 Open Access
 Total Downloads : 234
 Authors : Shankarachar M Sutar, M. Radhakrishna, P. Ramesh Babu
 Paper ID : IJERTV4IS100408
 Volume & Issue : Volume 04, Issue 10 (October 2015)
 DOI : http://dx.doi.org/10.17577/IJERTV4IS100408
 Published (First Online): 23102015
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Fluid Induced Piping Vibration with Elastically Restrained Different End Supports*

Shankarachar1 , M. Radhakrishna2 Senior Scientist, Chief Scientist, Design and Engineering Division,
CSIRIndian Institute of Chemical Technology, Uppal Road, Tarnaka, Hyderabad, India.
P. Rameshbabu3
Associate Professor, Department of Mechanical Engineering, University college of Engineering, Osmania University, Hyderabad, India
AbstractThe dynamic stability of elastically restrained pipe conveying fluid is investigated in this study. The frequency expression is derived for classical boundary conditions by considering the supports as compliant material with linear and rotational stiffness. A new transcendental frequency equation is developed by using EulerBernoulli beam theory; the equation of motion is derived from energy expressions using the Hamilton's Principle. The natural frequencies are presented for a wide range of restraint parameters. Cases are studied for different boundary conditions Linearly and Rotationally Restrained, Rotational Linearly & Linearly Restrained, RotationalLinearly and Fixed, Rotationally Restrained Guided, are computed and it is noted that as the flow velocity increases the first mode frequency decreases and by varying the mass ratio the frequency increases.
KeywordsElastically Restrained, Frequency, Linearly Restrained, Pipe, Guided Support

INTRODUCTION
The vibration analysis of piping systems is important from the view point of safeguarding the equipment's and pipelines from damage which is mostly applicable in chemical, petrochemical and other allied industries. It is well known that pipeline systems may undergo divergence and flutter type of instability due to fluidstructure interaction.
The dynamic behavior of fluid conveying pipes was predicted first by Ashley and Haviland in 1950 [1]. and later by Housner in 1952. Housner considered a simply supported beam model for the pipeline and analysed it using a series solution approach which showed that critical flow velocity could cause buckling [2]. S. S Rao developed a mathematical model for transverse vibration for elastically restrained conditions for beams [3]. Naguleswaran and Williams developed solutions for natural frequencies in axial mode for HingedHinged, FixedHinged and Fixed Fixed boundary conditions [4]. Chen and Paidoussis developed dynamic stiffness matrix for coupled fluid structure interaction [5], [6]. Huang Yimin considered Galerkins method and obtained natural frequencies for fluid conveying pipeline with different boundary conditions [7]. R.A. Stein and M.W.Tobriner discussed Vibration of pipes containing flowing fluid, in which the effects of foundation modulus, flow velocity and internal pressure on the dynamic stability, frequency response and wave
propagation characteristics of an undamped system was studied [8]. Wang Shizhong, Liu Yulan, Huang Wenhu, had conducted research on solid liquid coupling dynamics of pipe conveying fluid, where they studied the influence of flowing velocity, pressure, solidliquid coupling damping and solidliquid coupling stiffness on natural frequency for simply supported ends [9]. Weaver D.S and Unny T.E. studied the dynamic stability of finite length of pipe conveying fluid using FluggeKempner equation to find the critical flow velocities [10].
In most of the cases, the differential equation of motion of fluidconveyed pipe is deduced using the Galerkins method in Lagrange system. Subsequently, the solution of the differential equation is obtained by considering many numerical methods such as transfer matrix, finite element, perturbation, RungeKutta and differential quadrature.
It is the need to have a better understanding of the dynamics of the pipes conveying fluid. The various important factors that influence the dynamic behaviour of fluid conveying pipe are (i) Flow Velocity (ii) Support conditions and (iii) Interaction with supporting medium. Hence, the estimation of exact natural frequencies of pipes is presented with exact approach for finding the transverse vibration of elastically restrained pipes.
Nomenclature
EI bending stiffness of a pipe mp mass of pipe

velocity of the fluid

lateral deflection of pipe mass ratio
nondimensional parameter mf fluid mass
t time

the axial coordinate


nondimensional Velocity
& coefficient of trigonometric function
natural boundary conditions.


MATHEMATICAL MODEL

FORMULATION OF PROBLEM
The governing differential equation of motion and boundary conditions corresponding to the transverse vibration of pipe has been derived by considering the equilibrium approach. Consider a straight uniform single span pipe conveying fluid of length L where K1 and K2 are translational and kt1 and kt2 are rotational stiffness parameters as shown in Figure1. The displacement of the pipe is assumed to be restrained in the zx plane.
Fig.1. Elastically restrained pipe conveying fluid and with both ends supported
Where mp is the mass per unit length of pipe, mf is the mass of fluid per unit length, U is the flow velocity, E is the elastic modulus of the pipeline material, I is the moment of inertia of crosssection of the pipe and z is the lateral deflection of the pipeline; x and t are the axial coordinate and time, respectively. Considering only time dependent variables, and after simplification, we obtain the following transverse vibration equation of the pipeline conveying fluid.
From first principles and applying the EulerBernoulli beam theory and Hamilton's energy equations for the elastically restrained pipe conveying fluid, the differential equation of motion and boundary conditions are obtained as (3)
Where and L=Length of the pipe supports or span.
The equation of motion Eq. (1) can be written in the following nondimensional form:
(6)
Where,
= Mass ratio, = Natural boundary condition, =Natural frequency of pipe vibration,
V=Nondimensional velocity and =Wavelength
;
(7)
;
When the natural frequency of the pipe approaches zero the critical flow velocity has been computed for all the end conditions. When the flow velocity is equal to the critical velocity, the pipe bows out and buckles, as the forces required to make the fluid deform to the pipe curvature are greater than the stiffness of the pipe. The term Coriolis force represents the damping of the system, and its effect on the frequency of vibration is negligible and so is omitted, as the present work aims to obtain upper bounds for the frequencies of vibration of the pipe conveying fluid. The damping term is omitted and Eq. (7) is a non dimensional partial differential equation of higher order with boundary problem.
EI = Bending stiffness of a pipe;
= mass of pipe and mass of fluid; U= Velocity of the fluid; t = time;
;
(8)
(9)
, where c and s are constants and
;
The boundary conditions for the piping system are given below
(2)
(3)
(4)
(5)
substitution of eq. (8) in to (7) results.
(10)
The roots of equation (10) is given by
; ;
Considering first two and
Then ;
(11)
1.8
2.7464
3.0968
3.3162
7.7
2.2363
2.6561
2.9096
1.9
2.7434
3.0941
3.3137
7.8
2.2205
2.6428
2.8975
2.0
2.7402
3.0913
3.3110
7.9
2.2044
2.6293
2.8829
2.1
2.7368
3.0883
3.3082
8.0
2.1879
2.6156
2.8728
2.2
2.7332
3.0851
3.3053
8.1
2.1712
2.6016
2.8600
2.3
2.7295
3.0818
3.3022
8.2
2.1540
2.5874
2.8471
2.4
2.7256
3.0784
3.2990
8.3
2.1366
2.5729
2.8340
2.5
2.7215
3.0748
3.2957
8.4
2.1187
2.5581
2.8206
2.6
2.7173
3.0711
3.2922
8.5
2.1005
2.5431
2.8070
2.7
2.7128
3.0672
3.2885
8.6
2.0819
2.5278
2.7932
2.8
2.7082
3.0631
3.2848
8.7
2.0630
2.5122
2.7791
2.9
2.7035
3.0589
3.2809
8.8
2.0436
2.4964
2.7649
3.0
2.6985
3.0545
3.2768
8.9
2.0238
2.4803
2.7503
3.1
2.6934
3.0500
3.2726
9.0
2.0036
2.4638
2.7355
3.2
2.6881
3.0453
3.2682
9.1
1.9830
2.4471
2.7205
3.3
2.6826
3.0405
3.2638
9.2
1.9619
2.4301
2.7052
3.4
2.6770
3.0355
3.2591
9.3
1.9403
2.4128
2.6897
3.5
2.6711
3.0304
3.2543
9.4
1.9182
2.3951
2.6739
3.6
2.6651
3.0251
3.2494
9.5
1.8957
2.3771
2.6579
3.7
2.6589
3.0196
3.2443
9.6
1.8726
2.3588
2.6415
3.8
2.6525
3.0140
3.2391
9.7
1.8490
2.3404
2.6249
3.9
2.6459
3.0082
3.2338
9.8
1.8249
2.3212
2.6080
4.0
2.6391
3.0023
3.2282
9.9
1.8002
2.3019
2.5909
4.1
2.6322
2.9962
3.2226
10.0
1.7748
2.2822
2.5734
4.2
2.6250
2.9899
3.2168
10.1
1.7488
2.2621
2.5556
4.3
2.6177
2.9835
3.2108
10.2
1.7222
2.2416
2.5376
4.4
2.6101
2.9769
3.2047
10.3
1.6949
2.2207
2.5192
4.5
2.6024
2.9701
3.1984
10.4
1.6669
2.1995
2.5005
4.6
2.5944
2.9632
3.1920
10.5
1.6381
2.1778
2.4815
4.7
2.5863
2.9561
3.1854
10.6
1.6085
2.1556
2.4621
4.8
2.5780
2.9488
3.1786
10.7
1.5780
2.1331
2.4424
4.9
2.5694
2.9414
3.1718
10.8
1.5467
2.1100
2.4223
5.0
2.5607
2.9338
3.1647
.
.
.
.
5.1
2.5517
2.9260
3.1575
.
.
.
.
5.2
2.5425
2.9180
3.1501
12.9
–
0.11260
0.9678
5.3
2.5332
2.9098
3.1426
.
.
0.10649
0.8940
5.4
2.5236
2.9015
3.1349
.
.
.
.
5.5
2.5138
2.8930
3.1270
14.6
.
–
0.7223
5.6
2.5037
2.8843
3.1190
.
.
.
0.6177
5.7
2.4935
2.8754
3.1108
.
.
.
.
5.8
2.4830
2.8664
3.1024
.
.
.
.
5.9
2.4723
2.8571
3.0939
15.6
.
.
–

NATURAL FREQUENCY EVALUATION
Let the solution of the general equation (6) given as (12)
(13)
(14)
(15)
The above boundary conditions (2), (3), (4) and (5) are substituted in exact solution equations (12) to (15) to get the transcendental frequency equation can be written as follows
Table I. Natural Frequencies (N=1) for Mass Ratio and for Linearly and
Rotationally Restrained End Conditions
V
@
@
V
@
@
0.1
2.7729
3.1203
3.3381
6.0
2.4613
2.8477
3.0852
0.2
2.7727
3.1201
3.3379
6.1
2.4502
2.8381
3.0763
0.3
2.7723
3.1197
3.3375
6.2
2.4388
2.8282
3.0673
0.4
2.7717
3.1192
3.3371
6.3
2.4271
2.8182
3.0581
0.5
2.7710
3.1186
3.3365
6.4
2.4152
2.8080
3.0487
0.6
2.7701
3.1178
3.3357
6.5
2.4030
2.7976
3.0391
0.7
2.7690
3.1168
3.3348
6.6
2.3906
2.7870
3.0294
0.8
2.7678
3.1157
3.3338
6.7
2.3780
2.7761
3.0194
0.9
2.7664
3.1145
3.3327
6.8
2.3651
2.7651
3.0093
1.0
2.7648
3.1131
3.3314
6.9
2.3519
2.7539
2.9990
1.1
2.7631
3.1116
3.3300
7.0
2.3384
2.7424
2.9885
1.2
2.7612
3.1099
3.3284
7.1
2.3247
2.7307
2.9778
1.3
2.7592
3.1081
3.3267
7.2
2.3107
2.7189
2.9669
1.4
2.7570
3.1062
3.3249
7.3
2.2964
2.7068
2.9559
1.5
2.7546
3.1040
3.3229
7.4
2.2818
2.6944
2.9446
1.6
2.7520
3.1018
3.3208
7.5
2.2670
2.6819
2.9331
1.7
2.7493
3.0994
3.3186
7.6
2.2518
2.6691
2.9215
(16)
Equation (16) is the general frequency equation of elastically restrained pipe conveying fluid.
Assuming C = 0 then

Applying the B.C, , , in the general frequency equation (11), for
Linearly & Rotationally Restrained End Condition (Reference Fig.1) will result as
Fig.2. Linearly and Rotationally Restrained End Conditions

Applying the B.C, , , , , in the general frequency equation [16],
for Rotationally RestrainedLinearly Restrained and Linearly Restrained Condition will result as
Fig.3.Rotationally Restrained Linearly Restrained and Linearly Restrained End Condition
2.0
2.8303
2.8794
2.9118
3.4
2.4585
2.5320
2.5792
2.1
2.8083
2.8585
2.8916
3.5
2.4246
2.5011
2.5499
2.2
2.7857
2.8371
2.8709
3.6
2.3893
2.4689
2.5196
2.3
2.7626
2.8153
2.8499
3.7
2.3522
2.4355
2.4882
2.4
2.7389
2.7929
2.8283
3.8
2.3134
2.4006
2.45550
2.5
2.7146
2.7699
2.8062
3.9
2.2724
2.3641
2.4215
2.6
2.6895
2.7464
2.7836
4.0
2.2292
2.3258
2.3859
2.7
2.6638
2.7223
2.7605
4.1
2.1832
2.2856
2.3488
2.8
2.6373
2.6975
2.7367
4.2
2.1342
2.2431
2.3097
2.9
2.6373
2.6720
2.7123
4.3
2.0815
2.1980
2.2686
3.0
2.5818
2.6457
2.6872
4.4
2.0245
2.1500
2.2251
0.1
3.1754
3.2105
3.2340
4.5
1.9622
2.0985
2.1789
0.2
3.1599
3.1955
3.2193
4.6
1.8933
2.0429
2.1295
0.3
3.1441
3.1802
3.2044
4.7
1.8160
1.9824
2.0765
0.4
3.1281
3.1648
3.1893
4.8
1.7273
1.9158
2.0190
0.5
3.1119
3.1491
3.1740
4.9
1.6224
1.8414
1.9562
0.6
3.0954
3.1332
3.1585
5.0
1.4919
1.7567
1.8866
0.7
3.0786
3.1170
3.1427
5.1
1.3140
1.6576
1.8084
0.8
3.0615
3.1006
3.1267
5.2
1.0018
1.5367
1.7185
0.9
3.0442
3.0839
3.1104
5.3
–
1.3777
1.6117
1.0
3.0266
3.0669
3.0939
5.4
1.1297
1.4782
1.1
3.0086
3.0497
3.0771
5.5
–
1.2937
1.2
2.9903
3.0321
3.0600
5.6
0.9536
1.3
2.9717
3.0143
3.0426
5.7
–
1.4
2.9527
2.9961
3.0249
Fig.4. Rotationally RestrainedLinearly Restrained and Linearly Restrained

Applying the B.C, , , ,
Table II. Natural Frequencies (N=1) for Mass Ratio and for Rotationally Restrained Linearly Restrained and linearly Restrained End Condition
V
@
@
V
@
@
0.1
3.1754
3.2105
3.2340
1.5
2.9333
2.9776
3.0069
0.2
3.1599
3.1955
3.2193
1.6
2.9136
2.9587
2.9886
0.3
3.1441
3.1802
3.2044
1.7
2.8934
2.9395
2.9700
0.4
3.1281
3.1648
3.1893
1.8
2.8728
2.9199
2.9510
0.5
3.1119
3.1491
3.1740
1.9
2.8518
2.8998
2.9316
0.6
3.0954
3.1332
3.1585
2.0
2.8303
2.8794
2.9118
0.7
3.0786
3.1170
3.1427
2.1
2.8083
2.8585
2.8916
0.8
3.0615
3.1006
3.1267
2.2
2.7857
2.8371
2.8709
0.9
3.0442
3.0839
3.1104
2.3
2.7626
2.8153
2.8499
1.0
3.0266
3.0669
3.0939
2.4
2.7389
2.7929
2.8283
1.1
3.0086
3.0497
3.0771
2.5
2.7146
2.7699
2.8062
1.2
2.9903
3.0321
3.0600
2.6
2.6895
2.7464
2.7836
1.3
2.9717
3.0143
3.0426
2.7
2.6638
2.7223
2.7605
1.4
2.9527
2.9961
3.0249
2.8
2.6373
2.6975
2.7367
1.5
2.9333
2.9776
3.0069
2.9
2.6373
2.6720
2.7123
1.6
2.9136
2.9587
2.9886
3.0
2.5818
2.6457
2.6872
1.7
2.8934
2.9395
2.9700
3.1
2.5526
2.6187
2.6614
1.8
2.8728
2.9199
2.9510
3.2
2.5224
2.5908
2.6349
1.9
2.8518
2.8998
2.9316
3.3
2.4911
2.5619
2.6075
, in the general frequency equation [16],
for Rotationally RestrainedLinearly Restrained and Fixed End Condition will result as
Fig.5. Rotationally Restrained Linearly Restrained and Fixed End Condition
.
Table III. Natural Frequencies (N=1) for Mass Ratio and for Rotationally Restrained and Linearly Restrained and Fixed End Condition
V
@
@
V
@
@
0.1
3.1754
3.2105
3.2340
2.
7
2.6638
2.7223
2.7605
0.2
3.1599
3.1955
3.2193
2.
8
2.6373
2.6975
2.7367
0.3
3.1441
3.1802
3.2044
2.
9
2.610
2.6720
2.7123
0.4
3.1281
3.1648
3.1893
3.
0
2.5818
2.6457
2.6872
0.5
3.1119
3.1491
3.1740
3.
1
2.5526
2.6187
2.6614
0.6
3.0954
3.1332
3.1650
3.
2
2.5224
2.5908
2.6349
0.7
3.0786
3.1170
3.1427
3.
3
2.4911
2.5619
2.6075
0.8
3.0615
3.1006
3.1267
3.
4
2.4585
2.5320
2.5792
0.9
3.0442
3.0839
3.1104
3.
5
2.4246
2.5011
2.5499
1.0
3.0266
3.0669
3.0939
3.
6
2.3893
2.4689
2.5196
1.1
3.0086
3.0497
3.0771
3.
7
2.3522
2.4355
2.4882
1.2
2.9903
3.0321
3.0600
3.
8
2.3134
2.4006
2.4555
1.3
2.9717
3.0143
3.0426
3.
9
2.2724
2.3641
2.4215
1.4
2.9527
2.9961
3.0249
4.
0
2.2292
2.3258
2.3859
1.5
2.9333
2.9776
3.0069
4.
1
2.1832
2.2856
2.3488
1.6
2.9136
2.9587
2.9886
4.
2
2.1342
2.2856
2.3097
1.7
2.8934
2.9395
2.9700
4.
3
2.0815
2.1980
2.2686
1.8
2.8728
2.9199
2.9510
4.
4
2.0245
2.1500
2.2251
1.9
2.8518
2.8998
2.9316
4.
5
1.9622
2.0985
2.1789
2.0
2.8303
2.8794
2.9118
.
.
.
.
2.1
2.8083
2.8585
2.8916
5.
1
–
1.6042
1.7679
2.2
2.7857
2.8371
2.8709
5.
2
1.5367
1.7185
2.3
2.7626
2.8153
2.8499
.
.
.
2.4
2.7389
2.7929
2.8283
5.
3
–
1.5093
2.5
2.7146
2.7699
2.8062
5.
4
.
.
1.4782
2.6
2.6895
2.7464
2.7836
5.
5
–
Fig.6. Rotationally Restrained and Linearly Restrained and Fixed End Condition

Applying the B.C, , , , , in the general frequency equation [16],

for Rotationally Restrained and Guided End Condition will result as
Fig.7. Rotationally Restrained and Guided End Condition
Table IV. Natural Frequencies (N=1) for Mass Ratio and for
Rotationally Restrained and Guided End Condition
V
@
@
0.1
1.8577
1.8336
1.8156
0.2
1.8665
1.8445
1.8269
0.3
1.8740
1.8547
1.8381
0.4
1.8800
1.8639
1.8488
0.5
1.8843
1.8718
1.8586
0.6
1.8869
1.8782
1.8672
0.7
1.8875
1.8831
1.8746
0.8
1.8862
1.8863
1.8804
0.9
1.8827
1.8875
1.8846
1.0
1.8768
1.8868
1.8870
1.1
1.8685
1.8840
1.8875
1.2
1.8573
1.8789
1.8860
1.3
1.8429
1.8714
1.8823
1.4
1.8250
1.8611
1.8762
1.5
1.8026
1.8478
1.8676
1.6
–
1.8311
1.8561
1.7
1.8102
1.8415
1.8
–
1.8231
1.9
1.8003
2.0
–
Fig.8. Rotationally Restrained and Guided End Condition


RESULTS AND DISCUSSIONS
Table 1, shows the graph of nondimensional natural frequencies with nondimensional velocities. In the case of

Linearly and Rotationally Restrained end conditions the instability region lies in the range of 12.99435 to 15.66188. However, in other three cases like b) Rotationally Restrained Linearly Restrained and Linearly Restrained end conditions c) Rotationally Restrained and Linearly Restrained and Fixed end conditions d) Rotationally Restrained and Guided end conditions are shown in Tables 2, 3 & 4 the pipe flutters at a much lower velocity, in the flow region of b) V=5.3 to V=5.7, c) V=5.147 to V=5.338 and d) V=1.6 to V=2.0. Figures 2, 4, 6 and 8 shows the points of flutter for three mass ratios. The percentage reduction in frequency as velocity increases from (reference fig.2) V=0.1 to V=2.77298 is 72 %, For Tables 2 and 3 (reference fig.4 & 6) from V=0.1 to V=3.175468 shows 68.24%, reduction in frequency. Table 4 shows the frequency reduction (reference fig.8) from V=0.1 to V=1.85777 is 81.42%. It is found that the natural frequencies remain same for all the three mass ratios, which means that the instability condition is close with higher mass ratio and fluid velocity.


CONCLUSIONS


Exact method is developed for pipes conveying fluid for Linearly Restrained and Rotationally restrained end conditions, Rotationally Restrained
Linearly Restrained and Linearly Restrained end conditions, Rotationally Restrained Linearly Restrained and Fixed end conditions and Rotationally Restrained and Guided end conditions

The frequencies of the first mode of vibration are computed by varying the fluid velocity

Critical velocity for different mass ratios are found

A FORTRAN program is developed for computation of natural frequencies by using Muellers Iteration method for nonlinear equations (Bisection) and the iterated value of x (nondimensional) natural frequency is found by the Inverse Parabolic Interpolation method

The natural frequencies are obtained by varying the fluid velocities.
ACKNOWLEDGEMENTS
The authors gratefully acknowledge the Management of CSIRIndian Institute of Chemical Technology, Hyderabad, India and College of Engineering, Osmania University, Hyderabad, India for providing the infrastructural facilities.
REFERENCES

Ashley H, Haviland G, Bending Vibration of a Pipeline containing flowing fluid, Transactions of ASME, Journal of Applied Mechanics, Sept 1950, pp 229232

G.W. Housner, Bending vibration of a pipe line containing flowing fluid, J. Appl. Mech., 19, 6(1952), pp205208

Rao. SS., "Vibration of Continuous Systems" JohnWiley &Sons Inc. Publishing company USA 2007, pp. 317328.

S. Naguleswaran & C. J. H. Williams.," Lateral Vibration of a Pipe Conveying a Fluid" published in Journal of Mechanical Engineering Science June 1968 vol. 10 no. 3, pp 228238

Chen SS. ,"Free Vibration of a coupled fluid/structural system "Journal of Sound and Vibration (1972) 21(4) , pp387398

M.P Paidoussis., "Dynamic Stability of pipes conveying fluid"
Journal of Sound and Vibration (1974)33(3), pp 267 294

Huang Yimin, Liu Yongshou, Li Baohui, Li Yanjiang, Yue Zhu feng, Natural frequency analysis of fluid conveying pipeline with different boundary conditions, Nuclear Engineering and design 240(2010), pp 461467

Stein R. A., Tobriner M.W., Vibration of pipes containing flowing fluid, Transactions of the ASME, Dec 1970, pp 906916

Wang Shizhong, Liu Yulan, Huang Wenhu, Research on Solid Liquid coupling Dynamics of pipe conveying fluid, Journal of Applied Mathematics and Mechanics, Vol.19, No. 11, Nov 1998,
pp 10651071

Weaver D.S and Unny T.E., On the Dynamic Stability of Fluid conveying Pipes, Transactions of ASME, March 1973, 4852

Chaos, Solitons& Fractals, Class of analytical closedform polynomial solutions for clampedguided inhomogeneous beams, Volume 12, Issue 9, July 2001, Pages 16571678