THE INFLUENCE OF ANGULAR ACCELERATION THROUGH CRITICAL SPEEDS ON

80 Pages·2012·7.9 MB·English

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.
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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.
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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
• t.
(ù~ + bt)i
S•at + (
Appendix -- Evaluation of the Integrais dt
5•
~ta.
(--a--
and at+ bt)id t
67
Bibliography 75
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ILLUSTRATIONS
Figure(s) Page( s)
1 Rankine' s Theory of Whirling 3
2 Jeffcottr s Theory of Whirling 10
3 Effect of striking a rota ting membe r 13
4 Simple st rota ting system 16
5 Apparatus 38
6 - 23 Graphs of maximum amplitude of whirl versus
angular acceleration of shaft 53 - 60
24 Path of integration in the z, plane 69
25 Path of integration in the z&. plane 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
3
. f~th of rotating shaft.
Shaft certre line. ,' \ 1
.........
' 1 ---+1 ________ .,.-
J.. 1
___:::~~ ___B ~ari!}l{ axis ~-
~ (=Axis of sta\tionhry shaft) . ~"'
1
A B
Shaft deflection maintained
by centrifugal force due to
rotation about bearing axis
AB.
Fig.l.
Rankine' s The ory of W'hirling.
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.