THE INFLUENCE OF ANGULAR ACCELERATION

THROUGH CRITICAL SPEEDS ON AMPLITUDES OF

WHIRL OF A ROT A TING SHAFT

by

Francis Howitt

The sis submitted in partial fulfilment

of the requirements for the degree

of Master of Engineering.

Depa rtment of Mechanical Engineering

McGill University

January,1961

- i-

SUMMARY

A detailed analysis of the phenomenon of whirling and the

application of steady- state theory to transient conditions. The

equations of mot ion for a simple rotor system subjected to

uniform ac cele ration are set up and a method of solution demon­

strated. These solutions are shown to be applicable to the case

of simply supported, accelerating, vertical, slender shafts.

The theoretical results are found to be in satisfactory

agreement with those obtained experimentally and provide a

qualitative guide, not only to the effect of angular acceleration,

but also to the relative influences of shaft balance, mass,

stiffness, and viscous damping upon amplitudes of whirl.

-ii-

ACKNOWLEDGEMENTS

The author acknowledge s with gratitude the guidance and

encouragement of his research director, Professer A. R. Edis,

and the counsel of Dr. B. G. Newman, in this project , and thanks

Mr. F. G. Corrick and the staff of the McGill Mechanical Engineer­

ing Department Workshop for their assistance in the construction

of the apparatus.

- iii -

CONTENTS

Page

Acknowle dgement s ii

List of Illustrations iv

Introduction 1

Evolution of Whirling Theory 2

Application of Jeffcott' s Theory to an Accelerating Shaft 15

Solution of the Equations of Motion of an Accelerating Rotor 21

Analysis of the Solutions to the Equations of Motion . 28

Application of Simple Rotor Theory to a Real System 35

Apparatus 37

Procedure 41

Table of Observations 44

Table of Re sults 48

Graphs 52

Discussion of Re sults 61

Conclusions 65

Appendix -- Evaluation of the Integrais

~ta.

a nd 5• at+ (--a-- bt)i dt

Bibliography

S•at +

• t.

( (ù~ + bt)i dt

67

75

Figure(s)

1

2

3

4

5

6 - 23

24

25

- iv -

ILLUSTRATIONS

Rankine' s Theory of Whirling

Jeffcottr s Theory of Whirling

Effect of striking a rota ting membe r

Simple st rota ting system

Apparatus

Graphs of maximum amplitude of whirl versus

angular acceleration of shaft

Path of integration in the z, plane

Path of integration in the z&. plane

Page( s)

3

10

13

16

38

53 - 60

69

72

1

INTRODUCTION

Many modern machines are run above their minimum

critical speed and no difficulties are encountered providing the

operating speed is not near the whirling speed. Numerous

investigations have be en made in order that the c ritical speeds

of the most complex systems may be calculated with the maximum

accuracy, and account taken of system stiffness, inertia, and

damping.

Little effort however, has be en directed towards di sc ove ring

what happens when a machine passes through a critical speed,

although eve ry time it is started or stopped it may have to traverse

one. It is obvious that if the dangers of whirling are to be kept to

a minimum, not only the steady state conditions for whirl, but also

the effect of transient conditions, must be considered.

The object of this thesis is to derive an equation relating the

amplitude of whirl of a rotor to its angular acceleration, and to

compare this with the observed maximum amplitudes of whirl of

shafts accelerating through their critical speeds.

2

EVOLUTION OF WHIRLING THEORY

The basic theory of whirling is well-established and accepted

by most workers in this field. Unfortunately however, it is not

generally known by ether engineers due to the perpetuation in text­

books of earlier, but erroneous, explanations of the phenomenon.

It is the refere felt desirable to establish the validity of current

concepts before attempting to extend steady- state whirling theory

to transient conditions.

This is best achieved by tracing the history ofwhirling via

the original papers of major contributors, accurately noting their

as sumptions, theo ries and fin ding s. The se summa ries show the

limitations of the earlier ideas and provide convincing evidence in

support of current theories. No attempt has been made to reproduce

the analyses of the various writers however, since the important

ones may already be found in numerous secondary sources.

The phenomenon of whirling was first reported by Rankine(i)*,

who communicated the results of a mathematical investigation of the

action of the centrifugai force in long !ines of shafting. He considered

a shaft rotating about a straight line AB and reasoned that any small

deflection of the centre line of the shaft from the axis AB, would give

rise on the one hand to a centrifugai force tending to make the

deflection become greater, and on the other hand, to an elastic stress

resisting the deflection and tending to straighten the centre line again,

(Figure 1). Equating the two quantities he obtained, as be expected,

integrais identical in form with tho se obtained by Pois son in his

investigation of the transverse vibrations of elastic rods. He then

* Numbers in parenthesis refer to the Bibliography at the end of

the paper

Shaft .........

'

3

. f~th of rotating shaft. certre line. ,' \ 1

1 1 .. 1

___:::~~ ___ B~ari!}l{ axis J ---+ ________ .,.-~-~ (=Axis of sta\tionhry shaft) . ~"'

A

1

Shaft deflection maintained by centrifugal force due to rotation about bearing axis AB.

Fig.l.

Rankine' s The ory of W'hirling.

B

4

proceeded to de rive a size and strength relationship for a shaft

such that for an indefinitely small deflection the stiffness would

be grea ter than the centrifugai force and centrifugai whirling

would thus be impossible.

Rankine confined his attention to long, slender shafts, -­

simply supported and cantilevered, -- because, he argued, the

effect of an additional rotating load at a point not near a bearing

was not of much practical importance, since if "a shaft is so long

and so rapid in its rotation as to require precautions against cen­

trifugai whirling, the first precaution is to avoid loading it with

rotating masses which are not very near the bearings".

The next contributor to the subject was Greenhill (2) who

inve stigated the stability of a rota ting shaft subject to thrust and

twisting, such as the propeller shaft of a ship, assuming that under

the combined action of thrust, torque and centrifugai whirling, the

straight form of such a shaft would become un stable, and the central

line of the shaft would become slightly displaced into a spiral. Like

Rankine he assumed that a whirling shaft revolved bodily about the

axis of the bearings with the imparted angular velocity. He deduced

that "the number of revolutions which would make the straight form

of the shaft unstable by centrifugai whirling, is exactly the same as

the number of lateral vibrations the shaft would make if still and

slightly displaced, the same thrust and twisting couple acting on

the shaft in both cases", and he cited the work of Rayleigh {3) and

of Rankine {1) in support of his conclusions.

It is noteworthy that neither Rankine nor Greenhill attempted

to substantiate their theories with experimental evidence, thus

setting a lamentable precedent which has persisted until the present day.

5

The literature of whirling abounds with theoretical analyses but

there is a dearth of experimental results. Future researchers

should note tha:t the first inve stigator to correlate the ory and practice,

Dunkerley {4), is generally regarded as the founder of whirling theory.

In fact, Dunkerley claimed no great originality for his work but

quoted Rankine and Greenhill and attributed the theoretical analysis

in his paper to Professer Osborne Reynolds-- who presented the

paper to the Royal Society. Nevertheless, it is Dunkerley who is

usually credited with the ideas expounded in the work and it remains

of major importance, since it is frequently quoted and accepted by

many texts as a sound analysis of the phenomenon of whirling. * Dunkerley claimed that "every shaft, HOWEVER NEARLY

BALANCED, when driven at a particular speed bends and, unless

the amount of deflection be limited, might even break". He also

noted however, that above this "critical" speed the shaft would

again run true, -- a fact previously unreported. The critical

speed depended upon "the manner in which the shaft is supported,

its size and modulus of elasticity, and the size, weight and position

of any pulleys it carries". In many cases, observed Dunkerley,

the period of whirl of the shaft coincided with the natural pe riod of

lateral vibration, as might naturally be expected; however in the

case of a loaded shaft, pulleys could give rise to righting moments

* E. g., the explanation "The Whirling of Shafts" by R. V. Southwell

in "An Introduction to the Theory of Elasticity for Engineers and

Physicists", 2nd Ed., Oxford University Press, 1941, pp. 208-209,

is exactly the same as Dunkerley' s.

6

tending to straighten the shaft and "generally, in a loaded shaft,

the period of whirl is less than the natural period of vibration, to

an extent depending upon the size and positions of the pulleys".

It was also noted that, "as in the vibration of rods, so in the

whirling of shafts, there are a series of periods at which the shaft

whirls "·

The theory expounded in the paper was merely an extension

of the earlier work on unloaded shafts, using the ordinary elastic

solid equations, to the loaded cases. Two methods of analysis were

adopted. In the first, the period of whirl was calculated taking both

the shaft and pulleys into account together, but the solutions proved

too complicated for practical use. A second method was therefore

devised whereby the period of whirl was first calculated for the shaft

alone, using the Bernoulli-Eulerian method as before, and then for

each of the pulleys in turn, neglecting the shaft. These periods of

whirl were then combined, using an approximate formula, to give

the period of whirl of the whole system. Dunkerley claimed that a

shaft and pulley system with individual periods of whirl of N1

for the

shaft and Na. for the pulley, was analogous to a spring carrying weights

w, and w'l.. The period of whirl of the whole system could thus be

obtained in the same way as the frequency of a spring carrying the

combined weight (W1 + W2.}.

Therefore, period of whirl of {shaft + pulley} combined,

N 1 N2. Ne= . Similarly for a three-element system

JNa. + N~ 1 2.

N 1 Na.N3

Ne= /Na.Nz. + Nz.Na. + Na.Nz.." Al12. 2.3 31

Application of these formulae however

gave errors of up to +8. 6 °/o, (where percentage error

= iOOX (observed period of whirl-calculated period)) observed period ·

7

Dunkerley attributed this to the fact that the formula was only strictly

true for loads applied at the same point, and he therefore modified his

original equation to Ne= N,Nz. where "a" is an empirical constant JN~ + N~a

dependant upon the particular system and is the multiplier of the great-

est term in the denominator. Using a value of a=. 885, {determined

by trial and error), Dunkerley obtained an error range of + 6. 3 to

- 6. 1 °/o for the calculated periods of whirl of the systems tested.

However agreement between theory and observation was not by

itself sufficient evidence of the general applicability of Dunkerley' s

formulae declared Chree (5) in a detailed examination of the phenomenon

of whirling, and, although he described Dunkerley' s paper as "elaborate

and important", he dismissed the derivation of the formula

NN NG =

1 2. as mathematically unconvincing and in general not

,JN '- + Na. 1 2.

strictly true. Nor could he see any indication that Dunkerley bad

grasped the real nature of the connection between the speed of whirling

and the frequency of the lateral vibrations of the shaft when stationary.

Chree claimed that whirling was due, not to forced vibrations being set

up with a frequency equal to one of the na tura! pe riods, but to the

indirect action of the rotation in reducing to zero the righting forces

which naturally act on the shaft when displaced laterally. Thus when

a shaft rota ting about it s longitudinal axis wa s di splace d la te rail y, the

elastic stresses tended to return it to the undisturbed position, as

when the shaft was stationary, "but the centrifugai forces have exactly

the opposite tendency: they thus re duce the righting forces, and so

diminish the frequency of vibration". At the critical speed the frequency

of vibration becomes zero and the shaft !oses its stability, like a ship

whose center of gravity coïncides with its metacenter. This was

8

important said Chree, for a rotating shaft may be subjected to periodic

forces like any other shaft, and in considering their effect, it was

necessary that one should consider the frequency of the lateral

vibrations of the shaft AS REDUCED BY ROTATION, not the

frequency when stationary.

Chree also criticized Dunkerley' s applications of the Euler­

Bernoulli elastic theory as being 11cumbrous mathematically and

the formulae to which they lead, and which were employed by Dunkerley,

often admit of great simplification, without appreciable diminution

of accuracy under his experimental conditions". Nevertheless,

although Chree avoided using Dunkerley' s approximate formula,

his analysis was based on the same elastic solid equations and it

produced results for Dunkerley' s experiments which were rarely

better, and usually worse than Dunkerley' s predictions for the

period of whirl. The paper won considerable acceptance however,

and Chree' s analysis of whirling is the one given in sorne current

texts. * Much of its influence can be ascribed to the fact that Chree' s

theory of variation of lateral frequency of vibration with speed of

rotation, appeared to offer a simple explanation of Dunkerley' s

observations that, a rotor is stable above its first critical speed

and, it possesses more than one critical; whereas Dunkerley1 s

hypothe sis, that at the c ritical speed, the centripetal force due to

bodily rotation of the rotor is exactly equal to the elastic force

whatever the deflection, would appear to indicate that at higher

speeds the elastic forces would be inadequate and the shaft would

break, instead of running true as is actually observed.

* E. g .• J. E. Younger' s "Advanced Dynamics", The Ronald Press

Co., 1958, pp. 262-263.

9

Dunke rley' s 11mathematically unconvincing 11 formula for the

lowest whirling speed of a system in terms of the lowest whirling

speeds of its components, did gain general acceptance however,

despite Chree' s censure, for it manifestly gave good results, and

it was eventually shown to be fundamentally sound by a new inve stigator,

Jeffcott (6), in an analysis of the periods of lateral vibration of

loaded shafts.

The following year however, the new investigator presented

a paper {7) which challenged the assumptions of ali the earlier work.

Until now it had been taken as axiomatic by every writer that shafts

revolved bodily around the axis of their bearings with the imparted

angular velocity: Jeffcott disputed this and argued that 11when a

shaft whose centre of mass does not lie on its axis of figure (the

shaft then being said to be 'out of balance'), is rotated, its

geometrical axis ceases to remain straight and in coïncidence with

the axis of the bearings, but becomes bent and rotates round the

latter axis". No previous writer had considered unbalance as a

factor. Jeffcott however, attributed the bending of shafts upon

rotation solely to this cause, and stated that the bending was greatest

when the speed of rotation was close to the free lateral vibrations

of the shaft, which was then said to fwhirlt.

He applied his hypothesis to "the case of a light shaft supported

freely in bearings at its ends and carrying a mass 1 m' at the center

of its span, the mas s center however being slightly eccentric by a

distance 1 a' from the elastic center of the shaft", (Figure 2).

Assuming the weight of the shaft and the moment of the inertia of

the mass to be negligible, the motion of the cross-section at the

center of the span would, by his hypothesis, b e governed by three

10

/

Centre of mass

Bearing axis, (=Axis of stationar.y shaft).

" Elastic centre line,

' (=Axis of rotation).

Fig.2.

Jeffcott•s Theory of Whirling.

11

influences; the restoring force due to elastic deflection of the cross­

section from the bearing axis, a damping force proportional to the

velocity of the cross- section, and the disturbing effect produced by the

combination of the cross-section' s impressed rotation in its own plane

and the eccentricity of its center of mass with respect to the elastic

center.

Solving the equations of motion for this system, he showed that

under steady conditions, the elastic center of the shaft described a

circular path around the bearing axis, which is of maximum radius

when the impressed speed of rotation of the shaft coïncides with its

free lateral vibrations, as predicted. He also showed that "at speeds

appreciably below the whirling speed the shaft rota tes with mas s center

farther from the axis of rotation than the elastic center while at speeds

appreciably above the critical speed, the mass center is closer to the

axis". As Jeffcott pointed out, this explained the phenomenon observed

when balancing that, if a pencil is held near a shaft to mark the part of

the circumfe renee that is most deflected, at speeds below the c ritical

an unbalanced body will be marked on the heavy side, and conversely,

above the critical it will be marked on the light side.

His solution also showed that at any speed the deflection of the

shaft was directly proportional to the amount of unbalance, whilst

damping was only of much importance at or near a whirling speed.

If the critical speed could be avoided by a margin of 10°/o either side ,

then the maximum amplitude of vibration should not be more than about

five times the eccentricity of the mass center, even if there is no damping.

Thus although Jeffcott1 s paper included no experimental results his

theory could be readily substantiated. It was appreciated that balance was

an important factor in whirling as Jeffcott emphasi zed, and although the

12

concept of an elastic rather than a fixed axis seemed unusual it was

found that a rotating member when struck would vibrate in a fixed

plane, (Figure 3), a fact completely at variance with the old theories.

Nevertheless old ideas die hard and some writers have continued to

nurse that of a fixed axis, the early theories have be en modified to

take into account unbalance, and attempts made to explain the

balancing phenomena reported by Jeffcott. * Most workers in the field however, eventually accepted and

used the new theory and their labours have provided additional

confirmation of the soundness of Jeffcott' s ideas. Morris (8) has

forcefully expounded the limitations and objections to the earlier

works, showing that Chree 1 s theory is dynamically unsound and

stating that, contrary to Dunkerley 1 s ideas, a perfectly balanced

shaft bas no speed of instability. He also extended Jeffcott' s theory

to more complex rotor systems.

More recently Johnson {9) has demonstrated mathematically

the fact that a free vibration of a rotating shaft can be built up in

any fixed plane and that such mot ion is independant of the speed of

rotation and therefore identical with that of a stationary shaft. The

phenomenon has been clearly demonstrated in an apparatus described

by the same author in a later paper {10). This apparatus has been set

up at the Cambridge University Engineering Laboratory and incorporates

a shaft with a whirling speed of only 13 r. p. m.: at such low speeds the

fallacy of a fixed axis of rotation is convincingly apparent.

There is still a dearth of experimental data on whirling to draw

attention to the inadequacies of the original theories, but the situation

is gradually improving.

* E. g., see A. H. Church1 s "Mechanical Vibrations", John Wiley and

Sons Inc., 1957, pp. 91-92.

13

1 1

1

( Free vibrations due 1 to hanuner blow are in one _ ---\plane, ( e.s for e. ..... _,--- -\Bte.tione.ry she.ft)~ \ , , \ / , 'V,

Fig.J.

Effect of striking e. rote.ting member.

14

.An excellent paper by Downham (11), analyses a series of experi­

mental studies of the effects of large rotor inertia, and of asym­

metrical bea ring stiffne ss, upon whirling. He notes that using

Jeffcott1 s theory it may be shown that although increasing rotor

inertia decreases the natural frequency of a system, it increases

the value of the c ritical speed; a fact that may be confirmed by

expe riment. Dunke rley expected the cri tic al to be reduced below

the natural frequency but experimental limitations prevented him

from detecting his error.

Summarizing; Jeffcott' s theory appears to offer a complete

explanation of ali known whirling phenomena. The concept of an

elastic axis of rotation is less obvious than the original hypothesis,

but it can be demonstrated practically, and as will be seen later,

it is mathematically simple. A good augury for the future is that

the theory may now be found in a few textbooks. *

* E. g. , N. O. Mykle stad' s "Fundamentals of Vibration Analysis",

McGraw-Hill Book Co. Ltd. , 1956, pp. 56-59.

15

APPLICATION OF JEFFCOTT' S THEORY

TO AN ACCELERA TING SHAFT

The simplest rotating system is that analysed by Jeffcott,

namely a simply supported vertical shaft carrying a concentrated

mass M at the center of its span, the weight of the shaft and the

moment of inertia of the mass are assumed to be negligble.

Suppose the shaft is initially straight and balanced, then upon

rotation there will be no disturbing influence and the axis of the

shaft will remain coïncident with the axis of the bearings. If

however, the mass is eccentric by a distance t from the centre

of the shaft, it will give rise to a disturbing force when the shaft

is rotated and the shaft will be deflected from its initial axis.

Suppose rectangular co-ordinates be set up at the intersection

of the bearing axis with the plane through the centre of mass; x and

y are the co-ordinates of the elastic center E, and the line joining

the centre of mass G with the elastic center E makes an angle e

with the x-axis, {Figure 4). The co-ordinate s of G are evidently

{x+ t co se) and {y + E sine), thus the equation of motion of M

parallel to the x-axis will be

d~ dx M dt1.. {x+ Eco se) + R dt+ Kx = 0 Eq. 1.

where R is the coefficient of damping due to viscous resistance

(e. g. : air damping and bea ring friction) and K is the shaft stiffne ss

(the elastic restoring force per unit displacement}.

Similarly the equation of motion of M parallel to the y-axis

will be

d~ d Md 1.. {y+ E sine}+ R ~ + Ky= 0

t dt Eq. 2.

These two equations describe the motion of M at any instant UNDER

ANY CONDITIONS.

16

""1

x-axis.

y-axis.

1.

1

/

Axis of rotation.~ . Bearing axis.

Fig.4.

Simplest rotating system.

17

Suppose the shaft rotates about its elastic center with a steady

speed (1). Then at time t, 9 = (l)t and the equations become

d~ dx M dtt. {x+ E sos(l)t) + R dt+ Kx = 0

and

M :~{y + E sinC'ù\t) + R * + Ky = 0

But (a) =constant, d'- da.x 2. dta. {x+ E cos(l)t) = dtt. - E(l) coswt

and

Substituting these values in equations 3 and 4

dz.x dx 2. M--+ R-+ Kx =ME w COS(I)t

dtL dt and

da. y R~+ Ky= ME (1) a.

sin wt M dtL + dt

Eq. 3,

Eq. 4.

Eq. 5,

Eq. 6.

The solutions to these two differentiai equations are weil known:

each contains two terms, the first is an oscillatory motion whose

amplitude decreases with time and thus becomes negligible, the second

is a vibration of amplitude

MEwa.

,J(K - M(I)L )L + R (a)a.

Furthe r examination of the solutions reveals that the sum of their squares is constant,

i.e.: a. 2. 2..

x + y = { OE) = constant.

Therefore the path of the elastic center under steady conditions must be

a circle about 0 of radius equal to the vibration amplitude.

Obviously the radius will be maximum when K = Mwa. and thus

the critical speed wc = ~. Although the re suit in this simple case

is the same as that obtained using the olde r theories the anal y sis is

18

significantly different. Jeffcott has assum.ed a disturbing force of

Mew"- and shown that this will cause the elastic center to describe

a circular path around the bearing axis. The older theories however,

assumed circular motion of the center of mass about the bearing axis

and stated that this produced a disturbing force of M{OG)w L. Thus

although the theories may agree as to the cri tic al speed they will

predict different amplitudes of whirl.

It has already been noted that the equations

dL dx M dt,_(x + e cos 9) + R dt+ Kx = 0 Eq. 1

and dL dv ·

M dtL(y + e sin 9) + R ill+ Ky = 0 Eq. 2

are valid not only for constant speeds of rotation but for any con­

ditions. Suppose the system be accelerated from rest at a uniform

rate c.), then at time t, e = f e.it a..

Sub stituting this identity in equations 1 and 2

d& 1·'~- dx M-d L(x + f. cos zwt ) + R- + Kx = 0

t dt Eq. 7

and d'~- ~· ~ ~

M dtL(Y + f. sin 2 wt ) + R dt + Ky = 0 Eq. 8.

But ~=constant,

d'- 1 . a. d dx . 1 . t.) dtL(x + f. cos z wt ) = dt (dt - ec.lt sm z wt

d l.x . 1t. a. .1. . L . .1. . a. .= dtL - ew t cos 2 wt - ew sin 2 wt .

dL 2. d 1y t. Similarly - .. (y+ f. sin .1. ~t ) = - ec.l1 t'- sin+ ~ta.+ f.W cos t wt.

dt• 2 dtL

Sub stituting in equations 7 and 8

dl.x dx .t. t. ~tl. . ,;,ta. M dtL + R dt+ Kx =Me (w t cos -

2- + w sin -

2-] Eq. 9,

Mda.y ~ [ ... a. . ..ït,_ . ~t'~-dtL + R dt+ Ky = Me w t sm -

2- - wcos -

2- ] Eq. 10.

19

These are the equations of motion for constant acceleration of

the simplest of al! rotating systems, unfortunately they possess no

simple solutions. A possible approach however is indicated by

Lewis (12) in "Vibration During Acceleration Through a Critical

Speed". This paper was an attempt to describe the motion of a

single mass elastic system, with linear damping, such as this,

unde r the action of an exciting pe riodic force of constant amplitude

but changing frequency; the equation of the system being:-

w.. . ( 2. ) - x + dx + kx = P cos 1rht + cr g

where W, g, d, k, P, h and cr are ali constants, and x and t are the

variables, {x and x being the second and first order derivatives

respectively of x with respect to t). It was claimed that this

corresponded with the case of a machine with uniform acceleration.

In fact this is not correct, for the amplitude of the disturbance in-

creases with time, as well as the frequency, and the exciting force

should therefore have been .... not P cos (1rht~+ cr), but

P[(l1Tht)z.cos(1Thtt.+ cr)+ 21Th sin (1rht2. + cr)]-- as hasjust b e en

demonstrated.

Although the equation was wrong the solution was most ingenious

and Lewis' s method is closely followed in this analysis. The paper

is al so importa nt because it was the fi rst attempt to examine the

effects of acceleration, the results appeared reasonable, the error

therefore passed unnoticed, and the paper has remained the standard

work on the subj ect. Thus McCann a nd Bennett (13), who used an

electric-analog computer to inve stigate vibration du ring ac cele ration

through critical speeds, followed Lewis and assumed an exciting

force of constant amplitude. The only work that has been done on the

c o rrect equa t ion is that o f B a ker (1 4 ), at M. I. T ., who u sed a

differentia! analyser to solve it, and obtained curves of vibration

20

amplitudes for various rates of constant angular acceleration and

deceleration. None of these writers, incidentally, appear to have

attempted any experimental verification of their results.

It is evident therefore, that much work remains to be done in

this field. Although Baker has obt ained particular solutions to the

basic equation of an accelerating single rotor system, a need still

exists for a general solution which will display the relative influences

not only of acceleration, but also of rotor eccentricity, mas s, damping,

and stiffness, upon amplitude of whirl. The need for practical testing

of findings has already been emphasized.

A single rotor system remains the obvious choice for analysis,

since it is not only the simplest but, as will be seen later, its

behaviour should be characteristic of more complex systems. Its

equations of motion have been derived, all that remains is to solve

them.

21

SOLUTION OF THE EQUATIONS OF MOTION OF

AN ACCELERATING ROTOR

It has be en shown that the motion of a smoothly ac cele ra ting

rotor can be described by the two equations :

dt.x dx . a.t. ~ta. . ~tL M dt'-+ Rdt + Kx =Me: [w t cos -

2- + w sin-

2-] Eq. 9,

and ~ d .a. t. ci.t'" . t.>ta.

M dt&.+ R 7t + Ky= Me: [fa) t sin -2-- w cos2] Eq. 10.

Consider equation 9, this can be re-arranged in the form

d a. x R dx K . a. t. .dt t. . w t a.. dt a. + M dt+ Mx= e: (fa) t cos - 2- + fa) sin -

2-] Eq. 11.

To find the complementary function of this equation put

d'"x dx K dta. + R dt+ M x= 0 Eq. 12.

Then the roots of 12 are A and B where

-~+)(~)~- 4~ M A= -- a+ bi Eq. 13,

2

and

-j(~)a.-R K - 4-

B= M M

a - bi Eq. 14, --2

assuming (R)~ K - <4-M M'

a and b are real numbers and i=.f-1:

Thus R

a=--2M

Eq. 15,

and Eq. 16.

Therefore • 2. • ...

( )( [ • 2. 2. 6) t . 6) t ]

D - A D - B)x = e: w t cos -2

- + w sin -2

-

22

. t. . t. 1 [.z. t. (l)t .. (l)t ]

x= (D _ A)(D _ B) E (1) t cos - 2- + w s1n - 2-

. t. . t.. 1 { 1 [. 1. t.. (l)t . (l)t ]

=A _ B D _ A E (1) t cos - 2- + (1) sin - 2-

1 [.1. 2. ~tl. . ~tt. e: (1) t cos -

2- + (1) sin -

2- ] }

D-B

-1 At -1 -At But (D - A) f(t) = e D e f(t)

1 At J -At . a. 2.. ~ta.. . w t t.. x= A _ B { e e E [ (1) t cos -

2- + (1) sin -

2- ] dt

Bts -Bt .1. a.. ~tt. . wt"' - e e E [(1) t cos -

2- + w sin -

2- ] dt } Eq. 17,

and similarly,

1 AtJ-At ·&t.. wt2... ~ta.. y= A _ B { e e E [ w t sin -

2- - (1) cos -

2-] dt

Bt} -Bt .z. &. • ~tz. . ,.;t1 - e e E [(1) t sm -

2- - (1) cos -

2-] dt } Eq. 18.

Consider the integral terms in equation 17;

-At . a. z. w t . w t J . t.. • l.

e E [(1) t cos -2

- -r w sin -2-] dt

J

-At . . c.it t. • C:,t t. = e E [(l)t ((l)t cos -2-) + (1) sin -

2-] dt

-At . w t t. J -At . w t z. -At w t t.. = E { e (l)t sin -2-- e (1) sin -

2- dt - Ae cos -

2-

J 2 -At w t &. J -At . w t a. - A e cos-

2-dt+ e (1) sin-

2-dt}

-At . . wt a. A -At ,.;ta. f 2 -At wt t. = E { e fill t s1n -

2- - e cos -

2- - A e cos -

2- dt } .

Similarly

S -B t . a.. "' w ta. . c.»t .1. - B t . 6> t L B - B t w t 2.. e E [w t sin -

2-- (1) sin -

2-] dt = E { e falt sin -

2- - e cos-

2-

2 - Bt (l)t J • L

- B e cos -2

- dt }

23

• t... • t.. E { . wt wt x= wt sin--- Acos---

A- B 2 2 AtJA2 -At w.t Ld e e cos -

2- t

. wt ,_ wt" BtJ 2 -Bt ~t ,_ - wt sin -

2- + B cos -

2- + e B e cos -

2- dt }

€: wt,_ =A_ B }{- (A- B) c.os -

2--

AtJ 2 -At wt t... e A e cos -2

- dt

BtJ 2 -Bt wt t.. + e B e cos -

2- dt} Eq. 19.

In the same way equation 18 becomes:-

E wtt.. Ats 2 -At Y= A_ B {- (A- B) sin -

2-- e A e

• t. • (a) t

s1n -2

- dt

BtJ 2 -Bt ~ta. + e B e sin -2

- dt } Eq. 20.

But from equations 13 and 14 ...

1 1 1 -i _..::.._._= =-=-A - B - a + bi + a + bi 2bi 2b

At {-a+bi)t -at e = e = e {cos bt + i sin bt)

-At { a - bi )t at { 1.

e = e = e cos bt - sin bt)

Bt {- a - bi)t -at . e = e = e {cos bt - 1 sin bt)

-Bt ( a + bi )t at e = e = e {cos bt + i sin bt)

Substituting these values in 19 ...

wtt.. i -at x=E{- cos-

2-+ Zbe (cosbt+i sin bt) (a - 2iab - b )e {cos bt -J 2 2 at

. . wt1. i -at 1 sm bt) cos -

2- dt -

2b e (cos bt - i sin bt) (a + 2iab - b ) J 2 2

at · t &. e {cosbt+isinbt)cosw

2 dt} Eq. 21.

Z4

But x is a real deflection and therefore the imaginary terms must

equate to zero.

wt e at 1. a. . wt . a. -at J . 1. x= e: {- cos -Z- + Zb [cos bt e (zab cos bt +{a - b) smbycos -zdt

J at( a. L :\ wt 1. -sin bt e -Zab sin bt +{a - b) cos b!f cos-Z- dt

,( at( a. ,_ . ~ wta. - cos bJe - Zab cos bt- (a - b ) sm bt.J cos -Z- dt

f at a. 2. ~ wt 2. - sin bt e r- z ab sin bt + (a - b ) cos b!J cos -z- dt ] }

wta. e -at s at wt a. = E {- COS -Z- + -b- [cos bt e rz ab COS bt COS -Z-

2. .. wt a.) + (a - b ) sin bt cos -Z- dt

5 at wt L 1. 1 wtL) + sin bt e (zab sin bt cos -Z-- (a - b) cosbtcos2 dt]}

Eq. ZZ.

Now ....

wt'" wtL wt~ z cos bt cos -z- =cos ( -z- + bt) + cos ( -z-- bt)

wta. wtL (-z- + bt)i ( - - bt)i

[ e z ] =Real part of e +

Similarly .... wt.. wtL

. t. ( -z- + bt)i . { -z- - bt)i Zsinbtcosw~ =Realpartof[-ie +1e ]

In se rting the se values in equation ZZ .... . L . a.

wt . wt . . a. -at s ( -Z- + bt)l {-- bt)l wt e at Z

x= Real part of {e: {- cos -Z- + b [cos bt e (zab {e + e . 2. • t.

wt . wt . { -+ bt)l { - - bt)l

'&. ,_ z z ) - i(a - b )(e - e ) dt

25

• a.. • 'L

( (a); + bt)i ( (a): - bt)i

+sin btJea'(- 2iab(e - e )

wt~ wt~ &. 2. ( -2- + bt)i ( -2- - bt)i

- (a - b )( e + e ) ) dt ] } )

• 2. -at w t e /. 1. t.

=Real part of (E {- cos -2

- + b ['\.cos bt {2ab - i{a - b ))

wt'-

J at + ( -

2- + bt)i

+ sin bt (- 2iab - {a'" - ba.))) e dt .

1

~ r at + { (1): - bt)i +(cos bt (2ab + i(a'-- ba.))+ sin bt {2iab - {a2.- b'-)))J e dt]})

• 2.. -at =Real part of (E {- cos w: + T [{cos bt - i sin bt)(2ab - i(at.- ba.))

wt'l. 2 'l. 'l. J at + ( -- + bt)i

e dt + (cos bt + i sin bt)(2ab + i{a - b ))

wt2.

S at + ( -

2- - bt) i

e dt]} ) Eq. 23.

It may similarly be shown that ....

. t. -at . wt e 2. a.

y =Real part of (E {- sm -2

- - b [i(cos bt - i sin bt)(2ab - i(a - b ))

. ,.

S at + ( w; + bt)i

e dt+ i{cos bt + i sin bt)(2ab + i(a~ - b"'))

wta..

Seat+ (2- bt)i

dt]} )

Thus

wt .. x=Realpartof{E[- cos-

2-+ F]}

Eq. 24.

Eq. 25,

26

and

y= Real part of {E: [-• 2.

. wt l"F] ll s1n --- § 2

Eq. 26,

where -at

~tl. a. J at + ( - 2- + bt)i

F =~[{cos bt - i ,_

sin bt){2ab - i(a - b )) e ili

+ (cos bt + i sin bt)(2ab + i(a._- b'"nJ• at+

• 2.

( wt b)" --- t 1 2

dt]

Eq. 27.

• 1.

J at+ ( w; + bt)i s at + The integral s e dt and e

• 2..

( wt - bt)i 2

dt

may be evaluated by the method expounded in Lewis' s paper {12),

as demonstrated in the Appendix.

Summarising the results obtained;- if the simple rotor system

shown in Figure 4 is subjected to angular acceleration from rest for

time t at a uniform rate ~, its elastic center E will be deflected a

distance r from its initial equilibrium position.

r = Jx'- + y,_

~t,_ where x= Real part of {E: [- cos -

2- + F]}

y = Re al part of { E: [-

"2.

sin~- iF]} 2

Eq. 25,

Eq. 26,

E: = eccentricity of the center of mass of the rotor with

respect to the elastic center of the system.

i=Ff

-at e

F = b [(cos bt-

c.) t 1.

S at + ( -

2- + bt)i

i sin bt){2ab - i(a'"'- ~)) e dt

. 1

+{cos bt + i sin bt)(2ab + i(a'-- b1

)) e j at+ (wt - bt)i 2

dt]

Eq. 27,

27

Eq. 15,

Eq. 16,

R =coefficient of viscous damping,

M =mass of rotor,

K = shaft stiffne s s,

and from the Appendix ...

c.»t~ -v

e J at + (2 + bt)i Ri' loo e dt= -. [

2w ,/z• + v 0 1

z'- z loo e-v dv - e 1 1 -- dv]

0 ,./z,+ v

' ~

S at + { w ~ - bt)i ff. 1

e dt = -. [- 2Trl(.e z~ + 2w r -v

e """"F~=-dv

,./z;_ + v

where

' z .. - za. - e ..

iw ~ z =-- (t + u )

' 2 1

1 u = -:- (b - ai)

1 (a)

1 i (b . ) 1. z =- -. - a1 ' 2w

iw a. z =- - (t + u)

L 2 2.

u,_ = ~ (- b - ai) (o)

1 i ( b ')a. z =- -. - - a1 1. 2 (o)

0

-v e

;:== dv] ,Jz.,. + v

K.. = 0 when arg. z < Tr and K..= 1 when arg. z > Tr.

Eq. A6,

Eq. A11,

Eq. A2,

Eq. A3,

Eq. A4,

Eq. A7,

Eq. A8,

Eq. A9,

28

ANALYSIS OF THE SOLUTIONS

TO THE EQUATIONS OF MOTION

It is immediately apparent that although exact solutions to the

equations of motion have been ohtained , the labour involved in their

evaluation would be considerable.

An examination of the solutions however, reveals a number of

terms referring solely to the initial conditions, namely:-

Soo e-v dv in equation Ab

0 ,Jz,• + v

and in equation 11,

-- z: and z~ being the values of z, and zL re spectively when t = 0.

The se repre sent the constants of integration corre sponding to the

imposed initial conditions; x= 0, dx = 0, y= 0, and~= 0 at t =O. dt dt

In any practical system some damping will be pre sent and the

contribution to the amp l i tude of the damped harmonie vibrations due

to these terms will be negligible in the vicinity of the critical speed,

hence the se terms may be omitted when computing the maximum

amplitude du ring accele ration.

Therefore ....

F ~ e ~at [(cos bt - i sin bt)(2ab - i(a' - b' ))(-ji;, e z,'- z, f ../z: ~vv dv) 0

+ (c o s bt + i sin bt)(2ab + i(a • - b))(-ji;; e z~- z, [ .jz: ~vv dv)]

Eq. 28.

Z9

• 2.

. ~ But

1 z, - z, e =e

at+ ( wzt + bt)i . 2. at wt

= e {cos--+ z i sin w; )(cos bt + i sin bt)

• 1.

1 at+ z - z

and e 2. ~ = e ( wzt - bt}i . a.

at wt = e {cos--+ z • 1.

1. . Cl) t )( b . 1. . bt) s1n -Z- cos 1 - s1n .

Sub stituting in equation Z8

. e {( i ) at { wt . . wt )( b . b ) -at Jf . a. · z..

F :;: -b- - Z 6, e cos -Z- + 1 s 1n -Z- cos t - 1 s1n t

(cos bt + i sin bt)[(Zab- i(az.- 1{)) Soo e-v dv ,Jz, + v

0

+ (Zab+ i(a~- ~)) Soo e-v dv]} ,Jz~. +v

0

But (cos bt - i sin bt)(cos bt + i sin bt) = (cos 2.bt + sin a.bt) = 1

2. a.. + (Zab + i(a - b ))

roo e-v dv]

J ,Jz +v 0 1

Consider now the integral terms in equation Z9 ....

Eq. 29.

roo e-v dv and loo e-v dv where z =- izw [t + (b - ai) ]a. j_ . 1 z + v "z... + v 1 Cl) o IV 1 o •

d i~[ {b+ai)]a. F . . .. ' d h h an z~., =- - t - . rom 1nspect1on 1t 1S ev1 ent t at t ese definite int~grals ar~not cyclic terms but complex quantities whose

value varies smoothly with t ....

thus Soo e-v dv decreases as t increases, A/z,+v

0

whilst l roo e-v dv l increases to a maximum as lt - (b: ai) l j

0 ,.jzl. + v

approaches zero, and thereafter steadily decreases.

30

R a=--

2M bo-!-J~-(~)~ and

l 1 JKz 1 ( MR )a. b + ai = M - .s..

Thus the critical speed (a)c =ta) tc~ J~ -t ( ~ ).,_, since the critical

speed is that speed which give s the maximum amplitude. It is

obviously close to the value for constant speed conditions but it

cannot be exactly the same. Inspection of equation 29 reveals why

this is so.

Consider the expression

((2ab - i(aa. - bt. )) Joo -;==e -:::;::v~dv + {2ab + i(a;r.. - tf)) 100

e-v dv]

0 "z 1 + v

0 A/z,_ + v

the imaginary parts are of opposite sign, therefore the expression oo -v

is maximum NOT when 1 e dv is maximum, but a little 0

,/zt. + v

Soo -v

e dv has further decreased in value. ,./z

1 + v

0

later when

the critical speed of an ACCELERATING rotor is slightly HIGHER

than for a shaft running at constant speed. Similarly it may be shown

that the critical speed of a DECELERATING shaft is LOWER than the

constant speed c ritical.

Lewis (i2) has shown that if

31T 1T 2 > arg. z >- 2

then if z is large, the definite integral 500

e-v dv may be ,.jz+v

0

evaluated using the formula ....

ioo e-v -t{ 1 1.3 1.3.5 -:=::::;::=:.. dv = z 1 - - + -- + } ,.Jz + v 2z (2z)a.- (2z).J · · ·

0

Eq. 30.

31

For small values of z ....

Joo e-v r 3 f 2z (2z)a. -;==;==-dv = (1 - 2tq,v 71" e - 2z { 1 + - + + ... } ,Jz+v 1.3 1.3.5

0

Eq. 31,

where 1<.=0 if arg.z<TI

= 1 if arg. z) 71" as before.

It may be deduced from Lewis' s paper that in the region of the

critical speed the z1-expression is less than 6°/o of the z~.-expression.

Under these conditions equation 31 is obviously applicable and since

arg. z > 71" ••••

F =- ..!.. IT (cos ~t,_ + i . c:.»tz. . 2. ~. r= 3 s1n -

2- )(2ab + 1(a - b }hv 71" e -

c' b,J2~ 2

2 f { 1

2za. (2z,l za. + -- +

1. 3 1. 3. 5 + ... }) Eq. 32.

But i w [ (b + ai) ] a.

zz. =- 2 t- ~

and t _ ffi [ (b + ai} ] z2. - .,J - 2 t - ~ , Eq. 33,

also, at the critical speed, z,_ is small and therefore in a severe

approximation such as this, it and its higher terms may be neglected.

Thus 1 . t z.

F =-(cos~+ c . b 2

. . ~t ,_ }[2 b '{ 2.- b,_}] rffi.TI 3 [ - (b + ai} l) 1 s1n 2

a + 1 a 2

. e + t . fa) c fa)

Eq. 34 ,

whe re tc = time at the cri ti cal speed.

. ka. k3 Ev1dently F œ (k + -. + -. ) -- approximately -- where

c 1 fa) fa)

k1

, k,_ and k3

are COMPLEX constants for the particular system.

32

x = Real part of { E [-~t2..

F]} Eq. 25, But cos -2- +

y= Real part of {E [-~t2.

iF]} Eq. 26, sin---2

and amplitude r =J x 1 + ya..

"[ li! ~] Hence it may be deduced that r ::;: K + ,...-: + . whe re max 1 ,., w w

K1

, K2.. and K3

are REAL constants for the particular system.

Consider now the case of an undamped system; a= 0 and under K

steady conditions the critical speed, wc = M =b. When the undamped

system is accelerating, if the shift in the critical is neglected ....

tc (a) =-.-=b

c ti)

[t - (b +. a. i) ] __ 0 since a=O, c (a)

h. F a,...-: c "' (a)

. [ Ka.] r :;: K, + r;:,_ max JI w

bfi 3 and if -2

-:- e >> 1 then r w max

Thus the maximum amplitude of whirl of an accelerating

undamped system is inversely proportional to the square root of

the ac cele ration, providing ~ is small relative to the shaft stiff-

ness. For larger values of w •...

Ks. r ~ [K, +F] max .-1 fù

and in the case of a damped system ....

. ~ ~ r ::;: [K + ,.-:- + . max • ~ fù w

Eq. 35 .

The soundne s s of the se deductions is confirmed by the ir applicability

to the mechanical analog solutions of Baker {14) of the same equation:

these solutions also demonstrate the 11 shift in the critical" upon

acceleration that was predicted earlier, a convincing demonstration

of the validity of the analysis and the approximations.

33

The analysis also makes an assessment of the relative

importance of the system constants possible. It is evident from

the equations

wtl. x= Real part of {e: [- cos -

2- + F]}

y= Real part of {e: [-. a.

. w t 1. F]} Sln ---2

and r=Jx'~-+y'l.

That the amplitude of whirl at any irs tant is exactly proportional

to the eccentricity of the center of mass with respect to the elastic

center. The relation of shaft stiffness, viscous damping and rotor

R 1J K R 1. mass in the constants a,(= M ), and b, (= 2 4 M - ( M ) ), clearly

indicates that the effect ofboth the stiffness and the damping is

inversely proportional to the mass. This infers that the EFFECTIVE

stiffness of a hollow shaft in whirl is greater than that of the equivalent

solid shaft.

It may be deduced from the equation ....

·a. ·&, ~ ( . 1 wt wt . ~ z. i'rr 3 b + ai) F ::;: -{cos---tsin- )[2ab + 1(a - b )] { -. e + [t - . ] }

c b 2 2 2w c w

Eq. 34,

that the maximum amplitude of an undamped rotor is approximately

proportional to the stiffness term b, for a given acceleration.

J K R a. Since b (= -f- 4 M - ( M ) )is reduced when damping (R) is present,

it is reasonable to expect that the presence of damping will re duce

the amplitude and this is normally so. Inspection of equation 34

reveals however, that under certain circumstances heavy damping . . [2ab + i(a1

- b2.)] R may increase the amphtude, smce F œ and a=--.

c b M The se findings are confirmed by Baker 1 s re sults.

34

A qualitative guide to the effects of the different factors is

expecially valuable of course, as it is unlikely that ali constants

will be known in any given system. If they are known, then the

formulae given for the evaluation of 500

e-v dv in equations '*'z+v

0

30 and 31, permit computation of the amplitude of whirl at any

instant, to any degree of accuracy. This is extremely laborious

however, even if interest is restricted to critical conditions, and

transients are neglected. It must also be remembered that only

an extremely simple, idealized system has been considered thus

far, the exact analysis of a practical system would be even more

complex. The exact solutions are therefore solely of academie

interest and the importance of the analysis must reside in the

approximations and their pertinence to real systems.

35

APPLICATION OF SIMPLE ROTOR THEORY TO

A REAL SYSTEM

Any rota ting system which ope rates above its minimum

whirling speed is normally carefully balanced and adequately

damped in order to keep the amplitude of whirl to a minimum.

Suppose such a system was nevertheless found to vibrate excess-

ively at the critical speed, it would obviously be most helpful if

equation 35, -- r = [K, + ~ + ~3 ] -- were applicable, since max. Il/ w w

the measurement of whirl amplitude at three different accelerations

would permit the evaluation of the constants, and r would then max.

be known for any value of c.). This would show whether the vibration

could be kept within acceptable limits merely by increasing the

angular acceleration.

Equation 35 however, only related to a single rotor of negligible

inertia carried on a weightless shaft. Morris (8) has shown that

Jeffcott' s theory can be extended to a rotor of appreciable inertia

but the re sulting equations of motion are complex, even for steady

conditions. A bare shaft may have a negligible moment of inertia

but it is equivalent to an infinite number of thin disks, each rotating

about the e lastic axis connecting them. Such a system will have an

infinite number of possible modes of vibration, whereas the ideal

rotor with all its mass concentrated at the center of gravity has only

one mode of transverse vibration and one critical speed. Nevertheless,

since a bare shaft is essentially an infinit e number of connected ideal

rotors, its behaviour may be expected to be similar to that of the ideal

system. This view is supported by Robertson (15}, who states that the

unbalanced whirl theory of an ideal rotor is applicable to the primary

unbalance whirl of a rotor with di stribut e d inertia, e x cept that the

36

mass used in the equations of motion is then not the actual mass

of the rotor but an equivalent mas s, the as ses sment of which, he

admits, is difficult. A paper by Taylor (16), provides experimental

confirmation of the relevance of ideal rotor theory to actual rotor

behaviour. His graphs show that the deflection at the center of a

round uniform shaft corresponded closely with the predicted deflection

of an equivalent single rotor system, at all speeds throughout his

tests. It is also clear from his results that there is comparatively

little viscous damping in an actual system. Taylor quotes a damp­

ing coefficient of. 07 and Baker1 s graphs, which cover a range of

coefficients from . 1 to . 4, show that for an accelerating system

this is very small.

It must be remembered however that both Robertson' s and

Taylor' s papers are concerned with constant speed conditions and

that an accelerating shaft is subjected to additional torsional

stresses. Neverthless if the moment of inertia is small and there

is little friction these stresses will be very small, and it would

appear that the behaviour of a round, uniform shaft passing through

its primary critical speed will be approximately the same as that

of an equivalent undamped simple rotor. It was therefore decided

to conduct a series of tests upon a set of such shafts to confirm

that the predicted amplitude/acceleration relationship,

• ( Ka. ] -- r ~ K +~ -- was applicable. max. 1

"' w

37

APPARATUS

Funds for this project we re seve rely limited and it was therefore

necessary to restrict the experiments to the determination of whirl

amplitude and acceleration. The equivalent mass, stiffness, damp­

ing, and unbalance of the shafts, could have been found by measuring

the steady deflection of each shaft at the span center at four different

known speeds, and solving the four resultant equations. However,

since the deflection at non-critical speeds is very small, the

measuring instruments would have had to be extremely sensitive

and the apparatus would thus have had to be very rigidly mounted,

closely toleranced, and expensive, which was impractical. By

concentra ting upon the variation of whirl amplitude with ac cele ration,

it was hoped that it would be possible to keep the costs low and

design a very simple, yet reliable, apparatus that would give

reproducible re sults.

The system finally selected, (Figure 5), adequately fulfilled

these expectations. The shaft under test was vertically mounted in

self-aligning bail bearings. The vertical attitude of the shaft per­

mitted the deflection under its own weight to be neglected and the

self-aligning bearings approximated to simpJe supports. The pro­

vision of a simple, almost frictionless, universal joint at the driving

end of the shaft, re duced additional stiffne ss the re to a minimum,

whilst its location cio se to the bea rings ensured that any speed

variation across the joint would be negligible. Interchangeable

sleeve s in the self-aligning bea rings permitted the substitution of

various sizes of test shafts, although ali the shafts were slender

in order that the moments of inertia should be as small as possible.

Plate P.

38

A;-mounted in a single row ùeep-Lroove ball bearil16·

Shaft; -mounted in 8elf-ali~nir~ ball bearings.

Take-up spool B.

Unit mounted in s e li'-alit:ninc l.Jall bearint:;;;s.

.Bronze wire.

'ii erome ter t;auce t-1.

c.

'n'eight W.

Ayparutus .

39

Since the bea rings we re qui te free- running this meant that even at

high accelerations the torque in the shaft would be low and its

effects could be neglected, (the torque required to overcome shaft

inertia was never more than . 27 lb. -in. for maximum acceleration).

The falling weight W provided the motive force which was

transmitted to the shaft smoothly and positively by the system of

reels and take-up spools. Since the force was virtually constant

and the viscous friction in such a system is negligible, the shaft

reels and spool are subjected to a steady acceleration. Thus if

J shaft, J A' J B and J C we re the moments of ine rtia of the test

shaft, re el A, spool B and reel C re spectively and ca) f , w sha t A'

wB and wC were their corresponding angular accelerations then,

neglecting friction:-

where re = radius of reel C and x= linear acceleration of weight

W. The linen cord and the bronze wire were both extremely light

and flexible and could the refere be ignored.

In this system ca) h f = c.) , w = ca)C and x= re c.)C. s at A B

Suppose rA= radius of reel A and rB =radius of spool B, then

. . rA rA

wC = wB = rB w A = rB w shaft ·

Thus equation 36 be come s ....

rA re rA . (Jshaft + JA' + (JB + JC)-'] c.) h f =W. rC (i - -- w ).

rB s a t g rB shaft

Eq. 37.

40

Eq. 38 w = ----------------------------------~~---shaft r W. r t. r A C A

Jshaft + J A+ (JB +Je) rB + --g- rB

w = ---------k, + ka. w

Eq. 39

where k1

and ka. are constants for a given system.

It is evident that w f may be varied by changing W or the sha t

moment of inertia of any part of the system. In this case the

moments of ine rtia we re made fairly large and an adequate range

of values of acceleration could be obtained by merely altering W --

note that if the moments of inertia were negligible then g r

wshaft ~ r rB and variation of W would have no effect . CA

A stop-watch was used to measure the time taken by the

weight to accelerate the shaft from rest through a known number

of revolutions and hence the acceleration could be computed

directly. The maximum amplitude o f whirl was determined by

means of the micrometer gauge M, which was screw ed in until

the indicator paper clipped to the gauge plate P was marked by

the thin smear of eng ine er' s blue on the a ccelerating shaft.

41

PROCEDURE

The shaft to be tested was mounted in the apparatus, as

shown in Figure 5, and then accelerated through its critical speed

by allowing weight W to fall a distance of approximately 11 feet.

After each run , the micrometer gauge was adjusted until a blue mark

showed that the shaft had just touched the indicator paper on plate P

during its acceleration. An average of 12 runs was required to find

this position, but the results were consistent and reproducible, and

the gauge wa s found to me a sure the maximum amplitude with an

accuracy of.±.. 001 in. ; a change of gauge position of . 002 in.

producing either excessively heavy marking or no marking at ali ,

depending upon the direction of movement.

Subtraction of the gauge reading at this position, from the

reading when the paper was just touching the stationary shaft,

("zero reading"), gave the maximum amplitude directly and the

result was checked by the insertion of feeler gauges between the

paper and the stationary shaft. Equal mark density on the "zero

reading" and "maximum amplitude" papers served as an additional

check on the accuracy of the readings.

The weight was timed over every run and the standard deviation

of the times calculated for each value of W. It was found that a

deviation of less than . 25 secs. could be consistently achieved and

this was judged compatible with probable operator reaction times

and stop-watch accuracy. The latter was checked by comparing two

stop-watche s with one another ove r 1 sec. , 5 sec. , 20 sec. , 1 min.

and 5 min. intervals and against a rated watch over a period of a

quarter of an hour without detecting any appreciable error.

42

Since the transmission system was completely free from slip,

descent of the weight through a known distance would accelerate the

shaft through a known number of revolutions. The weights of the cord

and wire were negligible, and the two reels were separated by a

sufficiently large distance for the line of action of the linen cord

to be virtually constant, hence the angular acceleration of the shaft 2x number of revs. of shaft

could be assumed constant and equal to (time of fall)2 ·

Once the maximum amplitude at a given acceleration had been found

the weight W was changed and the maximum amplitude was determined

for the new ac cele ration.

Passage through the critical speed was readily detected by the

strong shaft vibrations experienced at that point and efforts were made

to find out whether sufficient energy was being absorbed by the shaft

at is critical speed to effectively alter the rate of acceleration. Tests

were run at accelerations raising the shaft to speeds just above or

below the critical, and the times and weights recorded were compared,

but no significant energy absorption could be detected:- inertias and

weights had been deliberately kept high when designing the apparatus

in arder to eliminate such effects.

When the maximum amplitude had been found for an adequate

range of accelerations, the shaft was changed for another of the same

length,(42 in. overall, 40 :6

in. between bearing centers), but

different size or material, and the whole procedure was repeated.

A total of twelve shafts was tested in this way consisting of three

shaft materials, (SAE 1020 steel, Alcan 65 ST aluminum alloy and

Tobin bronze), in four different sizes, {3/8, 1/2, 5/8 and 3/4 in.

diameter), the pairs of brass bushes for each shaft size enabling the

same bearings to be used throughout.

43

The weights used varied from approximately 3 lbs. to nearly

50 lbs. and gave shaft accelerations of from 5 to 50 revs. /sec. 2

.

Howeve r it proved impracticable to test any shaft ove r the full range

because the duration of each run was strictly limited by the dropping

height and only the smallest shafts attained their critical speeds at

the lower value of acceleration. In addition the number of runs at

high accelerations was kept as low as possible, since they placed

a severe strain on the apparatus and dropping a 50 lb. weight not

only involved severe handling difficultie s, but also required the

replacement of the bronze wire after every drop due to failure of

the wire on the over- run.

A further problem at high accelerations was the fact that the

maximum amplitude of the shaft on acceleration through the critical

speed was less than that on deceleration through the critical, hence

very careful observation was needed to tell whether the shaft was

marking the indicator paper on acceleration, or only on the over- run;

i f the amplitude w a s small it was virtually impossible.

44

TABLE OF OBSERVATIONS

Length of shafts = 42 in. Distance between bearing centers = 40 :6

in.

T = time taken to accelerate the shaft smoothly through N

revolutions (correct to nearest 1/10th of a second}.

r = maximum amplitude ( + . 001 in.} of center of shaft max. -

during pe riod of acceleration.

Ste el Shaft s

Shaft No. 1.

Nominal dia. = 3/4 in.

T secs. 2.8 2.6

N revs. 58~ 4

60

r in. . 050 . 049 max.

Shaft No. 2.

Nominal dia. = 5/8 in.

T secs.

N revs.

r in. max.

Shaft No. 3.

4. 1

63 ·

. 761

3.7

63

. 602

Nominal dia. = 1/2 in.

T secs.

N revs.

r in. max.

3.8

63

. 100

3. 6

63

. 091

2.4

60

.048

3.4

63

. 478

3. 1

63

.074

Actual dia. =. 749 in.

2.0 1.7

60 60

. 048 . 048

Actual dia. = . 624 in.

3. 1

63

. 398

2.9

63

. 305

2. 5

63

.126

Actual dia. = . 500 in.

2.8

63

. 047

2.4

63

. 039

45

Shaft No. 4.

Nominal dia . = 3/8 in. Actual dia. = . 374 in.

T secs. 4.9 4.2 3. 6 3.4 2.9 2.4

N revs. 63 63 63 63 63 63

r in. .213 . 179 . 160 . 137 .124 . 090 max.

Aluminum Shaft s

Shaft No. 5.

Nominal dia. = 3/4 in. Actual dia. = . 749 in.

T secs. 2.5 2.3 2.2 1.9 1.7

N revs. 59 59 60 60 60

r 1n. .084 . 081 . 078 . 072 .070 max.

Shaft No. 6.

Nominal dia. = 5/8 in . Actual dia. =. 624 in.

T secs. 2.9 2.6 2.3 2. 1 1.8

N revs. 63 63 63 63 63

r in. . 445 max.

. 405 . 348 . 346 . 339

Shaft No. 7.

Nominal dia. = 1/2 in. Actual dia. =. 499 in.

T secs. 3.4 3.0 2.8 2.6 2. 4 2.1

N revs. 63 63 63 63 63 63

r in. 1. 050 . 676 . 594 . 500 . 41 4 . 266 max.

Shaft No. 8.

Nominal dia. = 3/8 in.

T secs.

N revs.

r 1n. max.

5.0

63

. 464

3.7

63

. 365

46

3.2

63

. 275

Bronze Shafts

Shaft No. 9.

Nominal dia. = 3/4 in.

T secs. 2.8 2.6

N revs. 58~ 4

60

r in. . 050 . 049 max.

Shaft No. 10.

Nominal dia. = 5/8 in.

T secs.

N revs.

r in. max.

Shaft No. 1 i.

4. 1

63

. 761

3.7

63

. 602

Nominal dia. = 1/2 in.

T secs.

N revs.

r in. max.

3.8

63

.100

3.6

63

.091

2.4

60

. 048

3.4

63

. 478

3. 1

63

. 074

Actual dia. = . 37 6 in.

2.8

63

. 222

2.4

63

.174

2.2

63

.147

Actual dia. = . 749 in.

2.0 1.7

60 60

. 048 .048

Actual dia. = • 624 in.

3. 1

63

. 398

2.9

63

. 305

Actual dia. = . 499

2.8

63

. 047

2.4

63

. 039

2.5

63

.126

47

Shaft No. 12.

Nominal dia. = 3/8 in. Actual dia. =. 376 in.

T secs. 4.9 4.2 3. 6 3.4 2.9 2.4

N revs. 63 63 63 63 63 63

r 1n. .213 . 179 . 160 .137 . 124 . 090 max.

48

TABLE OF RESULTS

Comparisons of observed maximum shaft amplitudes and

those predicted by the equation for an undamped rotor ....

.K!:. r = K

1 + r-: ,

max. Il/ w

the constants K 1 and Ka. being determined by substitution of two

of the observed values of r max.

2N Angular acceleration of shaft, w =----,_

T

Shaft No. 1. . 2

w revs./sec. ;

r in. {observed); max.

Equation;

r in. (predicted); max.

Shaft No. 2.

. 1 2 ..., revs. sec. ;

r in. (observed); max.

Equation;

r in. (predicted); max.

Shaft No. 3. . 2

w revs. /sec. ;

r in. {observed); max.

Equation;

r in. (predicted); max.

Steel Shafts

15

.124

23

.113

31

.105

.:..ill. r =. 071 + r-:-max 11 (a)

.1.24

1.5

. 445

r max.

. 440

1.1.

1.. 050

r max.

. 882

.1.1.4 . 1. 08

1.9 24

. 405 . 348

1.340 =. 097 +j w . 405 . 370

14 1.6

. 676 • 594

5.32 =- .724+,r-;:;-

. 71.2 . 606

48

. 1.01.

. 1.01.

29

. 346

. 346

1.9

. 500

. 500

39

. 339

. 31.2

3 . -'' d1a 4 .

i" dia 8 .

1. . 2" d1a.

22 29

. 41.4 . 266

. 41.3 . 266

49

Shaft No. 4. ~" dia 8 .

C:, revs./sec. 2

5.0 9 . 2 12 16 22 26 ;

r in. {observed); . 464 . 365 . 275 . 222 .174 . 147 max.

1. 376 Equation; r =- . 123 +J max. (Il)

r in. (predicted); . 493 . 321 . 275 . 221 . 171 .147 max.

Aluminum Shafts

Shaft No. 5. 3 . 4

" d1a.

c.i revs. /sec. 2

19 22 25 42 ; 33

r in. (observed); . 084 . 081 . 078 . 072 .070 max.

Equation; = . 041 + J .1866 r . max. (Il)

r in. {predicted; . 084 . 081 . 078 . 07 3 . 070 max.

Shaft No. 6. %" dia.

c.i revs./sec . 2

13 15 22 26 ; 17

r in. (observed); . 604 . 527 . 478 . 369 . 303 max.

Equation; 3.70

r = - . 422 + r-;;-max.

r in. {predicted); . 604 . 534 . 476 . 372 . 303 max.

Shaft No. 7. 1 . -•• d1a 2 .

c.i revs./sec. 2

9 . 2 13 26 ; 15 22

r in. (observed); . 785 . 611 . 551 . 416 . 361 max.

Equation; ..LQ1.

r = - . 242 + ;-;;;-max.

r in. (predicted); . 773 . 611 . 552 . 414 . 361 max.

50

Shaft No. 8. 3 0 B 11 d1a.

~ revs./sec. 2

6.8 8.3 12 15 24 29 ;

r in. {ob se rved); .913 . 806 . 629 . 505 0 352 0 231 max.

Equation; ~ r = - . 432 + r-;:; max.

r in . {predicted); . 937 . 806 . 599 0 490 . 296 . 231 max.

Bronze Shafts

Shaft No. 9. ! " dia.

~ revs./sec. 2

; 15 18 21 30 42

r in. (observed); . 050 . 049 . 048 .048 .048 max.

Equation; =. 0432 \J . Q2~5 r . max. (a)

r in. {predicted); . 0500 . 0495 0 0490 . 0480 0 0473 max.

Shaft No. 10. 5 . -" d1a 8 .

~ revs. /sec. 2

; 7.5 9.2 11 13 15 20

r in. {observed); . 761 0 602 . 478 . 398 . 305 0126 max.

Equation; 4.4S

r =- .875+;-;:;-max.

r in. {predicted); 0 761 . 603 . 485 . 368 . 281 .126 max.

Shaft No. 11. 1 2" dia.

,;, revs. /sec. 2

8.7 ; 9.7 13 16 22

r in. {observed); . 100 max.

. 091 . 074 . 047 . 039

Equation; ~ r =-. 064 +r-;:; max.

r in. (predicted); .100 max.

. 092 .070 . 057 . 039

51

Shaft No. 12. 3 . -" d1a 8 .

w revs. /sec. 2

5.2 7.1 9.7 11 15 22 ;

r in. {observed); . 213 . 179 . 160 .137 . 124 . 090 max.

Equation; ~ r =- .026 +JT max.

r in. {predicted); . 213 .178 .150 .140 .115 . 090 max.

52

GRAPHS

The effect of angular acceleration upon maximum amplitude

of whirl is represented graphically in the following figures.

The values of K1

and K:z. derived for each shaft have been

sub stituted in the equation r = K + Jft:.; and in figures 6 - 17 max. 1 lit/ w

the observed results, shown in red, are compared with these

predicted curve s of amplitude against e.ngular acceleration.

Collation of the results for different shafts is further facilitated

by the provision ofboth dimensional and non-dimensional scales,

the symbol .A indicating the non-dimensional form of amplitude

maximum amplitude (r ), in. { = max. }

radius of gyration of shaft, in.

and A the non-dimensional form of angular acceleration

_ angular acceleration (w ), rads. 1 sec.2. {- (natural angular frequency, rads./ sec~) } '

whilst in figures 18- 23 the predicted curves are brought together

for each shaft material, figures 18 - 20 showing the non-dimensional,

and figures 21 - 23 the dimensional curves.

5 3

-l- li'k 1- l i-

'

_-t- t-~-l_ Ci 1

1-t-!

1 '\ J.

. • 1~ i-L- J\ i'' ±=-- _ j t-

!-'

+++-+-+ -+-t--+-t-1--+--t- t-

~ i

-

l l

i \ -

- 1 ~f'\ 1 1

'

- ~t.~ 1 1-

~t~ l-ll- 1--: '" : ~tlB.

f\ ! - i- i rh - - -

- - t- t-

-f+ _;l+--t-

['\ -t - 1 -

i- - t .

' - 1 1

- ~~ -t--

1ti 1-t-

rr 1- - - 1- -+

- t~ 1 .·

t- 1 1

~ . - 1-

t- - Î 1-

1 ..

l-1- -+-t-t-t-+-t-4 -t-1--+-t-H-11--+-t L! t-

i -r- - 1-+-- ++--r+·H-f-I--1L+++-,....t+·H-H-t---t--t---H·--t- l +r ÏT -1--' t- - t-1-H-i

r-- t++4+~~HH~~++~+r~rH~-r-G~+-.,~....,_~_tt~~~~_t~' ~ - î---r-r-r-t- t

1 r t ~ - +- ï 1:--+---1 1

r-+-'-t--r r J. ' 1 1 1 1 '

j - tl -H-t-t-t-t--t--t--r-t-li i+-+ -+--+---+ u ~ :_1-=t p-l±_j :o ~- li _rl-; ~ 1 -~--

1

[ 1 H-t- H -r, o · re 1r s ~Ta ~è :ii ~ W-q ++++HH-+H++++-++1--+-t-H-l (H J±! .htr~H-:-rr t.c11 1 ht1 ·- :nM 1--"--P iio_I

~~

1-t- r-1-1

J

1 ~.

1 1 ~ t '"'! -+

j.. •---r-.

• 1 • +--

t 1 , •

1-r -171 • 1 _ _,

-·~- ~ ' -T--t

~ j .

T--t--+- ·

1

- 1

A~ r"'ax. ;y,

. L

· p,n 1

. - 6 ~

1! . ~

1 •

1,__

f • -

1 •

. t -·

ï~:.-t ;- 1 ~ : ~~ 1--4. t 1---,-'

• .... 1- t r .... Î.,.

-·- i · ' r.~. ·

t ......... i

- ~ -,-"1' ,_ - i l

_, ' l-

!

~ t . L ~ . .

1

1 •

" l

1- -~ --t'" -·

r

~ 4.

1

3

1 1 • .

r-i . ! ( 1 i .

rmu. . i Y1

' . An

' (()

11 ·n7 .ln

-~ - -- -- J

Î 1

. !

\ - '

:\

54 J(

\ : f\ :.

1-

- t l -t

. . '

00? '

. Î

• 1 . 1

1\ "

0

•. 1

l ... --

1 . .. . .. r ' " ~

+ .

1

l/2't dia. stea!L l 'sha "t ... ; . ~-

J t- . ..;....

+ 1 ' 1

f • •

t • ' l 1 l

' . ~ -i j 1 i

. '

r • , t

, 1

1---t --r--- --t ... ! t - · .

' · · J - r ·-,

1 1 · 1 nn,;f .. . .....,

. -' 0 "' ; · b 2. .. ~:, . _ , re s . s pC • fA"

• 1

' li'iu q

• t ' l. e. •.•

_, 1

3/8'' dia. stee.,..· sha!ft.

:~~ . . . 1: "r-.:.

"":::::: . ··-~ : . !---

---r---

• 0 revs ./s~c: ~

( ·---+---+--+- ...u nlun •. A'H--_,._· -l---+-----0.-ln...t.lj.la- ----1- -·a~-

.

55

-t: 1- 1- -+ ;t= ~ -h-l.là .. 1 1 1

' _)_ """ . ~ -1-r- - Î f-

r- f-· r- -

1 ' 1 1

~l-

- f- 1- - r-T 1- 1--

- -

ïlllm~ -tJi 1 ' 111;-;rt PL Fl .

\. ,_ 1-

1 r-

1'--I Ll

r-

....

H-_±

· l 1 T'

~7a· ts 12.~ i . -. u ~ a.c · i lJ.

' ' ; lJill i - - bn r--

--

i l"" .

r-s 1-- ,h a.:·11: - - -- ,-- ,_

1- r- - ! a 1-- - f- - - r-- - r- r-- f- 1--. -.( t-- 1- r-- 1-. 1

- r- - ~ ..... !:-- 1-- r- -

\.-. 1"---' ~1 l -r-- - r-

t-J~rr -+ - r- h-- -t+

1 :-""=r +~--- 1 _j___ r r- ï ·r

1 1 1 j_ 1 : - -1 - 1 --~ :+ 1 -ir f-

1 1- 1 j-+-- -- - 1 .!_l H-- 1 l

J ' ll 1

- ~ ~ r-- 1 - 1- -, cr. -:rë &~-y;JJ ' 2. 1-j-~ rJ± 4-j- ' ilG,. - t -1

L 1 ' ~ m +- , ! !+ oo.a. 1 1 f+L li +-ri f-

-+-r- ±tL- -~~ Et 1

:_~t +i - 1 -- ' ! ! · ~; 1--+ I . t !- ..:...w l - j- r-- H-i-__.__

1 'l - . r,_t , 1 ;- h--Y+ -i 1 -~-~= ' • t

~-r~r 1 ' i . t ,_._.1_,

56

}\! ~.J.. f- 1-1

1 1 -:- ~ 1- 1- i 1- 1- 1-

_l 1-:..!1 ~: 1- f-t 1 .L h '

1 1 1 : : r.-· +

-t=' ~ -~ -- T :

tn R ~ 1

1-

1- - 1-

1-

l( 1 1-

1-

· 1- 1

- -

_(') '-!.re lr 6-!1 ~Fi 12.

Fi-P.~ -

ft"' lrtr:

- -

i

l' l\ +-

-1- · ir ·- 1- -

- 1\ t=r H~~~ ~ ~L . n· - 1-- - - 1 i i

_J_ L , • - Jr - - - -ri + - 1-1 - - t 1

11 ;rn 1 - - - 1- 1-

~ . ~ 1- . 1 - ~! . - 1-- -['\ ·- c-+ - i 1-

-i ~i . _j..

! - 1- 1- ._._ I T ït- -j 1

:jj_ ~ ± -+ 1 _] p._ W-i--

' 1- ......._

1- .-r + 1 1----- ·j . -1

- 1--t t-L -t---

1- - 1- 1- - i'-~·- -'11 H- -+--

J. r---.,..1 1 1 1

~: ·- • .j_ 1- 1-1 1 .

~i 1

i rLI- p ' + 1-- l - 1 1- .-::-~ j 1 .!. H ~- ~~--- q . tr - ( - r-- rr- - 1 t- -f 1 '

1-- - 1- - H -! 1 ' ' r Li ll~ JL 1 1

t- ~--~- a· ,.,+ ~~~ .

,-~< o·:rè ~s.~;.s ,_ ~ - ! - ~ c .. . 1 1 - --'-f-1 i r f- tL .. H. nn~ !+ ·:-+ i 1- f~-=+-H f- ( - iCIF ... ', ( Il? 1 1-

h -; 1-' [ tt-+ r-j r- '

=~ ~] J--,_j lliJ l:rJ J j ... -

- - f-'--f-- . __j j - -f-j- H- : : r-t. h · -~~ -~ ~ t:f· 1 .. t+ - w--f+ ~ -t= ·-r1 J_j_j l 1-- -

-+-1=-1 --r--f" - j - - L ' ' r

------~--------~l

57

'

1 t i 1. - -t •

j 1 l • 1 1

; 1 • ' -1 ~

' 1 + . t • . 1-- l j ••

-,-· ~ _;_ t- ~Î L. 1 · -· • - -+--,

11 ; • •· .• • T .

't-- + -1 i j . 1 . . ~ ,.... rf .

1 • . -.f- .-+1'

• 1 ·- · trJ t ~-t7- , r . i- ...- 1

- .. f i ~-· -rl-j 1 1 - . t~-- i

• t -,-J ·- • Î ' ' 1

. t t

1-j L : , 1 1

; j :-t _;_

' , ,..l..... - . 1 ~

t l 1 t 1 .

1 +

j .

j ••

()Q 1 1 •

.6 r ' • -

1 1 .

. ï ~ . --· ! . • •• -~ 1 • j t •

·-: H 't 1 . t .j.....; •• ' •

1"1 ' ..,{ C1~

t ~-...-- . '-l-- __ __. _ _._

• • 1

+--t--!-..... . ...J_

··t~t-; t-_J

1

·-+

' ' 1 . 1 •

' t

• .L f; 1

1 t : ! L . 1 ... +

(

(

 . . ' ; r"'~~-r 1 Yl

:

?()

t

.2,0

1

' 1~

1.5

t 1

.10 .l.U

•• t ,1 •

1 t .

·-~ -·-· -· -r

.. .•. 1 . . - ~

; t 1

• j • 1

; .

i ;

·\·: ..

1 • l . • 1 •

r•

. ' '

58

\ .

.. t· 1 .

...

• , L.

...

-t- 1 --.j'

• 1 ~ . . .

1- • • ·-

1-i-r-· . t-r- ..

'.,

1 t·•--

1·"""

1

-~Ti.i îil ;-r=

:' . 1 + ·- T : ' .

t 1 . 1 r

1-~---.--+ r- . ' ... --

~· . _, -t

- . . 1 l'iu

' , 1 1

/2'1 . (ii.

-· t j

1 . ~ t

- ..... - .. . ~- ·- . ~

.16 ~ . j .

proilz~ shalft ..

!

' t

• L 1--h- •! · r · . 1 • -· -- - 1. ,-~~- .. .

---1--j - ... ·-·- f-·- . . .. t. ,

1 •.• ' '1 , , .. 1--., -;-,-

. 1 '

' ~ .~ : . . . . ! .! -- ·--, ~ ~-: H-·-~""' . . , -. . .. . . .. L..J.. "'-.. 1 • J 1

. !..- . ' , - ' ·"-..: 1 t 1

·j t·• 1- H' T 1 ~- "} ... ' t - , 1-t.. r.· ·. 1 'Ç,___ : 1 1 L,- 1 i.. ,. . . . -......:. 1 --l-l. + --· ·-- ' 1 '

O. ri-; ·r··- l. o· · 1 2.. t ~. ,_ - ; , , - ~ • : re s . s ~ c ..

()(A

1 ... t-

- - . ; : . i

• 1 • 1 1 • ,. . '

\ . ·\·· . . .

' 1

' 1

' ' ' • 1

1 •. '

• t . t +-

· Fi.2' l7

1 • /8" . i$.. pronz ~ she!ft'. '

< 0 .0 re~s./s~o~ ~---+----4-----~~-+----4-----~---+~--4-----~~-+----4-•

~---+----+----4----*( ---~----~--· ~L~()----4----+----~·~·()~~.~~ ------ ---

. ~-r--

1

i I

59

1 + 1- Lf-+- ffir· H- f- i 1 . 1 r,' i -:i-:1

1 1

r-F 1 l-L d-'-h

1 1

\-Y+1 ·3 '1' 1- 1- - 1 · i .H~f 1'1-1-- Il 1

.!- ~-H

_! ~-~tt

1

1 1- -l- - L ~lt 1-ttt--~~{-- --t· 1 - .1~ 1

~~\ 1--' -~- 1 1 1 hlr :-- -~-+- - --~ -l-+ ti= ~ili~ t:: - - , ...

++ ! 1 1

1-+-1-++ f-H'-+-+-t-++++-1-H-l-+1"\-:-1-++ + l 1 1 _L rT1 +-1

++-H-+-H - j_.1

1-l--1--l-l-l--+-l-IT ...... ffi 1 L +-t-- .( -- r-l.

1 1 1 -ri-- ; 1 H-1-+1-1-1-- - -

t-+-1-+-t-H-++-1-++-H- -1-++~·1-f~ ~ff -t-+-t-fl-j 1-1-+-+-1-+-l +++-~- 1 1 1 1 1 ti- ~ .::..: t 1

H -+-H-+-H-++-

J-+-t--1-H-l-t--t-+++++-,,- t !- \ 1-l t--1 [\ "'.! --+t - t...; T n- H 1 j-.,.. ( t 1 1-i- !-_ '-l ~! ~ ;j ~~- - 1'-.._. ... _, : P -l,l·-- 1 H-- i

1-HH-+-+++~+~~·--H-1·-+-1_-+-1-l-ql 1 ~1---'N-t-t- N f- + =1)-: ... - 11l 1t , .~~ +t- r-=t ri tl r

60

lf:t r~-+t ; :_j ~ ~m dJ ~ !Tt 1

1

, P\ 1 l'i !- !'~ 1 - - 1 - 1 !- - - ~ f\ 1

1 [j 1 11 I.J \J L t " !- R~ I-l !-1

·!--~ 1

lili !i:~t t- !-ë_èf .-1

t-r -~ 'i..

. !- ~ ~ 1-h-x t-

~ !- 1+ +--ri- 1 1 )-.. t" l J+ 1\ t l - 1 ! 1

f- R-.t 1 t- t- -

' 1 " 1 1-l± 1 1 ....... _ J'.

k - ~"l_ !t-l r l rt lA.. 1--.-1 1 ' 1 ' ' J ' 1 h ._

1 H-~-1 '!-.. -;..... : nJ ~ ' k]~. ' i T 1 'lll v J th ' N-. '

h -i '7" 'f!:.:... _j __ j t-? . 1 ~ -f/i li':' . éli. p.. ~~ t- ++-- -t t-t- j-+--1-1- L:-~

t~ ·-dr ~l: ~h ~-+ r-,J i-+-+ t=f - .J __ - ï' -l-

i 1 ,_] LJ- j· 1 ' ~~ 1 1 1 ' '

H-i~-~ ::c ~ - 1 ~ : rer: -1 k-6 . f' ~· ~_s~ ~t}_... l ~ dt -i + 1 1 ~ 1

-r- mt= =i' ;- 1 j h-'t tJ-t- !- 'J 18= l 1-- hl/0 . _ru..a 1 1 --LLJ_

H- l'-# !

~-tt ~~-±-f 1~· +r --~\ -If_ f-~ f- 1

ri-: ·, r- ~~-- - 1 _...L 1 1 1

t-f++ -H+-~ 1- -1 ·1 ++ -W+ 'J~ -' 1 1 1

~ - 1

j ll' ~+t-- +l ~ 1 Hi! '1- r&- ' >- ·! + ~--+ l i ' " 1

!- -L +- -+-- 1- ~

~ -· ·-H- -f--,-+ -- t -,- .L . .j

1

+.:rr Jt -t + - lÎI ~m~ ~ùTn shr "'ta~ : I'T! 1 J_ T :_ ~ L . t r lr - ---l i..,. i

_. t ..- l t- i : 1 : 1 1 '

-'tt .tl - _, _ f-1-i-1 Î' ~~ ~~

l"~ d~~ · -r--i- ~+ · rr l,-~ Jti 1--!' _! ;_; +--- -.l

va~~ rf ·-l- - ' __.__ ·!-- --t-- !--;=-ct tf=t- ----t-.4- +-H + -i-f- ~~-=+= ~=R=

_J -1 '-t -?' · Ti- +; ~-r t ' 1 - :-+T 1 -j

!---1-- +--1 it - L;_. J : r f-- ; t-Lij _j__ 1-Ti ~-'- ~+ ~_. ~:t -p! -1 r± 1---h f- ;- ! --1 §! ~~~

~ -=t' j ~=_;_ ~1 !_ 1

sfJ l- ---1 t--LQ- ï -t-i--~ i 1 1 r-;- - i ---: -1-1-- T . . lî ' ,--~

1

Il 1

1 1 lt-l-- i-t--~ · ~( -f-1

tt! ~ ~d : rè' re~/s ~èi. ._ç ~~±î ,_, ' . - T Q_j~ 1 ' ~c . :J_ - ,-L- 1

-If ~ -!_ ___ J

l-- ' lj-r 4- li~ 1 1 j h- ! r 1 r i t-+ ' 1~ 1 ; t-j - -f-ffi -i i i 1 1 1 - - jT

-1- '~i . • ! ! f 1 ; -~ 1 ' 1 - l-' ~ i ~ -~ r -, f • ï r l l +-· :-r" . - - -

r.lll • . 1 '

\~ii ~~ 1 CL flCL f li1 -· 1 ~L 1--

H 1 :-i ~ -t 1 _j_.__~ -' i l- r--; 4iu 23

- - ll~ !~~ IJ_ t-T -T-R-- !-+ ~ . t--- . ;-t ~ Ft: ~-~rti 1

!- r 1 ~>onze shtl+ t-f- ll·t+ -ïrl-1 Î ~ 1 ·1-r ~~ 1-i-t-- 1 t ~llJ • 1 ti 1 i Il i Tt-1

i+-_L_,! -·-1 1 J - FE -+_Li__ ;~ f---i-- 1 ~- -1-t--L t i 1 --+---- · ~ ' +·TI ---': · -

ht J : 1 c:: ~- ~ -

1 . 3(:4 ~ ~~ ~-~rn -~ .tt-~ 1 tli ~ t t- t- - 1- ~- ti - (_L - ~-+t - r - +-

t 1

1 ~ii= ~!l p-_ ~ ~f- f-L! lf p j r-tm !- f -

llft- ti_!_ i i- j ' ~r- - 1r -----+-+-

~~+t Fh 1 1 1

1( . ~ ......... ...... - ~ Ir n T, -+--+-1

. ,_,___'b,

~ ~.1 1 - 1 "-1-- • . : Lt-- _, T T -f-+-t- ! --ii!_ - l++ +-- .

1 j ·-+

~ ~ J.

v.:Jit ~-dia l'T'-'-- ~ ~~ ~t!f - 1 . . r+ . J. - . - 1+-:P::t . '·]= t-1 -H i-r-1 1 _ _j_ ,---j- . .. -. . l H--i - .. . 1_Lj :

1 i c 1 ' 1 1 f2 'f. .iL , r -1-+~ 1 J H ~ ( 1

1 ~- 1 ~-~ +- H- H w.~ lrs-.-J~ ~c~___j__ ,~) +tt -1

1.---4 rl- 0 . ,-- fFi ·-~ ~d r . + r __ .__ 1 -.

1 1 - i 1 1 j

-- ; f -· ·-+ t-t= 1 ~- G+-r-- i 1 ; l n_f ,_ ~---

1 1 1 --. -+ 1 1 -i . 1--- !- !- 1- î r -j·

1 1 ~{-f-)- - 1 1 :1 :-f::-- LLj 1 1

1 1 -t it -t4 Ht - 1--

L 1---1~± ~-~ ~-+ - ~- - r--ft 'i - • l ---.--:----: ; !- -+ --1 -1-

- H.-- ~1---rl~-1 - - - H-1 1 - 1 !-j_ r-- ! : l 1 ·+ - +-1 + ·--+- 1

1 J.__j_

61

DISCUSSION OF RESULTS

lt is important when considering the results of any investigation

to bear in mind both the initial as sumptions in the theoretical analysis

and the accuracy of the experimental observations.

Ample evidence exists, as has already been shown, that

Jeffcott1 s theory correctly describes the mechanism of whirling for

a shaft under steady conditions and it seems reasonable therefore

that the theory should also be applicable to transient conditions.

By the same token, since the behaviour of a round uniform shaft,

in the region of its primary critical speed under steady conditions

is analogous to that of an ideal single rotor, the same relationship

should hold under transient conditions . It has also been observed

that the viscous damping of a typical rotor system is low compared

with its stiffness and, since only heavy damping has much effect on

an ACCELERATING rotor, undamped theory will be applicable.

Unfortunately, facilities were not available for determining

the mass, stiffness, and unbalance of each shaft in terms of its

ideal rotor equivalent, and the only measurements possible were

the maximum amplitudes and the a~celerations. However, by a

series of bold approximations in the theoretical analysis of an

accelerating ideal rotor, it has been shown that in the undamped

case, the maximum amplitude of whirl, r ~ K + ~ where K1 max. 1 Il/ w

and Kz. are constants. The los s in accuracy is justified since the

original solutions were too complex for general use, whereas the

new formula is attractively simple and clearly indicate s the effect

of acceleration upon whirl amplitude; obviously at low rates of

acceleration, r œ ~. max. Il/ w

62

The experimental observations that have been made, provide

the rneans to establish the validity of this relationship, and thus justify

the assumptions upon which it is based. The accuracy of these obser-

vations, within the lirnits of the apparatus, was ensured by using

simple techniques and by rnaking an average of twelve drops per

shaft with each weight. The maximum amplitude recorded was

readily checked, as described earlier, and the value of the weights

used provided a guide to the accuracy of the acceleration readings.

On the ba sis of this data it is e stirnated that, allowing for errors

due to flexure of the apparatus, the maximum amplitude was

recorded with an accuracy of+ 2°/o, and the accelerations are correct

to within + 10°/o. This is a satisfactory performance for such a

simple apparatus and adequately indicates the influence of acceleration

upon whirl amplitude.

The intuitive expectancy that increasing the acceleration decreases

the maximum amplitude of whirl is confirrned, although the experirnents

indicate that the change is greatest at low values of a c celeration and

there appears to be a limit below which the amplitude cannot be reduced

by increasing the acceleration. . Kz..

The equatlon r = K + ,-: also max. 1 Il/ (&)

predicts this re suit, and the figures 6 - 17 show that curve s obtained

f rom this equation give values of w a nd r that coincide with the max.

observed values within the lirnits of experimental accuracy. In sorne

cases K1

is negative, indicating that the amplitude w ill becorne zero

and then attain a negative minimum as the acceleration increases; this

seerns unlike ly and is c ontrary to Baker• s solution s , (1 4 ). It should b e

rernembered though that the formula for r cornpletely ignores the max.

rnovernent of the critical speed with change of acceleration and this Ieads

to appreciable erro r a t high values of ~. Neverthe less

63

at ail the accelerations te sted, the formula did hold and the exper-

iments thus offer convincing evidence of the validity of the analysis

and the utility of the approximation over a wide range, although

further work will be necessary before they are firmly established.

The analysis predicted that the amplitude of whirl of any

shaft was directly proportional to the initial unbalance E , thus the

deflection of a well-balanced shaft will be law irrespective of its

stiffness . Figures 18- 23 confirm that shaft amplitude at any

acceleration is not merely a function of material properties and/or

shaft dimensions. The small amplitudes obtained with the three

! 11 diameter shafts {Figures 21 - 23) indicate that balance is probably

the critical factor, since the absence of a sleeve between these shafts

and the bearings has eliminated a source of unbalance common to all the

others.

It was also shawn in the analysis of the solution that the max-

imum whirl amplitude of an undamped shaft is approximately

proportional to b =ft , thus for a shaft of given dimensions and

unbalance ....

r a max.

Modulus of elasticity of shaft material Shaft density

Considering the mate rials used in the se expe riments

b =b =1.1. b ' steel · aluminum · z bronze

therefore the amplitude of whirl of the bronze shafts s hould be

appreciably less than their steel a nd aluminum c ounte r parts, assum­

ing simila r d e grees of u nb a lance , and this hypothe sis is c onfirm e d

by figures 18 - 23. The increase of amplitude with stiffness may be

ascribed to the rai sing of the c r itical speed and the consequent

increase in the unbalanc e force s at the critical.

64

It will be observed however, that the amplitudes of the

aluminum shafts tend to be higher than that of their steel equivalents,

although b . ~ b , this discrepancy may be ascribed alum1num steel

to the neglect of damping. Geiger (i7) has ascertained by experiment

that when a steel crankshaft passes through a critical speed,

60 - 65 °/o of the total energy of torsional oscillation is conve rted

into heat by the internai damping of the material and that only the

remaining 40 - 35 °/o is accounted for by external damping at the

bearings and from windage. Although the proportions will vary

from material to material as the internai damping varies, the

damping characteristics of the individual metal will evidently have

a marked effect on whatever damping is present, and they may be

supposed to be of similar importance in a whirling shaft.

Kimball and Lovell (i8) place the metals aluminum, brass,

phosphor bronze, and mild steel, (ali cold rolled), in that arder

of increasing damping capacity, a result only partially confirmed

by Foppl {i9) however, who reports that aluminum and steel have

much stronger damping characteristics than bronze. Nevertheless

they ali agree that steel has the grea test damping capacity of the

materials named, and although the coefficient of viscous damping

of any practi cal rotor system is small, it is clearly sufficient for

the maximum amplitude of whirl of a steel shaft to be perceptibly

lower than that of a corresponding aluminum shaft at low accelerations.

The effect of damping however, is to reduce the sensitivity of a sys­

tem; therefore its response to a change of acceleration is also

reduced; hence the amplitude/acceleration curves are steeper for

aluminum than for steel, as shawn in figures 18, i9, 21 and 22.

65

CONCLUSIONS

It has been shown that angular acceleration has a strong

influence upon amplitudes of whirl at critical speeds. Increasing

the angular acceleration decreases the amplitude, the effect being

most marked at low accelerations.

Jeffcott' s theory of whirling, although difficult to apply, offers

an explanation of the phenomena observed and forecasts that the

maximum amplitude of whirl, r , is approximately equal to max.

(K1 +/!) for an ideal single undamped rotor; where ~ is the

angular acceleration, and K1

and K,_ are constants. Thus at low

ac cele rations ....

r œ ~ (approximately). max. Il/ w

The theory also predicts that the amplitude of whirl of such a

system is directly proportional to the eccentricity of the center

of mass with respect to the elastic center of the shaft; that the

amplitude increases as the stiffness increases; that damping reduces

the amplitude of whirl at low accelerations at the expense of reducing,

or even completely eliminating the effect, at high accelerations, and

that the effect of the stiffne s s and damping is inversely proportional

to the rotor mass.

It is claimed that undamped single rotor theory is applicable

to the practical case of a vertical simply supported, round, uniform

shaft and experiments carried out on a set of such systems have

provided results that accord with ideal rotor theory. Since only a

very simple apparatus was available for these tests however, further

work will be necessary to establish the validity of ali the predictions, and

toassess their applicability to more complex systems.

66

The complete solution is too complex for ready application

but its validity is sub stantiated by its ability to reproduce re sults

that agree with Baker' s (14) mathematical-machine solutions of

the same equation. It is suggested that it would be interesting to

repeat these experiments with both solid and round shafts, using

more sophisticated equipment which would allow the determination

of the equivalent shaft eccentricities, mass, stiffness, and damping.

The se re sults would not only permit verification of the relative

importance of each constant, but knowledge of typical figures would

enhance the value of Baker' s graphs.

67

APPENDIX

c.) t 2.

Je at+ { - 2- + bt)i dt EVALUATION OF THE INTEGRALS

AND

c.»t~

5 at + ( -

2- - bt)i

e dt

Consider the integral

wta.

St at + ( -

2- + bt)i

I = e dt. ' 0

Let z, =-iC:, 2 2 (t + u 1 )

where iwu t =(a+ bi)t 1

i.e. : i . 1

u = - -:- (a + b1) = -:- (b - ai) 1 (o) (o)

From equation A2 . . . . t = ~ - u1

dt ~ dZ

1 =,_j~

Suppose Z1

= Z 1t when t = 0, then

Thus

Zl 1

wta.

iw ~ -u =-2 '

i 2 2

C:, (b - ai)

Eq. A1.

Eq. A2.

Eq. A3.

Eq. A4.

I = 1 5t eat + ( -2

- + bt)i . dt=

0

rt -z + z , J6;. J e • ' . 2 . Z . dZ, 0 (o) 1

= jii•z,• l t e- 2 1

r-:7 dZ 1 IV z,

0

Let z, =x,+ iy1 where x1

and y1

are real numbers. Now

i w a. 2t 1 a. -2

[t + -. (b - ai) + "72: (b - 2abi - aa.)] . (o) (o)

z, --

Eq. A5.

68

Separating real and imaginary parts ....

ab x =- at--.-'

x, b But t = - - - -:­

a w

parabola {Figure 24).

and

Y, --. a. a&

6)[_.!.,_ ] h . f 2

a 1 - w., t e equat10n o a

This parabola is the path of integration in the z, -plane which

is equivalent to the real axis in the t-plane. Consider now the

integral r, = ./1i e2

•' J' ~ 0

dZ;; t is real and Z 1

is complex

Çt e -Z, hence ) ~ dZ, re presents an integration along the real axis

0 '

in a complex plane.

Point A on the parabola corresponds to the point z,t when

t = 0, whilst point B corresponds to Z 1

at time t. Now

It -Z

e • ,r2. dZ, only possesse s a singularity at the origin, thus

0 •

integrating along the path oo AB oo (shown in red), no singularity r r

is enclosed, hence by Cauchyt s theorem

S -z, Jz: dZ 1 = 0. c

Thus the integral may be written

St -z,

_e __ dZ =

0 JZ; Sz -z

_e __ dZ =

z ,JZ '

dZ- dZ.

y l

69

----------~------~~------~-------------------- x l

:a

~-----------1----------\-------------------- oo r

~--------------+---------------~-----------------~r

Pa th 01' in""Gl!u-re:. t.i.on in the z - plane . .L

70

If, in the integral 1 oor . . . , Z is put equal to {Z, f + V), whilst z, t

Soor

in the integral ... , Z is put equal to (Z1

+ V), where V is

z, a real number,then the equation becomes

1 -Z

= e 1 S oo e-V - Z soo e-V

Jz,' + V dV - e 1 ,J z, + v dV. 0 0

Thus

Let

r,=Jf [ io"",J z,~-:v dV- .z.•- z,J"" .-v 0 ,J z, +v

In the case of I~ ....

iéd 2 Za. =- T (t + u:L)

where iC:,u t =(a - bi)t, .... thus

i 1 u = - -:- {a - bi) = -:- {- b - ai) ~ (a) (o)

dV]

Eq. A6.

Eq. A7.

Eq. A8.

From equation A7 .... t = ~ - u . ·. dt ~ dZt--~ 2 c:, z,_ · ,J c.) 'l.

If Z2.. = Zi, when t = 0,

th en

Th us

z;_ = - i26, u: = - ;c.) (- b - ai)

2.

c.) tL

St at + ( -

2- - bt)i

r'l. = e dt 0

e 2. lt -Z

JZ: dZ1 .

0 2.

Eq. A9.

Eq. AfO.

71

Suppose z,_ =x~,.+ iy'L where x,_ and Y~,. are real nurnbers.

Th en ab x,_= - at + -.­(a)

• 2. 2.

and c.) t. 2bt 1 2. 2.

y,_ = - -2

[ t - -.- + -=- (b - a ) ] (a) (a)

(a) x,. a thus y2. =- 2 [ at. - c.)'- ] the equation of a parabola again {Figure 25).

In th~s case C and D represent the points Zi and Z2.' when the

time is 0 and t respectively, but this time the path of integration

oor CDoor {shown in red) may enclose the singularity at the origin

giving Hankel' s integral for the gamma-function.

. . . î e-::. dZ,. =- z,J; if the singularity is enclosed. c ""z'2.

-V -V -Z' 2J; + e 7.

e ----dV

-Z e a. e ----dV 1

00

0 ,jz~ +v 100

o Jz,_ +v

and

I = [ - 2J; '<.. e a. + F zr a. (a)

s oo __ e_-_v_ dV -

o .Jz~ +v J"" .-v dV]

o ,Jz,_ +v

Eq. A11.

where K.= 1 if the singularity is enclosed {i.e. if arg. Zz.> Tr)

and 1(.= 0 if it is not {i.e. if arg. Z2.. < Tr).

72

.Pa t h ol' i:nte~ration i n the z.J - p l ane . "-

73

BIBLIOGRAPHY

1.. W. J. Macquorn Rankine,

"On the Centrifugai Force of Rota ting Shafts",

The Engineer, Vol. 27, 9 April, 1869, p. 249.

2. A. G. Greenhill,

"On the Strength of Shafting When Exposed Both to Torsion

and to End Thrust11 ,

Proc. Inst. Mech. Eng. , April, 1883.

3. J. W. Strutt, Baron Rayleigh,

"The Theory of Sound", Vol. l, Chap. 8, i877.

4. S. Dunke rley,

"On the Whirling and Vibration of Shafts",

Phil. Trans., June, 1894.

5. C. Chree,

"The Whirling and Transverse Vibrations of Rotating Shafts 11 ,

Phil. Mag., Vol. 7, 1904.

6. H. H. Jeffcott,

"The Periods of Lateral Vibration of Loaded Shafts -- the

Rational Derivation of Dunkerleyt s Empirical Rule for

Determining Whirling Speeds 11 ,

Proc. Roy. Soc., A 95, June , 1918.

7. H. H. Jeffcott,

"The Lateral Vibration of Loaded Shafts in the Neighbourhood

of a Whi r ling Speed. The Effec t of Wan t of Balance ",

Phil. Mag. , Vol. 37 , March, 1919.

8. J. Morris ,

"The Strength of Sha fts in Vibration" ,

C r o sby, L o c kwood and Son, 1929.

,...

74

9. D. C. Johnson,

"Free Vibration of a Rotating Elastic Body",

Aircraft Engineering, Vol. 24, No. 282, August 1952, pp. 234-236.

10. D. C. Johnson,

"Apparatus for Demonstrating Shaft Whirl",

Engineering, Vol.178 , No. 4622, 27 August 1954, pp. 266-267.

11. E. Downham,

"The ory of Shaft Whirling", Parts l, Il, Ill, IV, and v.

The Enginee r, Vol. 204, No s. 5307-5311, 1957, pp. 518-522,

552-555, 588-591, 624-628, and 660-665.

12. F.M. Lewis,

13.

"Vibration During Acceleration Through a Critical Speed",

A. S.M.E. Trans. Vol. 54, No. 24, (App. Mechs. ), 15 December,

1932, pp.253-259.

G. D. McCann Jr. and R. R. Bennett,

"Vibration of Multifrequency Systems During Acceleration

Through Criti cal Speeds 11,

A. S.M.E. Trans., Vol. 71, Dec. 1949, pp. 375-382.

14. J. G. Baker,

"Mathematical-Machine Determination of the Vibration of an

Ac cele rated Unbalanced Rotor",

A. S.M.E. Trans. Vol. 61, 1939, pp. A-145- A-150.

15. D. Robertson,

"The Whirling of Shafts", Parts I and II.

The Engineer, Vol.158, 1934, pp . 216,217,228,229,231.

j

75

16. H. D. Taylor,

"Critical- Speed Behaviour of Unsymmetrical Shafts",

A. S.M.E. Trans. Vol. 62, 1940 ; pp. A-71 - A-79.

17. C. Geiger,

"Mitteilungen aus den Forschungsanstalten des GHH- Konzerns",

No. 6, 1934.

18. A. L. Kimball and D. E. Lovell,

"Internai Friction in Solids 11,

A. S.M.E. Trans. Vol. 48, 1926, p. 479.

19. o. Foppi,

"The Practical Importance of the Damping Capacity of Metals

Especially Steel s",

Journal of the Iron and Steel Institute, No. II, Vol. 134, 1936,

pp. 39 3-423.