APM-56-19
Dynamic Balancing of R otating Machinery
in the Field
By
E. L. THEARLE,1 SCHENECTADY, N. Y.
This paper describes a method and portable equipm ent
for balancing rotating machinery while running under
normal operating conditions. A direct and exact solution
of the problem is offered, which makes it possible to
balance a rotor completely by means of a definite pro­
cedure utilizing the data obtained from only three balanc­
ing runs of the machine. The system is so developed as to
require a m in im u m of m ental effort on the part of the
operator.
chinery other than unbalance. This paper will be limited to a
description of means of dealing with that component of vibration
which occurs at running-speed frequency, caused by mass un­
balance in the rotating member. This is the only component
of vibration which may be eliminated by the addition of balance
weights to the rotating mass. However, the system and equip­
ment here described will not only simplify the elimination of vi­
bration at running-speed frequency, but will aid in analysis to
determine the causes of vibration at other frequencies.
For the present let us simplify the problem by considering a
N RECENT years both the users and manufacturers of rota­ single, substantially rigid, rotating mass mounted in two sup­
tive apparatus have given increasing attention to the matter porting bearings. Many actual machines may be considered
as a combination of such units which may be treated separately.
of vibration elimination. This has resulted in considerable
progress in the development of machines and processes for theIt can be shown that, for the correct balance of such a rotor, two
production balancing of rotating parts in the factory. There weights placed in different radial planes of the rotor are in general
exists also the problem of balancing rotating machinery in the necessary and are always sufficient. For the purpose of this
field, with the rotor mounted in its own bearings and probably illustration, a rotor having a horizontal axis of rotation will be
assumed. Such a rotor is represented diagrammatically in Fig. 1.
running under normal operating conditions.
Machines which have been serviced must often be rebalanced
before use. Also, the state of unbalance of a rotor may change,
as time goes on, due to the slight shifting of its parts or to the
gradual relief of stresses in the shaft or body of the rotor. In
such cases, which occur frequently, the cost and delay of disas­
sembly, shipment of the rotor to and from a balancing machine,
and reassembly are usually prohibitive.
Many rotors change shape slightly with changes of running
speed, and will therefore show considerable unbalance at their
normal running speed even though carefully balanced at some
lower speed in a balancing machine.
There have recently been built several very large turbo-alter­
The vibratory motion of any point on either bearing or pedestal
nators, the alternator field alone weighing over 100 tons, which
may be represented by three components—the horizontal and
is too great a weight for any available balancing machine. Such
vertical radial components and the axial component. The
rotors must be balanced in their own bearings.
purpose of balancing, at any chosen running speed of the rotor,
Thus there is a real need for a direct and scientific method of is to reduce the greatest of these three components to a practical
dynamic balancing in the field—one simple of application and
minimum. When this is accomplished, the other two compo­
utilizing readily portable equipment. The system of balancing
nents will also have been reduced and will remain less than that
to be described was developed in view of this need. Experience component originally greatest. For the sake of example, it
with this system indicates that in many cases its use accomplishes
will here be assumed that the horizontal radial component is the
a better balance in a shorter time than is realized with the usual
greatest; therefore only this component will be dealt with in the
balancing machine.
analysis. If either one of the other components is the greatest,
There are many possible causes of vibration2 in rotating ma­
the procedure is the same except that'measurements are made
in the direction corresponding to this greatest component.
1 R esearch E ng ine e r, G e n e ra l E le c tric C o . A ssoc-M em . A .S .M .E .
It follows, in this ideal case, that if the horizontal components of
M r . T hearle w as g ra d u a te d fr o m C o r n e ll U n iv e r s ity , w here he
vibration of two points, one chosen on each pedestal, are re­
served as in s tru c to r of ap p lie d m e ch anics fro m 1920 u n t il 1925. H e
duced to zero, the purpose of balancing has been accomplished
was professor of m e c h a n ic a l engin eerin g, head of th e d e p a r tm e n t, a t
th e U n iv e rs ity of A rk ansas fro m 1925 u n t il 1928, w h e n he becam e as­
and no vibration is transmitted to the structure supporting the
sociated w ith th e R esearch L a b o r a to r y of th e G e n e ra l E le c tric
rotor.
Com pany.
When balancing any substantially rigid rotor there are four
1 “ T u rb in e V ib r a tio n a n d B a la n c in g ,” b y T . C . R a th b o n e ,
A .S .M .E . T ran s ., v o l. 51, 1929, p a p e r APM -51-23.
variables to be dealt with—the amount and position of each of
C o n tr ib u te d b y th e A p p lie d M e c h a n ic s D iv is io n for p re se n ta tio n
two corrective weights, to be placed in different radial planes of
a t th e A n n u a l M e e tin g , N e w Y o r k , N . Y ., D e ce m b e r 3 to 7, 1934, of
the rotor, usually one near each end of the rotor as shown in
T h e A m e r ic a n S o c ie t y o f M e c h a n ic a l E n g in e e r s .
Fig. 1. The farther apart these corrective planes are, the smaller,
D iscussion o f th is p a p e r s h o u ld be addressed to th e Secretary,
A .S .M .E ., 29 W e s t 3 9 th Street, N e w Y o r k , N . Y . , a n d w ill be ac­
in general, may be the corrective weights.
cepted u n til J a n u a r y 10, 1935, for p u b lic a tio n a t a la te r date.
When balancing such a rotor in a balancing machine, the rotor
N o t e : S ta te m e n ts a n d o p in io n s ad v a n c e d i n p apers are to be
may be mounted elastically in such a way that it is pivoted to
und ersto o d as in d iv id u a l expressions o f th e ir a u th o rs, a n d n o t tho se of
oscillate about some chosen radial axis. If this is done, only two
the Society.
I
TRANSACTIONS OF THE A M ER IC A N SOCIETY OF M ECHANICAL EN GIN EERS
746
of the four unknown variables (one weight and its position)
need be dealt with at once. When balancing a large rotor in its
own bearings in the field (or factory) it has been customary to
assume that a balance weight on any one end of the rotor in­
fluences the vibration of the corresponding pedestal only. This
is not the case. Thus when balancing by the method most
commonly used heretofore, a weight Wn (Fig. 1) will be found
and placed at a certain angular position on the rotor (deter­
mined largely by trial and error) such that the nearest pedestal
N does not vibrate more than a practical tolerance. Then,
similarly, a weight W/ is determined and placed on the rotor in
such a position that pedestal F is practically vibrationless.
Returning to pedestal N, it is usually found that the weight W/
has destroyed the previous work done there. Subsequently,
further correction at end N destroys the apparent balance ob­
tained at F by weight W/, and so on. With very long rotors it
is true that this series may converge fairly rapidly, but the
method still involves many expensive trial runs. With some
quite short rotors, this series may actually diverge, and balancing
by this method becomes impossible. A correct method of
balancing in the field must therefore deal simultaneously with
four variables—the amount and position of each of two weights.
The system to be described here does this by including the effect
of each weight on both pedestals or on any other two points
chosen on the machine.
R
e p r e s e n t a t io n o f
V ib r a t io n s
by
positions as they are considered to be rotating with constant
phase angles S and e measured from the line OR fixed to, or
marked on, the rotor. In Fig. 3b the sinusoidal displacements of
the vibrating points N and F are plotted against time, or angular
displacement of the rotor, showing how these generating vectors
N and F specify the motions of their corresponding points N and
F.
S im p l e “ S in g l e -Pl a n e ” B a l a n c in g
The simplest mass to balance is one which is relatively short
axially compared to its diameter, and which is known to exert
no dynamic couples. Such a mass may be represented by a
symmetrical flywheel on a true-running shaft which rotates in
two bearings. Under these assumed conditions, a single cor­
rective weight placed at some chosen radius in the radial plane
of symmetry of the wheel will be sufficient to produce a balance
and therefore to eliminate, simultaneously, vibrations of both
pedestals.
The simplest and most crude method of determining phase
angles of vibration is by marking the shaft as it rotates. A
stylus held near the rotating shaft, which has previously been
painted with chalk, will mark the “high side” of the shaft as it
rotates and vibrates, giving a rough indication of the phase
angle of vibration. The amplitude of vibration may be deter­
mined by means of any of the common vibration-amplitude in­
dicators held against the shaft or pedestal.
V ectors
The sinusoidal component of vibration of any point in any
chosen direction may be completely specified by a line, known
as its generating vector, such as ON in Fig. 2 where the length
ON specifies the amplitude of the vibration. When the vibra­
tion under consideration is produced by a rotating mass, such as
a machine rotor, the generating vector ON is thought of as ro­
tating about point 0 at the same speed as the machine rotor.
If OR represents some arbitrarily chosen zero line of reference
marked on and rotating with the rotor, then the angle 5 will be
known as the “phase angle” of the generating vector ON. The
projection of the vector ON on the fixed axis OX will then repre­
sent the displacement of the vibrating point from its mid-posi­
tion at any instant. Having fixed the zero reference line OR
on the rotor, the sinusoidal component of vibration of a point in
any chosen direction may be completely specified by the am­
plitude (length ON) and phase angle S of its generating vector.
It will later become evident that here we are interested mainly
in differences between phase angles, and that absolute phase
angles, measured from some known radial line on the rotor to be
balanced, do not enter into the calculations. For the sake of
uniformity in representing the generating vectors of any vibra­
tions, during calculation their phase angles will arbitrarily be
measured from the horizontal or X axis.
Fig. 3a shows the generating vectors N and F, specifying the
synchronous vibration of any two points N and F, in successive
F
ig
. 3
The unbalanced wheel is now rotated at some chosen speed, the
shaft marked as already described, and the amplitude of vibration
measured. Upon bringing the wheel to rest, the mark on the
shaft may appear as shown at a, Fig. 4(a). The vibration which
existed may be represented by its generating vector Oa, drawn
as shown by Fig. 4(6) in any direction and of a length propor­
tional to the measured amplitude.
A trial weight Wi, of any convenient size, is placed at any
convenient point at a chosen radius on the wheel, as shown in
Fig. 4a, the wheel again rotated at its former speed, the shaft
marked, and the amplitude of vibration observed. The second
mark on the shaft may now appear as at 6, Fig. 4(a), shifted
APM-56-19
APPLIED MECHANICS
through an angle 5, in a counter-clockwise direction, from its former
position. This vibration which took place with the trial weight
in place may now be represented by its generating vector 06
drawn, as in Fig. 4(6), displaced an angle S in a counter-clockwise
direction from the original vector Oa, and of a length proportional
to the last observed amplitude.
The vector ab then represents the change in the vibration
produced by the trial weight Wi. It is then apparent from the
diagram of Fig. 4 that if the trial weight Wi, and therefore its
effect (vector ab), are displaced through the angle y, and its
magnitude is increased to
its resultant effect will be equal in magnitude and opposite
in direction to the original vibration vector Oa, and will, therefore,
annul the original vibration. This treatment is based upon the
assumption that, at the same speed of rotation, vibration am­
plitudes are proportional to the unbalanced forces producing
them. The assumption is, in general, justified by both theoreti­
cal considerations and experience. -The determination of phase
angles by shaft marking is not sufficiently accurate to yield very
satisfactory results and a more accurate means of measurement
will be described later. The example given, illustrating a method
described in the literature before,2was inserted here simply as an
F ig .
4(6)
aid to the understanding of the complete system of balancing to
be presented.
Vector A
lgebra
The solution of the general balance problem is greatly facili­
tated by the use of vector algebra. Here we deal with the vec­
tors N, F, A, and B and the vector operators a, (}, 9, and 4>.
These vectors and vector operators each have two dimensions,
a magnitude and an angle. It is thus necessary to state both the
magnitude and the angle in order to specify any one of these
quantities.
Writing an operator before any vector indicates the operation
of rotating the vector through the angle of the operator and
multiplying the magnitude of the vector by the magnitude of the
operator. Thus writing the operator a before the vector A,
as aA , produces a new vector aA of magnitude equal to the
product of the magnitudes of a and A, and at an angle equal to
747
the sum of their angles. This new vector aA , therefore, bears
a definite angular relation and magnitude ratio to the original
vector A , fixed by the operator a.
For example, suppose the vector A is of magnitude 12 units,
at an angle of 30 deg, as shown in Fig. 5. The vector A may
then be written A = 12 units, 30 deg. The operator a = 0.6,
10 deg, for example, applied to the vector A, gives the new vector
aA , such that the magnitude of aA = 0.6 X 12 units = 7.2 units
and the angle of aA = 3 0 deg + 10 deg = 40 deg. Thus the
vector aA = 7.2 units, 40 deg, as shown in Fig. 5. This opera-
F ig . 5
tion is written as a multiplication and, as such, follows the com­
mon laws of algebra.
Also, a vector may be “divided” by an operator to give a new
vector, or a vector may be divided by another vector to deter­
mine the operator relating them. Thus for example, if A =
12 units, 30 deg, and a = 0.6, 10 deg, then A / a — C. Hence
the magnitude of 0 = 12/0.6 = 20 units, the angle of C = 30
deg — 10 deg = 20 deg, and C = 20 units, 20 deg.
With these same values of A and C, if A /C = a, then the
magnitude of a = V2/2G = 0.6, the angle of a = 30 deg — 20
deg = 10 deg, and a = 0.6, 10 deg. The operator A /A = 1 is
obviously of magnitude unity, at zero angle. These operations,
written as division, also follow the common laws of algebra.
It is felt that vector subtraction, which is the only other opera­
tion required here, may be most advantageously accomplished
graphically on a polar-coordinate chart prepared for the purpose.
For example, suppose A = 12 units, 30 deg, and B = 9 units, 120
deg.
To determine A-B, by subtracting vector B from vector A (see
Fig. 6), draw A, 12 units long, at 30 deg, and B, 9 units long, at
120 deg.
Then A-B is the vector drawn from B to A (indicated by
B -+ A ). Its magnitude may be determined by direct measure­
ment. Its angle is determined by drawing the line ba parallel to
BA, by means of a parallel ruler, and reading the angle directly at
a. Care must be exercised to read the angle corresponding to the
vector from B to A rather than that from A to B.
748
TRANSACTIONS OF THE A M ERICA N SOCIETY OF MECHANICAL EN GINEERS
D
u a l -P l a n e
B
a l a n c in g
As previously pointed out, the correct balance of a com­
paratively long and substantially rigid rotor supported in two
pedestals, as shown in Fig. 7, in general requires the addition of
two corrective weights, one in each of two separate radial planes.
The data necessary to determine the amounts and positions of
the two corrective weights is obtained in three runs, all at the
same chosen balancing speed, by measuring vibration amplitude
and phase angle at each of any two chosen points on the machine
under each of the following three conditions:
(1) No corrective weights on the rotor
(2) Any single known weight, of reasonable amount,
placed at any angular position on the first end of the
rotor
(3) Any single known weight of reasonable amount placed
at any angular position on the second end of the rotor.
we are dealing is linear, i.e., vibration amplitudes are proportional
to the forces causing them. This is an ideal state usually closely
approximated in practise.
Consider now the balancing of such a rotor as shown in Fig.
7a. The rotor is run at some pre-chosen balancing speed and the
vibration amplitudes and phase angles are measured at the near­
end and far-end points Pn and P/, respectively. These measure­
ments determine the generating vectors of the vibrations which
are to be annulled: N at an angle 5 and F at an angle e as laid
out in Fig. 7b. This constitutes the first run.
A trial weight W'n of any reasonable amount is then applied
at any angular position in the near-end balancing plane. The
amount and position of this weight are recorded. The second
run is then made, at the same speed as before, and measurements
of the amplitudes and phase angles at points P» and P/ are re­
peated. It will be found that the application of the trial weight
W'n has changed the generating vectors of the vibrations at these
two points to N 2 and F2 (at angles S2 and «2) as shown in Fig. 7c.
The effect of this trial weight on the vibration of the near-end
pedestal is then shown by the vector drawn from N to N 2,
(Fig. 7c). (AT — N i) = N 2 — N = A
Similarly, the effect of this same weight on the vibration of the
far-end pedestal is shown by the vector drawTi from F to Ft,
(Fig. 7c). (F — F2) - F2— F
The vectors N -*■N i and F — F2 represent the effects of the
same weight W'n. Assuming a linear system (effects propor­
tional to causes) if this weight be increased by any amount, the
vectors N —>■N 2 and F —> F 1 will be increased proportionately.
Also if the position of this weight on the rotor be shifted through
any angle, these vectors F -* Fi and N — Ni will be turned
through the same angle. Thus there is a definite ratio of mag­
nitudes of N —► N i and F —*■F2 and a fixed angle between them,
independent of the state of balance of the rotor. Therefore, if
the vector JV -*■N2is written
the vector F —*■F2may be written
F ig . 7
The first of these three runs determines the generating vectors
of the vibrations to be eliminated, and the latter two runs deter­
mine the susceptibility of the rotor vibration to individual weights
placed in each of its two chosen balancing planes separately.
The apparatus used in making vibration-amplitude and phaseangle measurements, to be described, includes a contactor which
is attached to one end of the machine, rotating synchronously
with it, and from which phase angles are read directly. (See
Fig. 7a.) All readings are made from this station. In this
work, the balancing planes, or ends of the rotor, and sides of the
machine are distinguished as the near end, the far end, and the
right and left sides, as they would appear to an observer stationed
at the contactor and facing the machine. Also, the positive
direction of measuring angles is taken as counter-clockwise as it
would appear to an observer in the above position. This nomen­
clature eliminates the confusion incident to the use of the terms
front, rear, collector end, commutator end, etc., which are not
universally applicable.
It must here be assumed that the vibrating system with which
where a is a vector operator which is a constant characteristic of
the machine and its mounting.
The weight W'n is now removed and a trial weight W't of
any reasonable amount is applied at any angular position in the
far-end balancing plane.
The third run is then made, at the same speed as before, ob­
serving the angle and amplitude readings which determine the
generating vectors Nz and F3 of the vibrations at the near-end
and far-end pedestals, respectively. These vectors are shown in
Fig. 7d. The effect of the weight W 't on the far-end pedestal
may then be represented by the vector
Since the effect N —*■Ns on the near-end pedestal is due to the
same weight W't, this vector may be written
where 0 is a vector operator similar to a.
The vectors A, aA, B, and /3B, redrawn in Fig. 7e, specify the
susceptibility of the machine to balance weights. Having de­
termined, in three runs of the machine, the vibrations which it is
desired to eliminate and the susceptibility of the machine to
weights, we now wish to calculate the final weights Wn and Wi
which will give a correct balance.
The trial weights W'n and W't and the final wreights Wn
and Wf require a statement of both magnitude and position
(direction) to specify each of them. They are therefore vector
quantities and may be treated as such. Each final weight may
A PPLIED MECHANICS
be derived from its corresponding trial weight by a shift in angle
and a multiplication. Thus the two new vector operators 6
and 0 may be introduced such that
APM-56-19
749
C a l c u l a t io n s
The calculation of 6 and <t> from [2] and [3] may be made in
the following series of steps:
and
Assuming vibration amplitudes to be proportional to the
forces causing them, any vector operation 9 on a trial weight
W'n, for example, will result in the same vector operation on the
effects A and aA of that weight.
The statement may now be written that the operations 0 and
4> performed on the trial weights W'n and W ’t, respectively, will
balance the rotor. This is equivalent to writing the statement
that the operators 6 and 4> applied to the effects A, aA, B, and
0B of these trial weights, will produce new effects equal and op­
posite to the generating vectors N and F of the original vibration.
and
Solving these equations for the operators 8 and 0,
and
It must be remembered that all quantities appearing in Equa­
tions [2] and [3] are vector quantities, and that all operations
indicated are vector operations as previously defined.
These calculations are most easily made on the standard sheet
devised for this purpose, as shown with sample calculations in
Fig. 8. On this sheet each step in the calculation is indicated.
Vector multiplications and divisions are replaced by their equiva­
lent components appearing only as arithmetic additions, sub­
tractions, multiplications, and divisions. For example, the angle
of a, item 13, is indicated as being found by subtracting item 5
from item 9, (9 — 5). The magnitude of a, item 14, is indicated
as (10 -5- 6), found by dividing item 10 by item 6. Vector sub­
tractions are performed graphically. For example, under con-
750
TRANSACTIONS OF THE A M ERICA N SOCIETY OF MECHANICAL ENGIN EERS
struction No. 1, Fig. 8, the values of angle and magnitude of A
items 5 and 6, are indicated as direct measurements of the vector
N —<• Ni, as drawn, by (5, 6 = N —► Ni). These measurements
are made as shown in the preceding section on vector algebra.
The operators 6 and 4>appear as the result of these calculations.
For example, item 31, 8, is the angle through which the near-end
trial weight should be shifted, in a counter-clockwise direction,
to be in the position of the correct final weight. Item 32 is the
number by which this trial weight should be multiplied to give the
amount of the correct final weight. Similarly, items 33 and 34
are the corresponding angle and amount of the operator <t>.
Thus operators 8 and <t> determine the corrective weights to be
applied to the rotor, and the application of these weights com­
pletes the process.
R e f in e m e n t o p B a l a n c e
Many actual machines will be found to be non-linear. That is,
changes in vibration amplitude will not be exactly proportional
to the balance weights producing them. For this reason, and
because of errors of measurement and calculation, it may be
found desirable to refine the balance of a machine after having
completed the process described above.
Providing no change in the machine has been made, such as of
speed or means of support, having once completed the above
balancing process, requiring three runs, any subsequent re­
balancing requires only a single run.
After having applied the corrective weights dic­
tated by any previous balancing, the machine is
again run and new measurements of the vectors
N and F are made and recorded on a new calcula­
tion sheet. All items on the previous calculation
sheet which are marked with an asterisk may be
transferred to the new sheet and the calculations
repeated as before.
in the gap of the magnetic circuit, and is caused to vibrate with
the machine pedestal, by means of the slender rod (g).
In order to maintain ruggedness of design, the flat springs
(e) must be so stiff axially that the natural frequency of vibration
of coil relative to magnetic field would not differ sufficiently from
the frequency of vibration of some low-speed machinery. If this
generator were used on low-speed machinery, the field structure
would not remain substantially stationary in space, and the mo­
tion of the coil relative to the field would not be a true
measure of the vibration of the machine pedestal.
To avoid this difficulty, the mechanism at the end of
the instrument serves as a “negative spring” to balance
the positive stiffness of the flat springs (e) and thus
lower the natural frequency of vibration of the instrument
without sacrifice of ruggedness. The notched blocks (j)
are fixed to the magnetic field of the generator by means
of the flat springs (h). The adjustable notched ring (m)
is fixed, to the central tube (d) carrying the coil. The stiff
struts (k) pivot in the notches of the blocks (j ) and the
ring (m), and are loaded by means of the adjustable springs (n).
An axial displacement of the tube (d) and coil, relative to the
field, thus tilts the struts and introduces a “negative” restoring
force, substantially proportional to the displacement. By ad­
justment of the tension in the springs (n) this negative restoring
force may be made approximately equal to the positive restoring
force exerted by the flat springs (e), thus reducing the natural
frequency of vibration of the instrument to a very low value and
making it suitable for use on low-speed machinery.
When one of these generators is attached to a vibrating ma­
chine, the field structure of the instrument remains practically
stationary in space while the coil vibrates with the machine and
generates an alternating electromotive force proportional to the
velocity of this vibration. If true vibration indications are to
be obtained, the suspension of the instrument as described is
quite necessary because of the great difficulty in finding a truly
stationary point on any building or structure surrounding a
vibrating machine.
M e a s u r e m e n t o f V ib r a t io n
In order to take full advantage of the balanc­
ing method here described, instruments have been
devised by means of which vibration amplitude and
phase angle can be accurately measured. This
balancing equipment consists of two generators, a
contactor, and a microammeter, with the necessary
adaptors and connections. The instruments are
set up on a machine to be balanced as shown diagrammatically
in Fig. 9. The two generators are identical and are attached
to any two points on the machine such as the bearing pedes­
tals, A and B.
Fig. 10 shows a cross-section through one of these generators,
as mounted on a machine pedestal (a) for horizontal vibration.
The main body of the generator is a permanent-magnetic-field
structure (6), annular in form, suspended by the springs (c).
The central tube (d) is carried by two flat springs (e) which main­
tain rigid radial alignment, but permit easy axial motion, of
the tube relative to the field (fe). The tube (d) carries the coil (/),
Fig. 11 shows the external appearance of a generator arranged
for horizontal vibration. Fig. 12 shows a generator mounted on
the pedestal of a 35,000-kw turbine. For dealing with the ver­
tical component of a vibration, the instrument is mounted as
shown in Fig. 13.
The contactor mechanism is shown diagrammatically in Fig.
14. An adapter (a) is fixed to the end of the machine shaft (6)
by means of three small screws. A clutch (c) is arranged so that
the contactor may be connected to or disconnected from the
machine, while it is running, without losing the angular relation
between machine and contactor. The cam (e), driven through
APPLIED MECHANICS
the flexible shaft (Vi), maintains the contact (/) closed during 180
deg of each revolution of the machine, and open the remaining
half revolution. The contact mechanism (J) may be turned
about the axis of the cam by means of the handwheel (g). The
pointer (h), from which phase angles of vibration are read di­
rectly on the scale (j), is fixed to the disk which carries the con­
tact mechanism (/), and moves with it. The pointer (fc), which
rotates with the cam, is only a convenience when attaching trial
weights to the machine and is not essential. The hand-wheel (g)
is geared to the contact mechanism (J) in the ratio of twelve to
one, so that three revolutions of the hand-wheel shifts the phase
angle of contact 90 deg. Since only angular differences are
used in the balancing calculations, no predetermined angular
relation between the cam and machine rotor need be observed
when attaching the contactor adapter to the machine shaft.
Fig. 15 shows the complete contactor and adapter, disconnected.
The face of the contactor is shown in Fig. 16.
APM-56-19
751
is zero, as shown by curve b. Since the electromotive force in­
duced in the generator coil is proportional to vibration velocity,
and since meter current is proportional to it, curve b may repre­
sent the meter current which would flow if the circuit through the
meter were continuously closed. However, the contactor main­
tains the circuit closed only during one half of each revolution of
o
F ig .
12
B a la n c e r
G e n e ra to r
o f 3 5 ,0 0 0 K w
F
ig
. 11
G e n e r a to r f o r P o r ta b le
M o u n te d
on
B e a r in g
P e d e s ta l
S te a m - T u r b in e G e n e r a t o r
D y n a m ic B a la n c e r
(M ounted for horizontal vibration.)
The meter is a highly damped, direct-current microammeter
as shown in Fig. 17. It is arranged with a tumbler switch for
connecting to either generator and a ratio switch giving meter
readings in the ratio of 1, 3, 10, and 30 so that the
most suitable scale may be used for any vibration
amplitude to be dealt with. Leads from the two
generators and the contactor plug into the meter
box.
Consider now the operation of these units when
brought together on a machine to be balanced, and
connected as shown in Fig. 9 with, for example, the
switch thrown to the generator on pedestal A. Re­
ferring to Fig. 18, curve a represents the sinusoidal
displacement of pedestal A during its vibration.
The vibration velocity is the first derivative of curve
a which is shown as curve b. Vibration velocity is
also sinusoidal; it is zero when displacement is a
maximum and is a maximum when displacement
F i g . 13
G e n e r a to r f o r P o r ta b le
D y n a m ic B a la n c e r
(M o unted for vertical vibration.)
752
TRANSACTIONS OF THE A M ERICA N SOCIETY OF MECHANICAL ENGIN EERS
the machine. If the circuit were closed at c, for example, it would
be opened at d, Fig. 18. The meter would then receive an im­
pulse, once per revolution of the machine, of magnitude repre­
sented by the shaded area under curve 6. Since the positive
and negative portions of this area are not equal, the meter will
read something other than zero.
If, now, the contact mechanism is shifted about the cam, by
turning the contactor hand-wheel until the meter reads zero
(and meter needle moves in same direction as hand-wheel),
then contact is being made at c and broken at d, Fig. 19, the
positive and negative areas under curve b being equal. The
pointer (h), Fig. 14, on the contactor then indicates the point
F
F
ig
. 16
C
ig
. 15
C
ontactor
W
ontactor
(e), Fig. 19, at the mid-point of the closed-circuit period, which
also locates the point / of extreme displacement of the pedestal.
The reading of the contactor pointer (h) on scale (j ), Fig. 14, is
then taken as the phase angle of the vibration of pedestal A.
Now if the contactor hand-wheel is turned three revolutions,
the contactor is turned through 90 deg, and the meter receives
an impulse equivalent to the full half-wave of electromotive force
or vibration velocity. The meter will then read a maximum, and
since this reading is proportional to the amplitude of vibration it
is recorded as such. This apparatus thus permits of accurate
measurement of both phase angle and amplitude of vibration.
Phase angles read directly from the contactor dial and meter
readings are recorded directly on the standard calculation sheet as
shown in Fig. 8.
The use of special cobalt-steel magnets in both the generators
and the meter gives high output from the generators and high
sensitivity in the meter without sacrifice of ruggedness. With
the meter ratio switch in the position for maximum sensitivity a
vibration of only one mil (double amplitude), at a frequency of
1200 cycles per min, produces approximately full-scale deflection of
the meter needle. Under these conditions phase angles can
easily be read to within one degree. Since the generator output
is proportional to vibration velocity, at 3600 rpm, for example,
three times this sensitivity would be obtained. Thus the greater
sensitivity is obtained at the higher speeds where accuracy of
balance is most desirable.
C o m p l e x M a c h in e s
Many machines are made up of several units running in more
than two bearings. In some such cases it may be convenient to
uncouple the units and balance each one separately. The driving
it h
Shaft
an d
A
dapter
F
ig
. 17
M
et er
and
C
ontrol
Box
unit may be run alone and balanced first. Then each of the other
units may be coupled to it and balanced, one at a time.
Where it is inconvenient to run any unit singly, such as, per­
haps, in a three-bearing, two-unit set, there are several methods of
attack. If such a machine is made up of rotors 1 and 2 running
in bearings A, B, and C, the balancer generators may first be
placed on pedestals A and B and rotor 1 balanced, neglecting the
effect on pedestal C. When this is completed, the generators
may be placed on pedestals B and C and rotor 2 balanced. This
procedure may be repeated for a refinement of balance. Or, a few
observations may indicate that some one pedestal is not very
susceptible to changes in balance weights or that balance weights
in certain corrective planes have but little influence on the vibra­
tion of the machine. In such cases this pedestal or these cor­
rective weights may be temporarily neglected in order to reduce
the problem to'its simplest form.
Some machines, certain steam turbines, for example, are so
arranged that in the field it is inconvenient to attach balance
weights to the rotor in two planes, only a single plane being
A PPLIED MECHANICS
available in which to apply a corrective weight. In such cases
a compromise balance is sought by applying the simple vector
method of calculation, as shown in Fig. 4, to vibration observa­
tions made at each end of the machine. In this way a single
F
iq
. 18
F
ig
. 19
balance weight may be determined which will result in minimum
vibration amplitudes.
Relatively long and slender rotors may cause some difficulty in
balancing due to distortion. This may usually be overcome by
treating the balancing corrections as two components—a couple
component and a force or so-called “static” component. The
latter is distributed along the length of the rotor, thus reducing its
deflection.
C o n c l u s io n
Only the basic system has been described.
Space will not per­
APM-56-19
753
mit a discussion of the many variations which are most suited to
particular types of machines.
Experience with this balancing system in both the factory and
the field has demonstrated its usefulness. In the factory, an
effective balancing machine may be had by attaching the balanc­
ing generators to almost any pair of bearings in which a rotor
can be run at constant speed. For example, in the manufacture
of some electrical machines, all rotors must be run at normal
F
iq
. 20
P
o r t a iil e
B
a l a n c in g
E
q u ip m e n t
in
C
a r r y in g
C
ase
speed in a machine for grinding their commutators. Applying
the balance generators to the grinder pedestals permits the bal­
ancing operation to be completed in the same machine.
High sensitivity of the instruments for measuring vibration
eliminates the necessity of balancing a rotor in very flexibly
mounted bearings which amplify the vibration amplitudes. This
high sensitivity, together with a sound method of interpreting
the data, provide a means of obtaining better results in less time
than can be realized with the ordinary balancing machine.