DOI : 10.17577/IJERTV4IS100408
- 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): 23-10-2015
- ISSN (Online) : 2278-0181
- 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,
CSIR-Indian 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 Euler-Bernoulli 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, Rotational-Linearly 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 fluid-structure 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 Hinged-Hinged, Fixed-Hinged and Fixed- Fixed boundary conditions [4]. Chen and Paidoussis developed dynamic stiffness matrix for coupled fluid structure interaction [5], [6]. Huang Yi-min 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 un-damped 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, solid-liquid coupling damping and solid-liquid 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 Flugge-Kempner equation to find the critical flow velocities [10].
In most of the cases, the differential equation of motion of fluid-conveyed 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, Runge-Kutta 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
non-dimensional parameter mf fluid mass
t time
-
the axial coordinate
-
-
non-dimensional 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 z-x 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 cross-section of the pipe and z is the lateral deflection of the pipeline; x and t are the axial co-ordinate 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 Euler-Bernoulli 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 non-dimensional form:
(6)
Where,
= Mass ratio, = Natural boundary condition, =Natural frequency of pipe vibration,
V=Non-dimensional 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 Restrained-Linearly 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 Restrained-Linearly 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 Restrained-Linearly 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
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-
RESULTS AND DISCUSSIONS
Table 1, shows the graph of non-dimensional natural frequencies with non-dimensional 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
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A FORTRAN program is developed for computation of natural frequencies by using Muellers Iteration method for non-linear equations (Bi-section) and the iterated value of x (non-dimensional) natural frequency is found by the Inverse Parabolic Interpolation method
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The natural frequencies are obtained by varying the fluid velocities.
ACKNOWLEDGEMENTS
The authors gratefully acknowledge the Management of CSIR-Indian Institute of Chemical Technology, Hyderabad, India and College of Engineering, Osmania University, Hyderabad, India for providing the infrastructural facilities.
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