Vibration problems in engineering by s timoshenko

Theprincipaladditionsare as follows: In the first chapteradiscussion of forced vibration with dampingnot proportional to velocity is included, vibra-tions of systems with non-linear char

NEW YORK

D VAN NOSTRAND COMPANY, INC.

FOURTH AVENUE

All Rights Reserved

written permission from the publisher,,

First Published October, 1928

Second Edition July,1937

RcpruiUd,AuyuU, 1^41, July, UL' t J, Auyu^t, t'^44, A/

PRINTED IN THEUSA

In the preparation of the manuscript for the second edition of thebook, the author's desire was not only to bring the book up to date by

includingsome new material butalso tomake itmore suitable forteachingpurposes With this in view, the first part of the book was entirely re-

written and considerablyenlarged. A number of examples and problemswith solutions or with answers were included, and in many places new

material wasadded

Theprincipaladditionsare as follows: In the first chapteradiscussion

of forced vibration with dampingnot proportional to velocity is included,

vibra-tions of systems with non-linear characteristics The third chapter is

of vibratory motion of systems with variable spring characteristics The

fourth chapter, dealingwith systems having several degrees of freedom,is

alsoConsiderably enlarged by adding a general discussion ofsystems with

viscous damping; an article on stability of motion with an application in

studying vibration ofa governorof asteam engine; an articleon whirling

ofarotating shaft dueto hysteresis; and anarticleonthe theoryof

damp-ing vibration absorbers There are also several additions in the chapter

on torsionalandlateralvibrations ofshafts.

The author takes this opportunity to thankhis friends who assisted in

various ways in the preparation of the manuscript* and particularly

ProfessorL S. Jacobsen,whoread over thecomplete manuscript and made

many valuable suggestions, and Dr J. A Wojtaszak, who checked

prob-lemsof the first chapter

STANFORD UNIVERSITY,

May29, 1937

PREFACE TO THE FIRST EDITION

With the increase of size and velocity in modern machines, theanalysis of vibration problems becomes more and more important in

practical significance, such as the balancing of machines, the torsional

vibration of shafts and of geared systems, the vibrations of turbinebladesand turbinediscs,the whirlingofrotatingshafts, the vibrationsof

railway track and bridges under the actionof rolling loads, the vibration

of foundations, can be thoroughly understood only on the basis of the

design proportions be found which will remove the working conditions

In the present book, the fundamentals ofthe theory of vibration aredeveloped, and their application to the solution of technical problemsis

illustrated by various examples, taken, in many cases, from actualexperience with vibration of machines and structures in service In

developing this book, the author has followed the lectures on vibrationgiven by him to the mechanical engineers of the Westinghouse Electric

chapters of his previously published book on the theoryof elasticity.*

The contents ofthe book ingeneral are as follows:

The first chapter is devoted to the discussion of harmonicvibrations

forced vibration is discussed, and the application of this theory tobalancing machines andvibration-recording instruments is shown The

com-plicatedsystems isalso discussed, andisappliedto the calculationofthewhirling speedsofrotating shaftsof variable cross-section

investi-gating the freeand forced vibrations of such systems are discussed A

particular caseinwhich theflexibilityof thesystemvarieswiththetimeisconsideredindetail, andtheresults of thistheoryare applied to the inves-

tigation ofvibrationsinelectriclocomotiveswith side-rod drive

*

Theoryof Elasticity, Vol II (1916) St Petersburg, Russia.

In chapter three, systems with several degrees of freedom are

con-sidered The general theory of vibration of such systems is developed,

and also its application in the solution of such engineeringproblems as:

the vibrationof vehicles,thetorsionalvibrationof shafts, whirlingspeeds

of shafts onseveral supports, and vibration absorbers

of prismatical bars; the vibration of bars of variable cross-section; thevibrations of bridges, turbineblades,and shiphulls; thetheoryof vibra-

tion of circularrings, membranes, plates, and turbine discs.

Brief descriptions of the most important vibration-recording

are given inthe appendix

made it possible for him to spend a considerable amount of time in thepreparationofthemanuscriptandto use asexamplesvarious actual cases

of vibration in machines which were investigated by the company's

engineers He takes this opportunity to thank, also, the numerous

friends who have assisted him in various waysin the preparation of the

many valuable suggestions.

Heisindebted,also,toMr. F C.Wilharmforthe preparationof

draw-ings, and to the VanNostrand Companyfortheir careinthe publication

oi the book

S. TIMOSHENKO

ANN ARBOR, MICHIGAN,

PAGE

4. Instrumentsfor Investigating Vibrations 19

6. OtherTechnical Applications 26

9. ForcedVibrationswithViscous Damping 38

12. ForcedVibrations with Coulomb's DampingandotherKindsof Damping 57

14. General Caseof DisturbingForce 64

v/15 Application ofEquationofEnergyin VibrationProblems 74

17 CriticalSpeedof aRotating Shaft 92

18. General Caseof DisturbingForce 98

19 Effect ofLow Spots onDeflection of Rails 107

VIBRATION OF SYSTEMS WITH NON-LINEARCHARACTERISTICS

21 ExamplesofNon-Linear Systems 114

25. Methodof SuccessiveApproximationsApplied toFreeVibrations 129

26. Forced Non-LinearVibrations 137

SYSTEMS WITH VARIABLE SPRING CHARACTERISTICS

27. Examplesof Variable Spring Characteristics 151

CHAPTER IV

PAGE

SYSTEMS HAVING SEVERAL DEGREES OF FREEDOM

TORSIONAL AND LATERAL VIBRATION OF SHAFTS

43. Approximate Methodsof Calculating Frequencies of NaturalVibrations 258

44. ForcedTorsional Vibration of a ShaftwithSeveral Discs 265

48. GyroscopicEffectsonthe CriticalSpeedsofRotatingShafts 290

VIBRATION OF ELASTIC BODIES

60 Effect of Axial ForcesonLateral Vibrations 364

CONTENTS ix

PAGE

2. Frequency Measuring Instruments 443

HARMONIC VIBRATIONS OF SYSTEMS HAVING ONE

ofequilibrium by an impact or by the sudden application and removalof

anadditional force, the elasticforces ofthememberinthe disturbed

posi-tion will no longer be in equilibrium with the loading, and vibrations will

ensue Generally an elastic system can perform vibrations of different

different shapesdepending onthe numberof nodessubdividing the length

systems having onedegree offreedom

Let us consider the case shown in Fig 1 If the arrangement be such

that only vertical displacements of the weight W are

possibleandthemassofthe springbesmallin

compari-son with that of the weight W, the system can be

considered as having one degree of freedom. The

configuration will be determined completely by the

vertical displacement of the weight

By animpulse ora suddenapplicationandremoval

of an external force vibrations of the system can be

produced Such vibrations which are maintained by

natural vibrations An analytical expression for these FIG 1.

vibrations can be found from the differential equation

of motion, which always can be written down ifthe forces acting on the

Let k denote the load necessary to produce a unit extension of the

spring This quantityiscalled spring constant Ifthe load ismeasured in

W willbe

2 VIBRATION PROBLEMS IN ENGINEERING

ofequilibrium byx and considering thisdisplacement as positive if it is in

a downward direction,the expressionforthe tensileforce inthe spring

Inderiving the differentialequation of motion we willuseNewton's

prin-ciple statingthat the product ofthe massof a particleanditsacceleration

is equaltotheforceactingin thedirection of acceleration Inourcase the

mass of the vibrating body is W/g, where g is the acceleration due to

gravity; theacceleration of the body W isgivenby the second derivative

of the displacement x with respect totime and will be denoted by x] the

forces actingon the vibratingbody are the gravityforce W, acting

down-wards,and the forceFof the spring (Eq a) which, forthe position of the

equa-tion ofmotion inthe caseunderconsiderationis

a

This equation holds for any position of the body W. If, for instance, the

bodyinitsvibratingmotiontakesa positionabovethe position of

equilib-riumandsuchthata compressivcforce inthespringisproducedthe

gravity force as it should be.

This equation will be satisfied if we put x = C\ cos pt or x = 2 sin pt,

general solution of equation (3) will be obtained:

x = Ci cos pt + 2 sinpt. (4)

It isseen that the verticalmotionofthe weight Whas a vibratory

charac-ter, since cos ptand sin pt are periodic functions which repeat themselves

It is seen that the period ofvibration depends only on themagnitudes of

mag-nitudeofoscillations. Wecansay alsothat the periodofoscillationofthe

lengthofwhichisequaltothe staticaldeflection 5^ Ifthe statical

calculated from eq. (5).

fre-quency of vibration Denoting it by/ we obtain

A vibratory motion represented by equation (4) is called a harmonic

motion In ordertodetermine the constantsof integration Ci and C2, the

0)the weightWhasadisplacementXQfromitsposition

of equilibrium and that its initial velocity is XQ. Substituting t = inequation (4) weobtain

in thisderivative t = 0, wehave

- = C

4 VIBRATION PROBLEMS IN ENGINEERING

Substitutingineq (4)the valuesof the constants (d) and (e),thefollowingexpressionforthe vibratorymotion ofthe weight Wwill be obtained:

, , -Ml

x = xo cos pt

-i sinpt.

It isseen that in this case the vibration consists of twoparts; avibration

*

The total displacement x of the oscillating weight W at any instant t isobtainedby adding together the ordinates of the two curves, (Fig.2a and

vectors Imagine a vector OA, Fig 3, of magnitude rr () rotating with aconstant angularvelocityp around afixed point,0. Thisvelocityiscalledcircularfrequencyof vibration Ifatthe initialmoment = the vector

coincides with x axis, the angle which it makes with the same axisat any

instant t is equal to pt. The projection OA\ of the vector on the x axis

isequal to xo cosptandrepresents the firsttermofexpression (7). Taking

now another vector OBequal toxo/p

and perpendicular to the vector OA,

its projection on the x axis gives

the second term of expression (7).

The total displacement x of the

oscillating load W is obtained now

axis ofthe two perpendicular vectors

~OAand OB,rotating withtheangular

velocity p.

if, instead of vectors ()A and OB, we

consider the vector ()C, equal to the

geometrical sum of the previous two

vectors, and take the projection of this vector on the x axis. The

magni-tude of thisvector, from Fig. 3, is

It isseen that in thismanner we added together the two simple harmonic

motions, one proportional to cospt and the other proportional to sinpt.

The result of this addition is a simple harmonic motion, proportional to

this curve, equal to Vjar + (x^/p)'

2

, represents the maximum

displace-ment of the vibrating body from the position of equilibrium and iscalled

the amplitude

6 VIBRATION PROBLEMS IN ENGINEERING

Due to the angle a between the two rotating vectors OA and OC the

maximum ordinate of the curve, Fig. 2c, is displaced with respect to the

maximum ordinate of the curve, Fig 2a, by the amount a/p. In such a

case it may be saidthat the vibration, represented by the curve, Fig. 2c,

iscalledthe phasedifferenceof thesetwovibrations

PROBLEMS

1. TheweightW = 30Ibs is verticallysuspended on asteel wire of length I 50in.

andof cross-sectional areaA 0.00 1 in 2

. Determine thefrequencyof free vibrations

of the weight if the modulus for steel is E = 30-10 6

Ibs per sq in. Determine the

amplitudeof this vibration if the initialdisplacementXQ = 0.01 in. andinitial velocity

Solution Static elongation of^the wire is 8 st = 30-50/(30-106

-0.001) = 0.05 in.

Then, from eq (6Q,/ = 3.13V'20 = 14.0 sec." 1

. The amplitude of vibration, from

eq (8), is Vz2

+ (Wp)2 = V(0.01)2+[l/(27r-14)]2 = .01513in.

in., the diameterof

will thefrequencyof vibrationbe changed if the spring

remainingthesame?

3. A load W is supported by a beam of length lt

Fig 4. Determinethe spring constantandthefrequencyFIG 4 of free vibration of the load in the vertical direction

Solution. Thestatical deflection of thebeamunderload is

rigidity of thebeam in the vertical plane It is assumedthat this plane contains one

onlyvertical deflections. Fromthe definition the spring constant in this case is

ZIEI

massof thebeamonthefrequencyof vibration willbediscussed later, see Art 16.

4. AloadWis verticallysuspended ontwosprings asshownin Fig 5a. Determine

spring constantsofthetwosprings are kiandfa. Determinethefrequencyof vibration

_W W . ^

dgt in eq. (6), the frequencyof vibrationbecomes

6. Determinethe period of horizontal vibrations of the frame,shownin Fig.6,

this calculation.

Solution. Webeginwith astaticalproblemanddetermine the horizontal deflection

6 of theframe whicha horizontal forceHacting at the point of application of the loadW

magnitude Hh/2 Thenthe angleaof rotation of thejointsA andBis

Hhl

Consideringnowthe verticalmembersof theframeas cantileversbentbythe horizontal

two onedue

8 VIBRATION PROBLEMS IN ENGINEERING

""

QEI \ '2hlJ

'

Wh*[ 1 + HA

2hlJ

QgEI

If the rigidity ofthe horizontal memberis large in comparison withthe rigidity ofthe

verticals,thetermcontaining theratio I/I\ is smallandcan beneglected. Then

IWh*

r==27r

andthefrequencyis

6. Assumingthat the loadWin Fig 1 represents the cage ofanelevatormovingdown

withaconstant velocity vandthe spring consists of asteel cable,determinethemaximum

stress in the cable if duringmotion theupper end Aof the cable is suddenlystopped.

Assume that the weight W = 10,000Ibs., I 60 ft., the cross-sectional area of the

ft per sec. Theweightof the cable is to be neglected.

Solution. During the uniform motion of the cage the tensile force in the cable is

equal toW = 10,000Ibs.andthe elongationofthe cable at the instantofthe accidentis

displace-mentof the cagefromtheposition ofequilibrium at that instant is zeroandits velocity

is v. Fromeq.(7)weconclude that theamplitudeof vibration willbeequalto v/p,where

maxi-mum stress is (10,000/2.5) (.995/.192) =20,750 Ibs per sq in It is seen that dueto

thesudden stoppageof the upper endof the cable the stress in the cable increased in this caseaboutfive times.

k =2000Ibs per in is insertedbetweenthe lowerendof the cableandthecage.

Solution. Thestatical deflection in this case is 5^ = .192

amplitude of vibration, varying as square root of the statical deflection, becomes

maximum dynamicalstress is (10,000/2.5)1.80 = 7,200Ibs per sq in It is seen that

2. Torsional Vibration Let us consider a vertical shaft to the lower

y///////////,

Fig.7 Ifa torque isappliedinthe plane ofthe disc

of the shaft with the disc will be produced. The

angular position of the disc at any instant can be

defined by the angle <pwhich a radius of the

vibrat-ing disc makes with thedirection of thesame radius

this case we take the torque kwhichisnecessary to

radian In thecase ofacircularshaft oflength Iand diameter dweobtain

from the known formula for the angle of twist

For any angle of twist <p during vibration the torque in the shaft is k<p.

The equationof motion in the case of a body rotating withrespect to an

to this axismultipliedwiththe angularaccelerationisequaltothemoment

of the external forces acting on thebody with respect tothe axis of

actingonthe shaft andthe equation ofmotionbecomes

of rotation, which inthis case coincideswith the axis ofthe shaft,and is

the angularacceleration ofthe disc. Introducing the notation

the equationofmotion (a) becomes

(11)

Thisequation has thesameformaseq (3) ofthe previousarticle, henceits

solutionhas the same form as solution (7) and weobtain

<f> = <pocospt + sinpi, (12)

P

10 VIBRATION PROBLEMS IN ENGINEERING

respec-tively ofthediscattheinitialinstantt 0. Proceedingasinthe previous

In the case of a circular disc ofuniform thickness and ofdiameter D,

andusing expression (9), we obtain

1WDH

diam-eter d. When the shaft consists of parts of different diameters it can be

readilyreducedtoanequivalent*shafthavinga constant diameter Assume,

for instance, that a shaft consists of two parts of lengths Zi and 1% and of

diametersd\ and dz respectively. If a torque Mt is applied to this shaftthe angleof twistproduced is

7

It is seen that the angle of twist of a shaft with two diameters d\ and d%

isthesameasthatofashaft ofconstant diameterd\ andofareducedlength

Lgivenbythe equation

The shaft oflength L and diameterd\ has thesamespring constant asthegivenshaft oftwodifferentdiametersandisanequivalent shaft in this case.

Ingeneral ifwe haveashaft consisting ofportions with different

diam-eterswecan,without changingthe spring constantofthe any

portion ofthe shaft oflength l n and ofdiameterdn bya portionofashaft

(15)

Theresultsobtainedforthecase showninFig. 7can beusedalso inthe

case of a shaft with two rotating masses at the ends as shown in Fig. 8.

Such a case is of practical importance since an arrangement of this kind

maybeencountered very ofteninmachine design. A propeller shaft withthe propelleronone end andthe engine on the otherisan example ofthis

kind.*(jf twoequal and opposite twistingcouples are appliedat the ends

of the shaft in Fig 8 and thensuddenly removed, torsionalvibrations will

opposite directions, f From this fact

itcanbe concludedatonce thatthere

is acertainintermediate crosssection

mn of the shaft which remains

cross section is called the nodalcross

from the condition that both

por-tions of the shaft, to the right and

to the left of thenodal cross section,

rotating in opposite directions will not be fulfilled.

por-tions of the shaft respectively Thesequantities, as seenfrom eq. (9), are

*Thisis the case inwhichengineers for the firsttimefoundit of practicalimportance

and must remain zero since the moment of external forces with respect to the same

axis is zero (friction forcesareneglected). Theequality to zeroofmomentof tum both masses

momen-12 VIBRATION PROBLEMS IN ENGINEERING

inversely proportionalto the lengths of the corresponding portions of the

shaftandfromeq. (c) follows

Fromtheseformulae the periodandthe frequencyof torsionalvibrationcan

be calculated provided the dimensionsofthe shaft, the modulus G andthe

momentsof inertia ofthe massesatthe ends are known The mass ofthe

shaft is neglected in our present discussion and its effecton the period of

vibration willbeconsidered later, seeArt 16.

momentof inertia incomparison with the other the nodalcross sectioncan

PROBLEMS

1. Determinethefrequencyof torsional vibration ofashaftwithtwocircular discs

of uniformthickness at the ends, Fig 8, if theweightsof the discs are W\ = 1000Ibs.

and Wz =2000Ibs.and their outerdiametersareD\ = 50in. and Dz= 75in

respec-tively. Thelengthof the shaft is I = 120 in. andits diameter d = 4in. Modulus in

Solution. Fromeqs.(d) the distance of thenodalcross sectionfromthe larger disc is

2.Inwhatproportionwill the frequencyof vibration of the shaft considered in the previousproblemincrease if along a length of64in thediameterof the shaft willbein-

creasedfrom 4in to 8 in.

Solution. The lengthof 64 in of 8 in. diametershaft can be replaced by a 4 in.

in.,whichis only one-half of the length of the shaft considered in the previousproblem

the shaft itsfrequencyincreases in the ratioV2 : 1.

3. A circular bar fixed at theupper end andsupporting acircular disc at the lower

end (Fig. 7) has a frequency of torsional vibration equal to/ = 10 oscillations per

diam-eterd = 0.5in.,theweightof the disc W - 10Ibs.,andits outerdiameterD = 12 in.

Solution. Fromeq. (b),G 12 -10 6

4. Determinethefrequencyof vibration of the ring, Fig 9,aboutthe axis 0,

by somebendingof thespokesindicated in thefigurebydotted lines. Assumethat the

totalmassof theringis distributed along the center line of therimandtake the length

of the spokesequal to the radius r of this center line. Assumealso that thebending

of the rim can be neglected so that the tangentsto the deflection curves of the spokes

rigidityBofspokesaregiven.

which a shearing forceQ and a bending moment M are actingand using the known

formulasfor bendingof a cantilever, the following expressions for the slope <f>and the deflection r<p at theendare obtained

Mr2

IfMtdenotesthetorqueapplied to therimwehave

-14 VIBRATION PROBLEMS IN ENGINEERING

Thetorque required toproduce an angle of rotation of therim equal tooneradian is

nowthe casewheninadditiontotheforce ofgravityandtotheforce inthespring (Fig 1) there is acting on the load W a periodical disturbing force

Psinut. The period of this force is r\ = 2?r/co and its frequency is

satisfyeq (18). Substituting (c) inthat equation wefind

This expression contains two constants of integration and represents the

solution ofthe It isseenthatthissolution consists oftwo

parts, the first two terms represent free vibrations which were discussedbefore and the third term, depending on the disturbing force, represents

theforced vibration of the system It is seen thatthis latervibration hasthe same period n =

27r/co as the disturbing force has Itsamplitude A,

is equal to the numerical value of the expression

/7>

2

The factor P/k is the deflection which the maximum disturbing force P

/p2) takes care

usually called themagnificationfactor. Weseethat it dependsonlyonthe

ratio o)/p which is obtained by dividing the frequency of the disturbing

force by the frequency of free vibration of the system In Fig. 10 thevalues of the magnification factor are plotted against the ratio co/p.

It is seen that for the small values of the ratio /p, i.e., for the case

when the frequency of the disturbing force is small in comparison with

unity, and deflectionsareaboutthesame asinthe case ofastaticalaction

ofthe force P

Whentheratio co/p approachesunity the magnification factorand the

co =

coincides with the frequency of free vibration of the system This is the

condition of resonance The infinite value obtained for the amplitude of

forced vibrationsindicates thatif the pulsatingforce actsonthe vibrating

ata propertime andina properdirectionthe of

16 VIBRATION PROBLEMS IN ENGINEERING

vibration increases indefinitelyprovidedthereisno damping In practical

forced vibration will be discussed later (see Art 9).

When the frequency of the disturbing force increases beyond the

Its absolute value diminishes with the increase of the ratio co/p and

a pulsating force of high frequency (u/p is large) acts on the vibrating

body it produces vibrations of very small amplitude and in many cases

the body may beconsidered asremaining immovable in space The

Considering the sign ofthe expression 1/(1 w'2/p2) it is seenthat for

the case w < pthis expression is positive and for o> > p it becomes

thanthat of the natural vibration of

the disturbing force are always in the

same phase, i.e., the vibrating load

the same moment that the disturbing

force assumes its maximum value in

the difference in phase between the

.forced vibration and the disturbing force becomes equal to IT. This

directionthe vibrating load reachesitsupper position This phenomenon

simplependulum AB (Fig. 11) forcedvibrationscan beproducedbygiving

in Fig 11-a, the motionsofthe pointsA and B willbe inthe samephase.

Iftheoscillatory motion ofthe point A has a higher frequency than that

pointsA and BinthiscaseisequaltoTT.

In the foregoing discussion the third term only of the general solution(19) has been considered In applying a disturbing force, however, notonly forced vibrations are produced but also free vibrations given by thefirsttwotermsinexpression(19). Afteratimethelattervibrationswill be

damped out due to different kinds of resistance * but at the beginningof

vibration can be found from the general solution (19) by taking into

consideration the initial conditions Let us assume that at the initial

instant (t = 0) the displacementandthe velocity ofthe vibratingbody are

equal to zero. The arbitrary constants of the solution (19) must then be

andforced vibration proportional to sin ut.

Let us consider the case when the frequency ofthe disturbing force is

very close to the frequencyof free vibrations ofthe system, i.e., co is close

q f . A 2? (co + p)t (co

-p)t

- f ut _ gm ~n = - cog- - gm -

cos- - - ~ - cosco*. (22)V '

Since Aisa small quantity the functionsin A variesslowlyanditsperiod,

equalto 27T/A,is large. Insuchacaseexpression (22) canbe consideredas

18 VIBRATION PROBLEMS IN ENGINEERING

representingyibrations ofa period 2?r/coand ofavariableamplitude equal

Fig 12. Theperiodof beating,equalto 27T/A, increases ascoapproachesp,

1. A loadWsuspendedverticallyon aspring, Fig. 1, produces astatical elongation

forcePsin cot, havingthefrequency5 cycles per sec is actingon the load. Determine

theamplitudeof forced vibration if W 10Ibs.,P = 2Ibs.

Solution. From eq. (2), p = 'V

/

~g/8 8i = X/386 = 19.6 sec."1

. We have also

w = 27T-5 = 31.4sec." 1

. Hence the magnification factor is l/(w2/P2 1) = 1/1.56.

2. Determinethe total displacement of the load Wof the previousproblemat the

3. Determinethe amplitudeof forced torsional vibration of a shaft in Fig 7

pro-ducedbya pulsatingtorqueMsin ut if the free torsional vibration of thesameshafthas

thefrequency/ = 10sec."1

, co = 10?r sec." 1andthe angle of twistproducedbytorque Af,

if acting onthe shaft statically, is equal to 01 of a radian.

Solution. Equationofmotionin this case is (see Art 2)

where<f> is the angle of twistandp2 = k/I. Theforced vibrationis

<p == ~ ~ ~ sin cot == "

sin cot.

/(p2 co2 /c(l co2/p2Notingthat the statical deflection isM/k -0.01andp = 2ir - 10weobtainthe required

amplitude equal to

001

vibrationsa weight Wsuspended on a springcan be used (Fig 14). Ifthepoint of suspension A is immovable and a vibration

in the vertical direction of the weight is produced, the A \

equation of motion (1) can be applied, in which x

denotes displacement of W from the position of

equilibrium Assume now that the box, containing

of suspension A vibrates also and due to this fact FIG 14.forced vibration of the weight will be produced. Let

x\ = asinco, (a)

so that the point of suspension A performs simple harmonic motion of

=*/

20 VIBRATION PROBLEMS IN ENGINEERING

corresponding force inthe springis k(x xi). Theequationof motion of

Thisequationcoincideswith equation (18)forforced vibrationsand wecan

vibrations of the load are damped out and considering only forced

q sin cot a sin cot

22'

It is seen that in the case when co is small in comparison with p, i.e., the

with the frequency of free vibration of the system, the displacement x is

oscillatorymotionasthe pointofsuspensionA does Whencoapproachesp

Consideringnowthecasewhenco isverylarge incomparison withp, i.e.,

the frequencyofvibration ofthebodyto whichtheinstrumentisattached

be considered as immovable in space Taking, for instance, co = lOp we

vibrations of the pointof suspensionA will scarcelybe transmittedto theload W.

recording vibrations Assume that a dial is attached to the box with itsplungerpressingagainst the loadWasshowninFig 209. Duringvibration

of relative motion of the weight W with respect to the box This tude is equalto the maximumvalue of the expression

tions also can be measured by tho same instrument The springs of the

free vibrations of tho weight W both in vortical and horizontal directions

the foundationandoftho bearingsof thesamefrequencywillbe produced

will give the amplitudes of vertical and horizontal vibrations with

suffi-cient accuracy since in this case co/p = 9 and tho difference between the

To got a rocord of vibrations a cylindrical

drum rotating with a constant spood can bo used

If such a drum with vortical axis is attached to

the box, Fig. 14, and a pencil attached to tho

weight presses against the drum, a complete rocord

of the relative motion (24) during vibration will

berecorded Onthis principle various vibrographs

in Fig. 213 and Geiger's vibrograph, shown in Fig. 214 A simple

weight WisattachedatpointA toa beam by a rubberband AC. During

vertical vibrations of the hull this weight remains practically immovable

of vibrations of the weight

22 VIBRATION PROBLEMS IN ENGINEERING

Then the pencil attached to it will record the vibrations of the hull on a

rotatingdrum B To get asatisfactory result the frequencyoffree tions of the weight must be small in comparison with that of the hull of

vibra-theship. Thisrequiresthat the statical elongation ofthe stringAC must

the elongationofthestringunderthestatical actionofthe weightW must

be nearly 3ft. Therequirementof largeextensionsisadefect in thistype

ofinstrument

A device analogous to that shown in Fig 14 can be applied also for

frequency of natural vibrations ofthe weight W must be made very large

instrumentisattached Then pislarge incomparisonwithcoinexpression.(24) and the relative motion of the load W is approximately equal to

oo?2sinut/p2 and proportional to the acceleration x\ of the bodyto which

displacements ofthe load W are usually small and require special devices

forrecordingthem Anelectrical methodforsuchrecording,usedintigating accelerations of vibrating parts in electric locomotives, is dis-

inves-cussed later (see page 459)

PROBLEMS

1. Awheelis rolling along awavysurfacewith aconstant horizontal speed v, Fig 16.

Determine theamplitude of the forced vertical vibrations of the load Wattached to

the axle of thewheelbya spring if the statical deflection of the springunderthe action

irX

equation y =asin- inwhich a = 1 in.andI=36in.

Solution Considering vertical vibrations of the loadWonthe springwefind,from

Dueto the wavy surface the center o of the rollingwheel makes vertical oscillations.Assumingthat at the initialmomentt = the pointof contact of thewheelis at a; =0

TTVi

a = 1 in., <o = = 20*-, p2 = 100. Then the amplitudeof forced vibration is

l/(47r2 1) = .026 in. At the givenspeed v the vertical oscillations of thewheel are

the wheel l/ as great weget o> = 5ir and the amplitude of forced vibration becomes

l/(7r

2

of resonancewhen vv/l p atwhichcondition heavy vibration of the load Wwill beproduced

2. For measuringvertical vibrations of a foundation theinstrument shownin Fig.

equal to 1 in.?

Solution. Fromthe dial readingweconcludethat theamplitudeof relativemotion,

seeeq 24, is 01 in. Thefrequencyof free vibrations of theweight W,fromeq (6), is

from eq 24,is

(30/3.14)2

cab ofa locomotive which makes, by moving up and down, 3 vertical oscillations per

of theweight Wis60per second. Whatis themaximumacceleration of thecabif the

Solution. Fromeq.24wehave

Hencethemaximum vertical acceleration of thecabis

Notingthatp = 27T-60andw =2?r-3, weobtain

aco 2 = .001-4ir 2

(602 -32 = 142 in and

see.-142

24 VIBRATION PROBLEMS IN ENGINEERING

5. Spring Mounting of Machines Rotating machines with some

resultofwhichundesirable vibrations offoundations and noisemayoccur

To reduce these bad effects a spring mounting of machines is sometimes

used Let a block of weight W in Fig 17 represent the machine and P

denote the cent 'fugal force due to unbalance when the angular velocity

the centrifugal force is Pco2 and, measuring the angle of rotationas shown

disturbingforce equal to Pco2

respectively If the

therewill be no motionof the block W and the total centrifugal force will

machine In this way a vibrating system consisting of the block W on

vertical springs, analogous to the system shown in Fig.

1, is obtained

To determine the pulsating vertical force transmitted through the springs

to the foundation the vertical vibration of the block under the action of

the disturbingforce Pco2sin co mustbeinvestigated.*

for forced vibrationsgiven in article 3 and substituting Pco2

It isassumedhere that vibrations are smalland donot effect appreciably the

mag-nitude of the disturbing force calculatedontheassumptionthat theunbalanced weightj>

expression hasbeenobtainedbefore in discussingthe theoryofvibrographs,

of forced vibration dependsonlyonthe valueof the ratio a/p The

abso-lute values of the second factor in expression (a) are plotted against thevalues of w/p in Fig. 18 It is seen that for large values of u/p thesequantities approach unity and the absolute value of expression (a)

W and multiplying it by the spring constant k, we obtain the maximum

pulsating force inthe spring which will be transmitted to the foundation

8 10 12

FIG 18.

1.6 2.0

only if 1 co 2

/p2 is numerically larger than one, i.e., when o> > p V2.

When co is very large in comparison with py i.e., when the machine is

Pp2/k and we have, due to spring mounting, a reduction of the vertical

disturbing force in the ratio p2/or. From this discussion we see that to

reduce disturbing forces transmitted to foundation the machine must be

block Wis smallincomparison with the numberofrevolutions per second

of the machine The effect of damping in supporting springs will be

dis-cussed later (seeArt 10). Tosimplifythe problemwehave discussed hereonly vertical vibrations of the block To reduce thehorizontal disturbingforce horizontal springs must be introduced and horizontal vibrations

must be investigated. We will again come to the conclusion that the

fre-ofvibrationmust be smallin with the numberofrcvo

26 VIBRATION PROBLEMS IN ENGINEERING

lutions per second of the machine in orderto reduce horizontal disturbing

forces.

PROBLEMS

1. Amachineofweight W = 1000Ibs. and making 1800revolutions perminuteis

supportedby four helical springs (Fig 176) madeof steel wire ofdiameter d = Hin.Thediametercorresponding to the centerline of the helix isD = 4in. andthenumber

of coilsn = 10. Determinethemaximum vertical disturbing force transmitted to the

foundationif the centrifugal force ofunbalancefor theangular speedequal to 1 radian per sec isP = I pound

Solution. The statical deflection of the springsunder the action of the load W is

2nDW _ 2.lO-43 -1000 _ .

5" "

fromwhichthe spring constant k = 1000/1.71 =585Ibs per in.and the square of the

circularfrequencyof free vibrationp2 g/8 Kt =225are obtained. By using equation

re-maining unchanged?

3. What magnitude mustthe spring constant inproblem 1 have in order tohave

centrifugal force Poo2?

consists of two discs rotating in a

vertical plane with constant speed

in opposite directions, as shown inFig 19. Thebearingsofthe discs are

be rigidly attached to the structure,the vibrations of which are studied

By attaching to the discs the

with respect to vertical axis mn, the centrifugal forces Po>2

the axis ran.f Such a pulsating force produces forced vibrations of the

*Suchanoscillator is described in apaperbyW.Spath,see V.D.I, vol 73, 1929.

f It isassumedthat the effect of vibrationsonthe inertia forces of theunbalanced

structure which can be recorded by a vibrograph Bygraduallychanging

the speed of the discs the number of revolutions per second at which the

established Assuming that this occurs at resonance,* the frequency of

free vibration of the structure is equal to the above found number ofrevolutionsper second ofthe discs.

mea-suring the frequency of vibrations is known as Frahm's tachometer.This consists ofasystemofsteel stripsbuiltinat theirlowerendsasshown

(a)

the frequencies of any two consecutive strips is usually equal to half a

vibration per second

In figuring the frequency a strip can be considered as a cantilever

quarterofthe weight Wi ofthestrip isadded fto the weightW, thelatter

being concentrated at the end. Then,

(W + 11

ZEI

the period of natural vibration of the strip. In service the instrument

is attached to the machine, the frequency vibrations of which is to be

considered(see Art.9).

tThis instrumentis describedbyF. Lux, E T. Z., 1905, pp 264-387.

28 VIBRATION PROBLEMS IN ENGINEERING

period of one revolution of the machine will be in a condition near

obtained

Instead of a series of strips of different lengths and having different

Thefrequencyof vibration ofthe machine canthen befound byadjustingthe length of the strip in this instrument so as to obtain resonance On

Indicator of Steam Engines. Steam engine indicators are used for

accuracy of the records of such indicators will depend on the ability ofthe indicator system, consisting of piston, spring and pencil, to followexactly the variation of the steam pressure. From the general discussion

with that of the steam pressure variation in the cylinder.

W = 133 Ib is weight of the piston, piston rod and reduced weight

ofotherparts connected with the piston,

s = .1 in. displacement ofthe pencil produced bythe pressure ofone

n = 4 is the ratio of the displacement of the

pencil to that of the

piston

From the condition that the pressure on the piston equal to 15 X .2

in., wefind that the spring constant is:

k = 3.00 : 025 = 120Ibs.in-1

.Thefrequency ofthe free vibrations ofthe indicatoris (seeeq (6))

= 94 persec.

the usual frequency of steam engines and the indicator's record of steam

will bo In the case of

FIG 21.

Locomotive Wheel Pressure on the Rail It is well known that inertiaforces of counter weights in locomotive wheels pro-

duce additional

rpressure on the track This effect

of counterweights can easily be obtained by using

the theory of forced vibrations Let W isthe weight

of the wheel and of all parts rigidly connected to

the wheel, Q is spring borne weight, P is centrifugal

force due to unbalance, co is angular velocity of

the wheel Considering then the problem as one

rail, Fig 21, willbe equal to

_ TF

*" ~

k

'

The period offree vibrations ofthe wheel on the rail isgiven by the

equa-tion f (see eq. (5)).

Micro-Indi-cator) is given in Engineering, Vol 113, p.716 (1922). Symposium ofPapers on

Indi-cators, see Proc. Meetingsof the Inst. Mech.,Eng., London, Jan. (1923).

comparison with the period of vibration of the wheel on the rail, therefore vibrations

ot tne wheel will not betransmitted to the cab andvariations in the compression of

be very

30 VIBRATION PROBLEMS IN ENGINEERING

Now, by usingeq (20), it can be concluded thatthe dynamical deflection

CO

I T \*>

-The pressure on the rail produced by the centrifugal force P will also

increase inthe sameratioandthemaximumwheelpressurewillbe givenby

Fora 100 Ib. rail, a modulus of the elastic foundation equal to 1500 Ibs.

persq in. and W = 6000 Ibs. we will have *

than that calculated statically.

9body As aresult of this assumptionit was found thatin the case offree

vibrations the amplitude of vibrations remains constant, while experience

gradually damped out In the case of forced vibrations at resonance it

vibration problems in better agreement with actual conditions damping

forcesmust be taken into consideration Thesedamping forces may arise

from several different sources such as friction between the dry slidingsurfaces of the bodies, friction between lubricated surfaces, air or fluid

resistance, electric damping, internalfriction duetoimperfectelasticity of

vibratingbodies, etc.

* Timoshenko and M

In the case offriction between dry surfaces the Coulomb-Morinlaw is

usually applied.* It isassumedthatinthe case ofdrysurfacesthe friction

force Fis proportional to the normal component N ofthe pressure acting

materialsofthe bodiesin contactand onthe roughnessoftheir surfaces

motion Thus usually larger values are assumed for the coefficients of

usually assumed also that the

coeffi-cient of friction during motion is

independent of the velocity so that

the point A in the same figure the

This law agrees satisfactorily with

ex-periments in the case of smooth

increase of the velocity as shown in Fig. 22bythe curveAD.]

In thecase offrictionbetweenlubricated surfacesthefrictionforcedoes

lubricantandonthevelocity ofmotion In thecase of perfectly lubricatedsurfaces in which there exists a continuous lubricating film between the

sliding surfacesitcanbeassumedthatfrictionforces are proportional both

to the viscosity of the lubricant and to the velocity. The coefficient of

the straight line OE.

* C A.Coulomb, M('moiresdeMath, etdePhys., Paris1785;see also his"Theorie

desmachinessimples," Paris,1821 A.Morin, Mlmoirespn's.p.div sav., vol. 4,Paris

1833andvol.6, Paris,1935. Fora review of the literatureonfriction, seeR v Mises,

Encyklopadied Math.Wissenschaften,vol. 4, p.153. Forreferences tonewliterature

Techn Mech Vol 1, p 751, 1929.

tThe coefficient of friction betweenthe locomotivewheel and the rail were

inves-"

FIG 22.

32 VIBRATION PROBLEMS IN ENGINEERING

We obtain also resisting forces proportional to the velocity if a body

causes fluid to be forced through narrow passages as in the case of dash

pots.* In further discussion of all cases in which friction forces are portional to velocity we will call these forces viscous damping

pro-In the case ofmotionof bodiesin airor in liquid with larger velocities

a resistance proportional to the square of velocity can be assumed with

tothe velocity canbe discussed inmany cases with sufficientaccuracyby

replacing actual resisting forces by an equivalent viscous damping which is

cycle as thatproduced bythe actual resistingforces. Inthis manner, the

necessary to know for the material of a vibrating body the amount of

considered

8. Free Vibration with Viscous Damping. Consider again the tion ofthe systemshownin Fig. 1 andassume that the vibratingbody W

vibra-encounters in its motion a resistance proportional to the velocity. In

suchcase, instead ofequationofmotion (1), weobtain

W

(a)

ff

The last term on the right side of this equation represents the damping

actinginthe directionopposite to thevelocity. The coefficient c isa stant depending on the kind of the damping device and numerically is

con-equaltothe magnitude ofthe dampingforce when the velocityis equal tounity Dividing equation (a) by W/g andusing notations

(25)

*

See experimentsbyA Stodola, Schweiz. Banzeitung,vol 23,p. 113, 1893.

during recent years. See O Foppl, V.D.I, vol 74, p 1391, 1930; Dr Dorey'spapei