Foundations for Machines-Analysis & Design_By Shamsher Prakash.pdf

Foundations for Machines

Foundations for Machines: Analysis and Design Shamsher Prakash University of Missq!)ri-Rolla

Vijay K. Puri Southern Illinois University, Carbondale

A Wiley-lnterscience Publication JOHN WILEY AND SONS New York Chichester Brisbane Toronto Singapore

Copyright 1988 by John Wiley & Sons, Inc. All rights reserved. Published simultaneously in Canada.

Reproduction or translation of any part of this work beyond that permitted by Section 107 or 108 of the 1976 United States Copyright Act without the permission of the copyright owner is unlawful. Requests for permission or further information should be addressed to

,., .:.t~~-~,ermissions Department, John Wiley & Sons, Inc.

{ibt-ary of'Congress Cataloging in Publication Data:

.- ~. ,' J?rakash~ Sham_s~~r. : Fo~ndations for machines.

(Wiley series in geotechnical engineering) "A Wiley-Interscience publication." Includes bibliographies and index. 1. Machinery-Foundations. I. Pori, Vijay.

II. Title. III. Series. TJ249.P618 1987 621.8 87-21678 ISBN 0-471-846864

Printed in the United State of America 10 9 8 7 6 5 4 3 2 1

To our friend the enlightened saint, humble philosopher, and friend of all mankind who speaks the language of the heart; whose religion is Jove; who always aspires to fill lives of one and all with spiritual bliss.

Preface

The design of machine foundations involves a systematic application of the principles of soil eljgineering, soil dynarn,,i~s, and theory of vibrations-a fact that has been well recognised during tlie last three decades. Since the classical work by Lamb in 1904 and the paper on "Foundation Vibrations" by Richart in 1962, the subject of vibratory response of foundations has attracted the attention of several researchers. The state of art on the subject has since made significant strides. Methods are now available not only for computing the response of machine foundations resting on the surface of the elastic half space but also for embedded foundations and foundations on piles. Elastic half space analogs have further simplified the computation process and are a convenient tool for the designer. The linear spring approach of Barkan, which could previously be used only for surface footings, has also been extended to account for the embedment effects. Recent advances dealing with the determination of the dynamic soil prop-

- erties and rational interpretation of the test data are of direct application to the design of machine foundations. Information on several aspects of machine foundation design such as design of embedded foundations and pile supported machine foundations is either unavailable or only inadequately treated in the presently available texts.

This text has been developed with the object of providing state-of-the-art ,,information on the analysis .,and design of machine foundations and is intended to cater to the interests of graduate students, senior under-graduates, and practicing engineers. Both authors have offered graduate-level courses on the subject in the United States and India. They also organized many short courses for practicing engineers, including four by the senior author at University of Missouri, Rolla. The authors have also been engaged in the design and performance evaluation of machine foundations. The feedback from the classroom and the professionals in the field has been of immense help in the planning and preparation of this text.

vii

viii PREFACE

The special features of this book are: (1) analysis of surface and embedded foundations by both the elastic half space method and the linear spring method; (2) analysis of pile supported machine foundations; (3) detailed discussion of the dynamic soil properties, methods for their de-termination, and evaluation of the test data; ( 4) detailed design procedure followed by examples; and (5) discussion of design of machine foundations on absorbers and vibration isolation.

Knowledge of soil mechanics and elementary mathematics or mechanics is needed to follow the text.

The reader is introduced to the problem of machine foundation and its special requirements in Chapter 1. In Chapter 2, the elementary theory of vibrations is discussed. Chapter 3 deals with the wave propagation in an elastic medium that provides an important basis for determination of dynamic soil properties as discussed in Chapter 4. Needless to say, soil properties play a critical role in the design of machine foundations. Chapter 4 thus forms a very important component of the text. Also included in this chapter is the procedure for rational selection of soil parameters for a given machine foundation problem. The determination of unbalanced forces and moments occasioned by the operation of a machine is reviewed in Chapter 5. The principal subject of the book, the analysis and design of machine foundations is introduced in Chapter 6, that deals with the design of rigid-block-type foundations for reciprocating machines. In this chapter the reader is made familiar with the concepts of elastic half space method and linear spring method for computing the vibratory response of surface footings. Foundations for impact-type machines such as hammers are dis-cussed in Chapter 7. Foundations for high-speed rotary machines are discussed in Chapter 8 and for miscellaneous machines in Chapter 9. The principles of vibration isolation and absorption are considered in Chapter 10. The design of embedded block foundations for machines is described in Chapter 11 followed by pile supported machine foundations in Chapter 12. A few case histories are discussed in Chapter 13 and construction aspects in Chapter 14.

Computer program for design of a block foundation based on principles discussed in Chapter 6 has been included in Appendix I, aud for design of a hammer foundation as in Chapter 7 has been included in Appendix II. A brief description of the commercially available programs PILAY for solution of piles and STRUDL for analysis of turbo-generator foundations is in-cluded in Appendix III.

The subject matter has been developed in a logical progression from one chapter to the next. Every effort has been made to make the text self-contained as far as possible. A comprehensive bibliography is included at the end of each chapter so that an interested reader may obtain additional information from published sources.

Development in certain areas, particularly the determination of dynamic soil properties and analysis of embedded foundations and piles under

PREFACE

dynamic loads, is taking place at a very rapid rate. Analysis and design procedures may therefore undergo modifications. This fact has also been brought to the attention of the reader"' at appropriate places in the text.

Thanks are due the American Society of Civil Engineers and National Research Council of Canada for permitting the use of materials from thefr publication. Acknowledgment to other copyrighted material is given at appropriate places in the text and figures.

In preparing this text, several of our colleagues and graduate students have helped in a variety of ways. The authors wish to express their sincere thanks to them. Special mention must be made of Dr. Krishen Kumar, who read the entire manuscript and made useful suggestions, particularly on Chapter 12, and Dr. A Syed for his useful comments and suggestions and of Mr. Murat Hazinedarogulu for assistance in writing the computer programs.

The manuscript was typed by Janet Pearson, Charlena Ousley, Allison Holdaway, and Mary Reynolds. The authors are most thankful to them for their care, painstaking efforts, and patience. John W. Koeing, technical editor at the University of Missouri, Rolla, provided editorial assistance and deserves our sincer~ !hanks. ->',

Contents

CHAPTER 1 INTRODUCTION

1.1 ;; Type of Machines 'l.~!l Foundations, 2 1.2 Design Criteria to Bi(Satisfied, 4 1.3 Relevant Codes, 9 1.4 Data Required for Design, 1 0 1.5 Significance of Soil Parameters, 1 0

References, 1 0

CHAPTER 2 THEORY OF VIBRATIONS

2.1 Definitions, 12 2.2 Simple Harmonic Motion, 14 2.3 Free Vibrations of a Spring-Mass System, Hi 2.4 Free Vibrations with Viscous Damping, 20 2.5 Forced Vibrations with Viscous Damping, 24 2.6 Frequency Dependent Excitation, 29 2.7 Systems under Transient Loads, 31 2.8 Rayleigh's Method, 34 2.9 Logarithmic Decrement, 38

2.10 Determiii!ltion of Viscous Damping, 39 2.11 Transmissibility, 41 2.12 Vibration Measuring Instruments, 42, 2.13 Systems with Two Degrees of Freedom, 44 2.14 Multidegree Freedom Systems, 50

Practice Problems, 58 References, 61

12

xii CONTENTS

CHAPTER 3 WAVE PROPAGATION IN AN ELASTIC MEDIUM 62 3.1 Wave Propagation in Elastic Rods, 63

3.1.1 Longitudinal Vibrations of Rods of Infinite Length, 63

3.1.2 Longitudinal Vibrations of Rods of Finite Length, 69

,3,13 Torsional Vibrations of Rods of Infinite Length, 74 3.1.4 Torsional Vibrations of Rods of Finite Length, 76

3.2 Wave Propagation in an Elastic Infinite Medium, 76 3.3 Wave Propagation in a Semi-infinite Elastic Half Space,

84 ' 'i . ' .

. ,, 3.4 .. .Wave""Generated .. by a.Surface Footing, 91 3.5 Final Comments, 93


CHAPTER 4 DYNAMIC SOIL PROPERTIES 4.1 Triaxial Compression Test under Static Loads, 96 4.2 Elastic Constants of Soils, 1 00 4.3 Factors Affecting Dynamic Shear Modulus, 104 4.4 Equivalent Soil Springs, 118 4.5 Laboratory Methods, 122

4.5.1 Resonant Column Test, 123 4.5.2 Ultrasonic Pulse Tests, 127 4.5.3 Cyclic Simple Shear Test 128 4.5.4 Cyclic Torsional Simple Shear Test, 131 4.5.5 Cyclic Triaxial Compression Test, 133

4.6 Field Methods, 135

95

4.6.1 Cross-Borehole Wave Propagation Test, 135 4.6.2 Up-Hole or Down-Hole Wave Propagation

Test, 136 4.6.3 Surface-Wave Propagation Test, 137 4.6.4 Vertical Footing Resonance Test, 140 4.6.5 Horizontal Footing Res6nance Test, 143 4.6.6 Free Vibration Test on Footings, 144 4.6.7 Cyclic Plate Load Test, 144 4.6.8 Standard Penetration Test, 145

4. 7 Evaluation of Test Data, 146 4.8 Damping in Soils, 147

CONTENTS

4.9 Examples, 156 4.10 Overview, 177


xiii

CHAPTER 5 UNBALANCED FORCES FOR DESIGN OF MACHINE FOUNDATIONS 189

5.1 Unbalanced Forces in Reciprocating Machines, 189 5.2 Unbalanced Forces in Rotary Machines, 201 5.3 Unbalanced Forces Due to Impact Loads, 205 5.4 Examples, 205

References, 211

CHAPTER 6 FOUNDATIONS FOR RECIPROCATING MACHINES 212

6.1;;, pesign Requirem~!'~s, 212 6.2 Modes of Vibration of a Rigid Foundation Block, 213 6.3 Methods of Analysis, 214 6.4 Elastic Half-Space Method, 214 6.5 Effect of Footing Shape on Vibratory Response, 234 6.6 Vibrations of a Rigid Circular Footing Supported by an

Elastic Layer, 236 6.7 Linear Elastic Weightless Spring Method, 240 6.8 Design Procedure for a Block Foundation, 260 6. 9 Examples, 268

6.10 Overview, 301 References, 303

CHAPTER 7 FOUNDATIONS FOR IMPACT MACHINES

7.1 7.2 7.3 7.4 7.5

Methods of Analysis, 307 Design Criteria, 318 Design Procedure for Hammer Foundations, 319

"ljq~' Examples> 323 Overview, 328 References, 329

CHAPTER 8 FOUNDATIONS FOR HIGH-SPEED ROTARY MACHINES

8.1 Layout of a Typical Turbogenerator Unit, 331,

306

330

xiv CONTENTS~

8.2 Loads on a Turbogenerator Foundation, 332 8.2.1 Loads Due to Normal Operation of Plant, 332 8.2.2 Loads Due to Emergency Conditions, 337

8.3 Design Criteria, 339 8.4 Design Concepts, 340 8.5 Methods of Analysis, 340

8.5.1 Simplified Methods, 341 8.5.2 Rigorous Methods, 357

8.6 Design Procedure, 363 8.6.1 Design Data, 364 8.6.2 Dynamic Analysis, 366

8.7 Examples, 371 Referentes, 374

CHAPTER 9 FOUNDATIONS FOR MISCELLANEOUS TYPES OF MACHINES 376 9.1 Foundations for Low-Speed Rotary Machines, 376 9.2 Foundations for Machine Tools, 391 9.3 Foundations for Stamping, Forging, and Punching

Presses, 392 9.4 Machines Supported on Floors, 394 9.5 Examples, 395

References, 398

CHAPTER 10 VIBRATION ABSORPTION AND ISOLATION 399 1 0.1 Principle of Vibration Absorption, 401 1 0.2 Common Vibration Absorbers, 404

1 0.2.1 Steel or Metal Springs, 404 1 0.2.2 Cork, 406 10.2.3 Rubber, 407 1 0.2.4 Timber, 408 1 0.2.5 Neoprene, 408 1 0.2.6 Pneumatic Absorber; 408

1 0.3 Design Procedure for Foundations on Absorbers, 41 0 1 0.4 Principles of Vibration Isolation with Wave Barriers,

413 10.4.1 Trench Barriers, 414 1 0.4.2 Pile Barriers, 420

CONTENTS

CHAPTER 11

10.5 Design Procedure for Wave Barriers, 423

. XV

1 0.6 Methods of Reducing Vibration Amplitudes in Existing Machine Foundations, 406

10.7 Examples, 431 1 0.8 Final Comments, 436

References, 436

DYNAMIC RESPONSE OF EMBEDDED BLOCK FOUNDATIONS. 438

11.1 Elastic Half-Space Method, 439 11 .1 .1 Vertical Vibrations, 440 11 .1 .2 Sliding Vibrations, 443 11 .1.3 Rocking Vibrations, 448 11.1.4 Coupled Rocking and Sliding Vibrations, 451 11.1.5 Torsional Vibrations, 456

11.:b Linear Elastic Weightless Spring Method, 459 11.2.1 Vertical Vibrations, 459 11.2.2 Sliding Vibrations, 462 11.2.3 Rocking Vibrations, 464 11.2.4 Coupled Rocking and Sliding Vibrations, 468 11.2.5 Torsional Vibrations, 469

11.3 Design Procedure for an Embedded Block Foundation, 471

11.4 Examples, 4 77 11.5 Compliance-Impedance Function Approach, 482 11.6 Overview, 448

References, 490

CHAPTER 12 MACHINE FOUNDATIONS ON PILES 493

12.1 Analysis of Piles under Vertical Vibrations, 495 12.1.1 End-Bearing Piles, 495 12.1.2 .Friction Piles, 497

'~~ 12.2 Analysis of Piles under Translation and Rocking, 517 12.3 Analysis of Piles under Torsion, 521 12.4 Design Procedure for a Pile-Supported Machine

Foundation, 529 12.5 Examples, 532 12.6 Comparison of Measured and Predicted Pile Response,

541

xvi CONTENTS

12.7 Final Comments, 547 Practice Problems, 550 References, 552

CHAPTER 13 CASE HISTORIES 554 13.1 Case History of a Compressor Foundation, 556 13.2 Case History of a Hammer Foundation, 569 13.3 Final Comments, 576

References, 576

CHAPTER 14 CONSTRUCTION OF MACHINE FOUNDATIONS 578 14.1 Construction Aspects of Block Foundations, 579 14.2 Construction Aspects of Frame Foundations, 580 14.3 Erection and Interfacing of a Machine to the

Foundation, 586 14.4 Gap around the Foundation, 589 14.5 Bonding of Fresh to Old Concrete, 589 14.6 Installation of Spring Absorbers, 589

References, 592

APPENDIXES 593 1 Computer Program for the De sign of a Block

Foundation, 595 2 Computer Program for the Design of a Hammer

Foundation, 61 0 3 Brief Description of Some Available Computer

Programs, 620 4 Computation of Moment of Inertia, 624 5 Conversion Factors, 629

NOTATION 631

AUTHOR INDEX 647

SUBJECT INDEX 651

1 Introduction

Machine foundations require the special attention of a foundation engineer. Unbalanced dynamic forces and momel),ts,are occasioned by the operation of a machine. The ;..achine foundation ilius transmits dynamic loads to the soil below in addition to the static loads due to the combined weight of the machine and the foundation. It is the consideration of the dynamic loads that distinguishes a machine foundation from an ordinary foundation and necessitates special design procedures. The foundation for the machine must therefore be designed to ensure stability under the combined effect of static and dynamic loads. In general, a foundation weighs several times as much as a machine, and the dynamic loads prod,ced by the machine's moving parts are relatively small compared to the combined weight of the machine and the foundation (Prakash and Puri, 1969). Even though the magnitude of the dynamic load is small, it is applied repetitively over long periods of time. The behavior of the supporting soil is generally considered elastic. For the range of vibration levels associated with a well-designed machine founda-tion, this assumption seems reasonable. The vibration response of the machine-foundation-soil system defined by its natural frequency and the amplitude of vibration under the normal operating conditions of the mach-ine are the two most important parameters to be determined in designing the foundation for any machine. In addition, the wave energy, which is transmitted through the underJ.:ing soil from the vibrating foundation, must not cause harmful effects on other machines, structures, or people io the immediate vicinity. This consideration and the operational requirements of the machine necessitate that the amplitudes of foundation vibration be limited to small values. Thus the local soil conditions and the foundation-soil interaction are important factors to be considered in the design of foundation for any machine. Satisfactory design of a machine foundation can be accomplished by systematic application of principles of soil mech-anics, soil dynamics, and theory of vibrations.

2 INTRODUCTION

The initial cost of construction of a machine foundation is generally a small fraction of the total cost of the machine, accessories, and the in-stallation, but the failure of the foundation as a result of poor design or construction can interrupt the machine's operation for long periods and cause heavy dollar losses. Great care should therefore be taken at all stages of the soil investigation and in the design and construction of these foundations to ensure their long-term satisfactory performance.

There are many types of machines and each may require a certain type of foundation. The different types of machines, their special features, and the types of foundations commonly used to support them are briefly described now. The criteria used in design of these foundations, the relevant codes of practice, and the data required for their design are also discussed sub-sequently.

1.1 TYPES OF MACHINES AND FOUNDATIONS

There are many types of machines. All generate unbalanced exciting loads. In general, the various machines may be classified into three categories:

1. Reciprocating machines: This category of machines includes internal combustion engines, steam engines, piston-type pumps and compressors, and other similar machines having a crank mechanism. The basic form of a reciprocating machine consists of a piston that moves within a cylinder, a connecting rod, a piston rod and a crank (Fig. 5.1). The crank rotates with a constant angular velocity. The crank mechanism converts the translatory motion into rotary motion and vice versa. The operating speeds of recip-rocating machines are usually smaller than 1200 rpm.

The operation of the reciprocating machine or the crank mechanism results in unbalanced forces both in the direction of piston motion and perpendicular to it (Section 5.1). The magnitude of forces and moments will depend upon the number of cylinders in the machine, their size, piston displacement, and the direction of mounting.

If one considers only the unbalanced force in the direction of piston motion in a machine with only one cylinder that is mounted centrally on a rigid foundation (Fig. l.la), the motion of the foundation will be only up and down. A two-cylinder reciprocating machine under similar conditions mounted centrally on a rigid foundation, will generate an oscillatory motion and no translation (Fig. 1.1b). Similarly, if a pistonis mounted horizontally, it will give rise to an unbalanced force and a moment on the foundation. The foundation will therefore undergo both translation and rotation simul-taneously (Fig. l.lc). In the case of a two-cylinder machine mounted horizontally, the unbalanced forces in a plane parallel to the base of the foundation generate a couple (Fig. l.ld). This results in a motion that is similar to the motion of a torsional pendulum. It therefore becomes clear

TYPES OF MACHINES AND FOUNDATIONS

/

/

------

Foundation

Ia)

--1_" /////// Maximum vertical amplitude

Maximum translation

.::.- ~lf..-

(c)

4 INTRODUCTION

the blows required in forging. The foundation block under the anvil serves as a support for the entire hammer.

The speeds of operation of both these hammers are usually low and range from 60 to 150 blows per minute. Their dynamic loads attain a peak in a very short period of time and then practically die out. The unbalanced force occasioned by the impact lasts only a fraction of a second. In between two successive blows, the foundation and anvil vibrate freely. The analysis of the hammer foundation, therefore, proceeds along lines that are different from those for the analysis of a reciprocating machine foundation. A massive block foundation is usually provided for impact machines. Vibration absor-ber pads are placed between the anvil and the foundation to absorb some of the vibrations.

3. Rotary machines: High-speed machines, such as turbogenerators, tur-bines, and rotary compressors, may have speeds that exceed 3000 rpm and may even reach 10,000 rpm.

Steam turbines have elevated pedestal foundations that may consist of an arrangement of a base-slab and vertical columns, which support at their tops a grid of beams on which skid-mounted machinery rests. Each element of such a foundation is relatively flexible (Fig. 8.1) as opposed to the rigid block-type foundation.

1.2 DESIGN CRITERIA TO BE SATISFIED

A machine foundation should meet the following requirements in order to be satisfactory (Prakash, 1981).

For static loads

1. The foundation should be safe against shear failure. 2. The foundation should not settle excessively.

These are standard requirements that are the same for all footings.

For dynamic loads

1. There should be no resonance. That is, the natural frequency of the machine-foundation-soil system should not coincide with the operating frequency of the machine. In fact, a zone of resonance is generally defined, and the natural frequency of the soil foundation system must lie outside this zone (Fig. 1.2). The foundation may thus be designated as "high tuned" when its fundamental frequency is greater than the operating speed or as "low tuned" when its fundamental frequency is lower than the operating speed. This concept of a high or low tuned foundation is illustrated in Fig. 1.2.

DESIGN CRITERIA TO BE SATISFIED

~ c 0 c ~0 Q

u. ~ i5 cS

~ ~ c 0

" u ~ " E "'

2.5

2.0 ~ tuned

1.5

1.0 )

0.5 -

0.0 0.0

I

I 1.0

I I I I

Low tuned

I 2.0 3.0 4.0 5.0 :~~~r~quency ratio ~ oper"~trig speed of machine

fundamentalfrequency of foundation

Figure 1.2. Tuning of a foundation.

5

-

-

-

-

6.0

2. The amplitudes of motion at the operating frequencies should not exceed the permissible values. These limiting amplitudes are generally specified by the machine's manufacturer.

3. The design should be such that the natural frequency of the foun-dation-soil system will not be a whole number multiple of the operating frequency of the machine to avoid resonance with higher harmonics (Section 5.1).

4. Vibrations occasioned by the machine operation should not be an-noying to persons or harmful to other precision equipment or machines in the vicinity or to adjoining structures.

In addition to the preceding criteria, geometrical layout of the foundation may be influenced by operational requirements of the machine.

The failure condition of vibril;(ing foundations is reached when the motion exceeds a limiting value, which thay be expressed in terms of the velocity or acceleration of the movement of the foundation. For steady-state vibrations, these may be expressed in terms of allowable displacements at specified frequencies (Richart, 1962). Figure 1.3 illustrates the order of magnitudes that are involved in the criteria for determining the dynamic response. Five curves delimit the zones of vibrations to which persons are sensitive when standing close to the vibrating machinery. These zones range from "not noticeable" to "severe." The boundary between "not noticeable" and

6 INTRODUCTION

+ From Reiher & Meister (1931)-(steady state vibrations) From Rausch (1943Hsteady state vibrations) 6 From Crandell (1949)-(due to blasting)

.OJ

1 r005 ~ 0.02

Frequency, cpm

Figure 1.3. Limiting amplitudes of vibrations for a particular frequency. (After Richart, 1962.)

"barely noticeable" in Fig. 1.3 is defined by a line that represents a peak velocity of about 0.01 in/sec (0.25 mm/sec), and the line separating the zones of "easily noticeable" and "troublesome" represents a peak velocity of 0.10 in/sec (2.5 mm/sec). The shaded area in Fig. 1.3 indicates the "limits for machines and machine foundations." This represents a peak velocity of 1.0 in/sec (25.5 mm/sec) below about 2000 cpm and corresponds to a peak acceleration of 0.5 g above about 2000 cpm. It should be noted that this shaded area indicates a limit for safety and is not a limit for the satisfactory operation of a machine.

The importance of a machine and its sensitivity to operational conditions along with the cost of installation and losses due to interruption (down time) determine the limit of the motion amplitudes for which the foundation must be designed (Richart, 1976). Permissible amplitudes at operating speed can be established from Fig. 1.4 (Blake, 1964). The vibration amplitudes are generally specified at bearing level of the machine.

The concept of "service factor" was introduced by Blake (1964). The

100 rpm

Figure 1.4. Criteria for vibrations of rotating machinery. Explanation of classes:

AA Dangerous. Shut it down now to avoid danger. A Failure is near. Correct within two days to avoid breakdown. B Faulty. Correct it within 1 0 days to save maintenence dollars. C Minor faults. Correction wastes dollars. D No faults. Typical new equipment.

"li:. This is a guide to aid judgment, not to replace it. Use common sense. Use with care. Take

account of all local circumstances. Consider: safety, labor costs, downtime costs. (After Blake, 1964.) Reproduced with permission from Hydrocarbon Processing, January 1964.

7

8 INTRODUCTION

service factor is an indication of the importance of a machine in an installation. Typical values of service factors are listed in Table 1.1. Using the concept of service factor, the criteria given in Fig. 1.4 can be used to define vibration limits for different classes of machines. Also with the introduction of the service factor, Fig. 1.4 can be used to evaluate the performance of a wide variety of machines. The concept of service factor is explained by the following examples.

A centrifuge has a 0.01 in (0.250 mm) double amplitude at 750 rpm. The value of the service factor from Table 1.1 is 2, and the effective vibration therefore is 2 x 0.01 = 0.02 in (0.50 mm). This point falls in Class A in Fig. 1.4. The vibrations, therefore, are excessive, and failure is imminent unless the corrective steps are taken immediately.

Another example is that of a link-suspended centrifuge operating at 1250 rpm that has 0.0030 in (0.075 mm) amplitude with the basket empty. The service factor is 0.3, and the effective vibration is 0.00090 in (0.0225 mm). This point falls in class C (Fig. 1.4) and indicates only minor fault.

General information for the operation of rotary machines is given in Table 1.2 (Baxter and Bernhard, 1967). These limits are based on peak-velocity criteria alone and are represented by straight lines on Fig. 1.4.

The maximum velocity for the lower limit of the "smooth" range is 0.01 in/sec (0.25 mm/sec) in Table 1.2 and the lower limit of the range "barely noticeable to persons" is also 0.01 in/sec (0.25 mm/sec) in Fig. 1.3. The lower limits of "slightly rough" for machines is 0.16 in/sec ( 4.0 mm/sec) in Table 1.2 whereas the value for "troublesome to persons" is 0.10 in/sec (2.5 mm/sec) in Fig. 1.3. Also the danger limit of "very rough" is 0.63 in/ sec (15.75 mm/sec) in Table 1.2 whereas Rausch's limit for machines is 1.0 in/sec (25.0mm/sec) in Fig. 1.3 (Rausch, 1973). Tbus Table 1.2 and Fig. 1.3 are similar (Richart, 1976).

Table 1.1. Service Factorsa

Single-stage centrifugal pump, electric motor, fan Typical chemical processing equipment, noncritical Turbine, turbogenerator, centrifugal compressor Centrifuge, stiff-shaftb; multistage centrifugal pump Miscellaneous equipment, characteristics unknown Centrifuge, shaft-suspended, on shaft near basket Centrifuge, link-suspended, slung

1 1 1.6 2 2 0.5 0.3

a Effective vibration= measured single amplitude vibration, in inches multiplied by the service factor. Machine tools are excluded. Values are for bolted-down equipment; when not bolted, multiply the service factor by 0.4 and use the product as the service factor. Caution: Vibration is measured on the bearing housing, except as stated. " Horizontal displacement on basket housing. Source: After Blake (1964). Reproduced with permission from Hydrocarbon Processing, January 1984.

RELEVANT CODES

Table 1.2. General Machinery-Vibration-Severity Data Horizontal Peak Velocity

(in/sec) 0.630

Machine Operation

Extremely smooth Very smooth Smooth Very good Good Fair Slightly rough Rough Very rough

9

Source: After Baxter and Bernhard (1967). Reproduced by permission of American Society of Mechanical Engineers, New York, NY.

1.3 RELEVANT CODES .. '""

The criteria for satisfactory design of a machine foundation are described in Section 1.2. Methods of analysis of foundations for different machines are described in Chapters 6 through 12. These enable the engineers to design safe and economical foundations.

Because installation of heavy machinery has assumed increased import-ance throughout the world, their foundations have to be specially designed to take into consideration both the vibrational characteristics of the load and the properties of the supporting soil, which is subject to dynamic conditions. Although many considerations relating to the design and construction of such machine foundations are specified by the machines' manufacturers, other details must comply with the general design principles that govern machine foundations. With this objective in view, codes for the design and construction of machine foundations have been written in West Germany (DIN 4024, 4025), Russia (CH-18-58), Hungary (MSZ 15009-64), and India (Indian Stardards Institution, 1966, 1967, 1968, 1969, 1970). Unfortunately, no such codes have been written in the United States (1987). For design of turbogenerator foundations, leading manufacturers such as Westinghouse, General Electric, and Honeywjfii have their own design criteria. The designer must familiarize himselr with the relevant standards (code of practice) prevalent in the country in which he works. t

t American Concrete Institute is working on the codes for design of foundations subjected to dynamic machinery. But no codes have been finalized so far (1987). Naval Facilities En-gineering Command (1983) describes only elementary criteria for design of machine foundations.

10 INTRODUCTION

1.4 DATA REQUIRED FOR DESIGN

To arrive at a satisfactory design for a machine foundation, all pertinent data must be procured. This data must include information on layout of the machine, operating speeds, unbalanced loads generated by the machine operation, point of application of the unbalanced loads, and permissible amplitudes of vibration. Details of the data required are discussed sepa-rately for each type of machine in Chapters 6 through 12.

Besides the preceding information about the machine, detailed infor-mation on the static and dynamic properties of the supporting soil should form an essential part of the data that must be procured.

1.5 SIGNIFICANCE OF SOIL PARAMETERS

The reader must have realized by now that the design of a machine foundation essentially involves determination of the vibration characteristics (natural frequencies and vibration amplitudes) of the machine-foundation-soil system. Besides the machine and the foundation data, the soil properties are a rather significant input parameter governing the computed. response, i.e., the predicted behavior of this system. Depending upon the method of, analysis (the elastic half space or the linear spring theory, Chapter 6), the mode of inputting the soil parameters may vary. It will be shown in Chapter 4 that the number of parameters affecting the relevant soil properties are large and sometimes quite complex.

Fortunately, the determination of soil properties for the design of ma-chine foundations has reached a stage where fairly precise evaluations can be made for given loading conditions. The soil parameters can be deter-mined in a realistic manner after a careful evaluation of the field or laboratory test data by following the procedure suggested in Chapter 4 (Section 4.7). The importance of soil parameters must always be kept in mind by an intelligent designer.

REFERENCES

Barkan, D. D. (1962). "Dynamics of Bases and Foundations." McGraw~Hill, New York. Baxter, R. L., and Bernhard, D. L. (1967). Vibration tolerances--for industry. Am. Soc. Mech.

Eng. [Pap.] 67-PME-14. Blake, M. P. (1964). New vibration standards for maintenance. Hydrocarbon Process. Pet.

Refiner 43 (!), 111-114. Crandell, F. J. (1949). Ground vibrations due to blasting and its effects on structures. J. Boston

Soc. Civ. Eng. 36 (2). Also reprinted in Contributions to Soil Mech. BSCE 1941-1953, pp. 206-229.

CH-18-58 Soviet Code of Practice for Foundations Subjected to Dynamic Effects.

REFERENCES 11

DIN 4024 Stutzkonstruktionen fiir rotierende Machinen (Supporting structures for rotary machines).

DIN 4025-1958 Fundamente fiir Ambo-Hiimmer (Schabotte-Hammer) Richtilinten fur die Konstruktionen-Bemessung und ausfuhrung (Criteria for the design and construction of foundations for anvil-hammer construction).

Indian Standards Institution (1966). "Indian Standard Code of Practice for Design and Construction of Machine Foundations," Part II, IS: 2974. lSI, New Delhi, India.

Indian Standards Institution (1967). "Indian Standard Code of Practice for Design and Construction of Machine Foundations," Part III, IS: 2974. lSI, New Delhi, India.

Indian Standards Institution (1968). "Indian Standard Code of Practice for. Design and Construction of Machine Foundations," Part IV, IS: 2974. lSI, New Delhi, India.

Indian Standards Institution (1969). "Indian Standard Code of Practice for Design and Constructi?n of Machine Foundations," Part I, IS: 2974 (rev.). lSI, New Delhi, India.

Indian Standards Institution (1970). "Code of Practice for Design and Construction of Machine Foundations," Part V, IS: 2974. lSI, New Delhi, India.

MSZ 15009-64 Hungarian Code for Design of Machine Foundations. Naval Facilities Engineering Command (1983). "Soil Dynamics, Deep Stabilization, and

Special Geotechnical Construction," Design Manual 7.3, NAVFAC DM-7.3. Dept. of the Navy, Naval Facilities Engineering Command, Alexandria, Virginia.

Prakash, S. (1981). "SoiliJ?ynamics." McGraw-Hi!l;;;)'~:cw York. Prakash, S., and Puri, V.K. (1969). Design of a "tYPical machine foundation by different

methods. Bull.-Indian Soc. Earthquake Techno/._6, 109-136. Rausch, E. (1943). ''Maschinenfundamente und andere dynamische Bauaufgaben." VDI

Verlag, Berlin. Reiher, H., and Meister, F. J. (1931). Die Empfindlinchkeit der Menschen gegen Erschiit-

terungen. Forsch. Geb. lngenieurwes. 2 (11), 381-386. Richart, F. E., Jr. (1962). Foundation vibrations. Trans. Am. Soc. Civ. Eng. 127, Part I,

863-898. Richart, F. E., Jr. (1976). Foundation vibrations. "Foundation Engineering Hand Book,"

Chapter 4. Van Nostrand-Reinhold, New York.

2 I Theory of Vibrations

It was mentioned in Chapter 1 that machine foundations may be subjected to either periodic loads or impact loads. A periodic load may be represented by a harmonic function, i.e., a sine or a cosine function. The problem of impact loads can be easily solved with an initial boundary value approach. For machine-foundation analysis it is only necessary to be familiar with the simple theoretical concepts of harmonic vibrations and with the methods needed to solve such problems. Although a block foundation may have six degrees of freedom, it is seldom necessary to solve for a system with more than two degrees of freedom. This simplifies the study of theory of vibra-tions. In analysis of flexible foundations, we have to use other solution techniques.

This chapter is tailored to provide basic concepts on vibration problems of simple systems such as spring-mass-dashpot systems. These concepts pro-vide the basis for attempting solutions to the machine-foundation problem.

2.1 DEFINITIONS

Period of Motion: If motion repeats itself in equal intervals of time, it is called periodic motion. The time that elapses when the motion is repeated once is called its period.

Aperiodic Motion: Motion that does not repeat itself at regular intervals of time is called aperiodic motion.

Cycle: Motion completed during a period is referred to as a cycle. Frequency: The number of cycles of motion in a unit of time is c.alled the

frequency of vibrations. Natural Frequency: If an oscillatory system vibrates under the action of

forces inherent in the system and no externally applied force acts, the frequency with which it vibrates is known as its natural frequency.

12

DEFINITIONS 13

Forced Vibrations: Vibrations that are developed by externally applied exciting forces are called forced vibrations. Forced vibrations occur at the frequency of the externally applied el

14 THEORY OF VIBRATIONS

position of this system is completely defined by the angle e only. Hence it is a system with one degree of freedom, that is, n is equal to 1. In Figs. 2.1 b and c, two and three independent coordinates are needed to fully describe the motion of the two systems respectively. Hence they consti-tute systems with two and three degrees of freedom.

The number of coordinates necessary to completely describe the motion of an elastic simply supported beam is infinite. Hence the beam in Fig. 2.1d constitutes an infinite degree of freedom system.

Resonance: If the frequency of excitation coincides with any one of the natural frequencies of the system, the condition of resonance is reached. The amplitudes of motion may be excessive at resonance. Hence, in the design of machine foundations, the determination of the natural fre-quencies of a system is important.

Frequency Ratio: The ratio of the forcing or operating frequency to the natural frequency of the system is referred to as the frequency ratio.

Principal Modes of Vibration: A system with n degrees of freedom vibrates in such a complex manner that the amplitude and frequencies do not appear to follow any definite pattern. Still, among such a disorderly array of motions, there is a special type of simple and orderly motion that has been termed the principal mode of vibration. In a principal mode, each point in the system vibrates with the same frequency, which is one of the system's natural frequencies.

Thus, a system with n degrees of freedom possesses n principal modes with n natural frequencies. More general types of motion can always be represented by the superposition of principal modes.

Normal Mode of Vibration: When the amplitude of motion of a point of the system vibrating in one of the principal modes is made equal to unity, the motion is called the normal mode of vibration.

Damping: Damping is associated with energy dissipation and opposes the free vibrations of a system. If the force of damping is constant, it is termed Coulomb damping. If the force of damping is proportional to its velocity, it is termed viscous damping. If the damping in a system is free from its material property and is contributed to by the geometry of the system, it is called geometricalt or radiation damping.

2.2 SIMPLE HARMONIC MOTION

The simplest form of periodic motion is harmonic motion, which is represen ted by sine or cosine functions. Let us consider the harmonic motion represented by the following equation:

z = Z sin wt (2.1)

I" For an explanation, see Section 3.4.

SIMPLE HARMONIC MOTION 15

in which w is the circular frequency in radians per unit time. We can represent z by the vertical projection of a rotating vector of length Z that rotates with a constant angular speed of w, onto a vertical diameter (Fig. 2.2). Because the motion repeats itself after 2Tr radians, a cycle of motion is completed when

or

wT=2Tr

T= 2Tr w

(2.2a)

(2.2b)

in which Tis the time period of motion. The frequency fis the inverse of the time period; hence

1 w f= I' =z'lr (2.3)

In order to determine the velocity and acceleration of motion, we differentiate Eq. (2.1) with respect. to time, t:

Velocity= i = wZ cos wt = wZ sin( wt + ~) (2.4) and

Acceleration= i = -w 2Z sin wt = w2Z sin(wt + Tr) (2.5) or

Acceleration= -w 2z (2.6)

Equations (2.4) and (2.5) show that both velocity and acceleration are also harmonic and can be represented by the vectors wZ and w2Z, which rotate at the same speed as Z, i.e., w rad/unit time. These, however, lead the displacement vector by Trl2 and Tr respectively.

r-1 cycle---1

wt

Figure 2.2. Vectorial representation of harmonic motion.

THEORY OF VIBRATIONS

wZ

I ' I \ I \ I / \ ,'A....

, ....

Figure 2.3. Vectorial representation of displacement (z), velocity (Z), and acceleration (i).

In Fig. 2.3, vertical projections of these vectors are plotted against the time axis t. The angles between the vectors are known as phase angles. Thus the velocity vector leads the displacement vector by 90; the acceleration vector leads the displacement vector by 180 and the velocity vector by 90.

2.3 FREE VIBRATIONS OF A SPRING-MASS SYSTEM

Figure 2.4a shows a spring of stiffness kin an unstretched position. If a mass m of weight W is attached at its lower end, the mass-spring system occupies the position shown in Fig. 2.4b. The deflection li""' of the spring from the undeflected position is

(2.7)

in which k is the spring constant, defined as force per unit deflection. The

lal (b) (c) (d) (e)

Sign convention z, z, z

t+

If)

Figure 2.4. Spring-mass system. (a) Unstretched spring; (b) equilibrium position; (c) mass in oscillating position; (d) mass in maximum downward position; (e) mass in maximum upward position; and (f) free-body diagram of mass corresponding to (c).

FREE VIBRATIONS OF A SPRING-MASS SYSTEM 17

position of the system corresponding to this state is referred to as the equilibrium position. In Fig. 2.4c, the m~ss IS shown displaced by a d~stance z in the downward direction; the maximum downward deflection IS. Zmax (Fig. 2.4d). The double amplitude at any time is shown in Fig. 2.4e. Figure 2.4f shows the free body diagram. .

If the mass is released from the extreme lower position (Fig. 2.4d), It starts to oscillate between the two extreme positions (Fig. 2.4e). If there IS no resistance to these oscillations, the mass will contmue to vibrate (theoretically) indefinitely.

If we neglect the mass of the spring, the equation of motion can be written as

2, F= mi (2.8a)

in which E F is the sum of all forces in the vertical direction. If th~ sign convention shown in Fig. 2.4 is used and the inertial force acts opposite to acceleration, the equation of motion becomes

-(kli"" + kz) +wc=mg) = mi (2.8b)

Because kli,"' is equal to W, we get

mi+kz=O (2.8c)

Equation (2.8c) is a second-order differential equation, and its general solution must contain two arbitrary constants, whtch can be evaluated from initial conditions.

The solution of this equation can be obtained by substituting

(2.9)

in which A and B are arbitrary constants, and wn is the natural circular frequency of the system. . .

If we substitute the preceding solutiOn mto Eq. (2.8c), we get

which gives

2- k w~~- m or w = [k n \J;

(2.10)

(2.11)

When w, Tn is equal to 2"1T, one cycle of motion is completed. This yields the following expression for natural period:


(2.12)

The natural frequency of vibration is the number of cycles completed in unit time and is the reciprocal of the time period T,. Therefore,

(2.13)

Equation (2.13) can also be written in the following form:

(2.14a)

Now

mg W k = k =Ostat (2.14b)

Therefore,

(2.15)

Equation (2.15) shows that natural frequency is a function of static deflection. When g is equal to 9810 mm/sec2 and i5 is expressed in

stat millimeters, the frequency in hertz can be shown in graphic form as in Fig. 2.5.

40

30

"N ~ 20 ...

10

0 0

~ 2

......_

r--

4 6 8 10 Ostat, (mm)

Figure 2.5. Relationship between natural frequency and static deflection.

FREE VIBRATIONS OF A SPRING-MASS SYSTEM 19

Arbitrary constants A and B in Eq. (2.9) can be determined from the initial conditions. Let the initial conditions be defined by the following values:

When t is equal to zero,

z = Z0 and i = V0 (2.16)

By substituting these values into Eq. (2.9), the solution can be obtained in the form

(2.17)

Other types of solutions of Eq. (2.8c) can be written in the following forms:

and

EXAMPLE 2.3.1

z = A exp(iw,t) + B exp(- iw,t) '';;:;:''~,-

(2.18)

(2.19)

A mass supported by a spring has a static deflection of 0.25 mm. Determine its natural frequency of oscillation.

Solution 1 rg- 1 ~9810 f,=-2 y-;:---=2 025 =31.541Hz 7r 0 stat 7T '

EXAMPLE 2.3.2 Determine the spring constant for the system of springs shown in Fig. 2.6.

(a) (b)

Figure 2.6. Equivalent spring constants: (a) springs in series; (b) springs in parallel.


Solution (a) On application of a unit load to the system of springs in Fig. 2.6a, the

total deflection is 1 1 kl + k, - + - = -7-c--'-k, kl klk2

Hence, the equivalent spring constant is given by klk2

k,,, = k + k I 2

If k 1 = k2 = k, the k,q, = k/2. (b) Let us consider that a unit load is applied at c to the system of springs

shown in Fig. 2.6b. It is shared at a and bin the ratios of x 2/(x 1 + x 2 ) and x 1 /(x 1 + x2 ). The deflection of points a and b are x2(x 1 + x2 ) x 1/k1 and x 11(x 1 + x2 ) X k 2 , respectively.

Therefore, the deflection of point c is

Hence, the resulting equivalent spring constant at c is

(x 1 + x 2 ) 2 k,,, = (x71k2 + x;lk1 )

If x 1 = x, = x and k 1 = k, = k, then k,,, = 2k.

2.4 FREE VIBRATIONS WITH VISCOUS DAMPING

All real systems exhibit damping. When the force of damping Fd is propor-tional to velocity, it is termed viscous damping. Thus

(2.20)

in which c is the damping constant or force per unit velocity, FL-IT. Figure 2.7a shows a spring-mass-dashpot system. If the mass is displaced

by a distance z below the position of static equilibrium, then the free-body diagram can be represented by Fig. 2.7b. By using the sign convention shown in this figure, the equation of motion can b_e written as

mi+ci+kz=O (2.21)

The solution to this equation may be written in the form

(2.22)

FREE VIBRATIONS WITH VISCOUS DAMPING 21

_l_ __ ' T

m m

(e) (b)

Figure 2.7. (a) Spring-mass-dashpot system; (b) free-body diagram.

in which s is !" ~onstant that will~;;;::determined later. By substituting this solution into Eq. (2.21), we obtain

which gives us

Therefore,

"(' c k)" 0 s+ms+me=

2 c k s +-s+-=0

m m

k m

"' and the general solution can be written as follows:

(2.23)

(2.24)

(2.25)

in which A and B are arbitrary constants depending upon the initial conditions of motion.

If the radical in Eq. (2.24) is zero, the damping is said to be critical damping c,, and we obtain

(!..s.)' = '5._ = w' 2m m n Le., (2.26)


The ratio of actual damping c to critical damping c, is defined as the damping factor g:

(2.27)

Now,

(2.28)

By substituting the preceding relationships into Eq. (2.24), we get

(2.29)

The nature of the ensuing motion depends upon the values of roots s 1 and s2 , and hence on the magnitude of damping (in terms of critical damping) present in the system. Three different cases of interest are considered here.

CASE I. /; > 1 When I;> 1, both s1 and s2 are real and negative and z (Eq. 2.25) decreases as t increases but it never changes sign. Such a system is called overdamped or nonoscillatory. A typical solution for g = 2 is shown in Fig. 2.8a. If an initial displacement is given to such a system, the mass is pulled back by the springs and dampers absorb all the energy by the time the mass returns to the initial position.

CASE II. /; = 1 When I;= 1, Eq. (2.29) gives s1 = s2 = -w,. The solution becomes

(2.30)

The values of z for g = 1 are shown in Fig: 2.8a, from which it is seen that z decreases as t increases but never changes sign. Hence such a system does not oscillate. This system is known as "critically" damped and g = 1 is the minimum value of damping for no oscillations in the system.

CASE III. g < 1 When damping is less than the critical damping (I; < 1), the values of s 1 and s, (Eq. 2.29) are obtained as

s12 = (-1; i~)wn (2.31) The general solution then becomes

z =A exp[(-g + i~)w.t] + B exp[(-g- i~)w,t] (2.32)

FREE VI ORATIONS WITH VISCOUS DAMPING 23

or

z =A exp(-gw,t) exp( +i~w,t) + B exp(-l;w,t) exp(-i~w,t) = exp- (gwnt)[A cos V(l- g')wnt + iA sin V(l- O'wnt)

+ B cos V (1- g')w,t- iB sin V (1- /; 2 )w,t] or

z = exp( -l;wnt)( C cos V (1 - 1;)2w,t + D sin V (1 - /;')w,t) (2,33) in which Cis equal to A+ B, and D is equal to i(A- B).

3

2

I

0

I

2

~ ' ~ 1---l:r --

I I

I

/ /

/

"'--.

2

'~ 2 N>" .. ,.,,_ r--

't ~ I r-:-. 3 4 5 6

w,t

(a)

(b)

7 8

Figure 2.8. (a) Free vibrations with g = 2, and g = 1.0. (b) Free damped oscillations for g


Thus, the natural circular frequency wnd in viscously damped vibrations equals

(2.34)

Equation (2.33) can then be written as

(2.35)

in which Z0 and are arbitrary constants depending upon the initial conditions.

Figure 2. Sb shows typical damped oscillations when g is less than 1. 0.

2.5 FORCED VIBRATIONS WITH VISCOUS DAMPING

Figure 2.9a shows a spring-mass-dashpot system under the action of a force of excitation, F, such that

F= F0 sin wt (2.36)

in which w is the frequency of excitation. The free-body diagram is shown in Fig. 2.9b and the equation of motion

is

mi + ci + kz = F0 sin w t (2.37)

The solution to this equation is

z = Z0 sin(wt- ) (2.38)

Then

i= wZ0 cos(wt- ) (2.39)

or

i = wZ0 sin(wt- + 7T/2) (2.40)

and

i = w2Z 0 sin(wt- + 7) (2.41)

Figure 2.9c shows z, i, and z vectors at any particular instant. The force

Equilibrium position

a

m

F0 sin wt

Ia I

z =

26 THEORY Of VIBRATIONS

in the spring is opposite to z, hence it can be represented by Oa in Fig. 2.9d. Similarly, the damping force, cwZ0 , acts in the opposite direction to that of the velocity and hence is represented by Ob. Similarly, Oc represents the inertial force, mw 2Z0 , which acts opposite to acceleration. The resultant of these forces is Fb, which is represented by Od, and must be equal in magnitude and opposite in sign to F0 Thus, the displacement vector lags behind the force vector by . From Fig. 2.9d, we get

or

zo=y 22 2 (k-mw) +(cw) (2.42)

and

-1 cw =tan 2 k-mw (2.43)

Equation (2.42) can be expressed in non-dimensional terms as follows: F0 /k Zo=-Y~(=1=-=m=w~~~k)~'=+=(~c=w=lk~)2 (2.44a)

Now, Fufk is equal to the static deflection ll,. of the system under the action of F0 . Also,

mw2

( w )' 2 --= - =r

k w"

in which r is the frequency ratio, and

Therefore,

(2.44b)

If there is no damping present, i.e., g = 0, undamped amplitude Z0

is given by

(2.44c)

FORCED VIBRATIONS WITH VISCOUS DAMPING

From Eq. (2.44b) we get

Hence,

Similarly,

Z 0 1 ll,. = [(1- r 2 ) 2 + (2tr) 2f 12

20 = N = magnification factor

ast 1

N = -Vr=(=1 -=r'""l""' =+""( 2=g=r l""'

_ 1 2gr =tan --2 1-r

27

(2.44d)

(2.45a)

(2.45b)

(2.46)

Figure 2. t~ is a plot of N anct""versus frequency ratio r, for r varying from 0 to 5.

3.0

2. 0

N

L 0

I

0

180" 0.05 I 0.15-fi/ f.-~0 0.375-----

w k', ~ LO 0.05 "" I ~ goo r----0.10 ~ / 0.15t_..'&. ~ (b) 1 -0.25 cc

'7 ~ 03:75 0 1 2 3 ~~ Frequency ratio~ 0.50 W 0 / (a) 1\ .............. I~ ~ ~ -.........__ ~

LO 2.0 3.0 4.0 Frequency ratio .:::_

W0

4 5

50

Figure 2.10. (a) Magnification factor and (b) phase angle, versus frequency ratio in forced vibrations. (After Thomson, 1972, p. 48. Reprinted by permission of Prentice-Hall, Englewood Cliffs, New Jersey.)


Effect of Frequency Ratio r for a Particular Case ( t = O) From Eq. (2.45b),

N=-1-1- r 2

When r=O N=1 When r = 1

When r->en N=O

. When r is equal to 1, resonance occurs, and the amplitude tends to be mfimte. Introduction of damping reduces the resonant amplitudes to finite values.

. The phase angle is zero if r is less than 1; the displacement z is in phase With the exc1tmg force, F0 , and is equal to 1800 if r is greater than 1.

Effect of Damping t

As the dampi~g increases, the peak of the magnification factor shifts slightly to the left. This IS due to the fact that maximum amplitudes occur in damped VIbratiOns when the forcing frequency w equals the system's damped natural frequency, wnd [Eq. (2.34)], which is slightly smaller than the undamped natural frequency, wn.

Figure 2.11. Vector diagram at resonance in a damped system under forced vibrations. (a) Dis-placement, velocity, and acceleration. (b) Forces.

0 Ia I

Fo

9 "" goo

cwZo = Fo

(b)

FREQUENCY-DEPENDENT EXCITATIONS 29

When r is equal to 1, the phase angle is 90' for all values of damping, except when I; is equal to 0. When r is less than 1, the phase angle is less than 90', and when r is greater than 1, the phase angle is greater than 90'.

The maximum amplitude of motion when r is equal to 1 and I; is greater than 0 is expressed by Eq. (2.47):

F, cw

The corresponding vector diagram is shown in Fig. 2.11.

(2.47)

The solution given by Eq. (2.44a) is a steady-state solution, which is important in most practical problems. However, there are transient vibra-tions initially that correspond to the solution given by Eq. (2.35). These vibrations, of course, die out in the first few cycles.

2.6 FREQUENCY-DEPENDENT EXCITATION

In many probl~Iils of machine fouii'iflitions, the exciting force depends upon the machine's operating frequency. Figure 2.12 shows such a system moun-ted on elastic supports with m 0 representing the unbalanced mass placed at eccentricity e from the center of the rotating shaft. The unbalanced force is F = (m 0ew 2 ) sin wt. Therefore, the equation of motion may be written as follows:

d 2z d 2 . dz (M- m ) - + m - (z + e sm wt) =-kz- c -0 dt2 0 dt2 dt

'

wt

z

M

0! 0! ~k/2 !' ~k/2 //,/#lW##/mrw####J#m/,7/

Figure 2.12. Force of excitation due to rotating unbalance.

(2.48a)


By rearranging the terms in the preceding equation, we get

Mi + ci + kz = m 0ew 2 sin wt (2.48b) In this system, M also includes m0 . Equations (2.48b) and (2.37) are

similar, except that m 0ew 2 appears in Eq. (2.48b) in place of F0

in Eq. (2.37). The solution of this equation may therefore be witten as,

2 m 0ew . z = sm wt Y (k- Mw 2 ) 2 + (cw )2 (2.49a)

Therefore, the maximum amplitude Z0

is given by

2 z = m 0ew 0 v 2 2 2 (k-Mw) +(cw) (2.49b)

and

cw tan= 2 k-Mw (2.50)

In nondimensional form, these equations can be arranged as follows:

or

and

Zo ,z

moe! M Y (1- r')' +Ag'rz

2gr tan 1> = --2 1-r

(2.51)

(2.52)

The value of MZ01m0e and 1> are plotted in Figs. 2.13a and 2.13b, respectively. These curves are similar in shape to those in Fig. 2.10 except that the peak amplitudes occur at

(2.53)

and the value of the ordinate when r is equal to 1 in Eq. (2.51) is Z0 1

m 0el M 21; (2.54)

SYSTEMS UNDER TRANSIENT lOADS 31

180" h 1 o.~5 0.05 tv 0.25 ~ ~ 0.50

K, ~ 1.0 t-0.10 M 0 90" ro t---t- 0.15 h ~ ro ~

3.0

2. 01 .

'"' 0 :;: "

~ 0. 0.25 ~ 0 ~ 0 1.0 2.0 3.0 4.0 5.0 ~ (b) Frequency ratio~ "" 0.375 ~ 1L k-:::0.50

.o

/Y ~ 1--\,1 ~ v I ~~~1.0

0 1.0 2.0 3.0 4.0 5. 0 -~}:ic}~quency ratio~

-t,. Wn

(a)

mZ, 1 . Figure 2.13. Response of a system with rotating unbalance. (a) me versus requency ratiO wlwn. (b) Phase angle tfJ versus frequency ratio wlwn. (After Thoms~n, 1972, p. 50. Reprinted with permission of PrenticeRHall, Englewood Cliffs, New Jersey.)

2.7 SYSTEMS UNDER TRANSIENT LOADS

Transient loads may be caused by hammers, earthquakes, blasts, and the sudden dropping of weights. In several such cases, the maximum motion may occur within a relatively short time after the application of the force. For this reason, damping may be of secondary importance m transient loads.

CASE I. SUDDENLY APPLIED LOAD Consider a spring-mass system (Fig. 2.14a) that is subjected to suddenly applied force represented by the forcing function F = F0 (Fig. 2.14b). The equation of motion of mass, m, is given by

.., . ..

mi + kz = F0 (2.55) The solution for displacement, z, is

(2.56)

Initially, if at t = 0, z and i are equal to zero, then A is equal to- Fofk,


F(t)

Fof-------

0

Ia I (b)

lcl

Figure 2.14. Dynamic amplification due to suddenly applied load. (a) Single degree of freedom system. (b) Suddenly applied load. (c) Magnification factor versus time.

and B is equal to 0. Thus, Eq. (2.56) becomes Fa

z = k (1- cos w"t) (2.57)

If the force F0 is applied gradually, the static deflection is F I k. Thus the magnification N of the deflection z is

0 '

z N = F lk = 1- COS wnt

0 (2.58)

Magnification N versus time is plotted in Fig. 2.14c. The magnification has a maxtmum value of 2, wh1ch occurs when cos w"t is equal to minus 1. The first peak ts reached when wnt is equal to 7T or tis equal to Tl2, in which Tis the natural period of vibration of the system.

CASE II. SQUARE PULSE OF FINITE DURATION Co.nsider. the system, shown in Fig. 2.15a, that is s~bjected to a pulse of umform mtenstty F(t) for a given duration r (Fig. 2.15b). . When tis less than r, the motion is governed by Eq. (2.55). The solution IS given by Eq. (2.57).

SYSTEMS UNDER TRANSIENT LOADS

Lk ?f Sboo (a)

(c)

33

F(t)

Fof---,

0 '

(b)

dT

Figure 2.15. Dynamic amplification due to a square pulse. (a) One degree of freedom system. (b) Square pulse,~.Jc) Magnification.

f," .

Wben t is equal to r, Fo ) Z = - (1 -COS W" T

' k

and F0 w" ( . ) i

7 = -k- Slll (t)n T

When t is greater than T, the equation of motion is

mi+kz=O

The solution for displacement z is

(2.59)

(2.60)

(2.61)

(2.62)

in which t' = t - r. The values of A and B in Eq. (2.62) are determined from the initial

conditions when tis eqOO.l to r. By equating z, and i, from Eq. (2.62), with those in Eqs. (2.59) and (2.60), respectively, we get A= (F,/k)(l- cos w"r) and B = (F01k) sin w" r. Therefore, Eq. (2.62) becomes

Fo ) ' Fo . . t' Z = k (1 -COS Wn T COS Wnt + k Sill W 11 T Sill Wn

or


in which

Therefore,

or

cjJ =tan -t _ 1- cos w11 r sin w11 'T

F Z = ; Y2(1- cos wn r) sin( wnt' - cf>)

F0 ( w T) z=k 2smz sin(wnt'-q,) The maximum value of z is

(z ) = Fa (z . wn T)- F, . 7TT m" k sm 2 - 2 k sm T

n

Hence, the dynamic magnification N is

_ Zmax _ . Wn 'T . 'TTT N- F/k-2sm-=2sm-o 2 Tn

(2.63a)

(2.63b)

(2.63c)

(2.64a)

(2.64b)

The maximum value of N is 2 when r I Tn is equal to ~. (Fig. 2.15c.) . Constd~r the bmttmg case in Eq. (2.64a), if r!Tn is very small so that

sm 7TT I Tn ts approximately equal to 7TT 1 T then n>

z = 2F0 7TT m" k Tn (2.64c)

Now,_k is equal to mw;, and Tis equal to 27T/w"" By substituting these quanl!hes mto Eq. (2.64c), we obtain

Now, F0 r is equal to the impulse I. Therefore,

I I

f

RAYLEIGH'S METHOD 35

2.8 RAYLEIGH'S METHOD

According to Rayleigh's method, the fundamental natural frequency, i.e., frequency in the first mode of vibrations, of a continuous elastic system with infinite degrees of freedom can be determined accurately by assuming a reasonable deflected curve for the elastic system. If the true deflected shape of the vibrating system is not known, the use of tbe static deflection curve of the elastic system gives a fairly accurate fundamental frequency.

In illustrating the application of this method, the energy principle is used. Expressions are developed for the kinetic energy KE, and potential energy PE. Because the total energy is constant, the sum of KE and PE is constant. Thus

:t (KE + PE) = 0 (2.66a) or

maximum KE = maximum PE . (2.66b) EXAMPLE 2.8.1

In Fig. 2.16, the weight of the spring of length L is w per unit length. Determine the natural frequency of the spring-mass system.

Solution Let the displacement of the mass from the equilibrium position be Yo and y = y0 cos w,t. If the extension of the spring is assumed to be linear, the displacement of the element of the spring at a distance z from the fixed support is y = (z I L )y0 cos wnt, and the velocity of element dz is y = -(zl L )wnYo sin w,t. The maximum KE of the element with mass (wig) dz is then d(KE)m, = (w/2g) dz ((z!L)wnYof

Figure 2.16. System with spring having weight.


By integrating this expression, we obtain the maximum kinetic energy of the spring

W WnYo 2 ( )21L (KE)m, = 2g L 0 Z dz or

The maximum kinetic energy of the rigid mass m is

and the total maximum kinetic energy is

1 (W+~wL) 2 2 2 g WnYo

(2.67)

(2.68)

The maximum potential energy of the spring can be computed as follows:

Maximum potential energy = LYo ky dy 1 2

= 2 kyo (2.69)

In a conservative system, the maximum kinetic energy equals the maximum potential energy. By equating the values of the two energies, we obtain

(2.70a)

Therefore, the natural frequency wn is given by

(2.70b)

The effect of the spring's weight can thus be accounted for by lumping one-third of its mass with the concentrated mass of the system.

EXAMPLE 2.8.2

By using Rayleigh's method, determine the fundamental frequency of the cantilever beam shown in Fig. 2.17.

RAYLEIGH'S METHOD 37

Figure 2.17. Fundamental frequency determination of cantilever beam.

Solution: d The deflected curve of the cantilever beam may be assumed to cor~espon to that of a weightless beam with the concentrated load P actmg at 1ts end. Then

PL 3 Yo= 3EI

P; -;,;i.f in which EI 'is the flexural stiffnesii~of the beam.

The stiffness k of beam at the free end is

P 3El k=-=-,-Yo L

The expression for the deflected shape of the cantilever is

and 1 2 3 EI 2 Maximum potential energy= 2 kyo= 2 L' Yo

(2.71)

(2.72)

(2. 73)

(2.74)

If the weight of the beam is w per unit length and if a harmonic motion is assumed,

Maximum KE of system=~ LL (wnYl 2 dx ~~ = ~ ( W;Yo )' r Hf)'- (f)T dx

_ ~ (33wL)w' 2 - 2 140g nYo (2.75)

By equating the two energies from Eqs. (2.74) and (2.75), we obtain the fundamental frequency of vibration of the cantliever beam as


PEl g /gEl Wn = - 3 ( 33 ) = 3.56 \I ~--4 L 140 wL wL

The exact solution is

/iEi Wn =3.515 \I~

wL

2.9 LOGARITHMIC DECREMENT

(2.76a)

(2.76b)

Logarithmic decrement is a measure of the decay of successive maximum amplitudes of free vibrations with viscous damping and is expressed (Fig. 2.8b) by

zl /5 =log -' z,

(2. 77)

in which z 1 and z2 are two successive peak amplitudes. If z 1 and z2 are determined at times t1 and (11 + 27T) from Eq. (2.35) and substituted into Eq. (2.77), we obtain

or

or

or

in which i; is small.

27Ti; /5 = log, exp , r:;--;z

v 1- i;

(2.78)

(2. 79)

(2.80a)

(2.80b)

If the damping is very small, it may be more conyenient to measure the difference in peak amplitudes for n cycles.

In such a case, if zn is the peak amplitude of the nth cycle, then

(2.81a) Also,

DETERMINATION OF VISCOUS DAMPING 39

(2.81b)

Therefore,

1 z0 /l=-log-n e zn

(2.82)

2.10 DETERMINATION OF VISCOUS DAMPING

Viscous damping may be determined from either a free-vibration or a forced-vibration test on a system.

In a free-vibration test, the system is displaced from its position of equilibrium, and a record of the amplitude of displacement is made. Then, from Eqs. (2.77) and (2.80b)

/5 1 z 1


or

ri., = H2(1- 2t;') V 4(1- 2eJ'- 4(1- Sg')] = ![2(1- 2t;') v 4 + 16t; 4 - 16t;'- 4 + 32t;'] = (1 - 2eJ 2t;'{l+T'

if t; is small. Now,

ri _ ri = ti -,r; =(I'- t, )(!, + f,) -= 2(1'- f,) .1 f n J,, fn /,, since (f, + [2 ) ![,,-=' 2. Therefore,

(2.86)

This method for determining viscous damping is known as the bandwidth method.

0.1 6

Vertic11 vibratioL Zmax = ~ 152 mm

0.1 4

0.1 2 1\ 'E .s ~ 0.1 .~ 0. E "' 0.0

0.0

0.0

f-- - ~\- '-0 J!

1\

8 I ~ 1/! I e =)oo ! " 6

/ I I 4 : I 10 14 18 22 26 30 34

Frequency (cps) Figure 2.18. Determination of viscous damping in forced vibrations by bandwidth method.

TRANSMISSIBILITY 41

2.11 TRANSMISSIBILITY

The system shown in Fig. 2.12 represents a practical case of a machine foundation that is subjected to rotating unbalance. The forces transmitted to the foundation through the spring and the dashpot can be easily computed.

The maximum force in the spring is kZ0 and the maximum force in the dashpot is cwZ0 , the two forces are put out of phase by 90 (Fig. 2.9d). Hence, the force transmitted F, to the base is

(2.87a) or

(2.87b)

If transmissibility T, is defined as the ratio of force transmitted F, to the force of excitation m 0ew

2, then by substituting for cw/ k = 2t;r and for Z0

from Eq. (2.49b), we obtain F, \C' '/1 + (2t;r)2

T, = -m-0

e_w_2 =, -Y-r(=l~-=r"'2c=)2'=+~(2=t;=r""')2 (2.87c) 4. 5

4. 0 '~ o- 1---E ~ 0

( ~ 0.125- 1--- ( ~ 0.125

3. 0

'r ~ = 0.25

0

1- I/E ~ o.5 v ,/' ~ 1.0 rx"' ~ 2.0 0 "~',":... (- 2.0-2~ 1'---t. ~ 1 0 1'--


The transmissibility T, versus the frequency ratio wlw" is plotted in Fig. 2.19. It will be seen that fort; equal to zero, the plot is the same as in Fig. 2.10a. Also, all curves pass through r = V2. When r is greater than V2, all the curves approach the x-axis asymptotically. The higher the frequency ratio, the better the isolation, and hence the smaller the force transmitted. But there may be excessive amplitudes at the time of starting and stopping a machine, because it will pass through the zone of resonance. Damping helps to reduce these amplitudes.

2.12 VIBRATION MEASURING INSTRUMENTS

Figure 2.20 shows the essential elements of a vibration measuring in-strument. It consists of a seismic mass m which is supported by springs and a dashpot inside a case, which is fastened to a vibrating base. The motion of the base is to be monitored. Let the motion of the base be represented by

x = X 0 sin wt (2.88a) The relative motion of the mass m of the instrument in relation to the

vibrating base is monitored. Thus, we can let the absolute motion of the mass m of the instrument be y and, by neglecting transients,

y = Y sin wt (2.88b) Then the equation of motion of m can be written as

my = - k( y - x) - c(.Y - i) (2.89) If we let y - x = z and y - i = i, we obtain

mi + ci + kz = mw 2X 0 sin wt (2.90)

Base Figure 2.20. A vibration measuring instrument (seismic instrument) mounted on a vibrating base.

VIBRATION MEASURING INSTRUMENTS 43

Equation (2.90) is similar to Eq. (2.48b). Hence the solution for Z0 can be written as in Eq. (2.49b):

and

2 mw X0 Zo = y 2 2 2 (k-mw) +(cw)

cw = tan- 1 -=~ k-mw 2

Equations (2.91) and (2.92) may be rewritten as

and

Z0 (w/wn)2 Xo Y (1- r 2 ) 2 + (2t;r)2

_,_ _ 1 2t;r "P =tan --2 1-r

(2.91)

(2.92)

(2.93a)

(2.93b)

Plots of Z0 / i,- ~nd the frequency'fi\lio and the phase angle and frequency ratio are shown in Fig. 2.21.

1.0 2.0

Frequency ratio ~ Wn

Ia I Figure 2.21. Response of a vibration measuring instrument to a vibrating base. (a) Amplitude. (b) Phase angle. (After Thomson, 1972, p. 60. Reprinted by permission of Prentice~Hall, Englewood Cliffs, New Jersey.)


Displacement Pickup

For large values of wlw", and for all values of damping, g, Z 01X0 is approximately equal to unity. Therefore, if the natural frequency of the instrument is low, such that the value of r is large, then the resulting relative motion Z0 equals X 0 . Therefore, the instrument functions as a displacement pickup.

One of the disadvantages of the displacement pickup is its large size. Because I Zl is equal to I Yl, the relative motion of the seismic mass will be as large as the amplitude of vibration to be measured.

Acceleration Pickup

By rearranging Eq. (2.93a), we obtain Z0 I 1

w2X 0 = w;y/[1- (w/w")2 ] 2 + [2gw!wJ 2 w;vc

(2.94)

When g is equal to 0.69, the values of VC in the denominator for different values of w I wn are

wlw, 0.0 0.1 0.2 0.3 0.4

1.000 0.9995 0.9989 1.000 1.0053

Thus Z0 is proportional to the absolute acceleration, w2X 0 , of the vibrating

base. The instrument thus functions as an accelerometer. The frequency ratio wlwn in an accelerometer must be small. Therefore, the natural frequency of the instrument must be high.

Phase Distortion

The instrument must be capable of reproducing a complex wave without changing its shape; that is, the phase of all harmonic components must be shifted equally along the time axis. This can be accomplished if the phase angle q, of the accelerometer output increases linearly with frequency. This condition is nearly satisfied when!; is equal to 0.70, and the phase distortion is practically eliminated.

2.13 SYSTEMS WITH TWO DEGREES OF FREEDOM

Figure 2.22 shows a two-mass-two-spring system, which has two degrees of freedom. Free-body diagrams of the masses are also shown. In a practical system, the spring k1 and the mass m 1 constitute the main system, and spring k, and mass m 2 a vibration absorber. The equations for motion of both the

SYSTEMS WITH TWO DEGREES OF FREEDOM 45

Sign conventions --:-::--]

z, Z, Z + t Zl > Z2 k1 ~ Fo sin wt

0!

Z)

Ia I (b)

Figure 2.22. (a) Two degrees of freedom system. (b) Free-body diagram.

masses may be written in the following form:

or

(2.95a) and

or

(2.95b)

The natural frequencies of this system are obtained by considering its free vibrations. Making F0 = 0 in Eq. (2.95a), we obtain

(2.95c)

Let

(2.96a)

and (2.96b)

By substituting the solutions from Eqs. (2.96) into Eqs. (2.95b and 2.95c), we obtain


(2.97a) and

(2.97b) From Eqs. (2. 97), we obtain

-m 1w:+k1 +k2 k2 k2 -m2 w: + k2

Simplifying this we obtain

or

or

or

Let

Therefore,

or

or

(2.98) in which


w _ I k, nll- 'V m +m

1 2 (2.99a)

(J)n/2 = f'-2

(2.99b)

The values of the two natural frequencies wn 1 and wn 2 for this system are obtained by solving Eq. (2.98) as a quadratic in w;.

Amplitudes of Vibrations For the force acting on mass m1 , the vibration amplitudes are obtained by assuming the following solution for the principal modes:

(2.100a)

and

z 2

~~;ji:J sin w t (2.100b) By substituting the solution from Eqs. (2.100) into Eqs. (2.95a) and (2.95b), we obtain

(2.10la)

and

(2.101b)

From Eq. (2.101b),

(2.102)

Substituting for Z2 from Eq. (2.102) into Eq. (2.101a), we obtain

or

[m 1m2 w 4 - k2 m 1 w 2 - k1m2 u/ + k 1k2 - k 2m 2 w 2 + k;- k;]z1 = F0 (k 2 - m 2 w2 )

or


or

or

(2.103a)

or

(2.103b) in which

Substituting for Z1 from Eq. (2.103a) into Eq. (2.102), we obtain

Zz = 4 m 1[w

Fo w !,z w

2(1 + ~t)(w~11 + w~12 ) + (1 + ~t)w~11 w!12] (2.105a) or

z = Fow ~12 2

m1 A(w 2 ) (2.105b)

From Eq. (2.103a), it is seen that z, = 0, if

(2.106) Then

or

(2.107)

Equations (2.103, 2.105, 2.106, and 2.107) explain the principle of vibration absorbers that will be used in Chapter 10 (Section 10.1). The amplitudes of motion of mass m 1 can be appreciably reduced by attaching to it, a


spring-mass system having its natural frequency given by Eq. (2.106). The negative sign in Eq. (2.107) indicates that Z2 and F0 are in phase opposition. In fact, the amplitude of the main mass Z1 , becomes zero at this frequency, because the force, k 2Z2 , exerted by spring 2 on mass m 1 is equal and opposite to the force of excitation F0 The size of the absorber mass m2 and its displacement, depend upon the magnitude of the disturbing force, F0(=k 2Z 2 ). For a given force F0 , the smaller stiffness of the absorber spring, the larger its amplitude Z2 and vice versa.

Figure 2.23a shows a plot of Z 1/Z,, versus w/w"12 (Eq. 2.103b) with

8

6

-~- 4 ~~-f_e

2

Iii lli_ II

I ! \ 11

J i \

1:: 11

i\ II

r'~ i \ ii p. = 0.20 I. 0~ V 1 I , I 018 1/ I \ Its .....

--t---0 0.5 1.0 1.5 2.0

w

Wn/2

Ia I 1.6

......

/ ..........

/ v 3 / / 2

1.5

1.4

I.

I.

1 /

I. 0 Wnll ~ 1 Wn/2

\i 9 " :-....... 8

7 -

r-.

-1-6

0 .1 .2 .3 .4 .5 .6 .7 .8 Mass Ratio P.

(b)

2.5

Figure 2.23. (a) Response versus frequency of a vibration absorber. (b) Natural frequencies versus ~J-(=m~ I m,).


JL = m 21m 1 of 0.20. Although the amplitude of the main mass m 1 becomes zero when w = W 1112 , there are two resonant frequencies at which the amplitude of mass m 1 becomes infinite. In Fig. 2.23b, (w/wn12 ) has been plotted versus !L (Eq. 2.98) for a particular case of wn11 = wn12

In Fig. 2.22a, if the forcing function F0 sin wt is acting on mass m 2 , instead of mass m 1 (as shown), it can be shown that the amplitudes of motion Z1 and Z2 are given by:

2 Z = Wnz2 F

I m,~(w2) o (2.108)

and

Z _ (1 + JL)W~ll + JLW~12 - w 2 2- m2~(w2) F0 (2.109)

2.14 MULTIDEGREE FREEDOM SYSTEMS

It has been shown in Section 2.1 that the number of independent coor-dinates of displacements in a vibrating system determines the degrees of freedom of the system.

In this section we will discuss the techniques applicable to the solution of vibrations of multidegree freedom systems.

Two approaches are commonly used for obtaining a solution: (1) stiffness matrix method; (2) flexibility influence coefficient method.

The stiffness coefficient k1i is defined as the force on the ith mass due to a unit displacement at the jth mass with all other masses held at their equilibrium position. With displacement x 1 , x2 , and x3 of points 1, 2, and 3, respectively, the principle of superposition can be applied to determine the forces in terms of stiffness coefficients as

ft = kuxt + kuxz + k13x3 f, = k21 x 1 + k22x2 + k23x 3 !3 = k31X1 + k32X2 + k33X3

In matrix notation, the equation is

in which

{f} = [k]{x}

kll k12 k13 [k] = k,, k,, k,,

k,, k,, k,, f={J,}.

(2.110)

(2.111)

(2.112)

MULTIDEGREE FREEDOM SYSTEMS 51

Thus in a system with n masses, if the displacements of masses are denoted by xi (where j can take integral values from 1 ton) the total spring force on mass i due to displacements xi of all masses is r:;~, k,h, the summation of the spring forces. Applying Newton's law for the ith mass,

n

m/ii + 2: kiixi = 0 j""l

(2.113)

There are n such equilibrium equations each corresponding to a mass. This procedure is called stiffness method.

Let the system vibrate in one of its principal modes of vibration. Then its motion will be sinusoidal with a natural frequency of wn corresponding to that mode. Since the amplitudes of the masses may be different from each other, the motion of any mass i may be expressed as

(2.114)

in which i = 1, 2, ... , n. Substituting Eq. (2.114) into Eq. (2.113) we get (p -- -;: ':r

n -~: :: -miw~Ai + 2:. kiiAi = 0,

j=l i=l,2, ... ,n

The frequency determinant corresponding to Eq. (2.115) is

(2.115)

=0 (2.116)

Expanding the determinant, we would get a frequency equation in the polynomial form as below:

2 2(n-l) + ( 1)" 2n 0 a +aw+ .. +a w - w = 0 1 n n-1 n n (2.117) Since the coefficient of w~ is not zero, Eq. (2.117) always has n roots. These roots would give the 1i~ natural frequencies of vibration, namely, w"" Wnz' . . . wnr ... wnn. Corresponding to each value of wn' there is an as-sociated mode shape with amplitudes Air), Ar), ... , A~), which can be obtained by solving Eq. (2.115).

When a system vibrates in a principal mode, all the masses attain max-imum displacements simultaneously and also pass through their equilibrium position simultaneously.

When the number of degrees of freedom exceeds three, the problem of forming the frequency equation and solving it for determination of frequen-


cies and mode shapes becomes tedious. Numerical techniques are invariably resorted to in such cases.

The flexibility influence coefficient a,i is defined as the displacement at ith mass due to a unit force applied at jth mass. With forces f 1 , [ 2 , and f3 acting at pmms 1, 2, and 3, the principle of superposition can be applied to determme the displacements m terms of the flexibility influence coefficients

x, = a,,[, + a32f, + a,,[, In matrix notation the equation is

{x} = [a]{f} in which

is the flexibility matrix.

(2.118a)

(2.118h)

(2.118c)

(2.119)

(2.120)

If Eq. (2.119) is multiplied by the inverse of the flexibility matrix [ar' we obtain the equation '

[ar'{x} = {f} = [k]{x} (2.121) It is thus seen that the inverse of the flexibility matrix is the stiffness matrix [k], i.e.,

[ar' = [kJ (2.122a) or

[a]= [kr' (2.122b)


vibration are multiplied are represented by the coordinates t;, then, for any mass 1,

or

X=A('lt +AC'lt ++AC'lc ++A(nlt I I ~J I !:.2 1 ~T t ~JJ

n

X="' AVlt I L..J I !:>r

r=l

Where r represents the rth mode. Then from Eq. (2.126), we get n n n L mA()i: +"' "'k A(')c = F(t)

1 1 br L.J L.J IJ 1 Sr i r=l r=t j=l

From Eq. (2.115),

Hence

n n

L mA(') < +"' w 2 mAC') t = F(t) 1 1 ~r L-1 nr t 1 Sr 1

r= 1 '"' 1

and

n

L m,A~')( {, + w;,t;,) = F,(t) r=I

(2.127)

(2.128)

(2.129)

Since the left-hand side is a summation involving different modes of vibration, the right-hand side should also be expressed as a summation of equivalent force contribution in the corresponding modes.

Let F, be expanded for convenience as n

F,(t) = L m,A~') f,(t) r=l

(2.130)

in whichf,(t), the modal force is derived subsequently as Eq. (2.134). Then from Eqs. (2.129) and (2.130),

where r = 1, 2, ... , n (2.131) This is a single degree of freedom equation and its solution can be written as

1 l' t;, =-- J,(r) sin w,(t- r) dr wnr 0

where 0 < r < t (2.132)

MULTIDEGREE FREEDOM SYSTEMS 55

It is seen that the coordinate g, uncouples the n degree of freedom system into n of single degree of freedom systems. The t;'s are termed as normal coordinates and this approach is known as normal mode method, i.e., the total solution is thus expressed as a sum of contribution of individual modes.

For determination of f,(t), multiply both sides of Eq. (2.130) by A~') (=A)'lT, the transpose of A)'l) and summing up for all the masses, we get

n n n n n

"' FAC'l ="' AC'l "' mAc'lt, ="' /, "' m.AC'l AC'l L.J I I LJ I LJ I I T L.J rL.i I l I (2.133) i=l i=J T"'l reel i=l

Using the orthogonality relationship, n

"' m.AC'l AC'l = 0 LJ l I I for r o;6 s i=1

the right-hand side of Eq. (2.133) reduces to n

f, L m,(A)~);, i=l ~

when r=s

. Hence

(2.134)

Using Eqs. (2.132) and (2.134), the complete solution from Eq. (2.127) is

" l' "" F ( )AC'l - (r) 1 L.. j= 1 j T j X,--~ A, -;;; 0 l::" (A('l)' sm w.,(t- r) dr r-1 r J=lm; 1 EXAMPLE 2.14.1

Figure 2.24 shows a three degrees of freedom system. Determine the stiffness matrix.

Figure 2.24. Computation of stiffness matrix.


Solution Let x 1 = 1.0 and x 2 = x 3 = 0. The forces required at 1, 2, and 3, considering forces to the left as positive, are

ft = kl + k, = k11 / 2 = -k2 = k 21 [, = 0 = k,l

Repeat with x 2 = 1, and x 1 = x 3 = 0. The forces are now

!1 = -k, = k1z [, = k, + k, = k,, [,=-k,=k,,

Repeat with x 3 = 1, and x 1 = x 2 = 0. The forces are

!I= 0 = k13 h = -k, = k,, [, = k, + k, = k,,

The stiffness matrix can now be written as

[(kl + k,) -k,

K = - k 2 (k2 + k3 ) 0 -k 3

0 l -k (k, + k.J

EXAMPLE 2.14.2

For the system shown in Fig. 2.25a, solve for natural frequencies and mode shapes.

Solution The stiffness coefficients K,i for the system are

k11 =2k, kl2 = ~k, k13 =0 k21 = -k, k,, = 2k, k,, = -k k31 = 0, k,, = -k, k,, d," k

The equations of motion of the system, from Eq. (2.113) are

m 1i 1 + 2kx1 - kx2 = 0

m 2i 2 - kx1 + 2kx2 - kx 3 = 0

MUlTIDEGREE FREEDOM SYSTEMS

0 +

0.445

0.802

First mode

Ia I

VERTICAL DISPLACEMENT - 0 +

1.247

Second mode

(b)

1.0 Third mode

Figur-?~:25. (a) Spring-mass sylt~dls for Example 2.14.2 (b) Mode shapes. The corresponding frequency

1 determinant from Eq. (2.116) is

(2k-m 1 w;) -k -k (2k- m 2 w!) 0 -k

0 -k =0

(k- m,w;) The frequency equation can be obtained by expanding the determinant.

Letting m 1 = m2 = m3 = m (for simplicity),

(2k- mw!)[(2k- mw;)(k- mw!)- (- k)( -k)]

57

- ( -k)[(- k)(k- mw;)- ( -k)(O)J + (0)[( -k)( -k)- (2k- mw;)(O)] = 0

which, on simplification, gives

- A' 3 + 5A' 2 - 6A' + 1 = 0

in which A' = mw ;; k (A' will be equal to 1.0 for a single degree of freedom). The roots of frequencfequation can be determined by any of the standard techniques. In this case, the trial-and-error method and the plotting of the graph of the function will be used. The roots are A;= 0.198, A;= 1.555, A;= 3.247, and since A'= mw;lk,

2 k Will= 0.198 m '

2 k w 112 = 1.sss m

2 k (I) n3 = 3.247 m

The amplitude coefficients for the three modes of vibration can be obtained from Eq. (2.125), from which we get


1 1 1 1 - -w-; A 1 + m k A 1 + m k A 2 + m k A 3 = 0

1 1 2 2 --,A,+m-kA 1 +m-A +m-A =0 w, k 2 k 3

1 1 2 3 --, A 3 +m k A, +m k A 2 +m -A 3 =0 (f)n k -

Letting A'= (mlk)w;,, the preceding equations can be rewritten as

(A' -1)A 1 + A'A 2 + A'A 3 = 0 A'A 1 +(2A'-l)A 2 +2A'A 3 =0 A'A 1 + 2A'A 2 + (3A' -l)A 3 = 0

It is more convenient to work with particular numerical values rather than with ratios. Therefore, let us arbitrarily set A 3 (1) = A 3(2) = A 3 (3) = 1. In th1s manner, we will obtain two simultaneous equations for A

1 and A

2 from

above:

(A' -1)A 1 + A'A 2 =-A' A'A 1 + (2A' -1)A 2 = -2A'

Substituting numerical values of A' determined above, we get

AI A, A, First mode 0.445 0.802 1.0 Second mode

-1.247 -0.555 1.0

Thrid mode 1.802 -2.247 1.0

These mode shapes have been plotted in Fig. 2.25b.

PRACTICE PROBLEMS

2.1 Determine the numerical value of viscous damping from the free vibrations record in Fig. 2.8b.

2.2 Show that, in frequency-dependent excitation, the damping factor t; is given by the following expression:

t;=Ht,-J;) !,,

in which [2 and J; are frequencies at which the amplitude is 1/"1/2 times the amplitude at r = 1.

PRACTICE PROBLEMS 59

2.3 For the system represented by Eq. (2.48b), show that the peak amplitude occurs at a frequency ratio of

and

1 r = -Y-,=1 ,;;_=2t;""'

1 z =-~~=

max Z~

If t; is greater than 0.707, r is imaginary. Discuss the significance of these values with the help of a diagram.

2.4 An unknown weight W is attached to the end of an unknown spring k and the natural frequency of the system is 1.5 Hz. If 1 kg weight is added to W, the natural frequency is lowered to 75 cpm. Determine the weight W and the spring constant k.

2.5 A body weighing 60 kg is suspended from a spring, which deflects 1.2 em under the load. It is subjected to a damping effect adjusted to a value O.QO times that requir~ct:for critical damping. Find the natural frequency of the undamped and damped vibrations, and, in the latter case determine the ratio of successive amplitudes.

If the body is subjected to a periodic disturbing force with a maximum value of 25 kg and a frequency equal to one-half its natural undamped frequency, determine the amplitude of forced vibrations and the phase differenc

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Foundations for Machines-Analysis & Design_By Shamsher Prakash.pdf