Timoshenko - Vibration Problems in Engineering

VIBRATION

PROBLEMS

IN

ENGINEERING

BY

S.

TIMOSHENKO

Professor of Theoretical andEngineering Mechanics StanfordUniversity

SECOND

EDITIONFIFTH

PRINTING

NEW

YORK

D.

VAN NOSTRAND

_COMPANY,

INC.

250

FOURTH

AVENUE

D.

VAN NOSTRAND

COMPANY,

INC.

All Rights Reserved

This book orany part thereofmaynot

be reproduced in any form without

written permission from the publisher,,

First Published . . . October, 1928

Second Edition . . July,1937

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

In the preparation of the _manuscript for the second edition of the book, the author's desire

was

not only to _{bring the}

_{book up}

to date

_by

including

some

new

material butalso to

make

it

more

suitable for_teaching

purposes.

With

this in view, the first _part of the book

was

entirely

re-written

and

_{considerablyenlarged.}

A

number

of _examples

and

_problems with solutions or with answers were included,

and

in

_many

_places

new

material

was

added.

The

principaladditionsare as follows: In the first _chapteradiscussion

of forced vibration with

_damping

_{not proportional to velocity} is _included,

and

an article

on

self-excited vibration. In the chapteron non-linear

sys-tems anarticleon the

method

ofsuccessive _{approximations}is

added and

it

is

shown

how

the

method

can be used in _discussingfree

and

forced

vibra-tions of _systems with non-linear characteristics.

The

third _chapter is

made

more

complete

by

includinginit_{a general} discussion of_{the equation}

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 of_systems with viscous _damping;

an

article on _stability of motion with an _application in

studying vibration ofa governorof asteam _engine; an articleon _whirling

ofa_{rotating shaft} dueto hysteresis; and anarticleon_{the theory}of

damp-ing vibration absorbers. There are also several additions in _{the chapter}

on torsional

and

lateralvibrations 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,

who

read over thecomplete manuscript

and

made

many

valuable suggestions, and Dr. J. A. Wojtaszak,

who

checked prob-lemsof the first chapter.

STEPHEN

TIMOSHENKO

STANFORD UNIVERSITY,

With

the increase of size

and

velocity in

modern

_machines, the analysis of vibration _problems

becomes more

and

more

important in

mechanical engineering design. Itis well

known

that _problemsof _great

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

vibration of shafts

and

of _{geared systems,} the vibrations of turbine blades

and

turbine_discs,the whirlingof_{rotatingshafts,} 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

theory of vibration.

Only

by

using this theory can the

most

favorable

design proportions be found which will

remove

the _working conditions

of the

machine

as far as possible

from

the critical conditions at which

heavy

vibrations

_may

occur.

In the present book, the fundamentals ofthe theory of vibration are developed,

and

their _application to the solution of technical _problemsis

illustrated

_by

various _{examples, taken,} in

_many

_cases,

from

actual experience with vibration of machines

and

structures in service. In

developing this _book, the author has followed the lectures

on

vibration given

by

him

to the mechanical _engineers of the _Westinghouse Electric

and

_{Manufacturing}

_Company

_during the _{year 1925,}

and

also certain

chapters of his _{previously published}

book on

the _theoryof _elasticity.*

The

contents ofthe

book

in_{general are as} follows:

The

first _chapter is devoted to the discussion of harmonicvibrations

of _{systems with one degree}of freedom.

The

_{generaltheory} of free

and

forced vibration is _discussed,

and

the _application of this _theory to balancing machines

and

vibration-recording instruments is shown.

The

Rayleigh approximate

method

of investigating vibrations of

more

com-plicatedsystems is_{also discussed,}

and

isappliedto the calculationofthe whirling speedsof_{rotating shafts}of variable cross-section.

Chapter

two

containsthe_theoryofthe non-harmonicvibration_of

sys-tems

with one degreeoffreedom.

The

_approximate

methods

for

investi-gating the free

and

forced vibrations of _{such systems} are discussed.

A

particular caseinwhich the_flexibilityof the_systemvarieswiththetimeis consideredin_detail,

and

theresults of this_{theoryare 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 vibration_{of vehicles,}thetorsionalvibrationof shafts, whirlingspeeds

of shafts

on

several supports,

and

vibration absorbers.

Chapterfour contains the _theoryofvibration ofelastic bodies.

The

problemsconsideredare: the_{longitudinal, torsional,}

and

lateralvibrations

of _prismatical bars; the vibration of bars of variable cross-section; the

vibrations _{of bridges,} turbineblades,

and

shiphulls; thetheoryof

vibra-tion of circular_rings,

_membranes,

plates,

and

turbine discs.

Brief descriptions of the

most

_important vibration-recording

instru-ments

which are of use in the _{experimental investigation} of vibration

are _given in_{the appendix.}

The

author

owes

a _verylarge debtof _{gratitude to the}

_management

of

the _WestinghouseElectric

and

Manufacturing

Company,

which

company

made

it _{possible for}

him

to _spend a considerable

amount

of time in the preparationofthe_manuscript

and

to use as_examplesvarious 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

_ways

in _{the preparation} of the

manuscript, particularly Messr. J.

M.

_{Lessells, J.}

_{Ormondroyd, and}

J. P.

Den

_Hartog,

who

havereadoverthe_{complete manuscript}

and

have

made

many

valuable _suggestions.

He

is_{indebted,also,}to

Mr.

F. C.

Wilharm

for_{the preparation}of

draw-ings,

and

to the

Van

Nostrand

Company

fortheir careinthe publication oi the book.

S.

TIMOSHENKO

ANN

ARBOR, MICHIGAN,

CHAPTER

I

PAGE

HARMONIC VIBRATIONS OF SYSTEMS

HA

VINO ONE DEGREE OF FREEDOM

1. FreeHarmonicVibrations 1

2. Torsional Vibration 4

3. ForcedVibrations 8

4. InstrumentsforInvestigating Vibrations 19

5. _Spring Mountingof Machines 24

6. OtherTechnical Applications 26

/V. Damping 30

^78. FreeVibration with ViscousDamping 32

9. ForcedVibrationswithViscous Damping 38

10. _{SpringMounting}of MachineswithDampingConsidered 51

11. FreeVibrations with Coulomb'sDamping 54

12. ForcedVibrations with Coulomb's _DampingandotherKindsof Damping. 57

13. MachinesforBalancing 62

14. General Caseof DisturbingForce 64

v/15. _ApplicationofEquationof_EnergyinVibrationProblems 74

16. _RayleighMethod 83

17. CriticalSpeedofaRotating Shaft 92

18. General Caseof _DisturbingForce 98

19. Effect ofLow Spots onDeflectionof Rails 107

20. Self-ExcitedVibration 110

CHAPTER

IT

VIBRATION OF SYSTEMS WITH NON-LINEARCHARACTERISTICS

21. ExamplesofNon-Linear Systems 114

22. VibrationsofSystemswith Non-linear Restoring Force 119

23. GraphicalSolution 121

24. NumericalSolution 126

25. MethodofSuccessiveApproximationsAppliedtoFreeVibrations 129

26. Forced Non-LinearVibrations 137

CHAPTER

111

SYSTEMS WITH VARIABLE SPRING CHARACTERISTICS

27. _ExamplesofVariable Spring Characteristics 151

28. Discussion of the Equation of _Vibratory Motion with Variable Spring

Characteristics 160

29. Vibrationsinthe SideRodDrive Systemof ElectricLocomotives 167

CHAPTER

IV

PAGE

SYSTEMS HAVING SEVERAL DEGREES OF FREEDOM

30. d'Alembert's Principleandthe PrincipleofVirtual_{Displacements} 182

31. GeneralizedCoordinatesand Generalized Forces 185

32. _Lagrange'sEquations 189

33. The SphericalPendulum 192

34. Free Vibrations. General Discussion 194

35. Particular Cases 206

36. ForcedVibrations 208

37. Vibrationwith Viscous_Damping 213

38. Stability ofMotion 216

39. Whirlingofa_RotatingShaft Caused byHysteresis 222

40. Vibrationof Vehicles 229

41. _DampingVibration Absorber 240

CHAPTER

V

TORSIONAL AND LATERAL VIBRATION OF SHAFTS

42. FreeTorsional VibrationsofShafts 253

43. Approximate MethodsofCalculating Frequenciesof NaturalVibrations .. 258

44. ForcedTorsionalVibrationofaShaftwithSeveralDiscs 265

45. TorsionalVibrationof DieselEngine Crankshafts 270

46. _Damperwith SolidFriction 274

47. LateralVibrationsofShaftson

_Many

_Supports 277

48. GyroscopicEffectsonthe CriticalSpeedsofRotatingShafts 290

49. EffectofWeightofShaftandDiscsontheCriticalSpeed 299

50. Effectof Flexibility ofShafts onthe BalancingofMachines 303

CHAPTER

VI

VIBRATION OF ELASTIC BODIES

51. _{Longitudinal Vibrations}ofPrismaticalBars 307

52. VibrationofaBar withaLoadat theEnd 317

53. TorsionalVibrationofCircular Shafts 325

54. LateralVibrationofPrismaticalBars 331

55. The EffectofShearingForce andRotatoryInertia 337

V_{56. Free VibrationofaBarwith}

Hinged Ends 338

Jbl. OtherEndFastenings 342

>/58. ForcedVibrationofa

Beam

with SupportedEnds 348

59. Vibrationof_Bridges 358

60. Effect ofAxial ForcesonLateral Vibrations 364

61. Vibrationof BeamsonElasticFoundation 368

62. RitzMethod 370

63. Vibration ofBarsofVariableCross Section 376

PAGE

65. Vibration ofHullsof_Ships 388

06. Lateral_ImpactofBars 392

67. LongitudinalImpactofPrismaticalBars 397

*8. VibrationofaCircularRing 405

'69. VibrationofMembranes 411

70. VibrationofPlates 421

71. VibrationofTurbineDiscs 435

APPENDIX

VIBRATION MEASURING INSTRUMENTS

1. General 443

2. _{Frequency Measuring Instruments} 443

3. The MeasurementofAmplitudes 444

4. SeismicVibrographs 448 5. _Torsiograph 452 6. Torsion Meters 453 7. Strain Recorders 457 AUTHOR INDEX 463 SUBJECT INDEX 467

HARMONIC

VIBRATIONS

OF SYSTEMS

HAVING

ONE

DEGREE

OF

FREEDOM

1. Free

Harmonic

Vibrations. If an elastic _system, such as a loaded

beam, a twisted shaft or a deformed spring, bedisturbedfrom its_position

of_equilibrium

_by

_{an impact} or

_by

the sudden application

and

removalof

an

additional force, the elasticforces ofthe

member

inthe 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

modes. For instance, a stringor a

beam

while vibrating

may

assume the

different _{shapesdepending}

on

the

number

of nodessubdividing the length

of the

member.

_{In the simplest} cases _{the configuration} of _{the vibrating}

system can be determined

by

one quantityonly. Such systems are called

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 possible

and

the

mass

ofthe 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

an

_impulse ora sudden_application

and

removal

of

an

external force vibrations of the _{system can} be produced.

Such

vibrations which are maintained

by

the elastic force in _{the spring alone} are called free or

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

moving

body

are known.

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

spring. This quantityis_{called spring constant.} Ifthe load ismeasured in

pounds

and

extensionininchesthe springconstant willbeobtainedinIbs.

perin.

The

static deflection of_{the spring} underthe action of the weight

Denoting avertical displacementof_{the vibrating weight} fromits _position

of_equilibrium

_by

x and _considering this_displacement as positive ifitis in

a

downward

direction,the expressionforthe tensileforce inthe spring

cor-responding to

_any

_{position of the weight} becomes

F =

W

+

kx. _(a)

Inderiving the differential_equation of motion

we

willuseNewton's prin-ciple statingthat the product ofthe

mass

of a _particle

and

itsacceleration

is _equaltotheforce_actingin 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

is_given

_by

the second derivative

of _{the displacement x with respect to}time

and

will be denoted

_by

x] the forces _acting

on

_{the vibrating}

_body

are the gravityforce

_W,

_acting

down-wards,

and

the force

F

of _{the spring (Eq.} a) which, forthe position of the

weight indicated in Fig. 1, is actingupwards.

Thus

thedifferential

equa-tion ofmotion inthe caseunderconsiderationis

x

=

W-(W

+

_{kx). (1)}

a

This _{equation holds} for

_any

position of the

body

W.

If, for instance, the

body

inits_vibratingmotiontakesa positionabovethe position of

equilib-rium

and

suchthat_{a compressivc}force inthe_springis_producedthe

expres-sion (a)

becomes

negative, and both terms

on

the right side of eq. (1)have

the

same

sign.

Thus

in this case the force in _{the spring} is added to the

gravity force as it should be.

Introducing notation

tf

=

^

=

4-

M

P

_W

.,'

(2)

differentialequation (1) can berepresented inthe followingform

x

+

p2x

=

0. ₍₃₎

This _equation will be satisfied if

we

_{put x}

=

C\ cos pt or x

=

2 sin pt,

where C\ and 2 are arbitrary constants.

By

_adding these solutions the

general solution of _{equation (3)} will be obtained:

x

=

Ci cos pt

+

2sinpt. (4) Itisseen that the verticalmotionofthe weight

W

has a vibratory

charac-ter, since cos pt

and

sin pt are periodic functions which repeat themselves

each time after

an

interval oftime r such that

p(r

+

t)-

pt

=

2*. ₍₆₎

This interval of time is called the _{period of vibration.} Its _magnitude,

fromeq. (6), is

^

/ \

r

-

-

_(c)

or,

by

using notation (2),

(5)

kg * g

It is _{seen that the period} ofvibration _depends only

on

the_magnitudes of

the weight

W

and

of_{the spring constant k}

and

is_independentofthe

mag-nitudeofoscillations.

We

cansay alsothat _{the period}ofoscillationofthe

suspended weight

W

isthe

same

as that ofa _{mathematical pendulum,}the

lengthofwhichis_equaltothe staticaldeflection 5^. Ifthe statical

deflec-tion 8st is determined _{theoretically or experimentally the period} r can be

calculated from _{eq. (5).}

The

number

of cycles per unit time, say per second, is called the

fre-quency of vibration. Denoting it

_by

_/

we

obtain

'-;-*>

<6>

or,

by

substituting g

=

386in. _persec.2

and

_expressing 8atin inches,

/

=

3.127

-v/_* cycles per second. (6') Qs t

A

vibratory motion represented

by

equation (4) is called a harmonic

motion. In ordertodetermine the constants_{of integration} Ci and C2, the

initial conditions

must

be considered. _{Assume, for instance,} that at the

initial

moment

_(t

=

_{0)the weight}

W

hasadisplacementXQfromits_position

of _equilibrium

and

that its initial velocity is XQ. _Substituting t

=

in

equation (4)

we

obtain

XQ

=

Ci. (d)

Taking

now

the _{derivative of eq. (4)} with_respecttotime

and

_substituting

in thisderivative t

=

_0,

we

have

io

_r

(\

Substitutingin_{eq. (4)}the valuesof the constants _(d) and (e),thefollowing

expressionfor_{the vibratory}motion of_{the weight}

W

will be obtained: , , -Ml .

x

=

xo cos pt -i sinpt.

P

(7)

It isseen that in this case the vibration consists of

two

_parts; avibration

which is _{proportionalto cos}

ptand depends on the initial _displacement of

the _{system and}another which is_proportional to_{sin pt and depends onthe}

FIG. 2. *

initial _velocity xo.

Each

of _{these parts can be represented}

graphically, as

shown

in Figs. 2a and 2b,

by

plotting the displacements against the time.

The

total _{displacement x} of the oscillating weight

W

at _any instant t is obtained

_by

_{adding together the ordinates} of the

two

curves, (Fig.2a and

Fig. 2b) for that instant.

Another

method

of _representing vibrations is

_{by means}

of _rotating

vectors. _Imagine a vector

_OA,

_{Fig. 3,} of _magnitude rr() _rotating with a

constant angularvelocityp around afixed _point,0. This_velocityiscalled circular_frequencyof vibration. Ifatthe initial

_moment

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

is_equal to xo cospt

and

represents the firsttermof_{expression (7). Taking}

now

another vector

OB

equal to_xo/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

by

adding the projections on the x

axis ofthe two perpendicular vectors ~OAand

OB,

rotating withtheangular

velocity p.

The same

result will be obtained

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

oe

=

and

the angle which it

makes

with thex axisis

pt

-

a,

where

Fio. 3.

Equating the projection of this vector on the x axis to _{expression (7)}

we

obtain \M*<r

+

( : ) cos (pi a) *

_\P

_/ icospt -\ sin_{pt. (8)} P

It isseen that in this

manner we

added _{together the} two _simple harmonic

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

The

result of this addition is a simple harmonic _{motion, proportional} to

cos _{(pt a),} which is _represented

_by

_Fig. 2c.

The

maximum

ordinate of

this _{curve, equal} to

V

jar

+

(x^/p)'

2

, represents the

maximum

displace-ment

of _{the vibrating}

_body

from the position of equilibrium and iscalled

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,

is behind the vibration represented

by

the _{curve, Fig. 2a,}

and

_{the angle}

a

iscalled_{the phasedifference}of these

two

vibrations.

PROBLEMS

1. The_weight

W

=

30Ibs.is_{verticallysuspended on a}steelwireof_lengthI 50in.

andof cross-sectionalarea

A

0.001 in.2

. Determine the_frequencyof freevibrations

of the _weight if the modulus for steel is

E

=

30-106

Ibs. _per sq. in. Determine the

amplitudeof thisvibrationiftheinitial_displacementXQ

=

0.01 in. andinitial _velocity

xo

=

1 in._persec.

Solution. Static _{elongation of^the wire} is 8st

=

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

Vz

2

+

(Wp)

2

=

_V(0.01)2

+

_[l/(27r-14)]2

=

.01513in.

2. Solve the previous problem assuming that instead of a vertical wire a helical

spring is usedfor _suspensionof the load W. Thediameterof the cylindrical surface

containing the centerlineof the wireformingthe spring is

D

=

1 in., the diameterof

the wire d

=

0.1 in., thenumberofcoilsn

=

20. Modulus ofmaterialof the wire in

shear

G =

12 -106

Ibs. _per sq. in. In what proportion

willthefrequencyofvibration_{be changed} if the _spring

has 10 coils, the other characteristics of the spring

remainingthesame?

3.

A

load

W

is _{supported by} a beam of length lt

Fig. 4. Determinethe spring constantandthefrequency

FIG. 4. _{of free} vibration of the load in the vertical direction

neglecting the massofthe beam.

Solution. Thestatical deflection ofthebeamunderloadis

-c)2

Herecisthe distanceofthe loadfromtheleftendofthebeamandEltheflexural

rigidity of thebeam in the vertical plane. Itis assumedthat thisplane contains one

of thetwoprincipal axesofthe cross sectionofthe_beam,sothatverticalloadsproduce onlyvertical deflections. Fromthedefinition the spring constantin thiscase is

ZIEI

Substituting B9tin eq. (6) the requiredfrequency can be calculated. Theeffect ofthe

massofthebeamonthefrequencyofvibrationwillbediscussedlater, see Art.16. 4.

A

load

W

isverticallysuspended ontwospringsasshownin Fig. 5a. Determine

spring constantsofthetwosprings arekiandfa. Determinethefrequencyofvibration

ofthe load

W

ifit issuspended ontwoequal springs asshown in Fig. 56.

Solution: In the case shown in Fig. 5a the statical

deflec-tionofthe load

W

is

_W

W

.

__^

The resultant spring constant is/Cifa/(fa -ffa). Substituting

dgt in eq. (6), the frequencyof vibrationbecomes

Inthe caseshownin Fig. 56

W

2~k and

+

=

_L /20*.

2*V

W

FIG. 5.

6. Determinethe period ofhorizontalvibrationsofthe frame,shownin Fig.6,

sup-portingaload

W

applied at thecenter. The massoftheframe should be neglected in

this calculation.

Solution.

We

beginwith astatical_problemanddetermine the horizontaldeflection

6oftheframe whicha horizontalforce

H

actingatthe pointofapplicationofthe load

W

will_produce. Neglectingdeformationsdue totension andcompressioninthemembers

FIG. 6.

and considering only bending, the horizontal bar

AB

is bentby two equal couplesof

magnitude Hh/2. Thenthe angleaofrotationofthejoints

A

and

B

is

Hhl

Consideringnowtheverticalmembersoftheframeas cantileversbentbythe horizontal forces///2,the horizontaldeflection6willconsist oftwoparts,oneduetobendingofthe

cantileversandthesecondduetothe rotationaofthejoints

A

and

B

calculated above. Hence

HhH

=

^-Vl

4- -l

-\

~

QEI

+

12#/i "" QEI _{\ '2hlJ} '

The_{spring constant}insuchcaseis

Substitutingin eq. (5), weobtain

Wh*[ 1

+

HA

2hlJ

QgEI

Ifthe rigidity ofthe horizontal memberis_{large in comparison with}the rigidity ofthe verticals,thetermcontaining theratioI/I\issmallandcan beneglected. Then

IWh*

r==27r

andthefrequencyis

6. Assumingthat the load

W

in Fig.1_{represents thecage}ofanelevatormovingdown

withaconstant velocityvandthe springconsists ofasteel cable,determinethe

maximum

stress in the cable if duringmotion theupper end

A

ofthe cableis _{suddenlystopped.}

Assume that the weight

W

=

10,000Ibs., I 60 _ft., the cross-sectional area of the cable

A

2.5 sq.in.,modulusof elasticity ofthe cable

E

=

15-106

Ibs._persq.in., v

=

3 ft. _persec. Theweightofthe cableis to be neglected.

Solution. _During the uniform motion ofthe cage the tensile force in the cableis

equalto

W

=

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

6at =

Wl/AE -

.192 in. Due to the velocityv the cage will not stop suddenly and

willvibrateonthe cable. Counting time fromthe instantofthe accident, the

displace-mentofthe cagefromtheposition ofequilibrium at that instantiszeroanditsvelocity

is v. From_eq.(7)weconclude that theamplitudeofvibrationwillbe_equalto v/p,where

p

=

vg/bst 44.8 sec."

1 _and

v

=

36 in. _per sec. Hencethe

maximum

elongation of

the cable is 5d = S8t

+

v/p

=

.192 -f-36/44.8

=

.192-f .803

=

.995 in. and the-

maxi-mum

stress is _{(10,000/2.5) (.995/.192)}

=

_20,750 Ibs._per sq. in. It isseen that dueto

thesudden stoppageofthe upper endofthe cable thestress inthe cable increased in thiscaseaboutfivetimes.

7. Solve the previous problem assuming that a spring having a spring constant

k

=

2000Ibs._perin.isinsertedbetweenthelowerendofthe cableandthecage. Solution. Thestatical deflection in thiscaseis_5^

=

.192 -f- 5 = 5.192in. and the

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

.803 A/5.192/.192. The maximum dynamical deflection is5.192 -f .803 _Vs.192/,192

maximum

dynamicalstress is (10,000/2.5)1.80

=

7,200Ibs. _persq. in. Itisseen that

byintroducing the spring a considerablereduction in the

maximum

stress isobtained.

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

end of which a circular horizontal disc is _attached,

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

Fig.7. If_{a torque} is_appliedin_{the plane} ofthe disc

]* {

and

_{then suddenly removed,} free torsional vibration

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 the

same

radius

when

the disc is at rest.

As

_{the spring constant} in

this case

we

_{take the torque k}whichis_necessary to

~

'

produce

an

angle of twist of the shaft _equal to one

radian. In thecase ofacircularshaft of_length Iand diameter d

we

obtain

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

immovable

axisstatesthatthe

moment

of inertia ofthe

_body

with respect

to this axismultipliedwiththe angularaccelerationis_equaltothe

moment

of the external forces acting on the

body

with respect tothe axis of

rota-tion. In our case this

moment

is _equal

and

_opposite to _{the torque} k<p

acting

on

the shaft

and

the equation ofmotionbecomes

lip

=

k<p (a)

where 7 denotes the

moment

of inertia ofthe disc with_{respect to} the axis

of rotation, which inthis case coincideswith the axis ofthe _shaft,

and

is

the angularacceleration ofthe disc. _{Introducing the notation}

P2

=

*, (10)

the equationofmotion (a) becomes

+

p2

?

=

0. ₍₁₁₎

Thisequation has the

same

formas_{eq. (3)} of_{the previousarticle,} henceits

solutionhas the

same

form as solution (7)

and

we

obtain

<f>

=

<pocospt

+

sinpi, (12)

where

w

and

>oare the angular displacement

and

angularvelocity respec-tively ofthediscattheinitialinstantt 0. _Proceedingasin_{the previous}

article

we

concludefromeq. (12) that the periodoftorsional vibrationis

T

=

=

2T

J-p

*k

(13)

P *_K

and

its_frequency is

/=

T

=

iv/'

(U)

In the case of a circular disc ofuniform thickness and of_{diameter D,}

where

W

is_{the weight} ofthe disc. _Substitutingthisin _{eqs. (13)} and _(14),

and

_{using expression (9),}

we

obtain

1WDH

It

was assumed

in our discussion that the shaft has a constant

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}

M

t is applied to this shaft

the angleof twist_produced is

Ut~WJ-l\, U~J.,M.I.

7

It is seen that the angle of twist of a shaft with

two

diameters d\ and d%

isthe

same

asthatofashaft ofconstant diameterd\

and

ofareduced_length

L

given

by

the equation

The

shaft of_length

L

and diameterd\ has the

same

spring constant asthe givenshaft of

two

differentdiametersandisanequivalent shaft in this case.

In_general if

we

havea_{shaft consisting ofportions with} different

portion ofthe shaft of_length l_n and ofdiameterdn

by

a portionofashaft

ofdiameterdand of_lengthIdeterminedfromthe equation

(15)

The

resultsobtainedforthecase

shown

in_Fig. 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

may

beencountered very ofteninmachine _design.

A

propeller shaft with

the propeller

on

one end andthe engine on the otheris_{an example} ofthis

kind.*(jf

two

equal and opposite twistingcouples are _appliedat the ends

of the shaft in Fig. 8 and then_{suddenly removed,} torsionalvibrations will

be produced during which the masses at the ends are _{always rotating in}

opposite directions, f

From

this fact

itcanbe concludedatonce thatthere

is acertainintermediate crosssection

mn

of the shaft which remains

im-movable

during vibrations. This

cross section is called the nodalcross

section,

and

itspositionwillbe found

from the condition that both por-tions of the shaft, to the right and to the left of thenodal cross section,

must

_{have thejsame} period of

vibra-tion, sinceotherwise the condition that the masses at the ends always are

rotating in opposite directions will not be fulfilled.

Applying eq. (13) to eachof the two portions ofthe shaft

we

obtain

or _(c)

where k\ and k% are the _springconstantsforthe left and forthe right

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

*_Thisis

the caseinwhichengineersforthefirsttimefounditof practicalimportance

togo into investigationof vibrations, see II. _{Frahm, V.D.I., 1902, p.} 797.

fThis follows from theprinciple of momentofmomentum. Attheinitial instant

themoment of

momentum

ofthe twodiscswithrespect to theaxis of the shaftiszero

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

axisiszero (friction forcesareneglected). Theequality to zeroofmomentof

inversely proportionalto the lengths of the corresponding portions of the

shaft

and

fromeq. (c) follows

a /2

and, sincea

+

b

=

Z,

we

obtain

, Z/2 ,/,

Hi

,

/I

+

/2 /I

+

*2

Applying

now

totheleft_portionoftheshaft eqs. (13) and ₍₁₄₎

we

obtain

,~

(d)

From

theseformulae the periodand_{the frequency}of torsionalvibrationcan

be calculated provided the dimensionsofthe shaft, the

modulus

G

and

the

moments

of inertia ofthe massesatthe ends are known.

The

mass

ofthe

shaft is _neglected in our _present discussion

and

its effect

on

the period of

vibration willbeconsidered later, seeArt. 16.

It can beseen (eq.d) thatifoneofthe_{rotatingmasses has a very}large

moment

of inertia in_comparison with the other the nodalcross sectioncan

betaken atthelargermass andthesystemwith

two

masses (Fig.8) reduces

to that with one

mass

(Fig. 7).

PROBLEMS

1. Determinethefrequencyof torsionalvibrationofashaftwithtwocirculardiscs of uniformthickness at the ends, Fig.8, iftheweightsof thediscsare

W\

=

1000Ibs.

and

Wz =

2000Ibs.and theirouterdiametersareD\

=

50in. and Dz

=

75in.

respec-tively. Thelengthof the shaftis I

=

120 in. andits diameter d

=

4in. Modulus in

shear

G

12-10*Ibs. _persq.in.

Solution. Fromeqs.(d)the distanceofthenodalcrosssectionfromthe largerdiscis

120-1000-502

120

=

21.8in.

1000-502

₊

2000-752 ₁

₊

_4.5

Substitutingin eq. (6)weobtain

1 7r-386-4<.12-106

.

f

=

=

9'8 O8Clllatlons

2.Inwhat_proportionwillthe frequencyofvibrationofthe shaft consideredin the previousproblemincreaseif_{along a length}of64in.thediameterofthe shaftwillbe

in-creasedfrom 4in.to 8in.

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

lengthof4in. diametershaft. Thusthe lengthof_{the equivalent shaft}is4

+

56

=

60

in.,whichis_{only one-half}ofthe lengthofthe shaft consideredin_{the previous}problem.

Since the _frequency of vibration is _{inversely proportional} to the square root of the

lengthof the shaft (see eq. 17), weconcludethat as theresult of the reinforcementof

the shaftits_frequencyincreasesintheratio

V

2: 1.

3.

A

circularbarfixedattheupper end andsupporting acircular discat the lower

end _{(Fig. 7)} has _{a frequency} of torsional _{vibration equal} to_/

=

10 oscillations per

second. Determinethemodulusinshear

G

if_{the length}ofthe barI

=

40in.,its

diam-eterd

=

0.5in.,theweightofthedisc

W

-

10_Ibs.,anditsouterdiameter

D

=

12 in.

Solution. Fromeq. (b),

G

12 -10

6

Ibs. _persq.in.

4. Determinethefrequencyofvibrationofthering, Fig.9,abouttheaxis0,

assum-ing that the centerof_{the ring}remainsfixedandthat rotationoftherimisaccompanied

Fia. 9.

by somebendingofthespokesindicatedinthe_figurebydottedlines. Assumethat the

totalmassofthe_ringis_{distributed along the center}lineoftherimandtake 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 tangentstothe deflection curvesof the spokes

have radial directions at the rim. The total _weightof _{the ring}

W

and the flexural

rigidity

B

of_spokesare_given.

Solution. _{Considering each spoke}asa cantilever of_lengthr,Fig. 96,attheendof

which a shearing force

_Q

and _{a bending} moment

M

are actingand _{using the} known

formulasfor bendingof a _{cantilever, the following expressions}forthe slope <f>and the

deflection r<pattheendareobtained

Qr2

_Mr

9 from which 2B

M

=

Qr

^Qr*

35 2B<t>

Mr

2 2B '

If

_Mt

denotesthe_{torqueapplied to the}rimwehave

-Thetorque required toproduce an angleof rotation of therim _{equal to}oneradian is

the spring constantandis_{equal to}k

=

16B/r. Substitutingin eq. (14),weobtain the

requiredfrequency

1 /16B 1 /1_60S

3. Forced Vibrations. In the

two

_previous articles free vibrations of

systems withone degree offreedom have been discussed. Let us consider

now

the case

when

inadditiontotheforce of_gravity

and

totheforce inthe spring (Fig. 1) there is _acting on the load

W

a _{periodical disturbing force}

P

sinut.

The

_period of this force is r\

=

2?r/co

and

its frequency is

/i

=

w/27T. Proceeding asbefore (see p. 2)

we

obtain the following

differ-ential equation

W

1L_'i

=

_W

-

_(W

₊

kx)

+

P

sinut, (a)

g

or,

by

usingeq. (2)

and

notation

we

obtain

x

+

_p2x

=

_qsincot. ₍₁₈₎

A

particular solution of this equation is obtained

_by

_assuming that x is proportional to sino^, i.e.,

by

taking

x

=

A

sinwt, (c)

where

A

is a constant, the magnitude of which

must

be chosen so as to

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

we

find

A

=

Thus

the required particular solutionis a sin ut

x

=

-p* M*

Adding

to this particular solutionexpression (4), representing thesolution

oftheeq. (3) for freevibration,

we

obtain

x

=

Ci cos pt

+

C

2_{sin pt}

+

Q '

8m

"f- (19) p* or

This expression contains

two

constants of integration and _represents the generalsolution oftheeq. (18). Itisseenthatthissolution consists of

two

parts, the first

two

terms _represent free vibrations which were discussed before and the third _{term, depending on the disturbing force, represents}

theforced vibration of the system. It is seen thatthis latervibration has the

same

_period

n

=

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

is _equal to the numerical value of _{the expression}

-JL_

.

L

_J

₍₂₀₎

p

2

-

a/2 k 1

-

co2_/7>2

The

factor

_P/k

is the deflection which the

maximum

disturbing force

P

would produceif _{actingstatically}

and

the factor 1/(1

w

2

/p2) takes care

ofthe _dynamical actionof this force.

The

absolute value of thisfactor is

usually called the_{magnificationfactor.}

We

seethat it _dependsonly

on

the

1.0 1.2 1.4- 1.6 1-8

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 the

values 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

the frequency of free vibration, the magnification factor is _{approximately}

unity,

and

deflectionsareaboutthe

same

asinthe case ofastaticalaction ofthe force P.

When

the_{ratio co/p approachesunity the magnification} factor

and

the

amplitude of forced vibration _rapidly increase and

become

infinite for

co

=

_p,

i.e., forthe case

when

the frequency ofthe disturbing forceexactly

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 pulsating}force actson_{the vibrating}

vibration _{increases indefinitely}providedthereisno damping. In _practical

problems

we

always have

damping

the effect ofwhich onthe amplitude of

forced vibration will be discussed _{later (see} Art. _9).

When

the _frequency of the _disturbing force increases

_beyond

the

frequency of free vibration _{the magnification} factor _again becomes finite.

Its absolute value diminishes with the increase of the ratio _co/p

and

approacheszero

when

this ratiobecomes_verylarge. This

means

that

when

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 as_remaining

immovable

_{in space.}

The

prac-tical _{significance of}this fact will be discussed in the next article.

Considering the sign ofthe expression 1/(1 w'2_/p2) itis seenthat for

the case

w

<

pthis _expression is _positive and for o>

>

_p

it becomes

nega-tive. Thisindicates that

when

the

fre-quency of _{the disturbing} force is less

thanthat of the natural vibration of

the _system the forced vibrations

and

the disturbing force are _always in the

same

phase, i.e., the vibrating load

(Fig. 1) reaches its lowest position at

the

same

moment

that the disturbing

force assumes its

maximum

value in

FIG. 11. a

downward

direction.

When

co

>p

the difference in _phase between the

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

means

that at the

moment when

theforceisa

maximum

in a

downward

directionthe vibrating load reachesits_{upper position.} This

_phenomenon

can be illustrated

_by

the _{following simple experiment.} In thecase of a

simple

pendulum

AB

(Fig. 11) forcedvibrationscan beproduced

by

giving

an oscillating motion in the horizontal direction to the pointA. Ifthis

oscillating motion has a frequency lower than that of the

_pendulum

the

extreme positions of the

pendulum

during suchvibrationswillbeas

shown

in Fig. 11-a, the motionsofthe _points

A

and

B

willbe inthe

same

_phase. Ifthe_oscillatory motion of_{the point}

A

_{has a higher frequency} than that

of the

_pendulum

the extreme positions ofthe

pendulum

during vibration

willbeas

shown

in Fig. 11-6.

The

phase difference of the motions of the

points

A

and

B

inthiscaseis_equaltoTT.

In the above discussion _{the disturbing} force

was

taken _proportional

to sinut.

The same

conclusions will be obtained if cos_o>, instead of

In the foregoing discussion the third term only of _{the general} solution (19) has been considered. In applying a disturbing force, however, not only forced vibrations are produced but also free _{vibrations given}

_by

the first

two

termsin_{expression(19).} Afteratimethelattervibrationswill be

damped

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

motion they

may

be _{of practical} importance.

The

amplitude of the free

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 displacement

and

the velocity ofthe vibrating

body

are

equal to zero.

The

_arbitrary constants of the solution (19)

must

then be

determined in such a

manner

that fort

=

x

=

and

x

=

0.

These conditionswillbe satisfied

_by

_taking

r

n

r

Ci

=

l), C-2

=

p2 co2

Substituting in_{expression (19),}

we

obtain

(21)

<l ( CO .

\

x

=

;I

sm

ut sin pt )

/r co- \ p /

Thus

themotion consists of

two

_parts, free_{vibration proportional} to sin_pt

and

forced _{vibration proportional} to sinut.

Let us consider the case

when

the frequency ofthe disturbing force is

very close to _{the frequency}of free vibrations of_{the system,} i.e., cois close

to p. Using notation

p co

=

_2A,

whore

A

is _{a small quantity,} and _neglecting a small torm with the factor

2A/p,

we

represent expression (21) inthe following form:

q f . . .

A

2? (co

+

p)t . (co

-

p)t

-

f

_u

t

_

gm

~n

=

-

cog

-

-

gm

-

-p

2 co2

sin

A

(co

+

p)t qsin

M

/f

^^

cos- -

-

~

-

_cos

co*. _{(22)V '}

p2

-

co2 2 2coA

Since

A

isa small quantity the functionsin

A

varies_slowly

and

its_period, equalto 27T/A,is_large. Insuchacase_{expression (22)} canbe consideredas

*

representingyibrations ofa period 2?r/coand ofavariableamplitude equal

to qsin

_A/2wA.

This kind of vibration is _{called beating} and is

shown

in

Fig. 12.

The

_{periodof beating,equalto 27T/A, increases as}co_approachesp,

FIG. 12.

i.e., as

we

approach the condition of resonance. For limiting condition

co

=

_p

we

can _{putin expression (22)}

_A,

instead ofsin

A

and

we

obtain

>

X

=

COSwt. ₍₂₃₎

The

_amplitude ofvibration_{in eq. (23) increases indefinitely} with the time

as

shown

in _Fig. 13.

FIG. 13.

PROBLEMS

1.

A

load

W

suspendedverticallyon aspring, Fig. 1, produces astaticalelongation

force

P

sincot, havingthefrequency5cyclespersec. isactingon theload. Determine

theamplitudeofforced vibrationif

W

10Ibs.,

P

=

2Ibs.

Solution. From _eq. (2), p

=

'V

/

~g/88i

=

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.

Deflection _{produced by}

P

if _{acting statically}is 0.2 in. and the amplitude of forced

vibrationis 0.2/1.56

=

0.128in.

2. Determinethe totaldisplacement ofthe load

W

of_{the previous}problemat the

instant t

=

1 sec. if at the initial moment (t

=

0) the load is at rest in _equilibrium

position.

2 31 4

Answer, x

-

~

-(sin lOx 1-sin 19.6)

=

+

.14inch.

1.56 19 6

3. Determinethe amplitudeof forced torsional vibration ofa shaft in _{Fig. 7}

pro-duced_{bya pulsatingtorque}

M

sinutifthefreetorsionalvibrationofthesameshafthas

thefrequency/

=

10sec."1

,co = 10?rsec." 1_and

the angleoftwist_{producedbytorque}Af,

if_acting onthe shaft statically, is_equal to .01 of aradian. Solution. Equationofmotioninthiscaseis_{(see Art. 2)}

where<f>isthe angleoftwistandp

2

=

k/I. Theforced vibrationis

M

_{_}

M

<p== ~ ~ ~ sincot == " sincot. /(p2 co2) /c(l co2_/p2)

Notingthatthestaticaldeflectionis

_M/k

-0.01andp

=

2ir-₁₀we_{obtainthe required}

amplitude equal to

001

=

0.0133 radian.

4. Instruments for _{Investigating Vibrations.}

For

_measuring vertical

vibrationsa weight

W

suspended on a springcan be used (Fig. 14). Ifthe

point 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

the _{suspended weight}

W

y is attached to a

body

per-formingvertical vibration. In such a case _{the point}

of _suspension

A

vibrates also

and

due to this fact FIG. 14. forced vibration of _{the weight} will be _produced. Let

us assume that vertical vibrations of the box are _given

_by

equation

x\

=

asinco, (a)

so that the point of _suspension

A

_{performs simple} harmonic motion of

amplitudea. In such case the elongation of _{the spring} isx x\

and

the

corresponding force in_{the spring}is _{k(x xi).}

The

_equationof motion of

the _{weight then becomes}

W

x

=

k(x xi),

or,

by

substitutingfor x\ its_{expression (a)}

and

_{using notations}

we

obtain

x

+

p2#

=

q sinco.

Thisequationcoincides_{with equation (18)}forforced vibrations

and

we

can

applyhere the conclusionsof _{the previous}article.

_Assuming

that the free

vibrations of the load are

_damped

out and considering only forced

vibra-tions,

we

obtain

q sin cot a sin cot

x

=

22

=

~

₂₂

'

It is seen that in the case

when

co is small in _{comparison with} p, i.e., the

frequencyofoscillationof_{the point} of_suspension

A

is smallin _comparison

with the frequency of free vibration of _{the system, the displacement x} is

approximately equal to x\

and

the load

W

_{performs practically} the

same

oscillatorymotionasthe pointofsuspension

A

does.

When

co_approaches

_p

the denominator in _expression (c) approaches zero and

we

approach

reso-nance condition at which

_heavy

forced vibrations arc _produced.

Considering

now

thecase

when

cois_{verylarge incomparison withp,}i.e.,

the frequencyofvibration ofthe

_body

to whichtheinstrumentisattached

is _veryhighin _{comparisonwith frequency} offree vibrations ofthe load

W

the_amplitudeofforced vibrations (c) becomessmallandthe weight

W

can

be considered as

immovable

in space. Taking, for instance, co

=

_lOp

we

find that the amplitude of forced vibrations is_{only a/99, i.e.,} in this case

vibrations of _{the point}of _suspension

A

will scarcelybe transmittedto the

load

W.

This fact is utilized in various instruments used for _measuring

and

recording vibrations.

Assume

that a dial is attached to the box with its plungerpressingagainst the load

W

as

shown

in_{Fig. 209.} Duringvibration

of relative motion of _{the weight}

W

with _{respect to} the box. This

ampli-tude is _equalto the

maximum

value of _{the expression}

V

1

A

x x\

=

asin

wU

11 VI co2_/p2 /

=

asinut __ g 2 * (24)

When

p

issmallin_comparisonwith

w

thisvalue is_verycloseto_the

ampli-tude a of _{the vibrating}

_body

to which the instrument is attached.

The

numerical values of the last factor in _{expression (24) are plotted against}

theratio

_u/p

_{in Fig.} 18.

The

instrument described has _{proved very} useful in

_power

_plants for

studying vibrations of turbo-generators. Introducing in addition to vertical also horizontal springs, as

shown

in Fig. 209, the horizontal vibra-tions also can be measured

by

tho

same

instrument.

The

_{springs of} the

instrument are _{usually chosen} in such a

manner

that the frequencies of

free vibrations of tho weight

W

both in vortical

and

horizontal directions

are about 200 _{por minute.} If _{a turbo-generator}

makes

1800 revolutions

per minute itcan be oxpoctedthat,owingto

some

_unbalance, vibrations of

the foundation

and

oftho bearingsof the

same

_frequencywillbe produced.

Then

thedialsoftheinstrument attachedtotho foundationor toa bearing

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

motionin which

wo

are interested

and

the relative

motion ₍₂₄₎ is a small one.

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.

On

this _principle various vibrographs

are _{built, for instance, the vibrograph constructed} FIG. 15.

by

the

Cambridge

Instrument

_Company,

shown

in _Fig. 213

and

_Geiger's vibrograph,

shown

in _Fig. 214.

A

_simple

arrangement for _recording vibrationsin _{ship hulls} is

shown

_{in Fig.} 15.

A

weight

W

isattachedat_point

A

toa

beam

_by

a rubber

band

AC.

During

vertical vibrations of the hull this weight remains practically

immovable

Then

the pencil attached to it will record the vibrations of the hull

on

a

rotating

drum

B.

To

get asatisfactory result the frequencyoffree vibra-tions of the weight

must

be small in _{comparison with} that of the hull of

the_ship. Thisrequiresthat the statical _elongation ofthe string

AC

must

belarge.

For

instance, to getafrequencyofJ^ofanoscillation_{per second}

the elongationofthe_stringunderthestatical actionof_{the weight}

W

must

be nearly 3ft.

The

requirementof largeextensionsisadefect in thistype

ofinstrument.

A

device analogous to that

shown

in Fig. 14 can _{be applied} also for

measuringaccelerations. Insuchacasea_rigidspring

must

be used andthe

frequency of natural vibrations of_{the weight}

W

must

be

made

_very large

in _{comparison with the frequency} of the _vibrating

_body

to which the

instrumentisattached.

Then

_p

is_{large incomparison}withcoin_expression

.(24)

and

the relative motion of the load

W

is approximately equal to

oo?2sin_ut/p2

and

proportional to the acceleration x\ of the

_body

to which

theinstrument is attached.

Due

tothe rigidity of the spring the relative

displacements ofthe load

W

are usually small

and

require special devices for_recordingthem.

An

electrical

method

forsuchrecording,usedin inves-tigating accelerations of vibrating parts in electric _locomotives, is

dis-cussed later (see page 459).

PROBLEMS

1.

A

wheelisrollingalong awavysurfacewith a_{constant horizontal speedv,Fig. 16.} Determine theamplitude of the forced verticalvibrations of the load

W

attached to

FIG. 16

the axleofthewheel_{bya spring}ifthestaticaldeflection ofthe springunderthe action

ofthe load

W

is5^

=

3.86ins.,v

=

60ft._persec.andthe_wavysurfaceis_givenbythe

irX

equation y

=

asin- inwhich a

=

1in.andI

=

36in.

Solution. _Consideringverticalvibrationsoftheload

W

on_{the spring}wefind,from

Due

to the _wavy surface the center o of the rollingwheel makes vertical oscillations. Assumingthat at theinitialmomentt

=

the pointofcontactofthewheelisata;

=0

TTVi

and _puttingx

=

vt, thesevertical oscillationsare givenbytheequation y

-

asin

Theforced vibrationofthe load

W

isnow_{obtained from equation}(c) substitutinginit

a

=

1 _in., <o

=

=

20*-, p2

=

_{100. Then the}

amplitudeof forced vibration is

l/(47r2 1)

=

.026in. At the givenspeed vthevertical oscillations ofthewheel are

transmitted to the load

W

onlyin a verysmall proportion. Ifwetakethespeed vof

the wheel l

/

as great weget o>

=

5ir and the amplitude of forced vibration becomes

l/(7r

2

/4 1)

=

0.68in.

_By

further decreasein_speedvwe_finallycometothe condition

of resonancewhen vv/l p atwhichcondition _heavy vibrationoftheload

W

will be produced.

2. _{For measuring}vertical vibrationsof a foundation theinstrument shownin Fig.

14 is used. What is the amplitude of these vibrations if their _frequency is" 1800 _per

minute, thehandofthedialfluctuatesbetween_{readings giving}deflections.100in. and

.120 in. and the springs are chosen so thatthestatical deflection of the weight

W

is

equal to 1 in.?

Solution. Fromthedialreadingweconcludethattheamplitudeofrelativemotion,

see_{eq. 24,} is.01 in. The_frequencyof freevibrationsoftheweight W,fromeq. (6), is

/

=

3.14 persec. Henceo>/p

=

30/3.14. Theamplitudeofvibrationofthe foundation,

from eq. 24,is

(30/3.14)*

-1

(30/3.14)2

3.

A

devicesuchasshown in Fig. 14isusedformeasuringverticalaccelerationofa

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

second. Thespringof theinstrument isso rigid that the frequencyof freevibrations

oftheweight

W

is60_{per second.}

What

isthe

maximum

accelerationofthecabifthe

vibrations recordedbytheinstrumentrepresenting therelativemotionoftheweight

W

withrespecttothebox have an amplitudeai

=

0.001 in.? Whatistheamplitude aof

vibrationof thecab?

Solution. Fromeq.24wehave

Hencethe

maximum

verticalaccelerationofthecabis

Notingthatp

=

27T-60andw

=

2?r-3, weobtain

aco2

=

_.001-4ir2

(602

-

32

)

=

142in.

see.-and