machinery machinery vibration vibration andand rotordynamicsrotordynamics

machinery vibration

machinery vibration andand rotordynam

icsrotordynam

ics

John Vance • Fouad Zeidan • Brian Murphy

VanceZeidanMurphy

Technology/Machinery

An in-depth analysis of machine vibration in rotating machineryWhether it’s a compressor on an off shore platform, a turbocharger in a truck or automobile, or a turbine in a jet airplane, rotating machinery is the driving force behind almost anything that produces or uses energy. Counted on daily to perform any number of vital societal tasks, turbomachinery uses high rotational speeds to produce amazing amounts of power effi ciently. The key to increasing its longevity, effi ciency, and reliability lies in the exami-nation of rotor vibration and bearing dynamics, a fi eld called rotordynamics.

A valuable textbook for beginners as well as a handy reference for experts, Machinery Vibration and Rotordynamics is teeming with rich technical detail and real-world exam-ples geared toward the study of machine vibration. A logical progression of information covers essential fundamentals, in-depth case studies, and the latest analytical tools used for predicting and preventing damage in rotating machinery. Machinery Vibration and Rotordynamics:

• Combines rotordynamics with the applications of machinery vibration in a single volume

• Includes case studies of vibration problems in several diff erent types of machines as well as computer simulation models used in industry

• Contains fundamental physical phenomena, mathematical and computational aspects, practical hardware considerations, troubleshooting, and instrumentation and measure-ment techniques

For students interested in entering this highly specialized fi eld of study, as well as profes-sionals seeking to expand their knowledge base, Machinery Vibration and Rotordynamics will serve as the one book they will come to rely upon consistently.

DR. JOHN M. VANCE was professor of mechanical engineering at Texas A&M University, retiring in 2007. He received his PhD (1967) degree from The University of Texas at Austin. His book Rotordynamics of Turbomachinery (Wiley) has sold more than 3,000 copies and is used by turbomachinery engineers around the world. He is an inventor on several patents relating to rotating machinery and vibration reduction. His patented TAMSEAL has been retrofi tted to solve vibration problems in a number of high-pressure industrial compres-sors. He is an ASME Fellow and a registered professional engineer in the state of Texas.

DR. FOUAD Y. ZEIDAN is the President of KMC, Inc., and Bearings Plus, Inc., two companies specializing in the supply of high-performance bearings, fl exible couplings, and seals. Dr. Zeidan holds nine U.S. patents for integral squeeze fi lm dampers and high-performance journal and thrust bearings. He has published more than thirty technical papers and ar-ticles on various turbomachinery topics and has been lecturing at the Annual Machinery Vibrations and Rotordynamics short course since 1991. Dr. Zeidan holds a BS, MS, and PhD degrees in mechanical engineering from Texas A&M University.

BRIAN T. MURPHY, PHD, PE, is a senior research scientist with the Center for Electrome-chanics at The University of Texas at Austin. He is also president of RMA, Inc., which develops and markets the Xlrotor suite of rotordynamic analysis software used worldwide by industry and academia. Dr. Murphy is the creator of the polynomial transfer matrix method, which is the fastest known method of performing rotordynamic calculations. He has authored numerous technical papers on rotordynamics and machinery vibration, and is also caretaker of the Web site www.rotordynamics.org.

9780471462132

Cover photo courtesy of Sulzer Turbo Services showing technicians working on a compressor rotor.

MACHINERY VIBRATION ANDROTORDYNAMICS

MACHINERY VIBRATIONAND ROTORDYNAMICS

John Vance, Fouad Zeidan, Brian Murphy

JOHN WILEY & SONS, INC.

This book is printed on acid-free paper.

Copyright © 2010 by John Wiley & Sons, Inc. All rights reserved.

Published by John Wiley & Sons, Inc., Hoboken, New JerseyPublished simultaneously in Canada

No part of this publication may be reproduced, stored in a retrieval system, ortransmitted in any form or by any means, electronic, mechanical, photocopying,recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the1976 United States Copyright Act, without either the prior written permission ofthe Publisher, or authorization through payment of the appropriate per-copy fee to theCopyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923, (978)750-8400, fax (978) 646-8600, or on the web at www.copyright.com. Requests to thePublisher for permission should be addressed to the Permissions Department, JohnWiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201)748-6008, or online at www.wiley.com/go/permissions.

Limit of Liability/Disclaimer of Warranty: While the publisher and the author have usedtheir best efforts in preparing this book, they make no representations or warranties withrespect to the accuracy or completeness of the contents of this book and specificallydisclaim any implied warranties of merchantability or fitness for a particular purpose. Nowarranty may be created or extended by sales representatives or written sales materials.The advice and strategies contained herein may not be suitable for your situation. Youshould consult with a professional where appropriate. Neither the publisher nor theauthor shall be liable for any loss of profit or any other commercial damages, includingbut not limited to special, incidental, consequential, or other damages.

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Wiley also publishes its books in a variety of electronic formats. Some content thatappears in print may not be available in electronic books. For more information aboutWiley products, visit our web site at www.wiley.com.

Library of Congress Cataloging-in-Publication Data:

Vance, John M.Machinery vibration and rotordynamics / John Vance, Brian Murphy, Fouad Zeidan.

p. cm.Includes bibliographical references and index.ISBN 978-0-471-46213-2 (cloth)

1. Rotors–Dynamics. 2. Rotors–Vibration. 3. Machinery–Vibration. 4.Turbomachines–Dynamics. I. Murphy, Brian, 1956- II. Zeidan, Fouad. III. Title.

TJ177.V36 2010621.8′11—dc22 2009045963

Printed in the United States of America

10 9 8 7 6 5 4 3 2 1

The first author gratefully dedicates his part in this book tohis loving wife Louise, who made the book possible by herunselfish support of the task and devotion to her husband

while it was being written.

John M. Vance

CONTENTS

PREFACE xiii

1 Fundamentals of Machine Vibration and ClassicalSolutions 1

The Main Sources of Vibration in Machinery / 1The Single Degree of Freedom (SDOF) Model / 4Using Simple Models for Analysis and Diagnostics / 6Six Techniques for Solving Vibration Problems with

Forced Excitation / 13Some Examples with Forced Excitation / 15

Illustrative Example 1 / 15Illustrative Example 2 / 17Illustrative Example 3 / 20Illustrative Example 4 / 24

Some Observations about Modeling / 27Unstable Vibration / 28

References / 30

Exercises / 30

2 Torsional Vibration 35

Torsional Vibration Indicators / 36Objectives of Torsional Vibration Analysis / 37Simplified Models / 38Computer Models / 45

Kinetic Energy Expression / 46Potential Energy / 46

Torsional Vibration Measurement / 51French’s Comparison Experiments / 53

Strain Gages / 53

vii

viii CONTENTS

Carrier Signal Transducers / 54Frequency-modulated Systems / 55Amplitude-modulated Systems / 56Frequency Analysis and the Sideband System / 57French’s Test Procedure and Results / 59

A Special Tape for Optical Transducers / 61Time-interval Measurement Systems / 62

Results from Toram’s Method / 65Results from the Barrios/Darlow Method / 67

References / 68

Exercises / 69

3 Introduction to Rotordynamics Analysis 71

Objectives of Rotordynamics Analysis / 72The Spring–Mass Model / 74Synchronous and Nonsynchronous Whirl / 77Analysis of the Jeffcott Rotor / 78

Polar Coordinates / 79Cartesian Coordinates / 80Physical Significance of the Solutions / 81Three Ways to Reduce Synchronous Whirl Amplitudes / 82

Some Damping Definitions / 83The “Gravity Critical” / 83Critical Speed Definitions / 84Effect of Flexible (Soft) Supports / 84Rotordynamic Effects of the Force Coefficients—A

Summary / 90The Direct Coefficients / 90The Cross-coupled Coefficients / 91

Rotordynamic Instability / 91Effect of Cross-Coupled Stiffness on Unbalance Response / 99Added Complexities / 100Gyroscopic Effects / 101Effect of Support Asymmetry on Synchronous Whirl / 107False Instabilities / 110

CONTENTS ix

References / 112

Exercises / 114

4 Computer Simulations of Rotordynamics 119

Different Types of Models / 119Bearing and Seal Matrices / 126Torsional and Axial Models / 127Different Types of Analyses / 128Eigenanalysis / 129Linear Forced Response (LFR) / 133Transient Response / 134Shaft Modeling Recommendations / 135

How Many Elements / 13545-Degree Rule / 137Interference Fits / 138Laminations / 139Trunnions / 140Impeller Inertias via CAD Software / 140Stations for Added Weights / 142Rap Test Verification of Models / 143Stations for Bearings and Seals / 143Flexible Couplings / 144

Example Simulations / 146Damped Natural Frequency Map (NDF) / 147Modal Damping Map / 149Root Locus Map / 151Undamped Critical Speed Map / 151Mode Shapes / 157Bode/Polar Response Plot / 160Orbit Response Plot / 163Bearing Load Response Plot / 164Operating Deflected Shape (ODS) / 165Housing Vibration (ips and g’s) / 168

References / 168

x CONTENTS

5 Bearings and Their Effect on Rotordynamics 171

Fluid Film Bearings / 171Fixed-geometry Sleeve Bearings / 174Variable-geometry Tilting Pad Bearings / 185Fluid Film Bearing Dynamic Coefficients and Methods of

Obtaining Them / 190Load Between Pivots Versus Load on Pivot / 195Influence of Preload on the Dynamic Coefficients in Tilt

Pad Bearings / 201Influence of the Bearing Length or Pad Length / 203Influence of the Pivot Offset / 204Influence of the Number of Pads / 205Ball and Rolling Element Bearings / 208Case Study: Bearing Support Design for a Rocket Engine

Turbopump / 209Ball Bearing Stiffness Measurements / 213Wire Mesh Damper Experiments and Computer

Simulations / 214Squeeze Film Dampers / 216

Squeeze Film Damper without a Centering Spring / 217O-ring Supported Dampers / 220Squirrel Cage Supported Dampers / 223Integral Squeeze Film Dampers / 224Squeeze Film Damper Rotordynamic Force

Coefficients / 225Applications of Squeeze Film Dampers / 226

Optimization for Improving Stability in aCentrifugal Process Compressor / 226

Using Dampers to Improve the SynchronousResponse / 232

Using the Damper to Shift a Critical Speed or aResonance / 236

Insights into the Rotor–Bearing Dynamic Interaction withSoft/Stiff Bearing Supports / 238

Influence on Natural Frequencies with Soft/Stiff BearingSupports / 240

Effects of Mass Distribution on the Critical Speeds withSoft/Stiff Bearing Supports / 243

CONTENTS xi

Influence of Overhung Mass on Natural Frequencies withSoft/Stiff Supports / 252

Influence of Gyroscopic Moments on Natural Frequencieswith Soft/Stiff Bearing Supports / 255

References / 264

Exercises / 267Appendix: Shaft With No Added Weight / 269

6 Fluid Seals and Their Effect on Rotordynamics 271

Function and Classification of Seals / 271Plain Smooth Seals / 274Floating Ring Seals / 276Conventional Gas Labyrinth Seals / 277Pocket Damper Seals / 283Honeycomb Seals / 285Hole-pattern Seals / 287Brush Seals / 289

Understanding and Modeling Damper Seal ForceCoefficients / 291

Alford’s Hypothesis of Labyrinth Seal Damping / 292Cross-coupled Stiffness Measurements / 295Invention of the Pocket Damper Seal / 295Pocket Damper Seal Theory / 299Rotordynamic Testing of Pocket Damper Seals / 300Impedance Measurements of Pocket Damper Seal Force

Coefficients (Stiffness and Damping) and Leakage atLow Pressures / 301

The Fully Partitioned PDS Design / 304Effects of Negative Stiffness / 310Frequency Dependence of Damper Seals / 313Laboratory Measurements of Stiffness and Damping from

Pocket Damper Seals at High Pressures / 317The Conventional Design / 317The Fully Partitioned Design / 319

Field Experience with Pocket Damper Seals / 325Two Back-to-Back Compressor Applications / 325

xii CONTENTS

Case 1 / 325Case 2 / 328

A Fully Partitioned Application / 332Designing for Desired Force Coefficient Characteristics / 336

The Conventional PDS Design / 337The Fully Partitioned Pocket Damper Seal / 340Leakage Considerations / 343

Some Comparisons of Different Types of Annular GasSeals / 347

References / 348

7 History of Machinery Rotordynamics 353

The Foundation Years, 1869–1941 / 354Shaft Dynamics / 355Bearings / 360

Refining and Expanding the Rotordynamic Model,1942–1963 / 363

Multistage Compressors and Turbines, Rocket EngineTurbopumps, and Damper Seals, 1964–Present / 368

Stability Problems with Multistage CentrifugalCompressors / 370

Kaybob, 1971–72 / 370Ekofisk, 1974–75 / 373Subsequent Developments / 381

New Frontiers of Speed and Power Density with RocketEngine Turbopumps / 382

The Space Shuttle Main Engine (SSME)High-pressure Fuel Turbopump (HPFTP)Rotordynamic Instability Problem / 382

Noncontacting Damper Seals / 385Shaft Differential Heating (The Morton Effect) / 386

References / 388

INDEX 393

PREFACE

This book follows the first author’s book Rotordynamics of Turbomachin-ery in its practical approach and style. Much of the material in that bookhas been updated and extended with new information, new examples,and a few corrections that reflect what has been learned since then. Ofparticular interest and significance are the new chapters (4, 5, and 6)on bearings, seals, and computer modeling contributed by the co-authorsDr. Fouad Zeidan and Dr. Brian Murphy. Dr. Zeidan is the president oftwo companies that design and manufacture high performance bearingsand seals. These products often require the design and modeling of thecomplete rotor-bearing system to ensure reliable operation and compati-bility. Dr. Murphy is the author of XLRotor™, one of the most widely usedcomputer programs for rotordynamic analysis. Chapters 1 and 7 are alsocompletely new. Chapter 1 describes the classical analytical techniquesused by engineers for troubleshooting vibration problems. Chapter 7 givesa history of the most important rotordynamics analysis and experimentssince 1869.

The authors have noted (with some surprise) for many years that thesubject material of this book is not taught in most engineering colleges,even though rotating machines are probably the most common applicationof mechanical engineering. The book is organized so that the first threeor four chapters could be used as a text for a senior or graduate collegeelective course. These chapters have exercises at the end that can beassigned to the students, which will greatly enhance their understandingof the chapter material. The later chapters will serve the same students wellafter graduation as reference source material with examples of analysisand test results for real machines, bearings, and seals. But for the majorityof engineers assigned to troubleshoot a rotating machine, or to design itfor reliability, and having no relevant technical background, this entirebook can be the substitute for the course they never had.

It is the author’s hope that this book will make a significant contributionto the improvement of rotating machines for the service of mankind inthe years to come.

John M. VanceFouad Y. ZeidanBrian T. Murphy

xiii

1FUNDAMENTALS OF

MACHINE VIBRATION ANDCLASSICAL SOLUTIONS

This chapter is focused on practical applications of mechanical vibrationstheory. The reader may want to supplement the chapter with one of thevibration textbooks in the reference list at the end of the chapter if he hasno background in the theory.

THE MAIN SOURCES OF VIBRATION IN MACHINERY

The most common sources of vibration in machinery are related to theinertia of moving parts in the machine. Some parts have a reciprocatingmotion, accelerating back and forth. In such a case Newton’s laws requirea force to accelerate the mass and also require that the force be reacted tothe frame of the machine. The forces are usually periodic and thereforeproduce periodic displacements observed as vibration. For example, thepiston motion in the slider-crank mechanism of Fig. 1-1 has a fundamentalfrequency equal to the crankshaft speed but also has higher frequencies(harmonics). The dominant harmonic is twice crankshaft speed (2nd har-monic). Figure 1-2a shows the displacement of the piston. It looks almostlike a sine wave but it is slightly distorted by higher-order harmonicsdue to the nonlinear kinematics of the mechanism. Fig. 1-2b shows theacceleration of the piston, where the 2nd harmonic is amplified sincethe acceleration amplitude is frequency-squared times the displacementamplitude.

Even without reciprocating parts, most machines have rotating shaftsand wheels that cannot be perfectly balanced, so according to Newton’slaws, there must be a rotating force vector at the bearing supports of each

1

2 FUNDAMENTALS OF MACHINE VIBRATION AND CLASSICAL SOLUTIONS

Y

Lb

RC

a

ω

Figure 1-1 Slider-crank mechanism.

rotor to produce the centripetal acceleration of the mass center. Most ofthese force vectors are rotating and therefore produce a rotating displace-ment vector (all real machine parts are elastic) that can be observed asan orbit if two orthogonal vibration transducers are employed. Each ofthe transducers will produce a time trace similar to Fig. 1-2a or 1-2b.Harmonics and resulting distortion similar to Fig. 1-2a and 1-2b can beproduced by shaft misalignment or by nonlinearity of the bearing stiffness.The fundamental frequency of the X and Y (orthogonal) vibration vec-tors is shaft speed ω, so the fundamental vibration is x(t) = X cos(ωt) andy(t) = Y sin(ωt). This type of vibration is referred to as forced response orsynchronous response to unbalance. The vibration amplitude can becomevery large if the excitation frequency (rotor speed for example) becomesclose to one of the natural frequencies of the machine structure. This iscalled a resonance or a critical speed , but it is not an unstable motionsince the amplitude does not grow with time (unless there is no damping).

Another type of machine vibration problem, less common but moredifficult to deal with, can come from the characteristic natural vibrationfrequencies (eigenvalues) of the machine structure and its supports, evenif no imbalance or excitation is present. Natural frequencies die out instatic structures due to the energy dissipated by damping, but in rotatingmachines they can grow larger with time. This is known as self-excitedinstability or rotordynamic instability . It is an innate potential characteris-tic of some rotating machines, especially when fluid pressures are present(e.g., bearings, impellers, turbine wheels, or seals).

Every real structure has an infinite number of natural frequencies, butmany machinery vibration problems involve just one of these frequencies.That is why the simple single degree of freedom (SDOF) model (withjust one natural frequency) presented in vibration textbooks [1–3] can be

THE MAIN SOURCES OF VIBRATION IN MACHINERY 3

0 0.02 0.04 0.060

0.05

0.1

0.15

0.2Translation of slider

(sec)

(a)

(b)

(sec)

(ft)

0.176

8.382 × 10−3

6.536 × 103

−5.686 × 103

Xp (t)

Xp″ (t)

Tmax0

l2

l1

0 0.02 0.04 0.06

0

5000

1⋅104

−5000

−1⋅104

Acceleration of slider

Angular acceleration of bar BC

(ft/s

ec2 )

Tmax0 t

t

Figure 1-2 (a) Displacement of the piston, and (b) acceleration of the piston.

useful for analyzing vibration in machines. In fact, a SDOF model, con-sisting of one rigid mass, one spring, and one damper can be constructedto represent the vibration characteristics of any real machine in the neigh-borhood of a particular natural frequency of interest. This is called a modalmodel . To make physical sense out of complex machinery vibration data,or from realistic computer simulations of machinery vibration, the details

4 FUNDAMENTALS OF MACHINE VIBRATION AND CLASSICAL SOLUTIONS

of the SDOF mathematical model, its variations, and its solutions mustbe burned indelibly into the mind of the vibration engineer.

THE SINGLE DEGREE OF FREEDOM (SDOF) MODEL

The SDOF model as seen in most vibration textbooks is shown inFig. 1-3. Here it will be referred to as system A. The stiffness, damping,and mass are k , c, and m , respectively. The undamped natural frequencyis given by

ωn =√

k

mrad/sec (1-1)

The circular frequency ωn can be converted to hertz (Hz) (cycles/sec) asfn = ωn/2π , or to revolutions per minute (rpm) as N = 60 fn .

With a sinusoidal force applied to the mass, the differential equationof motion

mx + cx + kx = F sin(ωt) (1-2)

has a solution made up of two parts: (1) the particular solution for xthat gives F sin(ωt) on the right-hand side, and (2) the homogeneoussolution for x that gives zero on the right-hand side. The sum of the twosolutions, of course, gives F sin(ωt), which satisfies the equality sign.The two solutions represent the two types of machine vibration describedin the previous section, that is, forced response and characteristic (free)vibration. The particular solution for forced response is

xp(t) = F sin(ωt + φ)/√(

k − mω2)2 + (cω)2 (1-3)

Fsin (ωt)

m

kc

x(t)

Figure 1-3 Single degree of freedom vibration model (system A).

THE SINGLE DEGREE OF FREEDOM (SDOF) MODEL 5

Notice that the frequency ω of the forced vibration response is the same asthe frequency of the excitation. The angle φ gives the time φ/ω by whichthe response x lags the excitation force F . For analyzing a vibration prob-lem it is important to understand how k , c, and m influence the responseamplitude. They have different effects depending on the frequency ratioω/ωn , as we shall see in the section to follow. Looking at Eq. 1-3 wecan see that the amplitude X of the forced vibration response is

X = F/√(

k − mω2)2 + (cω)2 (1-4)

which depends on k , c, m , ω, and F . Notice that the denominator getssmall when the exciting frequency ω is ωn (Eq. 1-1) unless the dampingcoefficient c is large. A plot of Eq. 1-4 is shown in Fig. 1-7. It is calledthe Bode amplitude plot or the frequency response plot for system A.

The homogeneous part of the solution (for free vibration) with F = 0is given by

xh(t) = Aest (1-5)

where s is a complex number, s = λ + iωd . s is called the eigenvalue.Using the law of exponents, Eq. 1-5 can be rewritten as

xh(t) = Aeλt eiωd t (1-6)

where

eiωd t = cos(ωd t) + i sin(ωd t) (1-7)

Equation 1-5 or 1-6 satisfies the differential Eq. 1-2 with F = 0 providedthat the real part of the eigenvalue is λ = −c/2m and the imaginary partis the square root of ω2

d = k/m − (c/2m)2. The amplitude A in Eq. 1-5is of little interest here since it is determined only by the initial conditionthat instigates the free vibration. In rotating machinery, the differentialequations are more complicated but still are of the same class as (1-2)and have the same form of homogeneous solution as (1-5). The imaginarypart of s , ωd , is the damped natural frequency. Notice that it becomesequal to ωn , Eq. 1-1, when the damping coefficient c = 0.

The real part λ of the eigenvalue s determines how fast thefree vibration dies out. It is often converted into a damping ratioζ = c/ccr, where the critical damping ccr = 2mωn . Critical dampingis the amount required to prevent free vibration (and no more). Theconversion equation is ζ = −λ/ωn . Figure 1-4a shows free vibration withζ = 0.05 (5% of critical damping); Fig. 1-4b shows the same systemwith ζ = 0.25 (25% of critical damping). If a free vibration is graphedlike Fig. 1-4, the damping can be expressed as the natural logarithm of

6 FUNDAMENTALS OF MACHINE VIBRATION AND CLASSICAL SOLUTIONS

X(t) vs. time

−1−0.8−0.6−0.4−0.2

00.20.40.60.8

1

0 0.01 0.02 0.03 0.04 0.05 0.06

Time - sec

X

(a) 5% damping

(b) 25% damping

X(t) vs. time

−1−0.8−0.6−0.4−0.2

00.20.40.60.8

1

0 0.01 0.02 0.03 0.04 0.05 0.06

Time - sec

X

Figure 1-4 (a) Free vibration with 0.05 damping ratio; (b) free vibration with0.25 damping ratio.

the ratio of successive amplitudes Xn/Xn+1. The logarithmic decrementδ = ln(Xn/Xn+1) = 2πζ

/(1 + ζ 2

)1/2. The inverse expression is often

useful: ζ = δ/[

(2π)2 + δ2]1/2

.The algebraic sign of the real part of the eigenvalue λ is the mathemati-

cal test for vibration stability, i.e., whether the free vibration of frequencyωd will die out or, in the unstable case, will grow with time. For example,in the simple system of Fig. 1-3, λ becomes positive if the damping c isnegative. Negative damping is possible in mechanical systems, especiallywhen fluid pressures are acting.

USING SIMPLE MODELS FOR ANALYSIS AND DIAGNOSTICS

Techniques and methods for solving vibration problems can often bedeveloped by using the simple one degree of freedom model even thoughthe real system is more complicated. The main purpose of the model is

USING SIMPLE MODELS FOR ANALYSIS AND DIAGNOSTICS 7

to provide an understanding of the type of problem being encountered sothat the most effective type of “fix” can be identified. Sometimes a simplemodel can even yield useful approximations for the optimum parametricvalues, such as stiffness and damping to be employed. In contrast to thelarge and detailed finite element models being promoted by some forall diagnostic vibration analysis, this approach suggests that the engineershould first use the simplest possible model that contains the relevantphysical characteristics and resort to the more detailed models only whenthe simple models do not yield sufficient guidance for modifications tothe design or when improved accuracy is desired.

In addition to system A of Fig. 1-3, two more single degree of freedommodels are shown in Figs. 1-5 and 1-6. All three of these systems havea single natural frequency determined by their modal mass and stiffness,but there are subtle differences between the three models that are relatedto the type of excitation.

The constant amplitude exciting force F in system A is generallyunrealistic. Inertia forces in rotating machinery are proportional to speedsquared. Model C in Fig. 1-6 has an unbalanced rotor so that the excitingforce F = mω2u, where u is the offset of the center of rotor mass m fromthe axis of rotation. Note that the mass m is the rotating mass, not thetotal mass, so m on the left side of differential equation (1-2) must bereplaced by the total mass M unless the nonrotating mass is negligible.

In some cases the excitation is a vibration displacement at the base,rather than a force. This is represented by system B in Fig. 1-5.

These small differences in the models produce different frequencyresponse curves. The differences are useful in diagnosing problems anddetermining solutions. Obviously, to use these differences, the engineermust have a complete and thorough knowledge of the three models andtheir responses. The three systems illustrated in Figs. 1-3, 1-5, and 1-6and their mathematical analyses are described in most vibration textbooks[1–3]. In some cases the damping should be included in the most real-istic way possible, i.e., as viscous, Coulomb, hysteretic, or aerodynamicdamping. However, if the damping is other than viscous, it may usually be

Ysin (wt)

m

kc

x(t)

SYSTEM B

Figure 1-5 SDOF model with base excitation.

8 FUNDAMENTALS OF MACHINE VIBRATION AND CLASSICAL SOLUTIONS

Rotor mass = m

Total mass = Mkc

x(t)u ωt

SYSTEM C

Unbalance = u

Figure 1-6 SDOF model with rotating unbalance.

represented by an equivalent viscous damping coefficient that varies withfrequency [1, page 73]. For purely steel structures, it is usually less than5% of the critical value. System B may have its predominant dampingeither (1) between the vibrating base and the modal mass, or (2) from themass to ground. It is important to recognize the difference and set up themodel correctly.

The frequency response curves for systems A, B, and C are plots of theamplitude of forced vibration versus the frequency. The response ampli-tude for system A is computed from Eq. 1-4 at each frequency, usingappropriate values for k , c, m , and F . Figure 1-7 shows the response curvefor system A with parameter values from Table 1-1. For plotting the curve,frequency ω (rad/sec) has been converted to rpm (cpm). X_static in thetable is F /k , the displacement at zero frequency, which is the deflectionof the spring under a static force F . Resonance is the undamped natural

System A Displacement Response

0.00E+00

5.00E−03

1.00E−02

1.50E−02

2.00E−02

2.50E−02

0

400

800

1200

1600

2000

2400

2800

3200

3600

4000

4400

4800

5200

5600

6000

6400

6800

7200

7600

8000

rpm

inch

es

Figure 1-7 Forced response of system A (constant amplitude excitation force F).

USING SIMPLE MODELS FOR ANALYSIS AND DIAGNOSTICS 9

Table 1-1 System A values for Fig. 1-7

Data Units

InputMass 100 lbKstiff 30,000 lb/inCdamp 20 lb-sec/inForce 150 lbFreqstart 0 rpmFreqstop 8000 rpmNpoints 101 use 101

OutputResonance 3251.252 rpmZeta 0.11349 noneX_static 5.00E-03 in

frequency ωn converted to cpm. Zeta is the critical damping ratio, i.e., thepercentage of critical damping divided by 100. The solid curve in Fig. 1-7has all the parametric values of Table 1-1.

The dashed curve in Fig. 1-7 has all the values of Table 1-1 except thatthe damping coefficient c has been increased from 20 lb-sec/in. (in thesolid curve) to 30 lb-sec/in. The main effect of the increased damping is toreduce the vibration amplitude at the critical speed. It has very little effectat frequencies away from the critical speed. The critical speed (where thepeak vibration occurs) is 3200 rpm for the solid curve and about 3150 rpmfor the dashed curve. These are both slightly below the undamped naturalfrequency of 3251 cpm. Thus, damping tends to lower the critical speed.(This effect is reversed in system C (below) when the constant shakingforce F is replaced with a rotating unbalance force mω2u). In Fig. 1-7, notice that the response amplitude X (= 5 mils at zero frequency)becomes large near the natural frequency, and approaches zero at veryhigh frequencies. Figure 1-8 shows how the vibration X (the dashed curve)lags the force F with a phase angle φ (see Eq. 1-3). Figure 1-9 shows howthe phase angle varies with frequency. More damping (the dashed curve)makes the phase angle change more gradually as the excitation frequencypasses through ωn . The phase angle is 90 degrees at the undamped naturalfrequency ωn , regardless of the amount of damping. This fact is useful indetermining the value of ωn , since the phase angle can be measured butωn cannot be measured.

Graphs like Figs. 1-7 and 1-9 are often referred to as the frequencyresponse curves, or Bode plots. If the parameter values (k , c, m) arechanged, then the response curves will look similar but will have different

10 FUNDAMENTALS OF MACHINE VIBRATION AND CLASSICAL SOLUTIONS

X(t) lagging F(t)

−10−8−6−4−202468

10

0 0.02 0.04 0.06 0.08 0.1

Time - sec

F a

nd

X

Figure 1-8 X (dashed) lagging force (solid).

0

30

60

90

120

150

180

0 2000 4000 6000 8000 10000rpm

Deg

rees

Lag

Figure 1-9 Phase lag response of system A.

values of response amplitude and phase. Increasing the damping generallybrings the peak amplitude down but has a negligible effect at frequenciesaway from the natural frequency.

The necessity to plot many different curves for different values of F ,k , and m is avoided by plotting the curve with dimensionless ratios asshown in Fig. 1-10. The abscissa in Fig. 1-10 is frequency ratio ω/ωn ;the ordinate Xk/F is X/X _static (the ratio of vibration amplitude tostatic displacement under the force F ).

The frequency response of system B (Fig. 1-5, base vibration excitation)is given by

X = Y

√√√√ k2 + (ωc)2(k − mω2

)2 + (cω)2(1-8)

Figure 1-11 shows the response amplitude X calculated with the paramet-ric values of Table 1-2. In the table, X_Base is the displacement amplitudeY of the vibrating support. Notice that damping in system B (the dashed

USING SIMPLE MODELS FOR ANALYSIS AND DIAGNOSTICS 11

0.00E+00

5.00E−01

1.00E+00

1.50E+00

2.00E+00

2.50E+00

3.00E+00

3.50E+00

4.00E+00

4.50E+00

5.00E+00

0.0 0.3 0.6 0.9 1.2 1.5 1.8 2.2 2.5 2.8 3.1

Frequency ratio

X*k

/F

Figure 1-10 Dimensionless response of system A.

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000rpm

inch

es

Figure 1-11 Response to base excitation of system B.

curve) actually increases the vibration response at high frequencies. Solv-ing the differential equation for system B [1, page 66] shows that thecrossover frequency is 1.4 times the undamped natural frequency. Allthe curves with different damping values cross at this frequency, and theamplitude there is the same as X_Base. The frequency range above thisis called the isolation range, since the response there is reduced belowwhat would be obtained with a hard support. A vibrating system with afixed excitation frequency can be put into the isolation range by softeningthe spring Kstiff between the vibrating base and the mass.

The frequency response of system C (Fig. 1-6) is given by Eq. 1-9,where u is the unbalance (C.G. offset of the rotor), m is the rotor mass,and M is the total mass:

X = mω2u/√(

k − M ω2)2 + (cω)2 (1-9)

12 FUNDAMENTALS OF MACHINE VIBRATION AND CLASSICAL SOLUTIONS

Table 1-2 System B parameters for Fig. 1-11

Data Units

InputMass 0.35 lbKstiff 100 lb/inCdamp 0.1 lb-sec/inCdamp2 0.4 lb-sec/inX_Base 0.1 inFreqstart 0 rpmFreqstop 10,000 rpmNpoints 101 use 101

OutputResonance 3172.897 rpmZeta 0.166132 noneZeta2 0.66453 none

0.00E+00

1.00E+00

2.00E+00

3.00E+00

4.00E+00

5.00E+00

6.00E+00

7.00E+00

8.00E+00

011

0022

0033

0044

0055

0066

0077

0088

0099

00

1100

0

rpm

XM

/mu

Figure 1-12 Response with an unbalanced rotor for three damping values.

The ratio X /u is often used and is sometimes called the magnificationfactor . The response calculated with the parametric values of Table 1-3is shown in Fig. 1-12 with dimensionless amplitude XM /mu. In the table,m = Rotrmass and M = Rotrmass + Housmass. Note that XM /mu isapproximately X /u in this case, since m/M = 0.98 (the housing massis negligible). Figure 1-12 shows that system C response starts out atzero and damping in system C reduces the peak amplitude of vibrationresponse and raises the critical speed. At very high frequencies the vibra-tion amplitude approaches a limiting value determined by the amount ofunbalance. Increasing the housing mass will reduce this value.

SIX TECHNIQUES FOR SOLVING VIBRATION PROBLEMS WITH FORCED EXCITATION 13

Table 1-3 System C parameters forFig. 1-12

Data Units

InputRotrmass 50 lbHousmass 1 lbKstiff 28,000 lb/inCdamp 8 lb-sec/inCdamp2 16 lb-sec/inCdamp3 32 lb-sec/inUnbalance 0.0015 inFreqstart 0 rpmFreqstop 11000 rpmNpoint 101 use 101

OutputResonance 4398.29 rpmZeta 0.065798 noneZeta2 0.131597 noneZeta3 0.263193 noneTotalmass 51 lbMassratio 0.980392 none

SIX TECHNIQUES FOR SOLVING VIBRATIONPROBLEMS WITH FORCED EXCITATION

When vibration measurements from the real system are compared andidentified with the theoretical response from the appropriate model (A,B, or C) one of the following techniques for reducing the vibration willoften become apparent.

1. Identify and reduce the excitation source. This most obvious solu-tion is also the one least likely to be possible in systems of type Aor type B, but it should be investigated first. In rotating machin-ery (system C), this technique is implemented by balancing therotating parts. Balancing will be effective only when the vibrationfrequency is equal to the speed of a rotating part or its integerharmonics, and this fact is the corollary of a diagnostic rule: Fre-quency components in a measured spectrum that are synchronouswith a rotating speed or one of its harmonics are often caused byrotating imbalance. In a reciprocating machine (Fig. 1-1), balanc-ing the 2nd harmonic often requires a separate unbalanced balance

14 FUNDAMENTALS OF MACHINE VIBRATION AND CLASSICAL SOLUTIONS

shaft rotating at twice crankshaft speed to cancel out the inertiaforces.

2. Tune the natural frequency to a value further away from the frequencyof excitation to avoid resonance. A study of the frequency responsecurves for any of the systems A, B, or C reveals that the vibra-tory excitation is highly magnified at frequencies near the naturalfrequency. This magnification factor R, or Q factor as it is some-times called, can typically range from 5 to 50 or more dependingon the amount of damping. The excitation frequency can seldom bechanged, but the natural frequency can sometimes be easily changedby changing the modal stiffness. This is one place where intelligentconstruction of the analytical model becomes important, since themodal stiffness may be made up of several real stiffnesses in par-allel or in series. In parallel combinations the very low stiffnesseshave little effect in determining the modal stiffness, while in seriescombinations the very high stiffnesses have little effect. The tuningmethod is effective only when the excitation frequency is constantor when it only varies over a narrow range.

3. Isolate the modal mass from the vibratory excitation by making themodal stiffness very low. Notice that all the response curves showa very low response to the vibratory excitation at frequencies muchhigher than the natural frequency (far to the right on the responsecurves). Once again, the excitation frequency usually cannot bechanged but the natural frequency can be brought far down by avery soft modal stiffness, thus placing the system response far tothe right of resonance on the response curve. This method is par-ticularly effective in systems of type B. A typical application isisolating an electronics box from a vibrating vehicle frame.

4. Add damping to the system. Damping is added by incorporatingmechanisms that dissipate vibratory energy into heat. When theywork, damping mechanisms produce forces that act in opposition tothe vibratory velocity. Contrary to popular belief, however, addingdamping indiscriminately does not always reduce vibration. Damp-ing does work well whenever operation is near resonance (and thisis the operating condition most likely to cause a problem). At fre-quencies away from resonance damping has very little effect, exceptto increase the forces transmitted to ground at high frequencies farabove resonance. In a system B application where isolation is used,damping added between the modal mass and the vibrating supportwill actually increase the vibration of the mass at high frequencies.In a system C (rotating machinery) application with rolling elementbearings, adding damping to the bearing supports will increase the