Harris'''' Shock and Vibration Handbook Part 1 ppt

HARRIS’
SHOCK AND
VIBRATION
HANDBOOK
Cyril M. Harris Editor
Charles Batchelor Professor Emeritus
of Electrical Engineering
Columbia University
New York, New York
Allan G. Piersol Editor
Consultant
Piersol Engineering Company
Woodland Hills, California
Fifth Edition
McGRAW-HILL
New York Chicago San Francisco Lisbon London Madrid
Mexico City Milan New Delhi San Juan Seoul
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8434_Harris_fm_b.qxd 09/20/2001 11:40 AM Page iii
Library of Congress Cataloging-in-Publication Data
Harris’ shock and vibration handbook / Cyril M. Harris, editor, Allan G.
Piersol, editor.—5th ed.
p. cm.
ISBN 0-07-137081-1
1. Vibration—Handbooks, manuals, etc. 2. Shock (Mechanics)—
Handbooks, manuals, etc. I. Harris, Cyril M., date. II. Piersol, Allan G.
TA355.H35 2002
620.3—dc21 2001044228
Copyright © 2002, 1996, 1988, 1976, 1961 by The McGraw-Hill Companies,
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1234567890 DOC/DOC 07654321
ISBN 0-07-137081-1
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8434_Harris_fm_b.qxd 09/20/2001 11:40 AM Page iv
ABOUT THE EDITORS
Cyril M. Harris, one of the world’s leading authorities on shock, vibration, and

noise control, currently lectures at Columbia University where he is the Charles
Batchelor Professor Emeritus of Electrical Engineering. Dr. Harris has received
many honors for his scientific and engineering achievements, including membership
in both the National Academy of Sciences and the National Academy of Engineer-
ing. He has been the recipient of the Gold Medal and the Sabine Medal of the
Acoustical Society of America, the Franklin Medal of the Franklin Institute, the
Gold Medal of the Audio Engineering Society, and the A.I.A. Medal of the Ameri-
can Institute of Architects.
He received his Ph.D. degree in physics from M.I.T. and has been awarded hon-
orary doctorates by Northwestern University and the New Jersey Institute of Tech-
nology.Among books written or edited by Dr. Harris are the following McGraw-Hill
publications: Handbook of Acoustical Measurements and Noise Control, Third Edi-
tion (1991); Noise Control in Buildings (1994); Dictionary of Architecture and Con-
struction, Third Edition (2000); and Handbook of Utilities and Services for Buildings
(1990).
Allan G. Piersol is a professional engineer in private practice specializing in the
analysis of and design for shock, vibration, and acoustical environments. He received
an M.S. degree in engineering from UCLA and is licensed in both mechanical and
safety engineering. Mr. Piersol is a Fellow of the Acoustical Society of America and
the Institute of Environmental Sciences and Technology, and a recipient of the latter
organization’s Irvin Vigness Memorial Award. He is the co-author with Julius S.
Bendat of several books published by John Wiley & Sons, the most recent being
Engineering Applications of Correlation and Spectral Analysis, Second Edition
(1993), and Random Data: Analysis and Measurement Procedures, Third Edition
(2000). He is also a co-author of NASA-HDBK-7005, Dynamic Environmental Cri-
teria (2001), and a contributor to numerous other engineering handbooks.
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PREFACE
The first edition of the Shock and Vibration Handbook in 1961 brought together for
the first time a comprehensive survey of classical shock and vibration theory and

current applications of that theory to contemporary engineering practice. Edited by
Cyril M. Harris and the late Charles E. Crede, the book was translated into several
languages and became the standard reference work throughout the world. The Sec-
ond Edition appeared in 1976, the Third Edition in 1988, and the Fourth Edition in
1996.
There have been many important developments in the field since the Fourth
Edition was published, including advances in theory, new applications of computer
technologies, new methods of shock and vibration control, new instrumentation,
and new materials and techniques used in controlling shock and vibration. Many
new standards and test codes have also been enacted. These developments have
necessitated this Fifth Edition, which covers them all and presents a thorough,
unified, state-of-the-art treatment of the field of shock and vibration in a single
volume that is approximately 10 percent longer than its predecessor edition.A new
co-editor, highly regarded as an author in his own right, has collaborated with an
original editor in this endeavor. The book brings together a wide variety of skills
and expertise, resulting in the most significant improvements in the Handbook
since the First Edition.
New chapters have been added and many other chapters updated, revised, or
expanded to incorporate the latest developments. Several chapters written by
authors who are now deceased have been revised and updated by the editors, but the
credits to the original authors are retained in recognition of their outstanding con-
tributions to shock and vibration technology. (For convenience, and to retain as
closely as possible the chapter sequence of prior editions, several chapters have been
designated Part II or III of an associated chapter.) The editors have avoided dupli-
cation of content between chapters except when such repetition is advisable for rea-
sons of clarity. In general, chapters in related areas are grouped together whenever
possible. The first group of chapters presents a theoretical basis for shock and vibra-
tion. The second group considers instrumentation and measurement techniques, as
well as procedures for analyzing and testing mechanical systems subjected to shock
and vibration.The third group discusses methods of controlling shock and vibration,

and the design of equipment for shock and vibration environments. A final chapter
presents the effects of shock and vibration on human beings, summarizing the latest
findings in this important area. Extensive cross-references enable the reader to
locate relevant material in other chapters.The Handbook uses uniform terminology,
symbols, and abbreviations throughout, and usually both the U.S. Customary System
of units and the International System of units.
The 42 chapters have been written by outstanding authorities, all of them experts
in their fields. These specialists come from industrial organizations, government and
university laboratories, or consulting firms, and all bring many years of experience to
their chapters. They have made a special effort to make their chapters as accessible
xi
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as possible to the nonspecialist, including the use of charts and written explanations
rather than highly technical formulas when appropriate.
Over the decades, the Handbook has proven to be a valuable working reference
for those engaged in many areas of engineering, among them aerospace, automotive,
air-conditioning, biomedical, civil, electrical, industrial, mechanical, ocean, and
safety engineering, as well as equipment design and equipment maintenance engi-
neering. Although this book is not intended primarily as a textbook, it has been
adopted for use in many universities and engineering schools because its rigorous
mathematical basis, combined with its solutions to practical problems, are valuable
supplements to classroom theory.
We thank the contributors to the Fifth Edition for their skill and dedication in the
preparation of their chapters and their diligence in pursuing our shared objective of
making each chapter the definitive treatment in its field; in particular, we thank
Harry Himelblau for his many helpful suggestions. We also wish to express our
appreciation to the industrial organizations and government agencies with which
many of our contributors are associated for clearing for publication the material
presented in their chapters. Finally, we are indebted to the standards organizations
of various countries—particularly the American National Standards Institute

(ANSI), the International Standards Organization (ISO), and the International
Electrotechnical Commission (IEC)—as well as to their many committee members
whose selfless efforts have led to the standards cited in this Handbook.
The staff members of the professional book group at McGraw-Hill have done an
outstanding job in producing this new edition. We thank them all, and express our
special appreciation to the production manager,Tom Kowalczyk, for his support.
Cyril M. Harris
Allan G. Piersol
xii PREFACE
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CONTENTS
Chapter 1. Introduction to the Handbook 1.1
Cyril M. Harris, Charles Batchelor Professor Emeritus of Electrical Engineering, Columbia
University, New York, NY 10027.
Chapter 2. Basic Vibration Theory 2.1
Ralph E. Blake, formerly Consultant,Technical Center of Silicon Valley, San Jose, CA.
Chapter 3. Vibration of a Resiliently Supported Rigid Body 3.1
Harry Himelblau, Consultant,The Boeing Company, Space and Communications Division,
Canoga Park, CA 91309-7922.
AND
Sheldon Rubin, Consultant, Rubin Engineering Company, Sherman Oaks, CA 91403-4708.
Chapter 4. Nonlinear Vibration 4.1
Fredric Ehrich, Senior Lecturer, Massachusetts Institute of Technology, Cambridge, MA 02139.
AND
H. Norman Abramson, Retired Executive Vice President, Southwest Research Institute,
San Jose,TX 78228.
Chapter 5. Self-Excited Vibration 5.1
Fredric Ehrich, Senior Lecturer, Massachusetts Institute of Technology, Cambridge, MA 02139.
Chapter 6. Dynamic Vibration Absorbers and Auxiliary Mass Dampers 6.1
F. Everett Reed, formerly President, Littleton Research and Engineering Corporation,

Littleton, MA 01460.
Chapter 7. Vibration of Systems Having Distributed Mass and Elasticity 7.1
William F. Stokey, Late Professor of Mechanical Engineering, Carnegie-Mellon University,
Pittsburgh, PA 15236.
Chapter 8. Transient Response to Step and Pulse Functions 8.1
Robert S. Ayre, Late Professor of Civil Engineering, University of Colorado, Boulder,
CO 80309.
Chapter 9. Effect of Impact on Structures 9.1
William H. Hoppman II, Late Professor of Engineering, University of South Carolina,
Columbia, SC 29208.
v
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Chapter 10. Mechanical Impedance 10.1
Elmer L. Hixson, Professor Emeritus of Electrical Engineering, University of Texas at Austin,
Austin,TX 78712.
Chapter 11. Statistical Methods for Analyzing Vibrating Systems 11.1
Richard G. DeJong, Professor of Engineering, Calvin College, Grand Rapids, MI 49546.
Chapter 12. Vibration Transducers 12.1
Anthony S. Chu, Director of Marketing, Test Instrumentation, Endevco Corporation, San Juan
Capistrano, CA 92675.
Chapter 13. Vibration Measurement Instrumentation 13.1
Robert B. Randall, Associate Professor, University of New South Wales, Sydney, NSW 2052,
Australia.
Chapter 14. Vibration Analyzers and Their Use 14.1
Robert B. Randall, Associate Professor, University of New South Wales, Sydney, NSW 2052,
Australia.
Chapter 15. Measurement Techniques 15.1
Cyril M. Harris, Charles Batchelor Professor Emeritus of Electrical Engineering, Columbia
University, New York, NY 10027.
Chapter 16. Condition Monitoring of Machinery 16.1

Joëlle Courrech, Area Sales Manager, Brüel & Kjaer, Sound and Vibration Measurement,
A/S Denmark.
AND
Ronald L. Eshleman, Director,Vibration Institute,Willowbrook, IL 60514.
Chapter 17. Strain-Gage Instrumentation 17.1
Earl J. Wilson, formerly Chief of Strain and Environmental Branch, National Aeronautics and
Space Administration, Flight Research Center, Edwards AFB, CA 93524.
Chapter 18. Calibration of Pickups 18.1
M. Roman Serbyn, Associate Professor, Morgan State University, Baltimore, MD 21251.
AND
Jeffrey Dosch, Technical Director, PCB Piezotronics, Depew, NY 14043-2495.
Chapter 19. Shock and Vibration Standards 19.1
David J. Evans, Mechanical Engineer, National Institute of Standards and Technology,
Gaithersburg, MD 20899-9221.
AND
Henry C. Pusey, Executive Director, Society for Machinery Failure Prevention Technology,
Winchester, VA 22601-6354.
Chapter 20. Test Criteria and Specifications 20.1
Allan G. Piersol, Consultant, Piersol Engineering Company, Woodland Hills, CA 91364-4830.
vi CONTENTS
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Chapter 21. Experimental Modal Analysis 21.1
Randall J. Allemang, Professor of Structural Dynamics Research Laboratory, University of
Cincinnati, Cincinnati, OH 45221.
AND
David L. Brown, Professor of Structural Dynamics Research Laboratory, University of
Cincinnati, Cincinnati, OH 45221.
Chapter 22. Concepts in Vibration Data Analysis 22.1
Allan G. Piersol, Consultant, Piersol Engineering Company, Woodland Hills, CA 91364-4830.
Chapter 23. Concepts in Shock Data Analysis 23.1

Sheldon Rubin, Consultant, Rubin Engineering Company, Sherman Oaks, CA 91403-4708.
Chapter 24. Vibration of Structures Induced by Ground Motion 24.1
William J. Hall, Professor Emeritus of Civil Engineering, University of Illinois at Urbana-
Champaign, Urbana, IL 61801.
Chapter 25. Vibration Testing Machines 25.1
David O. Smallwood, Distinguished Member of the Technical Staff, Sandia National
Laboratories, Albuquerque, NM 87185.
Chapter 26, Part I. Shock Testing Machines 26.1
Richard H. Chalmers, Late Consulting Engineer, Induced Environments Consultants,
San Diego, CA 92107.
Chapter 26, Part II. Pyroshock Testing 26.15
Neil T. Davie, Principal Member of the Technical Staff, Sandia National Laboratories,
Albuquerque, NM 87185.
AND
Vesta I. Bateman, Principal Member of the Technical Staff, Sandia National Laboratories,
Albuquerque, NM 87185.
Chapter 27. Application of Digital Computers 27.1
Marcos A. Underwood, President, Tu’tuli Enterprises, Gualala, CA 95445.
Chapter 28, Part I. Matrix Methods of Analysis 28.1
Stephen H. Crandall, Ford Professor of Engineering Emeritus, Massachusetts Institute of
Technology, Cambridge, MA 02139.
AND
Robert B. McCalley, Jr., Retired Engineering Manager, General Electric Company,
Schenectady, NY 12309.
Chapter 28, Part II. Finite Element Models 28.29
Robert N. Coppolino, Principal Engineer, Measurement Analysis Corporation, Torrence,
CA 90505.
Chapter 29, Part I. Vibration of Structures Induced by Fluid Flow 29.1
Robert D. Blevins, Consultant, San Diego, CA 92103.
CONTENTS vii

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Chapter 29, Part II. Vibration of Structures Induced by Wind 29.21
Alan G. Davenport, Founding Director, Boundary Layer Wind Tunnel Laboratory, and
Professor Emeritus of Civil Engineering, University of Western Ontario, London, ON N6A 5B9,
Canada.
AND
Milos Novak, Late Professor of Civil Engineering, University of Western Ontario, London, ON
N6A 5B9, Canada.
Chapter 29, Part III. Vibration of Structures Induced by Sound 29.47
John F. Wilby, Consultant,Wilby Associates, Calabasas, CA 91302.
Chapter 30. Theory of Vibration Isolation 30.1
Charles E. Crede, Late Professor of Mechanical Engineering and Applied Mechanics,
California Institute of Technology, Pasadena, CA 91125.
AND
Jerome E. Ruzicka, formerly Barry Controls, Brighton, MA 02135.
Chapter 31. Theory of Shock Isolation 31.1
R. E. Newton, Late Professor of Mechanical Engineering, United States Naval Postgraduate
School, Monterey, CA 93943.
Chapter 32. Shock and Vibration Isolators and Isolation Systems 32.1
Romulus H. Racca, formerly Senior Staff Engineer, Barry Controls, Brighton, MA 02135.
AND
Cyril M. Harris, Charles Batchelor Professor Emeritus of Electrical Engineering, Columbia
University, New York, NY 10027.
Chapter 33. Mechanical Properties of Rubber 33.1
Ronald J. Schaefer, President, Dynamic Rubber Technology, Wadsworth, OH 44281.
Chapter 34. Engineering Properties of Metals 34.1
James E. Stallmeyer, Professor Emeritus of Civil Engineering, University of Illinois at Urbana-
Champaign, Urbana, IL 61801.
Chapter 35. Engineering Properties of Composites 35.1
Keith T. Kedward, Professor of Mechanical Engineering, University of California at Santa

Barbara, Santa Barbara, CA 93106-5070.
Chapter 36. Material Damping and Slip Damping 36.1
Lawrence E. Goodman, Late Professor of Mechanics and Recorder Professor of Civil
Engineering, University of Minnesota, Minneapolis, MN 55455.
Chapter 37. Applied Damping Treatments 37.1
David I. G. Jones, Consultant, D/Tech Systems, Chandler, AZ 85226.
Chapter 38. Torsional Vibration in Reciprocating
and Rotating Machines 38.1
Ronald L. Eshleman, Director, Vibration Institute, Willowbrook, IL 60514.
viii CONTENTS
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Chapter 39, Part I. Balancing of Rotating Machinery 39.1
Douglas G. Stadelbauer, formerly Executive Vice President, Schenck-Trebel Corporation,
Deer Park, NY 11729.
Chapter 39, Part II. Shaft Misalignment of Rotating Machinery 39.37
John D. Piotrowski, President, Turvac, Inc., Oregonia, OH 45054.
Chapter 40. Machine-Tool Vibration 40.1
Eugene I. Rivin, Professor, Wayne State University, Detroit, MI 48202.
Chapter 41. Equipment Design 41.1
Karl A. Sweitzer, Senior Systems Engineer, Eastman Kodak Company, Rochester,
NY 14653-7214.
AND
Charles A. Hull, Staff Engineer, Lockheed Martin Corporation, Syracuse, NY 13221-4840.
AND
Allan G. Piersol, Consultant, Piersol Engineering Company, Woodland Hills, CA 91364-4830.
Chapter 42. Effects of Shock and Vibration on Humans 42.1
Henning E. von Gierke, Director Emeritus, Biodynamics and Bioengineering Division,
Armstrong Laboratory, Wright-Patterson AFB, OH 45433-7901.
AND
Anthony J. Brammer, Senior Research Officer, Institute for Microstructural Sciences, National

Research Council, Ottawa, ON K1A 0R6, Canada.
Index follows Chapter 42
CONTENTS ix
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CHAPTER 1
INTRODUCTION
TO THE HANDBOOK
Cyril M. Harris
CONCEPTS OF SHOCK AND VIBRATION
Vibration is a term that describes oscillation in a mechanical system. It is defined by
the frequency (or frequencies) and amplitude. Either the motion of a physical object
or structure or, alternatively, an oscillating force applied to a mechanical system is
vibration in a generic sense. Conceptually, the time-history of vibration may be con-
sidered to be sinusoidal or simple harmonic in form. The frequency is defined in
terms of cycles per unit time, and the magnitude in terms of amplitude (the maxi-
mum value of a sinusoidal quantity). The vibration encountered in practice often
does not have this regular pattern. It may be a combination of several sinusoidal
quantities, each having a different frequency and amplitude. If each frequency com-
ponent is an integral multiple of the lowest frequency, the vibration repeats itself
after a determined interval of time and is called periodic. If there is no integral rela-
tion among the frequency components, there is no periodicity and the vibration is
defined as complex.
Vibration may be described as deterministic or random. If it is deterministic, it
follows an established pattern so that the value of the vibration at any designated
future time is completely predictable from the past history. If the vibration is ran-
dom, its future value is unpredictable except on the basis of probability. Random
vibration is defined in statistical terms wherein the probability of occurrence of des-
ignated magnitudes and frequencies can be indicated.The analysis of random vibra-
tion involves certain physical concepts that are different from those applied to the
analysis of deterministic vibration.

Vibration of a physical structure often is thought of in terms of a model consist-
ing of a mass and a spring. The vibration of such a model, or system, may be “free”
or “forced.” In free vibration, there is no energy added to the system but rather the
vibration is the continuing result of an initial disturbance. An ideal system may be
considered undamped for mathematical purposes; in such a system the free vibra-
tion is assumed to continue indefinitely. In any real system, damping (i.e., energy dis-
sipation) causes the amplitude of free vibration to decay continuously to a negligible
value. Such free vibration sometimes is referred to as transient vibration. Forced
vibration, in contrast to free vibration, continues under “steady-state” conditions
1.1
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because energy is supplied to the system continuously to compensate for that dissi-
pated by damping in the system. In general, the frequency at which energy is sup-
plied (i.e., the forcing frequency) appears in the vibration of the system. Forced
vibration may be either deterministic or random. In either instance, the vibration of
the system depends upon the relation of the excitation or forcing function to the
properties of the system. This relationship is a prominent feature of the analytical
aspects of vibration.
Shock is a somewhat loosely defined aspect of vibration wherein the excitation is
nonperiodic, e.g., in the form of a pulse, a step, or transient vibration.The word shock
implies a degree of suddenness and severity. These terms are relative rather than
absolute measures of the characteristic; they are related to a popular notion of the
characteristics of shock and are not necessary in a fundamental analysis of the appli-
cable principles. From the analytical viewpoint, the important characteristic of shock
is that the motion of the system upon which the shock acts includes both the fre-
quency of the shock excitation and the natural frequency of the system. If the exci-
tation is brief, the continuing motion of the system is free vibration at its own natural
frequency.
The technology of shock and vibration embodies both theoretical and experi-
mental facets prominently. Thus, methods of analysis and instruments for the meas-

urement of shock and vibration are of primary significance. The results of analysis
and measurement are used to evaluate shock and vibration environments, to devise
testing procedures and testing machines, and to design and operate equipment and
machinery. Shock and/or vibration may be either wanted or unwanted, depending
upon circumstances. For example, vibration is involved in the primary mode of oper-
ation of such equipment as conveying and screening machines; the setting of rivets
depends upon the application of impact or shock. More frequently, however, shock
and vibration are unwanted.Then the objective is to eliminate or reduce their sever-
ity or, alternatively, to design equipment to withstand their influences. These proce-
dures are embodied in the control of shock and vibration. Methods of control are
emphasized throughout this Handbook.
CONTROL OF SHOCK AND VIBRATION
Methods of shock and vibration control may be grouped into three broad categories:
1. Reduction at the Source
a. Balancing of Moving Masses. Where the vibration originates in rotating or
reciprocating members, the magnitude of a vibratory force frequently can be
reduced or possibly eliminated by balancing or counterbalancing. For example,
during the manufacture of fans and blowers, it is common practice to rotate
each rotor and to add or subtract material as necessary to achieve balance.
b. Balancing of Magnetic Forces. Vibratory forces arising in magnetic effects of
electrical machinery sometimes can be reduced by modification of the mag-
netic path. For example, the vibration originating in an electric motor can be
reduced by skewing the slots in the armature laminations.
c. Control of Clearances. Vibration and shock frequently result from impacts
involved in operation of machinery. In some instances, the impacts result from
inferior design or manufacture, such as excessive clearances in bearings, and
can be reduced by closer attention to dimensions. In other instances, such as
the movable armature of a relay, the shock can be decreased by employing a
rubber bumper to cushion motion of the plunger at the limit of travel.
1.2 CHAPTER ONE

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2. Isolation
a. Isolation of Source. Where a machine creates significant shock or vibration
during its normal operation, it may be supported upon isolators to protect
other machinery and personnel from shock and vibration. For example, a forg-
ing hammer tends to create shock of a magnitude great enough to interfere
with the operation of delicate apparatus in the vicinity of the hammer. This
condition may be alleviated by mounting the forging hammer upon isolators.
b. Isolation of Sensitive Equipment. Equipment often is required to operate in
an environment characterized by severe shock or vibration. The equipment
may be protected from these environmental influences by mounting it upon
isolators. For example, equipment mounted in ships of the navy is subjected to
shock of great severity during naval warfare and may be protected from dam-
age by mounting it upon isolators.
3. Reduction of the Response
a. Alteration of Natural Frequency. If the natural frequency of the structure of
an equipment coincides with the frequency of the applied vibration, the vibra-
tion condition may be made much worse as a result of resonance. Under such
circumstances, if the frequency of the excitation is substantially constant, it
often is possible to alleviate the vibration by changing the natural frequency
of such structure. For example, the vibration of a fan blade was reduced sub-
stantially by modifying a stiffener on the blade, thereby changing its natural
frequency and avoiding resonance with the frequency of rotation of the blade.
Similar results are attainable by modifying the mass rather than the stiffness.
b. Energy Dissipation. If the vibration frequency is not constant or if the vibra-
tion involves a large number of frequencies, the desired reduction of vibration
may not be attainable by altering the natural frequency of the responding sys-
tem. It may be possible to achieve equivalent results by the dissipation of
energy to eliminate the severe effects of resonance. For example, the housing
of a washing machine may be made less susceptible to vibration by applying a

coating of damping material on the inner face of the housing.
c. Auxiliary Mass. Another method of reducing the vibration of the respond-
ing system is to attach an auxiliary mass to the system by a spring; with proper
tuning the mass vibrates and reduces the vibration of the system to which it is
attached. For example, the vibration of a textile-mill building subjected to the
influence of several hundred looms was reduced by attaching large masses to
a wall of the building by means of springs; then the masses vibrated with a
relatively large motion and the vibration of the wall was reduced. The incor-
poration of damping in this auxiliary mass system may further increase its
effectiveness.
CONTENT OF HANDBOOK
The chapters of this Handbook each deal with a discrete phase of the subject of
shock and vibration. Frequent references are made from one chapter to another, to
refer to basic theory in other chapters, to call attention to supplementary informa-
tion, and to give illustrations and examples. Therefore, each chapter when read with
other referenced chapters presents one complete facet of the subject of shock and
vibration.
Chapters dealing with similar subject matter are grouped together. The first 11
chapters following this introductory chapter deal with fundamental concepts of
shock and vibration. Chapter 2 discusses the free and forced vibration of linear sys-
INTRODUCTION TO THE HANDBOOK 1.3
8434_Harris_01_b.qxd 09/20/2001 11:38 AM Page 1.3
tems that can be defined by lumped parameters with similar types of coordinates.
The properties of rigid bodies are discussed in Chap. 3, together with the vibration
of resiliently supported rigid bodies wherein several modes of vibration are coupled.
Nonlinear vibration is discussed in Chap. 4, and self-excited vibration in Chap. 5.
Chapter 6 discusses two degree-of-freedom systems in detail—including both the
basic theory and the application of such theory to dynamic absorbers and auxiliary
mass dampers. The vibration of systems defined by distributed parameters, notably
beams and plates, is discussed in Chap. 7. Chapters 8 and 9 relate to shock; Chap. 8

discusses the response of lumped parameter systems to step- and pulse-type excita-
tion, and Chap. 9 discusses the effects of impact on structures. Chapter 10 discusses
applications of the use of mechanical impedance and mechanical admittance meth-
ods. Then Chap. 11 presents statistical methods of analyzing vibrating systems.
The second group of chapters is concerned with instrumentation for the measure-
ment of shock and vibration. Chapter 12 includes not only piezoelectric and piezo-
resistive transducers, but also other types such as force transducers (although strain
gages are described in Chap. 17).The electrical instruments to which such transducers
are connected (including various types of amplifiers, signal conditioners, and re-
corders) are considered in detail in Chap. 13. Chapter 14 is devoted to the important
topics of spectrum analysis instrumentation and techniques.The use of all such equip-
ment in making vibration measurements in the field is described in Chap. 15.There has
been increasing use of vibration measurement equipment for monitoring the mechan-
ical condition of machinery, as an aid in preventive maintenance; this is the subject of
Chap. 16. The calibration of transducers, Chap. 18, is followed by Chap. 19 on national
and international standards and test codes related to shock and vibration.
A discussion of test criteria and specifications is given in Chap. 20, followed by a
comprehensive chapter on modal analysis and testing in Chap. 21. Chapters 22 and
23 discuss data analysis, in conjunction with Chap. 14; the first of these two chapters
is primarily concerned with an analysis of vibration data and the second is concerned
with shock data. Vibration that is induced in buildings, as a result of ground motion,
is described in Chap. 24. Then Chap. 25 considers vibration testing machines, fol-
lowed by Chap. 26 on conventional shock testing and pyrotechnic shock testing
machines.
The next two chapters deal with computational methods. Chapter 27 is concerned
with applications of computers, presenting information that is useful in both analyt-
ical and experimental work. This is followed by Chap. 28, which is in two parts: Part
I describes modern matrix methods of analysis, dealing largely with the formulation
of matrices for use with digital computers and other numerical calculation methods;
the second part shows how finite element methods can be applied to the solution of

shock and vibration problems by the use of computer techniques.
Part I of Chap. 29 describes vibration that is induced as a result of air flow, the
second part discusses vibration that is induced by the flow of water, and the third
part is concerned with the response of structures to acoustic environments.
The theory of vibration isolation is discussed in detail in Chap. 30, and an analo-
gous presentation for the isolation of mechanical shock is given in Chap. 31. Various
types of isolators for shock and vibration are described in Chap. 32, along with the
selection and practical application of such isolators.The relatively new field of active
vibration control is described in Chap. 33. A presentation is given in Chap. 34 on the
engineering properties of rubber, followed by a presentation of the engineering
properties of metals (including conventional fatigue) and the engineering properties
of composite materials in Chap. 35.
An important method of controlling shock and vibration involves the addition of
damping or energy-dissipating means to structures that are susceptible to vibration.
Chapter 36 discusses the general concepts of damping together with the application
1.4 CHAPTER ONE
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of such concepts to hysteresis and slip damping.The application of damping materi-
als to devices and structures is described in Chap. 37.
The latter chapters of the Handbook deal with the specific application of the
fundamentals of analysis, methods of measurement, and control techniques—where
these are developed sufficiently to form a separate and discrete subject. Torsional
vibration is discussed in Chap. 38, with particular application to internal-combustion
engines.The balancing of rotating equipment is discussed in Chap. 39, and balancing
machines are described. Chapter 40 describes the special vibration problems associ-
ated with the design and operation of machine tools. Chapter 41 describes proce-
dures for the design of equipment to withstand shock and vibration—both the
design and practical aspects. A comprehensive up-to-date discussion of the human
aspects of shock and vibration is considered in Chap. 42, which describes the effects
of shock and vibration on people.

SYMBOLS AND ACRONYMS
This section includes a list of symbols and acronyms generally used in the Hand-
book. Symbols of special or limited application are defined in the respective chap-
ters as they are used.
Symbol Meaning
a radius
A/D analog-to-digital
ANSI American National Standards Institute
ASTM American Society for Testing and Materials
B bandwidth
B magnetic flux density
BSI British Standards Institution
c damping coefficient
c velocity of sound
c
c
critical damping coefficient
C capacitance
CPU central processing unit
CSIRO Commonwealth Scientific and Industrial Research Organisation
D diameter
D/A digital-to-analog
DFT discrete Fourier transform
DSP discrete signal processor
e electrical voltage
e eccentricity
E energy
E modulus of elasticity in tension and compression (Young’s modulus)
f frequency
f

n
undamped natural frequency
f
i
undamped natural frequencies in a multiple degree-of-freedom system,
where i = 1,2,
f
d
damped natural frequency
f
r
resonance frequency
F force
f
f
Coulomb friction force
FEM finite element method, finite element model
FFT fast Fourier transform
INTRODUCTION TO THE HANDBOOK 1.5
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g acceleration of gravity
G modulus of elasticity in shear
h height, depth
H magnetic field strength
Hz hertz
i electric current
I
i
area or mass moment of inertia (subscript indicates axis)
I

p
polar moment of inertia
I
ij
area or mass product of inertia (subscripts indicate axes)
IC integrated circuit
ISO International Standards Organization
I imaginary part of
j
͙
−
ෆ
1
ෆ
J inertia constant (weight moment of inertia)
J impulse
k spring constant, stiffness, stiffness constant
k
t
rotational (torsional) stiffness
l length
L inductance
m mass
m
u
unbalanced mass
M torque
M mutual inductance
ᑧ mobility
MIMO multiple input, multiple output

n number of coils, supports, etc.
NEMA National Electrical Manufacturers Association
NIST National Institute of Standards and Technology
p alternating pressure
p probability density
P probability distribution
P static pressure
q electric charge
Q resonance factor (also ratio of reactance to resistance)
r electrical resistance
R radius
ᑬ real part of
s arc length
S area of diaphragm, tube, etc.
SEA statistical energy analysis
SIMO single input, multiple output
SCC Standards Council of Canada
t thickness
t time
T transmissibility
T kinetic energy
v linear velocity
V potential energy
w width
W weight
W power
W
e
spectral density of the excitation
W

r
spectral density of the response
x linear displacement in direction of X axis
y linear displacement in direction of Y axis
z linear displacement in direction of Z axis
1.6 CHAPTER ONE
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Z impedance
α rotational displacement about X axis
β rotational displacement about Y axis
γ rotational displacement about Z axis
γ shear strain
γ weight density
δ deflection
δ
st
static deflection
∆ logarithmic decrement
⑀ tension or compression strain
ζ fraction of critical damping
η stiffness ratio, loss factor
θ phase angle
λ wavelength
µ coefficient of friction
µ mass density
µ mean value
␯ Poisson’s ratio
ρ mass density
ρ
i

radius of gyration (subscript indicates axis)
σ Poisson’s ratio
σ normal stress
σ root-mean-square (rms) value
τ period
τ shear stress
φ magnetic flux
Φ phase angle
␺ phase angle
⌿ standard deviation
ω forcing frequency—angular
ω
n
undamped natural frequency—angular
ω
i
undamped natural frequencies—angular—in a multiple degree-of-freedom
system, where i = 1,2,
ω
d
damped natural frequency—angular
ω
r
resonance frequency—angular
Ω rotational speed
Ӎ approximately equal to
CHARACTERISTICS OF HARMONIC MOTION
Harmonic functions are employed frequently in the analysis of shock and vibration.
A body that experiences simple harmonic motion follows a displacement pattern
defined by

x = x
0
sin (2πft) = x
0
sin ␻t (1.1)
where f is the frequency of the simple harmonic motion, ω=2πf is the corresponding
angular frequency, and x
0
is the amplitude of the displacement.
The velocity ˙x and acceleration ¨x of the body are found by differentiating the dis-
placement once and twice, respectively:
˙x = x
0
(2πf ) cos 2πft = x
0
ω cos ωt (1.2)
¨x =−x
0
(2πf )
2
sin 2πft =−x
0
ω
2
sin ωt (1.3)
INTRODUCTION TO THE HANDBOOK 1.7
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TABLE 1.2 Conversion Factors for Rotational Velocity and Acceleration
Multiply
Value in → rad/sec degree/sec rev/sec rev/min

or → rad/sec
2
degree/sec
2
rev/sec
2
rev/min/sec
By
To obtain
value in ↓
rad/sec 1 0.01745 6.283 0.1047
rad/sec
2
degree/sec 57.30 1 360 6.00
degree/sec
2
rev/sec 0.1592 0.00278 1 0.0167
rev/sec
2
rev/min 9.549 0.1667 60 1
rev/min/sec
TABLE 1.1 Conversion Factors for Translational Velocity and Acceleration
Multiply
Value in → g-sec, ft/sec in./sec cm/sec m/sec
or → g ft/sec
2
in./sec
2
cm/sec
2

m/sec
2
By
To obtain
value in ↓
g-sec, 1 0.0311 0.00259 0.00102 0.102
g
ft/sec 32.16 1 0.0833 0.0328 3.28
ft/sec
2
in./sec 386 12.0 1 0.3937 39.37
in./sec
2
cm/sec 980 30.48 2.540 1 100
cm/sec
2
m/sec 9.80 0.3048 0.0254 0.010 1
m/sec
2
The maximum absolute values of the displacement, velocity, and acceleration of a
body undergoing harmonic motion occur when the trigonometric functions in Eqs.
(1.1) to (1.3) are numerically equal to unity.These values are known, respectively, as
displacement, velocity, and acceleration amplitudes; they are defined mathemati-
cally as follows:
x
0
= x
0
˙x
0

= (2πf )x
0
¨x
0
= (2πf )
2
x
0
(1.4)
It is common to express the displacement amplitude x
0
in inches when the
English system of units is used and in centimeters or millimeters when the metric
system is used. Accordingly, the velocity amplitude x
0
is expressed in inches per sec-
ond in the English system (centimeters per second or millimeters per second in the
metric system). For example, consider a body that experiences simple harmonic
1.8 CHAPTER ONE
→
→
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TABLE 1.3 Conversion Factors for Simple Harmonic Motion
Multiply numerical
value in terms of → Amplitude Average Root-mean- Peak-to-peak
By value square (rms) value
To obtain value value
in terms of ↓
Amplitude 1 1.571 1.414 0.500
Average value 0.637 1 0.900 0.318

Root-mean-
square (rms) 0.707 1.111 1 0.354
value
Peak-to-peak 2.000 3.142 2.828 1
value
motion having a frequency f of 50 Hz and a displacement amplitude x
0
of 0.01 in.
(0.000254 m). According to Eq. (1.4), the velocity amplitude ˙x
0
= (2πf ) x
0
= 3.14
in./sec (0.0797 m/s). The acceleration amplitude ¨x
0
= (2πf )
2
x
0
in./sec
2
= 986 in./sec
2
(25.0 m/s
2
).The acceleration amplitude x
0
is often expressed as a dimensionless mul-
tiple of the gravitational acceleration g where g = 386 in./sec
2

(9.8 m/s
2
). Therefore
in this example, the acceleration amplitude may also be expressed as ¨x
0
= 2.55g.
Factors for converting values of rectilinear velocity and acceleration to different
units are given in Table 1.1; similar factors for angular velocity and acceleration are
given in Table 1.2.
For certain purposes in analysis, it is convenient to express the amplitude in terms
of the average value of the harmonic function, the root-mean-square (rms) value, or
2 times the amplitude (i.e., peak-to-peak value).These terms are defined mathemat-
ically in Chap. 22; numerical conversion factors are set forth in Table 1.3 for ready
reference.
INTRODUCTION TO THE HANDBOOK 1.9
→
APPENDIX 1.1 NATURAL FREQUENCIES
OF COMMONLY USED SYSTEMS
The most important aspect of vibration analysis often is the calculation or measure-
ment of the natural frequencies of mechanical systems. Natural frequencies are dis-
cussed prominently in many chapters of the Handbook. Appendix 1.1 includes in
tabular form, convenient for ready reference, a compilation of frequently used
expressions for the natural frequencies of common mechanical systems:
1. Mass-spring systems in translation
2. Rotor-shaft systems
3. Massless beams with concentrated mass loads
4. Beams of uniform section and uniformly distributed load
5. Thin flat plates of uniform thickness
6. Miscellaneous systems
The data for beams and plates are abstracted from Chap. 7.

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1.10 CHAPTER ONE
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INTRODUCTION TO THE HANDBOOK 1.11
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1.12 CHAPTER ONE
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1.13
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1.14 CHAPTER ONE
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1.15
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