large number of sampling points. Hence, the GDQR provides an acceptable precision in the calculations for the vibrations of fluid-conveying pipes, if only the number of sampling points is reasonable. It may be noted that, in Ref. [8], the effect of some key parameters (including the mass ratio, opening angles and etc.) on the natural frequency has not been investigated based on the modified inextensible theory. Hence, more extensive calculations have produced the natural frequencies with different mass ratios and opening angles. Figs. 6
138
W. Lin, N. Qiao / Computers and Structures 86 (2008) 133–139 30
Dimensionless frequency
25
20
15
10
5
0 0
0.1
0.2
0.3
0.4
0.5
0.6
Mass ratio Fig. 6. Natural frequencies of a clamped–clamped curved pipe as a function of mass ratio for h0 = p, N = 17, v = 2.0.
Numerical calculations and comparisons show that the GDQR behaves very satisfactorily for every value of the flow velocity and boundary conditions. Compared with the finite element method, a great reduction in the number of degrees of freedom is manifested. The GDQR makes it convenient to directly convert the differential equations to algebraic equations in conjunction with the convenient numerical programming. In the presented study, the GDQR natural frequencies are always slightly smaller than those obtained from the FEM solutions. As a result, the GDQR demonstrates itself as an efficient and accurate method to analysis the vibrations of curved pipes conveying fluid. Moreover, the effect of some key parameters on the natural frequency of the pipe system is also shown by GDQR solutions. To the authors’ best knowledge, this may be the first study of using the GDQR to investigate the vibrations of fluid-conveying pipes. However, the application of this method for analyzing the nonlinear vibrations of pipes conveying fluid will be the subject of further study. Acknowledgements
80
The research was supported by the National Nature Science Foundation of China (No. 10272051) and the Science Foundation of HUST (No. 2006Q003B). The comments and suggestions made by the anonymous referees are also gratefully appreciated.
Dimensionless frquency
60
References
40
20
0
2
3
4
5
Bend angle Fig. 7. Natural frequencies of a clamped–clamped curved pipe as a function of bend angle for b = 0.5, N = 17, v = 2.0.
and 7 show the corresponding results for clamped–clamped curved pipes. When the mass ratio increases, the natural frequencies vary slightly; however, the natural frequencies become smaller quickly as the opening angle increases. It should be noted that, similar results can be achieved for the clamped–hinged and hinged–hinged pipes. 6. Conclusions and discussion The dynamic analysis of curved pipes conveying fluid has been conducted by using a version of the generalized differential quadrature rule in which the boundary conditions can be satisfied exactly.
[1] Paidoussis MP, Li GX. Pipes conveying fluid: a model dynamical problem. J Fluid Struct 1993;7:137–204. [2] Chen SS. Flow-induced vibration of circular structures. Washington: Hemisphere; 1987. [3] Paidoussis MP. Fluid–structure interactions. Slender structures and axial flow, vol. 1. London: Academic Press; 1998. [4] Chen SS. Flow-induced in-plane instabilities of curved pipes. Nucl Eng Des 1972;23:29–38. [5] Chen SS. Vibration and stability of a uniformly curved tube conveying fluid. J Acoust Soc Am 1972;51:223–32. [6] Aithal R, Steven Gipson G. Instability of damped curved pipes. J Eng Mech Div–ASCE 1990;116:77–90. [7] Ko CL, Bert CW. A perturbation solution for non-linear vibration of uniformly curved pipes conveying fluid. Int J Nonlinear Mech 1986;21:315–25. [8] Misra AK, Paidoussis MP, Van KS. On the dynamics of curved pipes transporting fluid, part I: inextensible theory. J Fluid Struct 1988;2:221–44. [9] Misra AK, Paidoussis MP, Van KS. On the dynamics of curved pipes transporting fluid, part II: extensible theory. J Fluid Struct 1988;2:245–61. [10] Huang Y, Zeng G, Wei F. A new matrix method for solving vibration and stability of curved pipes conveying fluid. J Sound Vib 2002;251:215–25. [11] Qiao Ni, Yuying Huang. Differential quadrature method to stability analysis of pipes conveying fluid with spring support. Acta Mech Solida Sin 2000;13:320–7. [12] Lin Wang, Qiao Ni. Hopf bifurcation of a nonlinear restrained curved pipe conveying fluid by differential quadrature method. Acta Mech Solida Sin 2003;16:345–52.
W. Lin, N. Qiao / Computers and Structures 86 (2008) 133–139 [13] Qiao Ni, Lin Wang, Qin Qian. Chaotic transients in a curved fluid conveying tube. Acta Mech Solida Sin 2005;18:207–14. [14] Qiao Ni, Lin Wang, Qin Qian. Bifurcations and chaotic motions of a curved pipe conveying fluid with nonlinear constraints. Comput Struct 2006;84:708–17. [15] Wu TY, Liu GR. A generalization of differential quadrature method. In: Abstract book 20th international congress of theoretical and applied mechanics, Chicago, 2000, p. 119. [16] Wu TY, Liu GR. The differential quadrature as a numerical method to solve the differential equation. Comput Mech 1999;24: 197–205. [17] Wu TY, Liu GR. A generalized differential quadrature rule for initialvalue differential equations. J Sound Vib 2000;233:195–213. [18] Kang KJ, Bert CW, Striz AG. Vibration and buckling analysis of circular arches using DQM. Comput Struct 1996;60:49–57.
139
[19] Liu GR, Wu TY. Vibration analysis of beams using the generalized differential quadrature rule and domain decomposition. J Sound Vib 2001;246:461–81. [20] Wu TY, Liu GR. Application of the generalized differential quadrature rule to sixth-order differential equations. Commun Numer Meth Eng 2000;16:777–84. [21] Wu TY, Liu GR. Application of the generalized differential quadrature rule to eighth-order differential equations. Commun Numer Meth Eng 2001;17:355–64. [22] Wu TY, Liu GR. The generalized differential quadrature rule for fourth-order differential equations. Int J Numer Meth Eng 2001;50:1907–29. [23] Wu TY, Liu GR. Free vibration analysis of circular plates using the generalized differential quadrature rule. Comput Meth Appl Mech Eng 2002;191:5365–80.
Computers and Structures 86 (2008) 133–139 www.elsevier.com/locate/compstruc
In-plane vibration analyses of curved pipes conveying fluid using the generalized differential quadrature rule Wang Lin *, Ni Qiao Department of Mechanics, Huazhong University of Science and Technology, Wuhan 430074, Hubei, People’s Republic of China Received 24 November 2006; accepted 2 May 2007 Available online 18 June 2007
Abstract In-plane vibrations of curved pipes conveying fluid are investigated by using the generalized differential quadrature rule (GDQR) proposed. The ‘‘modified inextensible’’ theory for the curved pipes is considered, and the steady-state combined force is taken into account. Several examples of curved pipes conveying fluid with different boundaries are presented to illustrate the validity of the GDQR. The obtained natural frequencies compare quite well with those predicted by the finite element method. Based on the GDQR, the effect of some key parameters on the natural frequencies of the pipe system is further discussed. Compared with other methods, GDQR is more convenient to deal with the boundary conditions of the pipe and gives acceptable precision in the numerical results. This method can be useful for further study of the nonlinear dynamics of pipes conveying fluid. Ó 2007 Elsevier Ltd. All rights reserved. Keywords: Curved pipes conveying fluid; Generalized differential quadrature rule; Natural frequency; In-plane vibration; Natural frequency; DQM
1. Introduction Vibration and stability of curved pipes conveying fluid has been the subject of increasing attention in current engineering practice encountered often in hydroelectric and nuclear power plants, suction and pressure pipes, and fuel feeding lines in aerospace. Refs. [1–3] have presented a perspective review of the available research on this dynamical problem. If the curved pipe has a constant cross-section, one obtains a governing equation with constant coefficients. Based on this governing equation, one can study the dynamic behaviour of curved pipe conveying fluid. Hence, many methods, such as the finite element method (FEM), analytical method, Galerkin method, multiple scales method, transfer matrix method and differential quadrature method (DQM), have been applied to investigate this dynamic model during the previous decades. Chen [4,5] and Aithal and Steven Gipson [6] have studied curved pipes conveying fluid by an analytical method, where natural fre*
Corresponding author. E-mail address: wanglinfl[email protected] (W. Lin).
0045-7949/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruc.2007.05.011
quencies and critical flow velocities were obtained; however, the equations of motion they used neglect the steady-state combined force due to the flow in curved pipes. Ko and Bert [7] solved the nonlinear governing equation of a curved pipe conveying fluid by the method of multiple scales in conjunction with the Bubnov–Galerkin method, but his process proposed is very complex. Misra et al. [8,9] investigated the vibrations of curved pipes by finite element method in 1988, developed three models: (i) inextensible theory, (ii) extensible theory and (iii) a modified inextensible theory. For pipes with both ends supported, the results predicted by extensible theory are remarkably close to those of the modified inextensible one. Hence, the extensible (or the modified inextensible) theory, in which the real effects neglected by inextensible theory are taken into account, is the correct one. Next, Huang et al. [10] presented a transfer matrix method for solving vibration and stability of curved pipes conveying fluid. Quite recently, Ni and Huang [11], Wang et al. [12–14] employed the conventional DQM to study the dynamic vibrations of curved pipes, the boundary conditions were imposed by using the approximated double d technique. All these methods mentioned in the foregoing
134
W. Lin, N. Qiao / Computers and Structures 86 (2008) 133–139
have proved to be effective in analysis of certain curved pipes conveying fluid, but each method has its own disadvantages. For instance, the finite element method requires a great deal of degrees of freedom as the number of discrete nodes becomes relatively large; the convergence of DQM with d-point technique requires suitable computing parameters. The GDQR has been proposed recently as a general numerical method to solve high-order differential equations and applied to various problems [15–18]. As pointed out [15], the GDQR is a generalization of the conventional differential quadrature method since it can be applied to any finite order differential equations in a strict form. The GDQR employs the same number of independent variables as that of the conditions at a point, while the conventional DQM uses only one function value at one point. Thus the GDQR applications in mechanics include fourth-order differential equations of Bernoulli–Euler beams [16,19], sixth-order differential equations for structural arches [20], eighth-order differential equations for cylindrical roofs [21], and fourth-order equations for static and free vibrational analyses of rectangular and circular plates [22,23]. In this paper, the GDQR is presented to study the inplane vibrations of modified-inextensible curved pipes conveying fluid, which is governed by a sixth-order differential equation [8] and constrained by three boundary conditions at each end. The vibration analysis of curved pipe conveying fluid is much more complex than that of the circular rings or arches, because the curved pipe is conveying internal fluid and displays much richer dynamics. The study will present ample examples for the clamped–clamped, clamped–hinged and hinged–hinged pipes to illustrate the validity of the GDQR. The numerical results are compared with the FEM solutions developed by Misra et al. [8], and it will be seen that the results are quite satisfactory. Moreover, the effect of some other key parameters on the natural frequency of the system is discussed subsequently. 2. Governing equation
w
u
R θ0
V
Fig. 1. A curved pipe conveying fluid.
where prime and dot denote differentiation with respect to f and t, respectively; P0 is the steady-state combined force, which depends on the dimensionless fluid flow velocity, in addition to the gravity loading and the orientation of the pipe. If both ends of the pipe are supported and the gravity effect is neglected, then P0 ¼ u2 [8]. In Eq. (1), several nondimensional parameters are defined as follows: g ¼ w=L; b ¼ mf =ðmf þ mp Þ; t ¼ EI=ðmf þ mp Þ 1=2 t=L2 ;
u ¼ ðmf =EIÞ1=2 LV ; ð2Þ
f ¼ s=L; ba ¼ ma =ðmf þ mp Þ; a ¼ m a =ðmf þ mp Þ; h0 ¼ L=R; b 1=2 D ¼ cL2 = EIðmf þ mp Þ ; D ¼ Dðc=cÞ
in which s is the curvilinear co-ordinate along the deformed centerline; ma and c are the added mass per unit length and the coefficient of viscous damping due to the surrounding fluid, associated with the transverse motion; and c play similar roles in longitudinal motion. It is more convenient to use the GDQR for analyzing the equations of motion in another form by defining the following dimensionless quantities: b ¼ mf =ðmf þ mp Þ; 1=2
ð1Þ
V
O
n ¼ w=R;
Consider curved pipes conveying fluid as shown in Fig. 1. The system consists of a uniformly curved pipe of radius R, length L, mass per unit length mp, effective flexural rigidity EI, and conveying fluid of mass per unit length mf, flowing with a constant velocity V, and the opening angle is h0. Denote the displacements as w and u corresponding to the tangential and radial displacements, respectively. If the modified inextensible theory associated with Ref. [8] are employed, in which the steady-state combined force is taken into account, the non-dimensional equation of motion of the curved pipe can be expressed as [8]: vi u2 giv þ 2h20 g00 þ h40 g g þ 2h20 giv þ h40 g00 þ a Þ€ þ 2b1=2 g00 h20 ð1 þ b g u g_ 000 þ h20 g_ 0 þ ð1 þ ba Þ€ 00 2 2 2 2 0 0 00 00 00 þ Dg_ h0 D g_ þ P g þ h0 g þ h0 P g þ h20 g ¼ 0
θ
v ¼ ðmf =EIÞ
1=2
RV ;
2
s ¼ ½EI=ðmf þ mp Þ t=R ; h ¼ s=Rh0 ; ba ¼ ma =ðmf þ mp Þ; a ¼ m a =ðmf þ mp Þ; h0 ¼ L=R; b 1=2
D1 ¼ cR2 =½EIðmf þ mp Þ
;
ð3Þ
D1 ¼ D1 ðc=cÞ
Hence, Eq. (1) can be rewritten as 1 o6 n 1 o4 n 1 o2 n þ ð2 þ v2 Þ 4 4 þ ð1 þ 2v2 Þ 2 2 þ v2 n 6 6 h0 oh h0 oh h0 oh 1
þ 2b2 v
1 1 o4 n 1 o2 n 1 o4 n 2v þ þ 2b ð1 þ b Þ a h30 osoh3 h20 osoh h20 os2 oh2
2 3 a Þ o n þ 1 D1 o n D1 on ð1 þ b os2 h20 oh2 os os ! 4 2 1 on 1 on þ P0 4 4 þ 2 2 2 þ n ¼ 0 h0 oh h0 oh
ð4Þ
W. Lin, N. Qiao / Computers and Structures 86 (2008) 133–139
The boundary conditions which will be considered in the following are expressed by: A: for clamped–clamped pipes, 9 nðh; sÞ ¼ 0 > = /ðh; sÞ ¼ 0 at h ¼ 0 and at h ¼ 1; > ; o/ðh; sÞ=oh ¼ 0
N dr wðxi Þ X ðrÞ ¼ Eij wj dxr j¼1
ð5Þ
ð6Þ
C: for clamped–hinged pipes, 9 nðh; sÞ ¼ 0 > = /ðh; sÞ ¼ 0 at h ¼ 0 and > ; o/ðh; sÞ=oh ¼ 0 9 nðh; sÞ ¼ 0 > = at h ¼ 1 /ðh; sÞ ¼ 0 > ; 3 3 o nðh; sÞ=oh ¼ 0
ð7Þ
For h 2 [0, 1], the dimensionless domain h is divided into N points hi (i = 1, 2, . . ., N) using the cosine type sampling points proposed in Refs. [16,17]. At boundary point h1, one has three boundary conditions as described in Eqs. ð1Þ (5), (6) or (7). Thus, three independent variables n1, n1 ð2Þ and n1 , are used in the GDQR. At boundary point hN, one also has three boundary conditions and three indepenð1Þ ð2Þ dent variables nN, nN and nN . It can be seen that, the governing equation is to be implemented at the non-boundary points, and only their function values are the independent variables. Thus, one has n1 = nN = 3, n2 = n3 = = nN1 = 1. From the theory mentioned in the foregoing, the PN total number of independent variables is M ¼ i¼1 ni ¼ N þ 4. Then Eq. (8) can be written as nðrÞ ðxi Þ ¼
For completeness of the present work, the GDQR is briefed first. To solving high-order differential equation with given conditions at any point, the GDQR is formulated directly as [14,15] nj 1 N X M X dr wðxi Þ X ðrÞ ðlÞ ðrÞ ¼ E w ¼ Eik U k ijl j dxr j¼1 l¼0 k¼1
ði ¼ 1; 2; . . . ; N Þ ð8Þ
in which N is the number of sampling points, ni (i = 1, 2, . . ., N) the number of equations at point which the function has to satisfy, wið0Þ ¼ wðxi Þ ¼ wi function vaðrÞ ðkÞ lue, wðkÞ i ¼ w ðxi Þ (k = 1, 2, . . ., ni 1) its derivatives, E ik ðrÞ (the convenient expression of Eijl Þ the weightingPcoefficient N of the rth-order derivative at point xi, and M ¼ i¼1 ni . The total independent variables Uk can be expressed in series as
ðrÞ
ði ¼ 1; 2; . . . ; N Þ
Eik U k
ð11Þ
where
ðrÞ
þ ð1 þ ba Þ
T
ð0Þ
ð1Þ
ðn 1Þ
; . . . ; wN ; wN ; . . . ; wN N
N þ4 X 1 ð2Þ € a €ni þ P0 ni ¼ 0 E n 1þb 2 ij j j¼1 h0
ði ¼ 2; 3; . . . ; N 1Þ
g ð9Þ
As it is known that the DQM only chooses function values as independent variables, while in the GDQR, the independent variables are chosen as function values and their derivatives of possible lowest order wherever necessary. It can be seen that, if ni = 1 (i = 1, 2, . . ., N) are applied in Eq. (8), the GDQR becomes the DQM as
ð12Þ
It should be noted that the weighting coefficients Eik in Eq. (11) have been explicitly obtained and used in papers and will be directly applied in this study. Based on the GDQR, applying Eq. (11) to governing Eq. (4) leads to " N þ4 X 1 ð6Þ 1 1 ð4Þ ð2Þ E þ 4 ð2 þ v2 ÞEij þ 2 ð1 þ 2v2 ÞEij 6 ij h0 h0 j¼1 h0 # 1 0 ð4Þ 1 0 ð2Þ þ 4 P Eij þ 2 2P Eij nj þ v2 ni h0 h0 ! N þ4 N þ4 X X 1 ð3Þ 1 ð1Þ _ 1 ð2Þ _ 1=2 n þ 2b v E þ E þ D E n D1 n_ i j 1 ij ij 3 2 ij j h h h 0 j¼1 j¼1 0 0
fU g ¼ fU 1 ; U 2 ; . . . ; U k ; . . . ; U m g ðn 1Þ
N þ4 X
fU 1 ; U 2 ; . . . ; U k ; . . . ; U N þ4 g n o ð1Þ ð2Þ ð1Þ ð2Þ ¼ n1 ; n1 ; n1 ; n2 ; n3 ; . . . ; nN 1 ; nN ; nN ; nN
3. The generalized differential quadrature rule
ð1Þ
ð10Þ
k¼1
where / = u/R denotes the dimensionless radial displacement.
ð0Þ
ði ¼ 1; 2; . . . ; N Þ
4. Apply the GDQR to the curved pipes conveying fluid
B: for hinged–hinged pipes, 9 nðh; sÞ ¼ 0 > = at h ¼ 0 and at h ¼ 1; /ðh; sÞ ¼ 0 > ; 3 3 o nðh; sÞ=oh ¼ 0
¼ fw1 ; w1 ; . . . ; w1 1
135
ð13Þ
Similarly, the GDQR’s analogues of the boundary conditions Eqs. (5)–(7) are written according to Eq. (11) as follows: A: for clamped–clamped pipes, ni ¼ 0;
nið1Þ ¼ 0;
nið2Þ ¼ 0 ði ¼ 1 or N Þ
ð14Þ
136
W. Lin, N. Qiao / Computers and Structures 86 (2008) 133–139
B: for clamped–hinged pipes, ni ¼ 0;
nið1Þ
ni ¼ 0;
nið1Þ ¼ 0;
¼ 0;
nð2Þ i N þ4 X
¼ 0 ði ¼ 1Þ ð3Þ
Eij nj ¼ 0 ði ¼ N Þ
ð15aÞ ð15bÞ
j¼1
C: for hinged–hinged pipes, ni ¼ 0;
nið1Þ ¼ 0;
N þ4 X
ð3Þ
Eij nj ¼ 0 ði ¼ 1 or N Þ
ð16Þ
j¼1
where the subscript b denotes elements associated with the boundary points while d the remainder, such as n o ð1Þ ð2Þ ð1Þ ð2Þ fU b g ¼ n1 ; n1 ; n1 ; nN ; nN ; nN ; ð18Þ
Obviously, all the sub-matrixes in Eq. (17) can be determined by Eq. (13) and its corresponding equations of boundary conditions. Eq. (17) forms a dynamical motion equation and can be solved by Newmark method. It should be noted that, as the centerline of the curved pipe is inextensible, the tangential displacement is related with the radial displacement. Thus, if the status of tangential displacement is calculated by solving Eq. (17), the radial one can be obtained by the following relationship: /i ¼
N þ4 X
ð1Þ
Eik nk
ði ¼ 1; 2; . . . ; N Þ
ð19Þ
k¼1
For a self-excited vibration, the solution of Eq. (17) is written in the following form: fU g ¼ fU g expðXsÞ
ð20Þ
where
T T T fU g ¼ fU b g fU d g
ðX2 ½M þ X½G þ ½KÞfU d g ¼ f0g
ð21Þ
fU g is an undetermined function of amplitude, X is a dimensionless frequency related to the circular frequency of motion x by rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mf þ mp 2 X¼R x ð22Þ EI
ð23Þ
To obtain a non-trivial solution of the above equation, it is required that the determinant of the coefficient matrix vanish, namely, detðX2 ½M þ X½G þ ½KÞ ¼ 0
One has (N 2) equations from Eq. (13), and 6 equations from the equations of boundary conditions (Eqs. (14), (15) or (16)). Therefore, there are a total of N 2 + 6 = N + 4 equations corresponding to (N + 4) independent variables. By rearranging Eq. (13) and the equations of boundary conditions, an assembled form is given as follows: ( _ ) Ub ½K bb ½K bd ½0 ½0 fU b g þ
½K db ½K dd ½Gdb ½Gdd fU d g U_ d ( € ) Ub ½0 ½0 þ ð17Þ
¼0 €d ½M db ½M dd U
fU d g ¼ fn2 ; n3 ; . . . ; nN 2 ; nN 1 g
Substituting Eq. (20) into Eq. (17) and after having eliminated fU b g, one can obtain a homogeneous equation, which is corresponding to a generalized eigenvalue problem:
ð24Þ
where [M], [G] and [K] denote the structural mass matrix, damping matrix and stiffness matrix, respectively. The matrix elements of [G] and [K] are associated with some parameters of the system, such as dimensionless flow velocity and mass ratio. Therefore, one can compute the eigenvalue numerically from Eq. (24) and obtain the natural frequency of the curved pipe for different boundary conditions. It ought to be mentioned that, Eq. (24) forms a generalized eigenvalue equation and can be reduced to standard eigenvalue equation. For that purpose, we let
T
fUg ¼ ½fU d g fU d g
ð25Þ
and one obtains ½AfUg ¼ XfUg in which " 1 ½M ½G ½A ¼ ½I
ð26Þ
1
½M ½K ½0
# ð27Þ
5. Numerical results In this section, the first purpose of the numerical calculations is to check the validity of the GDQR applying to the curved pipe systems. As mentioned in the foregoing, the frequency equation is obtained by setting the determinant of coefficients in Eq. (23) equal to zero. It is a functional relation between the frequency and system parameters, such as the total angle, mass ratio and flow velocity. Therefore, if the total angle and mass ratio are chosen to be constant values, the natural frequencies of the curved pipes can be solved from Eq. (24). Consider the in-plane vibrations of a semi-circular pipe conveying a ¼ D1 ¼ D1 ¼ 0, which has been studied fluid with ba ¼ b by Misra et al. [8]. Some results are shown in Figs. 2–4. The natural frequencies obtained are for the three cases of support conditions specified in Eqs. (14), (15) and (16). In these three figures, seventeen sampling points (N = 17) were used to obtain convergence in the calculating scheme. It is shown that the natural frequencies vary slightly as the dimensionless flow velocity increases. It can be clearly observed that the lowest four natural frequencies are reasonably in agreement with the FEM results obtained by Misra et al. [8]. However, the GDQR natural frequencies
30
30
25
25 Dimensionless frequency
Dimensionless frequency
W. Lin, N. Qiao / Computers and Structures 86 (2008) 133–139
20
15
10
5
0
20
15
10
5
0
0.5
1
1.5
2
2.5
0
3
0
0.5
Dimensionless flow velocity Fig. 2. Natural frequencies of a clamped–clamped curved pipe as a function of dimensionless flow velocity for b = 0.5, h0 = p, N = 17 (the real lines denote the GDQR solutions and the dashed denote the FEM solutions).
30
30
25
25
20
15
10
1 1.5 2 2.5 Dimensionless flow velocity
3
Fig. 4. Natural frequencies of a clamped–hinged curved pipe as a function of dimensionless flow velocity for b = 0.5, h0 = p, N = 17 (the real lines denote the GDQR solutions and the dashed denote the FEM solutions).
Dimensionless frequency
Dimensionless frequency
137
20
15
10
5 5
0 0
0
0.5
1
1.5
2
2.5
3
Dimensionless flow velocity Fig. 3. Natural frequencies of a hinged–hinged curved pipe as a function of dimensionless flow velocity for b = 0.5, h0 = p, N = 17 (the real lines denote the GDQR solutions and the dashed denote the FEM solutions).
are always a little less than the FEM solutions, especially for the hinged–hinged support condition. Fig. 5 shows the convergence, with increasing number of sampling points, of the lowest four natural frequencies associated with v = 2. It may be seen that convergence is very fast. For a small number of sampling points (7 or so), the results are not reliable at all. However, for a rather large number of sampling points (e.g., N > 13), the obtained natural frequencies are fairly close to those
8
10
12
14
16
18
Sampling number Fig. 5. Natural frequencies of a clamped–clamped curved pipe as a function of sampling points for b = 0.5, h0 = p, v = 2.0.
obtained utilizing the FEM developed in Ref. [8]. Obviously, a much higher computational cost will be required with a very large number of sampling points. Hence, the GDQR provides an acceptable precision in the calculations for the vibrations of fluid-conveying pipes, if only the number of sampling points is reasonable. It may be noted that, in Ref. [8], the effect of some key parameters (including the mass ratio, opening angles and etc.) on the natural frequency has not been investigated based on the modified inextensible theory. Hence, more extensive calculations have produced the natural frequencies with different mass ratios and opening angles. Figs. 6
138
W. Lin, N. Qiao / Computers and Structures 86 (2008) 133–139 30
Dimensionless frequency
25
20
15
10
5
0 0
0.1
0.2
0.3
0.4
0.5
0.6
Mass ratio Fig. 6. Natural frequencies of a clamped–clamped curved pipe as a function of mass ratio for h0 = p, N = 17, v = 2.0.
Numerical calculations and comparisons show that the GDQR behaves very satisfactorily for every value of the flow velocity and boundary conditions. Compared with the finite element method, a great reduction in the number of degrees of freedom is manifested. The GDQR makes it convenient to directly convert the differential equations to algebraic equations in conjunction with the convenient numerical programming. In the presented study, the GDQR natural frequencies are always slightly smaller than those obtained from the FEM solutions. As a result, the GDQR demonstrates itself as an efficient and accurate method to analysis the vibrations of curved pipes conveying fluid. Moreover, the effect of some key parameters on the natural frequency of the pipe system is also shown by GDQR solutions. To the authors’ best knowledge, this may be the first study of using the GDQR to investigate the vibrations of fluid-conveying pipes. However, the application of this method for analyzing the nonlinear vibrations of pipes conveying fluid will be the subject of further study. Acknowledgements
80
The research was supported by the National Nature Science Foundation of China (No. 10272051) and the Science Foundation of HUST (No. 2006Q003B). The comments and suggestions made by the anonymous referees are also gratefully appreciated.
Dimensionless frquency
60
References
40
20
0
2
3
4
5
Bend angle Fig. 7. Natural frequencies of a clamped–clamped curved pipe as a function of bend angle for b = 0.5, N = 17, v = 2.0.
and 7 show the corresponding results for clamped–clamped curved pipes. When the mass ratio increases, the natural frequencies vary slightly; however, the natural frequencies become smaller quickly as the opening angle increases. It should be noted that, similar results can be achieved for the clamped–hinged and hinged–hinged pipes. 6. Conclusions and discussion The dynamic analysis of curved pipes conveying fluid has been conducted by using a version of the generalized differential quadrature rule in which the boundary conditions can be satisfied exactly.
[1] Paidoussis MP, Li GX. Pipes conveying fluid: a model dynamical problem. J Fluid Struct 1993;7:137–204. [2] Chen SS. Flow-induced vibration of circular structures. Washington: Hemisphere; 1987. [3] Paidoussis MP. Fluid–structure interactions. Slender structures and axial flow, vol. 1. London: Academic Press; 1998. [4] Chen SS. Flow-induced in-plane instabilities of curved pipes. Nucl Eng Des 1972;23:29–38. [5] Chen SS. Vibration and stability of a uniformly curved tube conveying fluid. J Acoust Soc Am 1972;51:223–32. [6] Aithal R, Steven Gipson G. Instability of damped curved pipes. J Eng Mech Div–ASCE 1990;116:77–90. [7] Ko CL, Bert CW. A perturbation solution for non-linear vibration of uniformly curved pipes conveying fluid. Int J Nonlinear Mech 1986;21:315–25. [8] Misra AK, Paidoussis MP, Van KS. On the dynamics of curved pipes transporting fluid, part I: inextensible theory. J Fluid Struct 1988;2:221–44. [9] Misra AK, Paidoussis MP, Van KS. On the dynamics of curved pipes transporting fluid, part II: extensible theory. J Fluid Struct 1988;2:245–61. [10] Huang Y, Zeng G, Wei F. A new matrix method for solving vibration and stability of curved pipes conveying fluid. J Sound Vib 2002;251:215–25. [11] Qiao Ni, Yuying Huang. Differential quadrature method to stability analysis of pipes conveying fluid with spring support. Acta Mech Solida Sin 2000;13:320–7. [12] Lin Wang, Qiao Ni. Hopf bifurcation of a nonlinear restrained curved pipe conveying fluid by differential quadrature method. Acta Mech Solida Sin 2003;16:345–52.
W. Lin, N. Qiao / Computers and Structures 86 (2008) 133–139 [13] Qiao Ni, Lin Wang, Qin Qian. Chaotic transients in a curved fluid conveying tube. Acta Mech Solida Sin 2005;18:207–14. [14] Qiao Ni, Lin Wang, Qin Qian. Bifurcations and chaotic motions of a curved pipe conveying fluid with nonlinear constraints. Comput Struct 2006;84:708–17. [15] Wu TY, Liu GR. A generalization of differential quadrature method. In: Abstract book 20th international congress of theoretical and applied mechanics, Chicago, 2000, p. 119. [16] Wu TY, Liu GR. The differential quadrature as a numerical method to solve the differential equation. Comput Mech 1999;24: 197–205. [17] Wu TY, Liu GR. A generalized differential quadrature rule for initialvalue differential equations. J Sound Vib 2000;233:195–213. [18] Kang KJ, Bert CW, Striz AG. Vibration and buckling analysis of circular arches using DQM. Comput Struct 1996;60:49–57.
139
[19] Liu GR, Wu TY. Vibration analysis of beams using the generalized differential quadrature rule and domain decomposition. J Sound Vib 2001;246:461–81. [20] Wu TY, Liu GR. Application of the generalized differential quadrature rule to sixth-order differential equations. Commun Numer Meth Eng 2000;16:777–84. [21] Wu TY, Liu GR. Application of the generalized differential quadrature rule to eighth-order differential equations. Commun Numer Meth Eng 2001;17:355–64. [22] Wu TY, Liu GR. The generalized differential quadrature rule for fourth-order differential equations. Int J Numer Meth Eng 2001;50:1907–29. [23] Wu TY, Liu GR. Free vibration analysis of circular plates using the generalized differential quadrature rule. Comput Meth Appl Mech Eng 2002;191:5365–80.