Machinery Malfunction
Diagnosis and Correction
Vibration Analysis and Troubleshooting
for the Process Industries
Robert C. Eisenmann, Sr., P.E.
President — MACHINERY DIAGNOSTICS, Inc. — Minden, Nevada and
Robert C. Eisenmann, Jr.
Manager of Rotating Equipment — HAHN & CLAY — Houston, Texas
PTR Prentice Hall, Englewood Cliffs, New Jersey 07632
The original Hard Copy format of this book was previously published by: Pearson Education, Inc. Copyright Assigned to Robert C. Eisenmann, Sr. by Hewlett-Packard effective June 6, 2005.
Global Machinery Diagnostics Services Manager - GE Energy - Sugar Land, Texas
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Machinery malfunction diagnosis and correction: vibration analysis and troubleshooting for the process industries / Robert C. Eisenmann, Sr., and Robert C. Eisenmann, Jr.
p. cm -- (Hewlett- Packard professional books) Includes bibliographical references and index.
ISBN 0-13-240946-1
1. Machinery -- Monitoring. 2. Machinery -- Vibration. I. Eisenmann, Robert C., II. Title. III. Series.
TJ153.E355 1997
621.8'16-dc21 97-31974 CIP .
To Mary Rawson Eisenmann, Wife and Mother
Who Always Kept The Home Fires Burning While The Boys Went Off To Play With Their Machines
iv
Table of Contents
Preface
. . . xi
Chapter 1 - Introduction
. . . 1
Machinery Categories 4 Chapter Descriptions 5 Bibliography 8Chapter 2 - Dynamic Motion
. . . 9
Malfunction Considerations and Classifications 9 Fundamental Concepts 10
Vector Manipulation 21 Undamped Free Vibration 28
Case History 1: Piping System Dynamic Absorber 31 Free Vibration with Damping 37
Forced Vibration 45
Case History 2: Steam Turbine End Cover Resonance 55 Torsional Vibration 58
Bibliography 66
Chapter 3 - Rotor Mode Shapes
. . . 67
Mass and Support Distribution 67
Case History 3: Two Stage Compressor Rotor Weight Distribution 72 Inertia Considerations and Calculations 74
Damping Influence 96 Stiffness Influence 105 Critical Speed Transition 120 Mode Shape Measurement 130
Case History 4: Vertical Generator Mode Shape 137 Analytical Results 142
Case History 5: Eight Stage Compressor Mode Shape Change 143 Bibliography 148
Chapter 4 - Bearings and Supports
. . . 149
Fluid Film Radial Journal Bearings 150
Case History 6: Shaft Position In Gas Turbine Elliptical Bearings 161 Fluid Film Radial Bearing Clearance Measurements 165
v
Bearing Supports — Measurements and Calculations 179
Case History 8: Measured Steam Turbine Bearing Housing Stiffness 181 Case History 9: Measured Gas Turbine Bearing Housing Stiffness 185 Bearing Housing Damping 187
Fluid Film Thrust Bearings 188 Rolling Element Bearings 193
Before Considering Bearing Redesign 196 Bibliography 198
Chapter 5 - Analytical Rotor Modeling
. . . 199
Modeling Overview 199 Undamped Critical Speed 201
Case History 10: Mode Shapes for Turbine Generator Set 206 Case History 11: Torsional Analysis of Power Turbine and Pump 208 Stability and Damped Critical Speed Calculations 213
Case History 12: Complex Rotor Damped Analysis 217 Forced Response Calculations 222
Case History 13: Gas Turbine Response Correlation 226
Case History 14: Charge Gas Compressor with Internal Fouling 230 Case History 15: Hybrid Approach To A Vertical Mixer 236
Bibliography 242
Chapter 6 - Transducer Characteristics
. . . 243
Basic Signal Attributes 244
Proximity Displacement Probes 253 Velocity Coils 272
Piezoelectric Accelerometers 278 Pressure Pulsation Transducers 285 Specialized Transducers 288 Aspects of Vibration Severity 294 Bibliography 302
Chapter 7 - Dynamic Signal Characteristics
. . . 303
Electronic Filters 303
Time and Orbital Domain 316 Time and Frequency Domain 333
Case History 16: Steam Turbine Exhaust End Bearing Dilemma 343 Signal Summation 347
Case History 17: Opposed Induced Draft Fans 349 Amplitude Modulation 353
Case History 18: Loose and Unbalanced Compressor Wheel 356 Frequency Modulation 359
Case History 19: Gear Box with Excessive Backlash 362 Bibliography 364
vi
Chapter 8 - Data Acquisition and Processing
. . . 365
Vibration Transducer Suite 365 Recording Instrumentation 369 Data Processing Instrumentation 379 Data Presentation Formats 383 Bibliography 394
Chapter 9 - Common Malfunctions
. . . 395
Synchronous Response 395 Mass Unbalance 398 Bent or Bowed Shaft 400
Case History 20: Repetitive Steam Turbine Rotor Bow 402 Eccentricity 406
Case History 21: Seven Element Gear Box Coupling Bore 407 Shaft Preloads 410
Resonant Response 416
Case History 22: Re-Excitation of Compressor Resonance 419 Machinery Stability 422
Case History 23: Warehouse Induced Steam Turbine Instability 429 Case History 24: Pinion Whirl During Coastdown 432
Mechanical Looseness 435
Case History 25: Loose Steam Turbine Bearing 438 Rotor Rubs 440
Cracked Shaft Behavior 443
Case History 26: Syngas Compressor with Cracked Shaft 446 Foundation Considerations 449
Case History 27: Floating Induced Draft Fan 451
Case History 28: Structural Influence of Insufficient Grout 454 Bibliography 458
Chapter 10 - Unique Behavior
. . . 459
Parallel Shaft - Two Element Gear Boxes 459
Case History 29: Herringbone Gear Box Tooth Failure 466 Epicyclic Gear Boxes 470
Case History 30: Star Gear Box Subsynchronous Motion 477 Process Fluid Excitations 483
Case History 31: Boiler Feed Water Pump Splitter Vane Failures 496 Case History 32: Hydro Turbine Draft Tube Vortex 499
Electrical Excitations 507
Case History 33: Motor With Unsupported Stator Midspan 515 Case History 34: Torsional Excitation From Synchronous Motor 519 Reciprocating Machines 522
Case History 35: Hyper Compressor Plunger Failures 526 Bibliography 534
vii
Chapter 11 - Rotor Balancing
. . . 535
Before Balancing 536
Standardized Measurements and Conventions 539 Combined Balancing Techniques 545
Linearity Requirements 547
Case History 36: Complex Rotor Non-Linearities 548 Single Plane Balance 552
Case History 37: Forced Draft Fan Field Balance 560 Two Plane Balance 564
Case History 38: Five Bearing, 120 MW Turbine Generator Set 575 Weight Sequence Variation 586
Case History 39: Three Bearing Turbine Generator at 3,600 RPM 588 Case History 40: Balancing A 36,330 RPM Pinion Assembly 597 Three Plane Balance 606
Static-Couple Corrections 616 Multiple Speed Calculations 618 Response Prediction 619 Trim Calculations 622
Balancing Force Calculations 623 Balance Weight Splitting 626 Weight Removal 628
Shop Balancing 629 Bibliography 636
Chapter 12 - Machinery Alignment
. . . 637
Pre-Alignment Considerations 638 Optical Position Alignment 649
Case History 41: Hyper Compressor Position Alignment 654 Laser Position Alignment 658
Optical and Laser Bore Alignment 660 Wire Bore Alignment 663
Case History 42: Hyper Compressor Bore Alignment 667 Shaft Alignment Concepts 669
Rim and Face Shaft Alignment 673 Reverse Indicator Shaft Alignment 681
Optics, Lasers, and Wires for Shaft Alignment 691 Hot Alignment Techniques 692
Case History 43: Motor to Hot Process Pump Alignment 697 Bibliography 702
viii
Chapter 13 - Applied Condition Monitoring
. . . 703
Maintenance Philosophies 703 Condition Monitoring 705 Machinery Performance 706 Vibration Response Data 708 Bearing Temperature Data 711 Data Trending 712
Case History 44: Four Pole Induction Motor Bearing Failure 714 Case History 45: Cracked Gas Compressor Intermittent Instability 718 Case History 46: High Stage Compressor Loose Thrust Collar 721 Pre-Startup Inspection and Testing 724
Startup Inspection and Testing 732
Case History 47: Turbine Solo Operation with Tapered Journal 735 Case History 48: Coupled Turbine Generator Startup 736
Case History 49: Heat Soak and Load Stabilization 739 Bibliography 742
Chapter 14 - Machinery Diagnostic Methodology
. . . 743
Diagnostic Objectives 744 Mechanical Inspection 744 Test Plan Development 745
Data Acquisition and Processing 746 Data Interpretation 749
Conclusions and Recommendations 750 Corrective Action Plan 750
Case History 50: Steam Turbine Electrostatic Voltage Discharge 751 Case History 51: Barrel Compressor Fluidic Excitation 758
Case History 52: High Speed Pinion Instability 766 Conclusions on Diagnostic Methodology 770
Bibliography 770
Chapter 15 - Closing Thoughts and Comments
. . . 771
Economic Reality 772
Corporate Considerations 773 Presentation of Results 778 Silver Bullets 780
Appendix
. . . 781
A — Machinery Diagnostic Glossary 781 B — Physical Properties 795
C — Conversion Factors 797 D — Index 801
xi
Preface
W
hen my son graduated from Texas A&M University, he was understandably eager to start working, and begin earn-ing a livable salary. He accepted a maintenance engineerearn-ing position at a large chemical complex, and embarked upon learning about process machinery. In the months and years that followed, he and his colleagues had many questions con-cerning a variety of machinery problems. From my perspective, most of these problems had been solved twenty or thirty years ago. However, it was clear that the new engineering graduates were devoting considerable effort attempting to unravel mysteries that had already been solved.The obvious question that arises might be stated as: How come the new engineers cannot refer to the history files instead of reworking these issues? A par-tial answer to this question is that the equipment files often do not provide any meaningful historical technical data. Major corporations are reluctant to spend money for documentation of engineering events and achievements. Unless the young engineers can find someone with previous experience with a specific mal-function, they are often destined to rework the entire scenario.
Although numerous volumes have been published on machinery malfunc-tions, there are very few technical references that address the reality of solving field machinery problems. This general lack of usable and easily accessible infor-mation was a primary force in the development of this text. The other significant driving force behind this book was the desire to coalesce over thirty-three years of experience and numerous technical notes into some type of structured order that my son, and others could use for solving machinery problems.
This is a book about the application of engineering principles towards the diagnosis and correction of machinery malfunctions. The machinery under dis-cussion operates within the heavy process industries such as oil refineries,
chem-xii
ical plants, power plants, and paper mills. This machinery consists of steam, gas and hydro turbines, motors, expanders, pumps, compressors, and generators, plus various gear box configurations. This mechanical equipment covers a wide variety of physical characteristics. The transmitted power varies from 50 horse-power, to units in excess of 150,000 horsepower. Rotational speeds range from 128 to more than 60,000 revolutions per minute. There is a corresponding wide range of operating conditions. Fluid temperatures vary from cryogenic levels of minus 150°F, to values in excess of plus 1,200°F. The operating pressures range from nearly perfect vacuums to levels greater than 40,000 pounds per square inch. Physically, the moving elements may be only a few feet long, and weigh less than 100 pounds — or they may exceed 200,000 pounds, and cover the length of a football field. In virtually all cases, these process machines are assembled with precision fits and tolerances. It is meaningful to note that the vibration severity criteria for many of these machines are less than the thickness of a human hair.
In some respects, it is amazing that this equipment can operate at all. When the number of individual mechanical components are considered, and the potential failure mechanisms are listed, the probabilities for failures are stagger-ing. Considerable credit must be given to the designers, builders, and innovators of this equipment. They have consistently produced machines that are con-stantly evolving towards units of improved efficiency, and extended reliability.
The majority of machinery problems that do occur fall into what I call the
ABC category. These common problems are generally related to Alignment, B al-ance, and incorrect Clearances (typically on bearings). Due to the continual appearance of these malfunctions, an entire chapter within this text has been devoted to each of these subjects. Machines also exhibit other types of failures, and a sampling of common plus unique problems are described within this book. Some people might view this document as a textbook. Others might con-sider this to be a reference manual, and still other individuals might use this book for troubleshooting. It has also been suggested that this book be categorized as a how to do it manual. Since 52 detailed case histories are combined with numerous sample calculations and examples, each of these descriptions are accu-rate and applicable. In the overview, the contents of this book cover a variety of machinery malfunctions, and it engages the multiple engineering disciplines that are required to solve real world problems. Regardless of the perception, or the final application, this is a book about the mechanics, measurements, calcula-tions, and diagnosis of machinery malfunctions. I sincerely hope that this text will provide some meaningful help for students, for new graduates entering this field, as well as provide a usable reference for seasoned professionals.
Finally, I would like to extend my deepest personal thanks to John Jensen of Hewlett Packard for the inspiration, encouragement, and opportunity to write this book. I am further indebted to John for his detailed and thorough review of much of the enclosed material. I would also like to thank Ron Bosmans, Dana Salamone, and Pamela Puckett for their constructive comments and corrections. Robert C. Eisenmann, Sr., P.E.
1 C H A P T E R Machinery Instrumentation Physical Behavior Knowledge ➜ E x p e ri e n ce ➜
1
Introduction
1
M
achinery development has been synon-ymous with technological progress. This growth has resulted in an evolutionary trend in industrial equipment that moves towards increased complexity, higher speeds, and greater sophistication. The water wheel has evolved into the hydro-electric plant, the rudimentary steam engine has grown into the gas turbine, and coarse mechanical devices have been replaced by elegant electronic circuits.Throughout this evolution in technology, new industries and vocations have developed. In recent decades, the Machinery Diagnostician has appeared within most maintenance engineering organizations. These individuals generally pos-sess an extensive knowledge of the machinery construction. They understand repair procedures, and they have a working knowledge of the peripheral equip-ment. This includes familiarity with the lube and seal oil system, the processing scheme, and the machine controls. Diagnosticians are generally knowledgeable of the machinery monitoring or surveillance instrumentation that covers every-thing from transducers to the data logging computers. Furthermore, when a problem does appear on a piece of equipment, it generally falls under the juris-diction of the machinery diagnostician to resolve the difficulty, and recommend an appropriate course of corrective action. This requirement imposes another set of demands. That is, these individuals must be familiar with problem solving techniques and proven methodology for correcting the machinery malfunction.
Clearly, the diagnostician must be qualified in many technical disciplines. As depicted in the adjacent diagram, the basic areas of expertise include knowl-edge of machinery, knowledge of physical behavior, plus
knowl-edge of instrumentation. The machinery background must be thorough, and it must allow the diagnostician to focus upon realistic failure mechanisms rather than esoteric theories. The category of physical behavior embraces technical fields such as: statics, dynamics, kinematics, mechanics of materials, fluid dynamics, heat transfer, mathematics, and rotordynamics. Knowledge in these areas must be fully integrated with the instrumentation aspects of the electronic measurements required to document and understand the machinery motion.
2 Chapter-1
Competence in these three areas is only achieved by a combination of knowledge and field experience. Acquiring knowledge often begins with specific technical training. For instance, all academic institutions provide the mathemat-ics and physmathemat-ics necessary to grasp many physical principles. A few universities provide an introduction to the world of analytical rotordynamics. Unfortunately, academia is often burdened by the necessity to obtain research grants, and gen-erate complex general solutions for publication. Certainly the college level con-tributions to this field are significant, and the global solutions are impressive. However, the working machinery diagnostician often cannot use generalized con-cepts for solving everyday problems. To state it another way, integral calculus is absolutely necessary for success in the classroom, but it is reasonably useless for most activities performed on the compressor deck.
Within the industrial community, a variety of training programs are avail-able. Instrumentation vendors provide courses on the application and operation of their particular devices. Similarly, machinery vendors and component suppli-ers have various courses for their clientele. Although these training courses are oriented towards solutions of field problems, they typically display shortcomings in three areas. First, the industrial courses are limited in scope to three or four days of training. This time frame is acceptable for simple topics, but it is inade-quate for addressing complex material. Second, industrial training courses are restricted to the instruments or devices sold by the vendor providing the train-ing. Although this approach is expected by the attendees, it does limit the depth and effectiveness of the training. The third problem with vendor training resides in the backgrounds of the training specialists. Although these people are usually well qualified to represent the products of the vendor, they often lack an under-standing of the realities within an operating plant. Clearly, the smooth presenta-tion of fifty computer generated slides has no relapresenta-tionship to the crucial decisions that have to be made at 2:00 AM regarding a shaking machine.
Another disturbing trend seems to permeate the specialized field of vibra-tion analysis. Within this technical area, there have been long-term efforts by some vendors to train people to solve problems based entirely on simplistic vibra-tory symptoms. This is extraordinarily dangerous, and the senior author has encountered numerous instances of people reaching the wrong conclusions based upon this approach. Many problems display similar vibratory symptoms, and additional information is usually required to sort out the differences. In all cases, the measurements must be supplemented with the logical application of physical laws. In addition, the machinery construction and operation must be examined and understood in order to develop an accurate assessment of the malfunction.
Very few professional organizations provide a comprehensive and inte-grated approach targeted to the topic of machinery diagnosis. The text contained herein attempts to provide a pragmatic and objective overview of machinery mal-function analysis. The three fundamental areas of physical behavior, machinery, and instrumentation knowledge are integrated throughout this book. The struc-ture of this text is directed towards developing a basic understanding of funda-mental principles. This includes the applicability of those principles towards machinery, plus the necessary instrumentation and computational systems to
3
describe and understand the actual behavior of the mechanical equipment. It should be recognized that acquiring basic knowledge does not guarantee that the diagnostician will be qualified to engage and solve machinery problems. As previously stated, experience is mandatory to become proficient in this field. Although the preliminary knowledge may be difficult to obtain, the experience portion may be even harder to acquire. This is particularly true for the individ-ual that works in an operating complex that contains a limited assortment of mechanical equipment. For this diagnostician, the ability to develop a well-rounded background may be hampered due to an absence of mixed machinery types, and associated problems. References such as the excellent series of books by Heinz Bloch1 provide detailed machinery descriptions, procedures, and guide-lines. If the diagnostician is not familiar with a particular machine, this is the one available source that will probably answer most mechanical questions.
In a further attempt to address the experience issue, this text was prepared with 52 field case histories interspersed throughout the chapters. These case studies are presented with substantial details and explanations. The logical steps of working through each particular problem are reviewed, and the encoun-tered errors as well as the final solutions are presented. It is the author’s hope that these field examples on major process machinery will provide additional insight, and enhance the experience level of the machinery diagnostician.
The equipment discussed in this text resides within process industries such as oil refining, pipeline, chemical processing, power generation, plus pulp and paper. The specific machines discussed include pumps, blowers, compressors, and generators that vary from slow reciprocating units to high speed centrifugal machines. The prime movers appear in various configurations from induction motors, to cryogenic and hot gas expanders, hydro-turbines, multistage steam turbines, and large industrial gas turbines. In some cases the driver is directly coupled to the driven equipment, and in other trains an intermediate gear box is included. Some of the discussed machinery was installed decades ago, and other mechanical equipment was examined during initial field commissioning.
It is an objective of this text to assist in understanding, and to demonstrate practical solutions to real world machinery problems. This book is not designed to be mathematically rigorous, but the presented mathematics is considered to be accurate. In all cases, the original sources of the mathematical derivations are identified. This will allow the reader to reference back to the original technical work for additional information. Significant equations in this text are numeri-cally identified, and highlighted with an outline box such as equation (2-1). Developmental and supportive equations are sequentially numbered in each chapter. In addition, intermediate results plus numeric sample calculations are also presented. These examples are not assigned equation numbers. In essence, this book is structured to supplement a formal training presentation, and to pro-vide an ongoing reference.
1 Heinz P. Bloch, Practical Machinery Management for Process Plants, Vol. 1 to 4 (Houston, TX:
4 Chapter-1
M
ACHINERYC
ATEGORIESIt is organizationally advantageous to divide process machinery into three categories. Typically, these individual machinery categories are administered under a singular condition monitoring program since they share a common tech-nology. However, the allocation of resources among the three segments varies in direct proportion to the process criticality of the mechanical equipment.
The first segment covers the large machinery within an operating plant. These main equipment trains are generally critical to the process. In most instances the plant cannot function without these machines. For example, the charge gas compressor in an ethylene plant, or a syngas compressor in an ammo-nia plant fall into this category. This equipment typically ranges between 5,000 and 50,000 horsepower. Operating speeds vary from 200 to 60,000 RPM, and fluid film bearings are normally employed. Most of the machinery problems pre-sented within this text reside within this critical category.
Machines of this class are typically equipped with permanently installed proximity probe transducer systems for vibration and position measurements, plus bearing temperature pickups, and specialized transducers such as torque sensors. Historically, the field transducers are hard wired to continuous monitor-ing systems that incorporate automated trip features for machinery protection. These monitoring systems are also connected to process and/or dedicated com-puter systems for acquisition of static and dynamic data at predetermined sam-ple rates. These data acquisition computer systems provide detailed information concerning the mechanical condition of the machinery.
The second major group of machines are categorized as essential units. They are physically smaller than the critical units, they normally have lower horsepower ratings, and they are usually installed with full backup or spare units. Machines within this category include trains such as product pumps, boiler feed water pumps, cooling water pumps, etc. Individual units in this cate-gory may not be critical to the process — but it is often necessary to keep one out of two, or perhaps two out of three units running at all times. It should be recog-nized that a particular service may be considered as essential equipment when a fully functional main and spare unit are in place. However, if one unit fails, plant operation then depends upon the reliability of the remaining train. In this man-ner, an essential train may be rapidly upgraded to the status of a critical unit.
These essential machinery trains are usually instrumented in a manner similar to the critical units previously discussed. Shaft sensing proximity probe systems, and thermocouples are hard wired to monitoring systems. These moni-toring systems may be integrated with computerized trending systems. Due to the similarity of construction and installation of the critical and the essential machines, the text contained herein is directly applicable to essential units.
The third group of machines are referred to as general purpose equip-ment. These units are physically smaller, and they generally contain rolling ele-ment bearings. These machines are often installed with full backups, or they are single units that are non-critical to the process. Machines within this category have minimal vibration or temperature measuring instrumentation
Chapter Descriptions 5
nently installed. This equipment is often monitored with portable data loggers, and the information tracked with dedicated personal computer systems. In many instances, small machines are not subjected to detailed analytical or diagnostic procedures. An in-depth analysis might cost more than the original purchase price of the equipment. Although there are not many direct references to small machinery within this book, the techniques and physical principles discussed for large machines are fully appropriate for these smaller units.
The technology necessary to understand the behavior of process machinery has been evolving for many years. For example, dedicated machinery monitoring systems are being replaced by direct interfaces into Distributed Control Systems (DCS) for trending of general information. Detailed dynamic data is simulta-neously acquired in a separate diagnostic computer system. This improvement in data trending and resolution allows a better assessment of machinery malfunc-tions. In addition, numerous developments in the areas of rotor dynamics, aero-dynamics, blade design, cascade mechanics, metallurgy, fabrication, testing, plus optimizing bearing and support designs have all combined to provide a wealth of knowledge. Understanding these individual topics and the interrelationship between design parameters, mechanical construction, vibratory behavior, posi-tion between elements, and the array of electronic measurements and data pro-cessing can be an intimidating endeavor.
In support of this complex requirement for knowledge plus experience, this book has been prepared. To provide continuity through the chapters, various fac-ets of several basic types of industrial machines are examined. It is understood that one text cannot fully cover all of the material requested by all of the readers. However, it is anticipated that the information presented within this text will provide a strong foundation of technical information, plus a source for future ref-erence. The specific topics covered in this book are summarized as follows.
C
HAPTERD
ESCRIPTIONSThe following chapter 2 on dynamic motion begins with a general classifi-cation of machinery vibration problems. A review of the fundamental concepts provides a foundation that extends into a description of a simple undamped mechanical system. The addition of damping, plus the influence of forced vibra-tion are discussed. Although the majority of the emphasis is placed upon lateral motion, the parallel environment of torsional vibration is introduced. Finally, the theoretical concepts are correlated with actual measured machinery vibratory characteristics for lateral and torsional behavior.
Rotor mode shapes are discussed in chapter 3. This topic begins with a review of static deflection, followed by the influence of rotor mass, and the distri-bution of mass and supports. Various aspects of inertia of mechanical systems are discussed, and critical distinctions are identified. Next, system damping, and effective support stiffness are discussed, and their influence upon the deflected mode shapes are demonstrated. The physical transition of a rotor across a criti-cal speed, or balance resonance region is thoroughly explained. These basic
con-6 Chapter-1
cepts are then extended into measured and calculated rotor mode shapes. In addition, the construction of interference maps are introduced, and a variety of illustrations are used to assist in a visualization of these important concepts.
Chapter 4 addresses machinery bearings and supports in rotating sys-tems. This includes an introduction to oil film bearing characteristics, and some computational techniques. This is followed by proven techniques for determina-tion of radial fluid film bearing clearances, plus the measurement of bearing housing coefficients. Fluid film thrust bearings are also discussed, and the char-acteristics of rolling element bearings are reviewed. Appropriate case histories are included within this chapter to assist in explanation of the main concepts.
Analytical rotor modeling is introduced in chapter 5. This is a continua-tion of the machinery behavior concepts initiated in the previous chapters. These concepts are applied to the development of an undamped critical speed analysis for lateral and torsional behavior. This is followed by the inclusion of damping to yield the damped response, plus a stability analysis of the rotating system. Fur-ther refinement of the machinery model allows the addition of dimensional forc-ing functions to yield a synchronous response analysis. This step provides quantification and evaluation of the transient and steady state vibration response characteristics of the machinery. Finally, the validity and applicability of these analytical techniques are demonstrated by six detailed case histories distributed throughout the chapter.
Chapter 6 provides a discussion of transducer characteristics for the common measurement probes. A traditional industrial suite of displacement, velocity, acceleration, and pressure pulsation probes are reviewed. The construc-tion, calibraconstruc-tion, and operating characteristics of each transducer type are sub-jected to a comprehensive discussion. In addition, the specific advantages and disadvantages of each standard transducer are summarized. Specialized trans-ducers are also identified, and their general applications are briefly discussed. Finally, the topic of vibration severity and the establishment of realistic vibra-tion limits is discussed.
Dynamic signal characteristics are presented in chapter 7. This section addresses the manipulation and examination of dynamic vibration signals with a full range of electronic filters. In addition, an explanation of combining time domain signals into orbits, and the interrelationship between the time and fre-quency domain characteristics are examined. Finally, common signal combina-tions such as signal summation, amplitude modulation, and frequency modulation are discussed. In all cases, appropriate examples are presented.
Chapter 8 covers data acquisition and processing in terms of the instrumentation systems required for accurate field data acquisition, plus the processing of the data into useful hard copy formats. Sample forms are included to facilitate documentation of field measurements. In addition, the functions and necessary compatibility issues between instruments and transducers are dis-cussed, and operational guidelines are offered. This chapter concludes with an overview of the most useful machinery data presentation formats.
Based upon the concepts discussed in the previous sections, chapter 9 dis-cusses the origin of many of the common malfunctions experienced by process
Chapter Descriptions 7
machinery. The topics include synchronous (rotational speed) excitations such as unbalance, bowed shafts, eccentricity, and resonant responses. The influence of preloads, machinery stability, mechanical looseness, rubs, and cracked shafts are discussed. In addition, foundation considerations are reviewed from several per-spectives. These general problems are applicable to all rotating machines, and several case histories are included to illustrate these fundamental mechanisms. Chapter 10 addresses the unique behavior of different types of machin-ery. Excitations associated with gear boxes, electrical frequencies, and fluid exci-tations are included. In addition, the behavioral characteristics of traditional reciprocating machines, plus hyper compressors are reviewed. Although this group does not cover all of the potential sources of excitation, it does provide a useful summary of problems that occur with regularity on many types of machines. Again, a series of fully descriptive field case histories are distributed throughout the chapter.
Rotational speed vibration is the dominant motion on most industrial machines. Chapter 11 is devoted to an in-depth discussion of this synchronous behavior, and the direct application of these concepts towards rotor balancing. This chapter begins with the initial thought process prior to balancing, and the standardized measurements and conventions. The concept of combined balanc-ing techniques are presented, and the machinery linearity requirements are identified. The development of balancing solutions are thoroughly discussed for single plane, two plane, and three plane solutions. In addition, static-couple solu-tions using two plane calculasolu-tions are presented, and multiple speed calculasolu-tions are discussed. The use of response prediction, and trim balance calculations are reviewed, and several types of supportive calculations are included. Again, field case histories are provided to demonstrate the applicability of the rotational speed analysis, and rotor balancing techniques on process machines.
The last portion of chapter 11 deals with shop balancing machines, tech-niques, and procedures. Although the fundamental concepts are often similar to field balancing, the shop balancing work is generally performed at low rotative speeds. This shop balancing discussion includes additional considerations for the various types of machinery rotors, and common balance specifications.
Machinery alignment persists as one of the leading problems on process machinery, and this topic is covered in chapter 12. Alignment is discussed in terms of the fundamental principles for casing position, casing bore, and shaft alignment. Each type of machinery alignment is discussed, and combined with explanations of several common types of measurements and calculations. This includes dial indicator readings, optical alignment, wire alignment, plus laser alignment, proximity probes, and tooling balls. The applicability of each tech-nique is addressed, and suitable case histories are provided to demonstrate the field use of various alignment techniques.
The concepts of applied condition monitoring within an operating plant are discussed in chapter 13 of this text. This chapter was based upon a tutorial by the senior author to the Texas A&M Turbomachinery Symposium in Dallas, Texas. The first portion of this chapter describes the logic and evolution of condi-tion monitoring, and the typical parameters involved. These concepts are
illus-8 Chapter-1
trated with machinery problems detected during normal operation. The second part of this chapter reviews the turnaround checks and calibrations that should be performed on the machinery control and protection systems. The third portion of this chapter covers the application of condition monitoring during a post-over-haul startup of a machinery train. Again, case studies are used to illustrate the main points of the transient vibratory characteristics.
Chapter 14 address a machinery diagnostic methodology that may be used for diagnosis of complex mechanical problems. This chapter was based upon a paper prepared by the senior author for an annual meeting of the Vibration Institute in New Orleans, Louisiana. This topic discusses the fundamental tools, successful techniques, and the seven-step process used for evaluation of machin-ery problems. Again, specific field case histories are included to illustrate some of the germane points of this topic.
The final chapter 15 is entitled closing thoughts and comments, and it addresses some of the other obstacles encountered when attempting to solve machinery problems. This includes candid observations concerning the problems of dealing with multiple corporate entities, plus the politics encountered within most operating plants. In many instances, an acceptable solution is fully depen-dent upon a proper presentation of results that combine economic feasibility with engineering credibility.
The appendix begins with a machinery diagnostic glossary for the spe-cialized language and terminology associated with this business. For reference purposes, a list of the physical properties of common metals and fluids, plus a table of conversion factors are included. The technical papers and books cited within this text are identified with footnotes, and summarized in a bibliography at the end of each chapter. In addition, a detailed index is provided in the last appendix section that includes technical topics, corporate references, and specific authors referenced throughout this book.
It is the authors’ hope that the material included within this book will be beneficial to the machinery diagnostician, and that this text will serve as an ongoing technical reference. To paraphrase the words of Donald E. Bently (circa 1968), founder and owner of Bently Nevada Corporation …we just want to make the machinery run better… To this objective, we have dedicated our professional careers and this manuscript.
B
IBLIOGRAPHY1. Bloch, Heinz P., Practical Machinery Management for Process Plants, Vol. 1 to 4, Houston, TX: Gulf Publishing Company, 1982-1989.
9
C H A P T E R
2
Dynamic Motion
2
M
any mechanical problems are initially recognized by a change in machinery vibration amplitudes. In order to under-stand, and correctly diagnose the vibratory characteristics of rotating machinery, it is essential for the machinery diagnostician to understand the physics of dynamic motion. This includes the influence of stiffness and damping on the quency of an oscillating mass — as well as the interrelationship between fre-quency, displacement, velocity, and acceleration of a body in motion.M
ALFUNCTIONC
ONSIDERATIONSANDC
LASSIFICATIONSBefore examining the intricacies of dynamic motion, it must be recognized that many facets of a mechanical problem must be considered to achieve a suc-cessful and acceptable diagnosis in a timely manner. For instance, the following list identifies some of the related considerations for addressing and realistically solving a machinery vibration problem:
❍ Economic Impact
❍ Machinery Type and Construction ❍ Machinery History — Trends — Failures ❍ Frequency Distribution
❍ Vibratory Motion Distribution and Direction ❍ Forced or Free Vibration
The economic impact is directly associated with the criticality of the machinery. A problem on a main process compressor would receive immediate attention, whereas a seal problem on a fully spared reflux pump would receive a lower priority. Clearly, the types of machinery, the historical trends, and failure histories are all important pieces of information. In addition, the frequency of the vibration, plus the location and direction of the motion are indicators of the problem type and severity. Traditionally, classifications of forced and free vibra-tion are used to identify the origin of the excitavibra-tion. This provides considerable insight into potential corrective actions. For purposes of explanation, the follow-ing lists identify some common forced and free vibration mechanisms.
10 Chapter-2
Forced Vibration Mechanisms Free Vibration Mechanisms
❍ Mass Unbalance ❑ Oil Whirl
❍ Misalignment ❑ Oil or Steam Whip
❍ Shaft Bow ❑ Internal Friction
❍ Gyroscopic ❑ Rotor Resonance
❍ Gear Contact ❑ Structural Resonances
❍ Rotor Rubs ❑ Acoustic Resonances
❍ Electrical Excitations ❑ Aerodynamic Excitations ❍ External Excitations ❑ Hydrodynamic Excitations
Forced vibration problems are generally solved by removing or reducing the exciting or driving force. These problems are typically easier to identify and solve than free vibration problems. Free vibration mechanisms are self-excited phe-nomena that are dependent upon the geometry, mass, stiffness, and damping of the mechanical system. Corrections to free vibration problems may require phys-ical modification of the machinery. As such, these types of problems are often dif-ficult to correct. Success in treating self-excited problems are directly related to the diagnostician’s ability to understand, and apply the appropriate physical principles. To address these fundamental concepts of dynamic motion, including free and forced vibration, the following chapter is presented for consideration.
It should be mentioned that much of the equation structure in this chapter was summarized from the classical textbook by William T. Thomson1, entitled Mechanical Vibrations. For more information, and detailed equation derivation, the reader is encouraged to reference this source directly. The same basic equa-tion structure is also described in his newer text entitled Theory of Vibration with Applications2. Regardless of the vintage, at least one copy of Thomson should be part of the reference library for every diagnostician.
F
UNDAMENTALC
ONCEPTSInitially, consider a simple system consisting of a one mass pendulum as shown in Fig. 2-1. Assume that the pendulum mass M is a concrete block sus-pended by a weightless and rigid cable of length L. Further assume that the sys-tem operates without frictional forces to dissipate syssys-tem energy. Intuitively, if the pendulum is displaced from the vertical equilibrium position, it will oscillate back and forth under the influence of gravity. The mass will move in the same path, and will require the same amount of time to return to any specified refer-ence point. Due to the frictionless environment, the amplitude of the motion will remain constant. The time required for one complete oscillation, or cycle, is called the Period of the motion. The total number of cycles completed per unit of 1 William Tyrell Thomson, Mechanical Vibrations, 2nd Edition, 9th Printing, (Englewood
Cliffs, New Jersey: Prentice Hall, Inc., 1962), pp.1-75
2 William T. Thomson, Theory of Vibration with Applications, 4th Edition, (Englewood Cliffs,
Fundamental Concepts 11
time is the Frequency of the oscillation. Hence, frequency is simply the reciprocal of the period as shown in the following expression:
(2-1)
The box around this equation identifies this expression as a significant or important concept. This same identification scheme will be used throughout this text. Within equation (2-1), period is a time measurement with units of hours, minutes or seconds. Frequency carries corresponding units such as Cycles per Hour, Cycles per Minute (CPM), or Cycles per Second (CPS or Hz). Understand-ably, the oscillatory motion of the pendulum is repetitive, and periodic. As shown in Marks’ Handbook3, Fourier proved that periodic functions can be expressed with circular functions (i.e., a series of sines and cosines) — where the frequency for each term in the equation is a multiple of the fundamental. It is common to refer to periodic motion as harmonic motion. Although many types of vibratory motions are harmonic, it should be recognized that harmonic motion must be periodic, but periodic motion does not necessarily have to be harmonic.
3 Eugene A. Avallone and Theodore Baumeister III, Marks’ Standard Handbook for
Mechani-cal Engineers, Tenth Edition, (New York: McGraw-Hill, 1996), pp. 2-36.
Fig. 2–1 Oscillating Pendu-lum Displaying Simple Har-monic Motion Frequency 1 Period ---= A C B Mass Negative Positive
Max. Neg. Displ. Zero Velocity Max. Pos. Accel.
Max. Pos. Displ. Zero Velocity Max. Neg. Accel. Zero Displacement Maximum Velocity Zero Acceleration φ W = M G φ Wsin φ W cos φ Equilibrium Stationary I-Beam Cable Length - L
12 Chapter-2
In a rotating system, such as a centrifugal machine, frequency is normally expressed as a circular rotational frequency ω. Since one complete cycle consists of one revolution, and one revolution is equal to 2π radians, the following conver-sion applies:
(2-2)
Combining (2-1) and (2-2), the rotational frequency ω may be expressed in terms of the Period as follows:
(2-3)
The frequency units for ω in equation (2-3) are Radians per Second, or Radi-ans per Minute. Again, this is dependent upon the time units selected for the period. Although these are simple concepts, they are continually used through-out this text. Hence, a clear and definitive understanding of period and fre-quency are mandatory for addressing virtually any vibration problem.
Returning to the pendulum of Fig. 2-1, a gravitational force is constantly acting on the mass. This vertical force is the weight of the block. From physics it is known that weight W is equal to the product of mass M, and the acceleration of gravity G. As the pendulum oscillates through an angular displacement φ, this force is resolved into two perpendicular components. The cosine term is equal and opposite to the tension in the string, and the sine component is the Restoring Force acting to bring the mass back to the vertical equilibrium position. For small values of angular displacement, sinφ is closely approximated by the angle φ expressed in radians. Hence, this restoring force may be represented as:
(2-4)
Similarly, the maximum distance traveled by the mass may also be deter-mined from plane geometry. As shown in Fig. 2-1, the cable length is known, and the angular displacement is specified by φ. The actual change in lateral position for the mass is the distance from A to B, or from B to C. In either case, this dis-tance is equal to Lsinφ. Once more, for small angles, sinφ≈φ in radians, and the total deflection from the equilibrium position may be stated as:
(2-5)
This repetitive restoring force acting over the same distance has a spring like quality. In actuality, this characteristic may be defined as the horizontal stiffness K of this simple mechanical system as follows:
(2-6)
If equations (2-4) and (2-5) are substituted into (2-6), and if the weight W is replaced by the equivalent mass M times the acceleration of gravity G, the fol-lowing expression is produced:
ω = 2π×Frequency = 2π×F ω 2π Period ---= Restoring Force ≈W ×φ Deflection ≈L ×φ Stiffness K Force Deflection ---= =
Fundamental Concepts 13
(2-7)
Later in this chapter it will be shown that the natural frequency of oscilla-tion for an undamped single degree of freedom system is determined by equaoscilla-tion (2-44) as a function of mass M and stiffness K. If equation (2-7) is used for the stiffness term within equation (2-44), the following relationship results:
(2-8)
Equation (2-8) is often presented within the literature for describing the natural frequency of a simple pendulum. A direct example of this concept may be illustrated by considering the motion of the pendulum in a grandfather’s clock. Typically, the pendulum requires 1.0 second to travel one half of a stroke, or 2.0 seconds to transverse a complete stroke (i.e., one complete cycle). The length L of the pendulum may be determined by combining equations (2-3) and (2-8):
If the period is represented in terms of the pendulum length L, the above expression may be stated as:
(2-9)
Equation (2-9) is a common expression for characterizing a simple pendu-lum. The validity of this equation may be verified in technical references such as Marks’ Handbook4. For the specific problem at hand, equation (2-9) may be solved for the pendulum length. Performing this manipulation, and inserting the gravitational constant G, plus the period of 2.0 seconds, the following is obtained:
Thus, the pendulum length in a grandfather’s clock should be 39.12 inches. This value is accurate for a concentrated mass, and a weightless support arm. In an actual clock, the pendulum is often ornate, and weight is distributed along the length of the support arm. This makes it difficult to accurately determine the location of the center of gravity of the pendulum mass. Nevertheless, even rough measurements reveal that the pendulum length is in the vicinity of 40 inches. In addition, clock makers normally provide a calibration screw at the bottom of the pendulum to allow the owner to adjust the clock accuracy. By turning this adjust-ment screw, the effective length of the pendulum may be altered. From the previ-4 Eugene A. Avallone and Theodore Baumeister III, Marks’ Standard Handbook for
Mechani-cal Engineers, Tenth Edition, (New York: McGraw-Hill, 1996), p. 3-15.
K Force Deflection --- W ×φ L ×φ ---≈ W L --- M×G L ---= = = ω K M --- M×G L --- 1 M ---× G L ----= = = ω 2π Period --- G L ----= = Period 2π L G ----× = L G Period 2 × 4π2 --- 386.1Inches/Second 2 ( )×(2.0Seconds)2 4π2 --- 39.12 Inches = = =
ous equations, it is clear that changing the pendulum length will alter the period of the pendulum. By moving the weight upward, and decreasing the arm length, the clock will run faster (i.e., higher frequency with a shorter period). Conversely, by lowering the main pendulum mass, the length of the arm will be increased, and the clock will run slower (i.e., a lower frequency with a longer period).
Although the grandfather clock is a simple application of periodic motion, it does provide a realistic example of the fundamental concepts. Additional com-plexity will be incorporated later in this text when the behavior of a compound pendulum is discussed. It should be noted that a compound pendulum is a mechanical system that normally contains two degrees of freedom. This addi-tional flexibility might be obtained by adding flexible members such as springs, or additional masses to a simple system. In a two mass system, each mass might be capable of moving independently of the other mass. For this type of arrange-ment, each mass must be tracked with an independent coordinate system, and this would be considered as a two degree of freedom system.
The number of independent coordinates required to accurately define the motion of a system is termed the Degree of Freedom of that system. Process machinery displays many degrees of freedom, and accurate mathematical description of these systems increases proportionally to the number of required coordinates. However, in the case of the simple pendulum, only one coordinate is required to describe the motion — and the pendulum is a single degree of free-dom system exhibiting harmonic motion. More specifically, this is an example of basic dynamic motion where the restoring force is proportional to the displace-ment. This is commonly referred to as Simple Harmonic Motion (SHM). Other devices such as the undamped spring mass (Fig. 2-7), the torsional pendulum (Fig. 2-25), the particle rotating in a circular path, and a floating cork bobbing up and down in the water at a constant rate are all examples of SHM.
Before expanding the discussion to more complex systems, it is desirable to conclude the discussion of the simple pendulum. Once again, the reader is referred back to the example of the oscillating pendulum depicted in Fig. 2-1. On this diagram, it is meaningful to mentally trace the position of the mass during one complete cycle. Starting at the vertical equilibrium position B, the displace-ment is zero at time equal to zero. One quarter of a cycle later, the mass has moved to the maximum positive position C. This is followed by a zero crossing at point B as the mass approaches the maximum negative value at position A. The last quarter cycle is completed as the mass returns from the A location back to the original equilibrium, or center rest point B.
Intuitively, the mass achieves zero velocity as it swings back and forth to the maximum displacement points A and C (i.e., the mass comes to a complete stop). In addition, the maximum positive velocity occurs as the mass moves through point B from left to right, combined with a maximum negative velocity as the mass moves through B going from right to left. Finally, the mass must de-accelerate going from B to C, and de-accelerate from C back to point A. Then the mass will de-accelerate as it moves from A back to the original equilibrium point
B that displays zero lateral acceleration.
Fundamental Concepts 15
eration characteristics of this pendulum would be a time domain examination. Although a meaningful visualization of the changes in displacement, velocity, and acceleration with respect to time may be difficult — a mathematical descrip-tion simplifies this task. For instance, assume that the periodic displacement of the mass may be described by the following fundamental equation relating dis-placement and time:
(2-10) where: Displacement = Instantaneous Displacement
D = Maximum Displacement (equal to pendulum position A or C) F = Frequency of Oscillation
t = Time
In a rotating system, such as a centrifugal machine, this expression can be simplified somewhat by substituting the rotational frequency ω that was previ-ously defined in equation (2-2) to yield:
(2-11)
The instantaneous velocity of this periodic motion is the time derivative of displacement. Velocity may now be determined as follows:
By converting the cosine to a sine function, expression (2-12) is derived:
(2-12)
Note that velocity leads displacement by π/2 or 90°. Another way to state the same concept is that displacement lags behind velocity by 90° in the time domain. The same procedure can now be repeated to examine the relationship between velocity and acceleration. Since acceleration is the time rate of change of velocity, the first time derivative of velocity will yield acceleration. The same result may be obtained by taking the second derivative of displacement with respect to time to obtain acceleration:
By adding π to the sine term, the negative sign is removed, and the follow-ing expression is obtained:
(2-13)
Acceleration leads displacement by π or 180°, and it leads velocity by 90°. It may also be stated that displacement lags acceleration by 180° in time. The rela-tionship between displacement, velocity, and acceleration may be viewed graphi-cally in the polar coordinate format of Fig. 2-2. This diagram reveals that
Displacement = D×sin(2π×F×t) Displacement = D×sin( )ωt Velocity t d d Displacement D×ω×cos( )ωt = = Velocity = D×ω×sin(ωt+π⁄2) Acceleration t2 2 d d Displacement –D ×ω2×sin( )ωt = = Acceleration = D ×ω2×sin(ωt+π)
mechanical systems in motion do display a consistent and definable relationship between frequency, and the respective displacement, velocity, and acceleration of the body in motion.
Understanding the timing between vectors is mandatory for diagnosing machinery behavior. It is very easy to become confused between terms such as leading and lagging, and the diagnostician might inadvertently make a 90° or a 180° mistake. In some instances, this type of error might go unnoticed. However, during rotor balancing, a 180° error in weight placement might result in exces-sive vibration or even physical damage to the machine. This type of error is totally unnecessary, and it may be prevented by establishing and maintaining a consistent timing or phase convention.
From Fig. 2-2, it is noted that time is shown to increase in a counterclock-wise direction. If this diagram represented a rotating shaft, time and rotation would move together in a counterclockwise direction. As discussed in succeeding chapters, phase is measured from the peak of a vibration signal backwards in time to the reference trigger point. This concept is illustrated in Fig. 2-3 that depicts a rotating disk with a series of angles marked off at 45° increments. Assume that the disk is turning counterclockwise on the axial centerline. If this rotating disk is observed from a stationary viewing position, the angles will move past the viewing point in consecutive order.
That is, as the disk turns, the angles progress in a 0-45-90-135-180-225-270-315° consecutive numeric order past the fixed viewing position. However, if the angles increased with rotation, the observed viewing order would be back-wards. Since this does not make good physical sense, the direction of numerically increasing angles are always set against shaft rotation as in Fig. 2-3. This angu-lar convention will be used throughout this text, and vector angles will always be considered as degrees of phase lag. This convention applies to shaft and casing vibration vectors, balance weight vectors, balance sensitivity vectors, plus all
Fig. 2–2 Timing Relationship Between Displacement Velocity, and Acceleration
Displacement Vector Displ. = D sin(ω t) Velocity Vector Vel. = Dω sin(ωt +π/2) Acceleration Vector Accel.= D ω2 sin(ωt + π) ω t + π ωt + π/2 Time Phase ω t
Fundamental Concepts 17
analytically calculated vectors. In short, angles and the associated phase are measured against rotation based upon this physical relationship.
For proper identification, phase angles should be specified as a phase lag, or provided with a negative sign. In most cases, it is convenient to ignore the nega-tive sign, and recognize that these angles are phase lag values. Using this con-vention, phase between the 3 vibration vectors in Fig. 2-2 may be converted by:
(2-14) (2-15) (2-16)
If a velocity phase angle occurs at 225°, it is determined from (2-14) that the displacement phase angle is computed by: 225°+90°=315°. Similarly, the velocity phase may be converted to an acceleration phase from equation (2-16) as: 225°-90°=135°. If the phase lag negative sign is used, the angle conversions in equations (2-14) to (2-16) must also be negative (i.e., -90° and -180°). In either case, consistency is necessary for accurate and repeatable results.
In addition to phase, the vibration magnitude of an object may be converted from displacement to velocity or acceleration at a constant frequency. This requires a conversion of units within the motion equations (2-12) and (2-13). For example, consider the following definition of English units for these parameters:
D = Displacement — Mils,peak to peak = Mils,p-p V = Velocity — Inches/Second,zero to peak = IPS,o-p A = Acceleration — G’s,zero to peak = G’s,o-p F = Frequency — Cycles/Second (Hz)
Reinstalling 2πF for the frequency ω, and considering the peak values of the
Fig. 2–3 Traditional Angle Designation On A Rotating Disk Stationary Viewing Position Angle or Phase Direction Time and Rotation 180° 0° 90° 270° 315° 45° 135° 225° Axis of Rotation
Phasedisplacement = Phasevelocity+90°
Phasedisplacement = Phaseacceleration+180°
terms (i.e., sin=1), equation (2-12) may be restated as follows:
Since velocity is generally defined as zero to peak (o-p), and displacement is typically considered as peak to peak (p-p), the displacement value must be halved to be consistent with the velocity wave. Applying the appropriate physical unit conversions, the following expression evolves:
Which simplifies to the following common equation:
(2-17)
Next, consider the relationship between acceleration and displacement as described by equation (2-13), and expanded with proper engineering units to the following expression:
Acceleration units for the above conversion are Inches/Second2. Measure-ment units of G‘s can be obtained by dividing this last expression by the acceler-ation of gravity as follows:
This conversion expression may be simplified to the following format:
(2-18)
The relationship between acceleration and velocity may be stated as:
Expanding this expression, and including dimensional units, the following equation for converting velocity at a specific frequency to acceleration evolves:
V = D ×ω = D ×2π×F V D 2 ----Mils 1Inch 1 000, Mils ---× 2πRadians Cycle --- FCycles Second ---× × = V D×F 318.31 ---= A D×ω2 D 2 ----Mils 1Inch 1 000, Mils ---× 2πRadians Cycle --- FCycles Second ---× × 2 = = A D F 2 × 50.661 --- Inches Second2 ---= A D F 2 × 50.661 --- Inches Second2 --- 1G 386.1Inches/Second2 ---× = A D F 2 × 19 560, --- D F 139.9 --- 2 × = = A = V ×ω = V ×2π×F A VInches Second --- 2πRadians Cycle --- FCycles Second ---× × 1G 386.1Inches/Second2 --- × =
Fundamental Concepts 19
Simplifying this expression, the following common equation is derived:
(2-19)
The last three equations allow conversion between displacement, velocity, and acceleration at a fixed frequency measured in Cycles per Second (Hz). A set of expressions for frequency measured in Cycles per Minute (CPM) may also be developed. Since machine speeds are measured in Revolutions per Minute (RPM), this additional conversion is quite useful in many instances. Performing this frequency conversion on equations (2-17), (2-18), and (2-19) produces the next three common conversion equations:
(2-20)
(2-21)
(2-22)
The vibration units for equations (2-20), (2-21), and (2-22) are identical to the English engineering units previously defined. However, the frequency for these last three equations carry the units of Revolutions per Minute (i.e., RPM or Cycles per Minute).
The simultaneous existence of three parameters (i.e., displacement, veloc-ity, and acceleration) to describe vibratory motion can be confusing. This is fur-ther complicated by the fact that instrumentation vendors are often specialized in the manufacture of a single type of transducer. Hence, one company may pro-mote the use of displacement probes, whereas another vendor may strongly endorse velocity coils, and a third supplier may cultivate the application of accel-erometers. The specific virtues and limitations of each of these types of trans-ducer systems are discussed in greater detail in chapter 6 of this text. However, for the purposes of this current discussion, it is necessary to recognize that dis-placement, velocity, and acceleration of a moving body are always related by the frequency of the motion.
This relationship between variables may be expressed in various ways. For example, consider an element vibrating at a frequency of 100 Hz (6,000 CPM) and a velocity of 0.3 IPS,o-p. From equation (2-17) the relationship between veloc-ity and displacement may be used to solve for the displacement as follows:
Similarly, the equivalent acceleration of this mechanical element may be determined from equation (2-19) in the following manner:
A V×F 61.45 ---= V D×RPM 19 099, ---= A D RPM 8 391, --- 2 × = A V×RPM 3 687, ---= D 318.31×V F --- 318.31×0.3IPSo-p 100Hz --- 0.955Milsp-p = = =
Thus, the displacement and acceleration amplitudes for this velocity may be computed for any given frequency. Another way to view this interrelationship between parameters is to extend this calculation procedure to a large range of frequencies, and plot the results as shown in Fig. 2-4. Within this diagram, the velocity is maintained at a constant magnitude of 0.3 IPS,o-p and the displace-ment and acceleration amplitudes calculated and plotted for several frequencies between 1 and 20,000 Hz (60 and 1,200,000 CPM).
Fig. 2-4 shows that displacement is large at low frequencies, and accelera-tion is larger at high frequencies. From a measurement standpoint, displace-ment would be used for lower frequencies, and acceleration would be desirable for high frequency data. Again, specific transducer characteristics must also be considered, and the reader is referred to chapter 6 for additional details on the actual operating ranges of transducers.
For purposes of completeness, it should be recognized that the circular func-tions previously discussed can be replaced by an exponential form. For instance, equation (2-23) is a normal format for these expressions:
(2-23)
In this equation, “i” is equal to the square root of minus 1 and “e” is the nat-ural log base that has a value of 2.71828. This expression will satisfy the same equations, and produce identical results to the circular formats. However, it is
Fig. 2–4 Equivalent Displacement, Velocity, and Acceleration Amplitudes V. Frequency
A V×F 61.45 --- 0.3IPSo-p×100Hz 61.45 --- 0.488 G’so-p = = = J J J J J J J J J J J J J J H H H H H H H H H H H H H H B B B B B B B B B B B B B B 0.001 0.01 0.1 1 10 100 1 10 100 1,000 10,000 20,000 1 10 100 1000 10000 20000
Vibration Amplitude (Mils, IPS, G's)
Frequency (Hertz) Velocity Acceleration 60 600 6,000 60,000 600 ,000 1,200,000 Frequency (Cycles/Minute) Displacement Displacement = D×eiωt
