Machinery malfunction diagnosis correction

9Malfunction Considerations and Classifications 9Fundamental Concepts 10 Vector Manipulation 21Undamped Free Vibration 28 Case History 1: Piping System Dynamic Absorber 31Free Vibration

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 07632The 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

Rotating Equipment Technical Authority - BP Products North America - Houston, Texas

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Original Library of Congress Cataloging in Publication Data Eisenmann, Robert C 1943-

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

Chapter 2 - Dynamic Motion 9

Malfunction Considerations and Classifications 9Fundamental Concepts 10

Vector Manipulation 21Undamped Free Vibration 28

Case History 1: Piping System Dynamic Absorber 31Free Vibration with Damping 37

Forced Vibration 45

Case History 2: Steam Turbine End Cover Resonance 55Torsional Vibration 58

Bibliography 66

Chapter 3 - Rotor Mode Shapes 67

Mass and Support Distribution 67

Case History 3: Two Stage Compressor Rotor Weight Distribution 72Inertia Considerations and Calculations 74

Damping Influence 96Stiffness Influence 105Critical Speed Transition 120Mode Shape Measurement 130

Case History 4: Vertical Generator Mode Shape 137Analytical Results 142

Case History 5: Eight Stage Compressor Mode Shape Change 143Bibliography 148

Chapter 4 - Bearings and Supports 149

Fluid Film Radial Journal Bearings 150

Case History 6: Shaft Position In Gas Turbine Elliptical Bearings 161Fluid Film Radial Bearing Clearance Measurements 165

Case History 7: Expander Journal Bearing Clearance 174

Bearing Supports — Measurements and Calculations 179

Case History 8: Measured Steam Turbine Bearing Housing Stiffness 181Case History 9: Measured Gas Turbine Bearing Housing Stiffness 185Bearing Housing Damping 187

Fluid Film Thrust Bearings 188Rolling Element Bearings 193Before Considering Bearing Redesign 196Bibliography 198

Chapter 5 - Analytical Rotor Modeling 199

Modeling Overview 199Undamped Critical Speed 201

Case History 10: Mode Shapes for Turbine Generator Set 206Case History 11: Torsional Analysis of Power Turbine and Pump 208Stability and Damped Critical Speed Calculations 213

Case History 12: Complex Rotor Damped Analysis 217Forced Response Calculations 222

Case History 13: Gas Turbine Response Correlation 226Case History 14: Charge Gas Compressor with Internal Fouling 230Case History 15: Hybrid Approach To A Vertical Mixer 236

Bibliography 242

Chapter 6 - Transducer Characteristics 243

Basic Signal Attributes 244Proximity Displacement Probes 253Velocity Coils 272

Piezoelectric Accelerometers 278Pressure Pulsation Transducers 285Specialized Transducers 288Aspects of Vibration Severity 294Bibliography 302

Chapter 7 - Dynamic Signal Characteristics 303

Electronic Filters 303Time and Orbital Domain 316Time and Frequency Domain 333

Case History 16: Steam Turbine Exhaust End Bearing Dilemma 343Signal Summation 347

Case History 17: Opposed Induced Draft Fans 349Amplitude Modulation 353

Case History 18: Loose and Unbalanced Compressor Wheel 356Frequency Modulation 359

Case History 19: Gear Box with Excessive Backlash 362Bibliography 364

Chapter 8 - Data Acquisition and Processing 365

Vibration Transducer Suite 365Recording Instrumentation 369Data Processing Instrumentation 379Data Presentation Formats 383Bibliography 394

Chapter 9 - Common Malfunctions 395

Synchronous Response 395Mass Unbalance 398Bent or Bowed Shaft 400

Case History 20: Repetitive Steam Turbine Rotor Bow 402Eccentricity 406

Case History 21: Seven Element Gear Box Coupling Bore 407Shaft Preloads 410

Cracked Shaft Behavior 443

Case History 26: Syngas Compressor with Cracked Shaft 446Foundation Considerations 449

Case History 27: Floating Induced Draft Fan 451Case History 28: Structural Influence of Insufficient Grout 454Bibliography 458

Chapter 10 - Unique Behavior 459

Parallel Shaft - Two Element Gear Boxes 459

Case History 29: Herringbone Gear Box Tooth Failure 466Epicyclic Gear Boxes 470

Case History 30: Star Gear Box Subsynchronous Motion 477Process Fluid Excitations 483

Case History 31: Boiler Feed Water Pump Splitter Vane Failures 496Case History 32: Hydro Turbine Draft Tube Vortex 499

Electrical Excitations 507

Case History 33: Motor With Unsupported Stator Midspan 515Case History 34: Torsional Excitation From Synchronous Motor 519Reciprocating Machines 522

Case History 35: Hyper Compressor Plunger Failures 526Bibliography 534

Chapter 11 - Rotor Balancing 535

Before Balancing 536

Standardized Measurements and Conventions 539

Combined Balancing Techniques 545

Static-Couple Corrections 616

Multiple Speed Calculations 618

Response Prediction 619

Trim Calculations 622

Balancing Force Calculations 623

Balance Weight Splitting 626

Optical Position Alignment 649

Case History 41: Hyper Compressor Position Alignment 654Laser Position Alignment 658

Optical and Laser Bore Alignment 660

Wire Bore Alignment 663

Case History 42: Hyper Compressor Bore Alignment 667Shaft 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 697Bibliography 702

Vibration Response Data 708

Bearing Temperature Data 711

Data Trending 712

Case History 44: Four Pole Induction Motor Bearing Failure 714Case History 45: Cracked Gas Compressor Intermittent Instability 718Case History 46: High Stage Compressor Loose Thrust Collar 721Pre-Startup Inspection and Testing 724

Startup Inspection and Testing 732

Case History 47: Turbine Solo Operation with Tapered Journal 735Case History 48: Coupled Turbine Generator Startup 736

Case History 49: Heat Soak and Load Stabilization 739Bibliography 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 751Case History 51: Barrel Compressor Fluidic Excitation 758

Case History 52: High Speed Pinion Instability 766Conclusions on Diagnostic Methodology 770

Preface

When my son graduated from TexasA&M University, he was understandably eager to start working, and begin earn-ing a livable salary He accepted a maintenance engineering position at a largechemical complex, and embarked upon learning about process machinery In themonths and years that followed, he and his colleagues had many questions con-cerning a variety of machinery problems From my perspective, most of theseproblems had been solved twenty or thirty years ago However, it was clear thatthe new engineering graduates were devoting considerable effort attempting tounravel 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 anymeaningful historical technical data Major corporations are reluctant to spendmoney for documentation of engineering events and achievements Unless theyoung 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 tions, there are very few technical references that address the reality of solvingfield machinery problems This general lack of usable and easily accessible infor-mation was a primary force in the development of this text The other significantdriving force behind this book was the desire to coalesce over thirty-three years

malfunc-of experience and numerous technical notes into some type malfunc-of structured orderthat my son, and others could use for solving machinery problems

This is a book about the application of engineering principles towards thediagnosis 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, gasand hydro turbines, motors, expanders, pumps, compressors, and generators,plus various gear box configurations This mechanical equipment covers a widevariety 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 widerange of operating conditions Fluid temperatures vary from cryogenic levels ofminus 150°F, to values in excess of plus 1,200°F The operating pressures rangefrom nearly perfect vacuums to levels greater than 40,000 pounds per squareinch Physically, the moving elements may be only a few feet long, and weigh lessthan 100 pounds — or they may exceed 200,000 pounds, and cover the length of afootball field In virtually all cases, these process machines are assembled withprecision fits and tolerances It is meaningful to note that the vibration severitycriteria 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 thepotential 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 stantly evolving towards units of improved efficiency, and extended reliability.The majority of machinery problems that do occur fall into what I call the

con-ABC category These common problems are generally related to Alignment, Bance, and incorrect Clearances (typically on bearings) Due to the continualappearance of these malfunctions, an entire chapter within this text has beendevoted 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 thisbook for troubleshooting It has also been suggested that this book be categorized

al-as a how to do it manual Since 52 detailed case histories are combined withnumerous sample calculations and examples, each of these descriptions are accu-rate and applicable In the overview, the contents of this book cover a variety ofmachinery malfunctions, and it engages the multiple engineering disciplinesthat are required to solve real world problems Regardless of the perception, orthe final application, this is a book about the mechanics, measurements, calcula-tions, and diagnosis of machinery malfunctions I sincerely hope that this textwill provide some meaningful help for students, for new graduates entering thisfield, 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 writethis book I am further indebted to John for his detailed and thorough review ofmuch of the enclosed material I would also like to thank Ron Bosmans, DanaSalamone, and Pamela Puckett for their constructive comments and corrections

Robert C Eisenmann, Sr., P.E.

October 1997

synon-of the machinery monitoring or surveillance instrumentation that covers thing from transducers to the data logging computers Furthermore, when aproblem does appear on a piece of equipment, it generally falls under the juris-diction of the machinery diagnostician to resolve the difficulty, and recommend

every-an appropriate course of corrective action This requirement imposes every-another set

of demands That is, these individuals must be familiar with problem solvingtechniques 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 edge of machinery, knowledge of physical behavior, plus knowl-

knowl-edge of instrumentation The machinery background must bethorough, and it must allow the diagnostician to focus uponrealistic failure mechanisms rather than esoteric theories

The category of physical behavior embraces technical fieldssuch as: statics, dynamics, kinematics, mechanics ofmaterials, fluid dynamics, heat transfer, mathematics,and rotordynamics Knowledge in these areas must befully integrated with the instrumentation aspects ofthe electronic measurements required to documentand 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 specifictechnical training For instance, all academic institutions provide the mathemat-ics and physics necessary to grasp many physical principles A few universitiesprovide 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 isabsolutely necessary for success in the classroom, but it is reasonably useless formost activities performed on the compressor deck

Within the industrial community, a variety of training programs are able Instrumentation vendors provide courses on the application and operation

avail-of their particular devices Similarly, machinery vendors and component ers have various courses for their clientele Although these training courses areoriented towards solutions of field problems, they typically display shortcomings

suppli-in three areas First, the suppli-industrial courses are limited suppli-in scope to three or fourdays of training This time frame is acceptable for simple topics, but it is inade-quate for addressing complex material Second, industrial training courses arerestricted 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 depthand effectiveness of the training The third problem with vendor training resides

in the backgrounds of the training specialists Although these people are usuallywell 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 relationship to the crucial decisionsthat have to be made at 2:00 AM regarding a shaking machine

Another disturbing trend seems to permeate the specialized field of tion analysis Within this technical area, there have been long-term efforts bysome vendors to train people to solve problems based entirely on simplistic vibra-tory symptoms This is extraordinarily dangerous, and the senior author hasencountered numerous instances of people reaching the wrong conclusions basedupon this approach Many problems display similar vibratory symptoms, andadditional information is usually required to sort out the differences In all cases,the measurements must be supplemented with the logical application of physicallaws In addition, the machinery construction and operation must be examinedand 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 containedherein 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 towardsmachinery, 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 guaranteethat 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 experienceportion 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 ofmechanical equipment For this diagnostician, the ability to develop a well-rounded background may be hampered due to an absence of mixed machinerytypes, and associated problems References such as the excellent series of books

by Heinz Bloch1 provide detailed machinery descriptions, procedures, and lines If the diagnostician is not familiar with a particular machine, this is theone available source that will probably answer most mechanical questions

guide-In a further attempt to address the experience issue, this text was preparedwith 52 field case histories interspersed throughout the chapters These casestudies are presented with substantial details and explanations The logicalsteps 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 hopethat these field examples on major process machinery will provide additionalinsight, 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 andpaper The specific machines discussed include pumps, blowers, compressors, andgenerators that vary from slow reciprocating units to high speed centrifugalmachines The prime movers appear in various configurations from inductionmotors, to cryogenic and hot gas expanders, hydro-turbines, multistage steamturbines, and large industrial gas turbines In some cases the driver is directlycoupled to the driven equipment, and in other trains an intermediate gear box isincluded Some of the discussed machinery was installed decades ago, and othermechanical equipment was examined during initial field commissioning

It is an objective of this text to assist in understanding, and to demonstratepractical 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 areidentified This will allow the reader to reference back to the original technicalwork 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 eachchapter In addition, intermediate results plus numeric sample calculations arealso 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: Gulf Publishing Company, 1982-1989).

4 Chapter-1

It is organizationally advantageous to divide process machinery into threecategories Typically, these individual machinery categories are administeredunder a singular condition monitoring program since they share a common tech-nology However, the allocation of resources among the three segments varies indirect 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 mostinstances the plant cannot function without these machines For example, thecharge gas compressor in an ethylene plant, or a syngas compressor in an ammo-nia plant fall into this category This equipment typically ranges between 5,000and 50,000 horsepower Operating speeds vary from 200 to 60,000 RPM, andfluid 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 installedproximity probe transducer systems for vibration and position measurements,plus bearing temperature pickups, and specialized transducers such as torquesensors 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 informationconcerning 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 lowerhorsepower ratings, and they are usually installed with full backup or spareunits 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 nized that a particular service may be considered as essential equipment when afully functional main and spare unit are in place However, if one unit fails, plantoperation 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 mannersimilar to the critical units previously discussed Shaft sensing proximity probesystems, and thermocouples are hard wired to monitoring systems These moni-toring systems may be integrated with computerized trending systems Due tothe similarity of construction and installation of the critical and the essentialmachines, 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 aresingle units that are non-critical to the process Machines within this categoryhave minimal vibration or temperature measuring instrumentation perma-

Chapter Descriptions 5

nently installed This equipment is often monitored with portable data loggers,and the information tracked with dedicated personal computer systems In manyinstances, small machines are not subjected to detailed analytical or diagnosticprocedures An in-depth analysis might cost more than the original purchaseprice of the equipment Although there are not many direct references to smallmachinery within this book, the techniques and physical principles discussed forlarge machines are fully appropriate for these smaller units

The technology necessary to understand the behavior of process machineryhas been evolving for many years For example, dedicated machinery monitoringsystems 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 indata 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, plusoptimizing bearing and support designs have all combined to provide a wealth ofknowledge Understanding these individual topics and the interrelationshipbetween 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, thisbook has been prepared To provide continuity through the chapters, various fac-ets of several basic types of industrial machines are examined It is understoodthat 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 willprovide a strong foundation of technical information, plus a source for future ref-erence The specific topics covered in this book are summarized as follows

The following chapter 2 on dynamic motion begins with a general cation of machinery vibration problems A review of the fundamental conceptsprovides a foundation that extends into a description of a simple undampedmechanical system The addition of damping, plus the influence of forced vibra-tion are discussed Although the majority of the emphasis is placed upon lateralmotion, the parallel environment of torsional vibration is introduced Finally, thetheoretical concepts are correlated with actual measured machinery vibratorycharacteristics for lateral and torsional behavior

classifi-Rotor mode shapes are discussed in chapter 3 This topic begins with areview of static deflection, followed by the influence of rotor mass, and the distri-bution of mass and supports Various aspects of inertia of mechanical systemsare discussed, and critical distinctions are identified Next, system damping, andeffective support stiffness are discussed, and their influence upon the deflectedmode 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 Inaddition, the construction of interference maps are introduced, and a variety ofillustrations 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 somecomputational techniques This is followed by proven techniques for determina-tion of radial fluid film bearing clearances, plus the measurement of bearinghousing coefficients Fluid film thrust bearings are also discussed, and the char-acteristics of rolling element bearings are reviewed Appropriate case historiesare included within this chapter to assist in explanation of the main concepts

Analytical rotor modeling is introduced in chapter 5 This is a tion of the machinery behavior concepts initiated in the previous chapters Theseconcepts are applied to the development of an undamped critical speed analysisfor lateral and torsional behavior This is followed by the inclusion of damping toyield 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 providesquantification and evaluation of the transient and steady state vibrationresponse characteristics of the machinery Finally, the validity and applicability

continua-of these analytical techniques are demonstrated by six detailed case historiesdistributed throughout the chapter

Chapter 6 provides a discussion of transducer characteristics for thecommon measurement probes A traditional industrial suite of displacement,velocity, acceleration, and pressure pulsation probes are reviewed The construc-tion, calibration, and operating characteristics of each transducer type are sub-jected to a comprehensive discussion In addition, the specific advantages anddisadvantages 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 sectionaddresses the manipulation and examination of dynamic vibration signals with afull range of electronic filters In addition, an explanation of combining timedomain 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 frequencymodulation are discussed In all cases, appropriate examples are presented Chapter 8 covers data acquisition and processing in terms of theinstrumentation systems required for accurate field data acquisition, plus theprocessing of the data into useful hard copy formats Sample forms are included

to facilitate documentation of field measurements In addition, the functions andnecessary compatibility issues between instruments and transducers are dis-cussed, and operational guidelines are offered This chapter concludes with anoverview of the most useful machinery data presentation formats

Based upon the concepts discussed in the previous sections, chapter 9 cusses the origin of many of the common malfunctions experienced by process

Chapter Descriptions 7

machinery The topics include synchronous (rotational speed) excitations such asunbalance, bowed shafts, eccentricity, and resonant responses The influence ofpreloads, machinery stability, mechanical looseness, rubs, and cracked shafts arediscussed In addition, foundation considerations are reviewed from several per-spectives These general problems are applicable to all rotating machines, andseveral 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 traditionalreciprocating machines, plus hyper compressors are reviewed Although thisgroup does not cover all of the potential sources of excitation, it does provide auseful summary of problems that occur with regularity on many types ofmachines Again, a series of fully descriptive field case histories are distributedthroughout the chapter

Rotational speed vibration is the dominant motion on most industrialmachines Chapter 11 is devoted to an in-depth discussion of this synchronousbehavior, and the direct application of these concepts towards rotor balancing.This chapter begins with the initial thought process prior to balancing, and thestandardized measurements and conventions The concept of combined balanc-ing techniques are presented, and the machinery linearity requirements areidentified The development of balancing solutions are thoroughly discussed forsingle plane, two plane, and three plane solutions In addition, static-couple solu-tions using two plane calculations are presented, and multiple speed calculationsare discussed The use of response prediction, and trim balance calculations arereviewed, and several types of supportive calculations are included Again, fieldcase histories are provided to demonstrate the applicability of the rotationalspeed analysis, and rotor balancing techniques on process machines

The last portion of chapter 11 deals with shop balancing machines, niques, and procedures Although the fundamental concepts are often similar tofield balancing, the shop balancing work is generally performed at low rotativespeeds This shop balancing discussion includes additional considerations for thevarious types of machinery rotors, and common balance specifications

tech-Machinery alignment persists as one of the leading problems on processmachinery, and this topic is covered in chapter 12 Alignment is discussed interms of the fundamental principles for casing position, casing bore, and shaftalignment Each type of machinery alignment is discussed, and combined withexplanations of several common types of measurements and calculations Thisincludes dial indicator readings, optical alignment, wire alignment, plus laseralignment, proximity probes, and tooling balls The applicability of each tech-nique is addressed, and suitable case histories are provided to demonstrate thefield use of various alignment techniques

The concepts of applied condition monitoring within an operating plantare 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 secondpart 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 haul startup of a machinery train Again, case studies are used to illustrate themain points of the transient vibratory characteristics

post-over-Chapter 14 address a machinery diagnostic methodology that may beused for diagnosis of complex mechanical problems This chapter was based upon

a paper prepared by the senior author for an annual meeting of the VibrationInstitute 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 ofthe germane points of this topic

The final chapter 15 is entitled closing thoughts and comments, and itaddresses some of the other obstacles encountered when attempting to solvemachinery problems This includes candid observations concerning the problems

of dealing with multiple corporate entities, plus the politics encountered withinmost operating plants In many instances, an acceptable solution is fully depen-dent upon a proper presentation of results that combine economic feasibilitywith engineering credibility

The appendix begins with a machinery diagnostic glossary for the cialized language and terminology associated with this business For referencepurposes, a list of the physical properties of common metals and fluids, plus atable of conversion factors are included The technical papers and books citedwithin this text are identified with footnotes, and summarized in a bibliography

spe-at the end of each chapter In addition, a detailed index is provided in the lastappendix section that includes technical topics, corporate references, and specificauthors referenced throughout this book

It is the authors’ hope that the material included within this book will bebeneficial to the machinery diagnostician, and that this text will serve as anongoing technical reference To paraphrase the words of Donald E Bently (circa1968), founder and owner of Bently Nevada Corporation …we just want to make the machinery run better… To this objective, we have dedicated our professionalcareers and this manuscript

1 Bloch, Heinz P., Practical Machinery Management for Process Plants, Vol 1 to 4, Houston, TX: Gulf Publishing Company, 1982-1989.

it is essential for the machinery diagnostician to understand the physics ofdynamic motion This includes the influence of stiffness and damping on the fre-quency of an oscillating mass — as well as the interrelationship between fre-quency, displacement, velocity, and acceleration of a body in motion.

Before examining the intricacies of dynamic motion, it must be recognizedthat many facets of a mechanical problem must be considered to achieve a suc-cessful and acceptable diagnosis in a timely manner For instance, the followinglist identifies some of the related considerations for addressing and realisticallysolving a machinery vibration problem:

❍ Economic Impact

❍ Machinery Type and Construction

❍ Machinery History — Trends — Failures

❍ Frequency Distribution

❍ Vibratory Motion Distribution and Direction

❍ Forced or Free VibrationThe economic impact is directly associated with the criticality of themachinery A problem on a main process compressor would receive immediateattention, whereas a seal problem on a fully spared reflux pump would receive alower priority Clearly, the types of machinery, the historical trends, and failurehistories are all important pieces of information In addition, the frequency ofthe vibration, plus the location and direction of the motion are indicators of theproblem type and severity Traditionally, classifications of forced and free vibra-tion are used to identify the origin of the excitation This provides considerableinsight into potential corrective actions For purposes of explanation, the follow-ing lists identify some common forced and free vibration mechanisms

10 Chapter-2

❍ Electrical Excitations ❑ Aerodynamic Excitations

❍ External Excitations ❑ Hydrodynamic ExcitationsForced vibration problems are generally solved by removing or reducing theexciting or driving force These problems are typically easier to identify and solvethan free vibration problems Free vibration mechanisms are self-excited phe-nomena that are dependent upon the geometry, mass, stiffness, and damping ofthe 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 tothe diagnostician’s ability to understand, and apply the appropriate physicalprinciples To address these fundamental concepts of dynamic motion, includingfree and forced vibration, the following chapter is presented for consideration

It should be mentioned that much of the equation structure in this chapterwas 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 Thomsonshould be part of the reference library for every diagnostician

Initially, consider a simple system consisting of a one mass pendulum asshown 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 system energy Intuitively, ifthe pendulum is displaced from the vertical equilibrium position, it will oscillateback and forth under the influence of gravity The mass will move in the samepath, 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 willremain constant The time required for one complete oscillation, or cycle, iscalled 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, New Jersey: Prentice Hall, 1993), pp 1-91.

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 orimportant concept This same identification scheme will be used throughout thistext Within equation (2-1), period is a time measurement with units of hours,minutes or seconds Frequency carries corresponding units such as Cycles perHour, 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 expressedwith circular functions (i.e., a series of sines and cosines) — where the frequencyfor each term in the equation is a multiple of the fundamental It is common torefer to periodic motion as harmonic motion Although many types of vibratorymotions are harmonic, it should be recognized that harmonic motion must beperiodic, but periodic motion does not necessarily have to be harmonic

3 Eugene A Avallone and Theodore Baumeister III, Marks’ Standard Handbook for cal Engineers, Tenth Edition, (New York: McGraw-Hill, 1996), pp 2-36.

Mechani-Fig 2–1 Oscillating

Pendu-lum Displaying Simple

Max Pos Displ Zero Velocity Max Neg Accel.

Zero Displacement Maximum Velocity Zero Acceleration

Radi-Returning to the pendulum of Fig 2-1, a gravitational force is constantlyacting 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 ofgravity G As the pendulum oscillates through an angular displacement φ, thisforce is resolved into two perpendicular components The cosine term is equaland 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 Forsmall 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 mined from plane geometry As shown in Fig 2-1, the cable length is known, andthe angular displacement is specified by φ The actual change in lateral positionfor 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 thetotal deflection from the equilibrium position may be stated as:

deter-(2-5)

This repetitive restoring force acting over the same distance has a springlike quality In actuality, this characteristic may be defined as the horizontalstiffness 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 isreplaced by the equivalent mass M times the acceleration of gravity G, the fol-lowing expression is produced:

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.

14 Chapter-2

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, itdoes provide a realistic example of the fundamental concepts Additional com-plexity will be incorporated later in this text when the behavior of a compoundpendulum is discussed It should be noted that a compound pendulum is amechanical 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 ment, each mass must be tracked with an independent coordinate system, andthis would be considered as a two degree of freedom system

arrange-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 mathematicaldescription of these systems increases proportionally to the number of requiredcoordinates However, in the case of the simple pendulum, only one coordinate isrequired to describe the motion — and the pendulum is a single degree of free-dom system exhibiting harmonic motion More specifically, this is an example ofbasic 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 upand down in the water at a constant rate are all examples of SHM

Before expanding the discussion to more complex systems, it is desirable toconclude the discussion of the simple pendulum Once again, the reader isreferred back to the example of the oscillating pendulum depicted in Fig 2-1 Onthis 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 accelerate going from B to C, and accelerate from C back to point A Then the mass will de-accelerate as it moves from A back to the original equilibrium point

de-B that displays zero lateral acceleration

Another way to compare and correlate the displacement, velocity, and

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 ofthe 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 besimplified somewhat by substituting the rotational frequency ω that was previ-ously defined in equation (2-2) to yield:

of velocity, the first time derivative of velocity will yield acceleration The sameresult may be obtained by taking the second derivative of displacement withrespect to time to obtain acceleration:

By adding π to the sine term, the negative sign is removed, and the ing expression is obtained:

follow-(2-13)

Acceleration leads displacement by π or 180°, and it leads velocity by 90° Itmay 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

16 Chapter-2

mechanical systems in motion do display a consistent and definable relationshipbetween frequency, and the respective displacement, velocity, and acceleration ofthe body in motion

Understanding the timing between vectors is mandatory for diagnosingmachinery 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 istotally unnecessary, and it may be prevented by establishing and maintaining aconsistent timing or phase convention

From Fig 2-2, it is noted that time is shown to increase in a wise direction If this diagram represented a rotating shaft, time and rotationwould move together in a counterclockwise direction As discussed in succeedingchapters, phase is measured from the peak of a vibration signal backwards intime to the reference trigger point This concept is illustrated in Fig 2-3 thatdepicts 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 thisrotating disk is observed from a stationary viewing position, the angles will movepast the viewing point in consecutive order

counterclock-That is, as the disk turns, the angles progress in a 270-315° consecutive numeric order past the fixed viewing position However, ifthe angles increased with rotation, the observed viewing order would be back-wards Since this does not make good physical sense, the direction of numerically

0-45-90-135-180-225-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 beconsidered as degrees of phase lag This convention applies to shaft and casingvibration vectors, balance weight vectors, balance sensitivity vectors, plus all

Fig 2–2 Timing Relationship Between Displacement Velocity, and Acceleration

(2-14) (2-15) (2-16)

If a velocity phase angle occurs at 225°, it is determined from (2-14) thatthe displacement phase angle is computed by: 225°+90°=315° Similarly, thevelocity 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 inequations (2-14) to (2-16) must also be negative (i.e., -90° and -180°) In eithercase, consistency is necessary for accurate and repeatable results

In addition to phase, the vibration magnitude of an object may be convertedfrom displacement to velocity or acceleration at a constant frequency Thisrequires a conversion of units within the motion equations (2-12) and (2-13) Forexample, 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

Phase displacement = Phase velocity+90°

Phase displacement = Phase acceleration+180°

Phase velocity = Phase acceleration+90°

18 Chapter-2

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 istypically considered as peak to peak (p-p), the displacement value must be halved

to be consistent with the velocity wave Applying the appropriate physical unitconversions, the following expression evolves:

Which simplifies to the following common equation:

(2-17)

Next, consider the relationship between acceleration and displacement asdescribed by equation (2-13), and expanded with proper engineering units to thefollowing expression:

Acceleration units for the above conversion are Inches/Second2 ment units of G‘s can be obtained by dividing this last expression by the acceler-ation of gravity as follows:

Measure-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 followingequation for converting velocity at a specific frequency to acceleration evolves:

The simultaneous existence of three parameters (i.e., displacement, ity, and acceleration) to describe vibratory motion can be confusing This is fur-ther complicated by the fact that instrumentation vendors are often specialized

veloc-in the manufacture of a sveloc-ingle type of transducer Hence, one company may mote the use of displacement probes, whereas another vendor may stronglyendorse 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 thefrequency of the motion

pro-This relationship between variables may be expressed in various ways Forexample, 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 bedetermined from equation (2-19) in the following manner:

20 Chapter-2

Thus, the displacement and acceleration amplitudes for this velocity may

be computed for any given frequency Another way to view this interrelationshipbetween parameters is to extend this calculation procedure to a large range offrequencies, and plot the results as shown in Fig 2-4 Within this diagram, thevelocity is maintained at a constant magnitude of 0.3 IPS,o-p and the displace-ment and acceleration amplitudes calculated and plotted for several frequenciesbetween 1 and 20,000 Hz (60 and 1,200,000 CPM)

Fig 2-4 shows that displacement is large at low frequencies, and tion is larger at high frequencies From a measurement standpoint, displace-ment would be used for lower frequencies, and acceleration would be desirablefor high frequency data Again, specific transducer characteristics must also beconsidered, and the reader is referred to chapter 6 for additional details on theactual operating ranges of transducers

accelera-For purposes of completeness, it should be recognized that the circular tions previously discussed can be replaced by an exponential form For instance,equation (2-23) is a normal format for these expressions:

func-(2-23)

In this equation, “i” is equal to the square root of minus 1 and “e” is the ural log base that has a value of 2.71828 This expression will satisfy the sameequations, and produce identical results to the circular formats However, it is

nat-Fig 2–4 Equivalent Displacement, Velocity, and Acceleration Amplitudes V Frequency

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

Vector Manipulation 21

sometimes easier to manipulate equations using an exponential form ratherthan a circular function For reference, the relationship between the exponentialand the circular function is shown as follows:

(2-24)

The “cos(ωt)” term is often referred to as the Real, or the In-Phase

compo-nent The “i sin(ωt)” term is the projection of the vector on the imaginary axis.

This is normally called the Imaginary, or the Quadrature component These

terms are used interchangeably It should be understood that the form, and notthe intent of the equations has been altered It should also be mentioned that

both the Real and Imaginary (In-Phase and Quadrature) components must

sat-isfy the equation of motion for the mechanical system

Many physical characteristics of machines are described with vectors Amagnitude is joined with a directional component to provide a parameter withreal physical significance These vector quantities are routinely subjected to var-ious types of mathematical operations More specifically, the addition, subtrac-tion, multiplication, and division of vectors must be performed as an integralpart of vibration and modal analysis, rotor balancing, analytical modeling, plusinstrumentation calibration

For reference purposes, it is necessary to define the methods used for vectormanipulation The different vector operations may be performed with a handheld calculator, they may be executed with the math tools incorporated inspreadsheets, or they may be included as subroutines into computer programs

In addition, some Dynamic Signal Analyzers (DSA) use vector math as part ofthe signal processing and computational capabilities In all cases, these funda-mental math operations must be performed in a consistent manner

From an explanatory standpoint, the specific vector equations will beshown, and a numeric example will be presented for each type of operation Theexamples will be performed with circular coordinates, however an exponentialform will provide an identical solution For consistency, the following pair ofpolar coordinate vectors will be used throughout this series of explanations:

(2-25) (2-26)

The first vector (2-25) has of a magnitude A, occurring at an angle α

Simi-larly, the second vector (2-26) has an amplitude of B, and an angle β As ously discussed, these vectors may be represented in a Cartesian coordinate (X-Y) system by the following pair of equations:

previ-eiωt = cos( )ωt +i×sin( )ωt

V a = A ∠α

V b = B ∠β

22 Chapter-2

Multiplying the amplitude by the cosine and sine of the associated anglewill allow conversion from polar to rectangular coordinates The cosine term rep-resents the magnitude on the X-Axis, and the sine term identifies the amplitude

on the Y-Axis From the last pair of equations, the individual Cartesian tudes for each vector component may be summarized as:

ampli-(2-27) (2-28) (2-29) (2-30)

This conversion of the initial vectors now provides the format to allow the

addition and subtraction of two vector quantities Vector addition is performed

by summing the individual X and Y components, and converting from Cartesianback to polar coordinates The summation of X-Axis components is achieved byadding equations (2-27) and (2-29) in the following manner:

Vector Manipulation 23

determined by vector addition of the two individual weight vectors For stration purposes, assume that 50 Grams was inserted into a hole at 60°, and 40Grams was installed at the 80° hole as described in Fig 2-5 The initial weightvectors are represented with equations (2-25) and (2-26) as:

demon-The summation of horizontal vector components in the X-Axis is mined with equation (2-31):

deter-Similarly, the summation of vertical vector components in the Y-Axis may

be computed with equation (2-32) as follows:

The calculated X and Y balance weights identify the combined effect of bothweights in the horizontal and vertical directions These weights are actually Xand Y coordinates that may be converted to a polar coordinate magnitude usingequation (2-33) in the following manner:

Fig 2–5 Vector Addition

Of Two Balance Weights

Effective Weight Vector

First Weight Vector Second Weight Vector

The same basic approach is used for vector subtraction, with one

signifi-cant difference Instead of adding Cartesian coordinates, the X and Y

compo-nents are subtracted That is, by subtracting the B vector from the A vector, the

X-Axis change is obtained by subtracting equation (2-29) from (2-27):

of 2.38 Mils,p-p at an angle of 134° Assume that the slow speed 1X runout was

x

V add y

Vector Manipulation 25

measured to be 0.94 Mils,p-p, at 78° Subtraction of the slow roll from the fullspeed vector yields a compensated, or a runout corrected vector

Mathematically, the initial vibration at running speed may be identified as

the A vector, and the slow roll runout may be represented by the B vector

Substi-tution of the defined vibration vectors into equations (2-25) and (2-26) providesthe following vectors for subtraction:

The difference between horizontal X-Axis vector components is determinedwith equation (2-35) in the following manner:

Similarly, the difference of vector components in the vertical Y-Axis may becomputed with equation (2-36):

The negative value for the horizontal component is perfectly normal, andacceptable This negative sign, combined with the positive sign on the verticalcomponent, identifies that the final vector will reside in the upper left polarquadrant (i.e., angle between 90° and 180°) The computed X and Y coordinatesmay now be converted to polar coordinates using equation (2-37) to determinethe magnitude of the runout corrected vector:

Fig 2–6 Vector

Subtrac-tion Of Shaft Runout From

Running Speed Vector

7, 8, and 11.

The major complexity associated with vector addition and subtraction isdue to the necessity for converting from polar to Cartesian coordinates, perform-ing a simple operation, and then converting from Cartesian back to polar coordi-nates Fortunately, this multiple conversion is not required for vectormultiplication and division

Vector multiplication of two vector quantities may be executed by simply

multiplying amplitudes, and adding the respective phase angles as follows:

Vector multiplication is necessary in the machinery diagnosis business Forexample, consider the situation of determining the required balance weight tocorrect the 1X vibration response of a machine Presuming that the unit has aproperly defined balance sensitivity vector, the required balance weight andangle can be determined from equation (2-39) This requires a vector multiplica-tion between the measured vibration, and the sensitivity vector For demonstra-tion purposes, assume that the measured vibration vector is 2.0 Mils,p-p at anangle of 40° Further assume that the rotor balance sensitivity vector is equal to

x

V sub y

Vector Manipulation 27

150.0 Grams/Mil,p-p at an angle of 190° Based on this data, the operable vectorsfor this vector manipulation are identified as:

Multiplication of these two vectors is performed with equation (2-39) as:

This vector product indicates that the installation of a 300 Gram weight at

an angle of 230° will balance the measured synchronous response of 2.0 Mils,p-p

at 40° Naturally, the accuracy of this value is dependent upon the correctness ofthe balance sensitivity vector

As described in further detail in chapter 11, a vector summation betweenthe calculated vibration from the weight, plus the current vibration vector willresult in a predicted vibration vector with the weight attached An additionalvector summation with the shaft runout will produce an uncompensated 1X vec-tor For a perfectly linear mechanical system, this would be the vibration ampli-tude and phase displayed by a synchronous tracking filter Although thisdiscussion is somewhat premature within the sequence of this text, the mainpoint is that vector calculations may involve a string of manipulations to achievethe necessary result

Vector division represents the final category of vector math Referring

back to the initial vectors, equations (2-25) and (2-26), vector division is formed by dividing the amplitudes, and subtracting the angles as follows:

per-(2-40)

This kind of manipulation is also easy to perform, and again a cautionarynote resides with the final value of the angle In many cases, this angle may dropbelow 0°, due to the relative size of angles α and β When the zero point is crossed(i.e., negative angle), the size of the angle may be increased by 360° to yield aphysically meaningful angle between 0° and 360°

Vector division is widely used for various types of machinery calculations.For instance, the computation of a balance sensitivity vector requires the divi-sion of a calibration weight vector by a differential vibration response vector Thetechnical details associated with this calculation are in chapter 11 However,from a pure computational standpoint, consider the following initial pair of vec-tors for division

28 Chapter-2

Division of these two vectors is performed with equation (2-40) as follows:

This calculation identifies a single balance sensitivity vector based upon ameasured differential response vector of 5.00 Mils,p-p at an angle of 60° This vec-tor change in shaft vibration response was due to the installation of a 400 Gramweight at an angle of 230° Vector division of the weight by the differential vibra-tion vector yields the balance sensitivity vector of 80.0 Grams/Mil,p-p at an angle

of 170° This unbalance sensitivity vector may now be used to compute balancecorrections in a manner similar to the earlier example of vector multiplication.These simplified rules for vector multiplication and division may be verified

by performing the same operations using exponential functions instead of thepresented polar coordinates The results will be identical, and this will reinforcethe concept that the vector math may be successfully executed using either expo-nential or circular functions In all cases, these vector manipulations are contin-ually used throughout the field of machinery analysis, and these proceduresmust be mastered to allow progression to the real machinery topics

Expanding upon the concepts of the previous section, again consider thesingle mass pendulum of Fig 2-1 Within this earlier mechanical system, themass of the concrete block was identified as the only significant element in thesystem If this concrete block remains constant, and if the weightless cable isreplaced by a coil spring, the simple spring mass system of Fig 2-7 is produced.Assume that the spring is suspended from a totally rigid I-Beam, and considerthe mass to be confined to movement only in the vertical direction Since damp-ing is not involved, this is considered as an undamped mechanical system Inaddition, there are no external forces applied to this system, so it must be classi-fied as a system that exhibits free vibration when it is displaced, and allowed tooscillate in the vertical plane The resultant motion is defined as undamped freevibration of this one degree of freedom mechanical system

If this physical example is converted into a traditional physics diagram, thesketches shown in Fig 2-8 evolve The left diagram shows the main mechanicalelements, and the right sketch displays the Free Body Diagram Normally, thismechanical system would remain at rest (i.e., no motion) For this system to

Undamped Free Vibration 29

move, some type of initial disturbance is required Furthermore, when thismechanical system is in motion, the free body diagram (Fig 2-8) reveals twoactive forces; a spring force, and the gravitational term The general equation ofmotion for this body is simply the equality of active forces as follows:

By rearranging terms, the following summation of forces is obtained:

Substituting a simpler alpha identification for each of the four variables,the equation of motion for this simple spring mass system may be stated in themanner that W T Thomson used:

(2-41)

If equation (2-41) is divided by the mass, the resultant expression contains

a system mechanical constant (i.e., K/M), plus the interrelated acceleration and

displacement of the body:

Fig 2–8 Equivalent Spring Mass Mechanical System And

Associated Free Body Diagram

Mass

Coil Spring

Stationary I-Beam

Spring withStiffness = K

30 Chapter-2

Extracting the common terms from this equation, the following is obtained:

(2-43)

Equation (2-43) is satisfied for all values of time t when the terms within

the brackets are equated to zero:

This may now be solved for the natural or critical frequency ωc as follows:

(2-44)

Another common form of this expression is obtained by converting the tional frequency ωc units of Radians per Second to Cycles per Second in accor-dance with equation (2-2) to yield the following:

Initially, the existence of a unique natural frequency that is a function ofthe mechanical system mass and stiffness may appear to be only of academicinterest In reality, there are field applications of this physical relationship thatmay be used to provide solutions for mechanical problems For instance, if amechanical system is excited by a periodic force at a frequency that approaches anatural resonant frequency of the mechanical system — the resultant vibratory