Three Dimensional Vibration Analysis Thesis 2009 Tmm&Ansys

ÇUKUROVA UNIVERSITY

INSTITUTE OF NATURAL AND APPLIED SCIENCES

PhD THESIS

Ahmet ÖZBAY

THREE DIMENSIONAL VIBRATION ANALYSIS

OF LIQUID-FILLED PIPING SYSTEMS

DEPARTMENT OF MECHANICAL ENGINEERING

ADANA, 2009

ÇUKUROVA ÜNİVERSİTESİ

FEN BİLİMLERİ ENSTİTÜSÜ

THREE DIMENSIONAL VIBRATION ANALYSIS OF LIQUID-FILLED PIPING SYSTEMS

Ahmet ÖZBAY

DOKTORA TEZİ

MAKİNA MÜHENDİSLİĞİ ANABİLİM DALI

Bu Tez ..../..../2009 Tarihinde Aşağıdaki Jüri Üyeleri Tarafından

Oybirliği/Oyçokluğu İle Kabul Edilmiştir.

İmza: …………………… İmza: …………………………. İmza: …………………………

Prof. Dr. Vebil YILDIRIM Prof. Dr. Naki TÜTÜNCÜ Doç. Dr. H. Murat Arslan

DANIŞMAN ÜYE ÜYE

İmza: ………………………

İmza: …………………

Doç. Dr. Ahmet PINARBAŞI Yrd.Doç. Dr. İbrahim KELEŞ

ÜYE ÜYE

Bu Tez Enstitümüz Makina Mühendisliği Anabilim Dalında Hazırlanmıştır.

Kod No:

Prof. Dr. Aziz ERTUNÇ Enstitü Müdürü

Not: Bu tezde kullanılan özgün ve başka kaynaktan yapılan bildirişlerin, çizelge, şekil ve fotoğrafların kaynak gösterilmeden kullanımı, 5846 sayılı Fikir ve Sanat Eserleri Kanunundaki hükümlere tabidir.

I

ABSTRACT

PhD THESIS

THREE DIMENSIONAL VIBRATION ANALYSIS OF LIQUID-FILLED PIPING SYSTEMS

Ahmet ÖZBAY

DEPARTMENT OF MECHANICAL ENGINEERING INSTITUTE OF NATURAL AND APPLIED SCIENCES

UNIVERSITY OF ÇUKUROVA

Supervisor

Year

: Prof. Dr. Vebil YILDIRIM

: 2009, Pages: 164

Jury : Prof. Dr. Vebil YILDIRIM

Prof. Dr. Naki TÜTÜNCÜ

Doç. Dr. H. Murat Arslan

Doç. Dr. Ahmet PINARBAŞI

Yrd.Doç. Dr. İbrahim KELEŞ

In the theoretical part of this work, the transfer matrix method (TMM) is employed

to study the free vibration analysis of liquid-filled (air/water) piping systems. The existing

governing equations which consist of a set of fourteen linear differential equations of first

degree are considered. Fixed-fixed and fixed-free ends are studied with five different basic

geometries of piping systems made of either copper or steel, such as single-span, L-bend, Z-

bend, U-bend and 3-D bend. A few experiments are also completed to support the theoretical

solutions. The effect of the elastic foundation on the natural frequencies is also studied.

Finally, a parametric study is carried out to understand correctly the vibrational behavior of

such systems. Present results are verified with the frequencies available in the literature.

Keywords:

Flow-Induced Vibration, Transfer Matrix Method, Three Dimensional Vibration Analysis, Liquid-Filled.

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II

ÖZ

DOKTORA TEZİ

AKIŞKAN TAŞIYAN BORU HATLARININ ÜÇ BOYUTLU TİTREŞİM ANALİZİ

Ahmet ÖZBAY

ÇUKUROVA ÜNİVERSİTESİ FEN BİLİMLERİ ENSTİTÜSÜ

MAKİNA MÜHENDİSLİĞİ ANABİLİM DALI

Danışman Yıl

: Prof. Dr. Vebil YILDIRIM

: 2008, Pages: 164

Jüri : Prof. Dr. Vebil YILDIRIM

Prof. Dr. Naki TÜTÜNCÜ

Doç. Dr. H. Murat Arslan

Doç. Dr. Ahmet PINARBAŞI

Yrd.Doç. Dr. İbrahim KELEŞ

Bu çalışmanın teorik kısmında, akışkan (hava/su) dolu boru

sistemlerinin serbest titreşim analizi için taşıma matrisi yöntemi (TMM)

kullanılmıştır. Literatürde mevcut on dört adet lineer birinci dereceden diferansiyel

denklemden oluşan denklem takımı göz önüne alınmıştır. Ankastre-ankastre ve

ankastre-serbest uçlar için, tek açıklıklı, L, Z, U ve üç boyutlu konfigürasyonlardan

oluşan bakır/çelik malzemeden yapılmış boru sistemleri ele alınmıştır. Teorik

sonuçları desteklemek amacı ile bazı deneyler gerçekleştirilmiştir. Elastik zemin

etkisi ayrıca çalışılmıştır. Son olarak parametrik bir çalışma gerçekleştirilmiştir. Bu

çalışmadan elde edilen sonuçlar, literatürde bulunan frekanslarla doğrulanmıştır.

Anahtar Kelimeler: Akış Kaynaklı Titreşim, Transfer Matris Metodu, Üç

Boyutlu Titreşim Analizi, Akışkan Dolu.

III

ACKNOWLEDGEMENTS

I am truly grateful to my research supervisor, Prof. Dr. Vebil YILDIRIM,

for his invaluable guidance and support throughout the preparation of this thesis and

during my graduate education.

I would like to express my special thanks to Advisory Committee Members,

Prof. Dr. Naki TÜTÜNCÜ, Assoct. Prof. Dr. Ahmet PINARBAŞI and Assoct. Prof.

Dr. H. Murat ARSLAN, for their devotion of invaluable time throughout my research

activities.

I would like to offer my cordial thanks to Assist. Prof. Dr. İbrahim KELEŞ

who have improved my morale with their encouraging advises during my thesis

study.

I would like to thank to all my research assistant friends at our Mechanical

Engineering Department and Colleague in Mersin Soda Ash Plant for their

continuous support and motivation.

Another point that should be emphasized here is the continuous moral

support, motivation, encouragement and patience of my wife Fügen ÖZBAY, my

daughter Derin ÖZBAY, and my family throughout my scientific efforts.

IV

CONTENTS

PAGE

ABSTRACT.…………………………………………...…………………....... I

ÖZ …………………………………………………………………................. II

ACKNOWLEDGEMENTS………………………………………………...… III

CONTENTS…………………………………………………........................... IV

LIST OF TABLES .…………………………………………………………... VII

LIST OF FIGURES…………………………………………………………... XII

NOMENCLATURE……………………………………………….................. XVIII

1. INTRODUCTION……………………………………………………….. 1

2. LITERATURE REVIEW........................................................................... 2

2.1. Flow Induced Vibrations in Pipelines ………………………………. 2

2.2. Transfer Matrix Method........……......………..................................... 8

3. MATERIAL AND METHOD.................................................................... 10

3.1. Material.................................................................................... 10

3.1.1. Pipe Materials................................................................... 11

3.1.2. Liquid …………………………………………………... 12

3.1.3. External Shaker ………………………………………… 12

3.1.4. Transducers....................................................................... 13

3.2. Method............................................................................................ 14

3.2.1. Governing Differential Equations ………….................... 14

3.2.1.1. Axial Vibration – Liquid and Pipe Wall …….… 15

3.2.1.2. Transverse Vibration in x-z Plane ……………. 27

3.2.1.3. Transverse Vibration in y-z Plan ……………... 33

3.2.1.4. Torsional Vibration …………………………... 34

3.2.2. Transfer Matrix Method.................................................... 38

3.2.2.1. Transfer Matrix Procedure ……………...……... 38

3.2.2.2. Field Transfer Matrices........................................ 42

3.2.2.2.(1). Liquid and Pipe Wall Vibration …................ 43

3.2.2.2.(2). Transverse Vibration in x-z Plane.................. 45

3.2.2.2.(3). Transverse Vibration in y-z Plane.................. 45

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3.2.2.2.(4). Torsional Vibration about z Axis ...………... 47

3.2.2.3. General Field Transfer Matrix............................. 47

3.2.2.4. Point Matrices...................................................... 49

3.2.2.4.(1). Bend Point Matrix.......................................... 49

3.2.2.4.(2). Spring Point Matrix........................................ 54

3.2.2.5. Boundary Conditions........................................... 56

3.2.2.6. Natural Frequencies............................................. 58

3.2.2.7. Vibration of a Pipe on Elastic Foundation .......... 59

4. RESULTS AND DISCUSSION................................................................. 60

4.1. Single Span Pipe with Various Conditions ……………………… 63

4.1.1. Fixed-Fixed Single Span Pipe ………………….…….… 64

4.1.2. Fixed-Free Single Span Pipe ………………….…….….. 69

4.1.3. Single Span Pipe with Rigid Support ………………….. 72

4.2. Two Pipe with 90 Degree Bend .………………………………… 74

4.2.1. L Bend with Fixed-Free End Conditions ………………. 75

4.2.2. L Bend with Fixed-Fixed End Conditions …………… 78

4.2.3. L Bend with Intermediate Conditions…………………... 89

4.3. Three Pipes in a Plane…………………………………………… 90

4.3.1. Z Bend with Fixed-Free End Conditions……………….. 90

4.3.2. Z Bend with Fixed-Fixed End Conditions……………… 92

4.3.3. U Bend with Fixed-Free End Conditions ……………… 102

4.3.4. U Bend with Fixed-Fixed End Conditions …………….. 103

4.4. Three Pipes in Two Planes ………………………………………. 115

4.4.1. 3D Bend with Fixed-Free End Conditions ……………. 115

4.4.2. 3D Bend with Fixed-Fixed End Conditions …………... 117

4.5. Elastic Foundation ………………………………………………. 125

4.5.1. Free Ended Single Span Pipe on an Elastic Foundation . 126

4.5.2. L Bend Free Ended Pipe on an Elastic Foundation …… 129

4.5.3. 3D Bend Free Ended Pipe on an Elastic Foundation … 132

4.6. Parametric Studies ……………………………………………… 135

4.6.1. Effect of Slenderness Ratio on the Natural Frequencies.. 135

VI

4.6.2. Effect of The Bend-Angle on The Natural Frequencies

of Planar Piping System……………………………….

148

5. CONCLUSIONS ……………………………………………………… 157

REFERENCES…………………………………………………….................. 159

CURRICULUM VITAE……………………………………………………… 162

APPENDIX ……………...……………………………………………………… 163

VII

LIST OF TABLES PAGE Table 3.1. Physical Properties of Copper Pipe ………………………… 11

Table 3.2. Physical Properties of Steel Pipe …………………………… 12

Table 3.3. Physical Properties of Liquid ………………………………. 12

Table 4.1. Comparison of the present theoretical natural frequencies

(rad/s) of 5m-length copper pipe filled by the air with the

literature(Fixed-Fixed and Open-Closed) ……………...……. 61

Table 4.2. Comparison of the present theoretical natural frequencies

(rad/s) of 5m-length copper pipe filled by the air with the

literature (Fixed-Fixed and Open-Closed) ..…...…………….. 61

Table 4.3. Natural frequencies (Hz) of 2m-length copper pipe with the

air (Fixed-Fixed and Open-Closed) ………………………… 65


water (Fixed-Fixed and Open-Closed) ……………………… 68

Table 4.5. Natural frequencies (Hz) of 3.5m-length steel pipe with the

air (Fixed-Fixed and Open-Closed) …….…………………… 70

Table 4.6. Natural frequencies (Hz) of 3.5m-length steel pipe with the

water (Fixed-Fixed and Open-Closed) ……………………… 70


air (Fixed-Free and Open-Closed) …………………………. 71


water (Fixed-Free and Open-Closed) ….…………………… 71

Table 4.9. Natural frequencies (Hz) of 3m-length steel pipe with the air

(Fixed-Free and Open-Closed) …………………………….. 71

Table 4.10. Natural frequencies (Hz) of 3m-length steel pipe with the

water (Fixed-Free and Open-Closed) .………………………. 72

Table 4.11. Natural frequencies (Hz) of 7m-length copper pipe filled by

the air for intermediate rigid support (Fixed-Fixed and Open-

Closed) . …………………………………………………. 73

Table 4.12. Natural frequencies (Hz) of 7m-length copper pipe filled by 73

VIII

the water for intermediate rigid support (Fixed-Fixed and

Open-Closed) ………………………………………………..

Table 4.13. Natural frequencies (Hz) of 6m-length steel pipe filled by the

air for intermediate rigid support (Fixed-Fixed and Open-

Closed) ……………………….……………………………... 74

Table 4.14. Natural frequencies (Hz) of 6m-length steel pipe filled by the

water for intermediate rigid support (Fixed-Fixed and Open-

Closed) …………………………….………………………… 74

Table 4.15. Natural frequencies (Hz) of L-bended steel pipe with the air

(Fixed-Closed / Free-Closed) ……………………………….. 76

Table 4.16. Natural frequencies (Hz) of L-bended steel pipe with the

water (Fixed-Closed / Free-Closed) …………………………. 76

Table 4.17. Natural Frequencies (Hz) of L-bended copper pipe with the

air (Fixed-Closed / Free-Closed) …………………………….. 77

Table 4.18. Natural Frequencies (Hz) of L-bended copper pipe with the

water (Fixed-Closed / Free-Closed) ………………………… 77

Table 4.19. Natural frequencies (Hz) of L-bended steel pipe with the air

(Fixed-Open/ Fixed-Closed) (L1 = L2 = 2.4 m) …………….. 78

Table 4.20. Natural frequencies (Hz) of L-bended steel pipe with the

water (Fixed-Open/ Fixed-Closed) (L1 = L2 = 2.4 m) ………. 81

Table 4.21. Natural frequencies (Hz) of L-bended copper pipe with the

air (Fixed-Open/ Fixed-Closed) (L1 = L2 = 1 m) ……..…….. 84


water (Fixed-Open/ Fixed-Closed) (L1 = L2 = 1 m) ..……….. 86


air (Fixed-Open/ Fixed-Closed) (L1 = L2 = 3.5 m) ..………... 87


water (Fixed-Open/ Fixed-Closed) (L1 = L2 = 3.5 m) .…….. 88

Table 4.25. Natural frequencies of L-bended copper pipe filled by the

water with intermediate rigid supports (Fixed-Open/ Fixed-

Closed) …………………………………….………………… 89

IX

Table 4.26. Natural frequencies (Hz) of Z-bended steel pipe with the air

(Fixed-Closed / Free-Closed) (L1 = L2 = L3=1.25m) ……...... 93

Table 4.27. Natural frequencies (Hz) of Z-bended steel pipe with the

water (Fixed-Closed / Free-Closed) (L1 = L2 = L3=1.25m) ..... 93

Table 4.28. Natural frequencies (Hz) of Z-bended steel pipe with the air

(Fixed-Open/ Fixed-Closed) (L1 = L2 = L3=1.25m) ….......... 94

Table 4.29. Natural frequencies (Hz) of Z-bended steel pipe with the

water (Fixed-Open/ Fixed-Closed) (L1 = L2 = L3=1.25m) ...... 96

Table 4.30. Natural frequencies (Hz) of Z-bended copper pipe with the

air (Fixed-Open/ Fixed-Closed) (L1 = L2 = L3=1m) ….……... 98

Table 4.31. Natural Frequencies (Hz) of Z-bended copper pipe with the

water (Fixed-Open/ Fixed-Closed) (L1 = L2 = L3=1m)……… 101


air (Fixed-Open/ Fixed-Closed) (L1 = L2 = L3=7/3m=2.33m) 102


water (Fixed-Open/ Fixed-Closed) (L1 = L2 = L3=2.333m) 103

Table 4.34. Natural Frequencies (Hz) of U-bended steel pipe with the air

(Fixed-Open/ Free-Closed) (L1 = L2 = L3=1.25m) ..……….... 105

Table 4.35. Natural Frequencies (Hz) of U-bended steel pipe with the

water (Fixed-Open / Free-Closed) (L1 = L2 = L3=1.25m)….... 105

Table 4.36. Natural Frequencies (Hz) of U-bended steel pipe with the air

(Fixed-Open / Fixed-Closed) (L1 = L2 = L3=1.25m) …..….... 106

Table 4.37. Natural Frequencies (Hz) of U-bended steel pipe with the

water (Fixed-Open / Fixed-Closed) (L1 = L2 = L3=1.25m)….. 108

Table 4.38. Natural Frequencies (Hz) of U-bended cooper pipe with the

air (Fixed-Open / Fixed-Closed) (L1 = L2 = L3=1m)………… 109


water (Fixed-Open / Fixed-Closed) (L1 = L2 = L3=1m) ..…… 111


air (Fixed-Open / Fixed-Closed) (L1 = L2 = L3=2.333m) ….... 112

X


air (Fixed-Open / Fixed-Closed) (L1 = L2 = L3=2.333m) ….. 113

Table 4.42. Natural Frequencies (Hz) of 3D-bended steel pipe with the

air (Fixed-Open / Free-Closed) (L1 = L2 = L3=1.25m) ……… 116


water (Fixed-Open / Free-Closed) (L1 = L2 = L3=1.25m) …... 116


air (Fixed-Open / Fixed-Closed) (L1 = L2 = L3=1.25m) .……. 117


water (Fixed-Open / Fixed-Closed) (L1 = L2 = L3=1.25m)…. 118

Table 4.46. Natural Frequencies (Hz) of 3D-bended copper pipe with the

air (Fixed-Open / Fixed-Closed) (L1 = L2 = L3=1m) ……….. 120


water (Fixed-Open / Fixed-Closed) (L1 = L2 = L3=1m) ….… 121


air (Fixed-Open / Fixed-Closed) (L1 = L2 = L3=2.333m) ….. 122


water (Fixed-Open / Fixed-Closed) (L1 = L2 = L3=2.333m) .. 123

Table 4.50. Natural Frequencies (Hz) of 6m length free ended steel pipe

with the air on elastic foundation (kf = 100000 N/m3, Δ = 1m)

(Free-Open/ Free-Closed)……………………..……………. 124

Table 4.51. Natural Frequencies (Hz) of 6m length free ended steel pipe

with the water on elastic foundation (kf = 100000 N/m3, Δ =

1m) (Free-Open/ Free-Closed) …………………..…………. 127

Table 4.52. Natural Frequencies (Hz) of 6m length free ended copper

pipe with the air on elastic foundation (kf = 100000 N/m3, Δ =

1m) (Free-Open/ Free-Closed)……………………………... 1228

Table 4.53. Natural Frequencies (Hz) of 6m length free ended copper

pipe with the water on elastic foundation (kf = 100000 N/m3,

Δ = 1m) (Free-Open/ Free-Closed)..…………………………. 129

XI

Table 4.54. Natural Frequencies (Hz) of 3m length L-Bended free ended

steel pipe with the air on elastic foundation (kf = 100000

N/m3, Δ = 0.5m) (Free-Open/ Free-Closed) (L1 = L2 = 1.5m) 130


steel pipe with the water on elastic foundation (kf = 100000



copper pipe with the air on elastic foundation (kf = 100000



copper pipe with the water on elastic foundation (kf = 100000


Table 4.58. Natural Frequencies (Hz) of 3m length 3D-Bended free ended

steel pipe with the air on elastic foundation (kf = 100000

N/m3, Δ = 1.25m) (Free-Open/ Free-Closed) (L1 = L2 =

1.25m) ………………….. ..………………..………………. 133


steel pipe with the water on elastic foundation kf = 100000


1.25m) ………………………….……………………………. 134


copper pipe with the air on elastic foundation kf = 100000


1.25m) ……………………..…………………………………. 134


copper pipe with the water on elastic foundation kf = 100000


1.25m) ……………………..…………………………………. 135

Table 4.62. Variation of the natural frequencies in Hz of a single-spanned

steel pipe with the slenderness ratio (Fixed-Fixed and Open-

Closed) ………………………………………………………. 136

XII


copper pipe with the slenderness ratio (Fixed-Fixed and

Open-Closed) ..………………………………………………. 137


steel pipe with the slenderness ratio (Fixed-Free and Closed-

Closed) ………………………………………………………. 138


copper pipe with the slenderness ratio (Fixed-Free and

Closed-Closed).………………………………………………. 139

Table 4.66. Variation of the natural frequencies in Hz of steel pipe

system with the bend angle (Fixed-Fixed) ………………….. 149

Table 4.67. Variation of the natural frequencies in Hz of copper pipe

system with the bend angle (Fixed-Fixed) …………………. 150

Table 4.68. Variation of the natural frequencies in Hz of steel pipe

system with the bend angle (Fixed-Free) ……………………. 151

Table 4.69. Variation of the natural frequencies in Hz of copper pipe

system with the bend angle (Fixed-Free) …………………… 152

XIII

LIST OF FIGURES PAGE Figure 3.1. Pipeline Test-Rig.…………………………......................…… 10

Figure 3.2. Crank Mechanism……………………………………………. 13

Figure 3.3. Fuji Electric Gauge Pressure Transmitter Model FKG ……… 14

Figure 3.4. The DLI Watchman® DCA-20 portable data collector ……... 14

Figure 3.5. Sign Convention for Internal Forces (Lesmez,1989)……..…. 15

Figure 3.6. Axial Pipe Element ………………………….…………….... 17

Figure 3.7. Radial Pipe Element …………………………….….……….. 18

Figure 3.8. Sign Convention for Pipe Element …………………………. 27

Figure 3.9. my and fx Forces in x-z Plane ……………………..………. 28

Figure 3.10. Sign Convention For Pipe Element …………………………. 33

Figure 3.11. mx and fy Forces in y-z Plane ……………………………….. 34

Figure 3.12. Pipe Reach Subjected to Torsion ……………………………. 35

Figure 3.13. Spring - Mass System ……………………………………….. 38

Figure 3.14. Free-Body Diagram of Spring …………………...…………. 39

Figure 3.15. Free-Body Diagram of a Mass ………………………...……. 40

Figure 3.16. General Pipe Element ..…………………………..…..……... 42

Figure 3.17. Sign Convention For Bend ………………………………….. 49

Figure 3.18. Forces at Spring ……………………………………………... 54

Figure 4.1. Experimental pressure-time history of 25m-length steel pipe

with fixed-fixed ends at different external excitation

frequencies................................................................................ 63

Figure 4.2. Experimental pressure-time history when external excitation

frequency is equal to the liquid frequency (6.0375 Hz) …....... 63

Figure 4.3. Single span pipe supported at two ends ……………………... 64

Figure 4.4. PIPE16 - Elastic straight pipe ……………………………….. 65

Figure 4.5. FFT Spectrum in tangential direction of fixed-fixed single

span 2m-length copper pipe with the air ……......................... 66

Figure 4.6. FFT Spectrum in radial direction of fixed-fixed single span

2m-length copper pipe with the air .......................................... 67

XIV

Figure 4.7. FFT Spectrum in axial direction of fixed-fixed single span

2m-length copper pipe with the air ………… ......................... 67

Figure 4.8. FFT spectrum in axial directions at two locations of fixed-

fixed single span 2m-length copper pipe with the water .......... 69

Figure 4.9. FFT Spectrum in tangential direction of fixed-fixed single

span 2m-length copper pipe with the water ….......................... 69

Figure 4.10. Single span pipe with rigid support ......................................... 72

Figure 4.11. L-Bended pipe supported at two ends …………………….... 75

Figure 4.12. FFT Spectrum in axial direction at two locations of L-bended

steel pipe with the air (Fixed-Open/ Fixed-Closed) …………. 79

Figure 4.13. FFT Spectrum in radial direction of L-bended steel pipe

with the air (Fixed-Open/ Fixed-Closed) (L1 = L2 = 2.4 m)..... 80

Figure 4.14. FFT Spectrum in tangential direction of L-bended steel pipe

with the water (Fixed-Open/ Fixed-Closed) ………………... 81

Figure 4.15. FFT Spectrums in radial direction at two locations of L-

bended steel pipe with the water (Fixed-Open/ Fixed-

Closed) (L1 = L2 = 2.4 m) ………………………………….. 82

Figure 4.16. Experimental pressure-time history of L-bended steel pipe

with the water (Fixed-Open/ Fixed-Closed) (L1 = L2 = 2.4 m)

at different external excitation frequencies….…...................... 83

Figure 4.17. Experimental pressure-time history of L-bended steel pipe


when external excitation frequency is equal to the liquid

frequency (76.53 Hz). ……………………………………….. 84

Figure 4.18. FFT Spectrums in axial and tangential directions of L-bended

copper pipe with the air (Fixed-Open/ Fixed-Closed) (L1 =

L2 = 1 m)….……………………………………...................... 85

Figure 4.19. FFT Spectrums in axial and radial directions of L-bended

copper pipe with the water (Fixed-Open/ Fixed-Closed) (L1

= L2 = 1 m) ……………………………………...................... 87

Figure 4.20. Experimental pressure-time history of L-bended copper pipe 89

XV


at different external excitation frequencies…………………...

Figure 4.21. Experimental pressure-time history of L-bended copper pipe

with the water (Fixed-Open/ Fixed-Closed) (L1 = L2 = 3.5m)

when external excitation frequency is equal to the liquid

frequency (44.55Hz) ……………………………..………...... 89

Figure 4.22. L-Bended pipe with intermediate rigid supports ............…... 90

Figure 4.23. U-Bended pipe supported at two ends ……....................…... 91

Figure 4.24. Z-Bended Pipe supported at two ends .………...................... 92

Figure 4.25. FFT Spectrums in axial and tangential directions of Z-bended

steel pipe with the air (Fixed-Open/ Fixed-Closed) (L1 = L2 =

L3=1.25m) ….…………………………………...................... 95

Figure 4.26. Experimental pressure-time history of Z-bended steel pipe

with the water (Fixed-Open/ Fixed-Closed) (L1 = L2 =

L3=1.25m) at different external excitation frequencies……… 96

Figure 4.27. Experimental pressure-time history of Z-bended steel pipe


L3=1.25m) when external excitation frequency is equal to the

liquid frequency (94.1176Hz)………………………............... 97

Figure 4.28. FFT Spectrums in tangential direction at two locations of Z-

bended steel pipe with the water (Fixed-Open/ Fixed-Closed)

(L1 = L2 = L3=1.25m) ………………….......…………….….... 98

Figure 4.29. FFT spectrums in tangential and axial directions of Z-bended

copper pipe with the air (Fixed-Open/ Fixed-Closed) (L1 =

L2 = L3=1m) ………………….......…………….…................ 102

Figure 4.30. Experimental pressure-time history of Z-bended copper pipe


L3=7/3m=2.333m) at different external excitation

frequencies. ………………….......…………….….................. 103

Figure 4.31. Z Bend Piping Configuration for Fixed-Free Boundary

Conditions ……………………………………….................... 104

XVI

Figure 4.32. Experimental pressure-time history of Z-bended copper pipe


L3=7/3m=2.333m) when external excitation frequency is

equal to the liquid frequency (46.666 Hz) ………………….. 104

Figure 4.33. FFT Spectrums in tangential and axial directions of U-bended

steel pipe with the air (Fixed-Open/ Fixed-Closed) (L1 = L2 =

L3=1.25m) ……………………………………...................... 107

Figure 4.34. FFT Spectrums radial directions at two locations of U-bended

steel pipe with the water (Fixed-Open/ Fixed-Closed) (L1 =

L2 = L3=1.25m) .………………………………...................... 109

Figure 4.35. FFT Spectrums in radial and axial directions of U-bended

cooper pipe with the air (Fixed-Open / Fixed-Closed) (L1 =

L2 = L3=1m) .………………………………........................... 110

Figure 4.36. FFT Spectrums in axial direction at two locations of U-

bended cooper pipe with the water (Fixed-Open / Fixed-

Closed) (L1 = L2 = L3=1m) …………………......................... 112

Figure 4.37. Experimental pressure-time history of U-bended copper pipe


L3=2.333m) at different external excitation frequencies...…… 114

Figure 4.38. Experimental pressure-time history of U-bended copper

pipe with the water (Fixed-Open/ Fixed-Closed) (L1 = L2 =

L3=2.333m) when external excitation frequency is equal to

the liquid frequency (46,623Hz) ………………...................... 114

Figure 4.39. 3D-bend piping configuration for fixed-free boundary

conditions …..…………………………………...................... 115

Figure 4.40. FFT Spectrums in tangential direction of 3D-bended steel

pipe with the air (Fixed-Open / Fixed-Closed) (L1 = L2 =

L3=1.25m) ….....………………………………...................... 118

Figure 4.41. FFT Spectrums in tangential direction of 3D-bended steel

pipe with the water (Fixed-Open / Fixed-Closed) (L1 = L2 =

L3=1.25m)...... 119

XVII

Figure 4.42. FFT Spectrums in tangential direction of 3D-bended copper

pipe with the air (Fixed-Open / Fixed-Closed) (L1 = L2 =

L3=1m). ............ ……………………………......................... 121

Figure 4.43. FFT Spectrums in radial direction of 3D-bended copper pipe

with the water (Fixed-Open / Fixed-Closed) (L1 = L2 =

L3=1m). ............ ……………………………......................... 121

Figure 4.44. Experimental pressure-time history of 3D-Bended copper


L3=2.333m) at different external excitation frequencies……... 124

Figure 4.45. Experimental pressure-time history of 3D-bended copper


L3=2.333m) when external excitation frequency is equal to

the liquid frequency (47.8 Hz) …..........……………….......... 125

Figure 4.46. Free ended pipe on elastic foundation …………...................... 125

Figure 4.47. L-Bended pipe on elastic foundation …………...................... 130

Figure 4.48. 3D-bended pipe on elastic foundation ……...…...................... 133

Figure 4.49. Variation of the natural frequencies of a single-spanned steel

pipe filled by the air with the slenderness ratio (Fixed-Fixed

and Open-Closed) a) Structural Modes b) Fluid Modes ....... 140


pipe filled by the water with the slenderness ratio (Fixed-

Fixed and Open-Closed) a) Structural Modes b) Fluid Modes 141


pipe filled by the air with the slenderness ratio (Fixed-Free

and Closed-Closed) a) Structural Modes b) Fluid Modes ..... 142


pipe filled by the water with the slenderness ratio (Fixed-Free

and Closed-Closed) a) Structural Modes b) Fluid Modes ........ 143

Figure 4.53. Variation of the natural frequencies of a single-spanned

copper pipe filled by the air with the slenderness ratio (Fixed-

Fixed and Open-Closed) a) Structural Modes b) Fluid Modes 144

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copper pipe filled by the water with the slenderness ratio

(Fixed-Fixed and Open-Closed) a) Structural Modes b) Fluid

Modes ……….…………………………………...................... 145


copper pipe filled by the air with the slenderness ratio (Fixed-

Free and Closed-Closed) a) Structural Modes b) Fluid Modes 146


copper pipe filled by the water with the slenderness ratio

(Fixed-Free and Closed-Closed) a) Structural Modes b) Fluid

Modes ……….…………………………………...................... 147

Figure 4.57. Bended Angle α ......................................……………….......... 148

Figure 4.58. Variation of the natural frequencies (Hz) in structural modes

of steel pipe system with the bend angle (Fixed-Fixed) …….. 153


of steel pipe system with the bend angle (Fixed-Free) ……... 154


of copper pipe system with the bend angle (Fixed-Fixed) ....... 155


of copper pipe system with the bend angle (Fixed-Free) ........ 156

XIX

NOMENCLATURE

A : Cross-sectional area (pipe, fluid)

A : Coefficients of integration

a : Wave speed

B : Matrix coefficients

b : Ratio of pipe radius to pipe wall thickness

C : Field transfer matrix coefficients

c : Coupled wave speed ratio

d : Ratio of pipe density to fluid density

E : Young’s modulus of elasticity

e : Pipe wall thickness

F : Force amplitude

f : Force

f : Natural frequency

G : Shear modulus of rigidity

g : Bend point matrix coefficient

h : Ratio of Young’s mdu1us to modified bulk modulus

I : Moment of inertia

J : Polar moment of inertia

K : Fluid isothermal bulk modulus of elasticity

k : Spring stiffness

L : Length of crank mechanism

l : Length of pipe reach

M : Moment amplitude

m : Moment

m : Mass

P : Fluid pressure amplitude

P : Fluid pressure

q : Ratio of fluid area to pipe area

r : Radius of pipe cross-section

XX

s : Coordinate along pipe axis

T : Torque

t : Time

U : Pipe displacement amplitude

u : Pipe displacement

V : Fluid displacement amplitude

v : Fluid displacement

w : Radial displacement

z : Axial displacement

α : Angle between incident pipe reaches

β : Angle of rotation due to shear

Δ : Field transfer matrix coefficient

Δ : Matrix determinant

δ : Differential element

θ : Angular direction

κ : Shape factor for shear

λ : Eigenvalues

υ : Poisson’s ratio

ρ : Mass density

σ : Stress

φ : Angle between local and global axes

Ω : Forcing frequency

ω : Natural circular frequency Γ : Distributed foundation stiffness

τ ,σ ,γ : Field transfer matrix coefficient

Φ : Foundation Modulus

XXI

Subscripts

f : Fluid

G : Global coordinate system

i : Pipe end

L : Local coordinate system

p : Pipe

p : Local axis index

q : Global axis index

R : Rows

s : Spring

t : Time

z : Direction along pipe axis

fp : Fluid and axial pipe wall field transfer matrix

tz : Torsion vibration about z-axis field transfer matrix

xz : Transverse vibration in X-Z plane field transfer matrix

θ : Angular direction

X,Y,Z :Global rectangular coordinate directions

x,y,z :Local rectangular coordinate directions

Superscripts

B : Point matrix for a bend

L : Left of discontinuity

M : Point matrix for a lumped mass

R : Right of discontinuity

S : Point matrix for a spring

T : Matrix transposition

-1 : Matrix inverse

1. INTRODUCTION Ahmet ÖZBAY

1

1. INTRODUCTION

Liquid-filled piping systems are very important for many industrial

applications. They are used for conveying gases and fluids over a wide range of

temperatures and pressures. The failure of piping systems in power or chemical

plants can cause severe economic losses and even loss of human lives. Some of the

design or operation factors that may cause failures in piping systems are: incorrect

support, transient pressure changes, thermal stresses, and flow induced vibration.

In general, a complete dynamic analysis of liquid-filled piping must consider

both the forces of liquid on the piping and the opposing forces of the piping on the

fluid, whether the source of excitation acts on the fluid or the pipe (Budny-1988). For

example, when a hydraulic transient is produced by a sudden valve operation a

pressure pulse is generated in the fluid and this pulse causes structural pipe vibration.

The study of liquid-filled pipes becomes more complicated when several

factors are taken into account. The five families of waves, tees and bends, supports of

various stiffness, structural restraints and hydraulic devices such as pumps, orifices

and valves must be considered. The speed of the wave components depends on pipe

material and fluid properties (Lesmez-1989). The frequencies at which the liquid and

pipe are vibrating are influenced by the structural support configurations of the pipe

and the hydraulic elements of the system.

In this study, the free vibration behavior of air/water-filled piping systems is

first studied with the help of the transfer matrix method (TMM). The existing

governing equations in analytical form which consider the axial, transverse and

torsional vibration of such piping systems are handled in the analysis. Some basic

configurations of copper/steel piping systems such as single span, L-bend, Z-bend, U-

bend and 3-D bend are studied for both fixed-fixed and fixed-free boundary conditions.

ANSYS software program and some experiments completed in this work are used to verify

the present theoretical results together with the results available in the literature. The effect

of the elastic foundation on the natural frequencies is also investigated. A parametric study

is, finally, carried out to understand correctly the vibrational behavior of such piping

systems.

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2. LITERATURE REVIEW Ahmet ÖZBAY

2

2. LITERATURE REVIEW

2.1. Flow Induced Vibrations in Pipelines

Flow-induced vibration phenomena have been treated by a variety of

engineering disciplines, each having its particular terminology. In an attempt to

provide a unified overview, let’s propose the following definition of basic elements

of flow-induced vibration:

a) Body oscillators;

b) Fluid oscillators; and

c) Source of excitation.

Oscillators are defied herein as systems of structural or fluid mass that acted

upon by restoring forces if deflected from their equilibrium positions and undergo

vibrations in conjunction with appropriate types of excitation. An engineering system

will usually possess several potential oscillators and several sources of excitation.

The first and most important task in the assessment of possible flow-induced

vibrations is therefore to identify them.

A body oscillator consist of either a rigid structure or part that is elastically

supported so that it can perform linear or angular movements or a structure or

structural part that is elastic in itself so that it can perform flexural movements.

A fluid oscillator consists of a passive mass of fluid that can undergo

oscillations usually governed either by fluid compressibility or by gravity. In both

cases, the oscillating fluid mass can be discrete or it can be distributed. Fluid-flow

systems may contain a number of oscillators. They may give rise to undesirable fluid

pulsations when excited; and they may amplify the vibration of a body oscillator if

one their natural frequencies coincides with the natural body-oscillator frequency.

Sources of excitation for either body or fluid oscillators are numerous and may be

difficult to detect. It is therefore useful to treat them within a basic framework.

• Extraneously induced excitation

• Instability-induced excitation

• Movement-induced excitation

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Extraneously induced excitation is caused by fluctuations in flow velocities or

pressures that are independent of any flow instability originating from the structure

considered and independent of structural movements except for added-mass and

fluid-damping effects.

Movement-induced excitation is due to fluctuating forces that arise from

movements of the vibrating body or fluid oscillator. Vibrations of the latter are thus

self-excited.

The fluid-structure interaction (FSI) in liquid-filled pipe systems has been

investigated extensively, because of its relevance to mechanical, civil, nuclear and

aeronautical engineering.

By using Bernoulli-Euler beam theory, Wilkinson (1978) showed that under

certain conditions the vibrations of the liquid column and that of the supporting

structure can interact.

Chaudhry (1979) and Wylie and Streeter (1982) give transfer matrices

corresponding to almost every element in hydraulic piping systems such as

oscillating valve, fixed orifice, pump, pipe bend accumulators, and uncoupled

straight pipe elements.

Otwell (1984) developed and verified a numerical model with experimental

data. The one-dimensional equations of continuity and momentum for the liquid and

pipe wall were solved by the method of characteristics. At an elbow, coupling was

introduced by continuity relationships. The translation of attached piping at an elbow

was represented by an added stiffness term, and solved simultaneously with the

characteristic equations. The equations were normalized and dimensionless

parameters were identified that describe the liquid-pipe interaction.

Budny (1988) evaluated the four equation model concerned with only axial

wave propagation and Poisson coupling to account for the fluid-structure interaction.

The developed model includes viscous damping and a fluid shear stress term to

account for the structural and liquid energy dissipation. If a piping system is to be

exposed to a steady state pipe vibration, fixing the piping with a rigid support will

limit the pressure rise to its minimum. However, if motion must be permitted, as in

the case of expansion loops, then installing a stiff damper is suggested.

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Wiggert et al. (1987) extended Wilkinson’s work by including the Poisson’s

effect and by using the Timoshenko beam theory. Experimental results with an L-

shaped pipe showed a good agreement with the numerical model.

Lesmez used the same model with a U-shaped bend for a variable length

piping system.

Baasri (1990) investigated the phenomenon of air release during hydraulic

transients in pipe flow with column separation. It was established that when line

pressure during a hydraulic transient drops to the vapor pressure of the liquid column

separation and cavitations bubbles occur throughout the system, and that air release

from saturated water is initiated towards these bubbles by the process of convective

diffusion. At the time of cavity collapse, the sudden increase in pressure causes the

system to agitate. This agitation significantly increases the rate of air release and

with disappearance of all vapor cavities small gas bubbles scattered throughout the

system are left behind. These gas bubbles cause a considerable decrease in both the

wave speed of the medium and the peak pressures.

Yakut (1996) investigated the flow acoustic coupling phenomenon

experimentally. Yakut (1996) observed from experiments that the flow-acoustic

coupling was realized if vortex-shedding frequency locked on the natural acoustic

frequency and its harmonics of pipeline.

Tusseling (1996) reviewed the literature on transient phenomena in liquid-

filled pipe systems up to 1996.

Li (1997) has studied; a specially designed multi-span tube array test rig was

used to investigate the effects of partial flow admission. Using this test rig the water

flow can pass across any location along the tube span. Various end supports were

used in the different experimental setups. Therefore, not only the first mode but also

the higher vibration modes can be excited, depending on the location of the flow and

tube-support configurations. It was been found that vibration modes higher than the

third mode do not have significant vibration displacement. The experiments show

that the fluid energy is additive along the span, regardless of the tube mode shape.

Response peaks were observed prior to the ultimate fluid-elastic instability. By

analyzing the corresponding Strouhal numbers, it was found that both vortex


5

shedding and secondary instability mechanisms exist. These two different

phenomena may interact and enhance each other. Therefore, high amplitude

displacement can be reached even before the ultimate fluid-elastic instability. The

previous and present experimental data suggest that the energy fraction is a

representative parameter in the analysis of the flow-induced vibration caused by non-

uniform flow velocity distribution.

Teng-yang (1997), investigated the measurements of the flow induced

vibration and flow velocities. Teng-yang (1997) analyzed the possibility fatigue

failure of the dog-leg pipe assembly of a Flixborough process plant. In this analysis,

the approximate axial, lateral, and rocking (angular) natural frequencies of the

assembly were determined. These frequencies were scaled for the model and

compared to flow frequencies measured on the model from proximitor displacement

measurements of the pipe. The results indicated that both axial and lateral resonance

of the prototype was probable.

Allison (1998) treated the effects of two types of flow-induced vibration on

structures of square cross-section under two-dimensional conditions: vortex-induced

vibration and galloping. The model incorporates the effects of the oscillating wake

by coupling the equation for the cylinder motion with an equation for the angular

displacement of the wake-oscillator. The model equations are examined by analytical

means in the quest for stability and bifurcation information. The effects of model

parameters are of primary interest. The analytical methods used are much more

efficient than numerical solutions.

Rungta (1998) presented the similarity between the dynamics of structural

systems and acoustic systems to show that structural uncoupling criteria were

applicable to acoustic systems. In the analysis uncoupling criteria is developed using

two degree of freedom lumped parameter structural systems. The uncoupling

criterion was applied to a continuous acoustic system, where the continuous system

was replaced with an equivalent lumped system for the first mode. Shifts of

frequencies were estimated using the uncoupling criteria. The criterion therefore

gives approximate shifts and was only applicable for assessment purposes.

Comparison of an acoustic system with a structural system shows that these


6

equilibrium equations in these two systems were exactly the same. Therefore, the

uncoupling criteria developed for structural systems are applicable for acoustic

systems.

Gidi’s (1999) investigations have focused on flow regime and two-phase flow

damping ratio. However, tube bundles in steam generators have vapor generated on

the surface of the tubes, which might affect the flow regime, void fraction

distribution, turbulence levels and tube-flow interaction, al1 of which have the

potential to change the tube vibration response. In Gidi’s (1999) study, flows regime

for bundle void generation was at al1 times bubbly and homogeneous, while the

upstream void friction generation cases showed a clear tendency to chum flow. A

change in flow regime from bubbly to chum flow will produce the same effect as an

increase in turbulence buffeting levels, and hence it seems difficult with the present

knowledge to distinguish between the two causes. In as much as turbulence levels are

related to flow regime, it is essential to have a clear knowledge of the flow regime in

steam generators in order to predict the fluid-elastic instability threshold of the tubes.

Evgin (2000) evaluated the effects of interface strength on the behavior of

buried flexible pipe.

Kartha (2000) studied experimentally explores the potential of different

active, passive and active/passive control methodologies for control of vibrational

power flow in fluid filled pipes. Circumferential modal decomposition and

measurements of vibrational power carried by individual wave types were carried out

experimentally. The importance of dominant structural bending waves and the need

to eliminate them in order to obtain meaningful experimental results has been

demonstrated. The effectiveness of the rubber isolator in reducing structural waves

has been demonstrated. Improved performance of the quarter wavelength tube and

Helmholtz resonator was obtained on implementation of the rubber isolator on the

experimental rig. Active control experiments using the side-branch actuator and 1/3

piezoelectric composite yielded significant dB reductions revealing their potential for

practical applications. A combined active/passive approach was also implemented as

part of this work. This approach yielded promising results, which proved that


7

combining advantages of both active and passive approaches was a feasible

alternative.

Durrani (2001) treated the dynamics of pipelines with a finite element

method. In his thesis a Finite Element Method (FEM) has developed for the

application of Coriolis force on a fluid filled Pipeline. He has calculated the

deflections and mode shape frequencies with the selected project data first using

standard textbook methods, second using the industrial methods and third using one

of the commercial software ANSYS. Nine cases were studied using this FEM and

actual industrial project data. The resultant data shows noticeable effects of Coriolis

force at relatively higher flow velocities.

Taking into account all the three major coupling mechanism, namely the

friction coupling, Poisson coupling and junction coupling, Li et al. (2002) studied the

vibration analysis of a liquid-filled pipe system by the transfer matrix method.

Evans (2004) presents a theory and experimental research relating the mass

flow rate within a pipe to pipe vibration. This approach has the potential to develop

into a non-intrusive, low-cost, flow rate measurement. Experimental results indicate

a nearly quadratic relationship between the signal noise and mass flow rate in the

pipe. This relationship is believed to be caused by friction coupling between the fluid

and the pipe. It is also shown that the signal noise-mass flow rate relationship is also

dependent on the pipe material and pipe diameter.

Nieves (2004) investigated the flow-induced vibration. Three finite elements

models for the pipe system were developed: a structural finite element analysis

model with multi support system for frequency analysis, a fluid structure interaction

(FSI) finite element model and a transient flow model for water hammer induced

vibration analysis in a fluid filled pipe. The natural frequencies, static, dynamic and

thermal stresses, and the limitation of the pipeline system were investigated. The

investigation demonstrates that a gap in a support at the segment k has a negative

effect on the entire piping system. In the water-hammer analysis, the limit maximum

flow rates were determined based on the rate of a rapid closure of the isolation valve.

A study of the fluid transient in a simple pipeline was performed. Results obtained

from FE model for fluid-structure interaction was compared with a model without

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considering fluid-structure interaction effects. The results show notable differences

in the velocities profile and deformation due to the fluid-structure interaction effects.

2.2. Transfer Matrix Method

Lesmez (1989) formulated the vibration of liquid-filled piping system by

using one-dimensional wave theory in both the liquid reaches and the pipe wall.

Considering both the junction coupling and Poisson coupling, he used the transfer

matrix method to study the motion of these systems

Akdoğan (1992) studied the transfer matrix method implemented to carry

dynamic analysis of piping systems with the Euler-Bernoulli beam theory. The

analysis involves determination of dynamic characteristics and steady state harmonic

response for such systems, especially at low frequencies. In his work the junction

coupling and Poisson coupling are considered.

Servaites (1996) analyzed the static and dynamic behavior of steel smoke

stacks subject to excitation by aerodynamic forces. A computer program created

modifying an existing analysis code, to be used specifically for stack analysis. This

analysis code utilizes the transfer matrix method to perform detailed bending and

vibration analyses. A detailed analysis was performed to demonstrate the validity of

approximating a tapered Timoshenko beam with a series of continuous, constant

cross-section beams.

Dolasa (1998) developed a design tool to analyze and design un-damped

beam and rotor systems in two dimensions. Systems modeled in two dimensions,

such as beams with different moments of inertia, could produce varying responses in

the each direction of motion. A coupling between the vertical and horizontal motions

also exists in rotor systems mounted of fluid film bearings. The transfer matrix

method has been used in the development of the software and an explanation of the

method is included in his thesis.

Fang Yu (2001) developed a method for exact vibration analysis of 3-D frame

structures. The transfer matrix for each beam element was rearranged in dynamic

stiffness matrix that was called dynamic stiffness matrix that related beam end forces

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and displacements. For each frequency, an eigenvector for displacement at the ends

of beam elements could be computed and the associated eigen function could be

determined by the eigenvector and the dynamic shape function based on the Euler-

Bernoulli and Timoshenko beam theories. Several examples were presented to

demonstrate the principles and algorithms and the results were compared and show

good agreement with those computed ANSYS or given in the references.

Daneshfaraz and Kaya (2007) presented an approach for the application of the

method of transfer matrix to the analysis of one dimensional flow problems hydraulic

branch of civil engineering, and to the lateral dynamic analysis of multi-storey

buildings. At their study various examples taken from the literature solved using

transfer matrix method. It was seen that the transfer matrix approach was in

sufficient agreement with other methods.

3. MATERIAL AND METHOD Ahmet ÖZBAY

10

3. MATERIALS AND METHOD

3.1. Material

The aim of the experiments is to support the theoretical solutions obtained

with the help of the transfer matrix method.

In order to measure the liquid natural frequencies; 4 tanks, valves and 2

pressure transducers were used. Two tanks were used as upstream tanks, the other

two were used as downstream tanks. The tanks were pressurized with the air that had

a maximum pressure of 4 Bar. Figure.3.1. shows the general piping setup used in this

study.

Figure 3.1. Pipeline Test-Rig

If the length of the pipe is known, the fundamental frequency and harmonics

are determined. An open-closed system results closure of the valve and excites the

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odd harmonics of the liquid. The fundamental frequency of an open-closed liquid

system is given by

lc

f ff 4= (3.1)

where ff is the fundamental frequency of the liquid, l is the length of the pipe, and cf

is the coupled wave speed which will described in Equations (3.22) and (3.23).

In order to measure the first structural natural frequency of the pipe

configurations, impact hammer test was applied. In this method, structure is excited

by hammer and causes it to vibrate while the structure is monitoring with

accelerometers and FFT analyzer. It gives significant peaks at its natural frequencies

in FFT spectrums. This method is tested by different samples before used in

experiments and proved its appropriateness. Also this method was used by Çınar

(1998) to define the natural frequencies of the piping system.

3.1.1. Pipe Materials

One inch nominal diameter cooper pipe and two inch nominal diameter steel

pipe types were used in the experiments. Table 3.1. and Table 3.2. list the physical

properties of the piping system.

Table 3.1. Physical properties of copper pipe

Young’s Modulus ( E ) 97 GPa

Density ( ρ ) 8350 kg/m3

Inside Radius ( r ) 14 mm

Thickness ( e ) 1 mm

Poisson’s Ratio (υ ) 0.35

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Table 3.2. Physical properties of steel pipe

Young’s Modulus ( E ) 157 GPa

Density ( ρ ) 7600 kg/m3

Inside Radius ( r ) 28.15 mm

Thickness ( e ) 3.6 mm

Poisson’s Ratio (υ ) 0.28

3.1.2. Liquid

The liquid that is used in the experiments is from the Soda Ash Plant water

supply system. Table 3.3. lists the physical properties of the water.

Table 3.3. Physical properties of liquid

Temperature 25.0 °C

Bulk Modulus ( K ) 2.2 GPa

Density ( ρ ) 997.0 kg/m3

3.1.3. External Shaker

The pipe was excited by an external shaker, which produces reciprocating

force. The shaker is a crank-slider mechanism that transfers rotary motion to

reciprocating motion. The shaker consists of a motor, pulley, crank and connecting

rod.

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Figure 3.2. Crank mechanism

Gamak Type AGM 3-phase cage induction motor with Altivar 58

Telemechanic variable speed controller was used in experiments. The speed was

increased by the pulley form 2840 rpm to 6090 rpm.

3.1.4. Transducers

Two pressure transducers and one portable vibration data collector were used

for pressure and vibration measurements.

Fuji electric gauge pressure transmitter model FKG was used for recording

the pressure.

The DLI Watchman® DCA-20 Portable Data Collector was used a single-

channel FFT analyzer and DLI Engineering’s ExpertALERT™ vibration analysis

software was used in experiments.


14

Figure 3.3. Fuji electric gauge pressure transmitter model FKG

Figure 3.4. The DLI Watchman® DCA-20 portable data collector


15

3.2. Method

3.2.1. Governing Equations

In this section differential equations available in the literature which govern

the free vibration behavior of liquid-filled piping systems are presented as in the

same in Lesmez ‘s study (1989).

Figure 3.5. shows a general pipe reach with the sign convention used in this

study. The z-axis is considered coincident with the centerline of the pipe reach.

zzf

f'f yyy δ

∂

∂+=

Figure 3.5. Sign convention for internal forces (Lesmez,1989)

3.2.1.1. Axial Vibration – Liquid and Pipe Wall

The fluid is assumed to be one-dimensional (the radial component of the fluid

velocity is zero and the flow is developed in only the radial direction), linear, and

X

Z

Y

v

f’x

fz

f’z

fy

fx

f’y

mz

mx my

m’z

m’x

m’y

δz

re

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homogeneous, with isotropic flow and uniform pressure and fluid velocity over the

cross-section. The pipe wall is assumed to be linearly elastic, isotropic, prismatic,

circular and thin-walled.

Two equations represent the axial continuity and momentum relations for the

liquid:

02 2

=⎥⎦

⎤⎢⎣

⎡∂∂

∂+

∂∂

+∂∂

ztv

tw

rK

tp (3.2)

02 02

2

=+∂∂

+∂∂

rtv

zp

fτρ (3.3.a)

in which p = p(z,t) is the fluid pressure, v = v(z,t) is the fluid displacement, and

w = w(z,t) is the pipe wall displacement. K and ρf are the fluid bulk modulus and

density, r is the inside radius of the pipe, and the shear stress along the pipe wall is

represented by τ0. In these equations it is assumed that the fluid density is constant

(the convective terms are ignored by assuming low Mach numbers, where the fluid

wave speed is much greater than the fluid velocity) and the radial component of the

fluid velocity is zero. The fluid friction term in the momentum equation can be

neglected for forced vibrations.

02

2

=∂∂

+∂∂

tv

zp

fρ (3.3.b)

Assuming an axisymmetric, linear elastic pipe walls with small deformations

and no buckling, the axial and circumferential stress-strain relationships for the pipe-

wall are

0* =⎥⎦⎤

⎢⎣⎡ +∂∂

−rw

zuE z

z υσ (3.4)

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0* =⎥⎦⎤

⎢⎣⎡

∂∂

+−zu

rwE zυσθ (3.5.a)

where the modified modulus of elasticity is defined as

( )2*

1 υ−=

EE (3.5.b)

in which σz = σz (z,t) and σθ = σθ (z,t) are the stresses in the axial and radial direction,

uz = uz (z,t) is the pipe wall displacement in the axial direction, and E and υ are the

Young's modulus and Poisson's ratio of the pipe wall, respectively. Figures 3.6 and

3.7 show a section of a pipe with stresses and displacements in the axial and radial

directions.

Figure 3.6. Axial pipe element

XY

Z

δz

v

σz

e

uz (axial) w (radial)

σz+ z

z

∂∂σ δz

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Figure 3.7. Radial pipe element

The equations of motion for the pipe wall are:

02

2

=∂∂

−∂∂

tu

zz

pz ρσ (3.6)

02 2

22

=∂∂

⎥⎦

⎤⎢⎣

⎡+−−

twrreepr fp ρρσθ (3.7)

in which ρp is the pipe wall density and e is the pipe wall thickness. The effect of the

radial fluid acceleration appears as an added mass in the last term of equation (3.7).

Equations (3.2)-(3.7) constitute the six-equation model.

By neglecting the radial inertia term in Equation 3.7 the radial stress, σθ , can

be evaluated in terms of the fluid pressure:

epr

=θσ (3.8)

Combining equations 3.5 and 3.8 and solving for the radial strain w/r can

eliminate the radial stress

zeδσθ

2δθ

θ∂ e

prδθδz

r

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zu

eEpr

rw z

∂∂

−= υ* (3.9)

Combining equations 3.4 and 3.9 give the expression for the axial stress

0=∂∂

−−zuEp

er z

z υσ (3.10.a)

multiply the above equation by the pipe cross-section area Ap, to obtain the axial

force, fz

0=∂∂

−−z

zppz

uEAperAf υ (3.10.b)

Differentiating Equation 3.9 with respect to time and combining it with

Equation 3.2 produces the expression for the fluid pressure

022

*2

* =∂∂

∂+

∂∂∂

−∂∂

ztvK

ztuK

zp zυ (3.11.a)

where

eErK

KK*

*

21+= (3.11.b)

Equations 3.3b, 3.6, 3.10.b and 3.11 constitute the four-equation model

presented by Otwell (1984), and Budny (1988). Differentiating Equations 3.3b and

3.6 with respect to the axial direction z, then differentiating Equations 3.10b and 3.11

with respect to time and combining them to solve for the axial force and fluid

pressure, can further reduce the followings.

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20

02

2

2

2

2

22 =

∂∂

+∂∂

−∂∂

tpbA

tf

zfa p

zzp υ (3.12)

02

2

22

2

2

2

22 =

∂∂

+∂∂

−∂∂

zf

dAa

tp

zpa z

p

ff

υ (3.13.a)

where

ff

Kaρ

*2 = (3.13.b)

pp

Eaρ

=2 (3.13.c)

erb = (3.13.d)

f

pdρρ

= (3.13.e)

In Equations 3.13.b and 3.13.c, af and ap are the non-coupled fluid and axial

pipe wall wave speeds, respectively, b is the pipe radius to wall thickness ratio and d

is the density ratio. Equations 3.12 and 3.13.a are second order partial differential

equations in the fluid pressure and axial pipe wall force. They may be expressed in

matrix form as:

010

12

0

2

2

2

2

22

2

=⎭⎬⎫

⎩⎨⎧

∂⎥⎦

⎤⎢⎣

⎡ −−

⎭⎬⎫

⎩⎨⎧

∂∂

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

pf

tdbA

pf

zaadA

azpz

ffp

p υυ (3.14a)

A similar equation can be obtained for the axial pipe wall and fluid

displacements by combining and solving Equations 3.3.b, 3.6, 3.10.b and 3.11.a.

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21

02

001

2

22

2

2

2

22

2222

=⎭⎬⎫

⎩⎨⎧

∂⎥⎥⎦

⎤

⎢⎢⎣

⎡−

⎭⎬⎫

⎩⎨⎧

∂∂

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−

−+

vu

td

db

vu

zadba

db

adbaa

db

zz

ff

fpf

υ

υυ (3.14b)

Poisson terms couple equation 3.12 and 3.13.a as shown by the off-diagonal

elements of the matrices in Equations 3.14.a and 3.14.b.

The separation of variables technique is used to solve for the force fz and fluid

pressure p in Equation 3.14a. Three steps are necessary to solve for the dependent

variables in the above equation: i) convert the partial differential equation into

ordinary differential equation, ii) find solutions for the ordinary differential equation,

and iii) find the constants of integration of the differential equation. The solution for

the constants of integration will be postponed to the next chapter since they depend

on the boundary conditions imposed on the piping system.

i) Separation of Variables

Assuming a harmonic oscillation for the time dependence, which is

appropriate for oscillatory flow and oscillatory structural motion in the axial

direction, we can write:

jwt

zz ezFtzf )(),( = (3.15)

jwt

zz ezptzp )(),( = (3.16)

where F(z) and P(z) are functions of z only, ω is the oscillatory frequency and

1−=j

Substituting the above equations into Equation 3.14 yields the ordinary

differential equation in Fz and P.

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22

010

12

02

22

2

=⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡ −+

⎭⎬⎫

⎩⎨⎧

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

PFbA

PF

aadA

azp

ıı

ıı

ffp

p υωυ (3.17)

where Fzıı and Pıı are the derivatives with respect to the axial direction z. The

elimination method can be used to reduce Equation 3.17 to a single dependent

variable. This procedure yields

0)(24 =+

+++

lF

lF ıııv τσγστ (3.18.a)

where l is the length of a pipe reach and

2

22

falωτ = (3.18.b)

2

22

palωσ = (3.18.c)

2

2222

pdalbωυγ = (3.18.d)

Equation 3.18.a is a fourth-order, ordinary differential equation with constant

coefficients.

ii) Solution of the Ordinary Differential Equation

The solutions for Fz in Equation 3.18a is of the form

lz

z eAzFλ

=)( (3.19)

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where A is a constant.

Substitution of Equation 3.18 into 3.17.a produces the characteristic equation

in λ:

0)( 24 =++++ στλγστλ (3.20)

where λ is the characteristic value. The roots of this equation are ±jλ1 and ±jλ2 ,

where

στγστγστλ 421 2

212 −++±++= )()(, (3.21)

This equation can also be expressed as:

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

−⎥⎦⎤

⎢⎣⎡ ++−⎥⎦

⎤⎢⎣⎡ ++== 22

222222222

21

222 422

21

pffpffpff aaadbaaa

dbaalc υυ

λω (3.22)

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

−⎥⎦⎤

⎢⎣⎡ +++⎥⎦

⎤⎢⎣⎡ ++== 22

222222222

22

222 422

21

pffpffpfp aaadbaaa

dbaalc υυ

λω (3.23)

The above equations give the expressions for the coupled wave speeds. These

coupled speeds are the same as those derived by Budny (1988) and Lesmez (1989) using

the method of characteristics. An inspection of Equations 3.22 and 3.22, assuming

no coupling between liquid and pipe wall by neglecting the second order Poisson terms,

yields

22

1fawl

=⎟⎟⎠

⎞⎜⎜⎝

⎛λ

(3.24.a)

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24

22

2pawl

=⎟⎟⎠

⎞⎜⎜⎝

⎛λ

(3.24.b)

Placing Equation 3.21 into 3.19, the solution for Fz (z) is:

lzj

lzj

lzj

lzj

z eAeAeAeAzF 2211

4321)(λλλλ −−−

+++= (3.25)

and using the relation

)lzsin(j)

lzcos(e

)lz(j

λλλ

±=±

(3.26)

Equation 3.24 can be written in the following form

)lzsin(A)

lzcos(A)

lzsin(A)

lzcos(A)z(Fz 24231211 λλλλ +++= (3.27.a)

where

211 AAA += (3.27.b) )( 212 AAjA −= (3.27.c) 433 AAA += (3.27.d) )( 434 AAjA −= (3.27.e)

iii) Solution for the Constants of Integration

The solutions for the pipe wall and fluid displacements and the fluid pressure

are of the same form as Equation 3.27a. To solve for the four dependent variables,

the constants of integration A1, A2, A3 and A4 must have known values. Expressing the

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axial and fluid displacements in similar forms as the force and fluid pressure in Equations

3.15 and 3.16 gives

jwt

zz ezUtzu )(),( = (3.28)

jwtzz ezVtzv )(),( = (3.29)

Placing Equation 3.28 into 3.6 and combining with Equation 3.27.a we obtain

the solution for the axial displacement:

⎭⎬⎫

⎩⎨⎧

⎥⎦⎤

⎢⎣⎡ −+⎥⎦

⎤⎢⎣⎡ −= )

lzcos(A)

lzsin(A)

lzcos(A)

lzsin(A

EAl)z(U

pz 2423212111 λλλλλλ

σ (3.30)

The fluid pressure is obtained by placing Equations 3.27.a and 3.30 into

3.11.b

( ) ( )⎭⎬⎫

⎩⎨⎧

⎥⎦⎤

⎢⎣⎡ +−+⎥⎦

⎤⎢⎣⎡ +−= )

lzsin(A)

lzcos(A)

lzsin(A)

lzcos(A

bA)z(P

pz 2423

221211

21

1 λλλσλλλσσυ

(3.31)

Finally, the fluid displacement is obtained by placing Equations 3.29 and

3.31 into 3.3b

( ) ( )⎭⎬⎫

⎩⎨⎧

⎥⎦⎤

⎢⎣⎡ −−+⎥⎦

⎤⎢⎣⎡ −−

−= )

lzcos(A)

lzsin(A)

lzcos(A)

lzsin(A

bKAl)z(V *

pz 24232

2212111

21 λλλλσλλλλσ

τσυ

(3.32)

Arranging Equations 3.27.a, 3.28, 3.31 and 3.32 into matrix form we obtain

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⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛−

⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛

=

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

4

3

2

1

2211

26261515

24241313

22221111

AAAA

lzsin

lzcos

lzsin

lzcos

lzcosB

lzsinB

lzcosB

lzsinB

lzsinB

lzcosB

lzsinB

lzcosB

lzcosB

lzsinB

lzcosB

lzsinB

FVP

U

Z

Z

λλλλ

λλλλ

λλλλ

λλλλ

(3.33.a)

where

σλEA

lBp

11 = (3.33.b)

σλEA

lBp

22 = (3.33.c)

υσλσ

bAB

p

21

3−

= (3.33.d)

υσλσ

bAB

p

22

4−

= (3.33.e)

υστλλσ

*1

21

5)(

bKAl

Bp

−= (3.33.f)

υστλλσ

*2

22

6)(

bKAl

Bp

−= (3.33.g)

3.2.1.2. Transverse Vibration in x-z Plane

Consider the free body diagram of an element of a beam bending in the x-z

plane shown in Figures 3.8.and 3.9. Where my is the internal bending moment, φy is

the rotation due to bending deformation, and fx is the internal shear force.

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Figure 3.8. Sign convention for pipe element

⎥⎦⎤

⎢⎣⎡ −∂∂

= yϕκz

uGAf x

spx yspGA βκ= (3.34.a)

)1(2 υ+=

EG (3.34.b)

υυκ

34)1(2

++

=s (3.34.c)

where G is the shear modulus, Apκs represents the effective shear area of the section

and κs is the shape factor for a thin-walled tube.

Y

Z

Original

Deformed

X

φy

βy

zux

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28

Figure 3.9. my and fx forces in x-z plane.

From the elementary beam theory

0=∂∂

−z

EIm Ypy

ϕ (3.35)

where Ip is the moment of inertia about the y-axis for the pipe wall. From equilibrium

conditions Figure 3.9.

02

2

=∂∂

−∂∂

tu

zf xx μ (3.36)

02

2

=∂

∂−+

∂

∂

tf

m yx

z

y ϕφ (3.37)

Y

Z

X

φy

my

fx

zz

xx

ff δ

∂∂

+

zz

yy

mm δ

∂

∂+

zδ


29

where

ffpp AA ρρμ += (3.38)

ffpp II ρρφ += (3.39)

and If is the moment of inertia about the y-axis for the fluid. Solving for φy and ux in

Equations (3.34.a) and (3.36), substituting the results in Equation (3.35) and

eliminating my from Equation (3.38), we obtain a fourth-order partial differential

equation in fx(z,t):

02

2

2

2

2

2

2

2

22

4

2

2

4

4

=⎥⎦

⎤⎢⎣

⎡∂∂

∂∂

+⎥⎦

⎤⎢⎣

⎡∂∂

∂∂

+∂∂

∂−

∂∂

+∂∂

tf

tGAtf

zGAEI

tzf

tf

zf

EI x

sp

x

sp

pxxxp μ

κμμ

κφμ

(3.40)

By neglecting rotatory inertia and shear deformation terms in equation 3.40,

we obtain the Euler- Bernoulli beam equation in the shear force fx,

02

2

4

4

=∂∂

+∂∂

tf

zf

EI xxp μ (3.41)

The separation of variables technique is used to solve for the dependent

variable fx, in time, t, and axial direction z.

tj

xx ezFtzf ω)(),( = (3.42)

Substitution of the above equation into Equation (3.39) we obtain

042 =⎥⎦⎤

⎢⎣⎡ −

−+

+ xıı

xiv

x Fl

Fl

F στγτσ (3.43)

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for the case of nonlinear transverse vibration equation single frequency input --> multiple frequency output we use Excitaion pulsating flow ( single frequency input) u(t)=u0+u1 e^(i w t) p(t)=p0 + p1 e^(i w t) and the solution (multiple frequency output) f(z,t) = Fx(z) A(n) e^(i n w t)


30

where

22lGA sp

ωκ

μσ = (3.44.a)

22l

EI p

ωφτ = (3.44.b)

42l

EI p

ωμγ = (3.44.c)

and l is the length of the pipe reach. The solution of a fourth-order ordinary

differential equation with constant coefficient shown in Equation (3.43) is given by

lz

x eA)z(Fλ

= (3.45)

where A is a constant. Substitution of Equation (3.45) into (3.43) produces the

characteristic equation in λ:

0)()( 24 =−+++ στγλστλ (3.46)

The roots of this equation are ±jλ1 and ±jλ2 , where

)(21))(

41(

22

2,12 τστσγλ +

⎭⎬⎫

⎩⎨⎧ −+= m (3.47)

From Equation (3.45) we may write

lzj

lzj

lz

lz

x eAeAeAeA)z(F 2211

4321

λλλλ −−−+++= (3.48)

and using the relations

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)lzsinh()

lzcosh(e

)lz(

λλλ

±=±

and )lzsin(j)

lzcos(e

)lz(j

λλλ

±=±

(3.49)

Equation (3.48) can be written in the following form

)lzsin(A)

lzcos(A)

lzsinh(A)

lzcosh(A)z(Fx 24231211 λλλλ +++= (3.50.a)

where

211 AAA += (3.50.b) 212 AAA −= (3.50.c) 433 AAA += (3.50.d) )( 434 AAjA −= (3.50.e)

Placing Equation (3.50.a) and combining with Equation (3.36) we obtain the

Ux(z)

⎥⎦⎤

⎢⎣⎡ +−+

−= )

lzcos(

lA)

lzsin(

lA)

lzcosh(

lA)

lzsinh(

lA

EIl)z(U

px 2

242

231

121

11

4

λλλλλλλλγ

(3.51)

The slope is obtained by placing Equations 3.50.a and 3.51 into 3.34.a

( ) ( )⎭⎬⎫

⎩⎨⎧

⎥⎦⎤

⎢⎣⎡ +−+⎥⎦

⎤⎢⎣⎡ ++−= )

lzsin(A)

lzcos(A)

lzsinh(A)

lzcosh(A

EIl)z(

py 2423

221211

21

2

λλλσλλλσγ

ψ

(3.52)

The bending moment is obtained by placing Equation (3.52) into (3.35)

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( ) ( )⎭⎬⎫

⎩⎨⎧

⎥⎦⎤

⎢⎣⎡ −−+⎥⎦

⎤⎢⎣⎡ ++

−= )

lzcos(A)

lzsin(A)

lzcosh(A)

lzsinh(Al)z(M y 24232

2212111

21 λλλσλλλλλσ

γ

(3.53)

Arranging Equations (3.50.a), through (3.53) into matrix form we obtain

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛−

⎟⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛−

⎟⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛−

=

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

4

3

2

1

2211

26261515

24241313

22221111

AAAA

lzsin

lzcos

lzsinh

lzcosh

lzcosB

lzsinB

lzcoshB

lzsinhB

lzsinB

lzcosB

lzsinhB

lzcoshB

lzcosB

lzsinB

lzcoshB

lzsinhB

FM

U

x

y

y

x

λλλλ

λλλλ

λλλλ

λλλλ

ψ

(3.54.a)

where

1

3

1 λγEI

lBp

= (3.54.b)

2

3

2 λγEI

lBp

= (3.54.c)

)( 21

2

3 λσγ

+=EI

lBp

(3.54.d)

)( 22

2

4 λσγ

−=EI

lBp

(3.54.e)

12

15 )( λλσγ

+=lB (3.54.f)

22

16 )( λσλγ

−=lB (3.54.g)

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3.2.1.3. Transverse Vibration in y-z Plane

The derivation of the governing equations for the pipe reach in Figures 3.10

and 3.11, vibrating in the y-z plane, is obtained by the same procedure as stated in

the previous section. The change in the sign of the shear angle βx determines sign

changes in the rotation and bending moment, whereas the shear force and lateral

displacement remain the same. Equation (3.54.a) becomes

Figure. 3.10. Sign convention for pipe element.

X Y

Z

Z

Y

Original Deformed

φx

βx

zu y

∂

∂

z∂

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Figure 3.11. mx and fy Forces in y-z Plane.

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛−

=

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

4

3

2

1

2211

26261515

24241313

22221111

AAAA

lzsin

lzcos

lzsinh

lzcosh

lzcosB

lzsinB

lzcoshB

lzsinhB

lzsinB

lzcosB

lzsinhB

lzcoshB

lzcosB

lzsinB

lzcoshB

lzsinhB

FM

U

y

x

x

y

λλλλ

λλλλ

λλλλ

λλλλ

ψ

(3.55)

where the coefficients of the matrix are given in equation (3.5.b) through (3.54.g).

3.2.4. Torsional Vibration

Consider the free body diagram of an element of a beam shown in the Figure

3.12. where mz is the internal torsional moment, and φz is the angle of rotation.

Z

Y X

z∂

Y

Z

φx

fy

mx

zzf

f yy δ

∂

∂+

zz

mm x

x δ∂∂

+

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35

Figure 3.12. Pipe reach subjected to torsion.

From the equilibrium condition we may write the following

02

2

=∂∂

−∂∂

tJ

zm z

ppz ϕ

ρ (3.56)

And from the elastic properties

zGJm z

pz ∂∂

=ϕ (3.57)

where G and Jp are the shear modulus and the polar moment of inertia for the pipe

wall, respectively. The wave equation for the moment mz (z,t) is:

02

2

2

2

=∂∂

−∂∂

tm

Gzm zpz ρ

(3.58)

φz

δz

zz

zz δ

ϕϕ

∂∂

+

Z z

zm

m zz δ

∂∂

+ Z

X

Y

mz


36

The separation of variables can be used to solve for mz in the above equation.

tj

zz ezMtzm ω)(),( = (3.59)

Substitution of the above equation into (3.56) yields

0" 2 =+ zz Ml

M γ (3.60.a)

where

22lG

p ωρ

γ = (3.60.b)

The solution of Equation (3.60.a) is of the form

lz

z eAzMλ

=)( (3.61)

Placing the above equation into (3.58.a) yields the characteristic equation in

λ:

02 =+ γλ (3.62)

The roots of this equation are λj± where

[ ]2/1

2/1⎟⎟⎠

⎞⎜⎜⎝

⎛±=±=

Gl pρωγλ (3.63)

Placing the characteristic value λ in Equation (3.61), the solution for Mz is


37

lzj

lzj

z eAeAzMλλ −

+= 21)( (3.64)

using the relation in Equation (3.26) the above equation becomes

)lzsin(A)

lzcos(A)z(M z λλ 21 += (3.65)

where A1 and A2 are given in Equation (3.27.b) and (3.27.c)

The solution of the rotation ψz about z–axis is found by placing Equation

(3.65) into (3.57) and using

tj

zz eztz ωψϕ )(),( = (3.66)

we obtain

⎥⎦⎤

⎢⎣⎡ −= )

lzcos(A)

lzsin(A

lJ)z(

ppz λλ

ρλψ 21 (3.67)

Equation (3.67) and (3.65) can be arranged in matrix form as

⎭⎬⎫

⎩⎨⎧

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡ −

=⎭⎬⎫

⎩⎨⎧

2

1

AA

)lzsin()

lzcos(

)lzcos(

lJ)

lzsin(

lJ)z(M)z(

pppp

z

z

λλ

λρ

λλρλ

ψ (3.68)

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38

3.2.2. Transfer Matrix Method

The basic principle behind the transfer matrix method is that of breaking up a

complicated system into individual parts with simple elastic and dynamic properties

that can be expressed in matrix form. When these individual matrices are arranged in

a prescribed fashion and multiplied together, the static and dynamic behavior of the

entire system is readily calculated.

Many structures encountered in practice consist of a number of elements

linked together end to end in a chain-like configuration. Examples of this type of

system include continuous beams, turbine rotors, crankshafts, pressure vessels, etc.

The transfer matrix method provides a quick and efficient analysis of such

systems simply by multiplying successive elemental transfer matrices together.

Therefore, this method is ideal for analyzing pipes.

3.2.2.1. Transfer Matrix Procedure

In order to analyze complex structures with the transfer matrix method, the

structure first needs to be divided into simple sections. For each of these simple

sections, a matrix is used to describe how the displacements and forces interact with

each other. For example, consider the following spring-mass system that is vibrating

with circular frequency ω in Figure 3.13.

Figure 3.13. Spring - mass system

ki-1

xi-1

zLi-1

mi-1

zRi-1 zL

i

ki mi

xi

zRi

ki+1

xi+1

mi+1

zLi+1 zR

i+1

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39

The mass mi-1 is attached to mass mi by a massless spring of stiffness ki. The

state vector immediately to the left of mass mi is labeled as ziL. The state vector

immediately to the right of mass mi-1 is labeled as zi-1R. The internal forces of spring

ki are shown in Figure 3.14.

Figure 3.14. Free-body diagram of spring

Because the spring is massless, the forces acting on either end of the spring

are equivalent.

Li

Ri NN =−1 (3.69)

From the stiffness definition for a linear spring, the internal force acting

through the spring is:

)( 11 −− −== iiiLi

Ri xxkNN (3.70)

Rewriting these equations yields

i

Ri

ii kNxx 1

1−

− += (3.71)

In matrix form,

R

i

iL

i Nxk

Nx

110/11

−⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡=

⎭⎬⎫

⎩⎨⎧

(3.72)


40

or

[ ] Rii

Li zTz 1−= (3.73)

This equation provides a relation between the two state vectors on either side

of the spring. The matrix [T]i is known as the field transfer matrix.

In a similar manner, the transfer matrix between two state vectors on either

side of the mass can be derived. From the free-body diagram in Figure 3.15, it can be

seen that the displacements on either side of the infinitely rigid, point mass will be

equal.

Figure 3.15. Free-body diagram of a mass

Li

Ri xx = (3.74)

Summing the forces and rearranging from the free body diagram,

iiLi

Ri xmNN 2ω−= (3.75)

In matrix form, these two equations can be represented as follows

L

ii

R

i Nx

mNx

⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡−

=⎭⎬⎫

⎩⎨⎧

101

2ω (3.76)

or

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41

[ ] Lii

Ri zPz = (3.77)

From these two transfer matrices, a variety of spring-mass problems can

easily be analyzed. As mentioned before, [Pi] is the point transfer matrix for mass

mi, and [Ti] is the field transfer matrix for spring ki. To find the relation between the

state vectors on either side of the series of the structure shown in Figure 3.13. The

internal transfer matrices are multiplied together in the following fashion.

[ ] Lii

Ri zPz 111 −−− = (3.78)

[ ] [ ] [ ] L

iiiRii

Li zPTzTz 111 −−− == (3.79)

[ ] [ ] [ ] [ ] L

iiiiLii

Ri zPTPzPz 11 −−== (3.80)

[ ] [ ] [ ] [ ] [ ] L

iiiiiRii

Li zPTPTzTz 11111 −−+++ == (3.81)

[ ] [ ] [ ] [ ] [ ] [ ] L

iiiiiiLii

Ri zPTPTPzPz 1111111 −−+++++ == (3.82)

The above transfer matrix multiplication can be written as:

Li

Ri zUz 11 ][ −+ = (3.83)

where U is the global transfer matrix.

3.2.2.2 Field transfer Matrices

The field transfer matrix expresses the forces and displacements at one

section of a chain-type structure in terms of the corresponding forces and

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42

displacements at an adjacent section. A general procedure that can be applied to

Equations (3.33.a), (3.54.a), (3.55), (3.67).

[ ] AzBzZ )()( = (3.84)

Where )(zZ is the state vector representing the dependent variables of any

one of the above equations, [ ])(zB is a matrix that depends on the geometry of the

pipe wall and material properties, and A is a vector containing the constants of

integration.

At point z=0 in Figure 3.16. , 1Z)( −= izZ the matrix equation (3.84)

becomes

Figure 3.16.General pipe element

ABZ i )]0([1 =− (3.85)

Solving for the column vector A in the above equation

11)]0([ −−= iZBA (3.86)

Substituting Equation (3.84) into Equation (3.82) yields

i-1 i

l

Y X

Z

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43

11)]0()][([ Z(z) −−= iZBzB (3.87)

At point z=l, izZ Z)( = , so Equation (3.87) becomes

111 ][)]0()][([ −−− == iii ZTZBlBZ (3.88)

where [T] is field transfer matrix.

3.2.2.2.(1). Liquid and Pipe Wall Vibration

Field transfer matrix for liquid and axial pipe wall vibration [Tfp] is

(Lesmez,1989):

( )[ ][ ]

( ) [ ][ ] ( )[ ]

⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−−−−

−++−+++−+

−−+−+

++−−++−

−

=

023231

130232

12

213023

3121302

212

22

CCChbC

hb)CC(

CCCC)(C)(CC

CCC)(CC)(C

C)(CChbCC

hbCC

]T[ fp

σστυσυσσ

γτσυγτσγγτγττ

υσ

υτγττγτυστ

γστυγτσυσ

(3.89)

where

2

22

falωτ = (3.90.a)

2

22

palωσ = (3.90.b)

dbσυγ 22= (3.90.c)

erb = (3.90.d)

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44

f

pdρρ

= (3.90.e)

*KEh = (3.90.f)

[ ])cos()cos(C 2

211

220 λλλλΔ −= (3.90.g)

⎥⎦

⎤⎢⎣

⎡−= )sin()sin(C 2

2

21

11

22

1 λλλ

λλλ

Δ (3.90.h)

[ ])cos()cos(C 212 λλΔ −= (3.90.i)

⎥⎦

⎤⎢⎣

⎡−= )sin()sin(C 2

21

13

11 λλ

λλ

Δ (3.90.j)

[ ] 12

221

−−=Δ λλ (3.90.k)

( ) ( ) στγστγστλ 421 22

1 −++−++= (3.90.l)

( ) ( ) στγστγστλ 421 22

2 −+++++= (3.90.m)

and the non-dimensional state vector at location i is:

T

ip

zzi EA

FlV

KP

lU

Z⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

= * (3.91)

3.2.2.2.(2). Transverse Vibration in x-z Plane

Same procedure applied for Equation (3.54.a) and field transfer matrix for x-z

plane [Txz] was found as (Pestel and Leckie, 1963):

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[ ]

( )[ ]

[ ]⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

−−−−−+−−−−+

−−−

++−−+−−

=

203231

3120132

2

231203

32

123120

)()()(

1)(

CCCCCCCCCCCCC

CCCCCC

CCCCCCC

Txz

σγγσγτστττγγ

ττγ

σγσγ

τσσ

(3.92)

22lGA

AA

sp

ffpp ωκρρ

σ+

= (3.93a)

22l

EIII

p

ffpp ωρρ

τ+

= (3.93.b)

42l

EIAA

p

ffpp ωρρ

γ+

= (3.93.c)

[ ])cos()cosh(C 2

211

220 λλλλΔ −= (3.93.d)

⎥⎦

⎤⎢⎣

⎡−= )sin()sinh(C 2

2

21

11

22

1 λλλ

λλλ

Δ (3.93.e)

[ ])cos()cosh(C 212 λλΔ −= (3.93.f)

⎥⎦

⎤⎢⎣

⎡−= )sin()sinh(C 2

21

13

11 λλ

λλ

Δ (3.93.g)

[ ] 12

221

−−=Δ λλ (3.93.h)

)(21)(

41 22

1 τστσγλ +−−+= (3.93.i)

)(21)(

41 22

2 τστσγλ ++−+= (3.93.j)

And state vector at location i in Figure 3.16 is:

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46

T

ip

x

p

yy

xi EI

lFEIM

lU

Z⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

=2

ψ (3.94)

3.2.2.2.(3). Transverse Vibration in y-z Plane

Field transfer matrix for the transverse vibration of a pipe reach in the y-z

plane [T yz] is (Pestel and Leckie, 1963).

[ ]( )[ ]

[ ]⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

−−−+−−−+−

−−−

++−−−++−−

=

203231

3120132

2

231203

32

123120

)()()(

1)(

CCCCCCCCCCCCC

CCCCCC

CCCCCCC

Tyz

σγγσγτστττγγ

ττγ

σγσγ

τσσ

(3.95)

where the coefficients are given in equation 3.93.a through 3.93.j. The state vector at

location i in Figure 3.15 is:

T

ip

y

p

xx

yi EI

lFEIM

lU

Z⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

=2

ψ (3.96)

3.2.2.2.(4). Torsional Vibration about z Axis

Field transfer matrix for the torsion about z axis [T tz] is (Pestel and Leckie,

1963).

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

−

−−=)cos()sin(

)sin()cos(]T[ tzλλλ

λλ

λ 1 (3.97)

where

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47

Gl pρωλ 222 = (3.98)

and the state vector Zi is

T

ip

zzi GJ

lMZ⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

= ψ (3.99)

3.2.2.3. General Field Transfer Matrix

The general field transfer matrix [TL], for a pipe reach of length is

(Lesmez,1989).

[ ]⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

][][

][][

tz

yz

xz

fp

L

TT

TT

T (3.100)

The state vector for equation (3.100) is

T

ip

zz

p

y

p

xx

y

p

x

p

yy

x

p

zzi GJ

lMEI

lFEIM

lU

EIlF

EIM

lU

EAF

lV

KP

lU

Z⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

= ψψψ22

*(3.101)

If we rearrange the overall state vector, Equation (3.101) becomes

T

ip

z

p

y

p

x

p

z

p

y

p

xzyxzyxi EA

FEI

lFEI

lFGJ

lMEIM

EIM

lV

lU

lU

lU

KPZ

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

=22

* ψψψ (3.102)

And the general field transfer matrix becomes:

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l

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14 13 12 11 10 9 8 7 6 5 4 3 2 1

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4 12 8 14 7 11 3 1 9 5 13 6 10 2

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48

[ ] 1−= iLi ZTZ (3.103)

or

[ ] [ ] [ ] [ ][ ] [ ] [ ] [ ]

[ ] [ ] [ ] [ ][ ] [ ]

[ ] [ ] [ ] [ ][ ] [ ] [ ] [ ]

[ ] [ ] [ ] [ ][ ] [ ] [ ] [ ]

[ ] [ ] [ ] [ ][ ] [ ] [ ] [ ]

[ ] [ ][ ] [ ] [ ] [ ]

[ ] [ ] [ ] [ ][ ] [ ] [ ] [ ]

1

2

2

*

44434142

44434142

44434142

2221

34333132

34333132

34333132

14131112

14131112

14131112

1211

24232122

24232122

24232122

2

2

*

00000000000000000000000000000000000000000000000000000000000000

00000000000000000000

0000000000000000000000000000000000000000000000000000

0000000000

−⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎭

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎬

⎫

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎨

⎧

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎭

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎬

⎫

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎨

⎧

ip

z

p

y

p

x

p

z

p

y

p

x

z

y

x

z

y

x

fpfpfpfp

yzyzyzyz

xzxzxzxz

tztz

xzxzxzxz

yzyzyzyz

fpfpfpfp

fpfpfpfp

yzyzyzyz

xzxzxzxz

tztz

xzxzxzxz

yxyzyzyz

fpfpfpfp

ip

z

p

y

p

x

p

z

p

y

p

x

z

y

x

z

y

x

EAF

EIlF

EIlF

GJlM

EIMEIMlVl

Ul

Ul

U

KP

TTTTTTTT

TTTTTT

TTTTTTTT

TTTTTTTT

TTTTTTTT

TTTTTT

TTTTTTTT

EAF

EIlF

EIlF

GJlM

EIMEIMlVl

Ul

Ul

U

KP

ψψψ

ψψψ

(3.104)

3.2.2.4. Point Matrices

Field transfer matrices for each pipe section are connected by three types of

point transfer matrices which are bend, spring and mass point matrices. Valves,

accumulators and control instrumentation can be modeled as concentrated or point

masses. In this section bend and spring point matrices will be outlined.

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49

3.2.2.4.(1). Bend Point Matrix

A piping system in two or three dimensional space can be treated as a

collection of straight pipe reaches, differing in orientation and joined end-to-end. The

difference in orientation generates junction coupling of the fluid pressure and of the

pipe wail moments and forces between the reaches. The junction itself is treated as a

discontinuity with negligible mass and length. Equilibrium and continuity

relationships constitute the basis for point matrices at bends (Lesmez,1989).

Pipe bends or elbows can be considered as a velocity discontinuity where the

direction of fluid flow changes and pressure inside the piping exerts a force on the

pipe wall.

Figure 3.17. Sign convention for a bend.

The sate vectors to the right and left of point i, ZiR and Zi

L can be related by a

bend point matrix.

[ ] Lii

BL

Ri ZPZ = (3.105)

Bend point matrix [ ]iBLP may be derived for rotation about z- axis

L

i-1 i

R

α

i+1

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50

(3.1

06)

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for rotation about y- axis

(3.1

07)

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52

for rotation about x- axis

(3.1

08)

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53

3.2.2.4.(2). Spring Point Matrix

Piping systems generally are supported at several locations, restricting motion

partially or totally, or they may be placed on an elastic foundation. The elastic

foundation can be represented by springs. Each spring can be modeled as a point

matrix (Lesmez, 1989).

Figure 3.18. shows the pipe reach has a spring support and is vibrating in the

y-z plane. The state vectors to the right and left of point i, RiZ and L

iZ can again

be related by a point matrix. Where ki is the stiffness of the spring.

Figure 3.18. Forces at spring

[ ] Lii

SL

Ri ZPZ = (3.109)

Spring point matrix is given as

x y

z

i-1 i

ki

RiZ

LiZ

RiFy

RiMx

RiFy

LiFy

LiMx

LiFy

i+1

kiUyi=Spring force

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54

(3.1

10)


55

3.2.2.5. Boundary Conditions

Fixed-Fixed and Fixed-Free boundary conditions for structural, open-closed

and closed-open boundary conditions for liquid are examined in this part. The state

vector becomes for fixed-open end

ip

z

p

y

p

x

p

z

p

y

p

x

ip

z

p

y

p

x

p

z

p

y

p

x

z

y

x

z

y

x

EAF

EIlF

EIlF

GJlM

EIMEIMlV

EAF

EIlF

EIlF

GJlM

EIMEIMlVl

Ul

Ul

U

KP

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎭

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎬

⎫

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎨

⎧

=

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎭

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎬

⎫

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎨

⎧

2

2

2

2

*

0

0

0

0000

ψψψ

(3.111)

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56

for fixed-closed end

ip

z

p

y

p

x

p

z

p

y

p

x

ip

z

p

y

p

x

p

z

p

y

p

x

z

y

x

z

y

x

EAF

EIlF

EIlF

GJlM

EIMEIM

KP

EAF

EIlF

EIlF

GJlM

EIMEIMlVl

Ul

Ul

U

KP

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎭

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎬

⎫

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎨

⎧

=

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎭

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎬

⎫

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎨

⎧

2

2

*

2

2

*

0

0

0

0

000

ψψψ

(3.112)

for free-closed end

i

z

y

x

z

y

x

ip

z

p

y

p

x

p

z

p

y

p

x

z

y

x

z

y

x

lUl

Ul

U

KP

EAF

EIlF

EIlF

GJlM

EIMEIMlVl

Ul

Ul

U

KP

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎭

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎬

⎫

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎨

⎧

=

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎭

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎬

⎫

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎨

⎧

0

0

0

0

0

0

0

*

2

2

*

ψψψ

ψψψ

(3.113)

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57

3.2.2.6. Natural Frequencies

The natural frequencies of the liquid-filled piping systems depend on the

boundary conditions. For example, fixed-free single span pipe’s, for open-closed

liquid boundary conditions, natural frequencies are determined as:

[ ] 01 ZTZ L= (3.114)

[ ] [ ] [ ] [ ][ ] [ ] [ ] [ ]

[ ] [ ] [ ] [ ][ ] [ ]

[ ] [ ] [ ] [ ][ ] [ ] [ ] [ ]

[ ] [ ] [ ] [ ][ ] [ ] [ ] [ ]

[ ] [ ] [ ] [ ][ ] [ ] [ ] [ ]

[ ] [ ][ ] [ ] [ ] [ ]

[ ] [ ] [ ] [ ][ ] [ ] [ ] [ ]

0

2

2

44434142

44434142

44434142

2221

34333132

34333132

34333132

14131112

14131112

14131112

1211

24232122

24232122

24232122

1

*

0

0

0

0000

00000000000000000000000000000000000000000000000000000000000000

00000000000000000000

0000000000000000000000000000000000000000000000000000

0000000000

0

0

0

0

0

0

0

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎭

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎬

⎫

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎨

⎧

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎭

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎬

⎫

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎨

⎧

EAF

EIlF

EIlF

GJlM

EIMEIMl

V

TTTTTTTT

TTTTTT

TTTTTTTT

TTTTTTTT

TTTTTTTT

TTTTTT

TTTTTTTT

lUl

Ul

U

KP

p

z

p

y

p

x

p

z

p

y

p

x

fpfpfpfp

yzyzyzyz

xzxzxzxz

tztz

xzxzxzxz

yzyzyzyz

fpfpfpfp

fpfpfpfp

yzyzyzyz

xzxzxzxz

tztz

xzxzxzxz

yxyzyzyz

fpfpfpfp

z

y

x

z

y

x

ψψψ

(3.115)

From here the following is written

[ ] [ ] 0*0 ZT= (3.116)

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free closed

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58

[ ] [ ][ ] [ ]

[ ] [ ][ ]

[ ] [ ][ ] [ ]

[ ] [ ]

⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎭

⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎬

⎫

⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎨

⎧

⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=

⎪⎪⎪⎪

⎭

⎪⎪⎪⎪

⎬

⎫

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

EAF

EIlF

EIlF

GJlM

EIMEIMlV

TTTT

TTT

TTTT

TT

p

z

p

y

p

x

p

z

p

y

p

x

fpfp

yzyz

xzxz

tz

xzxz

yzyz

fpfp

2

2

4443

4443

4443

22

3433

3433

3433

1 0000000000000000000000000000000

00000

0000000

(3.117)

[ ]*T is called eigen matrix and the frequencies satisfying the above condition

by making the determinant of the eigen matrix zero correspond to the natural

frequencies.

3.2.2.7. Vibration of a Pipe on Elastic Foundation

The field transfer matrices for pipe on elastic foundation are same with the

single straight pipe. Also the general field transfer matrix was composed of same

four sub matrices: longitudinal vibration of the liquid and pipe wall, transverse

vibration in the x-z as well as in the y-z planes and torsional vibration about the z-

axis. The general field transfer matrix expression is given in Equation (3.104).

Only the parameters σ, τ and γ are different in transverse vibration in the x-z and in

the y-z planes.

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59

22)(

lGA

AA

sp

ffpp

κωρρ

σΓ−+

= (3.118)

2*2)(

lEI

II

p

ffpp Γ−+=

ωρρτ (3.119)

42)(

lEI

AA

p

ffpp Γ−+=

ωρργ (3.120)

Where Γ is distributed foundation stiffness.

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4. RESULTS AND DISCUSSION Ahmet ÖZBAY

60

4. RESULTS AND DISCUSSION

The objective of this chapter is to present both the theoretical based on the

transfer matrix method (TMM) and experimental results of this study. The results

obtained in this work are verified by both finite element solution using ANSYS and

some results available in the literature. It may be noted that since ANSYS neglects

the effect of bulk modulus of the fluid, natural frequencies evaluated by the ANSYS

in just the structural modes are used for comparisons.

The fundamental frequency of the system in liquid mode at Open-Closed

boundary conditions is given by

l

cf f

f 4= (3.1)

where ff is the fundamental frequency of the liquid, cf is the coupled wave speed and l

is the length of the pipe. By using both Equations (3.1.) and (3.22.) coupled wave

speed cf may be calculated as 1405 m/s for steel pipe and as 1279 m/s for copper

pipe.

Akdoğan (1992) also studied a single-spanned pipe filled a light fluid at

various end conditions with the help of the transfer matrix method offered in this

study. Comparison of the present results with Akdoğan’s (1992) results is given in

Tables 4.1. and 4.2. From those tables an excellent agreement is observed between

the fundamental frequencies in both structural and fluid modes.

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Table 4.1. Comparison of the present theoretical natural frequencies (rad/s) of 5m-length copper pipe filled by the air with the literature(Fixed-Fixed and Open-Closed)

(E=117 GPa, ρp=8940 kg/m3, K=0.14 GPa, ρf:=1.2 kg/m3)

Modes Present Study

(TMM)

Akdoğan

(1992)

Equation

(3.1)

Structural 31.1377 31.138 -

Structural 85.7764 85.7764 -

Fluid 107.0931 106.86 106.869

Structural 168.01 168.01 -

Structural 277.425 277.43 -

Fluid 320.68 320.58 320.606

Structural 413.876 413.88 -

Fluid 534.378 534.52 534.343

Structural 577.163 577.16 -

Fluid 748.102 751.13 748.08

Table 4.2. Comparison of the present theoretical natural frequencies (rad/s) of 5m-length copper pipe filled by the air with the literature (Fixed-Fixed and Open-Closed)

(E=117 GPa, ρp=8940 kg/m3, K=0.14 GPa, ρf:=1.2 kg/m3)

Modes Present Study

(TMM)

Akdoğan

(1992) Equation (3.1)

Structural 4.89561 4.896 -

Structural 30.6702 30.670 -

Structural 85.8322 85.832 -

Fluid 106.865 106.86 106.869

Structural 168.068 168.07 -

Structural 277.551 277.55 -

Fluid 320.609 320.59 320.606

Structural 414.105 414.10 -

Fluid 534.34 534.52 534.343

Structural 577.539 577.54 -

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62

Earlier than the case studies considered in this work, the soundness of the

present experiments is confirmed by a few tests at just structural fixed-fixed ends.

As mentioned in the previous chapter, impact hammer test is used together

with FFT analyzer to determine the natural frequencies of the system in the structural

mode. The measurements are taken in different directions and locations. The liquid

natural frequencies are measured by using both the pressure transmitter and the

external shaker. The liquid inside the pipe is excited at the natural frequency in the

liquid mode by the external shaker. The pressure was fluctuated with large amplitude

when the shaker excited at liquid natural frequency. Comparison of the experimental

frequency measured in liquid mode is made with Equation (3.1.)

To explain the experimental procedure let’s consider 25m-length steel pipe.

The fundamental frequency of the liquid may be calculated in an analytical manner

as ff =5.936Hz by using Equation (3.1.) At normal conditions (without any external

disturbance) the pressure inside the pipe is about 2bar. When the pipe is excited by

an external harmonic force, then the amplitude of the inside pressure gets start to

fluctuate about the pressure of 2bar. As the amplitude of the harmonic force

increases, the amplitude of the inside pressure increases. When the external disturbed

frequency near the natural frequency of the liquid inside the pipe, due to the

resonance phenomena amplitudes become very large. This is clearly shown in Figure

4.1 and 4.2. shows the pressure values for definite time period while the excitation

frequency is near 6.0375 Hz.

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-6

-4

-2

0

2

4

6

8

10

1 125 249 373 497 621 745 869 993 1117 1241 1365 1489 1613

Time (s)

Pressure (1/10 Bar)

Figure 4.1. Experimental pressure-time history of 25m-length steel pipe with fixed-

fixed ends at different external excitation frequencies.

-6

-4

-2

0

2

4

6

8

10

580

595

610

625

640

655

670

685

700

715

730

745

760

775

790

805

Time (s)

Pressure (1/10 Bar)

Figure 4.2. Experimental pressure-time history when external excitation frequency

is equal to the liquid frequency (6.0375 Hz).


64

After verifying the present theoretical and experimental results a few case studies

are considered in this section. In the case studies considered five different

configurations of piping systems made of either one inch-nominal diameter copper or

two inch-nominal diameter steel such as

• Single-spanned

• L-Bended

• Z-Bended

• U-Bended

• 3-D Bended

are studied with two structural boundary conditions namely fixed-fixed and fixed-

free. Fluid boundary conditions are assumed to be closed at both ends. As a light

fluid both the air and the water are considered. Material and geometrical properties

of the pipes are presented in Tables 3.1 and 3.2. The physical properties of the water

considered in this study are given by Table 3.3.

4.1. Single-Spanned Pipe with Various End Conditions

Fixed-fixed, fixed-free, and intermediate rigid support applications are

examined for a single-spanned pipe.

Figure 4.3. Single-spanned pipe supported at two ends

L

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65

4.1.1. Fixed-Fixed Single-Spanned Pipe

In this section, the natural frequencies of 2m-length copper and 3.5m-length

steel pipes filled by both the air and the water are taken into consideration.

Table 4.3. Natural frequencies (Hz) of 2m-length copper pipe with the air (Fixed-Fixed and Open-Closed) Present Study

Modes TMM Experimental

ANSYS Equation

(3.1)

Structural 28.8841 27.06 29.004 -

Fluid 42.5994 - - 42.5217

Structural 79.3206 - 80.972 -

Fluid 127.591 - - 127.565

Structural 154.727 - 188.71 -

In ANSY solution of this problem the pipe is divided into ten elements which

are defined by element type Pipe 16. The source program and its numerical results

are presented in Appendix A.1.

Figure 4.4. PIPE16 - Elastic straight pipe

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66

Figures 4.5, 4.6, and 4.7 shows the FFT spectrum of fixed-fixed single-

spanned 2m-length copper pipe with the air in tangential, radial and axial directions,

respectively. The results taken from different directions are quite close to each other

and the distinct peak near 27 Hz verifies the tabulated TMM and ANSYS results.

Figure 4.5. FFT Spectrum in tangential direction of fixed-fixed single -spanned 2m-

length copper pipe with the air

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Figure 4.6. FFT Spectrum in radial direction of fixed-fixed single -spanned 2m-


Figure 4.7. FFT Spectrum in axial direction of fixed-fixed single -spanned 2m-


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The natural frequencies of 2m-length copper pipe with the water for Fixed-

Fixed/Open-Closed conditions are listed in Table 4.4. The FFT spectrums of this

example in axial and tangential directions are presented by Figures 4.8 and 4.9,

respectively.

Table 4.4. Natural frequencies (Hz) of 2m-length copper pipe with the water (Fixed-Fixed and Open-Closed)

Present Study Modes

TMM ExperimentalANSYS

Equation

(3.1)

Structural 21.8473 21.95 21.973 -

Structural 60.0014 - 61.243 -

Structural 157,057 - 142.73 -

Fluid 164 - - 160

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69

Figure 4.8. FFT spectrum in axial directions at two locations of fixed-fixed single -

spanned 2m-length copper pipe with the water

Figure 4.9. FFT Spectrum in tangential direction of fixed-fixed single -spanned 2m-

length copper pipe with the water


70

Tables 4.5 and 4.6 presents the natural frequencies of 3.5m-length steel pipe

with both the air and the water, respectively. The boundary conditions are assumed to

be fixed-fixed for the pipe and Open-Closed for the liquid.

Table 4.5. Natural frequencies (Hz) of 3.5m-length steel pipe with the air (Fixed-Fixed and Open-Closed)

Modes Present Study

(TMM) ANSYS Equation (3.1)

Structural 21.9154 22.007 -

Structural 60.102 61.369 -

Fluid 101.089 - 100.416

Table 4.6. Natural frequencies (Hz) of 3.5m-length steel pipe with the water (Fixed-Fixed and Open-Closed)

Modes Present Study


Fluid 24.3082 - 24.2981

Structural 26.4215 26.532 -

Fluid 72.4528 - 72.8944

Structural 72.8978 73.990 -

Fluid 121.492 - 121.491

4.1.2. Fixed-Free Single -Spanned Pipe

Here 3m-length copper and steel pipes filled by both the air and the water are

studied theoretically. Results are given by Tables 4.7-4.10. Present results show a

good harmony with the ANSYS’s results.

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Table 4.7. Natural frequencies (Hz) of 3m-length copper pipe with the air (Fixed-Free and Open-Closed)

Modes Present Study


Structural 2.02316 2.0234 -

Structural 12.6681 12.710 -

Fluid 28.347 - 28.348

Structural 35.4226 35.874 -

Fluid 85.0441 - 85.0433

Fluid 141.738 - 141.739

Table 4.8. Natural frequencies (Hz) of 3m-length copper pipe with the water (Fixed-Free and Open-Closed)

Modes Present Study


Structural 1.53023 1.5304 -

Structural 9.58195 9.6134 -

Structural 26.7944 27.134 -

Fluid 106.665 - 106.665

Table 4.9. Natural frequencies (Hz) of 3m-length steel pipe with the air (Fixed-Free and Open-Closed)

Modes Present Study


Structural 40.918 40.921 -

Fluid 178.114 - 178.057

Structural 255.504 256.39 -

Fluid 534.344 - 534.171

Structural 711.298 721.12 -

Fluid 890.571 - 890.285

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Table 4.10. Natural frequencies (Hz) of 3m-length steel pipe with the water (Fixed-Free and Open-Closed)

Modes Present Study


Structural 33.939 33.941 -

Structural 211.948 212.66 -

Structural 590.144 598.09 -

Fluid 745.591 - 745.405

4.1.3. Single -Spanned Pipe with Intermediate Rigid Support

Both copper and steel pipes with the air or the water are again examined for

fixed-fixed and Open-Closed boundary conditions. The total length of the pipe is 7m

for the copper pipe and 6m for the steel pipe. The pipe wall displacement at the

middle of the pipe is restricted in x and y directions.

Figure 4.10. Single -spanned pipe with rigid support

L/2

L/2

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Table 4.11. Natural frequencies (Hz) of 7m-length copper pipe filled by the air for intermediate rigid support (Fixed-Fixed and Open-Closed)

Modes Present Study


Structural 6.5154 6.5285 -

Structural 9.4505 9.4893 -

Fluid 12.1713 - 12.149

Structural 21.0942 21.468 -

Structural 26.0192 26.545 -

Fluid 36.4531 - 36.4471

Structural 43.9503 50.043 -

Fluid 60.7483 - 60.7452

Table 4.12. Natural frequencies (Hz) of 7m-length copper pipe filled by the water for intermediate rigid support (Fixed-Fixed and Open-Closed)

Modes Present Study


Structural 4.92799 4.9376 -

Structural 7.14797 7.1773 -

Structural 15.9552 16.237 -

Structural 19.6803 20.077 -

Structural 38.3385 37.850 -

Fluid 46.951 - 45.734


74

Table 4.13. Natural frequencies (Hz) of 6m-length steel pipe filled by the air for intermediate rigid support (Fixed-Fixed and Open-Closed)

Modes Present Study


Fluid 14.1784 - 14.1739

Structural 28.4835 28.541 -

Fluid 41.2327 - 42.5217

Structural 42.5231 41.408 -

Fluid 70.8704 - 70.8696

Structural 91.8079 93.547 -

Fluid 99.2181 - 99.2174

Table 4.14. Natural frequencies (Hz) of 6m-length steel pipe filled by the water for

intermediate rigid support (Fixed-Fixed and Open-Closed)

Modes Present Study


Structural 23.626 23.673 -

Structural 34.2012 34.34 -

Fluid 59.6176 - 59.3174

Structural 76.1604 77.590 -

Fluid 177.994 - 177.952

4.2. Two Pipes with 90o-Bended (L-Bended Pipe)

Free vibration behavior of two pipes which are connected with 90-degrees

Bended Figure 4.11 is worked out both theoretically and experimentally for different

pipe materials, different fluids and different boundary conditions including the

intermediate rigid support.

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Figure 4.11. L-Bended pipe supported at two ends

In this case study, the steel and the cooper are chosen as the pipe materials,

and the air and the water are used as the fluids. Open-Closed ends are used as fluid

boundary conditions.

4.2.1. L-Bended Pipe with Fixed-Free Ends

In these examples L1 = L2 =2.5m for the steel pipe and L1 = L2 =1m for the

copper pipe.

Natural frequencies of L-Bended fixed-free pipe made of steel are presented

in Table 4.15 for the air and Table 4.16 for the water, respectively.

L1 L2

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Table 4.15. Natural frequencies (Hz) of L-Bended steel pipe with the air (Fixed-Closed / Free-Closed)

Modes Present Study


Structural 2.45193 2.5768 -

Structural 2.5768 2.7227 -

Structural 6.84158 7.1050 -

Structural 7.10453 7.4044 -

Fluid 19.5947 - 17.0087

Structural 36.2799 36.354 -

Structural 36.44 36.531 -

Fluid 51.1022 - 51.0261

Structural 52.8358 53.022 -

Structural 54.1443 53.411 -

Fluid 85.5964 - 85.0435

Table 4.16. Natural frequencies (Hz) of L-Bended steel pipe with the water (Fixed-Closed / Free-Closed)

Modes Present Study


Structural 2.13731 2.1373 -

Structural 2.25819 2.2583 -

Structural 5.89353 5.8931 -

Structural 6.13976 6.1414 -

Structural 30.1096 30.153 -

Structural 30.2118 30.300 -

Structural 43.8803 43.978 -

Structural 43.9479 44.301 -

Fluid 79.2599 - 70.2914

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Natural frequencies of L-Bended fixed-free pipe made of copper are

presented in Table 4.17 for the air and Table 4.18 for the water, respectively.

Table 4.17. Natural Frequencies (Hz) of L-Bended copper pipe with the air (Fixed-Closed / Free-Closed)

Modes Present Study


Structural 5.15935 5.6669 -

Structural 5.66703 6.0667 -

Structural 14.8292 15.692 -

Structural 15.6908 16.487 -

Fluid 51.8677 - 42.5217

Structural 80.5901 80.754 -

Structural 81.1612 81.254 -

Table 4.18. Natural Frequencies (Hz) of L-Bended copper pipe with the water (Fixed-Closed / Free-Closed)

Modes Present Study


Structural 4.2864 4.2862 -

Structural 4.58839 4.5886 -

Structural 11.8711 11.868 -

Structural 12.4678 12.470 -

Structural 61.0249 61.079 -

Structural 61.2718 61.457 -

Structural 88.8322 88.928 -

Structural 89.0154 89.668 -

Fluid 184.568 - 159.997

Structural 193.246 195.81 -

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From Tables 4.17 and 4.18, for both the water and the air, a very good

harmony is observed between the present theoretical results and the finite element’s

solutions.

4.2.2. L-Bended Pipe with Fixed-Fixed Ends

A few of pipes with different lengths are studied in both theoretical and

experimental manner.

• L1 = L2 =2.4m for the steel pipe

• L1 = L2 =1m for the copper pipe.

• L1 = L2 =3.5m for the copper pipe.

The natural frequencies of L-Bended Steel Pipe (L1 = L2 = 2.4 m) with the

air (fixed-open / fixed-closed) are presented in Table 4.19. FFT spectrums in axial

directions at two locations are presented in Figure 4.12. Figure 4.13 shows the FFT

spectrum in radial direction of the same pipe.

Table 4.19. Natural frequencies (Hz) of L-Bended steel pipe with the air (Fixed-Open/ Fixed-Closed) (L1 = L2 = 2.4 m)

Present Study Modes


Equation

(3.1)

Structural 10.3506 10.00 10.352 -

Fluid 17.7185 - - 17.7174

Structural 38.6508 - 38.730 -

Structural 40.5088 - 40.600 -

Fluid 53.0942 - - 53.1522

Structural 55.7672 - 56.004 -

Structural 58.5156 - 58.757 -

Fluid 88.414 - - 88.587

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Figure 4.12. FFT Spectrum in axial direction at two locations of L-Bended steel

pipe with the air (Fixed-Open/ Fixed-Closed) (L1 = L2 = 2.4 m)


80

Figure 4.13. FFT Spectrum in radial direction of L-Bended steel pipe with the air

(Fixed-Open/ Fixed-Closed) (L1 = L2 = 2.4 m)

The natural frequencies of L-Bended Steel Pipe (L1 = L2 = 2.4 m) with the

water (Fixed-Open/ Fixed-Closed) are presented in Table 4.20. In this example, two

fundamental frequencies in both structural and fluid modes are also determined

experimentally. For the fundamental frequency in structural mode, Figure 4.14 shows

the FFT spectrum in tangential direction of the same pipe. FFT spectrums in radial

direction at two locations are presented in Figure 4.15. For this problem, the source

program and its numerical results are presented in Appendix A.2.

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Table 4.20. Natural frequencies (Hz) of L-Bended steel pipe with the water (Fixed-Open/ Fixed-Closed) (L1 = L2 = 2.4 m)

Present Study Modes


Equation

(3.1)

Structural 8.58553 9.16 8.5859 -

Structural 32.0602 - 32.124 -

Structural 33.608 - 33.675 -

Structural 46.257 - 46.451 -

Structural 48.5676 - 48.734 -

Fluid 73.3263 76.5333 - 73.2203

Structural 102.786 - 104.78 -

Figure 4.14. FFT Spectrum in tangential direction of L-Bended steel pipe with the

water (Fixed-Open/ Fixed-Closed) (L1 = L2 = 2.4 m)

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Figure 4.15. FFT Spectrums in radial direction at two locations of L-Bended steel

pipe with the water (Fixed-Open/ Fixed-Closed) (L1 = L2 = 2.4 m)


83

In this example to determine the fundamental frequency in fluid mode, the

pressure also get started to fluctuate at large amplitude at near the liquid natural

frequency 76.535 Hz as shown in Figure 4.16. In this figure, the fluctuation on

pressure amplitude is clearly observed while the excitation frequency is near the

liquid natural frequency. Figure 4.17 shows the pressure values for definite time

period while the excitation frequency is 76.535 Hz.

0

10

20

30

40

50

60

70

1 25 49 73 97 121 145 169 193 217 241 265 289 313 337 361

Time (s)

Pressure (Bar/10)

Figure 4.16. Experimental pressure-time history of L-Bended steel pipe with the

water (Fixed-Open/ Fixed-Closed) (L1 = L2 = 2.4 m) at different external excitation frequencies.


84

0

10

20

30

40

50

60

70

96 100 104 108 112 116 120 124 128 132 136 140 144

Time (s)

Pressure (Bar/10)

Figure 4.17. Experimental pressure-time history of L-Bended steel pipe with the

water (Fixed-Open/ Fixed-Closed) (L1 = L2 = 2.4 m) when external excitation frequency is equal to the liquid frequency (76.53 Hz).

As stated above, for fixed-fixed conditions, the L-Bended pipe made of the

copper is studied for two different lengths.

For L1 = L2 = 1 m and the air, both the theoretical and experimental results are

given in Table 4.21. The related FFT spectrums in both axial and tangential

directions are illustrated in Figure 4.18.

Table 4.21. Natural frequencies (Hz) of L-Bended copper pipe with the air (Fixed-Open/ Fixed-Closed) (L1 = L2 = 1 m)

Present Study Modes


Equation

(3.1)

Structural 21.1335 21.43 21.135 -

Fluid 42.5752 - - 42.5217

Structural 79.2158 - 79.376 -

Structural 82.8509 - 83.035 -

Structural 114.055 - 114.56 -

Structural 119.531 - 120.03 -

Fluid 127.231 - - 127.565

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Figure 4.18. FFT Spectrums in axial and tangential directions of L-Bended copper

pipe with the air (Fixed-Open/ Fixed-Closed) (L1 = L2 = 1 m)


86

For L1 = L2 = 1m and the water, both the theoretical and experimental results

are given in Table 4.22. The related FFT spectrums in both axial and radial directions

are illustrated in Figure 4.19.

Table 4.22. Natural frequencies (Hz) of L-Bended copper pipe with the water (Fixed-Open/ Fixed-Closed) (L1 = L2 = 1 m)

Present Study Modes


Equation

(3.1)

Structural 15.986 15.75 15.986 -

Structural 59.9224 - 60.036 -

Structural 62.694 - 62.804 -

Structural 86.2861 - 86.645 -

Structural 90.5175 - 90.782 -

Fluid 163.045 - - 159.997

Structural 194.661 - 195.23 -

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Figure 4.19. FFT Spectrums in axial and radial directions of L-Bended copper

pipe with the water (Fixed-Open/ Fixed-Closed) (L1 = L2 = 1 m)

Natural frequencies of L-Bended copper pipe with the air (Fixed-Open/

Fixed-Closed) (L1 = L2 = 3.5 m) are studied by just theoretically and the results are

tabulated in Table 4.23.

Table 4.23. Natural frequencies (Hz) of L-Bended copper pipe with the air (Fixed-Open/ Fixed-Closed) (L1 = L2 = 3.5 m)

Modes Present Study


Structural 1.72802 1.7283 -

Structural 6.51469 6.5275 -

Structural 6.81232 6.8270 -

Structural 9.44743 9.4862 -

Structural 9.85814 9.8955 -

Fluid 12.1644 - 12.149

Structural 21.086 21.460 -

Structural 21.3841 21.768 -


88

Natural frequencies of L-Bended copper pipe with the water (Fixed-Open/

Fixed-Closed) (L1 = L2 = 3.5 m) are studied by both theoretically and experimentally.

In the experimental study, the fundamental frequency in fluid mode is measured.

Those frequencies are presented in Table 4.24. Figures 4.20 and 4.21 illustrate the

experimental pressure-time history of this pipe system at different external excitation

frequencies and when the external excitation frequency is equal to the fundamental

liquid frequency (44.55Hz).

Table 4.24. Natural frequencies (Hz) of L-Bended copper pipe with the water (Fixed-Open/ Fixed-Closed) (L1 = L2 = 3.5 m)

Present Study Modes

TMM Experimental ANSYS

Equation

(3.1)

Structural 1.30701 - 1.3072 -

Structural 4.92745 - 4.9371 -

Structural 5.15272 - 5.1636 -

Structural 7.14567 - 7.1749 -

Structural 7.45698 - 7.4845 -

Structural 15.9493 - 16.231 -

Structural 16.1761 - 16.464 -

Structural 19.6615 - 20.059 -

Structural 20.1198 - 20.374 -

Fluid 46.5979 44.55 - 45.7134

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0

5

10

15

20

25

30

35

1 26 51 76 101 126 151 176 201 226 251 276 301 326 351 376

Time (s)

Pressure (Bar/10)

Figure 4.20. Experimental pressure-time history of L-Bended copper pipe with the

water (Fixed-Open/ Fixed-Closed) (L1 = L2 = 3.5 m) at different external excitation frequencies.

0

5

10

15

20

25

30

35

82 86 90 94 98 102

106

110

114

118

122

126

130

134

Time (s)

Pressure (Bar/10)

Figure 4.21. Experimental pressure-time history of L-Bended copper pipe with the

water (Fixed-Open/ Fixed-Closed) (L1 = L2 = 3.5m) when external

excitation frequency is equal to the liquid frequency (44.55Hz).


90

4.2.3. L-Bended Pipe with Intermediate Rigid Supports

In this case, Open-Closed fluid boundary conditions and fixed-fixed end

structural boundary conditions are considered for L-Bended pipe made of the

copper are considered. The intermediate rigid supports prevent translations in both x-

and y- directions and are located at the mid-span of each pipe. The lengths of each

pipe are assumed to be L1 = L2 = 2.5m. The fluid is chosen as the water. Theoretical

results are given in Table 4.25

Figure 4.22. L-Bended pipe with intermediate rigid supports

Table 4.25. Natural frequencies of L-Bended copper pipe filled by the water with intermediate rigid supports (Fixed-Open/ Fixed-Closed)

Modes Present Study


Structural 6.9293 6.9287 -

Structural 28.528 28.793 -

Structural 29.0541 29.521 -

Structural 38.4049 38.495 -

Structural 39.3536 39.412 -

Structural 49.241 49.778 -

Structural 49.827 49.983 -

Structural 55.200 55.758 -

Structural 56.1474 56.322 -

Fluid 62.927 - 63.9988

L1/2

L1/2 L2/2

L2/2

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4.3. Three Pipes in the Plane

Here both U-Bended Figure 4.23 and Z-Bended Figure 4.24 pipes made of

both the copper and the steel are studied for different fluids and structural boundary

conditions. Open-Closed boundary conditions are regarded as the fluid boundary

conditions.

Figure 4.23. U-Bended pipe supported at two ends

L3

L1

L2

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Figure 4.24. Z-Bended Pipe supported at two ends

4.3.1. Z-Bended Pipe with Fixed-Free Ends

In this case study, the pipe material is determined as the steel. The length of

each pipe as assumed to be equal (L1 = L2 = L3=1.25m). Both the air and the water

are used as the fluids. Open-Closed fluid boundary conditions and fixed-free

structural end conditions are carried out. The theoretical results for the air are

presented in Table 4.26 and for the water are presented in Table 4.27.

L3

L2

L1

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Table 4.26. Natural frequencies (Hz) of Z-Bended steel pipe with the air (Fixed-Closed / Free-Closed) (L1 = L2 = L3=1.25m)

Modes Present Study


Structural 5.16198 5.4328 -

Structural 5.43299 5.7595 -

Fluid 17.8572 - 22.6783

Structural 22.1406 22.141 -

Structural 25.141 24.008 -

Structural 29.6929 30.807 -

Structural 30.8186 31.810 -

Fluid 73.5498 - 68.0348

Structural 143.529 143.83 -

Table 4.27. Natural frequencies (Hz) of Z-Bended steel pipe with the water (Fixed-Closed / Free-Closed) (L1 = L2 = L3=1.25m)

Modes Present Study


Structural 4.50648 4.5061 -

Structural 4.77698 4.7770 -

Structural 18.3752 18.364 -

Structural 19.8433 19.913 -

Structural 26.3827 25.552 -

Fluid 102.021 - 94.9079

Structural 117.484 117.39 -


94

4.3.2. Z-Bended Pipe with Fixed-Fixed Ends

The theoretical and experimental results for the structural mode of Z-Bended

steel pipe with the air are presented in Table 4.28. For this example FFT spectrums in

axial and tangential directions are shown in Figure 4.26

Table 4.28. Natural frequencies (Hz) of Z-Bended steel pipe with the air (Fixed-Open/ Fixed-Closed) (L1 = L2 = L3=1.25m)

Present Study Modes


Equation

(3.1)

Fluid 19.6043 - - 22.6783

Structural 24.3879 25.72 24.385 -

Structural 26.3378 - 25.418 -

Structural 34.3461 - 34.348 -

Fluid 70.8841 - - 68.0348

Structural 114.42 - 116.91 -

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Figure 4.25. FFT Spectrums in axial and tangential directions of Z-Bended steel

pipe with the air (Fixed-Open/ Fixed-Closed) (L1 = L2 = L3=1.25m)

The theoretical and experimental results of Z-Bended steel pipe with the

water are presented in Table 4.29 . Figures 4.27 and 4.28 illustrate the experimental

pressure-time history of this pipe system at different external excitation frequencies

and when the external excitation frequency is equal to the fundamental liquid

frequency (94.12Hz). For the fundamental frequency in the structural mode, FFT

spectrums in tangential direction at two different locations are illustrated in Figure

4.29.


96

Table 4.29. Natural frequencies (Hz) of Z-Bended steel pipe with the water (Fixed-Open/ Fixed-Closed) (L1 = L2 = L3=1.25m)

Present Study Modes


Equation

(3.1)

Structural 20.229 23.90 20.226 -

Structural 21.0474 - 21.082 -

Structural 28.5025 - 28.490 -

Fluid 95.1599 94.1176 - 93.7219

Structural 98.776 - 96.965 -

Structural 107.068 - 107.16 -

0

5

10

15

20

25

30

35

40

45

1 38 75 112 149 186 223 260 297 334 371 408 445 482 519 556

Time (s)

Pressure (Bar/10)

Figure 4.26. Experimental pressure-time history of Z-Bended steel pipe with the

water (Fixed-Open/ Fixed-Closed) (L1 = L2 = L3=1.25m) at different

external excitation frequencies.

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05

1015202530354045

82 86 90 94 98 102

106

110

114

118

122

126

130

134

Time (s)

Pressure (Bar/10)

Figure 4.27. Experimental pressure-time history of Z-Bended steel pipe with the

water (Fixed-Open/ Fixed-Closed) (L1 = L2 = L3=1.25m) when external excitation frequency is equal to the liquid frequency (94.1176Hz).


98

Figure 4.28. FFT Spectrums in tangential direction at two locations of Z-Bended

steel pipe with the water (Fixed-Open/ Fixed-Closed) (L1 = L2 = L3=1.25m)

The total length of the Z-Bended copper pipe system is considered as either

3m or 7m.

The theoretical and experimental results of Z-Bended copper pipe (L1 = L2 =

L3=1m) with the air are presented in Table 4.30 . For the fundamental frequency in

the structural mode (13.05Hz), FFT spectrums in both tangential and axial directions

are illustrated in Figure 4.30


99

Table 4.30. Natural frequencies (Hz) of Z-Bended copper pipe with the air (Fixed-Open/ Fixed-Closed) (L1 = L2 = L3=1m)

Present Study Modes


Equation

(3.1)

Structural 11.2426 13.05 13.517 -

Structural 13.5169 - 14.200 -

Structural 18.9849 - 18.986 -

Fluid 30.4691 - - 28.3478

Structural 65.5427 - 65.934 -

Structural 72.2805 - 72.403 -

Fluid 89.3839 - - 85.0433

Structural 94.7097 - 93.895 -


100

Figure 4.29. FFT spectrums in tangential and axial directions of Z-Bended copper

pipe with the air (Fixed-Open/ Fixed-Closed) (L1 = L2 = L3=1m)

The theoretical and experimental results of Z-Bended copper pipe (L1 = L2 =

L3=1m) with the water are presented in Table 4.31. For the fundamental frequency in

the structural mode (10.35Hz), FFT spectrums in both tangential and axial directions

are illustrated in Figure 4.31.


101

Table 4.31. Natural Frequencies (Hz) of Z-Bended copper pipe with the water (Fixed-Open/ Fixed-Closed) (L1 = L2 = L3=1m)

Present Study Modes


Equation

(3.1)

Structural 10.2238 10.35 10.223 -

Structural 10.7357 - 10.740 -

Structural 14.3629 - 14.360 -

Structural 49.7901 - 49.869 -

Structural 54.7007 - 54.762 -

Structural 70.9215 - 71.018 -

Structural 71.6428 - 71.843 -

Structural 92.2388 - 94.131 -

Structural 94.9889 - 95.299 -

Fluid 117.056 - - 106.665


102

Figure 4.30. FFT Spectrums in tangential and axial directions of Z-Bended copper

pipe with the water (Fixed-Open/ Fixed-Closed) (L1 = L2 = L3=1m)

The theoretical natural frequencies of Z-Bended copper pipe (L1 = L2 =

L3=7/3m=2.333m) with the air are presented in Table 4.32.

Table 4.32. Natural Frequencies (Hz) of Z-Bended copper pipe with the air (Fixed-Open/ Fixed-Closed) (L1 = L2 = L3=7/3m=2.333m)

Modes Present Study

(TMM) ANSYS

Equation (3.1)

Structural 2.08803 2.4859 -

Structural 2.4858 2.6116 -

Structural 3.49142 3.4918 -

Fluid 12.0338 - 12.1492

Structural 13.0033 12.182 -

Fluid 13.3415 13.363 12.149

Structural 17.3428 - -



103

The theoretical and experimental natural frequencies of Z-Bended copper

pipe (L1 = L2 = L3=7/3m=2.333m) with the water are presented in Table 4.33 .

Table 4.33. Natural Frequencies (Hz) of Z-Bended copper pipe with the water (Fixed-Open/ Fixed-Closed) (L1 = L2 = L3=7/3m=2.333m)

Present Study Modes


Equation

(3.1)

Structural 1.88015 - 1.8802 -

Structural 1.97513 - 1.9753 -

Structural 2.64087 - 2.6410 -

Structural 9.20088 - 9.2135 -

Structural 10.0919 - 10.107 -

Structural 13.1205 - 13.151 -

Structural 13.2652 - 13.302 -

Structural 17.3886 - 17.484 -

Structural 17.5729 - 17.640 -

Structural 32.2449 - 32.737 -

Fluid 44.8799 46.666 - 45.7141


104

0

5

10

15

20

25

30

35

1 18 35 52 69 86 103 120 137 154 171 188 205 222 239 256

Time (s)

Pressure (Bar/10)

Figure 4.31. Experimental pressure-time history of Z-Bended copper pipe with the

water (Fixed-Open/ Fixed-Closed) (L1 = L2 = L3=7/3m=2.333m) at different external excitation frequencies.

0

5

10

15

20

25

30

35

33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71

Time (s)

Pressure (Bar/10)

Figure 4.32. Experimental pressure-time history of Z-Bended copper pipe with the

water (Fixed-Open/ Fixed-Closed) (L1 = L2 = L3=7/3m=2.333m) when external excitation frequency is equal to the liquid frequency (46.666 Hz).


105

4.3.3. U-Bended Pipe with Fixed-Free Ends

In this case study, steel pipe is examined in a theoretical manner for both the

air and the water. Open-Closed fluid boundary conditions and fixed-free structural

end conditions are studied. The theoretical natural frequencies are listed in Table

4.34 for the air and in Table 4.35 for the water.

Table 4.34. Natural Frequencies (Hz) of U-Bended steel pipe with the air (Fixed-Open/ Free-Closed) (L1 = L2 = L3=1.25m)

Modes Present Study

(TMM) ANSYS

Equation (3.1)

Structural 6.61737 6.6171 -

Structural 6.69652 7.2448 -

Structural 11.1277 14.648 -

Structural 14.6476 15.057 -

Fluid 24.3281 - 22.6783

Structural 37.7906 37.7906 -

Table 4.35. Natural Frequencies (Hz) of U-Bended steel pipe with the water (Fixed-Open / Free-Closed) (L1 = L2 = L3=1.25m)

Modes Present Study

(TMM) ANSYS

Equation (3.1)

Structural 5.48901 5.4884 -

Structural 6.43686 6.0090 -

Structural 11.0398 12.149 -

Structural 12.1526 12.489 -

Structural 31.3657 31.345 -

Fluid 96.6866 - 94.9079

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4.3.4. U-Bended Pipe with Fixed-Fixed Ends

Different lengths of pipes were connected as U Bended in this case. Open-

Closed fluid boundary conditions and fixed-fixed end conditions were applied to the

steel and copper pipes. Natural frequencies were found by transfer matrix method,

Ansys and experimentally and results listed in Tables.

Table 4.36. Natural Frequencies (Hz) of U-Bended steel pipe with the air (Fixed-Open / Fixed-Closed) (L1 = L2 = L3=1.25m)

Present Study Modes


Equation

(3.1)

Structural 18.5293 18.57 18.528 -

Fluid 15.0451 - - 22.6783

Structural 39.4704 - 29.548 -

Structural 40.6648 - 39.473 -

Fluid 71.8848 - - 68.0348

Structural 113.399 - 115.56 -

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Figure 4.33. FFT Spectrums in tangential and axial directions of U-Bended steel

pipe with the air (Fixed-Open/ Fixed-Closed) (L1 = L2 = L3=1.25m)


108

Table 4.37. Natural Frequencies (Hz) of U-Bended steel pipe with the water (Fixed-Open / Fixed-Closed) (L1 = L2 = L3=1.25m)

Present Study Modes


Equation

(3.1)

Structural 15.3709 15.45 15.368 -

Structural 18.2968 - 24.508 -

Structural 32.7491 - 32.740 -

Structural 95.726 - 95.852 -

Structural 104.44 - 104.50 -

Fluid 119.597 - - 93.7219

Structural 152.141 - 152.62 -

Copper pipes were studied 3 m and 7m length again in this case.

Table 4.38. Natural Frequencies (Hz) of U-Bended cooper pipe with the air (Fixed-Open / Fixed-Closed) (L1 = L2 = L3=1m)

Present Study Modes


Equation

(3.1)

Structural 8.3232 10.95 10.340 -

Structural 10.3401 - 16.547 -

Structural 21.8065 - 21.809 -

Fluid 28.6991 - - 28.3478

Structural 65.0579 - 65.146 -

Structural 70.6204 - 70.735 -

Fluid 87.9053 - - 85.0433


109

Figure 4.34. FFT Spectrums radial directions at two locations of U-Bended steel

pipe with the water (Fixed-Open/ Fixed-Closed) (L1 = L2 = L3=1.25m)


110

Figure 4.35. FFT Spectrums in radial and axial directions of U-Bended cooper pipe

with the air (Fixed-Open / Fixed-Closed) (L1 = L2 = L3=1m)


111

Table 4.39. Natural Frequencies (Hz) of U-Bended cooper pipe with the water (Fixed-Open / Fixed-Closed) (L1 = L2 = L3=1m)

Present Study Modes


Equation

(3.1)

Structural 7.82132 8.05 7.8207 -

Structural 8.7379 - 12.516 -

Structural 16.4962 - 16.495 -

Structural 49.2127 - 49.274 -

Structural 53.4471 - 53.500 -

Structural 74.1259 - 78.320 -

Structural 78.0921 - 80.327 -

Structural 86.2918 - 86.648 -

Structural 89.6348 - 89.779 -

Fluid 109.275 - - 106.665

Table 4.40. Natural Frequencies (Hz) of U-Bended cooper pipe with the air (Fixed-Open / Fixed-Closed) (L1 = L2 = L3=2.333m)

Modes Present Study


Structural 1.57107 1.9009 -

Structural 1.90083 3.0471 -

Structural 4.01121 4.0117 -

Fluid 12.0174 - 12.1492

Structural 13.0371 12.033 -

Structural 16.5752 13.057 -

Structural 16.581 19.150 -

Structural 19.0895 19.649 -


112

Figure 4.36. FFT Spectrums in axial direction at two locations of U-Bended cooper

pipe with the water (Fixed-Open / Fixed-Closed) (L1 = L2 = L3=1m)


113

Table 4.41. Natural Frequencies (Hz) of U-Bended cooper pipe with the air (Fixed-Open / Fixed-Closed) (L1 = L2 = L3=2.333m)

Present Study Modes


Equation

(3.1)

Structural 1.43772 - 1.4377 -

Structural 1.6113 - 2.3047 -

Structural 3.03398 - 3.0343 -

Structural 9.08962 - 9.1015 -

Structural 9.86178 - 9.8760 -

Structural 13.9154 - 14.484 -

Structural 14.4396 - 14.862 -

Structural 16.0525 - 16.118 -

Structural 16.5912 - 16.642 -

Structural 32.2892 - 32.882 -

Fluid 46.0184 46,6233 - 45.7141

In Figures 4.37 and 4.38, the gauge pressure and excitation frequencies are

plotted vs. time. It is also observed from these figures that the pressure react the

excitation frequency near the liquid natural frequency.


114

0

5

10

15

20

25

30

35

1 40 79 118 157 196 235 274 313 352 391 430 469 508 547 586

Time (s)

Pressure (Bar/10)

Figure 4.37. Experimental pressure-time history of U-Bended copper pipe with the

water (Fixed-Open/ Fixed-Closed) (L1 = L2 = L3=2.333m) at different external excitation frequencies.

0

5

10

15

20

25

30

35

267

271

275

279

283

287

291

295

299

303

307

311

315

319

Time (s)

Pressure (Bar/10)

Figure 4.38. Experimental pressure-time history of U-Bended copper pipe with the water (Fixed-Open/ Fixed-Closed) (L1 = L2 = L3=2.333m) when external excitation frequency is equal to the liquid frequency (46,623Hz).


115

4.4. Three Pipes in Two Planes

The pipe configuration considered in this section is shown in Figure 4.39.

For simplicity the length of each pipe is assumed to be equal in the following case

studies.

Figure 4.39. 3D-Bended pipe supported at two ends

4.4.1. 3D-Bended Pipe with Fixed-Free Ends

In this case study for 3-D pipe configuration, the pipe material is determined

as the steel. The length of each pipe is assumed to be equal (L1 = L2 = L3=1.25m).

Both the air and the water are used as the fluids. Closed-closed fluid boundary

conditions and fixed-free structural end conditions are handled. The theoretical

results for the air are presented in Table 4.42 and for the water are presented in Table

4.43.

1.25m

1.25m

1.25m

ITD

Highlight

ITD

Highlight


116

Table 4.42. Natural Frequencies (Hz) of 3D-Bended steel pipe with the air (Fixed-Open / Free-Closed) (L1 = L2 = L3=1.25m)

Modes Present Study


Structural 5.07353 5.8365 -

Structural 6.31677 6.4474 -

Structural 14.5468 16.121 -

Structural 17.5303 17.610 -

Fluid 26.0289 - 22.6783

Structural 32.1722 32.370 -

Structural 45.5397 47.326 -

Fluid 74.0333 - 68.0348

Fluid 116.818 - 113.391

Structural 132.501 132.69 -

Table 4.43. Natural Frequencies (Hz) of 3D-Bended steel pipe with the water (Fixed-Open / Free-Closed) (L1 = L2 = L3=1.25m)

Modes Present Study

(TMM) ANSYS

Equation (3.1)

Structural 4.84057 4.8409






Fluid 103.785 - 94.9079



117

4.4.2. 3D-Bended Pipe with Fixed-Fixed Ends

The theoretical and experimental results in the fundamental structural mode

of 3D-Bended steel pipe with the air are presented in Table 4.44. For this example

FFT spectrums in tangential direction are shown in Figure 4.40.

Table 4.44. Natural Frequencies (Hz) of 3D-Bended steel pipe with the air (Fixed-Open / Fixed-Closed) (L1 = L2 = L3=1.25m)

Present Study Modes

TMM ExperimentalANSYS Equation (3.1)

Structural 17.9003 20.10 20.699 -

Fluid 24.1988 - - 22.6783

Structural 33.8281 - 34.563 -

Structural 34.5657 - 34.839 -

Fluid 70.9162 - - 68.0348

Fluid 115.91 - - 113.391

Structural 122.079 - 122.09 -

Structural 122.249 - 122.30 -

Structural 174.064 - 174.72 -

ITD

Highlight


118

Figure 4.40. FFT Spectrums in tangential direction of 3D-Bended steel pipe with

the air (Fixed-Open / Fixed-Closed) (L1 = L2 = L3=1.25m)

The theoretical and experimental results of 3D-Bended steel pipe with the

water are presented in Table 4.45. In this example the fundamental frequency in the

structural mode is determined experimentally. For this example FFT spectrums in

tangential direction are also shown in Figure 4.41.


119

Table 4.45. Natural Frequencies (Hz) of 3D-Bended steel pipe with the water (Fixed-Open / Fixed-Closed) (L1 = L2 = L3=1.25m)

Present Study Modes


Equation

(3.1)

Structural 17.1596 17.55 17.168 -

Structural 28.6734 - 28.668 -

Structural 28.866 - 28.896 -

Fluid 96.4687 - - 93.7219

Structural 101.325 - 101.26 -

Structural 101.623 - 101.44 -

Structural 144.441 - 144.92 -

Figure 4.41. FFT Spectrums in tangential direction of 3D-Bended steel pipe with

the water (Fixed-Open / Fixed-Closed) (L1 = L2 = L3=1.25m)

The total length of the 3D-Bended copper pipe system is taken as either 3m or

7m.


120

The theoretical and experimental results of 3D-Bended copper pipe (L1 = L2 =

L3=1m) with the air are presented in Table 4.46. For the fundamental frequency in

the structural mode (10.71Hz), FFT spectrums in the axial direction are illustrated in

Figure 4.42.

Table 4.46. Natural Frequencies (Hz) of 3D-Bended copper pipe with the air

(Fixed-Open / Fixed-Closed) (L1 = L2 = L3=1m) Present Study

Modes TMM Experimental

ANSYS Equation

(3.1)

Structural 9.86233 10.71 11.541 -

Structural 18.0117 - 19.207 -

Structural 19.2065 - 19.456 -

Fluid 30.5497 - - 28.3478

Structural 68.3461 - 68.644 -

Structural 68.6145 - 68.719 -

Fluid 88.2313 - - 85.0433

Structural 98.4998 - 98.815 -

Structural 99.065 - 99.026 -


121

Figure 4.42. FFT Spectrums in tangential direction of 3D-Bended copper pipe with

the air (Fixed-Open / Fixed-Closed) (L1 = L2 = L3=1m)

The theoretical and experimental results of 3D-Bended copper pipe (L1 = L2 =

L3=1m) with the water are presented in Table 4.47. For the fundamental frequency in

the structural mode (7.65Hz), FFT spectrums in the radial direction are illustrated in

Figure 4.43.


122

Table 4.47. Natural Frequencies (Hz) of 3D-Bended copper pipe with the water (Fixed-Open / Fixed-Closed) (L1 = L2 = L3=1m)

Present Study Modes


Equation

(3.1)

Structural 8.72818 7.65 8.7290 -

Structural 14.5278 - 14.527 -

Structural 14.7123 - 14.716 -

Structural 51.8515 - 51.919 -

Structural 51.9149 - 51.976 -

Structural 74.4193 - 74.739 -

Structural 74.6579 - 74.739 -

Fluid 116.289 - - 106.665

Figure 4.43. FFT Spectrums in radial direction of 3D-Bended copper pipe with the

water (Fixed-Open / Fixed-Closed) (L1 = L2 = L3=1m)

The theoretical natural frequencies of 3D-Bended copper pipe (L1 = L2 =

L3=7/3m=2.333m) with the air are presented in Table 4.48.


123

Table 4.48. Natural Frequencies (Hz) of 3D-Bended copper pipe with the air (Fixed-Open / Fixed-Closed) (L1 = L2 = L3=2.333m)

Modes Present Study


Structural 1.82348 2.1218 -

Structural 3.33919 3.5341 -

Structural 3.53379 3.5809 -

Fluid 12.5046 - 12.1492

Structural 12.6687 12.675 -

Structural 13.0027 12.687 -

Structural 18.2772 18.295 -

The theoretical and experimental natural frequencies of 3D-Bended copper

pipe (L1 = L2 = L3=7/3m=2.333m) with the water are presented in Table 4.49. In this

example, the fundamental frequency in the fluid mode is measured experimentally.

Figures 4.44 and 4.45 illustrate the experimental pressure-time history of this pipe

system at different external excitation frequencies and when the external excitation

frequency is equal to the fundamental liquid frequency (47.806Hz).


124

Table 4.49. Natural Frequencies (Hz) of 3D-Bended copper pipe with the water (Fixed-Open / Fixed-Closed) (L1 = L2 = L3=2.333m)

Present Study Modes


Equation

(3.1)

Structural 1.60477 - 1.6050 -

Structural 2.67282 - 2.6733 -

Structural 2.7081 - 2.7086 -

Structural 9.57339 - 9.5875 -

Structural 9.58258 - 9.5966 -

Structural 13.7737 - 13.839 -

Structural 13.8412 - 13.866 -

Structural 16.7317 - 16.917 -

Structural 17.0937 - 17.141 -

Structural 32.7067 - 33.223 -

Fluid 43.6206 47.806 - 45.7141

-10

-5

0

5

10

15

20

25

30

35

40

1 44 87 130 173 216 259 302 345 388 431 474 517 560 603 646 689 732 775 818 861 904 947

Time (s)

Pressure (Bar/100)

Figure 4.44. Experimental pressure-time history of 3D-Bended copper pipe with the

water (Fixed-Open/ Fixed-Closed) (L1 = L2 = L3=2.333m) at different external excitation frequencies.


125

-10

-5

0

5

10

15

20

25

30

35

40

403

408

413

418

423

428

433

438

443

448

453

458

463

468

473

478

483

488

493

498

503

508

Time (s)

Pressure (Bar/10)

Figure 4.45. Experimental pressure-time history of 3D-Bended copper pipe with the

water (Fixed-Open/ Fixed-Closed) (L1 = L2 = L3=2.333m) when external excitation frequency is equal to the liquid frequency (47.8 Hz).

4.5. Pipes on Elastic Foundation

In this section the effect of the elastic foundation (Figure 4.46) on the natural

frequencies of the pipe system with free ends is worked out with the help of the

transfer matrix method. The pipe is rested on the elastic foundation along z-direction

as shown in Figure 4.46.

Figure 4.46. Free ended pipe on elastic foundation

l

Δ

kf

X

Y

Z

ITD

Highlight


126

The natural frequencies in structural modes of such single-spanned isotropic

beams with hollow section are given in an analytical manner by the followings

formulas (Pestel and Leckie, 1963).

p

y

ny

nn A

EI

EIl

l ρλπλ

ω21

4

4

2

2

12 ⎟

⎟⎠

⎞⎜⎜⎝

⎛ Φ+= (4.1.)

Δ=Φ /fk (4.2.)

πλ ⎟⎠⎞

⎜⎝⎛ +=

21nn (4.3.)

where ρ represents mass density of pipe, kf is foundation stiffness, Ap Cross-sectional

area, Iy Moment of inertia and Φ is the foundation modulus.

In this part of this study single-spanned , L-Bended and 3D-Bended pipes

rested on an elastic foundation with free ends are examined.

4.5.1. Free Ended Single -Spanned Pipe on an Elastic Foundation

The 6m-length steel and copper single-spanned pipes with free ends are

handled here. Both the air and water are used as fluids. The frequencies in structural

modes obtained by the transfer matrix approach are compared with the exact

frequencies evaluated by Equation (4.1). The results are listed in Tables 4.50-4.53.

It is observed from those tables that the transfer matrix method gives quite

reasonable results.


127

Table 4.50. Natural Frequencies (Hz) of 6m length free ended steel pipe with the air

on elastic foundation (kf = 100000 N/m3, Δ = 1m) (Free-Open/ Free-Closed)

Modes Present Study

(TMM) Theoretical Equation (3.1)

Structural 11.5036 11.5026 -

Fluid 14.1798 - 14.1739

Structural 25.8147 25.9200 -

Fluid 42.5237 - 42.5217

Structural 49.0373 49.3283 -

Fluid 70.8706 - 70.8695

Structural 80.2735 80.9887 -

Fluid 99.2181 - 99.2173

Structural 119.009 120.7183 -

Fluid 127.566 - 127.5652

Structural 155.913 155.9130 -

Table 4.51. Natural Frequencies (Hz) of 6m length free ended steel pipe with the

water on elastic foundation (kf = 100000 N/m3, Δ = 1m) (Free-Open/ Free-Closed)

Modes Present Study

(TMM) Theoretical

Equation (3.1)

Structural 9.54148 9.5379 -

Structural 21.4124 21.4929 -

Structural 40.6767 40.9030 -

Fluid 58.9688 - 58.5762

Structural 66.5919 67.1558 -

Structural 98.9105 100.0996 -

Structural 137.462 139.687 -


128

Table 4.52. Natural Frequencies (Hz) of 6m length free ended copper pipe with the

air on elastic foundation (kf = 100000 N/m3, Δ = 1m) (Free-Open/ Free-Closed)

Modes Present Study

(TMM) Theoretical

Equation (3.1)

Fluid 14.1998 - 14.1738

Structural 18.9776 19.1708 -

Structural 20.698 20.8824 -

Structural 25.526 25.6883 -

Structural 34.2529 34.4112 -

Fluid 42.5302 - 42.5216


Structural 62.6181 62.9014 -

Fluid 70.8733 - 70.869

Structural 81.6408 82.0799 -

Fluid 99.2195 - 99.2172

Structural 103.643 104.3201 -

Fluid 128.523 - 127.5649

Fluid 155.881 - 155.9127

Structural 156.586 157.6894 -


129

Table 4.53. Natural Frequencies (Hz) of 6m length free ended copper pipe with the

water on elastic foundation (kf = 100000 N/m3, Δ = 1m) (Free-Open/ Free-Closed)

Modes Present Study

(TMM) Theoretical

Equation (3.1)

Structural 14.3538 14.4999 -

Structural 15.6552 15.7944 -

Structural 19.3072 19.4294 -

Structural 25.9085 26.0270 -

Structural 35.3512 35.4953 -

Structural 47.3665 47.5757 -

Fluid 54.7762 - 53.3323

Structural 61.7582 62.0814 -

Structural 78.4048 78.9028 -

Structural 97.2265 97.9775 -

Structural 118.06 119.2688 -

Structural 143.247 142.7546 -

Fluid 159.888 - 159.996

Above, the results show that, transfer matrix method gives quite reasonable

results when compared with theoretical ones for free ended single-spanned pipe on

an elastic foundation.

4.5.2. L-Bended Free Ended Pipe on an Elastic Foundation

This example is studied here at the first time (Figure 4.47). The total length of

the pipe system made of either steel or copper is 3m. (L1 = L2 = 1.5m). The

theoretical results of TMM are listed in Tables 4.54- 4.57.


130

Figure 4.47. L-Bended pipe on elastic foundation

Table 4.54. Natural Frequencies (Hz) of 3m length L-Bended free ended steel pipe with the air on elastic foundation (kf = 100000 N/m3, Δ = 0.5m)

(Free-Open/ Free-Closed) (L1 = L2 = 1.5m)

Modes Present Study

(TMM) Equation (3.1)

Fluid 14.1748 14.1739

Structural 23.5469 -

Structural 33.543 -


Fluid 42.3477 42.5217

Structural 43.823 -

Af 70.7304 70.8695



Af 98.9495 99.2173



131

Table 4.55. Natural Frequencies (Hz) of 3m length L-Bended free ended steel pipe with the water on elastic foundation (kf = 100000 N/m3, Δ = 0.5m)


Modes Present Study

(TMM)

Equation (3.1)






Fluid 58.6578 58.5762


Structural 67.4916

Table 4.56. Natural Frequencies (Hz) of 3m length L-Bended free ended copper pipe with the air on elastic foundation (kf = 100000 N/m3, Δ = 0.5m)


Modes Present Study

(TMM)

Equation (3.1)

Fluid 14.1918 14.1738



Structural 23.012 -


Structural 34.5786



Fluid 42.4084 42.5216



Fluid 70.3662 70.8694


132

Table 4.57. Natural Frequencies (Hz) of 3m length L-Bended free ended copper pipe with the water on elastic foundation (kf = 100000 N/m3, Δ = 0.5m)


Modes Present Study

(TMM)

Equation (3.1)








Fluid 53.9701 53.3323


4.5.3. 3D-Bended Free Ended Pipe on an Elastic Foundation

This example is also studied here at the first time (Figure 4.48). The total

length of the pipe system made of either steel or copper is 3.75m. (L1 = L2 =L3=

1.25m). The theoretical results of TMM are listed in Tables 4.58- 4.61.


133

Figure 4.48. 3D-bended pipe on elastic foundation

Table 4.58. Natural Frequencies (Hz) of 3m length 3D-Bended free ended steel pipe with the air on elastic foundation (kf = 100000 N/m3, Δ = 1.25m)


Modes Present Study


Fluid 20.5961 22.6783




Fluid 71.0473 68.0348

Fluid 116.107 113.391


134

Table 4.59. Natural Frequencies (Hz) of 3m length 3D-Bended free ended steel pipe with the water on elastic foundation kf = 100000 N/m3, Δ = 1.25m)


Modes Present Study

(TMM)

Equation (3.1)



Fluid 96.4906 93.7219


Table 4.60. Natural Frequencies (Hz) of 3m length 3D-Bended free ended copper pipe with the air on elastic foundation kf = 100000 N/m3, Δ = 1.25m)


Modes Present Study




Fluid 22.385 22.6782



Fluid 65.9793 68.0347


135

Table 4.61. Natural Frequencies (Hz) of 3m length 3D-Bended free ended copper pipe with the water on elastic foundation kf = 100000 N/m3, Δ = 1.25m)


Modes Present Study

(TMM)

Equation (3.1)






Fluid 86.7394 85.3317

4.6. Parametric Studies

In this part of the present work, a parametric study is carried out to

understand correctly the vibrational behavior of the piping systems filled by either

the water or the air. 2” nominal diameter-steel and 1” nominal diameter-copper

pipes are studied. Either fixed-fixed or fixed-free structural boundary conditions are

considered with closed-closed fluid boundary conditions. The first four structural and

fluid natural frequencies are used to draw the diagrams which show the variation of

the natural frequencies with either the slenderness ratio, L/d, or the bend angleα .

4.6.1. Effect of the Slenderness Ratio on the Natural Frequencies

Here single-spanned pipe system is considered. Variation of the natural

frequencies of a single-spanned steel/copper pipe filled by the air/water with the

slenderness ratio for different boundary conditions are illustrated in Figures 4.49-

4.56. The theoretical results are also presented in Tables 4.62-4.65.

As guessed, increasing the slenderness ratio, L/d, decreases the natural

frequencies in both the structural and fluid modes. The fundamental frequency in the

fluid mode increases with the density of the fluid.

ITD

Highlight


136

Table 4.62. Variation of the natural frequencies in Hz of a single-spanned steel pipe with the slenderness ratio (Fixed-Fixed and Open-Closed)

Natural frequencies

Structural modes Fluid modes

Filled

by

L/d 1ω 2ω 3ω 4ω 1ω 2ω 3ω 4ω

1 121.300 169.938 235.516 271.901 1410 4231 7052 9872 2 57.609 84.969 111.979 169.938 705 2116 3526 4936 4 35.625 56.646 71.125 90.633 470 1410 2351 3291 6 24.447 42.485 50.596 67.975 353 1058 1763 2468 8 17.789 33.988 38.202 54.380 282 846 1410 1974 10 13.474 28.323 29.938 45.317 235 705 1175 1645 15 10.520 24.084 24.277 38.843 201 604 1007 1410

Air

20 8.417 19.766 21.242 33.443 176 529 881 1234 1 100625 169938 197159 271901 5902 17707 29511 41316

2 47807 84969 93576 135951 2951 8853 14756 20658

4 29574 56646 59303 90634 1967 5902 9837 13772

6 20298 42134 42484 67489 1476 4427 7378 10329

8 14771 31792 33988 51549 1180 3541 5902 8263

10 11188 24905 28323 40954 984 2951 4919 6886

15 8735 20029 24277 33429 843 2530 4216 5902

Water

20 6987 16434 21242 27834 738 2213 3689 5164


137

Table 4.63. Variation of the natural frequencies in Hz of a single-spanned copper pipe with the slenderness ratio (Fixed-Fixed and Open-Closed)

Natural frequencies


Filled

by

L/d

1ω 2ω 3ω 4ω 1ω 2ω 3ω 4ω

1 167.254 232.730 325.856 382.415 3037 9112 15186 21261

2 79.766 116.365 155.108 191.207 1519 4556 7593 10630

4 49.592 77.577 98.695 127.472 1012 3037 5062 7087

6 34.215 58.182 70.404 95.603 759 2278 3797 5315

8 25.018 46.546 53.330 76.483 607 1822 3037 4252

10 19.026 38.788 41.931 63.736 506 1519 2531 3543

15 14.904 33.247 33.837 54.631 434 1302 2169 3037

Air

20 11.956 27.849 29.091 46.891 380 1139 1898 2658

1 126.526 232.730 248.908 361.059 30425 91274 152123 212972

2 60.367 116.365 185.361 216.683 15212 45637 76062 106486

4 37.547 75.148 77.576 118.974 10142 30425 50708 70991

6 25.912 53.526 58.183 85.571 7606 22818 38031 53243

8 18.949 40.510 46.546 65.513 6085 18255 30425 42594

10 14.410 31.835 38.788 52.171 5071 15212 25354 35495

15 11.288 25.681 33.247 42.685 4346 13039 21732 30425

Water

20 9.054 21.130 29.091 35.622 3803 11409 19015 26621


138

Table 4.64. Variation of the natural frequencies in Hz of a single-spanned steel pipe with the slenderness ratio (Fixed-Free and Closed-Closed)

Natural frequencies


Filled

by

L/d

1ω 2ω 3ω 4ω 1ω 2ω 3ω 4ω

1 52.094 84.969 134.903 135.951 2821 8462 14103 19745

2 19.308 42.484 63.757 127.337 1410 4231 7052 9872

4 9.776 28.323 38.824 81.758 940 2821 4701 6582

6 5.817 21.242 26.195 33.987 705 2116 3526 4936

8 3.832 16.993 18.775 27.190 564 1692 2821 3949

10 2.705 14.044 14.161 22.658 470 1410 2351 3291

15 2.008 10.857 12.138 19.421 403 1209 2015 2821

Air

20 1.548 8.618 10.621 16.993 353 1058 1763 2468

1 43.375 84.969 117.141 135.951 11805 35414 59023 82632

2 16.082 42.485 53.950 67.975 5902 17707 29511 41316

4 8.133 28.323 32.588 45.317 3935 11804 19674 27544

6 4.835 21.242 21.912 33.987 2951 8853 14756 20658

8 3.183 15.674 16.994 27.190 2361 7083 11804 16526

10 2.246 11.710 14.162 22.658 1967 5902 9837 13772

15 1.667 9.043 12.138 19.421 1686 5059 8432 11804

Water

20 1.284 7.173 10.621 16.994 1476 4427 7378 10329


139

Table 4.65. Variation of the natural frequencies in Hz of a single-spanned copper pipe with the slenderness ratio (Fixed-Free and Closed-Closed)

Natural frequencies


Filled

by

L/d 1ω 2ω 3ω 4ω 1ω 2ω 3ω 4ω

1 72.668 116.365 188.084 191.207 6075 18224 30373 42522

2 27.314 58.183 88.809 95.604 3037 9112 15186 21261

4 13.932 38.788 54.319 63.736 2025 6075 10124 14174

6 8.322 29.091 36.838 47.802 1519 4556 7593 10630

8 5.493 23.273 26.523 38.242 1215 3645 6075 8504

10 3.883 19.394 19.913 31.868 1012 3037 5062 7087

15 2.884 15.438 16.624 27.315 868 2603 4339 6075

Air

20 2.225 12.281 14.545 23.901 759 2278 3797 5315

1 55.217 150.979 191.207 236.240 60849 182547 304246 425944

2 20.770 58.183 68.972 95.604 30425 91274 152123 212972

4 10.579 38.788 41.738 63.736 20283 60849 101415 141981

6 6.312 28.177 29.091 47.802 15212 45637 76062 106486

8 4.163 20.235 23.273 38.242 12170 36509 60849 85189

10 2.941 15.166 19.394 31.868 10142 30425 50708 70991

15 2.184 11.743 16.624 27.315 8693 26078 43464 60849

Water

20 1.684 9.333 14.546 22.657 7606 22818 38031 53243


140

0,0E+00

5,0E+04

1,0E+05

1,5E+05

2,0E+05

2,5E+05

3,0E+05

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

L/d

ω s (Hz)

ωs1

ωs2

ωs3

ωs4

(a)

0

2000

4000

6000

8000

10000

12000

1 3 5 7 9 11 13 15 17 19 21 23 25

L/d

wf

wf1wf2wf3wf4

(b)

Figure 4.49. Variation of the natural frequencies of a single-spanned steel pipe filled

by the air with the slenderness ratio (Fixed-Fixed and Open-Closed) a) Structural Modes b) Fluid Modes


141

0,0E+00

5,0E+04

1,0E+05

1,5E+05

2,0E+05

2,5E+05

3,0E+05

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25L/d

ω s (Hz)

ωs1

ωs2

ωs3

ωs4

(a)

05000

1000015000200002500030000350004000045000

1 3 5 7 9 11 13 15 17 19 21 23 25

L/d

wf (Hz)

(b)

Figure 4.50. Variation of the natural frequencies of a single-spanned steel pipe filled by the water with the slenderness ratio (Fixed-Fixed and Open-Closed) a) Structural Modes b) Fluid Modes


142

0,0E+00

2,0E+04

4,0E+04

6,0E+04

8,0E+04

1,0E+05

1,2E+05

1,4E+05

1,6E+05

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25L/d

ω s(Hz)

ωs1ωs2ωs4ωs4

(a)

0

5000

10000

15000

20000

25000

1 3 5 7 9 11 13 15 17 19 21 23 25L/d

ωf (Hz)

ωf1ωf2ωf3ωf4

(b)

Figure 4.51. Variation of the natural frequencies of a single-spanned steel pipe filled by the air with the slenderness ratio (Fixed-Free and Closed-Closed) a) Structural Modes b) Fluid Modes


143

0,0E+00

2,0E+04

4,0E+04

6,0E+04

8,0E+04

1,0E+05

1,2E+05

1,4E+05

1,6E+05

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25L/d

ω s(Hz)

ωs1

ωs2

ωs3

ωs4

(a)

0

10000

20000

30000

40000

50000

60000

70000

80000

90000

1 3 5 7 9 11 13 15 17 19 21 23 25

L/d

ωf (Hz)

ωf1ωf2ωf3ωf4

(b)

Figure 4.52. Variation of the natural frequencies of a single-spanned steel pipe filled by the water with the slenderness ratio (Fixed-Free and Closed-Closed) a) Structural Modes b) Fluid Modes


144

0,0E+00

5,0E+04

1,0E+05

1,5E+05

2,0E+05

2,5E+05

3,0E+05

3,5E+05

4,0E+05

4,5E+05

1 3 5 7 9 11 13 15 17 19 21 23 25

L/d

ω s (Hz)

ωs1

ωs2

ωs3

ωs4

(a)

0

5000

10000

15000

20000

25000

1 3 5 7 9 11 13 15 17 19 21 23 25

L/d

ωf (Hz)

ωf1ωf2ωf3ωf4

(b)

Figure 4.53. Variation of the natural frequencies of a single-spanned copper pipe filled by the air with the slenderness ratio (Fixed-Fixed and Open-Closed) a) Structural Modes b) Fluid Modes


145

0,0E+00

5,0E+04

1,0E+05

1,5E+05

2,0E+05

2,5E+05

3,0E+05

3,5E+05

4,0E+05

1 3 5 7 9 11 13 15 17 19 21 23 25L/d

s (Hz) ωs1ωs2ωs3ωs4

(a)

0

50000

100000

150000

200000

250000

1 3 5 7 9 11 13 15 17 19 21 23 25

L/d

ω f (Hz)

ωf1ωf2ωf3ωf4

(b)

Figure 4.54. Variation of the natural frequencies of a single-spanned copper pipe filled by the water with the slenderness ratio (Fixed-Fixed and Open-Closed) a) Structural Modes b) Fluid Modes


146

0,0E+00

5,0E+04

1,0E+05

1,5E+05

2,0E+05

2,5E+05

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25L/d

ω s(Hz )

ωs1

ωs2

ωs3

ωs4

(a)

0

5000

10000

15000

20000

25000

30000

35000

40000

45000

1 3 5 7 9 11 13 15 17 19 21 23 25

L/d

ωf (Hz)

ωf1ωf2ωf3ωf4

(b)

Figure 4.55. Variation of the natural frequencies of a single-spanned copper pipe

filled by the air with the slenderness ratio (Fixed-Free and Closed-Closed) a) Structural Modes b) Fluid Modes


147

0,0E+00

5,0E+04

1,0E+05

1,5E+05

2,0E+05

2,5E+05

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25L/d

ω s(Hz)

ωs1

ωs2

ωs3

ωs4

(a)

0

50000

100000

150000

200000

250000

300000

350000

400000

450000

1 3 5 7 9 11 13 15 17 19 21 23 25

L/d

ωf (Hz)

ωf1ωf2ωf3ωf4

(b)

Figure 4.56. Variation of the natural frequencies of a single-spanned copper pipe filled by the water with the slenderness ratio (Fixed-Free and Closed-Closed) a) Structural Modes b) Fluid Modes


148

4.6.2. Effect of the Bend-Angle on the Natural Frequencies

Here the bend angle between the axes of pipes is measured in counter-

clockwise direction as shown in Figure 4.57. Each section of pipe is assumed to be

equal (L1=L2=1m). The problem is studied for both fixed-fixed and fixed-free

structural boundary conditions. The pipe system is assumed to be made of either the

steel or the copper material. Only structured modes considered in this work

Figure 4.57. Bended Angle α

Variation of the natural frequencies in Hz of such pipe system with the bend

angle for different structural boundary conditions are demonstrated in Figures 4.58-

4.61. Some numerical results are tabulated in Tables 4.66-4.69.

α L1

L2

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Table 4.66. Variation of the natural frequencies in Hz of steel pipe system with the bend angle (Fixed-Fixed)

Natural frequencies

Structural modes

Filled

by

)( °α 1ω 2ω 3ω 4ω

15 570 1.571 1.572 1.162 30 563 1.571 1.574 1.398 45 555 1.571 1.575 1.596 90 546 1.571 1.577 1.750 105 536 1.570 1.579 1.865 120 526 1.570 1.581 1.950 135 515 1.570 1.584 2.011 150 504 1.569 1.587 2.057 165 493 1.569 1.591 2.092

Air

175 482 1.568 1.595 2.119 15 473 965 1.304 1.305 30 467 1.163 1.303 1.306 45 461 1.327 1.303 1.307 90 453 1.456 1.303 1.308 105 445 1.551 1.303 1.310 120 436 1.621 1.303 1.312 135 427 1.672 1.302 1.315 150 418 1.709 1.302 1.317 165 409 1.738 1.302 1.321

Water

175 400 1.759 1.301 1.324


150

Table 4.67. Variation of the natural frequencies in Hz of copper pipe system with the bend angle (Fixed-Fixed)

Natural frequencies

Structural modes

Filled

by

)( °α 1ω 2ω 3ω 4ω

15 178 566 498 499 30 176 628 498 499 45 174 661 498 500 90 171 680 498 500 105 167 691 498 501 120 164 698 498 502 135 160 703 498 503 150 157 706 498 504 165 153 709 498 505

Air

175 150 711 498 506 15 135 429 377 377 30 133 476 377 378 45 131 500 377 378 90 129 514 377 378 105 127 523 377 379 120 124 528 377 379 135 121 532 377 380 150 119 534 377 381 165 116 536 377 382

Water

175 113 538 377 383


151

Table 4.68. Variation of the natural frequencies in Hz of steel pipe system with the bend angle (Fixed-Free)

Natural frequencies

Structural modes

Filled

by

)( °α 1ω 2ω 3ω 4ω

15 92 93 549 554 30 93 93 533 541 45 93 94 514 526 90 94 95 494 509 105 95 96 475 491 120 96 97 455 473 135 97 99 436 456 150 98 100 418 438 165 99 102 402 422

Air

175 101 104 386 406 15 77 77 455 464 30 77 78 442 457 45 77 78 427 448 90 78 79 411 437 105 79 81 395 425 120 79 82 378 413 135 80 83 363 400 150 81 85 348 387 165 82 87 334 375

Water

175 84 89 321 362


152

Table 4.69. Variation of the natural frequencies in Hz of copper pipe system with the bend angle (Fixed-Free)

Natural frequencies

Structural modes

Filled

by

)( °α 1ω 2ω 3ω 4ω

15 29 29 171 173 30 29 29 166 169 45 29 29 160 165 90 29 29 154 159 105 29 30 148 154 120 30 30 141 148 135 30 31 135 143 150 30 31 130 137 165 31 32 124 132

Air

175 31 32 120 127 15 22 22 130 133 30 22 22 126 131 45 22 22 121 129 90 22 23 117 126 105 22 23 112 123 120 22 23 107 120 135 23 24 102 117 150 23 24 98 114 165 23 25 94 110

Water

175 24 26 90 107


153

0,E+00

5,E+02

1,E+03

2,E+03

2,E+03

3,E+03

5 15 25 35 45 55 65 75 85 95 105

115

125

135

145

155

165

175

Bend Angle (α)

ω s(Hz)

ωs1

ωs2

ωs3

ωs4

a) Filled by the Air

0,E+00

2,E+02

4,E+02

6,E+02

8,E+02

1,E+03

1,E+03

1,E+03

2,E+03

2,E+03

2,E+03

5 15 25 35 45 55 65 75 85 95 105

115

125

135

145

155

165

175

Bend Angle (α)

ωs ωs1

ωs2

ωs3

ωs4

b) Filled by the Water

Figure 4.58. Variation of the natural frequencies (Hz) in structural modes of steel

pipe system with the bend angle (Fixed-Fixed)


154

0,E+00

1,E+02

2,E+02

3,E+02

4,E+02

5,E+02

6,E+02

5 15 25 35 45 55 65 75 85 95 105

115

125

135

145

155

165

175

Bend Angle(α)

ωs (Hz)

ωs1ωs2ωs3ωs4


0

50

100

150

200

250

300

350

400

450

500

5 15 25 35 45 55 65 75 85 95 105

115

125

135

145

155

165

175

Bend Angle (α)

ω s(Hz)

ωs1ωs2ωs3ωs4


Figure 4.59. Variation of the natural frequencies (Hz) in structural modes of steel

pipe system with the bend angle (Fixed-Free)


155

0

100

200

300

400

500

600

700

800

0 10 20 30 40 50 60 70 80 90 100

110

120

130

140

150

160

170

180

Bend Angle (α)

ω s(Hz) ωs1

ωs2ωs3ωs4


0

100

200

300

400

500

600

5 15 25 35 45 55 65 75 85 95 105

115

125

135

145

155

165

175

Bend Angle (α)

ωs (Hz) ωs1

ωs2

ωs3

ωs4


. Figure 4.60. Variation of the natural frequencies (Hz) in structural modes of copper

pipe system with the bend angle (Fixed-Fixed)


156

0

20

40

60

80

100

120

140

160

180

200

5 15 25 35 45 55 65 75 85 95 105

115

125

135

145

155

165

175

Bend Agle (α)

ωs(Hz)

ωs1ωs2ωs3ωs4


0

20

40

60

80

100

120

140

160

0 10 20 30 40 50 60 70 80 90 100

110

120

130

140

150

160

170

180

Bend Angle (α)

ωs(Hz)

ωs1ωs2ωs3ωs4


Figure 4.61. Variation of the natural frequencies (Hz) in structural modes of copper

pipe system with the bend angle (Fixed-Free)

5. CONCLUSION Ahmet ÖZBAY

157

5. CONCLUSION

As is well known liquid-filled piping systems are very important for

many industrial applications. They are used for conveying gases and fluids over a

wide range of temperatures and pressures.

In this study, the free un-damped vibrational behavior of air/water-filled

piping systems is first studied with the help of the transfer matrix method (TMM).

The transfer matrix method provides a quick and efficient analysis of such systems.

The existence of bends, springs, orifices, valves, accumulators, pumps, and such

control instrumentations may be modeled easily in the transfer matrix method

without increasing the dimensions of the system matrices.

The closed-form governing equations available in the literature, which

consider the axial, transverse and torsional vibration of such piping systems, are

completely used in this work.

The fluid is assumed to be one-dimensional (the radial component of the fluid

velocity is zero and the flow is developed in only the axial direction), linear, and

homogeneous, with isotropic flow and uniform pressure and fluid velocity over the

cross-section. The fluid density is taken as constant (the convective terms are ignored

by assuming low Mach numbers, where the fluid wave speed is much greater than

the fluid velocity). The fluid friction term is neglected.

The pipe wall is assumed to be linearly elastic, isotropic, prismatic, circular

and thin-walled.

In the case studies considered in this thesis five different configurations of

piping systems made of either one inch-nominal diameter copper or two inch-

nominal diameter steel such as

• Single-spanned

• L-bended

• Z-bended

• U-bended

• 3-D bended

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5. CONCLUSION Ahmet ÖZBAY

158

are studied with two main structural boundary conditions namely fixed-fixed and

fixed-free. Intermediate rigid support is also studied in this work. Fluid boundary

conditions are assumed to be closed at both ends. As a light fluid both the air and the

water are considered.

The theoretical frequencies based on the transfer matrix method are supported

by some experiments performed in this study.

The effect of the elastic foundation on the natural frequencies is also

investigated.

A parametric study is, finally, carried out to understand correctly the

vibrational behavior of such piping systems. Variation of the natural frequencies of

the pipe system with the slenderness ratio and the bend angle are illustrated by

graphs.

The results obtained in this work are verified by both finite element solution

using ANSYS and some results available in the literature. A very good harmony is

observed among the literature, experimental and theoretical results.

159

REFERENCES

AKDOĞAN, İ., 1992, Dynamic Analysis Of Piping Systems Using Transfer Matrix

Method, MSc. Thesis, METU.

ALLISON, A. E., 1998, Analytical Investigation of a Semi-Empirical Flow Induced

Vibration Model, PhD Thesis, The University of Western Ontario.

AUXWORTHY, D. H., 1998, Water Distribution Network Modeling From

Waterhammer, PhD Thesis, University of Toronto.

BAASIRI, M. S., 1981, A Quantitative Study of Free Air Release During Transient

Flow, PhD Thesis, Utah State University.

BRENNEN C. E., and ACOSTA, A. J., 1976, The Dynamic Transfer Function for a

Cavitating Inducer, journal of Fluids Engineering, Vol. 98,182, pp. 182-191.

BRENNEN C. E., ACOSTA, A. J. and GAUGHEY, T. K., 1986, Impeller Fluid

Forces, Nasa Conference Publication, 2436, Vol. 1, pp. 270-295.

BUDNY, D. D., 1988, Influence of Structural Damping on The Internal Fluid

Pressure During a Fluid Transient Pipe Flow, PhD Thesis, Michigan State

University.

COLLINA, A., ZANABONI, P., and BELLOLI, M., 2005, On the Optimization of

Support Positioning and Stiffness for Piping Systems on Power Plants, 18th

International Conference on Structural Mechanics in Reactor Technology,

ÇINAR, A., 1998, Boru Hatlarının Dinamiği ve Sönüm Modelleri, PhD Thesis, İTÜ.

DOLASA, A. R., 1998, Computer-Aided Design Software for the Undamped Two-

Dimensional Static and Dynamic Analysis of Beams and Rotors, MSc.

Thesis, Virginia Polytechnic Institute.

DURRANI, J. N., 2001, Dynamics Of Pipelines With A Finite Element Method,

MSc. Thesis, University of Calgary.

EVANS, R. P., 2004, Mass Flow Measurement Trough Flow-Induced Pipe

Vibration, PhD Thesis, Idaho State University.

GIDI, A., 1999, Two- Phase Flow-Induced Vibrations in a Heated Tube Bundle, PhD

Thesis, McMaster University.

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HAKTANIR, V., 1990, Silindirik Helisel Çubukların Statik, Dinamik ve Burkulma

Davranışlarının Taşıma ve Rijitlik Matrisleri Metodu ile İncelenmesi”, PhD

Thesis, Ç.Ü.

HORNER, G. C., 1975, The Riccatti Transfer Matrix Method, PhD Thesis, The

Virginia University.

KARTHA, S. C., 2000, Active, Passive and Active Passive Control Techniques for

Reduction of Vibrational Power Flow in Fluid Filled Pipes, MSc. Thesis,

Virginia Polytechnic Institute.

KOO, G. H., and PARK, Y. S., 1998, Vibration Reduction By Using Periodic

Supports in a Piping System, Journal of Sound and Vibration, Vol. 210,

pp.53-68.

LESMEZ, M., W., 1989, Modal Analysis Of Vibrations In Liquid –Filled Piping

Systems, PhD Thesis, Michigan State University.

LI, M., 1997, An Experimental And Theoretical Study Of Fluidelastic Instability in

Cross Flow Multi-Span Heat Exchanger Tube Arrays, PhD Thesis, McMaster

University.

LIEW, A., FENG, N. S., and HAHN, E. J., 2004, On Using the Transfer Matrix

Formulation For Transient Analysis of Nonlinear Rotor Bearing Systems,

International Journal of Rotating Machinery, Vol. 10, pp. 425-431.

NIEVES, V. R., 2004, Static and Dynamic Analysis of a Piping System, MSc.

Thesis, University of Puerto Rico.

OTWELL, R. S., 1984, Efect Of Elbow Restraint on Pressure Transients, PhD

Thesis, Michigan State University.

PESTEL, E. C., and LECKIE, F. A., 1963, Matrix Methods in Elostomechanics,

McGraw-Hill, New York.

RUNGTA, R., 1998, Separation Criteria of Fluid Piping Systems Using a Lumped-

Parameter Method, MSc. Thesis, The University of Western Ontario.

SERVAITES, J. C., 1996, Computer Aided Analyses of Smoke Stack Designs, MSc.

Thesis, Virginia Polytechnic Institute.

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TENG-YANG, R., 1998, Measurement of Flow Induced Vibration And Flow

velocities in a Scaled Model of The Flixborough “Dog-Leg” Pipe, MSc.

Thesis, The University of New Brunswick.

YU, J. F., 2001, Vibration And Sensitivity Analysis of Structures With Optimal

Design Using Transfer Dynamic Stiffness Matrix, PhD Thesis, The

University of Texas Arlington.

YAKUT, K., 1996, Dirsek ve Vanalı Boru Hatlarında Akış Kaynaklı Titreşimlerin

Spektral ve Gürültü Analizi, PhD Thesis, E.Ü.

ZHANG, Q., 2001, Lateral Vibration of Non-Rotating Drill Pipe Conveying Fluid in

Vertical Holes, PhD Thesis, The University of Tulsa.

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CURRICILUM VITAE

Ahmet ÖZBAY was born in Adana, 1976. After graduating from high school

in 1994, he enrolled in the University of Çukurova, Adana, where he received a

Bachelors of Science degree in Mechanical Engineering in 2000. In Fall of 2000 he

enrolled the University of Çukurova, Adana, where he completed his Masters of

Science degree in September, 2002. While obtaining this degree he was employed as

research assistant in the same department. He started his Doctor of Philosophy

education in the same institute in 2002. He has been working as a maintenance

engineer in soda ash plant, Mersin Soda Sanayii since 2005.

163

APPENDIX

A.1. ANSYS Source Codes for Fixed-Fixed Single Span Copper Pipe with Air. Source: /PREP7 ET,1,16 R,1,0.028,0.001,,,,1.2 MP,EX,1,97e9 MP,NUXY,1,0.35 MP,DENS,1,8350 N,1,0,0,0 N,11,0,0,2 FILL,1,11 E,1,2 EGEN,10,1,1,1,1 D,1,ALL D,11,ALL Results: ***** INDEX OF DATA SETS ON RESULTS FILE ***** SET TIME/FREQ LOAD STEP SUBSTEP CUMULATIVE 1 29.004 1 1 1 2 80.972 1 2 2 3 188.71 1 3 3

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A.2. ANSYS Source Codes for L Bended Fixed-Fixed Steel Pipe with Water Source: /PREP7 ET,1,16 R,1,0.0635,0.0036,,,,997 MP,EX,1,157e9 MP,NUXY,1,0.28 MP,DENS,1,7600 N,1,0,0,0 N,11,0,0,2.4 FILL,1,11 N,21,2.4,0,2.4 FILL,11,21 E,1,2 EGEN,20,1,1,1,1 D,1,ALL D,21,ALL

Results:

SET TIME/FREQ LOAD STEP SUBSTEP CUMULATIVE 1 8.5859 1 1 1 2 32.124 1 2 2 3 33.675 1 3 3 4 46.451 1 4 4 5 48.734 1 5 5 6 104.78 1 6 6

Three Dimensional Vibration Analysis Thesis 2009 Tmm&Ansys

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