||l|llllllHIIllllllllllllilllllllllllllllllllljljlllzllljllll

3 1293 104916

This is to certify that the

thesis entitled

"Effect of Damping on the Accuracy of Dynammmeter
Measurement of a Cyclic Fluctuating Torque"

presented by

John Edward Nolan

has been accepted towards fulfillment
of the requirements for

_M_._§_n__degree in M E

Major professor

 

Date MW

 

 

 

 

)V1SSI.J RETURNING MATERIALS:
Place in book drop to
LJBRARJES remove this checkout from
“ your record. FINES will

be charged if book is
returned after the date
stamped below.

 

 

 

 

 

 

 

EFFECT OF VISCOUS DAMPING ON ACCURACY OF

DYNAMOMETER MEASUREMENT OF A CYCLIC FLUCTUATING TORQUE

By
JOHN EDWARD NOLAN

AN ABSTRACT
Submitted to the College of Engineering
Michigan State University of Agriculture and
Applied Science in partial fulfillment of
the requirements for the degree of

MASTER OF SCIENCE

Department of Mechanical Engineering

1960

 

John Edward Nolan

ABSTRACT

A dynamometer is a device to measure force or torque. When
the force or torque being measured fluctuates it becomes necessary
to damp the motion of the indicating mechanism to Obtain a reading.
Damping mechanisms modify the magnitude of the force transmitted to
the indicating.mechanism, altering the instantaneous indications.
This thesis describes the results of an investigation into the
accuracy of the indication for a cyclic fluctuating torque impressed
on a conventional dynamometer which uses viscous damping to producd
a readable indication. A single-cylinder four-stroke-cycle internal-
combustion engine, directly coupled to a cradled-field induction.
motor electric dynamometer is used to experimentally check the
analytical treatment of the variables involved.

The cylic-torque curve impressed on the dynamometer by the
engine was measured by a strain-gage rosette on the connecting shaft
between the two units, and recorded by photographing the trace on
an oscilloscOpe produced by the amplified strainpgage signal. The
average torque cycle thus obtained was treated by harmonic analysis
methods to obtain an analytical expression representing this torque
function.

1he actual damped-scale system.of the dynanometer was analyti-
cally represented as an equivalent simple damped springamass system.

Using the analytical torque function developed from the oscilloscope

ii

trace as a forcing function, the equation of motion of the scale

system was develOped.

The solution of this motion equation was obtained, leading to
the conclusion that a damped-scale system truly represents the aver-
age of a cyclic fluctuating torque, and that this situation is true

regardless of the amount of viscous damping involved.

John Edward Nolan

iii

EFFET OF VISCOUS DAMPING ON ACCURACY OF
DYNAMOMETER MEASUREMENT OF A CYCLIC FLUCTUATING TORQUE

By
JOHN ammo you»!

A 'IHESIS
Submitted to the College of Engineering
Michigan State University of Agriculture and
Applied Science in partial fulfillment of
the requirements for the degree of

MASTER OF SCIENCE

Department of Mechanical Engineering

1960

ACKNOWLEDGEhENTS

In the preparation of this thesis, and also during the experi-
mental portions of it, I have become indebted to many people for their
understanding, help, and encouragement.

First, I would like to thank my family for their patience during
some of the more trying times. I would also like to express my grati-
tude to Jean Dalrymple and Clement Tatro, of the Applied Mechanics
Department, for their assistance with the electronic apparatus used,
and to Carl Redman for his help and company throughout the many hours
that were spent in the laboratory. Especially, I would like to thank
Professor Louis L. Otto, who has been not only an understanding and
helpful friend, but also as inspiring a man as anyone 'could ever hepe
to have for a major professor.

John E. Nolan

BRIEF AU‘IUBIOCRAPHY

The author was born in Lansing, Michigan on January 7, 19%. He
has resided in Lansing all of his life, and obtained his elementary
education in the Lansing public schools. Upon graduating from‘Lansing
Eastern High School in June, 1951, he entered Michigan state University
the following September.

In June, 1957, he received a Bachelor of Science degree in
Mechanical Engineering. He was employed as a laboratory technician
for Ethyl Corporation during the summer of 1957, but returned to
Michigan State in the fall to continue his education. Upon completion
of requirements for a Master of Science degree, the author intends to
remain at the university until he has obtained the degree of Doctor

of PhilOSOphy in Mechanical Engineering.

vi

TABLEOFCONTENTS

Page

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . .
BRIEF AUTOBIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . .
LISTOFILLUSTRATIONS......................
mmmmnw ..........................
PROCEDURE . . . . . . . . . . . . . . . . . . . . . . . . . . . .
DISCUSSIONOFRESULTSOBTAINED .................

SUMMARYANDCONCLUSIONS.....................

sasiuéaw

APPENDICES O O O O O O O O O O O O O O O O O O O O O O O O O I O

A. Detailed Description of Test Equipment
B. Description of Test Equipment Assembly
C. Calibration of Measuring Equipment

D. Details of Calculations

B ELIOWH Y C O 0 O O O O O O O O C O O O O O O 0 O O O O O O 0 [+5

LIST OF ILLUSTRATIONS

Figure Page
1. Schematic Sketch of Equivalent System . . . . . . . . . . .. 6
2. Average Torque Curves over the Cycle . . . . . . . . . . . ll
3. Schematic Sketch of Torque-Pickup Circuit . . . . . . . . . 19
A. Orientation of the Strain Gages on the Shaft . . . . . . . 20
5. Curve used for Determination of the Spring Constant, k . . 30

6. Curve used for Determination of Effective Weight, W . . . . 31.

IN'IRODUCTICN

Throughout the field of engineering, and in other fields in
which it is of interest to know the amount of power produced by any
type of prime mover, such as an internal—combustion engine, steam
engine, steam turbine, etc. , a dynamometer is generally employed.
While there are a number of different types of dynamometere, the
purpose of any one of them is to measure the amount of force or
torque which is produced by the power source in question. This
measurement, along with a measurement of either angular or linear-
displacement rate, can be used to obtain an expression for the
horsepower output of the machine being tested. This paper will be
concerned only with the type of dynamometer that measures horsepower
output of a rotating shaft, this being the variety most comonly
encountered.

Included in the types of dynanometers which measure the
torque in a rotating shaft are the following: dry-friction dyna-
nometers including water brakes and fan (air) brakes, and electro—
magnetic dynanometers which include eddy-current dynanometers and
cradled-field electric dynanometers. All of these dynanometers
can be used for measuring input torque. In all types except the
last, the power which is transmitted to them from the source is
dissipated in the form of heat energy generated within the dyna-

mometer. The last type mentioned, 1e. , the cradled-field electric

.1.

3.

dynamometer, does not directly waste all of the power transmitted to
it, but instead converts a large percentage of the power into useful
electrical energy. Another feature of this type is that, with proper
connections, it can be used as a driving motor to supply power to
driven equipment. However, in many cases, the cost of this type of
dynamometer is prohibitive.

Mechanical energy is supplied to each type of dynamometer as
input torque impressed on the dynamometer shaft. The shaft is coupled
either mechanically (by friction), magnetically, or by viscous forces,
to an outer shell or field structure, which is mounted so that it is
able to rotate in nearly frictionless bearings. If a lever arm of
known length is attached to the side of this outer shell, and if the
outer end of this arm is made to react on a force-measuring scale,
for instance a Toledo scale, the torque that enters the dynamometer
Will be transmitted through the entire system until it meets an equal
and Opposite torque at the lever arm. The scale reading can then be
multiplied by the effective length of the lever arm to obtain torque.

The amount of fluctuation in the torque vs time curves of vari-
ous prime movers at a given average rotational speed varies from an
almost constant torque for a steam turbine or an electric motor to
the higily fluctuating torque of a one-cylinder internal-combustion
engine. For an electric motor or steam turbine, probably the dyna-
mometer arrangement as described above would result in an acceptable
torque measurement, because the scale reading would remain at nearly
the same value. However, with a one-cylinder gasoline engine, or
anything else which produces a varying torque curve, the above arrange-

ment would not be sufficient because the scale reading would fluctuate

_3_

with the torque, and it would be impossible to obtain an accurate
reading.

A certain amount of this fluctuation can be eliminated by using
a large flywheel, a heavy armature, and a heavy field or outer shell,
plus additional.heavy weights on the bottom.of the lever arm below
the scale, with all of these heavy masses serving to increase the
inertia of the system. However, the final answer for engines which
produce any more than a small variation in torque is some sort of
damping, usually in the form of dashpots.

It is the use of dashpots that brings up the main question of
this paper. Dashpots are devices which absorb energy and dissipate it
as heat, and in this case since the energy must come from the engine
being tested, does this energy go to the dashpot at the expense of the
desired torque reading and, as a result, is the reading on the scale

lower than it should be?

PROCEDURE

An experimental investigation of this question requires that
two things be known. The first of these is the torque as a function
of time at some location on the connecting shaft between the engine
and dynamcmeter, and the second is the physical characteristics of the
entire system from the point at which the torque is measured to the
end of the scale needle. This latter includes inertias, dampings, etc.

As a typical source of a cyclic fluctuating torque, a single-
cylinder spark-ignition engine operating on the four-strobe cycle was
used. The engine in question is better known as a CPR Octane-Rating
Test Engine. This engine was directly coupled to a cradled-field
three—phase induction motor by a short shaft and two flexible couplings.
The shaft was carefully machined to produce lmown elastic prOperties.
m the machined section a preperly-oriented sR-l. strain-gage rosette
was mounted. The signals produced by this strain-gage circuit in a
Wheatstone bridge could be used to measure with a high degree of
accuracy the torque carried by the shaft. The signal from the strain
gages'was carried through a set of slip rings to an amplifier, and
the amplified signal was displayed on a cathode-ray oscilloscOpe.
A Polaroid land camera was mounted on the oscilloscope so that the
trace could be recorded. A wire looped around the secondary ignition
wire and plugged into one channel of a dual-beam plug-in on the scepe

was found to produce sufficient signal so that the scope could be

A

triggered by the engine ignition.

After calibrating the torqueepickup by means of known weights,
the engine was started and twenty-three pictures were taken of the
torque curve. At the time that each picture was taken, the scale read-
ing and engine rpm were noted. These twenty-three curves were averaged
to give a single torque function, and from.this point on it was
assumed that the torque could be represented by a periodic repetition
of this function. This function was called f(t), where the t stands
for time.

In order to make use of f(t) in a mathematical equation, it was
necessary to obtain an analytical expression for it. Since the func-
tion was periodic, the twenty-four ordinate system of an harmonic
analysis was used to get an expression for f(t) in terms of a constant

term plus ten sins terms and ten cosine terms, as shown here;

10 10
f(t): a0 + a; an cos cunt + “2:1 bn sin cont ,

Thus the first requirement was fulfilled.

The dynamometer used was of the cradled-field electric type
which could be used as either a driving or absorption unit. The motor
was an induction motor and operated above or below a synchronous
speed of 1800 rpm. Protruding from one side of the motor housing was
the lever arm, which was connected through linkages, and a small stiff
spring to a "black box”. This box contained a complicated arrangement
of lever arms, one of which was connected to a small dashpot. The
output link of this black box was connected to the input of a Toledo
scale. The.mechanism inside the Toledo scale was even.more involved

than that inside the black box. In addition to being connected to the

é.

scale system, the lever arm from the dynamometer had two heavy weights
hanging from it at the same lever arm distance. Finally, there was
another lever arm extending from the opposite side of the dynamometer
housing, this arm being connected to a second and larger dashpot.
Needless to say, any attempt to incorporate the individual effect
of each of the many components of this system into an equation would
be a very difficult, if not impossible, task; and, even if it could
be done, the resulting equation would be more difficult to solve than
it was to derive. Therefore it was decided to develop the problem
in the form of an equivalent system.
file equivalent system prOposed was the one shown here, which
is as simple as possible. It consists of an equivalent mass m hanging
on a spring of equivalent spring constant
k. Any motion of the mass is through an
equivalent displacement 1:, this motion
It being retarded by a damping with coeffi-

cient c. f'(t) is the forcing function

 

on the equivalent system.

m
I Assuming for the moment that k and
, x m are constants, and that the damping is

 

 

Vf '(t) all viscous damping, which is the usual
assumption for dashpots, the equation of
FIGURE 1
motion for this system is
mt '9' Ox 4' kX‘f‘(t)e
If the effect of the small spring between the dynamometer lever
arm and the black box was neglected, everything from the dynamometer

field to the scale pointer would move together, and could therefore

1

be considered as the single equivalent mass m. This mass included

the effects of the dynamometer field, the large weights hung on the
lever arm, the weight of the part of the large dashpot that moved,

and everything that moved in the black box and in the Toledo scale.
It did not include the effect of the mass of the motor armature.

It was arbitrarily chosen that the displacement of this equiva-
lent system be measured at the end of the dynamometer lever arm. In
the equivalent system the equivalent mass would move through displace-
.ments as measured at this point. To determine the equivalent spring
constant of the system, a dial gage was connected so that its input
shaft rested on the top of one of the heavy weights below the black
box. Since these weights were at the correct torque arm distance,
the dial gage.measured equivalent displacement. The scale was loaded
to various readings and the dial gage reading was taken at each load-
ing. The indicated load force was plotted against displacement at
this point, and fortunately a straight-line plot resulted. This meant
that the equivalent spring constant R was actually a constant, its
value being equal to the slope of this straight line.

As is shown in the appendix.the damping coefficient c also
turned out to be a constant, at least within the limits of experimental
accuracy. The equivalent.mass.m was found not to be constant over the
whole range of displacements, but for the small variations in displace-
ment encountered in operation, the mass was assumed constant also.

The value for equivalent mass was picked from a curve once the range
of operation on the scale was known.
The only term in the equation of motion for the equivalent system

which has not been discussed is the forcing function, f'(t). Since the

é

forcing function must originate at the engine, the only way that it
can act on the equivalent system is through the coupling of the arm-
ature and field of the motor. 'mis coupling torque is a function of
time and is directly proportional to f'(t). Since the forcing function
must act at the point where equivalent displacement is measured, ie.,
at the end of the lever arm, f'(t) is equal to the coupling torque
divided by the length of the lever arm.

While f'(t) has now been defined, an expression for it which
can be used in the equation must still be found. For purposes of
discussion, it is convenient to give the coupling torque between the
armature and field a name; let it be called f"(t).

The armature of the motor experiences a varying torque f(t) on
its input shaft, and another varying torque f"(t) in the opposite
direction produced by the field coupling. Neglecting any friction
effects, application of Newton's second law of motion to the armature
results in the equation.

f"(t)=f(t) - ch,
where I is the moment of inertia of the armature, and o< is its angu-
lar acceleration. Since f(t) is now known, and I can be determined
experimentally, an expression for o(, Which would also be expected to be
a function of time, would facilitate finding an expression for f"(t).
Intuitively, one would expect f"(t) to be a periodic function with the
same frequency as f(t), but with much lower extremes of amplitude.
The average values of both f(t) and f"(t) would be expected to be the
same.

As is discussed in the appendix, it was found that it was im-

possible to determine an expression for (X, as the angular velocity

2.

over the cycle was constant within the limits of experimental accuracy.
Therefore, since the equation of motion for the equivalent system was
still lacking a forcing function, it was decided to assume that
f"(t) '—’- f(t). While it is obvious that this assumption is not valid,
it should not affect the answer to the main question of the paper, ie.,
the one concerning the dashpots. It is in fact conceivable that a
function could be produced at the strain gages which would result in
a function at the field-armature coupling that would exactly equal f(t).
Actually, all that is required here is a representative forcing function,
and f(t) is certainly that. f'(t) is now equal to f(t) divided by the
lever-arm length L.

Now everything in the equivalent system equation has been account-
ed for, and the equation is again,

mi+cx+kx=f'(t)= £194)-

f'(t) can be written, as was f(t), as a constant plus ten sine terms

and ten cosine terms, so the equation becomes

.. . _ , g,
m+cx+loc-a5+i[:ancoswnt+n=1bnsincunt.

This equation is solved in the appendix for the particular values
which applied when the data was taken. The resulting expression for x
is of the form
a6

0 O
xz-i-q-E lgxsinwnt-r Nncoswnt,
nxl n=

Each sine term and each cosine term in this expression will, over the
cycle, average out to zero. 'merefore the average displacement is

given by

xavg . =

“10”"
e

10

The force associated with this displacement is just a5. The torque
associated with this force is ab L, and a6 1.: a' which is the const-
ant term of f(t). As in the case of x, the sine and cosine terms of
f(t) also average out to zero, which means that the average value

of the torque is not affected by the dashpots. Therefore, the value
obtained by multiplication of the scale reading by the lever-arm
length should be a true representation of average torque.

Although a conclusion has already been reached, without the use
of any numerical data, it is interesting to see the results that were
obtained for a special case. While the details of the calculations
for this special case are left for the appendix, the results of
interest are contained in the two torque curves on the next page.

The curve with the large hump represents the torque curve as picked
up at the strain gages, while the other more-level curve (which to
the scale of this graph is practically a straight line) represents the
torque function which results from the solution of the equation. As
predicted, these two curves fluctuate about the same average torque
value, although there is considerable difference in the amounts of
fluctuation. During the time that the data was taken, it was noted
that the usual fluctuation of the scale needle was less than 0.2
pounds, except for occasional small unpredictable jumps. Since the
lever arm.was very nearly one foot long, the torque represented by
the scale reading varied less than 0.2 pound-feet. In looking at the
data for the more-level curve in figure 2, it was found that the
maximum variation was 0.15h pound-feet. This is very good agreement
considering that 0.2 pound-feet was the smallest graduation on the

dynamometer scale.

com N uncoomm .MSHB
am mm am am om me me ed as me ea ma as as as a m a o m a m m H o

 

 

 

 

 

 

 

 

 

 

 

 

 

 

TIII.IIIII.I:IIIJI- Jr Jvll ll [lllrlvl

l

 

 

 

A.aouoaosmcme an nepeowuow mecca;
mpmomouaomv .cOHuos Ho cowumewe no
n: soapsaom 5 35390 2:30 seduce IIIIIIII

.o>aso came one you oowmmonoxm
.3259.“ one son.“ umpgagm 350m 0. e o o o

A.mmdmnmouoma comma
Impose» one Esau moxmev .mommm
campus egg um o>neo nevus» mmmnm><

 

.caomo one ne>o mo>neo cameo» ommao>< II N mmDuHm

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

OOH

neeJ-punoa ‘anbuom

DISCUSSION OF RESULTS OBTAINED

The results of the problem as considered up to this point indi-
cate that the torque as determined from the dynamometer is a true
representation of average torque output of the engine. However, there
were a few simplifying assumptions made during the procedure, the
effects of which should be considered. The first of these assumptions
was that the damping coefficient c is a viscous damping coefficient
only. It is more than likely that there are other types of damping
in the system, and there should be some justification for neglecting
their effect. The main argument to this point is that their effect is
small compared to the viscous damping. That this is true is made
quite apparent by allowing the system to vibrate with and without the
dashpots being connected. With the dashpots connected, the system,
upon being released from a position diSplaced from its equilibrium
position, tends to return to equilibrium directly with no overtravel.
With the dashpots disconnected.the system, when subjected to the same
displacement, vibrated for many cycles before coming to rest. This
reasoning implies of course that the damping in the dashpots is truly
viscous. While this assumption might not be exactly correct either,
it is usually made in dealing with dashpots, and chances are that the
majority of their damping is viscous. TWO.more reasons for assuming
viscous damping only is that the magnitudes of other types of damping

are difficult to determine, and, even if they could be determined, the

.12.

32

resulting equation of motion would be non-linear and consequently much
more difficult to solve.

Another assumption made was that the dynamometer armature rotated
at a constant angular velocity. While it is shown in the appendix
that the variation is small, it is of interest to determine if the
final results of the problem would have been changed had this fluctua-
tion been included in the analysis. Including it would have produced
the same equation of motion to be solved except for the forcing
function. The forcing function would have had the same constant term,
but different coefficients for the sine and cosine terms. As with the
forcing function that was used, the only term which would have had any
effect on the average value of displacement would have been this same
constant term, so the conclusion is that this assumption had absolutely
no effect on the outcome of the problem.

The third assumptibn to be considered is the one that the fore-
ing function is periodic. It was apparent by examining the pictures
taken of the torque function that there were no two cycles exactly
alike, but still twenty-three representative possibilities were aver-
aged and the torque function used as f(t) was assumed to be this
average curve repeated periodically. The only justification for this
assumption was that it did allow the equation of motion to be solved.
If an attempt had been made to include variations in each different
cycle in the analysis, probably the first big problem would have been
to determine an expression for the forcing function, since it would no
longer have been periodic and harmonic analysis would no longer apply.
If an expression were found for f(t), it would not likely be a simple

sine and cosine relation, and the solution to the equation would be

l1:

much more difficult. Each new cycle, being different from the one
preceding, would start a new transient component in the response curve;
and before sufficient time could elapse for this transient to die out,
another different one would appear. In short, solution of the equation
by manual methods would be an impossibility. It is, however, conceiv-
able that a computer could be used to analyze the equation if the

mi.+ cx‘+ kx part were set up and if the forcing function were then
fed in directly from the amplified strain gage signal. Without the use
of a computer, the method chosen was felt to be the best available, and
the results produced by it were quite satisfactory.

A slight discrepancy which may be noticed is in the value of beam
load during the test. While the pictures were being taken, the average
beam.1oad as indicated by the Toledo scale was 11.3 pounds, correspond-
ing to an average torque of 11.865 pound-feet. In averaging the
torques from the pictures, the value was found to be 11.197 pound-feet.
This is a difference of 0.668 pound-feet. When the signal to the
ocilloscope was calibrated, each centimeter on the screen was made to
represent only 0.033h centimeter. The blame for this discrepancy was
put on the drift that was encountered in the amplifier and oscilloscope.
Even after allowing both instruments to warm up for a number of days
the drift was still present, and it occurred sometimes in a very short
time, easily in the time required to take twenty-three pictures. When
the discrepancy was first noticed it was felt that possibly another
set of pictures should be taken, but when the small amount of displace-
ment corresponding to 0.668 pound-feet on the screen was calculated,
it was decided that the pictures which had already been taken were

probably as good as would be obtained, since the drift had been a

.12

problem from the start. Therefore it was decided to use the average
torque obtained from the pictures in the calculations so that this
drift error would not appear as an error. On this basis, the question
of the paper could still be answered.

Possibly the amount of drift trouble could have been decreased
if the shaft upon which the strain gages were mounted had been of a
smaller diameter. This would have required less amplification from
Brush amplifier and oscilloscope and the drift might have been corre-

spondingly lessened.

SUMMARY AND CONCLUSIONS

In considering a damped dynamometer system upon which is imp
pressed a fluctuating torque, intuition leads one to suspect that
possibly the smoothing out of the curve for the scale by the damping
might be done at the expense of the average torque as indicated by
the scale. This paper has shown that the torque value which is deter-
minable by multiplying the beam load by the dynamometer lever-arm
length is the true value of average torque if the impressed torque
from.the engine is periodic, regardless of the amount of viscous

damping involved.

It?

APPENDICES

Appendix.A -- Detailed Description of Test Equipment

Dynamometer

Induction Type Dynamometer

Type HDM 365 Serial No. 263898

No load rpm l800 volts LAO cy. 60 ph. 3

As motor delivers 25 hp at 1750 rpm

As generator absorbs 27 hp at 1855 rpm

Phll load current as motor 30 amperes

Inrush current 170 amps at full voltage
Manufactured by
Harnischfeger Corporation
Milwaukee, Wisconsin

Scale

Toledo Springless Scale

Model 2081 Serial No. 250k

Factory No. 2081-0-5513712

Capacity 75 pounds
Manufactured by
Toledo Scale Company
Toledo, Ohio

Engine
CFR Octane-Rating Test Engine
No. 807998
Compression ratio variable from L to 10:1
Manufactured by
Waukesha Motor Company
Waukesha, Wisconsin

The dynamometer, Scale, and Engine are all included under
ME No. 2706.

StOp Clock

Type S-6 Inst. No. 219h1

115 volts .036 amps 60 cycle

Speed 10 rpm

Smallest graduation .001 minute

NE N0. 347k
Manufactured by
The Standard Electric Company
Springfield, Massachusetts

.11

Electronic Counter
Model 521 A ME No. 3Ah0
manufactured by
Hewlett Packard Company
Palo Alto, California

Strain Gages

SR-A Strain Gages

Four type 0-10

Resistance 10001 5 ohms

Gage factor 3.22 t 2%

Lot No. A
Manufactured by
The Baldwin-Lima-Hamilton Corporation
waltham 54, Massachusetts

Amplifier
Brush Amplifier
Model RD 561200 Serial No. 134
Amplifier ME No. 3634 with Bridge Balance ME No. 3635
Frequency Response 50 to 2000 cps.
Manufactured by
Brush Instruments
Division of Clevite Corporation
Cleveland 14, Ohio

Oscilloscope
Type 532-57 Oscillosc0pe MB No. 37lh, Serial No. 6325
Type CA Plug-in unit, MB No. 3703, Serial No. 002892
Manufactured by
Tektronix Incorporated
Portland, Oregon

Measuring Microscope
Pye TWO-Dimensional Measuring Microscope AM No. 708
Serial No. 39,566 Catalog No. 6147
Minimum graduation .01 mm.
Manufactured by
W. G. Pye and Company Ltd.
Grants Works, Cambridge, England

19

Appendix B -— Description of Test Equipment Assembly

The engine and dynamometer used are a permanent installation,
and were modified for this test only to the extent of installing a
different driveshaft which would accommodate slip-rings and strain
gages. The shaft used was a one and one-half inch diameter steel
shaft and it was connected to both the engine and dynamometer by means
of gear couplings producing a direct drive.

The slip—rings and brushes were built up from a kit directly on
the driveshaft. The four strain gages were mounted on the shaft, and
then were connected in a Wheatstone bridge circuit, the corners of the
bridge being connected to the slip-ring leads. This method of mounting
the gages and connecting them into the circuit eliminates any response
in the bridge due to either bending or compressing and pulling the
shaft. Any signal from the bridge should then represent torque only.
This is discussed more thoroughly in.Perry and Lissner.(l). The bridge
output was taken from the slip-rings through brushes and connected to
the bridge balance input on the front of the Brush Amplifier. After
amplification the signal was then sent to the oscilloscope. A schematic

sketch of the torque pickup circuit is shown in Figure 3.

‘{--.Slip-rings

 

 

 

r’l—\

 

Brush

1 l

Oscilloscope

 

Amplifier

 

 

 

 

 

 

 

 

FIGUREB

20

In order to show just how the four strain gages were mounted on the
shaft, suppose that the outside cylindrical surface of the shaft were
unrolled and laid out flat; then the orientation of the gages is easier

to see as shown here.

 

\1 /§\} 117/
b [:52 33F L [:51

The centers of gages R1 and R3 were diametrically opposite on the shaft,
as were those of R2 and RA'

The electronic tachometer received its signal from.a sending
device which was attached to the shaft on the front of the engine.

To assure that all of the pictures taken started at the same spot
in the cycle, the oscilloscOpe was triggered by the ignition pulse of
the engine. It was found that a piece of ordinary wire 100ped once
around either the secondary or the primary ignition wire and plugged
into the oscilloscope produced sufficient signal to trigger the sweep
circuit. To obtain the trigger signal and the torque signal simultane-
ously, a dual beam plug-in unit was used in the oscilloscope. The
triggering level was set so high that it could not be triggered by the
torque signal, and the ignition signal was adjusted in magnitude so
that it triggered consistently. When the pictures were taken the
triggering signal was raised until it no longer showed on the screen,

but only served as a trigger. Only the torque curves were photographed.

21

Appendix C -— Calibration of Measuring Equipment
Qalibration of the torque pickup.

The torque signal indicated on the screen of the oscilloscope
could be interpreted most easily if the amplifiers in the torque signal
circuit were adjusted to yield a known torque level for each centimeter
of signal rise. This adjustment could have been accomplished in either
of two methods. The first method requires an accurate knowledge of
the values of Poisson's ratio and modulus of elasticity of the shaft
on which the strain gages were mounted. If these two values are known
accurately, they can be used, along with the shaft diameter and the
strain gage resistance and gage factor, to compute the amount of torque
that would produce the same deflection of the scOpe trace as would a
known resistance shunted across one leg of the strain gage bridge.
Thenywhenever a resistor with this value of resistance is shunted into
the proper place in the circuit, the resulting scOpe deflection could
be adjusted in amplitude by means of the amplification controls on
either the scope or the Brush amplifier until the torque represented
by this deflection was on a convenient scale. When the shunt resistor
is removed from the circuit the deflection should return to zero.

There was a resistor built into the Brush amplifier which was meant to
be used for this purpose. However, its value (390,000 ohms) was too
low for this particular application, and when the scape was adjusted
to give a satisfactory deflection for the torque picture, the calibra-
tion signal produced by this resistor was off the screen. A resistor
with about 800,000 ohms resistance would have produced an acceptable
deflection; however a precision resistor would have been required, and

these are not readily available. This difficulty, along with the

3:2.

improbability that Poisson's ratio and the modulus of elasticity could
be determined accurately, resulted in the second method of calibration
being used.

This second method was much more direct and depended upon fewer
factors than the first. A lever arm.was made which could be attached
to the free end of the dynamometer shaft. A hole was drilled in this
lever arm exactly eighteen inches from the center of the shaft, and
a weight hanger suspended in this hole. The system was rotated until
this lever arm was horizontal,and the engine flywheel was then clamped
in position. With the lever arm and weight hanger on the shaft, the
scope trace was brought to the desired zero line. Eighty pounds of
weight were placed on the hanger to produce a deflection corresponding
to 120 pound-feet of torque. The amplification of the signal was then
adjusted to a convenient scale (1 cm.:= 20 lb. ft.). Then the weights
were removed one at a time and the trace displacement checked between
weights in order to assure that the response was linear. The first
time that this was done, the scope did not return to zero. After
sufficient repetitions, however, the system settled down and finally
returned to zero with each unloading. After this calibration was com-
pleted, the lever arm and weight hanger were removed. This of course
caused the SCOpe trace to drop, but since the displacement vs torque
plot was linear, the trace could be brought back to zero with the
vertical position control on the oscilloscope with no loss in accuracy.

There was a considerable problem of drift in both the Brush
amplifier and in the oscilloscope even over relatively short periods
of time. Therefore, once the system is calibrated as described above

the data should be taken as soon as possible, or the calibration should

_2_3_

be repeated. Calibration by this method is somewhat time consuming,and
it was felt that it would be desireable to have a quicker way of bring-
ing the system back into calibration once it had been done accurately
with the weights. As was mentioned before, a resistor of about 800,000
ohms was calculated to be the size which would produce a desireable
trace deflection when shunted across one leg of the bridge. It was also
stated that a precision resistor would be required to calibrate With.
However, the system could be calibrated with the weights, and once it
is calibrated a nonpprecision resistor of about 800,000 ohms could be
shunted across the bridge to produce a deflection. The amount of this
deflection could then be noted, and for subsequent calibrations it
would only be necessary to zero the signal, shunt the resistor and
adjust its deflection to the noted value, and then check to see that
the signal returns to zero when the shunt is removed. This is the
method which actually was used.
Other calibrations

The calibration of the torque picture is the only one that re-
quires any discussion. The preliminary calibration and adjustment of
the Brush amplifier is described thoroughly in the instruction manual,
and adjustments such as the method of zeroing the needleon the Toledo

scale are so obvious as to need no instruction.

3.1:

Appendix D-- Details of calculations
Analytical expgession for the torque curve.

After the twenty-three pictures of representative torque curves
were taken, each curve was divided into twenty-four equal segments.
The height of each curve at the left end of each of these segments was
then.measured accurately with the aid of a two-dimensional measuring
.microscope. The torques represented by each of these measurements
were then computed,and the curves were averaged to obtain the curve for
f(t) which is shown in Figure 2.

To determine an analytical expression for f(t) the method of
harmonic analysis was used as is outlined in wylie (2). The expression

produced for f(t) by using this method is in the form of a Fourier

series, ’
f(t)=a+§,a coswt+§ b sin wt,
0 n= n n n= n n

and harmonic analysis is simply the means of computing the coefficients
an and bn’ The number of terms that will result depends on the parti-
cular harmonic analysis system chosen. The one used here was the
twenty-four ordinate system.

Tb use this system, the curve for f(t) is divided into twenty-
four equal sections. Then the torque at the left end of each of these
sections is determined. This was done when the curves were averaged.
Let the torque at the beginning of the first section be called yo,
that at the beginning of the second section yl, and so on up to y23 at
the beginning of the last section. These values were, in pound-feet;

yo: -1.020 y6 = 1.2.805 312: -l.593 Flare-l. 753

y1= -O.375 :7 -: 88.621 y = -l.568 y1 = 4.1.1.7

y = -0-076 - 98.979 y = -1.6z.9 y = 4.012

y =
Ni: 0e520 ylo“: 5031+]- y16= -1e69l& y2 =-Ot993
y5= 8.328 5’11: - 1.192 y17= -1.679 y23z-l.181

E2

Once these values were known, the coefficients were found by
using the tabular outline described in Wylie. Since the operations in
this table are self-explanatory, no additional discussion is necessary.

The table for this problem is shown on the following four pages. The

resulting an and bn coefficients through n=10 are:

81:— 8.522 a7 =‘ 0.211 b2: -1J...638 D7 = 2.839
a =-1A.6Al. a8 = -l.220 b3=- 6.131. b8 = -0.87A
a =- 0.658 alO=-0.L.80 b5: - 3.827 b10= 0.51.7
35: ‘ 70576

Now “1 is equal to 27" f, where f is the frequency at which the
cycle is repeated.
f = 1800 rev./min. X l cycle/2 rev. X l min./60 sec.
=15 cycles/second

Then col: 307T and wn=30n7T , so everything in

10
f(t): a0 . Em; [an cos wnt + bn sin cant]

has been defined. t is time in seconds.

On page 11 is shown the torque function f(t) as plotted from the
results of averaging the twenty-three representative pictures. Plotted
as circles along this curve are the values of torque that result from
evaluating the Fourier expression for f(t) at each of these points.

The close proximity of these points to the curve substantiates the
accuracy of the method of harmonic analysis.
'petermination of k, the equivalent spring constant.

The procedure used to determine k,is described in.detail in the
main part of this paper. A plot of force vs deflection is shown on
page 30, and the slope of this curve is k. Its value was found to be

k —- 172.5 pounds/inch .

Tabular Determination of Fourier Coefficients

-l.020

 

 

 

 

 

 

 

 

 

 

 

yo to y - 0375 - 0076 0167 0520 80328

ya to 19 . . -1.18l - .993 -.891 -l.012 -l.1.1.7

Stuns ( -05) -10020 -10556 -10069 -0721; " 01+92 60881

Diffs. dl-d5) . . .806 .917 1.058 1.532 9.775

5'6 toy 1.2.805 88.621 98.979 1.3.335 5.31.1 -l.l92 -l.593
3718 to 2&3 4.753 4.679 -l.691. 4.729 4.619 4.563 . .
Sums (C ‘C ) #10052 8609h2 970285 #20106 30692 -20760 -10593
Diffs. fdéiall) 14.558 90.300 100.673 65.564 6.990 .376 . .
co to 06 4.020 4.556 4.069 -.721. -.z.92 6.881 1.1.052
312 to C7 -10593 -20760 30692 #20106 970285 8609h2 0 0
Sums (3 .06) .20613 ’L0316 20623 h10382 960793 930823 hl0052
ours. Qty-£5) .573 1.201. -1..761 42.830 -97.777 -80.061 . .
d1 to d .806 .917 1.058 1.532 9.775 11.558

an to 37 .376 6.990 15.561. 100.673 90.300 . .

Sums is -36) 1.182 7.907 16.622 102.205 100.075 1.1.558
nun. Jami-65) .630 4.073 41.506 49.111 430.525 . .

80 to 03 -20613 -h0316 20623 #10382

96 to CA 410052 930823 960793 0 0

SW3 (3 .13) 33.139 89.507 99.1116 1.1.382

mm. 5:042) 43.665 -9s.139 41.170 . .

hl to h3 .430 - 6.073 -hh.506

hs t0 hh -800525 -9901Ll 0 0

Suns (1 43) 40.095 405.211. 4.1.506

Diffs. imlfmg) 80.955 93.068 . .

 

27

Determination of Fourier Coefficients-Continued

I 31= 89.507 32: 99.116 k2: -91..170 11: -80.095

 

x .500 I Ji= 1.1.751. 55=19.708 Isis-1.7.085 1i=-1.0.01.8

I k1: -98.139 12= 405.211. m1= 80.955 m2= 93.068

 

'

x .866 [1.1: -81..988 15:- 91.115

10= 33.1.39 31: 89.507

mi= 70.107 mé= 80.597

3: ‘LB 0665 kl: .840988

 

 

 

 

 

 

 

 

 

 

.12: 99011-16 .13: 1010382 = ”1170085
Sum 00].. 1 137.855 - 90.750
Sum col. 2 130.889 - 81.988
Sum $8.7M: 21.30 -175.738 =12a2
Difference 6.966= 21.8.12 - 5 .762 8 12am
JQ= 38.1.39 11= 1.1.751. k0= 4.3 .665
- 2: -1.9.708 -33: 4.1.382 k2: -91..170
3'.“ C010 1 -110269 0 0
Sum 3010 2 30372 0 0
Sum - 708979-128.“ 0 0
Difference -11..61.1= 128.8 50.505 = 12af>
1
11: -1.0.01.8 1%: -91.ll5 = 70.107 11: -80.095
13=-1.1..506 =800597 l3=-1.1..506
Sum 0010 1 - 81105514. 0 0 0 0
Sum 30].. 2 - 910115 0 0 0 0
Difference 6 0561 = 12blo -1001-{90= 12b8 -350589 = 12136

l :1: 1.201. . r3: -1.2.830
x .707 l ti: .851 :5: -30.281

 

 

 

‘ f = -80.061

5

fa:

-560603

28

Determination of Fourier Coefficients-—Continued

l 31: 1.182 33: 1.6.622 gs= 100.075

 

x .707 [31: .836 gg=32.962 gg= 70.753

 

 

 

 

 

 

 

 

 

 

 

 

 

 

fi= 0851 8]": 0836
Q‘s-56.603 81: 70.753
Sum -55.752=P1 71.589=81
Difference 574.51.: p2 -69.9l7 = 82
f2: -150761 ‘31: -550752 3“: 1020205 81: 71.589
x 0866 £5: -40123 pi: -1580281 811: 880510 81: 610996
£14: '970777 Pg: 5701154 82: 70907 32: '690917
1 .500 £11: 4.8.889 p5: 28.727 g§= 3.951+ 8:2: -31..959
1": 4.8.889 pi: 4.8.281 g'= 32.962 1 g6= 1.1..558
f3= - 11.123 pé= 28.727 82= 411-959 3%: 3.9511
Sum (11: '530012 1'1: '1905514‘ 0 0 u = 148.512
Difference q2= -1.1..766 r2: -77.008 t= 67.921 . . ‘
f0= .573 f5= -30-281 f0: .573 p2: 57.11511
C11: ‘53 0012 r1: -190551+ ‘11,: 970777 'ffiz' 300281
Sum col. 1 - 52.1.39 98.350
Sum col. 2 - 1.9.835 87.735
Sum ~102.271.= 12al 186.085 = 120.3
Difference - 2.60M-= lZall 10.615= 128.9

 

 

Determination of

29

Fourier 0 0e ffic ients—C ontinued

 

 

 

 

 

 

ro= .573 r = -77.008 3i: 61.996 u = 1.8.512
q2= -1..1..766 -f§= 30.281 t= 67.921 gL= 88.510
Sum col. 1 -1.1..l93 129.917
Sum col. 2 4.6.727 137.022
Sum -90.920= 12a5 266.939 = le1
Difference 2.531.= 12a7 7.105 = 12b11
.2: -69.917 32: 7.907 3i: 61.996 u = 1.8.512
g5: 32.962 -g6=-1.1..558 -t =-67.921 -gL‘-= -88.510
Sum col. 1 -36.955 - 5.925
Sum col. 2 ~36.65l -39.998
Difference - .301.= 12b9 31..073= 12b7

 

 

Force at scale connection, Pounds

FIGURE 5 -- Curve used for determination
of the spring constant, k.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

80 ' ’ J
.1
60 _ .—
10 -—
§
20 ‘—
0 .—
k = 5133841000) _—. 172.1. 53:"
.20 /. .L 1 1 .--———
L: 580 fi-fi
-1. l J I J. .1 l
-200 400 0 100 200 300 1.00 500

Deflection at scale connection, Inches X 1000

3_1_

Determination of m,_equivalent mass.

The equivalent system which was chosen to represent the actual
system from the dynamometer field to the scale pointer is shown in
Figure l on page 6. If the forcing function is removed, the damping
disconnected, and the remaining parts of the system allowed to vibrate

freely, the frequency of this vibration is, from elementary vibrations,
l k
f_ZWJm .

= k
(21ft)?-

Solving this for m,

 

The equivalent weight of this mass would be

= =_58__=.L112461112§él=1688
w mg (217:)2 (27021-2 7

Since the character of the moving parts which were included in
the equivalent mass was quite complicated, it was felt that probably
the equivalent mass would not be the same for all ranges of the scale.
Therefore it was decided to determine the equivalent weight at intervals
of five pounds from zero to thirty pounds, and to plot a curve of equi-
valent weight versus range on the scale. Then, after the range of
operation during the experiment was known, the value of equivalent
weight to be used in the differential equation could be picked from
this curve.

For oscillations about the zero position on the scale, the equa-

 

tion I: 1688 is valid as it stands. However, in order to make the
f

system oscillate about other load levels, it was necessary to add
weights so that the scale when at rest indicated the load which was

to be oscillated about. These additional weights were placed on top

2
of the large weights which were underneath the "black box". Since dis-
placements of these large weights were the same value as equivalent
displacements in the equivalent system, adding the extra weights at
this point served to increase equivalent weight by the same amount
added. These additional weights should not be included in the value of
equivalent weight which is finally used in the equation. Tb compensate
for them, let W" represent the amount added, and let W1 represent the
total of effective weight plus added weights, or W'= W + W". When.the
system.is allowed to oscillate freely about some equilibrium position

other than zero, the equation to be used for determining W is

w' = w . 1.1.939 ,

or , 1688

or the equivalent weight to be used in the equation is determined by
dividing the square of the observed frequency into 1688 and subtracting
the amount of additional weight which was used.

In order to obtain an exact value of frequency for a given equi-
librium position of the scale, it would be necessary to let the system
oscillate through very small amplitudes about this position, since mass
does vary with displacement. However for very small amplitudes, even
with the dashpots disconnected, the amount of damping present in the
system due to friction is sufficient to practically stop the motion
completely before enough cycles are completed to obtain data which
‘would result in an accurate value of frequency. Theoretically, if
everything involved in the method of taking data is working perfectly,
one cycle of oscillation should be sufficient. However, the clock that

‘was used to time the oscillations was accurate to only .001 minute, and

3_3_

the order of magnitude of the frequencies was about one cycle per
second, or one oscillation would take place in about .017 minute.

For time intervals of this small value there is chance for error not
only in the reading of the clock but also in the human reaction time
involved in both starting and stopping the clock. For this reason
then it was felt that the time for a number of oscillations should be
used instead of the time for just one.

To overcome the problem of varying mass about a point, the pro-
cedure was as follows. The system was allowed to oscillate about each
point with starting amplitudes of five, four, three and two pounds
scale deflection. For each starting amplitude the frequency was then
plotted versus original amplitude for each range. These curves were
then extrapolated to the point where starting position is zero. The
resulting frequencies were the ones assumed to be correct about their
respective equilibrium positions. These resulting frequencies were
then plotted against their respective equilibrium scale locations.
Because this plot of points was somewhat scattered, it was decided
that a straight line would be as good a representation of the general
trend as could be chosen. After this line was drawn, it was used as
being accurate, and the value of frequencies used in subsequent cal-
culations were those picked from this curve.

To determine the variation in equivalent weight W with scale

position, the formula W = 33.3.8. - W" was used, the values of f2 being
f

determined from the plot of f2 versus scale position, W" being the
scale position. This plot of W versus scale position was the one used
to determine the value of W to put into the differential equation, and

is presented on the following page. Since the average beam load during

on «N

messed «soapanoa mason esfiundadsom

OH

 

8 am
_

l\“
853 «.3 _

 

 

-\

\dD-—‘-—h——

\

‘1'

season 39 1|\

 

 

\

 

 

 

 

 

 

 

 

.3 3330: 3.33.80 no 5435533 no.4 some visa 1.. 0 go:

003”

coma

8m.”

spunod ‘ 11.121911 quoten'gnbe 9111300533 = M

25

the test run was 11.3 pounds, the corresponding equivalent weight from
this curve is found to be w== 1325 pounds. This is the value of equiv-
alent weight to be used in the equation. Since the equation calls for
mass instead of weight, the value of mass corresponding to this weight
is

1325 lb. 860.2 ___ 3,1,3 lb. soc.2 .

z“!—
m g- 386in. in.

Determination of c, coefficient of damping for the equivalentgsystem.

With the dashpots connected, when the system was displaced from
equilibrium and released, it tended to return to equilibrium; however
the length of time that was required for its return indicated that the
equivalent damping coefficient was greater than critical. Jacobsen and
Ayre (3) give the following relation for systems with damping coeffi-
cients appreciably greater than the critical:

__ 27v l
56-715“?

where: 1/== damping factor== ratio of damping coefficient to critical
damping coefficient.

p == natural frequency of the system in radians per second.
6 == fraction of starting displacement at time t5 after start.
t

6:: time for the system to move from its starting displacement
to 6 X starting displacement.

This formula is meant for use in cases of large 11 and small. 6. Since
damping coefficient c is the unknown in this case, the above equation

can be rearranged using the following identities:

”V _, c

 

cc
cc:= Zmp

k
132:?

It“;

 

21’ 1 2c 1 2c 1 cm 1 c l
t .=r—_—-ln —w= ——— 1n —:= 1n _u=: __ 1n _¢= .— 1n —
épeccpepmzemkeke
SO:
c=—_IEI—té.
1n._

It has already been shown that k = 172.1. lbs./in., and to simplify the
above relation it was decided to make 6' a constant, that is to let the
amplitude decay to the same fraction of its initial value for every run.
The value of this fraction chosen was 1/5, so 1n %= 1n 5 = 1.161. Then

c = 157%? t6 = 107.1 té lb. sec./in. if té is measured in seconds. How-

ever, since the clock was graduated in minutes, it was found more con—
venient to use

c'=.6L26 t6 lb.sec./in.
where té is measured in minutes.

This equation indicates that the damping coefficient should be
independent of mass and of range of operation on the scale. Since c is
a characteristic of the dashpots, this should have been expected to be
the case. As a check, however, data was taken around the scale position
of zero pounds as well as around ten, twenty, and thirty pounds by
adding weights as was done in determining equivalent mass. For each run,
the system was displaced and released at the instant that the clock was
started. The clock was stopped when the displacement dropped to one-
fifth of its initial value. A number of starting amplitudes were used
at each range, and it was hoped that a plot of time versus starting amp
plitude could be made for each range, and that each of these curves
could have been extrapolated to zero starting amplitude, as was done in
determining m. However, since, as the above equation predicted, c did

not depend on scale range or.mass, there was not sufficient spread in

1'1
the values of té for various starting positions to gain anything by
doing this. Therefore, it was decided to simply average all of the
values of t6 and to use this average value in computing c. The average
time for 179 trials was t6 = 0.0978 minutes. Putting this into the
equation for c gives
c = 61.26(C1.0978) = 628.2 lb.sec./in.

This is the value of c that was used in the equation.

For a damped spring-mass system of the type chosen as the equiv-

alent system, the critical damping coefficient cc is given by
k Wk
c’c 2m n 2m1\’m 2dmk= g .

At the time that the test run was made, the equivalent weight was, from

the preceding section, 1325 pounds, and k equals 172.1. lbs./in. Then,

C _, 2F 2 1720‘.j ‘3 “80% lb056C0/1n0
0" 3%;

The damping factor during the run was then

 

_ _c_ _ 628.2 _
V‘- cc "" b.8066 -12090

Therefore, the assumption made that ‘V was large was valid.
Determination of the forcing function for the equation of motion.

In the main body of this paper it was pointed out that the forc-
ing function f'(t) which was finally used in the equation of motion was
the function f(t) which was picked up by the strain gages divided by
the length of the dynamometer lever arm. It was also mentioned that
the reason for using this as the forcing function was that it was
impossible to obtain a function which would accurately represent the
angular acceleration of the dynamometer armature over the cycle, at

least with the instrumentation which was available. This section is

intended to describe the procedure which was attempted to determine
this acceleration.

A magnetic pickup was mounted so that it pointed at the outside
cylindrical surface of the engine flywheel. The engine was turned by
hand to its top dead center position and a piece of steel wire was
then taped on the flywheel opposite the magnetic pickup. Then nine
other pieces of smaller diameter steel wire were also taped on the fly—
wheel at 36 degree intervals so that they all would pass underneath
the pickup as the engine rotated. The magnetic pickup was connected to
the vertical input of a cathode ray oscilloscope. The engine was then
started and the resulting trace on the screen was a horizontal line
except for a number of pips which were created whenever one of the
pieces of steel wire passed underneath the magnetic pickup. Since the
space between each consecutive pair of pips represented 36 degrees of
crank rotation, and, assuming that the horizontal internal sweep of the
oscilloscope was linear with time, it should have been possible to
determine an average AG/At over twenty intervals of the cycle of the
engine. Then, since dG/dt=w , plotting these values of AO/At
against t should have approximated a function which would represent to
over the cycle. This curve could then have been analyzed by harmonic
analysis as was the one for f(t). The resulting analytical expression
for a) could then have been differentiated once with respect to time
to give an analytical expression for o< . When this method was tried,
the pips produced were all separated by the same amount as closely as
could be determined, and even after being magnified five times by using
the 5X multiplier on the oscilloscope, there were no apparent consistent

differences in separations. This meant that, within the accuracy of

32

the method, 6) was constant, and that consequently 0L was equal to
zero over the whole cycle.

Because it was obvious that the above conclusion was not exactly
correct, it was decided to try to determine just how much (.1) did vary
from peak to peak over the cycle. A standard method of flywheel anal-
ysis was used to determine this variation in 6..) . The torque function
f(t) was used and it was supposed that this amount of fluctuation was
being impressed upon the dynamometer armature and that the field was
disconnected so that the armature acted only as a flywheel. If this
condition could be set up, the maximum variation in w from its mini-
mum to its maximum would be 1.15 percent of its average. Therefore,
it is not surprising that the pip method did not result in a function
for 01 . Actually, if the armature were free-wheeling as a flywheel,
the impressed torque from the engine required to keep it turning at
1800 rpm would not be nearly as rough as the curve represented by f(t).
The fluctuation of w in this case would even be less than 1.15 percent
of wavg' Also if the field was connected and if f(t) was the forcing
function, it would seem likely that the fluctuation in c.) would again

be less than .0115 because the field reaction would probably

ang
tend to dampen out the fluctuation rather than to amplify it. There-
fore, it was decided that the analysis made was on the safe side.
Because. of this small fluctuation in w , it was concluded that
it would be very difficult to obtain an accurate expression for ex
even with more refined equipment, and that the only alternative was to
assume OL = 0, and to use f(t)/L for the forcing function f'(t).

The eguation of motion and its solution.

It was shown earlier that the form of the equation of motion for

the equivalent system is

mx + c5: + kx=f’(t) = f(t)/L .
In the few sections preceding this one, values of m, c, k, and f(t)
which applied during the test were determined. Using these values and
recalling that L== 1.050 feet, the above equation becomes:

3.43 x.+ 628.2 x + 172.4 x ‘ 10.664 - 8.116 cos (307rt)

- 13.947 cos (601Tt) + 14.708 cos (901rt) - 0.627 cos (1201Tt)

- 7.215 cos (1507rt) + 4.009 cos (1801rt) + 0.201 cos (2101rt)

- 1.162 cos (2401rt) + 0.843 cos (27017t) - 0.457 cos (3001rt)

+21.185 sin (307Tt) - 13.941 sin (607Vt) - 5.842 sin (901Yt)

+ 11.960 sin (1207Tt) - 3.645 sin (1507Vt) - 2.825 sin (1807ft)

+ 2.704 sin (2107rt) - 0.83; sin (2407Tt) - 0.024 sin (2707Tt)

+ 0.521 sin (3001‘t) .

At first this equation appears quite unwieldy, but since it is a
linear equation it can be solved in parts and all of these parts can
then be superimposed to give the total solution. In other words,
twenty-one equations can be set up, all of them the same as the big
equation on the left side of the equals sign, but each with a different
term of f'(t) as its forcing function. Each of these twenty-cue equa-
tions can then be solved for x, and the solutions can then all be
superimposed to obtain the solution to the big equation.

There are two different types of small equations to be solved;
the ones with sine term forcing functions, and those with cosine term
forcing functions The one with the constant forcing function is a
special case of the cosine type with w = 0. Consider the cosine
function type

.. . _ 1
mx + cx + kx.. an cos QJnt .

1:1.

The general solution of this equation would consist of a solution to
the homogeneous equation mi'c + cx + kx s 0, plus a particular solution
which would satisfy the equation with the forcing function. The solu-
tion to the homogeneous equation would, however, represent the tran-
sient part of the response, and since it is supposed that the system
had settled down to steady state when the data was taken, only the
particular solution is of interest here. Assume then that the partic-
ular solution is of the form

x = An sin wnt + 8n cos cont
Then ° = Ana.) n cos cunt - ann sin cont , and

3E: - Ana)?1 sin wnt' - anfi cos cont .
Substituting these values into the equation mx + 01': + kx = a1; cos cont
gives

[0: - mwi) An - cwan] sin cont + [cwnAn + (k - mwfi) Bu]
°coswnt=aAcoswnt .
Equating coefficients of sin cunt and cos cunt gives
(k - mwfi) An - cwan= 0 ,

and 2
060,14n + (k - mwn) En: 8,3, .

These equations can be solved simultaneously for An and En to give:

A _. coins},
n- (k - mwfifi +?Zw7n
and aflk - mm?!)

B":
(k - mw%)2 + czwn

 

The particular solution of a typical equation with a cosine forcing
function is then

1 ' 2
— china“ in 6.) t an(k ' moan) cos c.) t
x- 3 n ‘7 n
(k - mw%)2 -1- c2411?1 (k - mooni)2 + 020%

 

E

For the special case of the constant forcing function, can: 0 and the

'
above expression for x reduces to x=._a.Q- The beam load which corre—

sponds to this displacement is just F = kx: a!) = 10.664 pounds. It
will be shown that solutions to the remaining equations will be combi—
nations of sine and cosine functions which average out to zero over the
period of one cycle. This should already be obvious for the equations
with cosine forcing functions (except can == 0) through an inspection of
the general particular solution above for these equations.
The form of the equations with sine forcing functions is
xm'é-rcx-c-kiv-rnb;1 sincont .

The procedure for finding the required particular solution of this
equation is the same as used before. First assume a solution of the
form

x = Cn sin cont + Dn cos cunt .
Then i: ann cos cant - ann sin cont ’ and
x: - anfoi sin cont - anfi cos cont .

substitute these into the above equation as before to obtain

[(k - mwfi) Cn - cwnDn] sin cont + [cwncn + (k - mwfi) Dn]
° cos cont = 131.1 sin cent ,
Equate coefficients of sine and cosine terms;
(k — mwfi) Cn - coonDn = b;1
cconcn + (k - mwfi) on: 0
Solve for Cn and Du:

6,10. - mwfa)

Cn" (k — mw%)2 + czefi

_bgcunc
" _ 2 2 2 2
(k mwn) + c con

 

Dn

Then the general form of the solution to the equations with sine term
forcing functions is
bn(k " mwfi) béwnc

X: "—7“ - mw ) A}. (3sz sin cont - (k - mwg)2 + c2“)? COS wnt 0

The combined solution of all twenty-one equations is:

 

 

x= [61860 - 227.887 cos (307Yt) + 116.081 cos (60fl't)
— 28.233 cos (90171:) + 94.413 cos (1201rt) + 9.842 cos (1501\'t)
- 2.552 cos (1807M) - 0.591 cos (210’A’t) + 0.660 cos (2401Tt)
- 0.324 cos (2701Tt) + 0.112 cos (3007ft) - 253.729 sin (30¢Tt)
+ 1.598 sin (601rt) + 39.582 sin (90fl’t) - 20.352 sin (1207Tt)
+ 0.956 sin (1507'\' t) + 3.402 sin (1807! t) -l.611.6 sin (ZlOfi't)
+ 0.226 sin (24077t) + 0.079 sin (2707Yt) - 0.193 sin (3207It)]
X 10'6 inches.

t is time in seconds.

It should be remembered that this displacement is measured at the
end of the dynamometer lever arm. Multiplying this expression for
x times k gives an expression which represents the fluctuation of the
Toledo scale needle, and multiplying it again by the dynamometer lever
arm length L gives an expression for the corresponding torque fluctua-
tion as indicated by the dynamometer. If both of these steps are made,
and if the resulting expression for torque is evaluated at a number of
different values of time over the period of one cycle, the resulting
curve is the smoother one shown on page eleven. The other curve shown
on this page is the input torque f(t) as measured at the strain gages.

Returning to the above long expression for x, in each of the

111+.

cos can t terms and in each of the sin can t terms, can is an integral
multiple of 30%. Therefore in every 1/15 second, each sine and cosine
term completes an integral number of complete cycles. The average
value of a sine or a cosine over a complete cycle is equal to zero, so
all of these sine and cosine terms in the expression for at average out
to zero. This leaves only the constant term to represent average
displacement,

x 0. 06186 inches .

avg. =

The average beam load corresponding to this displacement is
Fan; = kxavg. : 10.664 pounds,
and the average torque is

T = 1.05 F

avg. = 11.197 pound feet.

avg .

l.

2.

BIBLIOGRAPHY
Perry, C. C. and Lissner, H. R. , The Strain Gage Primer, McGraw-
Hill Book Company, Inc., New Y0rk, Toronto, London, 1955.

Wylie, C. R. Jr., Advanced Egineering Mathematics, McCraw-Hill
Book Company, Inc., New York, Toronto, London, 1951.

Jacobsen, L. S. and Ayre, R. 3., Engineering Vibrations, McGraw-
Hill Book Company, Inc., New York, Toronto, London, 1958.

Timshenko, S. and Young, D. H., Vibration Problems in Egineerigg,
Third Edition, D. Van Nostrand Company, Inc. , Pr ceton, New
Jersey, New York, Toronto, London, 1955.

ROOM we ONLY

HICHIGQN STQTE UNIV. LIBRQR

llllllll1ll|llelllIllI”!!!WllllllllllllllhllllllllllllIllll

 

04916121