ZEOLISIVM UNIVERSITY LIBRARIES Emmi" lllllmllllwill mum Michigan State 3 1293 00574 0042 University ll This is to certify that the dissertation entitled MODAL ANALYSIS OF VIBRATIONS IN LIQUID:FILLED PIPING SYSTEMS presented by Marlio William Lesmez has been accepted towards fulfillment of the requirements for Doctor of Philosophy degreein Civil Engineering DflWW/M/ “professor Date fiéZ// /7’(? MS U is an Affirmative Action/Equal Opportunity Insn'tun'on 0- 12771 PLACE IN RETURN BOX to remove this checkout from your record. TO AVOID FINES return on or boron duo duo. DATE DUE DATE DUE DATE DUE l l=|l l.- MSU Is An Affirmative ActionlEqual Opportunity Institution MODAL ANALYSIS OF VIBRATIONS IN LIQUID-FILLED PIPING SYSTEMS By Marlio William Lesmez A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Civil and Environmental Engineering 1989 ABSTRACT NODAL ANALYSIS OF VIBRATIONS IN LIQUID-FILLED PIPING SYSTEMS BY Marlio William Lesmez The vibration of liquid-filled piping systems is formulated using one- dimensional wave theory in both the liquid reaches and the pipe wall. Five families of waves and fourteen variables are considered and the effects of shear defamation and rotary inertia on the lateral vibration of the pipe reaches are included. A numerical model is described which includes both Poisson and junction coupling, thereby providing com- prehensive interaction between the fluid reaches and the piping. The transfer matrix method is used to study the motion of these systems. The motion is represented by an overall transfer matrix. This matrix is assembled by combining field transfer matrices representing the motion of single pipe reaches with point matrices describing specified boundary conditions . A one-inch (25 mm) diameter variable length piping system with a U-type 'bend is used.to obtain.the experimental data. Various fluid and struc- tural frequencies are excited by using a crank mechanism which vibrates the piping. Fluid pressure and pipe displacement responses for various forcing frequencies are obtained and compared with analytical results. Larger fluid pressure responses occur at higher harmonics than at the first fundamental frequency. Mode shapes for the liquid pressure and pipe motion are also presented. Good agreement of natural frequencies is found between predictions and observations. A mi familia y mi esposa Becky iv AW I would like to thank Professor David Wiggert, chairman of my graduate committee and the other members Frank Hatfield, Clark Radcliffe, Reinier Boumeester, and David Yen. My sincere appreciation goes to Professor Hatfield for his suggestions, assistance and discussion related to my research . I am indebted to Dr. Robert Otwell and Dr. Sidney Stuckenbruck for their work in this area which constitutes the basis of my research. My gratitude is also extended to Dr. Dan Budny for his assistance in the construction and design of the experimental apparatus and data acquisi- tion system. I found valuable the suggestions of my fellow graduate student David Fanson in the design of the crank mechanism. A word of gratitude is extended to Mr. Conan Doyle for his assistance with the graphics. The financial support provided by the National Science Foundation is also greatly appreciated. Finally, I am especially indebted to my family for their support and understanding during this time. I express my deepest gratitude to my wife Becky for her encouragement, inspiration, and support during this research. Her suggestions and patience during the preparation of the manuscript were invaluable. Her help and dedication made possible the completion of this project. TABLE OF CONTENTS LIST OF TABLES xii LIST OF FIGURES xv lfllflflKfllflflHfl! xix CHAPTER 1 OBJECTIVE AND SCOPE 1 1.1 Introduction 1 1.2 Objective 5 1.3 Scope 5 2 LITERATURE REVIEW 8 2.1 Introduction 8 2.2 Unsteady Flow in Pipelines 9 2.3 Fluid-Structure Interaction 10 2.3.1 Poisson Coupling 11 2.3.2 Junction Coupling 13 2.4 Oscillatory Motion 18 2.4.1 Impedance Method 19 2.4.2 Transfer Matrix Method 20 vi ANALITICAL.DEVELOPMENT 3.1 Introduction 3.2 Governing Differential Equations 3.2.1 Axial Waves - Liquid and Pipe Wall 3.2.2 Transverse Waves - Shear and Bending 3.2.2.1 Shear and Bending in x-z Plane 3.2.2.2 Shear and Bending in y-z Plane 3.2.3 Torsion About z-axis NUMERICAL SIMULATION 4.1 Introduction 4.2 Transfer Matrix Method 4.2.1 Description of Transfer Matrix Approach 4.2.2 Field Transfer Matrices 4.2.2.1 Liquid and Axial Pipe Wall Vibration 4.2.2.2 Transverse Vibration in x-z Plane 4.2.2.3 Transverse Vibration in y-z Plane 4.2.2.4 Torsional Vibration About z-Axis 4.2.2.5 General Field Transfer Matrix 4.2.3 Point Matrices 4.2.3.1 Bend Point Matrix 4.2.3.2 Spring Point Matrix 4.2.3.3 Mass Point Matrix vii 23 23 24 24 38 39 45 48 56 56 57 58 63 65 67 69 71 72 73 73 75 77 4.2.4 Overall Transfer Matrix 4.2.4.1 Overall Transfer Matrix Rearrangement 4.2.4.2 Coordinate Transformation 4.2.5 Boundary and Intermediate Conditions 4.2.5.1 Boundary Conditions 4.2.5.2 Intermediate Conditions 4.2.6 Natural Frequencies 4.2.7 Mode Shapes 4.2.8 Frequency Response 4.3 Comparison with Other Methods 4.3.1 Method of Characteristics 4.3.2 Component Synthesis Method and Experimental Data EXPERIMENTAL.APPARATUS 5.1 Introduction 5.2 Description of Experimental Apparatus 5.2.1 Liquid Components 5.2.1.1 Liquid 5.2.1.2 Constant Pressure Reservoirs 5.2.1.3 Valve 5.2.2 Pipe Components 5.2.2.1 Pipe Material viii 79 80 83 86 87 88 92 93 94 95 96 98 111 111 112 112 112 113 114 114 114 5.2.2.2 Pipe Supports 5.2.2.3 External Shaker 5.2.2.4 Spring 5.2.3 Experimental Configurations 5.2.4 Transducers 5.2.5 Dynamic Forces and Natural Frequencies of Shaker 5.2.5.1 Shaker Loads 5.2.5.2 Spring Loads 5.2.5.3 Natural Frequencies of Shaker Components 5.3 Experimental Procedure and Analysis 5.3.1 Frequency Range of Excitation 5.3.1.1 Liquid Frequencies 5.3.1.2 U-Bend Frequencies 5.3.2 Sampling Frequency and Sampling Time 5.3.3 Sampling Procedure 5.3.4 Analysis Procedure 5.4 Experimental Uncertainty EXPERIMENTAL.RESULIS AND COMPARISONS 6.1 Introduction 6.2 Transient Tests 6.2.1 Snap-Back Test ix 115 116 117 120 121 123 123 125 127 127 128 128 129 131 131 132 133 142 142 143 143 6.2.2 Valve Closure Test 6.2.2.1 Fixed U-Bend 6.2.2.2 Free U-Bend 6.2.2.3 U-Bend.with Spring 6.3 Harmonic Tests 6.3.1 U-Bend Response 6.3.2 Spectral Response of Liquid-Filled Piping 6.3.3 Liquid Mode Shapes 6.3.3.1 Liquid Mode Shapes at Liquid Natural Frequencies 6.3.3.2 Liquid Mode Shapes at U-Bend Natural Frequencies 7 SUMMARI'AND CONCLUSIONS APPENDICES A. LIQUIDeAXIAL.PIPB'NALL.TRANSFER.MATRIX A.1 Introduction A.2 Uncoupled Analysis A.3 Coupled Analysis B DATA.ACQUISITION B.1 Introduction 144 145 146 148 148 150 151 152 153 154 190 193 193 194 194 206 206 3.2 System Components B.2.l Piezoelectric Pressure Transducers B.2.2 Quartz Accelerometers B.2.3 Computer Hardware and Accessories 3.2.3.1 Analog-to-Digital Converter B.2.3.2 Programmable Real-Time Clock 3.2.3.3 Patch Panel B.2.4 Data Acquisition Software LIST OF REFERENCES xi 206 207 208 210 211 212 212 213 214 UIUlUIUIU'IUI mUkU-DNH ‘LIST OP‘TABLES Properties of Straight Liquid-Filled Pipe Natural Frequencies for Straight Pipe Properties of L-Shaped Liquid-Filled Pipe Physical Properties of Liquid Physical Properties of Piping System Shaker Components Spring Properties Piping System Configurations Location of Transducers and U-Bend Relative to Valve Fluid Harmonics for Pipe Configurations Natural Frequencies of U-Bend Experimental and Computed U-Bend Response to Snap-Back Test Experimental and Computed Frequencies of Liquid for Fixed Condition Experimental and Computed Frequencies of Liquid for Free Condition xii 97 98 99 113 115 118 118 120 122 129 130 144 145 146 .10 .11 .12 .13 .14 .15 .16 Experimental and Computed Frequencies of Natural Frequencies of U-Bend to Harmonic Excitation Liquid Pressure Mode Shapes at Liquid Natural Frequencies Liquid Pressure Mode Shapes at U-Bend Natural Frequencies Experimental and Configuration 1 Experimental and Configuration 2 Experimental and Configuration 3 Experimental and Configuration 4 Experimental and Configuration 5 Experimental and Configuration 6 Experimental and Configuration 7 Experimental and Configuration 8 Experimental and Configuration 9 lLiquid for Spring Condition Computed Results Computed Results Computed Results Computed Results Computed Results Computed Results Computed Results Computed Results Computed Results xiii for for for for for for for for for 149 151 154 156 157 158 159 160 161 162 163 164 165 A.l A.2 B.1 3.2 Pipe Material Properties Transfer Matrix Parameters Properties of Pressure Transducers Properties of Accelerometer Transducers xiv 194 196 205 208 wwwwu UIJ-‘wNH #bb99bJ-‘9 o: -u as in a- t» h) ta us'rorrIs-Is Liquid-Filled Piping System Sign Convention for Internal Forces Axial Pipe Element Radial Pipe Element Deformations for Transverse Vibration in x-z Plane Internal Forces for Transverse Vibration in x-z Plane Deformations for Transverse Vibration in y-x Plane Internal Forces for Transverse Vibration in y-z Plane Pipe Reach Subjected to Torsion Generalized Spring-Mass System Free Body Diagrams of Spring and Mass Systems Simple Spring-Mass System General Piping System General Straight Liquid-Filled Pipe Reach Forces and Displacements at Bend Forces at Spring Forces at Concentrated Mass 51 52 52 53 53 54 54 55 101 101 101 102 102 103 104 104 bkfiv§ 43> UIUIUIUIUIUIUIUIUIUI 4.9 .10 .11 .12 .13 .14 .15 *DQNO‘U‘IpUNH H o .4 Definition of Local and Global Axes Coordinate Transformation of Straight Pipe Reach Boundary Conditions Intermediate Boundary Conditions Pressure Amplitude Response for Straight Pipe Reach L-Shaped Liquid-Filled Pipe Mobility Diagrams for L-Shaped Pipe Experimental Liquid-Filled Pipe Set-Up Liquid Boundaries Crank Mechanism Crank Diagram Transducers and U-Bend Locations One Degree-of-Freedom Representation of U-Bend Experimental and Analysis Procedures Input and Output Displacements Input and Rated Torque at Shaft of Motor Low-Pass Filter Experimental Results of Snap-Back Test, U-Bend Empty, Frequency of Free Bend is 4.4 Hz Experimental Results of Snap-Back Test, U-Bend Filled, Frequency of Free Band is 3.9 Hz Experimental Time Pressure Response to Sudden Valve Closure at Closed End, U-Bend Fixed FFT for Sudden Valve Closure, U-Bend Fixed xvi 105 105 106 106 107 108 109 136 137 138 139 139 139 ' 140 141 141 141 166 167 168 169 .10 .11 .12 .13 .14 .15 .16 .17 .18 .19 Experimental Time Responses to Sudden Valve Closure at Closed End, U-Bend Free FFT for Sudden Valve Closure, U-Bend Free Experimental Time Responses to Sudden Valve Closure at Closed End, U-Bend with Spring FFT For Sudden Valve Closure, U-Bend with Spring Experimental U-Bend Displacements for Harmonic Excitation Computed U-Bend Mode Shapes, Computed U-Bend Mode Shapes, Spring Liquid Mode Shapes Liquid Mode Shapes Configuration 1 Liquid Mode Shapes Configuration 2 Liquid Mode Shapes Configuration 3 Liquid Mode Shapes at Liquid at U-Bend at U-Bend at U-Bend at U-Bend Configurations 4 and 5 Liquid Mode Shapes Configuration 6 Liquid Mode Shapes at U-Bend at U-Bend Configurations 7 and 8 Liquid Mode Shapes at U-Bend Configuration 9 xvii Free Bend Bend Attached to Natural Natural Natural Natural Natural Natural Natural Natural Frequencies Frequencies, Frequencies, Frequencies, Frequencies, Frequencies, Frequencies, Frequencies, 170 171 172 173 174 175 176 177 179 180 181 182 183 184 185 Liquid and Axial Pipe Wall Wave Speeds Versus Pipe Cross Sectional Ratio Parameters for Liquid and Aixal Pipe Wall Transfer Matrix Elements of Field Transfer Matrix in Axial Pipe Wall Submatrix Elements of Field Transfer Matrix in Upper Coupling Submatrix Elements of Field Transfer Matrix in Lower Coupling submatrix Elements of Field Transfer Matrix in Liquid Submatrix Displacement Calibration Curve for PCB Accelerometers xviii .198 199 200 201 202 203 213 Symbols Description Units A Cross-sectional area (pipe, fluid) m2 A Coefficients of integration a Wave speed m/s B Matrix coefficients b Ratio of pipe radius to pipe wall thickness C Field transfer matrix coefficients c Coupled wave speed ratio D Mean spring diameter mm d Wire spring diameter mm d Ratio of pipe density to fluid density E Young's modulus of elasticity . Pa e Pipe wall thickness m F Force amplitude N f Force N f Natural frequency Hz G Shear modulus of rigidity Pa g Bend point matrix coefficient xix 3 NEW 7: 'O ”.0013 Ratio of YOung's modulus to modified bulk modulus Moment of inertia Polar moment of inertia (-1)“ Fluid isothermal bulk modulus of elasticity Spring stiffness Length of crank mechanism Length of pipe reach Moment amplitude Moment Mass Number of active coils in spring Fluid pressure amplitude Fluid pressure External force amplitude Ratio of fluid area to pipe area Crank radius Radius of pipe cross-section Coordinate along pipe axis Torque Time Pipe displacement amplitude Pipe displacement Fluid displacement amplitude Pa N/m N-m N-m ks Pa Pa N-m wmbb'tb M DS‘S Fluid displacement m Radial displacement rad Axial displacement m Angle between incident pipe reaches rad Angle of rotation due to shear Field transfer matrix coefficient Matrix determinant Differential element Frequency parameter of a Bernoulli- Euler beam Flexural stiffness correction factor for a pipe bend Angular direction Shape factor for shear Eigenvalues Poisson's ratio Mass density kg/m3 Stress Pa Summation of forces Angle between local and global axes rad Rotation amplitude due to bending Rotation due to bending deformation rad Forcing frequency rad/s Natural circular frequency, rad/s Field transfer matrix coefficients xxi subscripts t‘ H- C) H: C) In ”A'U'O fp tz XZ yz X,Y,Z x,y,z U-bend Columns Fluid Global coordinate system Pipe and Local coordinate system Pipe Local axis index Global axis index Rows Spring Time Direction along pipe axis Fluid and axial pipe wall field transfer matrix Torsion vibration about z-axis field transfer matrix Transverse vibration in x-z plane field transfer matrix Transverse vibration in Y-Z plane field transfer matrix Global rectangular coordinate directions Local rectangular coordinate directions Angular direction xxii Superscripts mwzrw v-i Point matrix for a bend Left of discontinuity Point matrix for a lumped mass Right of discontinuity Point matrix for a spring Matrix transposition Matrix inverse Modified modulus Matrices and vectors FINN? d Vector, integration coefficients Solution matrix Point transfer matrix Field transfer matrix Transformation matrix Global transfer matrix State vector xxiii Chapter 1 Objective and Scope 1.1 1113251393192 Vibrations that occur in liquid-filled piping systems are of interest in a variety of industrial, water supply, hydraulic, nuclear power, aircraft and automotive applications. The dynamic behavior of these systems includes both transient and steady-state vibrations caused by rapid valve closures and unbalanced rotating machinery. This dynamic behavior has usually been modeled by uncoupled analyses. The analyses of liquid columns contained in pipes are based in part on the assump- tions that the piping system is sufficiently rigid to remain immobile and that it does not interact dynamically with either fluid oscillations or external loads. Possible sources of the external loads are machine vibrations and seismic motion. Analytical developments for free vibra- tions and resonance of liquid columns are presented by Jaeger [5] , Wylie and Streeter [6] and Chaudhry [7]. On the other hand, well-known modal analysis techniques, such as those described by Clough and Penzien [52], can be used to analyze vibrations of piping structures if the motion of the contained liquid is neglected. Chapter 1 Objective and Scope Recently, coupling analyses have been called to the attention of re- searchers. Experimental results in laboratories [19,24,25,64] have clearly shown that under certain conditions, vibrations of liquid columns and piping structural elements interact and respond differently than if the two components are treated independently. The interaction can be attributed in part to two coupling mechanisms, Poisson and junc- tion coupling. Poisson coupling takes place along a pipe length. Axial strain of the pipe is induced by circumferential strain caused by a change in the fluid pressure. Junction coupling occurs at locations where the flow area and/or flow direction changes. At junctions, varia- tions in fluid pressure create force resultants. It is necessary for these pressure forces to be balanced by axial forces in the pipe wall. The axial forces generate subsequent pipe motion that may excite flexural and torsional modes of vibration of the pipe. Poisson and junction coupling generate forces and displacements in the fluid and in the pipe wall that are transmitted and reflected back and forth along the length of the pipe. Wilkinson [64] identified five families of waves: tension, torsion, and two families of transverse bending waves in the pipe wall and pressure waves. in the liquid. A three-legged liquid-filled pipe in a three-dimensional space shown in Figure 1.1 is used to describe the transmission process of these waves. The pipes connect two reservoirs. A valve, location D, is located at the downstream reservoir where water is flowing at velocity V0. Junctions B and C are unrestrained allowing displacements and rotations. The pipe is rigidly supported at the ends. An instantaneous valve Chapter 1 Objective and Scope closure creates the system excitation. The following events take place as the valve begins to close. A pressure wave is generated and propagates in the fluid while an axial wave is transmitted along the pipe wall in leg 3, that is, Poisson coupling occurs. For most commer- cial pipe material, the axial wave (precursor wave) travels faster than the fluid pressure wave [21]. The precursor wave produces a displace- ment at junction C in the positive X direction. This motion, which is generated by Poisson coupling, induces a change in the fluid velocity and creates an unbalanced axial force at this location. The change in fluid velocity produces an increase in the fluid pressure that is trans- mitted towards the valve and location B. The unbalanced axial force is transmitted as a shear force and bending moment along leg 2. The motion at junction C is maximum when the fluid pressure wave from the valve reaches this point. Junction coupling is present at this point due to the unbalanced axial force produced by the change in pressure. Moreover, the motion at junction C and moment in leg 2 produce a rota- tion at junction B about the Y-axis. This generates a torsional moment that is transmitted along leg 1. The boundary conditions at D reflect the five waves described above. 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 stiffnesses, structural restraints and hydraulic devices such as pumps, orifices and valves must be con- sidered. The speed of the wave components depends on the pipe material and fluid properties. The frequencies at which the liquid and pipe are Chapter 1 Objective and Scope vibrating are influenced by the structural support configurations of the pipe and the hydraulic elements of the system. The frequencies of the system may also be affected by the interaction between the fluid and the structure due to Poisson and junction cou- pling. Some incidents of hydraulic resonance at various pumped storage sites and power plants are presented by Jaeger [5] , Wylie and Streeter [6] and Chaudrhy [7]. Jaeger [41] also reported several of these inci- dents and points out the importance and danger of vibrations from higher fluid harmonics. These harmonics may be excited by the motion of struc- tural components of the system. The analysis and design of piping systems can be performed ineither the time or the frequency domain. The method of characteristics has been used to model the propagation of acoustic waves in liquids [5,6,7], beams as described by the Timoshenko's theory [89] and in fluid- structure interaction systems [l7,l9,20,21,22]. Unfortunately, numerical limitations have thwarted the evolution of a generalized solution methodology. In the frequency domain the transfer matrix method has been used to model distributed parameter systems [50] . Wilkinson [64] and Wiggert, Lesmez and Hatfield [65] use this method to model liquid-filled piping systems. Wilkinson's model uses the Bernoulli-Euler'beam theory to describe the piping flexure, but does not include Poisson's coupling. His model then, does not account for the axial liquid pipe wall coupling. Chapter 1 Objective and Scope The transfer matrix method is appropriate to model piping systems be- cause it directly relates the force and displacement variables in the pipe wall and in the fluid at one end of the system to the corresponding variables at the other end. As a designing model, the method may be used to compute the natural frequencies, mode shapes and frequency responses of the system including the structural supports and hydraulic devices. It can also be used to compute the response of the system for free or forced vibration analysis. The variables are related by using a global transfer matrix. Elastic liquid-filled pipe reaches can be analyzed with an appropriate transfer matrix. Point matrices describe joints, such as tees, bends, point masses, and hydraulic elements. 1.2 Quartile The objective of the present research is to incorporate the flexural and torsional modes of vibration in an existing coupled liquid-axial pipe wall model. The proposed model accounts for Poisson and junction cou- pling and allows the inclusion of structural and hydraulic devices. The model represents an improvement over the previous model by Wilkinson [64] . In addition, an experimental apparatus was designed and built to provide experimental data collection to verify the analytical model. 1.3 5.222: This report is organized into two sections. Chapters 3 and 4 are in- cluded in the first section which is concerned with the development and Chapter 1 Objective and Scope verification of the numerical analysis technique. Chapter 3 presents the equations of motion that describe the coupled liquid and axial pipe wall model and the equations that describe the transverse and torsional modes of vibrations. Chapter 4 describes the numerical technique that accounts for the five families of waves propagating in the pipe and the liquid. The modeling of bends, masses, springs and rigid supports is also presented. Forced vibration is also incorporated into the model. The proposed model is compared with two numerical techniques and with experimental data available in the literature. The second part of the research is an experimental study of a piping system of variable pipe length and with a U-type bend that is excited by an external shaker. The variable pipe length allows changes in fluid frequency. The U-bend is free to move in one plane. The shaker excites the piping over a range of frequencies that includes the first natural frequencies of both the fluid and the U-bend. Chapter 5 describes the experimental apparatus and procedures. The experimental results and comparison with the analytical model are presented in Chapter 6. Chapter 1 Objective and Scope 7 :2 Mam. A Y x 1 z B 2 3 D c | g 4. V 0 Figure 1.1 Liquid-Filled Piping System Chapter 2 Literature Review m2 2.1 Mien The objective of this study is to incorporate the flexural and torsional modes of vibration in an existing coupled axial pipe wall and liquid model. A review of the evolution of the axial model and these modes of vibration is necessary to incorporate the appropriate coupling mechanisms. This section is devoted to a review of the previous works in these areas. This review will be divided into three sections. The first section reviews the studies of unsteady flow in closed pipes. The second one relates the works on the interaction between fluid and struc- ture in liquid-filled piping systems. The last section relates the studies of oscillatory motion in piping systems and the applications of the transfer matrix method to solve these problems. Chapter 2 . Literature Review 2.2 new The study of unsteady flow in pipelines, or waterhammer as this phenomenon is more commonly known, has been of interest since the middle of the 1800's [1]. Among the early significant contributions to the solution of waterhammer problems are those of Joukowsky [2] , Lamb [3] , and Allievi [4] . Their findings predicted with accurately predicted the liquid wave speed and its associated pressure rise. With the exception of Lamb, who included the effect of longitudinal stresses in the pipe wall by considering the pipe an elastic membrane, the others predicted the existence of only one wave propagation. Joukowsky conducted exten- sive experiments and found that the speed at which disturbances propagates in the water is related to the relative circumferential stiffness of the pipe. His study concluded that the speed of propaga- tion for the liquid in pipelines is less than the propagation speed in an infinite liquid. In his research, Joukowsky assumed that pressure is uniform across any given pipe section. He also neglected the mass of the pipe wall, the radial inertia of the liquid and the axial and bend- ing stresses in the pipe wall. Based on these assumptions he derived a modified wave speed for the fluid in which the liquid bulk modulus is adjusted by the structural properties of the pipe wall. The analysis of the waterhammer problem has produced much research after these early works. Some analytical solutions of unsteady flow problems with various boundary conditions are outlined in textbooks by Jaeger [5] , Wylie and Streeter [6] and Chaudhry [7]. Waterhammer problems are Chapter 2 Literature Review 10 still being researched today. The basic equations are being inves- tigated [8] and new numerical techniques are being developed [9,10] . 2.3 Winn Research in the area of fluid-structure interaction has identified four main forms of dynamic liquid-pipe forces: 1. Lateral momentum forces. Blevins [11] reported some of the research into this mechanism which has been extensively investigated. The lateral momentum forces induced by high, steady flow rates through curved pipes can reduce flexural stiffness and may produce buckling of initially straight pipes. 2. Transverse pressure variation. This phenomenon occurs in cases where the inside diameter of the pipe is a multiple of the length of the transverse acoustic wave in the liquid. This may result in the excita- tion of higher symmetric lobar modes of the pipe cross section. Leissa [12] presented estimates of natural frequencies of lobar modes for infinitely long cylindrical shells. 3. Dilation pressures. This mechanism is related to the Poisson ratio [13] in which an axial elongation of a straight pipe causes a decrease of its inside diameter or ,conversely, an axial contraction of the pipe causes a dilation of the inside diameter. This axial elongation or contraction of the pipe wall may be caused by a rapid change in the Chapter 2 ' Literature Review 11 fluid velocity creating a decrease or increase in the fluid pressure. This pressure change gives rise to an axial stress wave in the pipe. For most piping systems the propagation of the stress wave in the pipe wall is faster than that of the acoustic wave in the fluid. The result of this wave interaction is called the "precursor wave" and the mechanism by which it occurs is termed Poisson coupling. 4. Axial resultants at fittings. Variations in fluid pressure create pressure resultants that act at locations where flow changes area or direction, such as at bends, tees and orifices. These differential pressure forces have to be balanced by axial forces in the pipe wall to maintain equilibrium conditions. The axial forces generate subsequent pipe motion that may excite the flexural and torsional modes of vibra- tion of the pipe. This phenomenon is known as junction coupling. Poisson coupling and junction coupling are the phenomena to be studied in this research. The literature review concerning these follows. 2.3.1 Poisson Coupling In liquid-filled pipes, Poisson coupling results from the transformation of the circumferential strain, caused by internal pressure, to axial strain and is proportional to Poisson's ratio. Skalak [14] was among the first to extend .Joukowski's method to include Poisson coupling. His results identified the precursor wave for a sudden valve closure. The analytical model that he developed treated the pipe wall as an elastic Chapter 2 Literature Review 12 membrane to include the axial stresses and axial inertia of the pipe. Thorley [l] conducted experimental validation of Skalak's theory. Williams [85] conducted a similar study. He found that structural damping caused by longitudinal and flexural motion of the pipe was greater than the viscous damping in the liquid. In fact, Williams states that “mechanical damping can be more important for water hammer decay than viscous friction”. These researchers did not include the radial inertia of the liquid or the pipe wall. Lin and Morgan [15,16] included the pipe inertia term and the transverse shear in their equa- tions of motion. Their study was restricted to waves which have axial symmetry and purely sinusoidal variation along the axis. Walker and Phillips [17] extended the study by Lin and Morgan to include both the radial inertia of the pipe wall in the fluid and the axial equations of motion. Their interest in short duration, transient events produced a one-dimensional, axisymmetric system of six equations. Wilkinson and Curtis [18] developed a non-linear, twenty-one equation model for the axial and radial pipe wall deformations in both elastic and plastic zones. Vardy and Fan [75] conducted experiments on a straight pipe, generating a pressure wave by dropping the pipe onto a massive base. Their results showed good agreement with the analytical model by Wilkinson and Curtis [18] . They concluded that the fluid friction does not influence the pressure response and that the axial waves in the straight pipe are non-dispersive for a first order accuracy. Otwell [l9], Wiggert, Otwell and Hatfield [20] and Stuckenbruck, Wiggert and Otwell [21] neglected the radial acceleration in their studies using the six-equation model of Walker and Phillips [17] . This simplification reduced the mathematical model to four equations. They presented Chapter 2 Literature Review 13 numerical examples for various combinations of liquids and piping materials and for various coupling constraints. Budny [22] also reduced the six-equation model, but he included viscous damping and a fluid shear stress term to account for the structural and liquid energy dis- sipation. Experimental tests verified that the model satisfactorily predicts the wave speeds, fluid pressure, and structural velocity of a straight pipeline for several fluid periods after a transient has ex- cited the fluid. The aforementioned researchers have helped in the understanding of the Poisson coupling mechanism in fluid-structure interaction problems. They identified two important waves that propagate in a straight pipe reach, one in the liquid and one in the pipe wall. However, none of these studies, with the exception of Otwell [l9] and Wiggert et a1. [20], considered the possibility that a fitting, such as an elbow, may move in response to the precursor wave, thereby, altering the transient response of the liquid. The following section discusses the models that have included the junction coupling mechanism. 2.3.2 Junction Coupling Piping systems can be thought of as straight pipes joined at localized points by elbows, reducers, tees, orifices and the like. Pressure resultants at these points act as localized forces on the pipe, gener- ating the junction coupling mechanism. For pipes with only a few bends, a continuous representation of the piping was devised by Blade, Lewis Chapter 2 Literature Review 14 and Goodykoontz [23]. Experimental tests were conducted to analyze the response of an L-shaped pipe to harmonic loading. The experimental setup included.a.restricting orifice plate at the downstream and of the pipe. Their experimental results validated their model. They concluded that an uncoupled analysis does not produce accurate estimates of natural frequencies, and that the elbow, which provides coupling between the pipe motion and liquid motion, causes no appreciable reflection, attenuation, or phase shift in the fluid waves. Davidson and Smith [24] conducted a similar investigation. Their analytical model was based on a vibration transmission matrix and it showed good agreement with ex- ‘perimental results. As an extension of that work, Davidson and Samsury [25] developed a more accurate solution to analyze a pipe assembly comprised of straight sections and uniform bends arranged in a nonplanar configuration. Experimental results indicated a significant level of coupling between the plane compressional wave in the liquid and that in the pipe wall. Comparison of numerical and experimental results, however, indicated a need for further refinement. Wiggert, Hatfield and Stuckenbruck [26], and Wiggert, Hatfield and Lesmez [27] used a one- dimensional wave formulation in both the liquid reaches and the piping structure resulting in five wave components and fourteen variables. The five families of waves are pressure waves in the liquid, axial tension waves in the pipe wall, two families of transverse shear and bending waves, and torsional waves. The method of characteristics was used to solve for the fourteen variables and to find the expressions for the wave speeds. The authors showed a comparison of the predicted fluid pressures and structural velocities with experimental data to provide a. partial validation of the model. However, their model showed that Chapter 2 Literature Review 15 numerical errors are introduced by time-line interpolations and by numerical integration of the coupled transverse shear force and bending moment. Experimental validation of the transverse vibration was not included. Joung and Shin [28] developed a model that takes into account the shear and flexural waves of an elastic axisymmetric tube. The method of characteristics was used in the solution for four families of propagating waves: the extensional, transverse, and symmetric bending waves of the elastic tube and the acoustic wave of the fluid medium. Their results compared closely to Walker and Phillips' results [17] for relatively small pipe deformations. The above models used a continuous representation of the piping system. Another approach for complicated geometric configurations is to ap- proximate the system as a set of discrete connected masses. Several techniques have been applied to a variety of models. A basic technique uses spring and point masses to represent the pipe structure. Wood [29] studied a pipe structure loaded with a harmonic excitation. He found that the natural frequencies of liquid were shifted, especially when the frequency of the harmonic load is near one of the natural frequencies of the supporting structure. Ellis [301 reduced a piping structure to equivalent springs and masses by selectively lumping mass and stiffness at fittings and releasing specific force components at bends, valves and tees. His formulation of axial response was a modification of the method of characteristics and included pipe stresses and velocities. Otwell [19] and Wiggert et al. , [20] modeled a pipe elbow as two or- thogonal springs. The stiffness of the springs corresponded to the Chapter 2 Literature Review 16 flexural stiffness of the upstream and downstream pipe reaches connect- ing to the elbow. Their investigation also included experimental data for a rapid valve closure. Their results showed that the motion of the elbow, driven by the axial stress in the pipe and by the liquid pres- sure, caused appreciable alteration of the pressure. The pressure response was 33% greater than the response for an immobile elbow. Otwell [31] used a spring-mass oscillator to represent each mode of the structural response. This. approach, however, is limited to simple, orthogonal configurations because it provides only one liquid-structure coupling point and only one degree of freedom for each mode. A second method uses the finite element method to model the structure, treating each pipe element as a beam. Schwirian and Karabin [32] generalized this approach by using a finite element representation of the liquid and the piping. Their studies imposed coupling at fittings only. The effect of the supports and piping stiffness was shown to be significant. Wiggert and Hatfield [33] used the method of characteris- tics to model the fluid. They coupled the results at pipe junctions with a finite element code to solve for the structure. Hatfield, Wiggert and Otwell [34] used the modal synthesis technique [35,36] to analyze fluid- structure systems with harmonic loading. The model responses of the supporting structure were obtained from an existing finite element program. These responses were then coupled to the liquid analyses. Hatfield, Wiggert and Davidson [37] presented a validation of this methodology based on comparison to earlier experiments. Chapter 2 Literature Review 17 The previous models used the beam theory to represent the pipe reaches. These models, however, cannot represent the precursor wave because classic beam theory neglects deformations of cross sections. The reduc- tion in flexural stiffness at bends is also inappropriately treated by these models. To avoid these difficulties, Quezon and Everstine [38] used shell elements to represent the pipe wall. While providing useful estimates of flexural stiffness of a single bend, this method is com- putationally feasible for only short lengths of pipe. The investigation of these two coupling mechanisms, Poisson and junc- tion, in liquid-filled piping systems is continuing. Wiggert [39] presented a survey of the latest work in this area. The study of piping systems in industrial plants and experimental testing of large scale models as well as the inclusion of non-linearities such as cavitation, structural damping and fluid friction is necessary to gain a better understanding of these systems. Rothe and Wiggert [40] outlined some of these considerations when modeling condensation-induced waterhammer in power plant systems. Jaeger [41] reported incidents of hydraulic resonance caused by structural vibration at various pumped storage sites and power plants. The understanding of these mechanisms has been useful also in the study of seismic motion of pipelines. Hatfield and Wiggert [42] applied a response spectrum analysis to an elastically-supported, liquid-filled pipe aligned in the direction of ground motion. The same authors [43] described a technique for determining pressure and dis- placement responses of liquid-filled piping in the time domain. Both mathematical models included waves in the liquid and pipe wall coupled by the Poisson effect. Ogawa [44] conducted experiments on the dynamic Chapter 2 Literature Review 18 response of a real-scale piping system using a large scale shaking table to investigate earthquake induced hydraulic transient effects. The system was excited by a harmonic motion in the axial direction of pipe and showed a sharp resonance for a closed, liquid-filled pipeline. The system was analyzed using a simple model of a rigidly supported pipe- water column system. The analytical and experimental results suggested that the coupling of a closed low pressurized liquid contained in a piping system is an important factor for seismic response estimations of liquid- filled pipelines . 2.4 W Resonance in power conduits has been the cause of many severe and spec- tacular accidents as mentioned by Jaeger [41] in his remarkable discussion of incidents in hydropower systems. His discussion points out the importance and danger of vibrations from higher fluid harmonics. Jaeger [5] , Wylie and Streeter [6] and Chaudhry [7] included extensive discussions of this phenomenon in their textbooks. Resonance, which is an oscillating condition that leads to a pressure amplification in the piping system, develops when there is an exciter present at some point in the system. The piping can be excited in two ways. First, a device may act as a forcing function, exciting the system at one of its natural frequencies. Second, self-excited oscillations occur when a component of the system acts as an exciter. These two actions may occur simul- taneously or independently. Resonance, due to a forcing function, takes place when the interactive response occurs at or near one of the natural Chapter 2 Literature Review 19 periods of the system. A forcing excitation will be used in the current research to find the resonant frequencies of a liquid-filled pipe. Self-excited oscillations are caused by certain features of the piping system. Some of these features include a malfunctioning valve seal [45] , cavitating pump [41] or interactive structural and fluid com- ponents [46 ,47 , 22] . The analysis of resonating conditions in liquid-filled piping systems can be studied in either the time domain, by the method of characteris- tics, or the frequency domain, by the impedance method or transfer matrix method. The frequency domain method of analysis will be the focus of this research. 2.4.1 Impedance Method The impedance method was systemized for the analysis of complex liquid systems by Wylie [48] . The method computes the ratio of the oscillatory pressure and the discharge, known as the terminal impedance, by using known boundary conditions. Then, the terminal impedance is plotted .as a function of frequency to find the natural frequencies of the system and the extreme terminal impedances. This method has been used by Zielke and Hack [49] for the frequency response analysis of pumped storage systems . Chapter 2 Literature Review 20 2.4.2 Transfer Matrix Method The transfer matrix method has been widely used for analyzing structural and mechanical vibrations [50,51,52,53,54] and for analyzing electrical systems [55]. This method is an extension of Miklesta's and Holzer's methods [52,53,54] . Dawson and Davies [56] improved these methods by giving them an automatic natural frequency search capability for ideal- ized lumped property models. Pestel and Leckie [50] detailed the work of many authors dealing with lumped and distributed property models. The transfer matrix method was used by Chaudhry [57,58] for analysis of steady-oscillatory flows and for determining the frequency responses of hydraulic systems. Classic fluid transient textbooks such as those by Wylie and Streeter [6] and Chaudhry [7] describe the application of this methodology to hydraulic systems. To [59,60] used this.method to simu- late and analyze complicated reciprocating compressor piping systems. He developed nineteen parameter matrices for acoustic elements [59] and presented a description of a digital computer program and its applica- tions [60]. The method has also been used in solving fluid-structure interaction problems. Keskinen [61], To and Kaladi [62] and Dupuis and Rousselet [63] developed methodologies to study non-conservative systems involving fluid flow in pipes. Keskinen's method [61] involved numeri- cally solving a system of differential equations, expressed in matrix form, to obtain the transfer matrix for a pipe element which is treated as a discrete parameter model. To and Kaladi's model [62] differed from the previous work in that the transfer matrix was derived from a direct solution of the differential equation of motion of the pipe, which was Chapter 2 ‘ Literature Review 21 considered a distributed parameter model. They presented a method of analysis for complicated piping networks with moving mediums involving bends, piping components of various diameter and lumped masses such as valves. Experimental validation of their model was presented. Dupuis and Rousselet [63] formulated the transfer matrix by using equations of motion that included shear and extensional deformations, rotatory inero tie and variable pipe curvature. They applied the method to straight and curved cantilevered pipes containing a flowing fluid. Another application of this method has been in the study of liquid- filled piping systems such as is the topic of this research. Wilkinson [64] showed that under certain conditions the vibrations of the liquid column and that of the supporting structure can interact. This causes the coupled system to respond dynamically in a manner different from the response of either of the independent components. He used the transfer matrix approach with the vibration state at a point described by a fourteen element vector representing five wave families. These five families are: pressure waves in the liquid, tension waves in the pipe wall, two families of transverse bending waves and a torsion wave. The Poisson effect between the liquid pressure and the axial tension wave in the pipe wall was not included. The equation of motion for the transverse vibration was based on the Bernoulli-Euler beam theory. His results were compared with the experimental results for an L-shaped pipe. They indicated good agreement, but the author concluded that further study of this topic is needed. Wiggert, Lesmez and Hatfield [65] extended Wilkinson's work by including the Poisson effect and by using the Timoshenko beam theory to model the transverse vibration in Chapter 2 Literature Review 22 the piping. The Timoshenko beam accounted for the secondary effects of rotatory inertia and shear defamation. Experimental results with an L- shaped pipe showed good agreement with the numerical model. Lesmez, Wiggert and Hatfield [66] used the same model with a U-shaped bend for a variable length piping system. The variation of the pipe length allowed for different acoustic natural frequencies in the liquid. Excellent agreement in the natural frequencies for both experimental and computed results indicated that the method accounted for the appropiate coupling mechanisms. This dissertation describes in a more detailed fashion the work reported in the two previous studies. Chapter 3 Analytical Development m3 3.1 W The equations of motion for the vibration of a liquid and the axial, transverse and torsional vibrations of the pipe wall in a liquid-filled piping system are presented inithis chapter. The development includes simplifications of the radial properties of the fluid and pipe wall in the axial direction. Although these equations are well documented [l7,l9,20,21,22] , they will be described below since they constitute the starting point of this research. The major contribution of this study is the incorporation of the transverse and torsional vibrations of the pipe wall to an existing axial model that couples the liquid and pipe wall. This model includes Poisson and junction coupling and accounts for the effects of rotary inertia and shear deformation of the pipe wall. The addition of these mechanisms provides an improvement in accuracy over the previous axial- coupled models such as those by Otwell [l9] , Wiggert et a1. , [20] and Budny [22] and the junction coupled model described by Wilkinson [64] . The Timoshenko beam equation is used to represent the transverse 23 Chapter 3 Analytical Development 24 vibration of the pipe. The solution for the constants of integration constitute the connecting point between this chapter and Chapter 4. 3.2 W This study is concerned with piping systems in which the inside diameter of the pipe is much smaller than the pipe length, limiting the equations of motion that describe the system to a one-dimensional wave approach [67,68] . The junction mechanism guarantees that a pipe element can transmit axial, torsion and transverse shear and bending waves in the pipe wall. Therefore, the inclusion of these waves is necessary to more realistically represent the motion of the piping system. The next section considers the axial liquid and axial pipe wall wave equations. The equations that describe the transverse vibration of the piping in the in-plane and out-of-plane modes are then developed. Finally, the equations describing the torsional pipe waves are presented. Figure 3.1 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. 3.2.1 Axial Waves - Liquid and Pipe Wall The six-equation model by Walker and Phillips [17], which consists of one-dimensional continuity and momentum equations for the liquid, axial and radial momentum equations and two constitutive equations for the Chapter 3 Analytical Development 25 pipe wall, constitutes the basis for the axial coupled model. Otwell [19] , Wiggert et a1. , [20], Stuckenbruck et al.,[21], and Budny [22] reduced the six-equation model to a four-equation model by neglecting the radial inertia of the pipe wall. The fluid in the model is assumed to be one-dimensional, linear, 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 . Tim equations represent the axial continuity and momentum relations for the liquid: 2 in 2:2! LL _ ac+K [rat+ataz] 0 (3'1) 12 fi 2” a. + Pfatz + T - ° (“3’ 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 pf 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 10 [22] . Previous authors arrived at these equations by making a number of stan- dard assumptions [l7,l9,20,21,22]. First, the convective terms are ignored by assuming low Mach numbers, where the fluid wave speed is much greater than the fluid velocity. This implies that the fluid density in Chapter 3 Analytical Development 26 Equation 3.2a is constant. Second, the one—dimensional flow assumption implies that the radial component of the fluid velocity is zero and that the flow is developed in only the radial direction. The fluid friction term in the momentum equation can be neglected for forced vibrations [5,6] as is the case in this study. However, as noted by Williams [85] and Budny [22] the damping terms in both the liquid and the axial pipe wall equations of motion should be considered for transient events. Equation 3.2a reduces to: 2 £2 62 + pf .. o (3.2b) 2 at Assuming an axisymmetric, linear elastic pipe wall and small deforma- tions, the axial and circumferential stress-strain relationships for the pipe wall are: oz-E*[a_u_;+u}g] -o (3.3) dz r ' au * E __Z _ ao-E [r+yaz] 0 (3.4a) where E*- —-L2—- (3.4b) (1-v ) in which oz - oz(z,t) and do - 00(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 u are the Young's modulus and Poisson's ratio Chapter 3 Analytical Development 27 of the pipe wall. Figures 3.2 and 3.3 show a section of a pipe with stresses and displacements in the axial and radial direction. The equations of motion for the pipe wall are: _z - pp ‘7‘ - 0 (3.5) 32 at 2 2 pr-aoe-[ppre+pf§—]‘g—¥-0 (3.5) t in which pp 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 3.6. Equations 3.1 through 3.6 constitute the six-equation model. The radial inertia term is important when an excitation is approaching the first lobar mode of the cross section. Everstine, Marcus and Quezon [86] compared a one-dimensional finite element formulation with a three- dimensional one. The study showed that a one-dimensional finite element formulation of coupled pipe and liquid accurately predicts the dynamic responses up to the frequency of the first lobar mode. Therefore, neglecting the radial inertia term is accurate for frequencies below this mode. The expression for this lobar mode frequency is: BE ’1 “’0 " :7 [5pp(1-v2)(l+pfr/2ppe) (3'7) Chapter 3 Analytical Development 28 By neglecting the radial inertia term in Equation 3.6 the radial stress, 00, can be evaluated in terms of the fluid pressure: «[3 do (3.8) The radial stress can be eliminated by combining Equations 3.4 and 3.8 and solving for the radial strain w/r au 2 _ Di— , ”—1 (3.9) r E e 82 Combining Equations 3.9 and 3.3 give the expression for the axial stress a -—1 - o (3.10s) Multiply the above equation by the pipe cross-sectional area AP, to obtain the axial force, fz au - I - —'z' - fz uAp e p EAp 32 o (3.10b) Differentiating Equation 3.9 with respect to time and combining it with Equation 3.1 produces the expression for the fluid pressure Chapter 3 Analytical Development 29 2 dz ' ” dtdz + dtdz ' ( ° a) where * _K_ K _ (3.11b) 1 +JE§L11 E e Equations 3.2b, 3.5, 3.10b and 3.11 constitute the four-equation model presented by Otwell [l9], Wiggert et a1. , [20], Stuckenbruck et a1. , [21] and Budny [22] . These equations can be further reduced by dif- ferentiating Equations 3.2b and 3.5 with respect to the axial direction 2, then differentiating Equations 3.10b and 3.11 with respect to time and combining them to solve for the axial force and fluid pressure. 2 2 2 a f a f 2 ap —-;i - -—;Z + Apub 2—9 - o (3.12) dz dt dt 2 2 2 2 a f 2 V8 aft)- 3—2+A—&f~—;‘-o (3.13s) dz dt dz - where * 2 a - 3— (3.13b) f pf a2 - 5— (3.13c) pp Chapter 3 Analytical Development 30 r b - e (3.13a) p d - J (3.13a) Pf In Equations 3.13b and 3.13c, 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 3.13a are second order partial differential equations in the fluid pressure and axial pipe wall force. They may be expressed in matrix form as: 0 2 f 1 - bA 2 f 2 2 L2 2 - 0 ”1 P .2—2 2 - O (3.143.) “r “f dz P at P A similar equation can be obtained for the axial pipe wall and fluid .FF a. displacements by combining and solving Equations 3.2b, 3.5, 3.10b and 3.11. 2h 2 2 2 2" daf+ap -Vldaf 2 u l l 0 2 u L z i L z - o 3 14b) ”h 2 L 2 2 V " o b— 2 v ( - Poisson terms couple Equations 3.12 and 3.13a as shown by the off- diagonal elements of the matrices in Equations 3.14s and 3.14b. Chapter 3 Analytical Development 31 The separation of variables technique [76] is used to solve for the force f2 and fluid pressure p in Equation 3.14a. Three steps are neces- sary to solve for the dependent variables in the above equation: 1) convert the partial differential equation into ordinary differential equation, 2) find solutions for the ordinary differential equation, and 3) 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. 1) Separation of Variables Assuming a harmonic oscillation for the time dependence, which is ap- propriate for oscillatory flow [41,71] and oscillatory structural motion in the axial direction [68], we can write: fz(z,t) - Fz(z) ejwt (3.15) p(z,t) - P(z) ejwt (3.16) where Fz(z) and P(z) are functions of 2 only, to is the oscillatory frequency and j - (4)8. Substituting the above equations into Equation 3.14 yields the ordinary differential equation in F2 and P: Chapter 3 ' Analytical Development 32 2 . ap 0 ,, ‘ F 2 1 'VbA F 2» a2 a2 f + w o 1 P z - o (3.17) d_A_ f f P P P where F; and P" are the derivatives with respect to the axial direction 2. The elimination method [92] can be used to reduce Equation 3.17 to a single dependent variable. This procedure yields F2” + " + ”,+ 7) F" + 1% - o (3.18a) 2 2 where 1 is the length of a pipe reach and 2 2 r - 2;; (3.18b) “f 2 2 a -5‘%& (3.18c) a p 2 2 2 7 - 2y P-Pgl . (3.18d) d ap Equation 3.18s is a fourth-order, ordinary differential equation with constant coefficients . Chapter 3 Analytical Development 33 2) Solution of the Ordinary Differential Equation The solutions for F2 in Equation 3.18s is of the form - J[2 122(2) - A e 2 (3.19) where A is a constant. Substitution of Equation 3.19 into 3.18s produces the characteristic equation in A: 4 2 A + (r + a + 1) A + or - 0 (3.20) where A is the characteristic value. The roots of this equation are ijA, and isz, where 2 l 2 h A1,2 - 2 (r + a + 1) ;[(r + a + 1) - 407] (3.21) This equation can also be expressed as: 8 2 2 2 2 2 c; - Q—é— - % {[ a;+32+2y2 gag] - [[a;+a;+2v 33f] - 4afap] } (3.22) Chapter 3 ’ Analytical Development 34 O N I F» n I NP r-‘—\ r——1 m n n 2222+ 2222h22422‘2 323 p A: f+ap+ v daf af+ap+ V daf - afap ( . ) The above equations .give the expressions for the coupled wave speeds. These coupled speeds are the same as those derived by Budny [22] using the method of characteristics. An inspection of Equations 3.22 and 3.23, assuming no coupling between liquid and pipe wall by neglecting the second order Poisson terms, yields ‘01 2 2 1—1 - af (3.24s) (01 2 2 r2- - ap (3.24b) As noted by Stuckenbruck, et al. , [21] , Equation 3.24s is the classical fluid wave speed prediction [6] for a pipe anchored throughout against axial motion. Placing Equation 3.21 into 3.19, the solution for Fz(z) is: th _ an: - 112% _ 4);} e Fz(z) - A1 e + A, e + A; e + A, (3.25) Chapter 3 Analytical Development 35 and using the relation i (15) e j i - cosuf) i 35mm?) Equation 3.25 can be written in the following form Fz(z) - A;cos(A1§) + A,sin(A1§) + A,cos(A2§) + A‘sin(A2%) where A1 "' K1 ‘1' X2 A2 ‘ J (3.33s) , U P V i - -Bssin(A,§) B5c0s(A1f) -B.sin(A,§) B.cos(A,§) 1A, F L cos(A,§) sin(A1§) cos(A,§) sin(A,§)J A, where B, - 2‘1 (3.33b) A E0 9 32 - 21, (3.33¢) A Ea p 2 B, - a ' *1 (3.33d) A who 2 B, - a ' ‘2 (3.33e) Chapter 3 Analytical Development 38 2 (a - AI)1A1 (3.332) A va*ra P 2 (a ' A2)£A2 BO _ (3.33g) * A bit a p" 1 Equation 3.33a will be the starting point of the next chapter. Solutions for the constants of integration of the axial waves equations will be derived. 3.2.2 Transverse Waves - Shear and Bending The classical one-dimensional Bernoulli-Euler theory of flexural motions in elastic beams is inadequate to describe vibrations of higher modes as well as those beams where the effect of the cross-sectional dimensions on frequencies cannot be neglected [69]. Rayleigh [70] introduced the effect of rotatory inertia and Timoshenko [68,71,72] extended it to include the effect of transverse shear deformation. The equations that include these effects are generally referred to as the Timoshenko beam equations. The derivation of the equations for transverse vibrations in the x-z and y-z planes are presented following the sign convention in Figures 3.4 through 3.7. Chapter 3 Analytical Development 39 3 . 2 . 2 . 1 Shear and Bending in x-z Plane The derivation of the Timoshenko equations for the pipe reach in Figure 3.1, vibrating in the x-z plane, are based on the diagrams shown in Figures 3.4 and 3.5. Figure 3.4 shows that the slope of the center line of the pipe reach, dux is affected by both the bending moment and the dz shear force [50]. The action of the internal bending moment, my, rotates the face of the cross section through angle ivy. From there, the internal shearing force, fx’ turns the center line to adopt the slope a“; . The angle of the face of the pipe element remains unchanged. dz Inspection of Figure 3.4 shows that the angle between the line perpen- dicular to the face and the center line of the pipe element which is caused by the shear force is the shear angle, fly: 2:. fl - 32 ' '1’ (3°34) du . _ 2x - GApas [ea—23 - (6),] o (3.35a) where c - __a_ (3.35b) 2(1+V) Chapter 3 Analytical Development 40 The shear modulus is G, the product Apr: s represents the effective shear area of the section and ‘s is a factor depending on the shape of the cross section [73,74] . For a thin-walled tube ”3 is given by 2114.1). (3.36) s - 4+3u The relation between the bending moment and cross-sectional rotation is given by elementary beam theory as: 81/: - EI - 0 3.37 “‘2 Pal ‘ ) in which Ip is the second moment of inertia about the y—axis for the pipe wall. Equations 3.35a and 3.37 constitute elastic laws relating the deformations to the internal loading. Equilibrium considerations (Fig. 3.5) give the equations 2 2.2.. - (p A ”£192.. — o (3.3.) 32 P P 2 at 3 32¢ m + f - I + - 0 3. 9 at in which If is the second moment of inertia about the y-axis for the fluid, Equations 3.38 and 3.39 describe the translation and rotational Chapter 3 Analytical Development 41 equilibrium, respectively. Solving for )6}, and ux in Equations 3.35a and 3.38, substituting the results in Equation 3.37 and eliminating my from Equation 3.39, we obtain a fourth-order partial differential equation in fx(z,t): a‘f 82f a‘f E1 3 82f £1 + A + - I + I + A + p--:‘ (”p p ”fAf)—'33 (pp p pf f)“;3 2 ax-E -—3[(pp p pfAf)-—53] dz dt dz dt p 5 dz dt (9 A +p ) 2 2 + P P fAf ‘9 [on A egg—2" f ] - o (3.40) CA a 2 P P 2 p s at dt The third term in the above equation corresponds to rotatory inertia, the fourth to shear defamation and the last term represents the com- bined influence of shear deformation and rotatory inertia [52]. By neglecting these three terms, we obtain the Bernoulli-Euler beam equa- tion in the shear force fx’ 4 2 d f d f 81 + A + - 0 3.41 pf ‘ (Pp P PfAf)__—Kat2 ( ) z The separation of variables technique is used to solve for the dependent Variable fx’ in time, t, and axial direction 2 [50] . fx(z,t) - Fx(z) e3”t (3.42) Chapter 3 Analytical Development 42 Substitution of the above equation into Equation 3.40 we obtain + 2.2.2 F; - [ 1.2.2: ] Fx - 0 (3.43s) where a - (PgAp+Pff£) A222 (3.43b) CAI: ps 22 r - (PpIp+P£If) w 2 (3.43c) 21p 2‘ 1 - (PnAn+P£Ar) w 2 (3.43d) 21p and 2 is the length of the pipe reach. Equation 3.43s is a fourth-order ordinary differential equatian'with constant coefficients whose solution is of the fom - 11 Fx(z) - A e 2 (3.44) where A is a constant. Substitution of Equation 3.44 into 3.43s produces the characteristic equation in A: A‘. + (a + r) A2 - (1 - or) - 0 (3.45) Chapter 3 Analytical Development 43 The roots of this equation are iA, and ijA,, where li’z - [1+%(0-r)2]k;12‘(a+r) The solution of Equation 3.43s is hf _ 41% - an? _ an»? Fx(z) - A + A, e + A, e + A, e 18 and using the relations :01) s: (,2) e I - cosh(A§) i sinh(A§) and e j I - cos(Af) i jsin(A%) Equation 3.47 can be written in the following form _ z. z z. a Fx(z) Alcosh(A1£) + A,sinh(A12) + Ascos(A,2) + A.sin(A,£) where (3 (3. (3 (3 (3. (3 (3 .46) 47) .48) .49a) .49b) 49c) .49d) .49e) Chapter 3 ' Analytical Development 44 The solution for the other three dependent variables, My’ p), and Ux is based on the solution for Fx' The solution for Ux(z) can be found by substituting Equation 3.49s in Equation 3.38 4 Ux(2) - '-_L [A1 2 Slnh(Ali’) ‘2' A2 2 COSh(A1§) ' A3 1 2 E1 1,7 A” sin(A,f) + A, :3 cos(A,§) ] (3.50) 2 The solution for \Fy(z) is obtained by placing Equations 3.49s and 3.50 into 3.35a 2 wy(z) .. -—.L{ (a + 1:) [A,cosh(A,§) + A,sinh(1,f) ] EI p7 2 A + (a - A,) [A,cos(A,i-) + A‘sin(A,§) ] J (3.51) The expression for My(z) is found placing Equation 3.51 into 3.37 My(Z) _ -‘&{ (a + A?) A1 [A181m(xli) + AQCOSh(A1§) ] T 2 + (A, - a) A, [A,sin(A,§) - A4cos(A,§) ] } (3.52) Chapter 3 Analytical Development 45 Arranging equations 3.49a through 3.52 into matrix form we obtain r w P ‘ r ‘ Ux -n,sinh(x1f) -Blcosh(xlf) B,s1n(x,§) -B,cos(,\,§) A1 0y -Bscosh(A1§) -Bssinh(A1f) -B‘cos(A2§) -B.sin(A2§) A, ‘My” -35s1nh(xlf) -Bscosh(x1§) 4.311102%) B.cos(,\2f) ‘A,> (3'53” £ij _ cosh(A1%) sinhulf) cosuzf) sin(/\21)d ~A44 3 . where B, - iii; A1 (3.53b) p 8 32 - ‘1‘“ 7 A2 (3.53c) p 2 2 B, - fig? (0 + A1) (3.53d) p 2 2 B. - EL», (0 - x2) (3.53e) p 2 35 -f (a + A1) A1 (3.5313) 2 3. -$ (A2 - a) A: (3.533) The solution for the constants of integration is given in the next chapter. 3.2.2.2 Shear and Bending in y-z Plane The procedure to derive Timoshenko's beam equation in the y-z plane for the pipe element shown in Figures 3.6 and 3.7 is the same as described Chapter 3 Analytical Development 46 in the previous section. The only differences arise in the sign conven- tion. The shear angle is given by: au )3 - 342! +4, (3.54) au fy-GAK [J+¢x] -o (3.55) The bending relation is: 3" - 51 x - 0 3.56 mx paz ( ) Equilibrium considerations (Fig 3.7) give the equations 2 35: - (p A +pfAf)a_ux - 0 (3.57) 32 P P 2 at 2 1'3 - f - (p I ”£19253 - o (3.53) 32 Y P P 3:2 Combining Equations 3.55 through 3.58 gives the fourth-order partial differential equation in the lateral displacement fy(z,t): Chapter 3 Analytical Development 47 a‘f 62f a‘f EI a2 62f £1 + A+ - 1+ I + A+ 9-4. (”pppfAf)—12 (”pppf f)__22 2‘41“,c —2[(ppppfAf)42] 62 at 62 at psaz at (pA+p ) 2 2 + PP fAfa [(pA+pfAf)af1]-O (3.59) CAR 2 PP 2 ps at at The solution for the constants of integration is of the same form as for the shear and bending in the x-z plane. The change in the sign of the shear angle fix determines sign changes in the rotation and bending moment, whereas the shear force and lateral displacement remain the same. Equation 3.53a becomes: 1 ' r ‘ fUy {-Blsinhuxf) -B,cosh(x,‘}) B,s1n(x,§) -B,cos(x,f) A, ix B,cosh(A1‘;‘) B,sinh(A1‘;‘) B4cos(A2f) B‘sinugf) A2 ‘M "' Bssinhulz) escoshuf) 3.31:1023) -B.cos(A2z') ‘A,’ (3'60) x 1 z 1 2 z z z 1 (FY) . cosh(;\11) Sinhalz) c0302!) sinunw (A5 where the coefficients of the matrix are given in Equations 3.53b through 3.53g. Chapter 3 Analytical Development 48 3.2.3 Torsion About z-axis Figure 3.8 illustrates the internal moment, mz, acting on the pipe section and the rotation about the z-axis $2. The equilibrium condition is given by: Ez-iniz-o (3.61) 62 P P at: :11 -GJ 2; -o (3.62) in which Jp is the polar moment of inertia for the pipe wall. Combining Equations 3.61 and 3.62 the wave equation for the moment mz(z,t) is: a m; - £2 a ”z - o (3.63) 2 G 2 82 at The separation of variables can be used to solve for mzin the above equation. mz(z,t) - Mz(z)e3“’t (3.64) Chapter 3 Analytical Development 49 substitution of the above equation into 3.63 yields Mz+33_Mz -o (3.65a) 2 where 2 2 7-162 (3.65b) c The solution of Equation 3.65a is of the form - A; 1512(2) - A e 2 (3.66) Placing the above equation into 3.65a yields the characteristic equation in A: 2 The roots of this equation are ijA where .\-:[~,]”- :62[£p_]”’ (3.68) G Placing the characteristic value A in Equation 3.66, the solution for M2 is Chapter 3 Analytical Development 50 Mz(z) - K,e3*§ + Xze'3*§ (3.69) using the relation in Equation 3.26, the above equation becomes Mz(z) - A,cos(A§) + A,sin(A§) (3.70) Where A, and A, are given in Equations 3.27b and 3.27c. The solution of the rotation W2 about the z-axis is found by placing Equation 3.70 into 3.61 and using jwt ¢z<2.t) - *z(2)e (3.71) we obtain W (2) - ‘-A- A,sin(Az) - A,cos(Az) (3.72) z p J 1 2 2 P P Equation 3.70 and 3.72 can be arranged in matrix form as __A__ z _;A__ 1 92(2) p J i sin(A£) p J 2 cos(A£) A1 - P P P P (3.73) Mz(z) cos(Af) sin(A§) A, The matrix Equations 3.33a, 3.53a, 3.60 and 3.73 constitute the starting point to derive the field transfer matrices in the next chapter. Chapter 3 Analytical Development 51 Figure 3.1 Sign Convention for Internal Forces 2. X . Bf f3! == f)"F"i;EEL(5]Z Chapter 3 Analytical Development 52 Y X i 2 U2 (Axial) w (Radial) 60 02 «— //[///////// —«--02 + 73—21- 6 z e ———D- —-)-—————————-n)-— V ////////////// 62 Figure 3.2 Axial Pipe Element Figure 3.3 Radial Pipe Element Chapter 3 Analytical Development 3 _.. 5 Y Z 5 / l ‘ Z an (Original 7 “ts ' X \ ‘ a ‘_ Deformed \‘_ / 2 I vx \ ‘ WY 5 ‘33 ' \ 6 .‘ ‘ ‘§\ aux V. 62 32 ‘3 Y Z F2 fx / x ‘6 ‘1.» ‘\~ -~“- //lfiai=5- ‘ ““~. ‘1 + m’ ‘ “ ’ X Wd 5 “’ 9:2 ‘6-\ my+ 62 62 ‘3 1.» dz ‘3 5f ““‘.i fx 4' 63;: 6h! Figure 3.5 Internal Forces for Transverse Vibration in x-z Plane Chapter 3 Analytical Development 54 ' 4’ 2 Original 2 \ L IT— "l~'l!:ln / w 6. “”x Figure 3.6 Deformations for Transverse Vibration in y-x Plane --Z .\/ ”‘72 Y f "5112111 "3‘ + a 62 y .a” v ’ . ‘wx 1 mx . / ' Y ’ \ / / 8f V'”’4"IIIA f y + —a_zl 6 z V/b‘L/A Figure 3.7 Internal Forces for Transverse Vibration in y—z Plane Analytical Development Chapter 3 55 cowuuoh ou nauaannsm sowed omwm can shaman Chapter 4 3 Numerical Simulation MA 4.1 Introduction The equations that describe the motion of five families of waves were derived in the previous chapter. The separation of variables technique was used to solve for forces and displacements. The solutions for the constants of integration of the equations of motion are the connecting point between this chapter and the previous one. The solution for the constants of integration are derived in a general form'based on the end. points of a pipe reach. Once these constants are known for each family of waves, the transfer matrix is assembled. The transfer matrix method. is used to find the frequency response of liquid-filled pipe systems. This chapter gives a description of this method and presents comparisons with other numerical methods such as the method of characteristics and the component synthesis method. 56 Chapter 4 Numerical Simulation S7 4-2 W The transfer matrix method is the systemization of the Holzer and Myklestad procedures [50,52,53,54]. Holzer applied the method to tor- sional systems, whereas Myklestad applied it to bending vibrations of bars. Both methods calculate the natural frequencies and mode shapes of the system. This is done by assuming a frequency and starting with a unit amplitude at one end of the system and systematically calculating the responses at the other end. The frequencies that satisfy the re- quired boundary condition at the other end are the natural frequencies of the system. Their findings are based on the fact that when an un- damped system is vibrating freely at any one of its natural frequencies, no external force, torque or moment is necessary to maintain the vibra- tion. Also, the amplitude of the mode shape is immaterial to the vibration [53]. These two methods have been applied to lumped mass systems. The masses are lumped at discrete points of the system called stations and the portion between the lumped masses is assumed massless and of uniform stiffness. The transfer matrix method is suitable for the analysis of large liquid- filled piping systems made up of subsystems such as pipe links, snubbers, springs, concentrated masses, rigid supports, valves, pumps, orifices, and the like. Each subsystem is modeled as simple elastic and dynamic elements described by a field matrix and a point matrix. At the stations, the displacements and internal forces of the systems are arranged in a state vector. The overall transfer matrix is assembled by the systematic multiplication of the field and point matrices. Chapter 4 Numerical Simulation 58 The advantages of this method are: l. The assembly of complex, branched, parallel, and series systems is suitable for digital computation, . Boundary conditions at the ends and at intermediate points in a system are easy to identify and model, . The method can be applied to piping systems of non-uniform cross sections, . The stability of a system can be checked by the root locus technique [7], . Systems with more than two dependent variables can be analyzed, since the size of the matrices does not depend on the number of subsystems but rather on the order of the differential equations governing the systems behavior, . The method can be extended to stability problems such as flow induced vibrations and damped vibrations, . External excitation of the system can be modeled by the extended field, point, and global matrix and extended state vector, and . The method can be used to model systems as discrete parameter or distributed parameter systems. 4.2.1 Description of Transfer Matrix Approach The procedure to implement the transfer matrix method is illustrated using a spring-mass system [50] in Figure 4.1. The state vector at point i is a column vector whose elements are the various displacements, linear or rotary, of the point i and the corresponding internal forces. Chapter 4 Numerical Simulation 59 In the system shown in Figure 4.1, the state vector (Z)i at point i is comprised of the linear displacement 21 and the spring force f1: {mi-{21W}. A field transfer matrix relates the state vectors at two locations in a system. Equilibrium conditions can be used to obtain the field matrix of the system. Thus, the displacements and forces to the left of mass m denoted as (2);": are related to the forces and displacements to the i right of mass m1_1 denoted as (2)1;1 by means of the matrix [T]1, called the field matrix. Equilibrium of the massless spring (Figure 4.2a) is expressed L 3L1 - £1 (4.2) Also, £3 _ _i-_l 21 21-1 + k1 (4.3) l... {éF-[WJW i i-l or Chapter 4 Numerical Simulation 60 (21‘; - Ingmil (4.1m) In the same form, the state vectors to the left and right of mass mi are related by matrix [P]i called the point matrix. This is possible be- cause the transfer between the two adjacent state vectors is over.a point. Assuming the system is vibrating with a frequency w the follow- ing equation of motion is obtained, (Fig. 4.2b). f? - fL 2 (4.5) R - (4.6) R ; L . z 1 0 z {.5},- 1 1 Hf}. (2)1: - [Plimli‘ (4.7b) or Combining Equations 4.4b and 4.7b renders Chapter 4 Numerical Simulation 61 (2)1; - [P1,III,{21§_1 (4.8) Following a similar procedure, the state vector at the end of a system (Figure 4.1) consisting of n number of springs and masses, joined end- to-end can be related by multiplying together the various field and point matrices in the proper order. Thus, m“- [P] [I] [P] [I] [P] [I] (2)“ - [BMW (4 9a) n n n n-l n-l'” l 1 0 0 ' where [U] is the overall transfer matrix of the system [U] - nlplilrli (4.91)) Once the overall transfer matrix of the system is obtained, the natural frequencies can be found by applying the boundary conditions. This is shown for a simple spring-mass system in Figure 4.3. The global trans- fer matrix of this system is: 1— 1 UR 1 o 1 k 1 [U] - mlml‘ [ “‘1‘”: 1] [° 11 ] ' [ «.162 [1-m2/k1]] (4'10) The boundary conditions, f, - 0 and z, - 0 and the overall transfer matrix form the general equation for the system: Chapter 4 Numerical Simulation 62 o { g }1 - [a] {f }o (4.11) (4.12) The well-known frequency of a single spring-mass system is, thus, derived as: w - [§]“ (4.13) This procedure can be used for more complicated systems providing the field and point matrices for each subsystem are known. Figure 4.4 shows a liquid-filled piping system composed of straight links and subsystems such as point masses, springs, and supports that suppress the pipe motion partially or totally. These subsystems connect to and have an effect on the response of the pipe wall. A constant- pressure reservoir and a closed valve affect the behavior of the liquid. To find the natural frequencies of this system, the field matrices for straight pipe links, and the point matrices defined where there are bends, springs, point masses, and supports, must be known. The follow- ing sections will explain the derivation of these matrices. Chapter 4 Numerical Simulation 63 The field matrix for a straight pipe reach is composed of four sub- matrices representing the vibrations of the liquid and axial pipe wall, shear and bending in the x-z plane and in the y-z plane and torsion about the z-axis. Expressions for each submatrix are given. The field transfer matrix derivation for the liquid and axial pipe wall vibration is presented in detail. The field matrices for the transverse and torsional vibrations were obtained by Pestel and Leckie [50] . Their derivation is presented for completeness. Point matrices for springs, concentrated masses and bends are also presented. Point and field matrices for each subsystem are derived with respect to a local rectan- gular coordinate system, x,y,z. Transformation matrices to express these matrices in a global coordinate system, X,Y,Z are also presented. Supports, reservoirs and closed valves are treated as boundary condi- tions . 4.2.2 Field Transfer Matrices As mentioned earlier, the field transfer matrix expresses the forces and displacements at one section of a chain-type structure in terms of the corresponding forces and displacements at an adjacent section. For a discrete system, field matrices provide for transfer across the elas- tic segments between the masses [52,53,54]. To and Kaladi [62] and Wilkinson [64] use a distributed parameter approach to derive the field transfer matrices for fluid-structure interaction problems. Wylie and Streeter [8], and Chaudhry [7] use this approach for oscillatory flow problems. Chapter 4 Numerical Simulation 64 The derivation of the field matrices for a distributed parameter model involves three steps: 1) converting the partial differential equations of motion into one ordinary differential equation, 2) finding solutions for the differential equation, and 3) finding the constants of integra- tion of the differential equation. Steps one and two are developed in Chapter 3. The constants of integration are left in matrix form for each family of waves in section 3.2. These constants are solved as function of the state vector at the end points of the pipe reach of length 1 shown in Figure 4.5. A general procedure that can be applied to any of the matrices of the previous chapter is presented. Either one of the matrix Equations 3.33a, 3.53a, 3.60 or 3.73 can be represented as: 2(2) - [3(2)] A (4.14) where 2(2) is the state vector representing the dependent variables of any one of the above equations, 3(2) 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 2 - 0 in Fig. 4.5, 2(2) - 2i_1, the matrix Equation 4.14 be- comes 21-1 - [3(0)] A (4.15) Solving for the column vector A in the above equation Chapter 4 Numerical Simulation 65 -l A.- [3(0)] 2L1 (4.16) substituting Equation 4.16 into Equation 4.14 yields 2(2) - [3(2)] [3(0)1‘1 z,_1 (4.17) At point 2 - 1, 2(2) - Z1 , so Equation 4.17 becomes -1 2i - [3(1)] [3(0)] 21_1 - [T] 2L1 (4.18) where [T] is the field transfer matrix. The field transfer matrix for each one of the family of waves is presented next. 4.2.2.1 Liquid and Axial Pipe wall Vibration Equation 4.18 for the liquid and axial pipe wall vibration becomes (4.19a) 21 ' [Tfplizi-l where [rfp] - [3(2)] [3(0>1'1 (4.19b) Chapter 4 Numerical Simulation 66 The matrices [3(0)] and [3(2)] are obtained from the matrix of Equation 3.33a evaluated at IOCations i-l and i in Figure 4.5, respectively. The non-dimensional representation of the field transfer matrix and state vector are [r where fpl- 0C2’Co halal-(mama h '02 2V0'C3 (7”)02'Co T[(T"."1)C3‘cll 2 2V002 %[(7+‘Y)CI'[(T+‘7) +01103] (TH)C2“C° 2V[(0+T+‘1)C3‘C1] a(C,-aC,) - fihac, -%harC3 2 2 r - 9%! af 2 2 a - “La-1 a P 1 - 2v2 5; b _ I e d - 52 Pf h - :; 2h Co - A[A:cos(A1) - AZcos(A,)] A2 *1 C1 - A x—sin(A1) - K—sin(A,) 1 2 'C1+(0+7)C3 -2vrC, 0'02 '00 (4. (4. (4. (4. (4. (4. (4. (4. (4. 20) 213) 21b) 21c) 21d) 21e) 21f) 218) 21h) Chapter 4 - Numerical S imulat ion 67 C, - A[cos(A1) - cos(A,)] (4.211) c, - 63:51:10,) - tangy] (4.213) 2 2 -1 A - [1, - 1,] (4.21k) 2 l 2 *1 A, - 2 (r + a + 1) -[(r + a + 1) - 401] (4.212) A: - 12‘ {(1 + a + 1) +[(r + a + 7)2 - 4048} (4.21m) and the non-dimensional state vector at location i in Figure 4.5 is: T U F .2 2 Y. _2 z - (4.22) The matrix in Equation 4.20 is valid providing that the coupled wave speeds ratio are different from one another. This condition guarantees that the eigen values in Equations 4.212 and 4.21m are different from one another, therefore, avoiding the undeterminate form of Equation 4.21k. 4. 2 . 2 . 2 Transverse Vibration in x-z Plane Equation 4.18 for the transverse vibration of a pipe reach in the x-z plane becomes Chapter 4 Numerical Simulation 68 21 - [szlizi-l (4.23a) where [1,21 - [3(1)] tn<0>1‘1 (4.236) The matrices [3(0)] and [3(2)] are obtained from the matrix of Equation 3.53a evaluated at locations i-l and i in Figure 4.5, respectively. The non-dimensional representation of the field transfer matrix [50] and 8 tate VBCCOI are q 2 Co'002 C1'(U+T)Cs C2 ‘%['0C1+(‘Y+U )Cs] 7C3 60-762 C,-sz '02 - 2 [ngl 102 (7+, )Cs'761 60.,62 -[c,-(a+r)c,] (4.24) 47(01'0C3) 3732 '1C3 Co-aC, where 2 2 a - (pnAD+pfAf) w 2 (4.25a) GAP»s 2 2 f - (”212+Pflfi) w 2 (4.25b) EI P 2 4 1 - ffnfniffffl w 2 (4.256) EIp Co - A[A:cosh(A1) + Aicos(A,)] (4.25d) Chapter 4 ' Numerical Simulation 69 2 2 (A, A, C, - A T;sinh(A,) + :sin(A,)] (4.25e) F C, - ALcosh(A,) - cos(A,)] (4.25f) F c, - A §Isinh(A,) - §;sin(x,)] (4.253) 2 2 -1 A - [1, + 1,] (4.25h) A: - [ 1 + i (a - ¢)2]” - % (a + 1) (4.251) A, - [ 1 + i (a - 7):)” + % (a + r) (4.253) and the state vector in the x-z plane at location i in Figure 4.5 is given by 2 [.1233 525.5“: 21 ' 1 Y 21p 21p 1 (“'26) The matrix in Equation 4.24 differs from the one by Pestel and Lackie [50] in that the mass of the contained liquid is included in the parameters a, r and 1. 4. 2 . 2 . 3 Transverse Vibration in y-z Plane Equation 4.18 for the transverse vibration of a pipe reach in the y-z plane can be represented by Chapter 4 Numerical Simulation 70 21 - ['I'yz]121_1 (4.27a) where [1,21 - [3(2)] [3(0)1'1 (4.27b) The matrices [3(0)] and [3(2)] are obtained from the matrix of Equation 3.60 evaluated at locations i-l and i in Figure 4.5, respectively. The non-dimensional representation of the field transfer matrix and state VGC‘COI are q F c0'002 '[Ci‘(0+')cs] ‘02 “$1'001+(7+02)Cs] ~1C3 Co-rC, C1-rC, C, [1&2] - ~1C, (1+72)Ca-7C1 Co-yc2 [c1-(a+,)03] (4.28) L°7(C1'0C3) 162 16, 60-00, where the coefficients are given in Equations 4.25a through 4.25j. ‘The state vector in the y-z plane at location i in Figure 4.5 is 2 U w M 2 F 2 T 21" f x 4‘— L (4.29) Chapter 4 _ Numerical Simulation 71 4.2.2.4 Torsional Vibration About 2-Axis Equation 4.18 for the torsion about the z-axis is 21' [thlizi-l (4.30a) where [rm]? [3(2)] [B(O)I'1 (4.3015) The matrices [3(0)] and [3(2)] are obtained from the matrix of Equation 3.73 evaluated at locations i-l and i in Figure 4.5, respectively. The non-dimensional representation of the field transfer matrix and state VGCtOI' are -cos(A) -‘1‘81n(a\) [r (4.31) ] .- ‘z 1 A sin(A) -cos(A) where 2 2 2 P A - w 2 52 (4.32) and the state vector, 21 in Equation 4.30a is u 2 T 21 - 92 63‘ (4.33) p Chapter 4 Numerical Simulation 72 4.2.2.5 General Field Transfer Matrix The field transfer matrix for a single straight pipe reach shown in Figure 4.5 is composed of four submatrices: 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 2-axis. Their expres- sions were given in Equations 4.20, 4.24, 4.28 and 4.31, respectively. The state vectors have fourteen dependent variables: three for each of forces, moments, displacements and rotations of the pipe wall and pres- sure and displacement of the liquid. The equation below shows these arrangements : zi - [TL] 21.1 (4.34) where [TL] is the field transfer matrix for a pipe reach of length 2 in the local coordinate system. The fourteen by fourteen element matrix is given below: (4.35) 2 * AB 2 2 T n F U n: 2 2 2112144 314.251 gxwxmnga},,,,, K p p p p p p Chapter 4 Numerical Simulation 73 4.2.3 Point Matrices Three types of point matrices for the pipe wall will be discussed: bends, springs and point masses. Point matrices for structures that affect the liquid directly, such as orifices, accumulators and oscil- latory valves'were developed by Chaudhry [7] , and Wylie and Streeter [8] . Wilkinson [64] developed point matrices for T-junctions, curved bends and pumps as sources of excitation. 4 . 2 . 3 . 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 wall 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. The point matrix is derived for two reaches in a two-dimensional space and is shown in Figure 4.6. The reaches are connected so that a is the angle between the axis of each pipe. Figure 4.6a shows the internal forces in the local coordinate system and Figure 4.6b shows the dis- placements. The equilibrium and continuity conditions that relate the Chapter 4 Numerical Simulation 74 state vector, 21; to the right and the state vector, 2;: to the left of point i are as follows [26,27]: Equilibrium of fluid displacement, pipe moments and forces (Figure 4.6a): . L _ 22x. M: - MK 0 (4.37a) 2F : PRA cosa - FRcosa + l-‘Rsina - PLA 4» FL - 0 (4.37b) y f z y f 2 EF : -PRA sine + FRsina + FRcosa - FL - o (4.37e) 2 f 2 y y Displacements: -(VL - U2) + (VR - 01;) - 0 (4.37d) Continuity of fluid pressure, pipe displacements and rotations (Figure 4.6b): Rotations: iv: - 1!: (4.37e) Displacements: U; - U§cosa + Ui‘sina (4.37f) - 2' Disina - Ui‘cosa (4.37g) Pressures: PL - PR (4.37b) These equations are assembled in matrix form as: Chapter 4 Numerical Simulation 75 ' U ‘R r- . ' U ‘L 2; case 0 0 0 sina O 0 0 2; 1’- o 1 o o o o o o 1’— K* 18" ii -(l-cosa) o 1 o sina o o o ‘3? F F .2. a _ 1 .2. ApE 0 b(1 case) 0 cosa 0 0 0 gsina ApE .11.- «U >(4.38) I! ~sina o o o cosa o o o 723 9x 0 o o o o 1 o 0 1x Hg 0 o o o o o 1 0 up 2 F 2 F 2 E¥_— 0 gqb sina 0 ~g sina 0 0 0 case 1 E¥—- 1 911 ‘ . . 911 or 2‘; - [2,2112% (4.39a) and 2 A 2 g - '11)— (4.3%) p A q - A: (4.39c) 9 4.2.3.2 Spring Point Matrix Piping systems generally are supported at several locations, restricting motion partially or totally, or they may be placed on an elastic founda- tion. The elastic foundation can be represented by springs. Each spring can be modeled as a point matrix. Chapter 4 Numerical Simulation 76 Suppose the pipe reach in Figure 4.7a has a spring support and is vibrating in the y-z plane and in the axial direction z. The state vectors to the right and left of point i, II; and 21;, can again be related by a point matrix. The lateral displacement, rotation, moment and axial variables are continuous over point 1. But because of the spring restoring force, a discontinuity occurs in the shear force. When the spring is deflected by.an amount Uy’ the restoring force is kin, where k1 is the stiffness of the spring (Fig. 4.7b). The relations between the state vector elements to the left and right of the spring are then 0: - U: (4.40s) PR - PL (4.40b) (vR - 0:) - (vL - U2) (4.40c) F: - F: (4.40d) U; - U; (4.40e) wfi - w; (4.40f) M: - M: (4.40g) F; - F; - kin; (4.40h) Chapter 4 Numerical Simulation 77 Here the subscript i of the dependent variables has been dropped for clarity. In matrix notation, and in non-dimensional form the equations become: rU ‘R _ . rU ‘L 3‘ 1 o o o o o o o 31 3— o 1 o o o o o o 2- x* K* 1 .Y. 1 o o 1 o o o o o 2 F F -‘— o o o 1 o o o o -5— APE APE < U > - < U + (4.41) :1 o o o o 1 o o o I! tx o o o o o 1 o o wx 2.5 1.: El 0 o o o o o 1 0 E1? F 23 k 23 22 _x__ __i. _x__ RI 0 ° ° 0 HI 0 ° 1 1 HI . 9 J1 . p - L .1 or 2% - [ri112§ (4.42) 4.2.3.3 Mass Point Matrix Valves, accumulators and control instrumentation can be modeled as concentrated or point masses. For example, consider the pipe reach in Figure 4.8a which has a valve of mass 1111 in the mid-span. Assume that the radius of gyration of the mass of the valve is zero about the x-axis and that the system is vibrating in the y-z plane as well as in the Chapter 4 Numerical Simulation 78 axial direction 2. The point mass matrix connecting Z? and Zi‘ is found by noting that the lateral displacement, rotation and moment as well as the axial variables are continuous across m1,so that U: - U: (4.43a) PR - PL (4.43b) (vR - 0:) - (vL - vi) (4.43c) U? - U; , (4.43d) w: - w: (4.43e) Mi - M: (4.43f) An inertia force causes a discontinuity in the shear and axial force due to the vibrating mass. Equilibrium considerations in the free-body diagram shown in Figure 4.8b yield 2 F; - F; - miw U; (4.433) 2 FR - FL - m UL (4.43h) z z 2 In non-dimensional matrix notation Equations 4.43a through 4.43h become Chapter 4 Numerical Simulation 79 'U ‘R p « ’U ‘L 31 1 o o o o 0 <> 0 31 2— o 1 o o o o o o 3- K* x* X E 2 02 o 1 o o o o o 1 F mwl F 4L. __i__ .JL ‘AE ‘AE o o 1. o o c) 0 ‘AE p p p + U > - 1 U L (4.44) I! o o o o 1 o o o 31 2x 0 o o o o 1 o 0 ex M2 M2 -3- o o o o o o 1 o -3— E1 E1 p p F 22 m «.1228 F 22 J— o 0 0 o__1.__o o 1 L— E1 E1 1 E1 Lin b P deJ1 or 2% - [PE]12§ (4.45) 4.2.4 Overall Transfer Matrix The overall transfer matrix relates the state vector at one end of a system to that at the other end. The matrix is obtained by an ordered multiplication of all the intermediate field and point matrices [7]. Once the field and point matrices for each subsystem have been obtained, three steps are necessary to form the overall transfer matrix. The first step consists of rearranging the terms of the matrices. The second step transforms the matrices from a local to a global coordinate system by using transformation matrices. The final step is the ordered. multiplication of the transformed matrices. The first two steps are considered next. Chapter 4 ' Numerical Simulation 80 4.2.4.1 Overall Transfer Matrix Rearrangement The state vector in Equation 4.36 has fourteen elements corresponding to fourteen dependent variables that are necessary to describe a liquid- filled piping system in a three dimensional space. Assume that the piping system is in the y-z plane. This piping set-up allows motion in the axial direction, 2 and in the y-z plane. The state vectors and transfer matrix are given by: (4.46a) or 21 - [11121.1 (4.46b) where [Tfp] and [Tyz] are given in Equations 4.20 and 4.28, respec- tively. The state vector 21 is given by: U U F 1 {15355: skis-41L} «mm X p p p 1 In order to keep the variables subjected to continuity and equilibrium conditions separate from one another [26,27,50], the state vector is rearranged Chapter 4 Numerical Simulation 81 UU 1421:2er z_£Wx_2_z!Js__y___z (443) 1 K* 2 2 2 EIp 31p APE ‘ To have the transfer matrix compatible with the state vector in Equation 4.48, the columns and rows of the matrix in Equation 4.46a must be rearranged. Comparing the state vectors in the two previous equations * one can see, for example, that the second element, P/K , is now the first and the sixth element, fix, is now the second. In a similar manner the transfer matrix is rearranged. Row and column shifting can be achieved in two steps. First, the columns are rearranged by postmul- tiplying the transfer matrix by a square matrix [to]. Then, rearrangement of rows is accomplished by premultiplying by [tR] so that: [IL] - [5,] [r] [to] (4.49) where [tR] is given by: ‘I l (4.50) Ital - OOOOD—‘OOO OOOOOOOH OOOHOOOO HOOOOOOO COOOOHOO OOOOOOHO OOHOOOOO OHOOOOOO I l and [cc] - [ck]t (4.51) ‘Chapter 4 Numerical Simulation 82 The final field transfer matrix, in local coordinates, is given in the equation below: F ‘ f f f f q r w 2; Tag 0 0 T2? Tag 0 0 T2? 2; K K 2 2 ix 0 1;"; TX? 0 0 IX, 1);, 0 ix U U f o 11’? Ti"? 0 o 131'? 11'? o 31 U U f f f f 3‘ TI? 0 o 1.? 1,1; 0 0 n? 3‘ 1 32; »- 1f? 0 o '15? TE? 0 o if? < % L (4.52) in: o Tyz T3" 0 o Tyz 1‘2" 0 535 32 31 33 C BIB 31? 2 5i. yz yz yz yz 5i 0 T42 T41 0 0 T43 T44 0 El E1 p P 1'.- F {‘3 If? 0 0 15? TE? 0 0 1'59 fig L P a 1 L ‘ 1 L P ‘ 1-1 or where the elements of the matrix [TL], ng and Ti: are given in Equations 4.20 and 4.28, respectively. The subscripts R and C in each element of [TL] refer to rows and columns. The procedure described can be generalized to a three-dimensional space. The size of the overall transfer. matrix will be fourteen by fourteen. The state vector will be composed of three rotations, displacements, moments and forces, in addition to the liquid pressure and displacement. Chapter 4 Numerical Simulation 83 4.2.4.2 Coordinate Transformation State vectors and field and point matrices are developed based on the local coordinate system, x,y,z, associated with each particular subsys- tem. In order to relate the state vectors at the end of a pipe reach to those at the end of an adjacent pipe, the state vectors are expressed with respect to a set of global coordinates, X,Y,Z, fixed with respect to the piping system. The state vector 21. with respect to the local coordinates, x,y,z, and the state vector 26 with respect to the global coordinates, X,Y,Z, are related by the transformation matrix It]: 2L - [1:126 (4.54) where U 5215 F22 F T _ 2 ‘1' .1 .4 2 .1. _z ZL. {x* x 2 2 2 El E1 AE} (“553) p p p ,_ 2 4911411111253" (455,, c 18" 2 2 2 up up APE ‘ To obtain the transformation matrix, the orientation of the pipe reach with respect to the global axis must be defined. The orientation is Chapter 4 Numerical Simulation 84 defined by using the direction cosines of the local axis with respect to the global axis. This is expressed in matrix form as: x cos(¢pu) coup”) cos(cp13) X y - cos<¢21> com") cos(¢2s) 32! (4.56) 2 cos (w; 1) COS (9: 2) cos (‘P3 3) where (p p q is the angle between the local and global axes. The subscript p refers to the local axis, whereas q refers to the global axis. This is shown in Figure 4.9. As an example, Figure 4.10 shows a pipe reach in both the local and global coordinate systems. The local coordinates have been rotated about the X-axis, which is perpendicular to the plane of the paper. The local y-axis and global Z-axis coincide. The follow- ing relations can be observed: X Y (4.57) 2 or 2L- it] 2c (4.58) The transformation matrix for this example is: Chapter 4 1 Numerical Simulation 8S ‘ ' F F2; 1 o o o o o o o 1’; l K K wx o 1 o o o o o o ”x U U f o o o 1 o o o o f1 U U f o o -1 o o o o o 32 4 %'> - 0 0 0 0 1 0 O 0 < %’ > (4.59) M 2 2 J‘- o o o o o 1 o o 555- EI 31 p P F 22 F 12 l— o o o o o o o 1 3— E1 E1 p P F F 212'? o o o o o o -1 o XZE L p JL L 4L P JG The relation between the local state vector at locations 1 and i-l is: {2L}i - [Ti]1{zL}i-1 (4°60) Combining Equations 4.54 and 4.60 and remembering that [t]-11- [t]t the following relation is obtained t (2C), - [t1 [T111I‘1‘Zc’1-1 (4.61a) or (zb); - II¢I}(ZC)§_1 (4.61b) and [T411 - [tlt [Tililt] (4.61c) Chapter 4 Numerical Simulation 86 where [TG]1 is the field transfer matrix in global coordinates relating the state vectors at the two ends of a pipe reach. The state vectors are defined with respect to the global coordinate system. Similar procedures for rearranging and transforming the coordinates from local into global can be applied to the point matrices defined in Equations 4.38, 4.41 and 4.44. The overall transfer matrix for the system shown in Figure 4.4 between points 7 and S is given by: [U] - [Pg]7ITG]7[P§]5[IG]6IP31SITGJS... (4.62) The number of columns of the overall matrix may be reduced according to the boundary conditions at one end of the system or increased, in the case of a rigid support, to account for the new unknown introduced by the intermediate boundary condition. These special conditions will be defined in the next section. 4.2.5 Boundary and Intermediate Conditions Boundary and intermediate conditions are restrictions imposed on the system affecting the degrees of freedom of the pipe or liquid. The natural frequencies of the system are dependent upon these conditions. Chapter 4 Numerical Simulation 87 4 . 2 . 5 . 1 Boundary Conditions Figure 4.11 shows a piping system vibrating in both the axial direction 2 and in the Y-Z plane. The boundary conditions at location 0 are a reservoir of constant pressure for the liquid and restrictions on rota- tions and displacements of the pipe. The reservoir represents an open— end condition in the liquid, whereas the restriction of pipe motion is a fixed-end condition. Since the reservoir level is constant, a pressure node always exists at this end. This boundary will be referred to as an open-fixed end condition. The state vector, in global coordinates, at this end is given by: rP ‘ r0 1 ix 0 UY 0 U2 0 Jv» - 4v» (4.63) FY FY LFZJO LFZJO At location 1 in Fig. 4.13 there is a closed valve and a release (for example, a flexible hose of negligible length) that allow pipe motion. Moment and forces in the pipe vanish due to the release. A liquid pressure antinode exists at this end due to the presence of the closed valve. Assuming that the release does not affect the fluid pressure and that the valve is rigid so that fluid displacement is not allowed, the state vector at this closed-free end condition is given by: Chapter 4 Numerical Simulation 88 ’1) ‘ FP ) “'11 'x UY UY U2 U2 «v > - (o > (4.64) MK 0 FY o F o L ZJ 1 L J 1 4.2.5.2 Intermediate Conditions Rigid supports and external excitations of the piping are the inter- mediate conditions studied in this research. Figure 4.12 shows these two intermediate conditions. Lesmez, et a1. [66] presented a descrip- tion of these conditions. 1) Rigid Supports A rigid support restricts all motion of piping at a given location. This support condition may be represented by increasing the number of columns of the transfer matrix. The increase in the number of columns accounts for the reaction at the location and increases the number of unknowns. This boundary condition can be illustrated by a simple ex- ample. The clamped-clamped beam in Figure 4.12a has a support at location 2. Let T1 and T2 be the field transfer matrices between points 1 and 2, and 2 and 3, respectively. The matrices T1 and T2 can be written as follows: Chapter 4 Numerical Simulation 89 T111 T112 T113 T116 T211 T212 T213 T214 T121 T122 T123 T124 T221 T222 T223 T224 [Tl] - T131 T132 T133 Tls‘ , [T2] - T231 T232 T233 T234 (4.65) T141 T142 T143 T144 T241 T242 T243 T244 The state vector for shear and bending in the Y-Z plane is given by: z-(Uw F}T (466) YXMXY ° Applying the fixed boundary conditions UY - 0, Wx - 0 at l, the first and second columns of the field transfer matrix [T1] are dropped and the state vector becomes { Mx FY )T. The rigid support at point 2 causes a discontinuity in the shear force according to the relation L - FY + Q (4.67) This restraint also introduces the relation - O - T113Mx1+ TII‘FY (4.68) U Y 1 2 In order to introduce the reaction Q, a column is added in the field transfer matrix. It has unit value in the row corresponding to the discontinuity, as shown in the equation below. Chapter 4 Numerical Simulation R. 11,, T1,, UY 0 w 11,, 11,, 0 fix -- T133 Tls‘ O Y (4.69) Y 2 Tl‘s Tl“ 1 Q 1 Point 3 is reached by multiplication of the modified matrix T1 by the field transfer matrix [T2]: T211 T212 T213 T214 ~ T113 T114 H000 [I] - T231 T232 T233 TZS‘ T133 Tls‘ (4.70) T2“ T2“ T2“ T2“ T1“ T1“ Equation 4.66 can be added to the overall transfer matrix: (UY‘ PU“ 012 U1: . ‘1' U21 ":2 U2: “X1 1 _ U31 U32 Us: 4 1 U U FY ( .71) y n 42 cs 1 0 J3 T1,, T1“ 0 Q or 23 .. [a] 21 (4.72) where the state vector at location 1 includes the reaction at location 2 and the state vector at location 3 has now five elements .. 2) External Force A static or harmonic force can be represented by an extended state vector and an extended transfer matrix. Figure 4.12b shows the same beam as the previous figure, but now at point 2 a harmonic force of Chapter 4 Numerical Simulation 91 amplitude Q and frequency 0 is being applied. The force at location 2 is given by the relation F? _ FL 1 Q (4.73) This equation is introduced in the transfer matrix by adding one row and one column to matrix [T1] in the following manner:‘ UY R '11,, '11,, o w 11,, 11,, o :5, ' 1:: T134 0 Y (4.74) Y 1,, '11,, Q 1 1 l 2 0 0 1 where one more equation has been added: 1 - l. The overall extended transfer matrix is given by the product of the matrix in Equation 4.74 and that in [T2] . The final matrix equation that expresses the state vector at 3 is: U U11 U12 Q1 WY 321 U22 32 _ si 32 a 4.75 gfi U41 U42 Q4 {1Y}1 ( ) 1Y 3 _ o o 1 . or 23 - [u] 21 (4.76) where the Q1, Q2, Q3 and Q, represent the forcing terms. Chapter 4 Numerical Simulation 92 4.2.6 Natural Frequencies The natural frequencies of liquid-filled piping systems are important in the design process. Let [U] and 2 be the global transfer matrix and state vector of the single pipe reach vibrating in the Y-Z plane shown in Figure 4.11. The global transfer matrix relates the state vectors at the end points 21 - [U] 20 (4.77) The natural frequencies of this system depend on the boundary conditions described in Equations 4.63 and 4.64. Because some of the variables are zero at the boundaries, the number of elements of the state vectors at locations 0 and l is reduced. Therefore, the order of the global trans- fer matrix is also reduced. For example, the plane vibration of the pipe reach has eight variables. Four variables are known at each end reducing the number of elements of the state vector to four. This can be represented by 0 055 U56 U57 Us: V . 0 nos 66 U67 us: fix 0 ' 15 76 17 U1: y (4-73) 0 1 as so 31 Use F2 0 or (011- [01120) (4.79) Chapter 4 Numerical S imulat ion 93 where the vector 2 represents the non-zero variables and the order of the matrix [II] has been reduced by the boundary conditions that are zero. To have a non-trivial solution the determinant of the reduced matrix must be zero A-|[U]| (4.80) This generates an equation for the circular natural frequency m. In practice [50] , the procedure adopted is to choose certain values for w and compute the corresponding values of the frequency determinant 11(4)). The values of the determinant are then plotted against the frequency w. The values of m at which the determinant equals zero are the natural frequencies of the system. 4 . 2 . 7 Mode Shapes After the natural frequencies have been determined, the mode shapes of an elastic system can be found in terms of one variable. For example, assume that the fluid displacement at location 0 in Figure 4.11 has a unit amplitude. Equation 4.78 then becomes U55 U56 U81 U53 U65 U66 U67 Use ' U75 U16 U71 U" 1 Us; so U31 Us: (4.81) 000 0 («HAEI “ Chapter 4 Numerical Simulation 94 where the partitioning corresponds to the unknown variables. Once the variables at 0 are known, the variables at specified locations of the pipe reach may be found. This is accomplished by evaluating the trans- fer matrix at the natural frequency and at the given location. Then, the transfer matrix is multiplied by the state vector at the initial location. This is represented by r? ‘ [T15 T16 T11 T18. wx T25 T23 T21 T23 ' ‘ UY T35 Tao T31 T33 V U2 T45 T46 T41 T43 ”X 1V * - T55 T55 T51 T58 1FY> (4.82) "X T65 T66 T87 Too Fz FY T15 T76 T11 T73 L .0 (F2) 1 st5 Tu Tu Tu‘ where i represents a location along the pipe reach. This procedure is followed until the other boundary is reached. 4 . 2 . 8 Frequency Response The transfer matrix method can be used to determine the frequency response of systems having one or more periodic forcing functions. Chaudhry [7] describes a method to determine the frequency response of these systems. The extended matrix and state vector concepts and the method of superposition can be used to find the total response. The response of the system to each forcing function is evaluated and the results are then superimposed to determine the total response of the Chapter 4 Numerical Simulation 95 system. Determining the response for a forcing function of frequency 0 involves evaluating the global transfer matrix at that frequency. Then, the response of the dependent variable such as forces and displacements at the starting point are found by following a similar procedure to the one used to compute the mode shapes. For example, the moment and force of the system shown in Figure 4.12b and represented in Equation 4.75, become U U Q 0 11 12 1 0 - U21 U22 Q2 :XY (4.33) 1 3 0 0 1 1 1 Once the conditions at location 1 (Figure 4.12b) are known the response at the desired locations can be found by evaluating the transfer matrix from the starting point up to that location. This procedure is followed for each frequency of the frequency range at a specified frequency interval. 4.3 W The transfer matrix method is compared in this section with the method of characteristics (MOC) [5,6,7], the component synthesis method (CSM) [34,35,36] , and with experimental data available in the literature. Two piping systems are used to make the comparisons. The first system is a one-dimensional liquid-filled pipeline with a reservoir that has a closed end, either free or fixed. The second system is an L-shaped pipe Chapter 4 Numerical Simulation 96 connected to a reservoir and free at the other end. The L-shaped pipe was tested at the David W. Taylor Naval Research and Development Center in Maryland [24] . This pipe has been extensively studied. The previous modeling efforts included direct analytical solution of simultaneous differential equations [24], component synthesis using finite element discretization (Nastran) for the structural elements [37] and finite element analysis in three dimensions [86] . 4. 3. 1 Method of Characteristics The method of characteristics has been widely used to estimate the response of liquid systems to transient events. A description of the method is presented by Wylie and Streeter [6] and Chaudhry [7] . The MOC has also been used to model waves propagating in beams including rotary inertia and shear deformation [89] . Numerical difficulties in modeling beams have made the MOC unattractive in modeling liquid-filled pipings for plane vibration [65] . This method was used by Otwell [l9] , Wiggert et a1. [20] , and Budny [22] to compare experimental data in which fluid- structure interaction was allowed. Budny [22] incorporated damping in both the pipe and the liquid. His model is used here to predict the pressure response of the one-dimensional pipe with no energy dissipa- tion. Table 4.1 shows the characteristics of a straight copper pipe filled with water . Chapter 4 Numerical S imulation 97 m1 PROPERTIES OF STRAIGHT LIQUID-FILLED PIPE Pipe ' Liquid Property Value Property Value Young's Modulus 117 GPa 3 Bulk Modulus 2.2 Gpa 3 Density 8940 kg/m Density 1000 kg/m Poisson's Ratio 0.45 Inside Radius 13 mm Thickness 1.2 mm Wave Speed 3744 m/s Wave Speed 1248 m/s Boundary Conditions Case a Fixed-Fixed Open-Closed Case b Fixed-Free Open-Closed The Poisson's ratio has been adjusted to 0.45 so that the ratio of the coupled wave speeds is three. Taking the pipe wave speed as a mutiple of the liquid avoids any interpolations that may introduce numerical errors in the MOC [6,90]. The relative displacement as well as the net force must be zero at the free end condition as described by Budny [22]. Inertia forces are not included at the free end. A sinusoidal function applied to the liquid at the open-end is the source of excitation in both cases. A fast Fourier transform (FFT) analysis of the time history generated by the MOC is performed to obtain the frequency response. The pressure amplitude response at the closed end is plotted in Figure 4.13. The natural frequencies are the same regardless of the method of com- putation. Table 4.2 shows the natural frequencies for both cases. The results for Case a are shown in Figure 4.13a. Case b shows that when the pipe wall is free in the axial direction, the third liquid and first pipe natural frequencies coincides at 12 Hz. The frequencies of both Chapter 4 Numerical Simulation 98 the liquid and pipe are split apart to 9.0 and 14.4 Hz. The same result occurs at the ninth liquid harmonic which coincide with the second natural frequency of the axial pipe wall at 36 Hz. Figure 4.13b shows this result . was RAMA]. WES ma STRAIGHT PIPE Case a Case b Frequency Type Harmonic Frequency Type Harmonic - (HZ) (Hz) 4.0 F l 4.0 F 1 11.7 F 3 9.0 P 1 19.4 F 5 14.4 F 3 23.4 P 1 19.4 F 5 27.4 F 7 27.3 F 7 35.1 F 9 32.4 P 2 42.8 F 11 37.8 F 9 46.8 P 2 42.8 F 11 50.8 F 13 50.7 F 13 F - Fluid, P - Pipe 4.3.2 Component Synthesis Method and Experimental Data The second comparison of the transfer matrix method is with the com- ponent synthesis method and experimental data. Hatfield et a1. [37] devised the component synthesis approach which is an extension of the modal synthesis technique. [36] for dynamic analysis of structures. The L-shaped pipe in Figure 4.14 was used to validate the CSM. Table 4.3 describes the properties of the pipe, 70% copper and 30% nickel and filled with oil [24] . Chapter 4 Numerical Simulation 99 MAJ PROPERTIES OF LPSHAPED LIQUID-FILLED PIPE Pipe Liquid Property Value Property Value Young's Modulus 157 GPa3 Bulk Modulus 2 Gpa 3 Density 9000 kg/m Density 872 kg/m Poisson's Ratio 0.34 Outside Diameter 114 mm Inside Diameter 102 mm Radius of Bend 102 mm Sound Speed In-Situ 1372 m/s Boundary Conditions Fixed-Free Closed-Open A correction for the flexural stiffness, E1, of the bend was used for both numerical methods. A curved pipe subject to bending is less stiff' than would be indicated by elementary theory of bending [87]. The correction formula developed by von Harman [88] is 2 n _ W (4.34) 10 + 12(eR/r ) where n is the correction factor for the flexural stiffness, R is the radius of the bend, r is the inside radius of the pipe and e is the thickness of the pipe. The CSM does not include the Poisson's coupling; because in the computation of the normal modes of vibration of the pipe the interaction with the liquid is not considered. However, the mass of the contained liquid is included as part of the total mass. The first lobar mode of the cross section is 850 Hz. Chapter 4 Numerical Simulation 100 The pipe frequencies for the CSM are computed by using the finite ele- ment program Nastran. Ten modes and 25 elements are used.in the computations. Seven pipe reaches are used to model the pipe by using the transfer matrix method. The 90 bend is modeled using three pipe reaches each with a relative change of 30. in orientation. The liquid in the pipe was excited by a harmonic oscillator. Neither structural nor fluid damping are considered in the computations. The liquid is free to move at the free end. The mobility (ratio of velocity over force) of the liquid at the excitation point and free end, as well as the mobility of the pipe in the Y and 2 directions are shown in Figure 4.15. Discrepancies in predicted and measured responses in the vicinity of the amplitude peaks are because damping was not included in the computed analysis. In non-resonant frequency ranges, differences in the responses predicted by the two analyses are minor compared to the devia- tions of both predictions from observed responses, particularly in the structural responses (Figures 4.15c and 4.15d). The discrepancy in the structural mobility may be caused by the flexibility factor which is significant in accurately modeling elbows [86] . Chapter 4 Numerical Simulation 101 R L R zi-1 Zi—1 ZI ZI zi+1 zI+1 } i 3 l k: I ‘ I kaI-H: l k,_1 : i-1 | ”11.1 mi mi+1 ZI-I Zi zi+1 IZ P-1 Zii 12 i """"'<)--\/~V/\/A‘--C>--'-"' R l- L fi-1 fi 1‘; fR i a) Spring i b) Moss i Figure 4.2 Free Body Diagrams of Spring and Mass Systems I I k1 m1 .23-— l I , I | I I I 1 Figure 4.3 Simple Spring-Mass System Chapter 4 Numerical Simulation 102 '||<1 Figure 4.4 General Piping System ~01 0* H I———z Figure 4.5 General Straight Liquid-Filled Pipe Reach Numerical Simulation Chapter 4 103 scan as euseaooeaaeua use neouoh M.e shaman 35500235 3 mootou Ac .\..\ \. U a. l»- . f \..>\ , a ._ u N 2.4.8.. ._ a. «so so: 4 a .3 (an. Chapter 4 Numerical Simulation 104 L ._ FR Ski Sn 1 I. R Fyi Fyi Shflnglflmflnflhglfiwoe ““04 a) Liquid-Filled Pipe b) Free Body Diagram With Spring Figure 4.7 Forces at Spring L Fyi I. I-1 ' I+1 g“ 21 Z, m i L R Fyi i Fyi m. C02 I I a) Liquid-Filled Pipe b) Free Body Diagram With a Point Moss Figure 4.8 Forces at Concentrated Mass Chapter 4 ’ - Numerical Simulation 105 Figure 4.9 Definition of Local and Global Axes Y Figure 4.10 Coordinate Transformation of Straight Pipe Reach Numerical Simulation Chapter 4 106 Y g L 0 Z 1 21% I 0 1 Figure 4.11 Boundary Conditions Y 1 3 | [T11 2 [12]] I g! 77); I52 1 3 0) Rigid Support Y 1 3 [T1] 2 [12] l | 21 IE 2 3 1 Q*sin(0t) b) Harmonic Force Figure 4.12 Intermediate Boundary Conditions Chapter 4 .p I Preeeure Amplitude no I A Numerical Simulation 107 a) Coee a. Preeeure at Fixed-Cloud end .p L Preeeure Amplitude N . l b) Ceee b. Preeeure et Free-Cloud end I1 l Tronefer Matrix ”/0! -—--—-—- Method of Characterietice Figure h.13 Pressure Amplitude Response for Straight Pipe Reach Chapter 4 Numerical Simulation 108 2 7 Y E * Z 6— 3 0 go 1 + "“5 ¢ | 1 3 4 ¢ ¢——— 0.914m ——-f / Figure 4.14 L-Shaped Liquid-Filled Pipe Chapter 4 Numerical Simulation 109 a) Amplitude of mobility of liquid at O I I I I l I A O C 4'. L < "0e. . :5 km a O a: In a -60 | I i I I -ao ' a a . 20 50 100 200 300 500 Frequency (Hz) b) Amplitude oi mobility of liquid at 7 20 - l I g I o ' i I. " i e I I I . 9. -20 } .eeee..."....... e a .:'P-m .0 ‘ . 3 ~40 t -—-- ’ a: l l a l _ I I I ~50 1 » 20 50 100 200 300 500 - Frequency (Hz) Experimental —--—--— Tranefer Matrix -------------- Component Syntheeie Figure 4.15 Mobility Diagrams for L-Shaped Pipe Chapter 4 Numerical Simulation 110 C) Amplitude of mobility of pipe at 7. in Y direction 10 ‘ i ’ I ~10 A .1 I II .5 V ..‘L.. a 5 < J\::i0eeeee..... : - o 1 . :~~‘--1_ o -70 -90 i 20 100 200 300 500 Frequency (Hz) d) Amplitude of mobility of pipe at 7. in 2 direction 10 ‘ ‘ I ,I -10 '1] | . I “I? \2- ' 1. e -30 "nu-ah \ . .5 32 -50 In a -70 -90 20 Figure 4.15 50 100 200 300 500 Frequency (Hz) Experimental —---—--- Tranefer Matrix Component Syntheeie (Continuation) Chapter 5 Experimental Apparatus 51W The analytical model derived in Chapter 3 and the methodology described in Chapter 4 incorporate the flexural and torsional modes of vibration into an existing coupled axial pipe wall andliquid model. This model represents five families of waves. Four of the waves propagate in the pipe wall and one in the liquid. The previous section compared the frequency responses of the proposed model to the method of characteris- tics and the component synthesis method. The model was also compared with experimental results available in the literature. This chapter describes the experimental apparatus that was designed and built to validate the proposed model. The comparisons of the previous chapter validate the model only partially because the excitation was applied to the liquid column and the natural frequencies of the liquid were unchanged. The experimental apparatus is designed to excite the natural frequencies of a piping system. The excitation is harmonic. lll Chapter 5 Experimental Apparatus 112 The experimental apparatus was located in the basement of the Engineering Building on the campus of Michigan State University. The piping system, the experimental procedure and the sources of experimen- tal error are described in this chapter. Appendix B describes the data acquisition equipment utilized, including the hardware and software. 5.2 W This section examines the components considered in the design of the apparatus. These components will be discussed in reference to either the pipe or the contained liquid. Liquid medium, constant pressure reservoirs and a valve are the components related to the liquid. Pipe material, pipe supports and the external shaker are associated with the pipe. The final component of the experimental apparatus is the data acquisition equipment. Figure 5.1 shows the piping system set-up. 5.2.1 Liquid Components 5.2.1.1 Liquid The liquid used in the experiments was water from the university water supply system. Table 5.1 [78,79] lists the physical properties of the water . Chapter 5 Experimental Apparatus 113 m1. PHYSICAL HOMES 0F LIQUID Property 1 Value Temperature 25.0 'C 77. °F Bulk Modulus (K) 2.2 GPa s 320. kpsi 3 Density (p) 997.0 kg/m 1.93 slugs/ft 5.2.1.2 Constant Pressure Reservoirs The upstream and downstream reservoirs each consist of two 454 liter vertical F.E. Myers Model V1206 l48OOC8 well tanks. Each tank is rated for 517 kPa (75 Psi). The set-up of the tanks is shown in Figure 5.2a. A one inch U.S. nominal diameter pipe connects the bases of the pair of tanks, allowing them to act as a single reservoir. The tanks are pressurized with Engineering Building air supply that has a maximum pressure of approximately 650 kPa (94 Psi). The air pressure allows a constant liquid pressure at point C as shown in Figure 5.1. The air supply passes through a Schrader Model 3564-2000 pressure regulator, and is directed through an orifice on top of the tanks with a common header. The header is used to maintain equal air pressure in both tanks . The tanks are filled with water through a hose connection on the one inch nominal diameter transfer line at the base of the tanks. This transfer line is connected to the one inch diameter pipe at the base of each reservoir and has a shutoff valve at each end. In addition to Chapter 5 Experimental Apparatus 114 allowing filling and emptying of the tanks, this transfer line is used to transfer water between the reservoirs. Water level in the tank is monitored using a sight glass connected to the orifices on the side of one of the tanks in each reservoir. These tanks are also used to purge the air from the system. The purging procedure described by Budny [22] was used. The open boundary condition defined in section 4.2.5.1 is simulated by the set of tanks at one end of the piping system. 5.2.1.3 Valve A fast closing valve, described by Budny [22], was placed at one end of the piping system, as shown at point A in Figure 5.1. Figure 5.2b shows two views of the valve. For the purposes of this study, the valve was kept in the closed position simulating a dead-end [6,7] . 5 . 2 . 2 Pipe Components 5.2.2.1 Pipe Material The pipeline used is a one inch U.S. nominal diameter type L copper pipe with standard soldered fittings. Unions are installed at intervals of approximately 7.3 meters (24 ft.) along the pipeline to allow for changes of the total pipe length. Figure 5.1 shows the pipeline set- up. Table 5.2 [78,80,811 lists the physical properties of the piping sys tem . Chapter 5 Experimental Apparatus 115 MILL} PHYSICAL PROPERTIES-0F PIPING SYSTD! Property Value Young' 8 Modulus (E) 117 . 0 GPa 17 . 00 Mpsi s s Dens 1cy* (p) 8900.0 Kg/m 17.30 slugs/ft Inside Radius (r) 13.0 mm 0.51 in Outside Radius 14 . 3 mm 0 . 56 in Thickness (e) l . 3 mm 0 . 05 in Poisson's Ratio (u) 0.35 * Determined by water displacement [22] 5.2.2.2 Pipe Supports Unistrut model P2031 pipe clamps mounted approximately 3.7 meters (12 ft.) apart are used to provide support of the pipe to the wall. In addition, rigid supports are connected to the piping at each elbow to eliminate axial motion . Each rigid support is an aluminum block bolted to the wall. Each block has a hole matching the CD of the pipe drilled in its center. Each block is cut in half through this hole and bolt holes are drilled through both sections. Bolts are then used to hold the two pieces together enabling the block to act as a vise, squeezing the pipe around its entire circumference. The entire support is then bolted to the wall, using 3/8 inch Red Head anchor bolts. Rope hangers are used to hold the piping at locations where neither one of the above supports are placed. The locations of the supports are Chapter 5 ' Experimental Apparatus 116 displayed in Figure 5.1. The Unistrut supports may be replaced by rigid supports to vary the length of the piping. The procedure to change these supports is discussed in section 5.2.3. 5 . 2 . 2 . 3 External Shaker The excitation of the pipe is induced by a reciprocating force produced by an external shaker. The shaker is a crank-slider mechanism that transfers rotary motion to reciprocating motion. Top and side views of the mechanism are shown in Figure 5.3. Figure 5.4 shows a sketch of the same mechanism. The mechanism consists of five major elements: a motor, crank, connecting rod, linear bearing structure and connecting spring. These elements will be described next. The motor is a Dayton model 42140, permanent magnet DC variable-speed motor with a Dayton SCR control which allows changes of the rotational speed. The rated frequency and torque of the motor are 1800 RPM and 0.5 N-m (4.38 in-lb), respectively. The motor is mounted on a structure made of 102 mm (4 inch) L-shaped steel bars bolted to the wall. This structure prevents any vibration that may interfere with the experi- ments. A flywheel is attached to the end of the shaft of the motor. Bolted to the flywheel is an aluminum disk. A distance of 1.5 mm be- tween the center of the aluminum disk and the center of the flywheel forms the crank of the slider mechanism. A 1/2 inch Heim Unibal Spherical Rod End Bearing is joined to the aluminum disk by a 6.4 mm screw. This joint is a pin-type connection, as shown at point C in Chapter 5 Experimental Apparatus 117 Figure 5.4. The pin connection, which follows a circular path, trans- fers the rotary motion of the crank to the spherical bearing. The spherical bearing is screwed to a one inch diameter aluminum rod. At the other end of the rod there is a similar spherical bearing that connects to a linear bearing structure by a wrist-type connection. The motion at point P in Figure 5.4 oscillates along a linear path. The transfer of rotary motion to reciprocating motion occurs along the aluminum rod, whose points follow elliptical paths. The total length of the link is 880 mm (Figure 5.3). Two pairs of 1/4 inch linear self- aligning, super—ball bushing bearings slide on two stainless steel rods, simulating piston-type motion of the linear bearing structure. The static coefficient of friction is 0.2% [82]. These rods are supported on acrylic blocks which are glued to an acrylic base. The base is mounted on a L-shaped steel bar bolted to the wall. A 1/2 inch steel rod anchored to the bearing structure, in the same plane as the link, connects to the spring. Table 5.3 gives the technical information concerning these shaker components. 5.2.2.4 Spring The reciprocating motion of the linear bearing structure is transmitted to the pipe by a round-wire helical compression spring. The spring allows a linear relationship between the reciprocating displacement at the bearing structure and the force that is transmitted to the pipe. The technical information regarding the spring is shown in Table 5.4 [13]. Chapter 5 Experimental Apparatus 118 mm SHAKER COMPONENTS Component Material Diameter Thickness Length Mass (an) (Inn) (M) (3m) Motor Shaft 12.7 Flywheel Steel 127.0 32.0 2980 Disk Aluminum 61.0 22.9 150 Crank Off-Center 6.4 1.5 Heim Uniball Spherical Rod End Bearing 12.7 76.2 Rod Aluminum 25.4 825.0 Link 880.0* 1350 Linear Bearings 6.4 20.3 Linear Bearing Structure 1510 Rod Connector Steel 12.7 140.0 * Total distance between point C and P in Figure 5.4. (See Figure 5.3) m SPRING PROPERTIES Property value Material Hard-drawn steel wire, zinc plated* Mass (mg) 20.3 gm Spring Constant (kg) 7.0 kN/m 40 lb/in* Modulus of Rigidity (G) 79.3 Gpa 11.5 Mpsi Mean Spring Diameter (D) 15.9 mm 0.63 in Wire Diameter (d) 2.4 mm 0.09 in Active Coils (N) 11* Natural Frequency (fs) 290.0 Hz Spring Ends Both ends squared * Obtained from the manufacturer Chapter 5 Experimental Apparatus 119 The spring is connected to the bearing structure by a 1/2 inch steel rod. A screw rigidly connects the spring to the rod. Another screw connects the other end of the spring to an acrylic collar that embraces the copper pipe, as shown in Figure 5.3. These connections reduce the number of active coils to seven, thereby increasing the stiffness of the spring to 11.7 kN/m (67 lb/in). This value was obtained from Equation 5.1 [13]. k - -4—9— (5.1) where k8 is the spring constant and D and d are the mean spring diameter and wire diameter, respectively. The modulus of rigidity is G and the number of active coils is represented by N. The reduction of the number of active coils also increases the natural frequency of the spring to 460 Hz. The natural frequency of the spring can be obtained from Equation 5. 2 , [13] fs - % [ks/ms] a (5.2) where ms represents the active mass of the spring. Chapter 5 Experimental Apparatus 120 5 . 2 . 3 Experimental Configurations The general piping set-up in Figure 5.1 will be used to describe the pipe configurations. Each pipe configuration varies in total length, therefore, the frequency of the liquid varies. A U-type bend is placed between the valve and the reservoir. The legs of the U-bend are 1.83 nI (6 ft) each. The Unistrut and rigid supports restrict the motion of the pipe whereas the U-bend is free to move in the Y-Z plane. The external shaker is attached at the mid-point of the vertical leg of the U-bend, location D. The elbows of the U-bend are reinforced with a steel plate. Brass blocks are soldered to the copper pipe at both sides of'each elbow. The steel plates then are screwed to the brass blocks. The steel plates, screws and the brass blocks add 0.5 Kg of mass to the U- bend, localized at each elbow. m PIPING SYSTEM.CONFIGURATIONS Configuration Location Length Valve Movable Reservoir Rigid Support Meters Feet 1 B 4 4' C 40 99 134 45 2 B 5 5' C 55 62 182 43 3 A 1 1' 4 4' C 65 51 214 87 4 A l 1' 5 5' C 80 14 262 93 5 A 2 2' 4 4' C 80 16 262 92 6 B 6 6' C 91 89 301 40 7 A 3 3' 4 4' C 94 77 310 85 8 A 2 2' 5 5' C 94 79 310 99 9 A 3 3' 5 5' C 109 40 358 92 10 A 3 3' 6 6' C 145 67 477 80 (See Figure 5.1) Chapter 5 Experimental Apparatus 121 A total of 10 different pipe lengths can be obtained. The movable rigid supports are placed at locations where two pipe unions are aligned in the Y direction, as shown in Figure 5.1. Table 5.5 shows the total length of the system for each configuration. The change in pipe length allows variance of several parameters: frequency of the liquid, loca- tion of the U-bend, external excitation and data acquisition transducers. 5.2.4 Transducers The responses of the liquid pressure and pipe motion to the harmonic excitation are recorded as functions of time. These recordings were accomplished using PCB pressure and acceleration transducers interfaced with either a Digital PDP-ll/73 computer or a Tektronix D13 dual beam storage oscilloscope. A description of the components ofthe data acquisition equipment is presented in Appendix B. Two pressure transducers and two accelerometers were used in the recording. Another accelerometer monitored the motion of the linear bearing structure. Their locations are shown in Figures 5.1 and 5.5. One pressure transducer is located at the closed end where a liquid pressure antinode occurs. The other is located after the U-bend to monitor the effect of the motion of the pipe during the pressure response. The accelerometers are located at the spring and the elbow of each leg. At these loca- tions, large displacements are expected to occur for the first and second natural frequencies of the U-bend [83]. Chapter 5 Experimental Apparatus 122 m LOCATION OF'TRANSDUCERS AND'U-BENN RELATIVE TO VALVE Location Configuration 1 2 3 4 5 Valve 0.0 0.0 0.0 0.0 0.0 P, 0.3 0.3 0.2 0.2 0.2 B1 1.2 0.9 38.2 31.2 49.5 D, 3.4 2.5 39.6 32.4 50.6 D2 5.5 4.0 40.8 33.4 51.7 B, 5.7 4.2 41.0 33.5 51.8 B3 10.1 7.5 43.8 35.8 54.1 B, 14.6 10.8 46.6 38.1 56.3 P2 16.1 11.9 47.5 38.8 57.1 Reservoir 100.0 100.0 100.0 100.0 100.0 Location Configuration 6 7 8 9 10 Valve 0.0 0.0 0.0 0.0 0.0 P, 0.2 0.1 0.1 0.1 0.1 81 0.5 57.3 41.9 49.6 37.3 D1 1.5 58.2 42.8 50.4 37.9 D2 2.4 59.1 43.7 51.2 38.5 B; 2.5 59.2 43.8 51.3 .38.5 B; 4.5 61.1 45.7 53.0 39.8 B, 6.5 63.1 47.6 54.6 41.0 P, 7.2 63.7 48.3 55.2 41.4 Reservoir 100.0 100.0 100.0 100.0 100.0 (See Figure 5.5) P - Pressure Transducer D - Accelerometer B - U-Bend Chapter 5 Experimental Apparatus 123 Table 5.6 shows the relative locations of the transducers, U-bend elbows and rigid supports with respect to the total length of each experimental configuration. Distances are measured from the closed end (valve) to the open end (reservoir). 5.2.5 Dynamic Forces and Natural Frequencies of Shaker The magnitude and characteristics of the dynamic loads that the shaker puts into the spring and, therefore, into the pipe, must be known before experiments are performed. The dynamic forces are studied in the form of the reciprocating force induced by the crank mechanism and the force that the spring transmits to the pipe. Also, the shaker may introduce noise into the signals if a natural frequency of the component coincides with the frequency of excitation. 5.2.5.1 Shaker Loads The reciprocating force at point P in Figure 5.4 can be defined when the acceleration at this point is known. Given that O is the frequency of oscillation in radians per second, then the displacement of the piston z: measured from the dead-center position, at which at is zero, is [84]: 2 z: - [Rd-EL] - R cos(0t) + fif‘cos(20t)] (5.3) Chapter 5 Experimental Apparatus 124 The reciprocating force is: fp - (m + )an cos(0t) + R'cos(20t) (5 4) z 1 n‘b L ‘ The inertia torque, Tx exerted by the motor on the crank is: (m + ) ' T; - 42:11—32:22 [sin(20t)-§E[sin(0t)-351n(0t)]] (5.5) where m1 and 1111) are the mass of the link and linear bearing structure, respectively. The first term in Equation 5.4 is called the primary term. Its frequency varies with the frequency of the motor. The other term is called the secondary term because its frequency varies with twice the frequency of the motor. The importance of the secondary term is established by the ratio R/L. In the case of an infinitely long connecting rod, the secondary term may be neglected and the piston follows a harmonic motion. For a connecting rod of finite length the motion of the piston is periodic but not harmonic. This ratio is less than 0.2% for the crank mechanism in Figures 5.3 and 5.4 (see Table 5.3). When the effect of the secondary term is neglected, the motion that the piston induces in the spring is harmonic. The reciprocating force and torque, then, depend on the mass of the link and linear bear- ing structure, the crank radius and the frequency of oscillation. Neglecting the secondary terms, Equations 5.3 through 5.5 become: Chapter 5 ' Experimental Apparatus 125 z: - R [1 - cos(0t)] (5.6) £1; - (m1 + mb) ancosalt) (5.7) r° - M R2O2s1n(20t) (5.8) x 2 Placing the values for the mass of the link and bearing structure in Equation 5.8 at the rated frequency shows that the torque on the motor shaft is 0.1 N-m, 20% of the rated torque of the motor. The torque induced by the reaction force of the spring on the shaft of the motor should also be included. This torque depends on the displacement at both sides of the spring, the stiffness of the spring and the crank radius. Because the displacement of the spring at the U-bend is fre- quency dependent the torque also depends on the frequency of oscillation. The curve that describes this relationship is presented in section 5.3.3. 5.2.5.2 Spring Loads The shaker mechanism inputs a specified displacement at one end of the spring as given in Equation 5.6. The influence of the spring on the U- bend as shown in Figure 5.5 can be analyzed by studying a simplified structure. The U-bend can be considered a single mass-spring system connected to a spring with specified displacement 0° and a force F as Chapter 5 Experimental Apparatus 126 shown in Figure 5.6. Let InB and kB be the mass and stiffness of the mass-spring system, and k8 the spring constant. A transfer matrix analysis of this system yields: At location.2 the displacement U2 is zero, therefore, the frequency equation is [1-9:]u,+[%[1-9:]+&B]F-o (5.10) where w is the frequency of the spring-mass system and fl is the fre- quency at which the crank-slider mechanism is oscillating. This equation suggests that the dynamic loads from the shaker can be divided into two loads: one associated with the displacement, Va and one as- sociated with the force, F. This force is the same as the spring force described in the previous section. Its description is also presented in Section 5.3.3. Chapter 5 ‘ Experimental Apparatus 127 5 . 2 . 5 . 3 Natural Frequencies of Shaker Components The components that may induce high frequency vibration are the link, the spring and the steel rods on which the linear bearings slide. The link that connects the crank to the slider bearing may bend rather than having a purely rigid body motion, inducing lateral vibration to the mechanism. A transfer matrix analysis for a free-pinned solid beam (locations C and P in Figure 5.4) demonstrates that the first natural frequency occurs at 100 Hz which is 70 Hz above the rated frequency of the motor. The natural frequency of the spring is 460 Hz. The natural frequencies of the stainless-steel rods are 88 and 550 Hz for vibra- tions in the axial and transversal directions, respectively. The natural frequency of these components are higher than the rated fre- quency of the motor, therefore, the dynamic components of the shaker do not interfere with the harmonic motion induced onto the U-bend. 5.3 W The objective of the experiments was to excite the natural frequencies of the U-bend and the liquid contained in the pipe and to record the pressure response and pipe motion. Several sampling parameters had to be determined before collecting data for each pipe configuration. These included the frequency range of excitation, the sampling frequency and the duration of the sampling process. . Figure 5.7 is a diagram of the experimental procedure and analysis. Four transducers were used to collect the time series of the liquid pressure and pipe displacements. Chapter 5 Experimental Apparatus 128 These were compared with results from the analytical model developed in Chapters 3 and 4. This section describes the experimental procedures and analyses that were used for each pipe configuration. 5.3.1 Frequency Range of Excitation The determination of the frequency range of excitation depends on the rated frequency of the motor and the natural frequencies of the liquid and the U-bend. The rated frequency of the motor is 30 Hz which is the upper bound of the range. The lower bound of the frequency range is determined by the frequency at which the inertia forces of the shaker mechanism are overcome producing a harmonic oscillation of the linear bearing structure. This lower bound frequency was found to be at 3.4 Hz. Several natural frequencies of the liquid and U-bend are excited over the frequency range. 5 . 3 . l . 1 Liquid Frequencies The fundamental frequency of an open-closed liquid system is [5,6,7,4l] C if --f (5.11) Where ff is the fundamental frequency of the liquid, cf is the coupled wave speed and I is the length of the pipe. A pressure node is formed at the open end and a pressure antinode is formed at the closed end. Chapter 5 . Experimental Apparatus 129 The higher harmonics of the system are determined by the odd harmonics of the fundamental frequency computed from Equation 5.11. Table 5.7 lists the first through the ninth harmonic for the first nine pipe configurations. The coupled wave speed cf, which was determined for this piping system by Budny [22], is 1265 m/s. The results in Table 5.7 show that the ninth harmonic of configurations 7, 8 and 9 can be ex- cited. The first harmonic of configurations 7, 8 and 9 is periodic but not harmonic . W FLUID HARmNICS FOR PIPE CONFIGURATIONS Configuration Harmonic (HZ) First Third Fifth Seventh Ninth l 7.7 23.1 2 5.7 17.1 28.4 3 4.8 14.5 24.1 4 3.9 11.8 19.7 27.6 5 3.9 11.8 19.7 27.6 6 3.4 10.3 17.2 24.1 7 3.3 10.0 16.7 23.4 30.0 8 3.3 10.0 16.7 23.4 30.0 9 2.9 8.7 14.5 20.2 26.0 5 . 3 . l . 2 U-Bend Frequencies The U-bend may be thought of as a plane frame clamped at the columns and having rigid joints at the elbows. Chang [83] developed frequency charts for identical columns and cross-beam plane frames. The frequency Chapter 5 Experimental Apparatus 130 equation, which is based on the Bernoulli-Euler beam theory, includes the effect of axial vibration. It is given by: 3 El *2 f__C._ _.D_ 512 P 2 pA (.) 211 pp where g‘ is the frequency parameter and l is the length of a column. Table 5.8 shows the estimated frequencies, using Chang's development, for the U-bend shown in Figure 5.5 and the values presented in Tables 5.1 and 5.2. The mass of the liquid can be added to the denominator of the radical term as: ppAp-I-pfAf. The modes of vibration are asymmetrical and symmetrical. In the asymmetrical modes the elbows of the U-bend translates, simulating a rigid motion of the horizontal leg. The elbows in the symmetrical modes do not translate. W NATURAL F'RmUENCIES OF U-BEND Frequency parameter Natural frequency Mode of vibration f (HZ) Empty Liquid- Filled 1.790 5.1 4.1 asymmetrical 3 .553 21.0 16 . 9 symmetrical 4 . 541 34 . 3 27 . 6 asymmetrical 6 . 693 74. 5 60 .0 asymmetrical Chapter 5 Experimental Apparatus 131 The frequency range of 3.4 Hz through 30 Hz excites two natural fre- quencies of the empty U-bend. The same frequency range also excites three natural frequencies of the liquid-filled U-bend. 5.3.2 Sampling Frequency. and Sampling Time The frequency range of 3.4 Hz to 30 Hz used for the experiments was determined as shown in the previous section. The sampling frequency is 1000 Hz and the duration of the sampling process is 4096 milliseconds. These parameters minimize the sampling errors due to aliasing and leakage. The upper limit of the frequency range is 30 Hz. This fre- quency is more than ten times less than the Nyquist frequency which is 500 Hz [93] . In the frequency range, the number of sinusoidal cycles for the duration of the sampling varies from 14 to 123. The error introduced, due to leakage when computing the frequencies, is less than 0.12 Hz, which is the resolution of the fast Fourier transform (FFT). 5.3.3 Sampling Procedure The SCR motor control was calibrated to input the same frequency incre- ments for all pipe configurations. The spacing between forcing frequencies was 0.41 Hz, yielding a total of 65 discrete frequencies. An input-output calibration curve for the frequency range was obtained between an accelerometer located at the linear bearing structure and another located at the spring. These displacements are presented in Figure 5.8. The input displacement at the bearing structure slightly Chapter 5 Experimental Apparatus 132 increases as the circular frequency of the motor increases, this was caused by a small bending of the shaft of the motor. This increase was 0.4 mm over the frequency range from 3.4 Hz to 30.0 Hz. The spring force at D1 in Figure 5.5 is of the same form as the displacement at this location. The torque described in Section 5.2.5.1 and the rated torque of the motor are shown in Figure 5.9. The reaction force of the spring onto the shaft of the motor was included. The response spectra for the four transducers were obtained by sweeping through the frequency range. The signals from the pressure transducer located at the closed end and the accelerometer at the spring were monitored on an oscilloscope during the sweeping process. The oscillo- scope monitoring had two purposes. First, before each sampling the system was allowed to reach a steady-state condition. Second, it al- 1owed determination of the natural frequencies of the system. After the sweeping process was finished, the responses of the transducers at the natural frequencies were sampled. 5.3.4 Analysis Procedure The time series for the 65 discrete frequencies and the natural fre- quencies were stored in a PDP 11/73 microcomputer. A fast Fourier transform FFT of these series was performed to obtain the magnitude and frequency. Before the FFT analysis, each time series was low pass filtered. The filter, which is shown in Figure 5.10, had a cut-off frequency of 80 Hz. This cut-off minimizes the noise introduced to the Chapter 5 Experimental Apparatus 133 signals by the shaker components. The sampling parameters described in the previous section determined that the resolution of the FFT was 10.12 Hz. 5.4 W The uncertainty of the experiments performed arises from three sources: the transducers, the A/D conversion and the pipe and fluid characteris- tics. Appendix B describes the characetristics of the data acquisition equipment used . Tables B.l and 8.2 of Appendix B list the characteristics provided by the manufacturer for the two types of transducers used in the experi- ments and the error in the conversion of the analog voltage to digital format by the A/D converter. . Pressure transducers models 111A26 and 113A24 were used. Both models have a linear error of 2%. The linearity of the error means the error is a constant 2% along the entire operating range of the transducer. Thus, a reading of 50 psi (345 kPa) would have an error of $1.0 psi ($7.0 kPa). The resolution is a measure of the ability to distinguish between nearly equal values of a quantity. It is also referred to as "threshold", that is, the lowest level of valid measurement . The accelerometers used were manufacturer model 302A. The error is based on the type of power unit that is connected to the transducer. Double integrator units were used in all the experiments to obtain Chapter 5 . Experimental Apparatus 134 displacements. The error associated with this type of unit is 5%. Another source of error of the accelerometers is the nonlinearity of the response at frequencies less than 10 Hz. A calibration was necessary to find the conversion from volts to millimeters in the range from 3.4 Hz to 10 Hz. The procedure used is described in Appendix B. The error due to the A/D conversion is controlled by the 11 bit resolu- tion of the input data. Because the A/D board is configured for bipolar inputs of $10.0 volts, the 11 bit resolution is equivalent to an error of $ 9.7 millivolts. The base frequency for the programmable realtime clock is 10 Mhz, thus the accuracy of the time measurement is $0.1 microseconds . Thus, the maximun experimental error associated with the pressure read- ings due to the propagation of both the transducer and conversion error based on root mean square (rms) estimates for a 500 kPa reading is $12.0 kPa. The sources of error that arise from the fluid and pipe properties come from the measurement of these properties. The copper pipe was manufac- tured by American Brass Company. The tolerances for the inside diameter and wall thickness of the pipe were 0.4% (0.1 mm) and 3% (0.04 mm), respectively. The fluid temperature was measured with an accuracy of 0.5 ‘C. The length of each pipe configuration was taken with a lOOrfoot tape with 100 divisions per foot. The error associated with the total pipe length depends on the number of measurements taken. For example, 17 measurements were taken for configuration 10 and 9 for configuration Chapter 5 Experimental Apparatus 135 1. Then, the error in pipe length measurement varies from $0.04 ft (0.01 m) to $0.03 ft (0.01 m). A 12 foot tape with 1-6 divisions per inch was used to measure pipe lengths less than 10 feet. For example, the legs of the U-bend were measured with this tape. The error in the measurement is $0.06 inch (2 mm). Possible errors due to temperature changes and sagging of the tape were not estimated. and.“ 25 3:36:63 Haucufluaea an as»: ntal Apparatus a 83.8: :83 it u U I :55; .1 r . M. I. g a 2... __. 5...: on... .333 «cocoa—8a tone—m in. coo: ’ . £8 , @ II._II 9.8» 33.2.. m. SilllI . 1 4| 4" h< I an a e .n n u E . wlilelilelm J almlmlmimlimilim n .u .. w a e 5 N 18. a .. . t. _.._.. inns... . m. r A Loan: 2.81.. r238 .396: Chapter 5 Experimental Apparatus 137 1/4hdlAilietPut ma”) ‘flh \\\\“m~ Fummmecmne I/slmntwlkomr -—JSL—- tCIn “I. £111 1|mh1hmdhrAmqrmli a) Open End: Reservoir 1)) Closed End: Vdve (After Budny) Figure 5.2 Liquid Boundaries lacunae: scone n.n and»: Experimental Apparatus 138 5. E *l r _ 82.. 25.2 ”fin/L. \ 13m 82565 I a ”G B _ I oib< I . U = Him eoctoom boa... is «\— WL Eavoeccoo V Com Ens—Es? .855 a... 8&8 2.3m 6... Q. 6... «\— 251 :2. . \ / _ EE E 3:5 .5 a E: assoc 2m 3m satin . . I_I m E c. 2 6.... =38 6:: 6:. N} O I «WWI / , ha 3a .5583 Chapter 5 Chapter 5 Experimental Apparatus 139 Figure 5.4 Crank Diagram P1 P2 3:} at Open End Y Closed End 34 . l o-Inmt) z .._.. D‘i \. 02 L, AIJ El as 2 I 31914 50990” :2 Aluminum Collar I Pressure Transducer .. Accelerometer Figure 5.5 Transducers and U-Bend Locations Figure 5.6 One Dregree-of—Freedom Representation of U-Bend Experimental Apparatus Chapter 5 140 P P‘ Y 2 Motor Control Experimental Oscilloscope wr- Procedure m 9.0 Sampling Pmetere Number of Channels - 4 Samplhq Frequency - 1000 Hz ' Sompilnq Tine - 4096 me Store Time Series Filter 80 Hz. Cut-off Freq. Analysis Procedure I. L Frequency Amplitude Ratio Figure 5.7 Experimental and Analysis Procedures Chapter 5 Experimental Apparatus 141 10" ’5‘ 8- Displacement at D‘ s . l a s- Dleplacement at D g . .1 ./ = s 44 j: / .I 2 ' ‘l‘ ' o. 1 . l I | -'-' 2- I | A O l 1 II | I \ o" Tl --d~"-’J '\ ""/ 3 9 15 21 27 Frequency (Hz) Figure 5.8 Input and Output Displacements 0.0-I Rated torque i— --_--_--._--_--__--- A $0.4- 2 V d 3 U’ '5 0.2- _ Input torque .— I 0'1 'fl"T'rF"l'—f1 3 S 1 6 21 27 Frequency (Hz) Figure 5.9 Input and Rated Torque at Shaft of Motor 1 .o. q 0.8-I 3 ‘ , and 3 a i o 0.4- a J 0.2“ ‘ 0.0 v i f I i I f 1 ' fl 0 100 200 300 400 500 Frequency (Hz) Figure 5.10 Low-Pass Filter Chapter 6 Experimental Results and Comparisons 6.1 W The purpose of the experiments was to validate the analytical model derived in Chapters 3 and 4 and gain further physical understanding of the phenomena. Transient and harmonic tests were conducted to find the natural frequencies of the system. Fluid pressure and pipe displacement responses were monitored. Transient tests were used to calibrate the apparatus and measure the natural frequency] of the U-bend and natural frequencies of the contained liquid. The transient tests were rapid valve closure and snap-back of the piping. The U-bend was excited harmonically by the crank mechanism that inputs a harmonic load to the pipe, as discussed in Chapter 5. The first part of this chapter gives the results of the transient tests. The second part presents the experimental results of the harmonic test and compares them with the computed results from the analytical model. 142 Chapter 6 Experimental Results and Comparisons 143 6.2 W Transient tests allow measurement of the natural frequencies of the system. Snap-back of the U-bend and rapid valve closure were the tests performed. The description of the tests and the results are presented. 6.2.1 Snap-Back Test This test was used to measure the first natural frequency of the U-bend and calibrate the analytical model. The test consisted in displacing the elbows of the U-bend 12.7 mm (0.5 inch) from the equilibrium posi- tion and then releasing them. The calibration consisted of determining the effective stiffness of the spring between the linear bearing struc- ture and the U-bend. Tests were conducted on an empty U-bend and a . liquid-filled U-bend. The U-bend can be either free from or attached to a spring. Table 6.1 shows the results of the experimental tests and the computed results using either the transfer matrix method (TM) or the Bernoulli-Euler beam theory developed by Chang [83] as shown in Table 5.8. Figures 6.1 and 6.2 show the time responses and the FFT's of the time series for the various U-bend conditions mentioned above. The added masses from the accelerometers, aluminum collar and' steel bars at the elbows of the U-bend were included in the computed results for the TM. The stiffness of the spring was found to be 8 KN/m which is greater than the stiffness provided by the manufacturer, as shown in Table 5.4. The results, in Table 6.1, show that the natural frequency Chapter 6 Experimental Results and Comparisons 144 increases when the U-bend is attached to the spring. This result sug- gests that the stiffness of the U-bend - spring system increases. Including the mass of the liquid decreased the natural frequency of the U-bend, as expected . mu.“ E!PERIHENTAL.ARD COMPUTED'U-BERD RESPONSE TO SlAP-BACK.TEST Free Spring Condition Experimental TMM Chang [83] Experimental TMM (HZ) (H2) (H2) (H2) (HZ) Empty 4.4 4.4 5.1 5.1 5.1 Liquid-Filled 3.9 3.9 4.1 4.4 4.4 6. 2 .2 Valve Closure Test A rapid valve closure induces excitement of the liquid pressure. The experimental procedures as well as the software and hardware used for these tests are described by Budny [22]. These tests also allow measurement of the liquid wave speed if the length of the pipe is known as shown in Equation 5.11. An‘ open-closed system results upon closure of the valve and excites the odd harmonics of the liquid. The first harmonic corresponds to the first or fundamental frequency of the liq- uid. According to the state of the U-bend, three cases were studied for configurations 4, 8, 9 and 10: fixed, free and with spring. The total length of the pipe and relative location for each configuration are given in Tables 5.4 and 5.5. Chapter 6 Experimental Results and Comparisons 145 6 . 2 . 2 . 1 Fixed U-Bend A Unistrut was placed at each free elbow of the U-bend, locations 82 and B3 in Figure 5.5. The time series for the pressure at the closed-end for the four configurations are presented in Figure 6.3. The FFT's of the time series are shown in Figure 6.4. The FFT's were normalized with respect to the largest pressure or displacement response for all three cases. The natural frequencies from the experiment were compared with the frequencies computed by the TMM. A straight pipe of variable length with a harmonic oscillation at the open end was used to compute the natural frequencies of the system. The results, shown in Table 6.2 demonstrate the ability of the TMM to predict the natural frequencies of an axially coupled system. The computed wave speed is within 0.5% of the experimental . m2 f WWWWE OFLIQUIDFORFIXEDWDITION Configuration Wave Speed Harmonics (In/s) (H2) First Third Fifth E C E C E C E C 4 1267 1260 3.9 4.0 12.0 11.8 19.8 19.6 8 1266 1260 3.4 3.4 10.0 10.0 16.6 16.5 9 1265 ‘1260 2.9 2.9 8.5 8.7 14.4 14.3 10 1267 1260 2.2 2.2 6.6 6.5 11.0 10.8 E - Experimental C - Computed Chapter 6 Experimental Results and Comparisons 146 6 . 2 . 2 . 2 Free U-Bend For this case, the U-bend is free to vibrate in the Y-Z plane, as shown in Figure 5.5. The time series of the liquid pressure at the closed- end, P1, and the U-bend elbow displacement, Dz, were recorded; the results are shown in Figure 6.5. The FFT's of the time series are shown in Figure 6.6. Table 6.3 shows the experimental and computed results for the natural frequencies of the liquid and the experimental com- ‘pliance. Compliance [34] is defined as ratio of the elbow displacement over pressure at the closed-end, Da/Pl. MILL: mummmmmctnoruqummmmmon Fluid Harmonics Configuration First Third Fifth . E C E C E C 4 Frequency (Hz) 3.9 4.0 12.0 12.0 19.8 19.7 Dz/Pl (mm/Pa) 69.4 1.8 0.4 8 Frequency (Hz) 3.2 3.4 10.0 10.1 16.4 16.7 Dz/Pl (mm/Pa) 2.8 3.0 0.4 9 Frequency (Hz) 2.9 2.9 8.5 8.8 14.4 14.5 D2/P1 (min/Pa) 1.1 3.1 1.2 10 Frequency (Hz) 2.2 2.2 6.6 6.6 11.0 10.8 D2/P1 (mm/Pa) 0 1 8.6 0.3 E - Experimental C - Computed D, - Accelerometer response at U-bend elbow P1 - Pressure transducer response at closed end Chapter 6 Experimental Results and Comparisons 147 The computed frequencies show good agreement with the experimental frequencies of the system. The following observations can be made concerning the experimental compliance: 1) Configuration 4 shows the largest magnitude. The frequency of the U-bend and the first natural frequency of the liquid nearly coincide. The proximity of the frequencies results in a beat as shown in Figure 6.5a. 2) The magnitude of the compliance at the first fluid frequency decreases as both the frequency of the fluid and the U-bend move apart from each other. 3) The opposite occurs at the third liquid harmonic. Configuration 10 shows a larger compliance than configuration 4. This phenomenon takes place because the third harmonic of configuration 10 (6.6 Hz) is closer to the natural frequency of the U-bend than the other configurations. 4) The compliance at the third liquid harmonic is greater than that at the first harmonic for configurations 8, 9 and 10. The third liquid harmonic for configuration 10 constitute the dominant frequency of the U-bend. The compliance is greater than the compliance at the natural frequency of the U-bend as shown in Figure 6.6d. The pipe line from the closed-end to B1 and from B, to the open-end, Figure 5.5, was treated as a straight pipe. Only the axial modes of vibration for the pipe and the liquid modes were included in these reaches. This simplification is based on the previous experimental results of this research. The pressure response at the closed-end is Chapter 6 Experimental Results and Comparisons 148 affected by the status of the bend (as shown in Figure 6.5a for con- figuration 4). The reponse of the U-bend is affected by the closing of the valve. Only the liquid and U-bend frequencies were found to be of significance in the results. This simplification also reduces the numerical difficulties associated with the TMM as pointed out by Pestel and Leckie [50] and as described in Chapter 4. 6.2.2.3 U-bend with Spring The U-bend was attached to the spring, as described in Chapter 5, at location D1 in Figure 5.5. The experimental time series and FFT's are shown in Figures 6.7 and 6.8, respectively. The experimental and com- puted results are shown in Table 6.4. The same observations as for the previous case can be made. In the present situation the frequency of the U-bend is 0.5 Hz higher than the previous case, because of the additional stiffness provided by the spring. This causes the third fluid frequency for configuration 10 to be closer to the U-bend fre- quency, thus, resulting in a larger ratio than in the previous case. 6.3 W The harmonic tests consisted of obtaining the liquid pressure and pipe displacement responses of the liquid-filled pipe when a harmonic dis- placement is induced at the U-bend. Two liquid pressure readings at the Chapter 6 Experimental Results and Comparisons 149 closed end and at the U-bend (locations P1 and P2 in Figure 5.5) , and two displacement readings at the spring and at U-bend elbow (locations DI and D2 in Figure 5.5) were collected at each frequency. The fre- quency range of excitation varied from 3.4 Hz to 30 Hz. A preliminary evaluation of the U-bend subjected to harmonic displacement is presented. Then, the response spectra and liquid mode shapes follow. m5 EXPERIMENTAL.AND COMPUTED FREQUENCIES OF LIQUID FOR SPRING CONDITION . Fluid Harmonics Configuration First Third Fifth E C E C E C 4 Frequency (Hz) 3.9 4.0 12.0 12.0 19.8 19.7 D2/P1 (mm/Pa) 10.1 2.2 0.4 8 Frequency (Hz) 3.4 3 4 10.0 10.1 16.6 16.7 D2/P1 (mm/Pa) 6.4 2.4 -0.5 9 Frequency (Hz) 2.9 2.9 8.6 8.8 14.4 14.5 D2/P1 (mm/Pa) 0.6 3.4 1.3 10 Frequency (Hz) 2.2 2.2 6.4 6.6 10.7 10.8 D2/P1 (mm/Pa) 0.1 11.3 0.1 E - Experimental C - Computed D, - Accelerometer response at U-bend elbow P1 - Pressure transducer response at closed end Chapter 6 Experimental Results and Comparisons 150 6 . 3 . 1 U-Bend Response Displacements at locations D, and D, were recorded to find the natural frequencies of the U-bend. Two cases were considered, when the U-bend is empty and when it is liquid-filled. Figure 6.9 shows the responses at the spring and the elbow for the two cases. The added mass of the liquid reduces the natural frequencies of the U-bend. Table 6.5 shows the experimental and computed results at the natural frequencies. The computed mode shapes for the liquid-filled U-bend are displayed in Figures 6.10 and 6.11. Figure 6.10 shows the mode shapes of the U-bend without the spring. Figure 6.11 shows the mode shapes with the spring. The odd natural frequencies correspond to asymetrical modes and the even frequencies to symmetrical modes of vibration. The inclusion of the spring affects the first and second modes of vibration. The other mode shapes do not show any appreciable change. It can be noted that the inclusion of the spring allows for larger displacements of the leg where the spring is attached. The largest discrepancy between the experimental and computed natural frequencies occurs at the second mode. The computed frequencies for this mode are 0.9 Hz and 0.6 Hz lower than the experimental when the U-bend is empty and liquid-filled respec- tively . Chapter 6 Experimental Results and Comparisons 151 IIBLE_§.§ NATURAL FREIUENCIES 0? mm 1'0 WC EXCIIAI'ION Natural Frequencies (HZ) Empty Liquid-Filled Frequency Mode of Spring Free Spring Free Number Vibration E C E* C E C E* C l Asymmetrical 5.1 5.1 4.4 4.4 4.4 4.4 3.9 3.9 2 Symmetrical 22.7 21.8 21.2 18.1 17.5 17.0 3 Asymmetrical 35.5 35.0 28.3 28.6 28.0 4 Symmetrical 44.4 41.4 34.3 32.1 5 Asymmetrical 73.3 73.1 58.9 58.8 6 Symmetrical 89.9 89.8 72.9 72.9 * Results for the free case were obtained from the snap-back test, see Table 6.1. 6.3.2 Spectral Response of Liquid-Filled Piping Tables 6.9 and 6.11, at the end of this chapter, show the results at the natural frequencies of the system. The largest experimental pressure response occurred at the fifth and seventh harmonic of configurations 2 (28.3 Hz and 521 kPa) and 4 (27.8 Hz and 290 kPa). The large pressure responses are associated with the third natural frequency of the U-bend. This frequency, 28.3 Hz, corresponds to an asymmetrical mode, in which the elbows show a small displacement, Figure 6.11. This small displace- ment generates the large liquid pressure responses through the junction coupling mechanism. Chapter 6 Experimental Results and Comparisons 152 6 . 3 . 3 Liquid Mode Shapes The first pressure mode shape of the liquid in an open-closed pipe corresponds to 1/4 of a sinusoidal wave [5,6,7]. The maximum response occurs at the closed-end (s/l - 0), where a pressure loop develops. A pressure node, where the pressure is zero, develops at the open-end (s/t - 1), where s is a coordinate along the pipe length 1. The other mode shapes correspond to the odd multiples of the 1/4 sinusoidal wave. The response of the pressure transducers P1 and P, were used to obtain the mode shapes for each natural frequency of the system. The distance from these transducers to the closed and varies for each pipe configura- tion. In this way, a point of the liquid mode shape was obtained for each configuration. The relative location of P1 and P, with respect to the closed-end is given in Table 5.6. Tables 6.8 through 6.16 show the experimental and computed results at the natural frequencies of the system for configurations 1 through 9. The behavior of the mode shapes of the liquid is different if the frequency of the harmonic excitation is oscillating at a natural frequency of the liquid or the U-bend. Thus, the liquid mode shapes will be studied at the liquid frequencies and at the natural frequencies of the U-bend. The pressure ratio Pz/P1 will be used to compare the computed and experimental liquid mode shapes . Chapter 6 Experimental Results and Comparisons 153 6.3.3.1 Liquid Mode Shapes at Liquid Natural Frequencies Figure 6.12 shows the normalized pressure mode shapes at the natural frequencies of the liquid. The mode shapes were normalized with respect to the pressure at the closed end. The first through ninth odd har- monics are shown in this figure; the solid line represents the computed mode shape and the dots represent experimental points. The encircled numbers correspond to the pipe configuration. They are placed at the relative location of P2 with respect to the closed-end. Table 6.6 shows the experimental and computed results depicted in Figure 6.12. Good agreement between the experimental points and the computed mode shapes is noted in this figure. The correlation coefficient [92] between computed and experimental results is unity for all harmonics, except for the first. The first liquid mode shape, which is associated with the first harmonic, shows the largest discrepancies for configurations 1, 2, and 3. These configurations show that the first natural frequency of the liquid is higher than the first natural frequency of the U-bend. The minimal motion of the elbows of the U-bend at the liquid frequencies causes a minimal response of the pressure, increasing the experimental error. Pressure readings lower than 15 KPa are only four times greater than the resolution of the A/D converter board, as mentioned in the previous chapter . Chapter 6 Experimental Results and Comparisons 154 M LIQUID PRESSURE.NDDE SHAPES AI LIQUIDINAIURAL.FREQUENCIES Harmonic Conf. * First Third Fifth Seventh Ninth No. P2 E C E C E C E C E C l 0.15 0.83 0.97 0.71 0.74 2 0.11 0.73 0.98 0.89 0.86 0.60 0.57 3 0.47 0.89 0.75 0.56 0.60 0.94 0.83 4 0.38 0.78 0.81 0.24 0.24 0.94 0.97 0.41 0.46 5 0.56 0.60 0.61 0.85 0.88 0.23 0.22 0.99 1.00 6 0.07 -- 1.00 -- 0.95 -- 0.86 0.71 0.74 7 0.63 0.51 0.54 0.92 0.98 -- 0.32 -- 0.75 0.87 0.84 8 0.48 0.69 0.72 0.60 0.63 -- 0.80 0.52 0.54 0.89 0.89 9 0.55 0.63 0.65 0.82 0.84 0.35 0.36 0.90 0.99 0.00 0.09 Correlation 0.77 1.00 1.00 1.00 1.00 E - Experimental C - Computed * P2 is the relative location of the pressure transducer with respect to the closed-end, see Table 5.6. 6.3.3.2 Liquid Node Shapes at U-Bend Natural Frequencies The experimental liquid mode shapes at the frequency of the U-bend were obtained as in the previous section. The behavior of the liquid pres- sure at the natural frequencies of the U-bend follows the same trend as the behavior at the natural frequencies of the liquid. However, an abrupt change of the mode shape at the location of the U-bend is produced due to the motion of the elbows. Three of the natural fre- quencies of the U-bend were excited by harmonic oscillation. The first and third, which correspond to asymmetrical modes, as shown in Figure 6.10, allow motion of the elbows in the Y-Z plane. The second natural frequency, which is a symmetrical mode, allows no motion at the elbows. Chapter 6 Experimental Results and Comparisons 155 This lack of motion at the elbows prohibits the development of junction coupling. Therefore, the liquid pressure response is minimum at this mode. Tables 6.8 through 6.16, at the end of this chapter, show the experimental and computed results at the natural frequencies of the system . Figures 6.13 through 6.19 show the liquid mode shapes at the two asym- metrical U-bend frequencies for configurations 1 through 9. In addition to the features described in the previous section for Figure 6.12, the location of the horizontal leg of the U-bend is shown for each con- figuration (locations B, and B, in Figure 5.5). The location is marked by two parallel vertical lines. Figure 6.16 shows the liquid mode shapes for configurations 4 and 5 and Figure 6.18 for configurations 7 and 8. Table 6.7 shows the results depicted in Figures 6.13 through 6.19. The experimental and computed results show good correlation at the first natural frequency of the U-bend. The largest discrepancies at the third U-bend frequency occur for configurations 4 and 9. The liquid is oscil- lating between the ninth and eleventh harmonic for both configurations. The pressure gradient at P2 shown in Figures 6.16 and 6.19 are higher than for any other configuration. Thus, any change of the frequency of oscillation may cause considerable changes in the magnitude of the pressures . Chapter 6 Experimental Results and Comparisons 156 Ml LIQUID PRESSURE MODE SHAPES AI U-BEND NAIURAL.FREQUENCIES U-bend Frequency (HZ) 4.4 28.3 Configuration P2* First Third Number E C E C 1 0.15 0.00 0.07 0.76 0.74 2 0.11 0.16 0.14 0.94 0.97 3 0.47 0.55 0.55 1.18 1.22 4 0.38 1.20 1.18 0.94 1.53 5 0.56 0.82 0.80 0.53 0.35 6 0.07 0.00 0.19 0.22 0.20 7 0.63 0.92 0.98 1.80 1.54 8 0.48 1.66 1.65 0.28 0.43 9 0.55 2.05 1.95 1.09 1.83 Correlation 1.00 0.82 E - Experimental C - Computed * P2 is the relative location of the pressure transducer with respect to the closed-end. Chapter 6 Experimental Results and Comparisons 157 m EXPERIMENTAL.AND COMPUTED RESULTS FOR.CONFIGURATION 1 Displacement Pressure (mm) (kPa) Freq. Type Location Location Ratio Location Location Ratio (Hz) D1 02 DZ/Dl P1 P2 P2/P1 4.4 E 81 13.6 33.2, 2.4 49 0 0.0 4.4 C 2.8 0.07 7.6 E F1 0.2 0.5 2.5 12 10 0.83 7.8 C 2.6 0.97 18.1 E 82 3.8 0.7 0.2 0 0 -- 17.5 C 0.5 0.03 23.2 E F3 0.4 0.2 0.5 56 40 0.71 23.3 C 0.7 0.74 28.6 E S3 6.0 1.3 0.2 59 45 0.76 28.6 C 0.2 0.74 E - experimental, C - computed, F - fluid, 8 - structural frequency Chapter 6 Experimental Results and Comparisons 158 W2 EXPERIHENTAL.AND COMPUTED RESULTS FOR CONFIGURATION 2 Displacement Pressure (mm) (kPa) Freq. Type Location Location Ratio Location Location Ratio (Hz) D1 02 D2/Dl P1 P2 P2/P1 4.4 E 81 11.8 30.2 2.1 49 8 0.16 4.4 C 2.8 0.14 5.6 E Fl 0.3 1.8 6.0 11 8 0.73 5.7 C 2.8 0.98 17.1 E F3 1.2 0.1 0.1 18 16 0.89 17.1 C 0.0 0.86 18.1 E 82 3.9 0.7 0.2 0 0 -- 17.5 C 0.5 3.94 28.3 E F5 3.1 0.6 0.2 521 310 0.60 28.1 C 0.4 0.57 28.8 E $3 5.2 1.3 0.3 239 225 0.94 28.7 C 0.2 0.97 E - experimental, C - computed, F - fluid, S - structural frequency Chapter 6 Experimental Results and Comparisons 159 M EXPERIMENTAL.AND COMPUTED RESULTS FUR.CDNFICURATION 3 Displacement Pressure (mm) (kPa) Freq. . Type Location Location Ratio Location Location Ratio (Hz) D1 D2 D2/Dl P1 P2 P2/Pl 4.4 E 81 13.4 39.2 2.9 214 117 0.55 4.4 C 2.8 0.55 5.1 E F1 0.5 2.1 4.4 16 14 0.89 4.9 C 2.7 0.75 14.6 E F3 0.5 0.1 0.3 52 29 0.56 14.7 2.2 0.60 18.1 2 52 12.5 ' 2.5 0.2 7 o 0.00 17.5 C 0.1 0.14 24.4 E F5 0.4 0.2 0.5 18 17 0.94 24.2 C 0.4 0.83 28.3 E 83 4.9 1.1 0.2 43 50 1.18 28.6 C 0.2 1.22 E - experimental, C - computed, F - fluid, 8 - structural frequency Chapter 6 Experimental Results and Comparisons "sj H 0 E 160 W EXPERIMENTAL.AND COMPUTED RESULTS FOR CONFIGURATION 4 Displacement Pressure (Inn) (kPa) q. Type Location Location Ratio Location Location Ratio ) D1 D2 D2/D1 P1 P2 P2/P1 9 E F1 1.7 3.5 2.1 102 80 0.78 0 C 2.8 0.81 4 E 81 12.9 38.5 3.0 99 119 1.20 .4 C 2.8 1.18 .0 E F3 0.4 0.1 0.3 41 10 0.24 .0 C 2.3 0.24 .1 E 82 9.3 2.0 0.2 0 7 -- .5 C 0.5 18.53 .8 E F5 0.3 0.2 0.7 36 34 0.94 .7 C 0.8 0.97 .8 3 F9 '1.7 0.3 0.2 290 118 0.41 4 C 0.2 0.46 6 E 83 6.4 1.6 0.3 154 145 0.94 .6 C 0.2 1.53 - experimental, C - computed, F - fluid, 8 - structural frequency Chapter 6 Experimental Results and Comparisons 161 W12 EEPERIMENTAL.AND COMPUTED RESULTS FOR CONFIGURATION 5 Displacement Pressure (MI) (kPa) Freq. Type Location Location Ratio Location Location Ratio (Hz) D1 D2 DZ/Dl P1 P2 P2/P1 3.9 E F1 1.7 3.5 2.1 144 87 0.60 4.0 C 2.8 0.61 4.4 E 51 13.8 40.5' 2.9 160 131 0.82 4.4 C 2.8 0.80 12.0 E F3 0.3 0.1 0.3 26 22 0.85 12.0 C 2.3 0.88 18.1 E 52 12.6 2.5 0.2 10 0 0.00 17.5 C 0.1 0.08 19.8 E F5 0.2 0.2 1.0 39 9 0.23 19.7 C 0.8 0.22 27.6 E F9 1.9 0.4 0.2 231 228 0.99 27.5 C 0.2 1.00 28.3 E 83 5.4 1.3 0.2 188 100 0.53 28.6 C 0.2 0.35 E - experimental, C - computed, F - fluid, S - structural frequency Chapter 6 Experimental Results and Comparisons I21 '1 O E 17. 18. 17. 23. 23. 28. 28. NO 99 how 162 W EXPERIMENTAL.AND COMPUTED RESULTS FOR.CONFICURATION 6 Displacement Pressure (mm) (kPa) q. Type Location Location Ratio Location Location Ratio ) 01 D2 DZ/Dl P1 P2 P2/P1 .4 E Fl 1.1 0.9 0.8 0 0 - - 4 C 0.20 4 E 31 11.5 26.0 2.3 36 0 0.00 4 C 2.8 0.19 5 E F3 0.3 0.2 0.7 6 0 0.00 8 C 2.4 0.95 .1 E F5 0.9 0.1 0.1 5 0 -- 5 C 0.5 0.86 1 E 82 3.8 0.7 0.2 0 0 -- 5 C 0.5 0.03 2 E F7 0.4 0.2 0.5 56 40 0.71 3 C 0.7 0.74 6 E 83 6.0 1.3 0.2 59 45 0.76 6 C 0.2 0.74 - experimental, C - computed, F - fluid, 8 - structural frequency Chapter 6 Experimental Results and Comparisons 163 W EEPERIMENTLL.AND COMPUTED RESULTS FOR.CONFICURATION 7 Displacement Pressure (Inn) (kPa) Freq. Type Location Location Ratio Location Location Ratio (Hz) D1 D2 D2/Dl P1 P2 P2/Pl 3.4 E F1 0.9 1.6 1.9 39 20 0.51 3.4 C 2.8 0.54 4.4 E 81 13.2 38.5 2.9 65 60 0.92 4.4 C 2.8 0.98 10.0 E F3 0.2 0.2 0.8 12 11 0.92 10.1 C 2.5 0.98 16.8 E F5 0.7 0.1 0.1 22 O 0.00 16.8 C 10.8 0.32 18.1 E 82 12.2 2.4 0.2 11 0 -- 17.5 C 0.1 0.20 22.7 E F7 0.2 0.1 0.5 7 0 0.01 23.4 C 0.5 0.75 28.1 E 83 2.9 0.6 0.2 30 54 1.80 28.5 C 0.2 1.54 30.0 E F9 1.2 0.4 0.3 263 228 0.87 29.9 C 1.3 0.84 E - experimental, C - computed, F - fluid, 8 - structural frequency Chapter 6 I Experimental Results and Comparisons a: H (D E 164 mums: EEPERIMENTAL.AND COMPUTED RESULTS FOR CONFIGURATION 8 Displacement Pressure (mm) (kPa) q. Type Location Location Ratio Location Location Ratio ) D1 D2 D2/D1 P1 P2 P2/P1 .4 E Fl 1.1 2.0 1.8 39 27 0.69 .4 C 2.8 0.72 .4 E 81 13.6 40.4 3.0 41 68 1.66 .4 C 2.8 1.65 O E F3 0.2 0.2 1.0 35 21 0.60 1 C 2.4 0.63 .6 E F5 0.8 0.0 0.0 6 0 0.00 .7 2.7 0.80 .l E 82 11.2 2.3 0.2 0 0 - - .5 C 0.5 .7 E F7 0.3 0.2 0.5 87 45 0.52 .2 C 0.7 0.54 .3 E S3 5.0 1.2 0.2 56 16 0.30 .6 C 0.2 0.40 3 E F9 0.9 0.2 0.3 29 26 0.89 .9 C 0.1 0.89 - experimental, C - computed, F - fluid, S - structural frequency Chapter 6 Experimental Results and Comparisons 165 was INEERIMENTAL.AND COMPUTED RESULTS FOR.CONFICURATION 9 Displacement Pressure (Inn) (kPa) Freq. Type Location Location Ratio Location Location Ratio (Hz) D1 D2 D2/D1 P1 P2 P2/P1 2.9 E Fl 0.2 0.0 0.0 22 14 0.63 2.9 C 2.8 0.65 4.4 E 81 13.8 39.8 2.9 26 54 2.05 4.4 C 2.8 1.95 8.5 E F3 0.1 0.2 2.0 14 11 0.82 8.8 C 2.5 0.84 14.4 E F5 0.3 0.1 0.2 21 7 0.35 14.5 C 2.2 0.36 18.1 E 82 10.5 2.1 0.2 0 O -- 17.5 C 0.5 20.3 E F7 0.1 0.2 2.0 37 33 0.90 20.2 C 1.0 0.99 26.1 E F9 0.6 0.1 0.2 99 0 0.00 25.8 C 0.2 0.09 28.3 E 83 4.5 1.0 0.2 36 39 1.09 28.6 C 0.2 1.83 E - experimental, C - computed, F - fluid, S - structural frequency Chapter 6 Experimental Results and Comparisons 166 a) Tlrne response 0 on ’ ~ ._——— fl- - -- _——— _—— __. ’— Relatlve Displacement o I I ' ‘7 vvvvvvvvvvvvvv 0.8 - 0.8 - 0.4 l 0.2 ~ Amplltude 0.0 1 1 l O 3 U 9 (II (I, / B U-bond fro. —--—--U-b0nd Ilth lpflng Figure 6.1 Experimental Results of Snap-Back Test, U-Bend Empty, Frequency of Free Bend is 4.4 Hz Experimental Results and Comparisons Chapter 6 167 a) Time response aceEeoEaeE 320.8. o \\ 12. ‘I‘ II\|| \ \Iu r "I' ll'- III‘II‘ I‘lllsI 5 as l IIiiII 1. 1 I "I'-lel" I |‘\|I\II e 1" ' V I'Illll'l “' v ) ‘III I. . 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Jd . g :2. ._ n. . a c . u ... . . _ ...,. . a W a \ a . o m m e . . . e m m C v v ‘ ‘ o - ~ \ O m c u - c . 1.. e \I e! _ . _ — . \I . .a \I _d MW e. o m .3 ~ . _ 2 "..——_—_.. . mv nevrx .. 1 ._. e .23 m‘ N . g .. g _ . .. a .. ¢ . : on on . ‘ c a no. use a 5:83:50 3 t cozocsuzcoo as Experimental Results and Comparisons Chapter 6 «mph unom-b .ouoman «swab cunnsm you Hum m.m auswum cam pomoau um enummcum .:..l.i.l...i.| 303m nsomé um uses—003930 *6 DJ 1]. w/ .1 Mr u.N xucoaoet 3:: .2 cozocsuzcoo an 0.0 N6 ‘0 a: n.n xoceavot 2:: .n cozocauzcoo Ac opnmdwv opnmdmv f «.0 . to 1 0.0 j a 0.0 has a: o.+ xocoaoot 2:: J. cozocsutcoo an opnmdwv opmudwv Experimental Results and Comparisons Chapter 6 172 C e- I (mm) zueuleooldsm aniluill 3on5. asses as use-3039.3 means—m £33 .53-: .95 manage on ousuono 25.5 souusm ou none—oases aura House-Susanna 56 snow: Aev 25... a P D P D # D b y: r . z) r 2 E. s on n. 2 5:85:50 a. . 3 .5: « .m...im...-w....o on... .r m f s 1 M2... 1 9 t . v w o\ a a s a. .s a h. w A < . re \ml 2 1 (mi _ ... a c. o coast-3.230 an 8.1 3 g 0 (ad) eJnseu ° 5 5 (od) unseeJd' ' 4 can pesos us 35395 Aev es...— 3.... can. m C We... 8...“. o u m m m . . u 1. \l m .. m m 63 on A one o cozocsnccoo 50 any 25... e a a . o bDPprPPDD-D-PDPPbpb 3.1 on... not. 0 p I .f :... ::_. (ad) DJIIIIOJd . I ' V ' 1 I U I V " r (um) zusmeoomem § t sozocsmzcoo as Experimental Results and Comparisons Chapter 6 173 wanna—m so“: noon-= .ousuoao o>aa> cocoon uou E «6 shaman IIIIiIII. son—Hm neon-.— ua unassuming n 3 a m 0 \ a 0 j <-§- ..... . «.0 . v.0 .._ .. 2 _ .. a: «.N success: 2:: .3 cozocsuzcoo A“. ‘~ b . 0.0 I . _ .2 _ . _ . N... n.n xoceaeot Ba: .0 cozocsazcoo An... opnuldwv opmudwv USN OQmOHU ufi Ghfimmmwhm .a\a 0 b h D D b . 0.— 5.. ad hoses—vet 2:: .a cases—3:50 Au r 0.0 . 0.0 . 0.0 U . 0.. NI OJ. museums... 2:: J. cozocauzcoo an opnmdwv opnmdwv Chapter 6 Experimental Results and Comparisons 174 a) Liquid-filled U-bend U C l ...-"— A E E . V a 5 620‘ O 0 D T). .2 o 10- ‘ J A / \‘_ A‘ 1X? .. ........., : ' ' ' 1'5 ' 2'1 27 Frequency (Hz) b) Empty U-bend 40W 4 30- A E E v U 5 1 62°" 3 .9. J O. .2 I °1o '\ 1 1 ‘ \x- A c. ...j. ..13 ...2,1..,..2,7.., Frequency (Hz) Spring. D1 —--—---Eibow. Dz Figure 6.9 Experimental U-Bend Displacements for Harmonic Excitation Chapter 6 ' Experimental Results and Comparisons 175 a) First natural frequency 8) Second natural frequency 3 4 Asymmetrical D I D s Symmetrical B I f.- s.e Its f.- In II: i. : i : . ! I I l___ __ ...l a! a: a) a) Third natural frequency d) Fourth natural frequency D Asymmetrical I as Symmetrical 3I f.- 32.1 Ms D D D -—--—--—-- a s a s a e) Fifth natural frequency 0 Sixth natural frequency as Asymmetrical . 3I as Symmetrical B I f.- n.s its Normal mode shape —--—- Undlsturbed Figure 6.10 Computed U-Bend Mode Shapes, Free Bend Chapter 6 Experimental Results and Comparisons 176 a) First natural frequency 5) Second natural frequency B 4 Asymmetrical I D 4 Symmetrical B I f.- 4.4 II: f.- I'M II: i : i : . ! i I l_ _ _l B D D 3 z a c) Third natural frequency d) Fourth natural frequency 3 4 Asymmetrical I B 4 Symmetrical I f.- ss.sII: f.- sun: 3 '_"—" a s 3 a s e) Fifth natural frequency 0 Sixth natural frequency D 4 Asymmetrical I B s Symmetrical B I f.-II.0lfs f.- new: 8 --—-- --_-- D D --—-- B a z a 2 Normal mode shape —--—- Undlsturbed Figure 6.11 Computed U-Bend Mode Shapes, Bend Attached to Spring Chapter 6' 1.0 0.8 0.4 0.2 Pressure Amplitude Pressure Amplitude Pressure Amplitude on Io Figure Experimental Results and Comparisons 177 a) First Harmonic 0 0 . q———— Experimental 0.0 032 ' ' 0.4 0.6 018 ' ' ' 1:0 ' Location (s/f.) b) Third harmonic ' 0 (I) " O .flL W ' I ' I V I V I ' 030* ’ ‘oTz ' 'o.4 0.5 0.3 1.0 Location (e/t) c) Fifth harmonic \ 0 0 /\ qH—r I ‘ I J V 'o.4 0%" 033" Location (s/t) 6.12 Liquid Mode Shapes at Liquid Natural Frequencies Chapter 6 Experimental Results and Comparisons 178 d) Seventh harmonic d O O . a ‘ Experimental 1Lo.'o' ' ‘032' ' ‘o.'4' 'l 010' ‘ 'ofs' "130' Location (s/l) Pressure Amplitude .é .o Io Io l ... O O e) ninth harmonic 33\ /\ /\ V m m m OIO‘t‘0:2"'O:4' OTOV"O:Brr'1:O Location (e/l) 1 Pressure Amplitude P N A J L l l I d C o 1 Figure 6.12 (Continuation) Chapter 6 1.0 0.4 Pressure Amplitude 0.2 0.0 1.0 9 N I .° N i Pressure Amplitude -0.0 Experimental Results and Comparisons 179 a) First U-bend frequency - 4.4 Hz .l 7,—— U-ben'd location ..I cl ‘ (D i . Experimental 1 ' r I I ' I ‘ I‘ j I ' I ' I T I ' I ' 1 ‘ I ‘ 0.0 0.2 0.4 0.6 0.8 1.0 Location (ell) b) Third U-bend frequency - 28.3 H: d l ‘ (D j 1 q I ' I 1 T "V I ‘ I ffi ' I ‘ I I I ' I ‘ 0.0 0.2 0.4 0.0 0.8 1.0 Location (s/l) Figure 6.13 Liquid Mode Shapes at U-Bend Natural Frequencies, Configuration 1 Chapter 6 ' Experimental Results and Comparisons 180 1 a) First U-bend frequency - 4.4 Hz 1.0 ~ ' '8 ‘ ll be 4 i ti 3 0.8 ., - n oca on U 3 0.. . e .3. ‘ 0 E 0.4 - a. I Experimental 0.2 4 / 0.0 d \ ofo' ' '012' 1 '014' ' rofe' ' 'ofa' r 'Co‘ Lcccflon (e/t) b) Third U-bend frequency - 28.3 Hz 1.5 I 1.0 ‘ -~/l ' l 3 0.5 - 1‘." 1i 5; OJ! 4 2 I 3 3 8-0e5 4 a. -1.0 4 O -15 OIOr"0:2"'O:4r'VOTG'IVOIB I 1:0 Location (e/ 1) Figure 6.14 Liquid Mode Shapes at U-Bend Natural Frequencies, Configuration 2 Chapter 6 1.0 0.8 0.4 Pressure Amplitude 0.2 0.0 1.5 1.0 Pressure Amplitude o o l e Experimental Results and Comparisons 181 a) First U-bend frequency - 4.4 Hz Experimental / U-lIend location \ I 0 010' ' 'ofz' ' 'of4t ' 'ofc' ' '0st ' 'I.o' Location (e/l) b) Third U-hend frequency - 28.3 Hz J i O I r I ' I ' I I ' I ' I I I I f I ' I ' 0.0 0.2 0.4 0.5 0.8 1.0 Location (s/t) Figure 6.15 Liquid Mode Shapes at U-Bend Natural Frequencies, Configuration 3 Chapter 6 Experimental Results and Comparisons 182 1.4 - a) First U-hend frequency - 4.4 Hz 1.2 4 j \, 0.8 - Experimental 0.6 n Pressure Amplitude 0.4 - 0.2 d U-bend location/ . \ 0.0 1 I0 'I'I'Iifi'I'rfi I 0.0 0.2 0.4 0.6 0.8 1.0 Location (III) 2.5 1 h) Third U-hend frequency - 28.3 Hz Pressure Amplitude I ' I ' I ‘ I I I I I 1 I j I ‘ 0.0 0.2 0.4 0.6 0.8 1.0 Location (ell) Configuration 4 —--—- Configuration 5 Figure 6.16 Liquid Mode Shapes at U-Bend Natural Frequencies, Configurations 4 and 5 Chapter 6 1.0 P a Pressure Amplitude .0 9 Io a 9 o I P n 0.0 o 1: :3 '56. 0.2 § . -002 n E O. -0.0 -100 Experimental Results and Comparisons 183 a) First U-hend frequency - 4.4 Hz / U-bend location 0 Experimental . / AlAlL‘LI. .lalal lAiLlAl ' ' 'ofz' "'0141 ' ‘ofo' ' 'ofa' "130' Location (s/l) b) Third U-bend frequency - 28.3 Hz 0 I T I I ‘ I ' I r r ' I T I I I I 0.2 O 4 0.6 0.8 1 0 Location (e/t) Figure 6.17 Liquid Mode Shapes at U-Bend Natural Frequencies, Configuration 6 Chapter 6 1.8 1.5 Pressure Amplitude P o 0.3 2.0 Pressure Amplitude -2.0 Experimental Results and Comparisons 184 a) First U-bend frequency - 4.4 Hz .1 i cl 4 cl 4 - U-bend location I'I'I'T‘I'I'I‘I‘I'I'Ifi 0.0 0.2 0.4 0.6 0.8 1.0 Location (s/f.) 1 b) Third U-bend frequency - 28.3 Hz * 0 0 -i 4 cl 1 1 I I ' I ' I ' I '7T I I ' I ' I ' I ' I ' I I 0.0 0.2 0.4 0.6 0.8 i .0 Location (s/l) Configuration 7 —--—- Configuration 8 Figure 6.18 Liquid Mode Shapes at U-Bend Natural Frequencies, Configurations 7 and 8 Chapter 6 2.0 1.5 1.0 Pressure Amplitude 0.5 0.0 3.0 2.0 ‘00 Pressure Amplitude o lalmLel‘ Experimental Results and Comparisons 185 a) First U-bend frequency - 4.4 Hz Experimental i < U-bend location/ ' P I'fi'I'I'I‘I'r'III‘I'I' 0.0 0.2 0.4 0.6 0.8 1.0 Location (s/l) 1 h) Third U-bend frequency - 28.3 Hz 1 . .. '1 \ d T 0 036' "012' ' ‘034' ' 'oIo' ' 'oIs' ' '130' Location (s/l) Figure 6.19 Liquid Mode Shapes at U-Bend Natural Frequencies, Configuration 9 Chapter 7 SumnIary and Conclusions The primary objective of this study was to incorporate the flexural and torsional modes of vibration of liquid-filled pipe systems to an exist- ing axially coupled model. The motion of the pipe wall and the contained liquid was represented by using a one-dimensional approxima- tion. This approximation has been proved valid for the first lobar mode of the pipe cross section. A system of fourteen equations and fourteen dependent variables described the motion of the piping. Five families of waves that propagate in the pipe wall and in the liquid were iden- tified. The analytical model incorporated the Poisson and junction coupling mechanisms and included the effect of shear deformation and rotary inertia of the lateral motion of the pipe. The inclusion of these mechanisms represents appropriately the motion of the systems and constitutes an improvement over the previous model by Wilkinson [64] . The transfer matrix method was the numerical model used for the analysis of these systems. The method can predict the pipe wall displacements and forces as well as the pressure and displacement of the liquid. The model provides an alternative to other numerical and analytical methods. 186 Chapter 7 Summary and Conclusions 187 In addition,the Poisson and junction coupling were properly treated. The methodology to incorporate pipe constraints, such as rigid supports, springs, and inertia and external forces, was presented. The inclusion of hydraulic devices, such as orifices and pumps, may be easily ac- complished by the use of point matrices. The field transfer matrices for the flexural modes, developed by Pestel and Leckie [50], were modified to include the mass of the contained liquid. The field transfer matrix for the liquid-axial pipe wall was derived based on the model developed by Wiggert et al. [20]. Four submatrices were identified. The magnitude of the terms of this matrix in the analysis of liquid-filled pipes depends on the frequency at which the system oscillates. The compliance terms may be neglected for high frequency analysis. However, the main diagonal terms of each submatrix are important for low frequency studies. The results from the transfer matrix method (TMM) were compared with numerical methods such as the method of characteristics (MOC) and the component synthesis method (CSM). The TMM exhibited advantages over the other two methods. The run is a one-step computation, whereas the CSM requires two steps for the analysis. In contrast to the MOC, the TMM does not require interpolations for the analysis of systems subjected to harmonic oscillations. Experimental data, available in the literature were also used to provide validation of the transfer matrix method. An experimental apparatus was designed and built to validate the numeri- cal method. A one-inch, water-filled, copper pipe with a U-type bend Chapter 7 Summary and Conclusions 188 was excited with either a transient or harmonic loading to study the response of the system. The experimental tests were conducted on a liquid-filled pipe system with closed-open conditions for the liquid and fixed-fixed conditions for the U-bend. The natural frequencies of the liquid were varied by changing the length of the pipe. The harmonic excitation was applied to a U-bend that was allowed to vibrate in one plane. Numerical analysis results were compared to experimental results. The following conclusions were drawn from the experimental tests: 1) The snap-back transient test was used to calibrate the numerical model. The addition of the spring increased the stiffness of the U- bend. This additional stiffness increased the natural frequencies of the bend. Computed results showed that the spring increased by 0.5 Hz the natural frequencies of the U-bend. 2) The other transient test used was rapid valve closure. The closure of the valve excited the liquid odd harmonics, but only the first fre- quency of the U-bend. The time response of the liquid pressure and the displacement of the U-bend were presented. A fast Fourier transform analysis of the time series was performed to obtain the natural fre- quencies of the system. Good agreement between the experimental liquid harmonics and the corresponding computed results was obtained. The computed frequencies were obtained by assuming no shear or bending for the pipe legs between the valve and U-bend. The same assumption was made for the pipe legs and between the U-bend and the reservoir. Only the axial pipe wall and liquid modes were considered in these pipe Chapter 7 Summary and Conclusions 189 reaches. This test was also used to measure and compare the liquid wave speeds. The experimental results were compared with the wave speed obtained by Budny [22] . The computed results showed that the wave speeds were within 0.5% of the experimental values. 3) The harmonic test was used to excite the U-bend, thereby, exciting the liquid. Three U-bend frequencies and nine harmonics of the liquid were excited. Spectral analysis showed that large pressure responses occured at frequencies near the asyInetrical modes of the U-bend. These modes allowed motion of the elbows which generated the junction cou~ pling mechanism. This coupling mechanism was the primary factor to magnify the pressure. The second mode of the U-bend, which corresponds to a symetrical mode, did not excite the liquid pressure. The computed results predicted the natural frequencies of the system. The magnitude of the pressure response was increased when a liquid frequency was near one of the asymmetrical modes of the U-bend. 4) Variations of the pipe length changed the relative location of the U-bend and transducers with respect to the closed-end. A total of nine pipe configurations were studied. This allowed measurement of the liquid pressure mode shapes at discrete points for each pipe configura- tion. Computed mode shapes showed good correlation with the experimental points. The pressure mode shapes at the natural fre- quencies of the liquid corresponded to the odd harmonics of a l/lI sine wave. The mode shapes at the U-bend frequencies show an abrupt change at the horizontal leg of the U-bend. The results showed that at these frequencies the magnitude of the pressure at the bend can increase as Chapter 7 Summary and Conclusions 190 much as 100‘ from the pressure at the closed end. The magnitudes of the pressure responses at the fluid frequencies are larger than those at the U-bend frequencies . In summary, the tranfer matrix method is appropriate to predict the natural frequencies of liquid-filled pipesf Poisson and junction cou- pling are modeled with the use of this method. The experimental results showed that the larger pressure responses occured at higher harmonics and that the responses were magnified when the liquid frequency was near one of the asymmetrical modes of the U-bend. These modes allow motion at the elbows generating the junction coupling. This mechanism amplifies the pressure response of the system. Natural frequencies of complicated piping systems can be estimated by including the flexural, liquid and axial modes at locations where these modes may affect the response of the system. Other reaches can be analyzed by including the appropriate modes, for example only liquid or both liquid and axial. The model used in this study allowed motion in only one plane. It did not include fluid friction or structural damping. The extension to a three dimensional space can be accomplished by incorporating the torsion mode as well as the flexural mode of the out of plane motion. Experiments are necessary to estimate the influence of these modes on the responses of the system. The incorporation of energy dissipation into the model is necessary to estimate the magnitude of the responses to an excitation. APPENDIXA Appendix A Liquid-Axial Pipe Wall Transfer Matrix APPENDIXA A.1 Intreducticn The field transfer matrix [Tfp] shown in Equation 4.20 will be analyzed for two cases. First, Poisson's ratio is set to zero, thereby decou- pling the axial pipe wall and liquid vibration. The results are then compared with those of other authors. Second, Poisson's ratio is taken nonzero; in this case the analysis will focus on the orders of mag- nitude of the matrix terms when the frequency of oscillation varies. The analysis is facilitated by arranging the matrix in Equation 4.20 into four submatrices. The arrangement yields: P . g1; 0C2'Co ’C1+(U+‘Y)Cs 5 $702 11 [C1‘(U+T+‘1)C3] a(c,-ac,) acz-co : -§Parc, - figec, [Tfpl- 2uaC 2v[(a+r+1)C -c 1' (I+1)c -c l[(r+1)C -[(r+1)2+a ]c ] 2 31' so 1 ‘73 T 21’0703 '2VTC2 :T[(f+1)C3'CI] (TH)C:‘CO (A.l) 191 Appendix A Liquid-Axial Pipe Wall Transfer Matrix 192 The state vector associated with the above matrix is AE T U F 2 - { 33 -3 ¥ 2* } (A.2) p K The matrix in Equation A.1 can also be written as [rfp1- (A 3) I l =1;- Notice that the coupled submatrices T2: and T2; contain the factors U and 2v, respectively. Also, the main diagonal terms of the four su matrices are functions of the cosine of the eigen values A1 and A2, whereas the other nondiagonal terms depend on the sine function of the same eigen values. The expressions for these coefficients are given in Equations 4.21g through 4.20j . A-2 unseunled_Analx§ia Setting Poisson's ratio to zero results in uncoupling the axial pipe wall and liquid variables. The transfer matrix becomes separated into two sub-matrices. The liquid matrix is P cos (wt/at.) -j 8111((01/8f) p ... (A. 4) pfafi'l i -jsin(w£/af) casual/sf) i Pfaf0 1,1 _._—- up Appendix A Liquid-Axial Pipe Wall Transfer Matrix 193 where 9 represents the liquid velocity amplitude. This result agrees with that of Wilkinson [64], Chaudhry [7] and Wylie and Streeter [6]. The axial pipe wall matrix is a cos(w£/a ) jsin(o)l/a ) a { "oH . PH ‘0} pfap z 1 jsin(w£/ap) cos(w£/ap) 1 pfap z 14 where Oz represents the axial pipe wall velocity amplitude and oz is the axial stress. This matrix agrees with the matrix presented by Wilkinson [64]. A3 We The importance of the coupling terms of the transfer matrix can be studied by using an order of magnitude analysis. An inspection of the matrix terms in Equation A.l shows that the coefficients and the eigen value parameters a, r and 1 are function of the frequency in addition to the liquid and pipe material properties. The terms u, b and h depend on the liquid and pipe material properties. The radius to thickness ratio, b, and the frequency of oscillation, w, are the parameters varied. The Young's modulus to modified bulk modulus ratio, h, and Poisson's ratio, u, also affect the order of magnitude of the matrix terms, but they will be kept constant in this analysis. For comparison, a discussion based on numerical evaluations will be presented for pipes of five different Appendix A Liquid-Axial Pipe Wall Transfer Matrix 194 3 materials. The liquid is water with K - 2.2 GPa and pf - 1000 kg/m . A similar analysis was performed by Otwell [19] and Stuckenbruck [21] . The physical properties used for the pipe material are shown in Table A.l [21]. IABLLAJ PIPE MATERIAL PROPERTIES Material Young's Modulus Density Poisson's Ratio 3 GPa kg/m Steel 210.0 7600 0.27 Cast Iron 80.0 7600 0.25 Copper 115.0 8800 0.34 Aluminum 70.0 2700 0.33 Polyethylene . 0.8 1000 0.46 The influence of the pipe cross-section geometry ratio, b, can be facilitated by defining the ratio of coupled wave speeds as c c - :9 (A.6) f This ratio can also be defined as the eigen value ratio A1 c - x; (A.7) The relations between the coupled wave speed and eigen value for the liquid and axial pipe wall are defined in Equations 3.22 and 3.23. The Appendix A Liquid-Axial Pipe Wall Transfer Matrix 195 variation of c with respect to the pipe cross section ratio is shown in Figure A.1a for values of b between 10 and 160. Figures A.1b and A.1c show the coupled wave speeds for both the liquid and pipe in the axial direction for each of the five pipe materials studied and for the same pipe cross section range. The wave speed ratio ranges within one order of magnitude for values of b less than 160 (Figure A.1a), except for the polyethylene pipe. For example, for the copper pipe the variation of c is 4 (between 3 and 7), whereas for the polyethylene it is 17 (between 5 and 22). The variation of the wave speed ratio is due to a faster decrease of the liquid speed over the axial speed as b increases (Figure A.1b and A.1c). Therefore, the radius to thickness ratio does not introduce appreciable changes in the order of magnitude of the terms in the trans fer matrix . The variation of the matrix terms with respect to the frequency of oscillation, w, is analyzed for a copper pipe with b - 10 and liquid natural frequency of 4 Hz. Table A.2 shows the value of the coeffi- cients as the frequency increases. The characteristic parameters a, r and 7 are also shown.1\\l'1'he uppervlimiflt frequency is given by the first Vi lobar mode of the pipe cross section, “’0' As shown by Everstine et al. [86], a one-dimensional analysis is not valid for frequencies greater than “9' The frequency expression for the first lobar mode is given in Equation 3.7. This equation may also be written as Appendix A Liquid-Axial Pipe Wall Transfer Matrix 196 3 *2 a .2 ' r 5b2(1-u2)(l+b/2d) (A'a) “o For example, for the copper pipe of Table A.l with r - 0.1 m and b - 10 the first lobar mode frequency ”9 is 2380 rad/s or 380 Hz. IBELE‘AIZ MSFERHAIRIXPWS Coefficient Equation Amplitude Value Value at w -wo 2 2 Co 4.20g i(l+c )/(c -l) 1.25 1.3 s 2 C1 4.20h i(l+c )/A,(c -l) 3.50/A1 0.0 2 2 2 2 C2 4.201 12c /A1(c -1) 2.25/11 0.0 s 2 2 C, 4.20j ic(l+c)/A,(c -l) 1.50/A1 0.0 2 2 2 I 4.20a cfi,/a§ 0.96.\1 9075 2 2 2 a 4.20b c A,/a2 1.04;, 27220 2 p 2P2 2 2 1 4.20c 2v b/d cpxz/ap 0.27), 7070 The results given in Table A.2 show that the trigonometric coefficients depend on l/ArlI where n - 0,1,2 or 3 as (9 increases, whereas the eigen- value parameters depend on the square of the eigenvalues. The coefficients are shown in Figure A.2a for the copper water-filled pipe when c is 2.8. Figure A.2b shows the eigen values for the same piping system. The sixteen terms of the transfer matrix are plotted in Figures A.3 through A.6 for varying ratios of oscillation to the liquid natural frequency. Figure A.3 shows the terms for the axial pipe wall sub- matrix. The coupling submatrices are shown in Figures A.4 and A.5. Appendix A Liquid-Axial Pipe Wall Transfer Matrix 197 Finally, the liquid submatrix is shown in Figure A.6. The following observations can be made 1) The main diagonal terms of the liquid and pipe wall submatrices fluctuate between -1 and l. 2) At low frequencies, the main diagonal terms of the matrix, which depend on Co start at a value of l. The other terms are one order of magnitude lower than the main diagonal terms. This result is also shown in Figure A.2a. 3) At high frequencies, the amplitude of the matrix terms (2,1), (2,3), (4,1) and (4,3) increase as the frequency increases. These terms are associated with the apparent stiffness ratio, force or pressure over displacement. The amplitude of the terms (1,2), (1,4), (3,2) and (3,4) decrease as the frequency increases. These terms are associated with the compliance ratio, displacement over force or pressure. The above observations and the results in Figures A.3 through A.6 show that the compliance terms may be neglected for high frequency analysis. The main diagonal terms can be use for frequencies less than the first liquid frequency and the other terms may be neglected at low fre- quencies. Appendix A Liquid-Axial Pipe Wall Transfer Matrix 198 a) Wave Speed Ratio b its“ Copper Cast Iron Aluminum PclyetMene ‘ 5000 4000 m/e 3000 2000 1000 I Figure A. 1 Liquid and Axial Pipe Wall Wave Speeds Versus Pipe Cross Sectional Ratio Appendix A Liquid-Axial Pipe Wall Transfer Matrix 199 a) Trigonometric Function Coefficients A A A A A_AAL1 A A A A AAA. A A A AAAALL _--—--j 1. —-—- c —_ c . 1 t 3 a 0. ________o 5 ‘ g / r *4 -—-¢"’ P F -1. . 0.1 ' ' "nfil ' ' 'In'i'o ' ' "T'l O m / a f b) Eigen Value 1 h I 100 : :' i E . 10 ‘ g :I 1 I a . ' > 4 I . 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Appendix B Data Acquisition APPENDIX! 8.1 Wan To obtain the required information on harmonic behavior, the dependent variables of liquid pressure and structural displacement must be re- corded as a function of time. This was accomplished by using PCB pressure and acceleration transducers interfaced with either a Digital PDP-ll/73 computer, or a Tektronix D13 dual beam storage oscilloscope. 3-2 W The analog output signals of the transducers, which are directed to the computer, are converted to a digital format by an analog/digital board, with the sampling rate controlled by a programmable clock board. The software required to perform this conversion and data storage is described below in the Data Acquisition Software section. A schematic of the components of the data acquisition system is shown in Figure 5.7. Each component is described below. 204 Appendix B Data Acquis ition 205 8.2.1 Piezoelectric Pressure Transducers The principle of a piezoelectric transducer is that a charge is produced across the piezoelectric crystal, which is proportional to the applied pressure. Since this type of transducer is designed to measure dynamic and short term static pressure measurements, all pressure readings taken are dynamic pressure variations about a steady state static pressure. For this study PCB Piezotronics Models lllA26 and 113A24 Dynamic Pressure transducers with built-in unity gain voltage amplifiers were used to measure the liquid pressure within the pipeline at the harmonic and fast acting valve. These units were selected because of their high resonant frequency, acceleration compensated quartz element, and the fact that the signal quality is nearly independent of cable length and motion. Table 8.1 lists the published properties and the calibration properties as determined by the manufacturer. The calibration procedure was in compliance with MIL-STD-45662. m1 ROPER‘I'IES 0? PRESSURE TRANSDUCERS Property Units Value PCB Serial No. 111.52g 1.1329. Range (5 volts output) psi 500.0 1000.0 MPa 3.447 6.894 Resolution (min. value) psi 0.1 0.01 Pa 689.4 69.9 Sensitivity (output) mV/psi 9.71 5.23 mV/kPa 1.41 0.76 Resonant Frequency KHz 400.0 425.0 A/D Error @ Gain of 1 psi 0.97 0.97 kPa 6.73 6.73 Linearity (error) %bsl 2 . 0 2 . 0 Appendix B Data Acquisition 206 Connected to each pressure transducer is a PCB Battery Power Unit. The units are PCB Model 480DO6 with 1,10, and 100 range signal amplifiers. The function of each battery power unit is to power the transducer electronics, amplify the signal, remove bias from the output signal and indicate normal or faulty system operation. It is a combination power supply and signal amplifier. The transducers were mounted by tapping a brass block as per PCB specifications. The block was designed so that the end of the transducer would be flush mounted with the inside diameter of the pipeline. Since the end of the transducer is flat and the block was tapped with a circular hole, the mounting is not flush mounted. There is a small deviation due to the curvature of the hole. B.2.2 Quartz Accelerometers The principle of a quartz crystal accelerometer is that a charge is produced across the crystal in proportion to the applied acceleration. For this study PCB Piezotronics Model 302A Low Impedance Voltage Mode quartz accelerometers were used to monitor the desired motion. These units were selected because of their ability to measure the acceleration aspect of shock and vibration motion from lg to 500g, over a wide fre- quency range. They also offer exceptionally sensitive low frequency response, can follow long duration shock events, and have built-in amplifiers. These types of accelerometers are not linear for fre- quencies less than 10 Hz. A calibration curve was obtained for each Appendix B Data Acquisition 207 accelerometer to find the conversion from volts to millimeters. These curves are shown in Figure 8.1. A direct displacement measurement technique was used. The accelerometers were attached to the shaker mechanism shown in Figure 5.3 in place of the spring. The circular frequency of the motor was increased and the displacement output from the accelerometers was compared with observed readings from a displace- ment meter for the same frequency. The range of the displacement meter was one inch, with 1000 divisions per inch. Table 8.2 lists the pub- lished and calibration properties of the transducers as determined by the manufacturer. The calibration procedure was in compliance with MIL- STD-45662. The accelerometers were installed by clamping the base of the transducer to the test object with an elastic beryllium-copper threaded stud. To accomplish this, a mounting collar was designed and used to install the transducer at any point along the pipeline. The collar is made of a 63 mm square, 19 mm thick aluminum block. The block is tapped to accept the mounting stud, and a hole is drilled through the center of the block to match the OD of the one inch nominal copper pipe. The block is then cut through the center of the hole and bolt holes are drilled through both sections. Bolts are then used to hold the two pieces together enabling the block to act as a vise squeezing the pipe around its entire circumference . Appendix 8 Data Acquisition 208 mu PROPERTIES 0F.ACCELEROHETER.TRANSDUCERS Property units value 302A Serial No. Range FS (5 volt output) g 500.0 500.0 500.0 Resolution g 0.01 0.01 0.01 Sensitivity mv/g 10.04 10.03 10.04 mv/ft/s 0.831 0.832 0.831 mv/m/s 0.253 0.253 0.253 * mv/mm 322.5 322.5 322.5 Resonant Frequency kHz 45.0 45.0 45.0 Frequency Range (:58) Hz 1 - 5000 A/D Conversion Error g 0.976 0.976 0.976 Linearity %FS 1.0 1.0 1.0 Integration Error % 5.0 5.0 5.0 * For frequencies below 10 Hz, the conversion factors for displacement are obtained from Figure 8.1 Connected to each transducer is a PCB Dual Integrating Power Unit Model 48OAlO. The function of this unit is to supply constant current excita- tion to power ICP sensors over signal lead, eliminate DC bias voltage on output signal by capacitive decoupling, monitor bias voltage on sensor lead for normal or faulty operation by meter indication, and provide either acceleration or velocity output signals. In addition to the above features, it also provides a displacement output signal. 8.2.3 Computer Hardware and Accessories The computer used for the data collection was a Digital Equipment Corporation DEC PDP-ll/73. The installed operating system was RSX-llM- PLUS version 3.0. In addition to the standard equipment present within Appendix B Data Acquisition 209 a PDP-ll/73 system, an analog-to-digital converter and a programmable real-time clock board were installed to facilitate data acquisition. To direct the input and output signals to their appropriate locations, a patch panel was constructed and mounted on the face of the computer cabinet. 3.2.3.1 Analog-to-Digital Converter The AXVll-C is an LSI-ll analog input/output printed circuit board. The board accepts up to sixteen single-ended inputs, or up to eight dif- ferential inputs, either unipolar or bipolar. A unipolar input can range from 0 volts to +10 volts DC. The bipolar input range is :10 volts DC. The analog-to-digital (A/D) output resolution is 12 bit unipolar, or 11 bit bipolar plus sign, with output data notation in octal coding of binary, offset binary, or 2's complement. The A/D converter performance has a system throughput of 25K channel samples per second, with a system accuracy input voltage to digitized value of plus or minus 0.03% full scale. The board also has two separate digital-to-analog converters (DAC). Each DAC has a write-only register that provides lZ-bit input data resolution, with an accuraqy¢of plus or minus 0.02% full scale. By setting the required jumpers on the board, the AXVll-C was configured for bipolar differential inputs with the external trigger set to the I/O connector. The I/O connector was then hardwired to the KWVll-C program- mable real-time clock overflow. Appendix B Data Acquisition 210 B . 2 . 3 . 2 Programmable Real-Time Clock The KWVll-C is a sixteen bit resolution programmable real-time clock printed circuit board. It can be programmed to count from one to five crystal-controlled frequencies, from an external input frequency or event, or from the 50/60 Hz line frequency on the LSI-ll bus. The five internal crystal frequencies are 1 MHz, 100 kHz, 10 kHz, 1 kHz, and 100 Hz. The base frequency for the clock is 10 MHz, thus the accuracy of the time measurement is i 0.1 microseconds. The clock also has a counter that can be programmed to operate in either a single interval, repeated interval, external event timing, or external event timing from zero base mode. In addition to its clock functions, the KWll-C also has two Schmitt triggers. The triggers can be set to operate at any level between i 12 volts DC on either a positive or negative slope of the external input signal. In response to external events, the Schmitt trigger can start the clock, start A/D conversions in an A/D input board, or generate program interrupts to the processor. 8.2.3.3 Patch Panel To facilitate use of these data acquisition computer boards, a patch panel was installed on the front of the computer cabinet. It has BNC connectors installed which allow access to the eight differential A/D inputs, the two D/A outputs, and both Schmitt triggers. Switches and potentiometers for each Schmitt trigger were also installed to allow Appendix B Data Acquisition 211 external control of both the slope and triggering level. In addition, the panel also contains a three volt DC power supply with a connection to the KWVll-C board. 3.2.4 Data Acquisition Software Digital's K-Series Peripheral Support Routines were used for data ac- quisition. These machine language routines perform input and output operations through the Connect to Interrupt Vector Executive directive. The routines are highly modular, that is they are designed to perform specific operations. Thus, to complete the sampling, a user program is required to call each routine as various functions. are to be performed. A Fortran computer program was developed to facilitate the data acquisi- tion process. The program accessed the routines for computing and setting the clock rate, setting the A/D channel sampling information, creating and maintaining buffers to store the sampled data, and starting and stopping the sampling. The program was divided into two parts. The first part of the preprocessor is an interactive program. This program allows the user to select the sampling rate, number of channels to be sampled, number of samples per channel, the data acquisition device connected to each channel, and the range of frequencies to be sampled. The second part is the actual sampling routine. This program is designed so that the sampling process is started upon indication of the user. After the sampling process is finished the program requests the user to change the frequency of the motor. At this time the program Appendix B Data Acquisition 212 allows 30 seconds for the system to reach steady-state conditions. This process continues until all the frequencies of the frequency range have been sampled. The experimental procedure as well as the hardware and sofware com- ponents for rapid valve closure are described by Budny [22] . Appendix B Data Acquisition 213 70 -u. v- 50 -v- E E " 5 -~ L d- 3 5 so -- c d- .2 g -_ e > c all- 8 10 -"- -10 _ 0 6 12 1B 24 30 Frequency (Hz) Figure B.l Displacement Calibration Curve for PCB Accelerometers LIST DEW LIST 0? REFERENCES l) Thorley, A. R. D., "Pressure Transients in Hydraulic Pipelines," WWW. Vol. 91. Sept. 1969. pp. 453-461- 2) Joukowsky, N. E., translated by 0. Simin as "Water Hammer," - , Vol. 24, 1904, pp. 341- 424. 3) Lamb, 1-1. , "0n the Velocity of Sound in a Tube, as Affected by the Elasticity of the Walls," W. 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