NASA Technical Reports Server (NTRS) 19910018256: Dynamic measurements of gear tooth friction and load

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Technical Memorandum 103281 Technical Report TR- 90- C- 023
Dynamic Measurements of Gear
Tooth Friction and Load
Brian Rebbechi, Fred B Oswald, and Dennis P. Townsend
Lewis Research Center
Cleveland, Ohio
Prepared for the
Fall Technical Meeting of the American Gear
Manufactures Association
Detroit, Michigan, October 21-23, 1991

US ARMY
NASA
AVIATION
SYSTEMS COMMAND
Dynamic Measurements of Gear Tooth Friction
and Load
Brian RebbechiR, Fred B. Oswald, and
Dennis P. Townsend
National Aeronautics and Space
Administration
Lewis Research Center
Cleveland, Ohio 44135
1. INTRODUCTION problems. Anderson and Lowenthal [4] com-
puted overall losses due to friction and
The dynamic forces at the point of found good agreement between theoretical
tooth contact are of considerable interest predictions and experimental data. Krantz
to the designers of high-speed, light- and Handschuh (5] applied a similar tech-
weight gearing. Accurate prediction of the nique to an epicyclic gear rig, obtaining
dynamic loads can assist in minimizing the good correlation at low oil temperatures,
size and weight of a transmission. In a but poorer correlation at higher oil tem-
helicopter application, where the transmis- peratures. However, this technique cannot
sion is a significant fraction of vehicle detect the variation in friction during the
weight, such a reduction would be an impor- tooth engagement cycle. There is also the
tant factor in overall vehicle performance. problem of separating the power loss due to
A program to experimentally and theo- gear tooth friction from power losses due
retically study fundamental mechanisms of to other sources such as bearings, windage,
gear dynamic behavior is being undertaken and so forth.
at the NASA Lewis Research Center in sup- Extensive measurements of lubrication
port of a joint research program between conditions at a sliding-rolling contact
NASA and the U.S. Army. This paper pre- have been carried out on disk machines [6].
sents the results of dynamic tooth-fillet These experiments are of considerable value
strain gage measurements from the NASA in confirming the existence of elastohydro-
gear-noise rig, and it introduces a tech- dynamic lubrication and in identifying the
nique for using these measurements to separate regimes of lubrication that pre-
separate the normal and tangential vail under the various slide-to-roll
(friction) components of the load at the ratios. However, the usefulness of the
tooth contact. Resolution of the contact modes of behavior and friction coefficients
force is desirable for several reasons. in predicting lubrication conditions at an
Two of these reasons are the following: actual tooth contact, where the degree of
(1) A primary output of analytical sliding changes throughout the tooth
models of gear dynamic behavior is typi- engagement cycle (typical duration,
cally the normal force at the point of 250 µsec), needs to be verified. In this
contact (e.g., [1] and [2]). short period of time, large changes occur
(2) The measurement of dynamic in the lubricant temperature, shear, and
friction of meshing gears does not appear viscosity atpressures up to 1.4 GPa
to have yet been carried out successfully. (200 000 lbf-in. ). Dyson (7] reported
An interesting trial was carried out temperatures up to 400 °C and oscillatory
by Benedict and Kelly [3], but it was dis- shear rates up to 107 sec-1. These con-
continued because of dynamic response ditions cannot readily be produced outside
of an actual tooth mesh.
Visiting scientist from Australian
Aeronautical Research Laboratory.
Friction at the tooth contact is tooth-root fillets on both the loaded
important for determining not only power (tensile) and unloaded (compression) side
loss and efficiency, but also for under- of two adjacent teeth on the output
standing gear-tooth scoring and wear. An (driven) gear (Fig. 4). To measure maximum
important parameter in scoring is the fric- tooth bending stress, the gages were placed
tion coefficient [3]. Friction greatly at the 300 tangency location [9].
affects the heat input to the lubricant Strain gage signals were conditioned
when sliding velocities are high. by two methods: for static calibration and
This report presents dynamic, gear- measurement, a strain gage (Wheatstone)
tooth strain measurements from low-contact- bridge was used; for dynamic measurements,
ratio spur gears tested in the NASA gear- the strain gages were connected via a slip-
noise rig. The technique used to convert ring assembly to a set of constant-current
these strain measurements into normal and strain gage amplifiers.
tangential (friction) tooth loads is A 4-channel, 14-bit digital data
described. Plots of normal and tangential acquisition system was used to record the
forces, for both static and dynamic condi- dynamic strain data. Sample rates of 20 to
tions, are presented for a representative 50 kHz per channel were used, depending on
range of loads and speeds. The normal test gear speed.
force and dynamic strain data have been An optical encoder was mounted on the
used to verify a gear dynamics code in input shaft to measure roll angle and hence
another related report [8]. determine load location; the position of
the encoder was adjusted so it would pro-
2. APPARATUS duce 1 pulse/revolution at a known roll
angle.
2.1 Test Facility
3. TEST PROCEDURE
These tests were conducted in the NASA
Lewis gear-noise rig (Fig. 1). This rig 3.1 Calibration
comprises a simple gearbox powered by a
150-kW (200-hp) variable speed electric Calibration of the strain gages on the
motor, with an eddy-current dynamometer instrumented (driven) gear was conducted to
that loads the output shaft. The gearbox provide a matrix of strain output versus
can be operated at speeds up to 6000 rpm. applied load. Before commencing the strain
The rig was built to carry out fundamental gage calibration, the gears were demagne-
studies of gear noise and of dynamic tized. This demagnetization reduced the
behavior of gear systems. It was designed apparent strain resulting from the gages
to allow testing of various configurations moving through the magnetic field of the
of gears, bearings, dampers, and supports. adjacent gear. At normal gear operating
A poly-V belt drive served as a speed speeds, magnetic effects can induce an
increaser between the motor and input error signal in the gage.
shaft. A soft coupling was installed on For calibration, the instrumented gear
the input shaft to reduce input torque was meshed with a special gear whose adja-
fluctuations caused by a nonuniformity of cent teeth had been ground away; this per-
the belt at the splice. mitted loading of a single tooth only. The
Test gear parameters are shown in calibration was carried out for each of the
Table 1, test rig parameters in Table 2, two instrumented teeth for roll angles
and gear tooth profile traces in Fig. 2. ranging from 12° to 30°. At each test po-
The tooth surface roughness was measured by sition (roll angle) the torque was applied
using an involute-gear-checking machine at three levels - 45 percent, 88.5 percent,
with a diamond stylus of approximately and 132 percent of the nominal value of
10-µm (0.0003-in.) radius. The surface 71.8 N-m (635 in.-lb). At each of these
roughness varied along the length of the load levels the sliding direction was
tooth, with the region near the root reversed (by reversing roll direction), and
appearing to be lightly polished. The a linear curve was fit to the data for each
maximum surface roughness was estimated to sliding direction. By reversing the roll
be 1.34 µm, (34 Ain.) peak-to-peak, or an direction, the instrumented gear was effec-
average of 0.43 µm (11 Ain.) (Fig. 3). The tively tested as both the driven gear (out-
gear rig was operated at an oil fling-off put) and driving gear (input). In each
temperature of 54±2 °C (130±5 °F). At the instance the gear was rotated a small angle
mean temperature of 54 °C, the viscosity of (approximately 10 ) in the intended direc-
the synthetic oil (Table 2) used in the tion of roll until the desired roll angle
tests was 14 cSt (11.6 cP). Natural fre- was reached, so as to definitely establish
quencies from a four degrees-of-freedom a sliding direction.
eigensolution [8] are also shown in The strain gage calibration apparatus
Table 2. is shown in Fig 5. The results of the cal-
ibration for gages 1 to 4 are given in
2.2 Instrumentation Figs. 6 and 7, for loading on tooth 1. The
arrows indicate roll direction. The
General-purpose, constantan foil, results for loading on tooth 2 were very
resistance strain gages (gage length, similar.
0.38 mm (0.015 in.)) were installed in the
Dynamometer -^
Optional
Test output flywheel ^.
C
n
(a) Layout.
(b) Detail of gearbox.
Figuret.—NASA gear-noise rig.
TABLE 1. - TEST GEAR PARAMETERS
Gear type . . . . . . . . standard involute, full-depth tooth
Number of teeth . . . . . . 28
Module, mm (diametrial pitch in.^l) . . . . . . . . 3.175(8)
Face width, mm (in.) . . . . . . . . . . . . . . 6.35 (0.25)
Pressure angle, deg . . . . . . . . . . . . . . . 20
Nominal (100-percent) torque, N-m (in.-lb) 71.77 (635.25)
Theoretical contact ratio . . . 1.64
Driver modification amount, mm (in.) 0.023 (0.0009)
Driven modification amount, mm (in.) . . . . . 0.025 (0.0010)
Driver modification start, deg . . . . . . . . . . . . . . 24
Driven modification start, deg . . . . . . . . . . . . . . 24
Tooth root radius, mm (in.) . . . . . . . . . . . . 1.35(0.053)
Average surface roughness, µm (,dn.) 0.43(11)
3
TABLE II. - TEST RIG PARAMETERS
Input inertia, J1, kg-m' (lb-sec`-in.) . . . . . . 0.0237 (2.10)
Gear inertia, J,, J , kg-m' (lb-sec`-in.) 0.0000364 (0.00322)
3
Load inertia, J,, kg-m' (lb-sec`-in.) . . . . . . . 0.085 (7.5)
Input stiffness, K,, N-m/rad (1b-in./rad) . . . . . 341 (3017)
Gearbox stiffness,'K,, N-m/rad (lb-in./rad) . . . 6158 (54 500)
Load stiffness, K,, N-m/rad (1b-in./rad) 12 700 (112 300)
Synthetic turbine oil . . . . . . . . . . . . . . MIL-L-23699B
O
Viscosity at 130 C, cSt, (cP) . . . . . . . . . . 14 (11.6)
Natural frequencies (eigensolution), Hz 6.56, 52.5, 1220
1
8
1s 0 0002,n
22 i
T
^ (a) Driving gear.
E
c
L
O
1- 22
15
8
1
6 9 12 15 18 21 24 27 30 33
Roll angle. deg
(b) Driven gear.
Figure 2.—Test gear profile traces.
(a) Gage instanation.
Tooth 1 Tooth 2
30° tangency \ r i
to root fillet
(location of
gages 1 to 4) 7 2^
,
,30 \^Q3
LO
i LO \
Gage
Tooth 0 J
Rotation
Zero reset i (b) Gage location.
Root Tip
Figure 4. Straingage installation and location on test gear.
Figure 3.--Surface roughness measurements of driven gear.
4
3.2 Data Acquisition
3.2.1 Static strain data. - Strain
data were recorded under static (nonrotat-
ing) conditions for the gear set assembled
in its normal (running) configuration with
the standard running gear replacing the
calibration gear. The measurements were
made for two reasons: first, as a check on
the accuracy of the method used to resolve
tooth force into normal and tangential com-
ponents; and second, to provide information
on load sharing characteristics of the gear
assembly. A strain gage bridge circuit was
used to record strains for roll angles from
12° to 40° relative to tooth 2. Torque
levels of 37, 88, 100, and 132 percent were
applied, but unlike the single-tooth case,
linear curve-fitting of these data was not
appropriate because of the kinematic non-
linearities introduced by load sharing when
more than one pair of teeth are in contact.
As for the single-tooth case, these meas-
urements were carried out for the instru-
Figure 5—Strain gage calibration apparatus. mented gear acting as both the driven and
driving gear, thus reversing the sliding
direction.
2000 Driving gear 3.2.2 Dynamic strain data. - Dynamic
— — Dnven gear strains were recorded for the 4 gages, for
1500
a speed-load matrix of 28 points: 4 speeds
(800, 2000, 4000, and 6000 rpm) and 7
1000 —
m torque levels (16, 31, 47, 63, 79, 94,
Q
s soo and 110 percent of the nominal value of
71.8 N-m (635 in.-lb)). The data were
o
0 recorded by 14-bit data recorders via a
a (a) Gage 2, tensile strain. slip-ring assembly. Sample rates used were
0
50 000 Hz per channel for the 2000-, 4000-,
-500 and 6000-rpm speeds, and 20 000 Hz per
channel for the 800-rpm speed. A continu-
o -1000 ous record, consisting of 10 000 data
scans, was made at each speed so as to give
-1500 /
a record length of 0.2 sec at 50 000 Hz,
and 0.5 sec at 20 000 Hz. Because of the
-2000
interest in comparing tensile and com-
-2500 pressive strains on each tooth, data from
30 28 26 24 22 20 18 16 14 12 these two gages were simultaneously
Roll angle, deg recorded along with the encoder signal.
(b) Gage 1, compressive strain, This procedure was repeated for the second
Figure 6 —Static strain gage data, single tooth loading instrumented tooth.
on tooth 1 (arrows show roll direction). The data were then digitally
resampled, by using linear interpolation,
at either 1000 or 2000 samples per revolu-
40 Driving gear tion (depending on speed) and synchronously
m -- Driven gear averaged. Time domain synchronous averag-
Q ing, a technique now in wide use in gear
diagnostics (10], was used here to reduce
noise effects (especially from the torque
1 1
fluctuation caused by the belt drive). Its
n 0
oO (a) Gage 4, tensile strain implementation requires two data channels -
one for timing signal data and one for
0 strain data. The timing signal data pro-
MO vided resample intervals for exactly one
revolution.
_0 28 26 24 22 20 18 16 14 12 ANALYSIS
Roll angle, deg
(b) Gage 3, compressive strain. For a single tooth, measurement of the
Figure 7.—Static strain gage data measured at tooth 2 strain outputs Sr, and .S of gages
for single-tooth loading on tooth 1 (arrows show roll mounted on the copressive and tensile
direction). sides of the tooth respectively (Fig. 4)
will, in principle, enable resolution of
the tooth forces F. (normal) and Fr
(tangential), provided that the response of 5. RESULTS AND DISCUSSION
these two gages to the two forces is
linearly independent. The response of the 5.1 Calibration
gages can then be expressed as
Tooth-fillet strains for 100-percent
So = a11Fn + a12Ff (4.1) torque were evaluated by fitting a linear
curve to the calibration data for the three
22Ff (4.2) torque levels. These strains at gages 1 to
S, = a21Fn + a
4 are plotted in Figs. 6 and 7 as a func-
tion of roll angle, for loading of tooth 1.
or simply as Notable from these curves is the signifi-
cant influence of static friction on strain
(S} _ [a]{F} (4.3) output; the tensile gage (see Fig. 6(a))
shows a difference in strain between the
driving- and driven-gear cases (when slid-
ing direction reverses) that is 27 percent
of the mean strain reading. The signifi-
S
cance of this is twofold: first, it is
where {S} = S`
difficult to establish a "no-friction',
z
curve; and second, and possibly more impor-
tant, these curves (particularly the ten-
sile curve) illustrate the effect that
tooth friction has on the results. It is
l F apparent from Fig. 6 that the compressive
{F} = Ff gage is much less influenced by friction
and, thus, would be expected to give the
best indication of normal force if only one
and is the strain influence coeffi-
all gage were used. This is further confirmed
cient; that is, the strain at i due to a
by the tooth strain influence coefficients
unit normal force (j = 1) or a unit fric-
(see Appendix).
tion force (j = 2).
The strain influence coefficients all
5.2 Static Meshing
are evaluated by alternately setting Fn
and F. in equations (4.1) and (4.2) to
Measured strain is plotted in Fig. 8
zero. In practice, neither Fn nor F
as a function of roll angle for static
can actually be zero because a normal force
meshing of the gears (i.e., for multiple-
between the teeth is a prerequisite for a
tooth contact). This figure shows the
sliding force to develop. However, because
average strain (mean of driving- and
strain values were recorded for both direc-
driven-gear values) for 37-, 88-, 100-, and
tions of sliding (that is, for the instru-
132-percent torque. Figure 9 shows in
mented gear acting as both driving and
greater detail the tooth-fillet strains for
driven gear) at each roll angle value, we
inferred that the average of these two Gage 2 Gage 4
strain values is equivalent to the fric- 2000 tooth 1 tooth 2 Torque level,
percent
tionless case, and that the effect of fric-
tion alone will be one-half the difference 1500 " 132
100
between the two values. Thus, the coef- A.
ficients a12 and a22 (which relate to loco , ^•\^^;^ j 88—
friction) are evaluated from half the
difference between the driving gear and 500 37
driven gear curves of Fig. 6. Likewise,
0
the strain coefficients all and a21
I I I I I I I I I
(which relate to normal force) are eval-
(a) Loaded tensile strain side of tooth.
uated from the average of these two curves. 2
The solution for F and F is found by
premultiplying bothn sides of' equation (4.3) 2 Soo
by [ a ] -1; hence
{F} _ [a]-'(Si (4.4) -500
—1000
The analysis presented above ignores
the influence on strains S and St due —1500
to loading on adjacent teeA . In the case
—2000
of thin-rim gears [11], this effect can be 32 30 28 26 24 22 20 18 16 14 12
on the order of 12 percent. For the thick- Roll angle for tooth 2, deg
IIIIIIIII
rim gears used here, however, the influence
from adjacent teeth is at most 3 percent
28 26 24 22 20 18 16 14 12
(compare Figs. 6 and 7). In the data pre- Roll angle for tooth 1, deg
sented in this paper, the influence of
(b) Unloaded compressive strain side of tooth.
adjoining teeth has been included. The
computational procedure is outlined in the Figure 8.—Averaged static strain data on two successive
teeth.
Appendix.
gages 1 to 4 at 100-percent torque, with The total normal force between the
the instrumented gear acting as both driven one- or two-tooth pairs in mesh should be
and driving gear. The curves of Fig. 9 are equal to 1718 N (386 lbf). This value is
the averaged result of three trials. From the torque divided by the base circle
the results of Fig. 9, and the influence radius. The normal-force component of the
coefficient matrix previously described, plots shows agreement within 1.5 percent of
plots of normal and friction forces the expected value.
(Fig. 10) have been derived from the static An absolute value for the friction
data for the 100-percent-torque case. force cannot be determined during calibra-
tion since the coefficient of friction at
2000 Driving gear the tooth contact point is unknown. If an
-- Driven gear arbitrary value of unity is assigned to the
1500 maximum frictional force developed at
100-percent torque, then the friction value
1000 \\ should be either +l or -1 (depending on the
direction of sliding) in the single-tooth
Soo
IN contact region. This ideal is nearly
achieved in the static measurements for
0IIIIIIIII;J tooth 1 in Fig. 10(b). For tooth 2, the
b friction force is offset by about -0.4 from
o (a) Tensile strain, gages 2 and 4. the +1 values. Outside the single-tooth
contact region, the friction force
decreases in approximately linear fashion
with the normal tooth load. This implies a
-500
constant friction coefficient under these
-1000 static meshing conditions.
It is interesting to note the location
-1500 of the zero-crossing of the friction force
in Fig. 10 when tooth sliding changes di-
-2000 rection. This zero-crossing differs from
30 28 26 24 22 20 18 16 14 12 10 8
IIIIIIIIIR1oll anIgle on tooth 2, deg the pitch point by nearly 10 of roll. Some
of this difference may be due to deflection
of the gear shaft, which causes a shift in
3028 26 24 22 20 18 16 1412 10 the operating pitch point.
Roll angle on tooth 1, deg
(b) Compressive strain, gages 1 and 3. 5.3 Dynamic Case
Figure 9.--Static strain data, multiple tooth loading
(arrows show roll direction). The dynamic tooth strains for the 28
Pitch point Pitch point speed-load conditions are shown in Fig. 11.
tooth i tooth 2 To allow direct comparison, the compressive
500 1720 N = r Normal - 3
2000 aoo --` 386 lb , force strain data are inverted (shown as posi-
- ---- -------- - --------- tive) and overlayed on the tensile curves.
300 Tooth 1 Tooth 2 - 2 Notable features of these curves include
1000 200 the peak tooth strain corresponding with
1 the high point of single-tooth contact
100
r Frictional - (which occurs at about 231 roll angle), a
0 0 / force 0 dip or notch in the tensile tooth strain
i -curves near the pitch point (where the
Z° --------------'y"------------ -1 sliding force reverses), and dynamic
`o ° I I I' I I I J -2 effectTsh eb edcyonmaimnigc aepfpfaercetn ti sa tp ahritgihceurl asrpleyeds.
I I ( v
€ £ 0 notable in the curves for 4000 rpm at the
Z Z (a) Driving gear. lowest torque (16 percent). Here, the
force vanishes, thereby indicating tooth
2000 540000 `1-1-37-82-60- l -bN- ------ ,- Nfoorrcmeal 3 LL pseerpcaernatt itoonr qoucec utrhse.r eB yi sc ovnetrrya slti,t talte 1d1y0n-amic
300 Tooth t Tooth 2 2 effect, as evidenced by little difference
1000 200 ____ -------r_ -_ - 1 a20m0o0n,g 4t0h0e0 ,c uarnvde s6 0f0o0r rtphme) .four speeds (800,
100 In Fig. 12 the computed normal and
0 0 Frictional 0 friction forces are shown for four speeds
force at the highest torque (110 percent). Note
---------- -- -;---------- ---- -1 the very good agreement with expected
results at the low speed of 800 rpm
I I I! I I I
-2 (Fig. 12(a)), where we would expect to
30 26 22 18 14 10 approach a static case. Here, the normal
Roll angle on tooth 1, deg force is very close to the static nominal
value (a function of the torque divided by
30 26 22 18 14 1C the base circle radius). The friction
Roll angle on tooth 2, deg results show a marked transition in force
(b) Driven gear. from negative to positive as the tooth con-
Figure 10.—Normal and frictional forces at 100-percent torque under static tact passes through the pitch point, where
conditions.
there is pure rolling. Also the friction
Tensile tooth strain
coefficient appears to be less than that of -- Compressive tooth strain (inverted)
the static calibration case, as can be seen Pitch Pitch
by comparing Fig. 10(a), where the friction point for point for
force has a value of unity for a normal tooth 1 tooth 2
force of 386 lbf, and Fig. 12(a), where the 2000 Tooth 1 Tooth 2
friction force is a maximum of approxi- I Ii Torque levels,
I t percent
mately t0.75.
1500 /' \{ _-110
5.4 Accuracy —94
/ \ y \\ ' \ \r. \ ,
The results obtained herein for the
\y
static and dynamic tests indicate the 1000 /^\ \\ / \ \+ \ ^^ X63
feasibility of using multiple gages to
separate the tooth friction and normal
\
forces. The results of the static case are \\ ,_31
500
particularly encouraging. The value for ./!r//' N \ I .^.\ \\ \y \\ 116
the normal force is generally within
1.5 percent of the expected value. The
friction force, whilst at times much less o' \
accurate, nonetheless demonstrates the
c
trends we expected to see - that is, the o ^L_L_ I I I I I I I I
(c) 4000 rpm.
2000
Tensile tooth strain / \ Torque levels,
-- Compressive tooth strain (inverted) / \^ /\\^ percent
Pitch Pitch 1500 / // /^\ ___110
point for point for
P000 tooth 1 tooth 2 —94
Tooth 1 Tooth 2 Torqpueerc leenvtels, / // \\i^ \\\ ^/ // \\ \^-^ --79
1000 / / /^\ \ \ \ / / ^\ \ \
1500 /^ / / \yVi\\ \ /// //^^ IVI \ \ i 91410 // / /^\ \I \\\\\\ ^' _-'47
--79 500 /! //j / \\\ .} i / ^^ \\ \ II x-31
/I^ / l Ill l^ r —16
1000 j/lj^ \^J ^ ^^' X63
l/ \ 111 / J _47
I -I
500 / jr%. ^\ .I \ `\ `^^ x/31 p30 28 2'6 24 22 20 18 16 14 12 10
_16 Roll angle on tooth 1, deg
I I I I I I I I I I I
30 28 26 24 22 20 18 16 14 12 10
Roll angle on tooth 2, deg
c
(d)6000 rpm.
^ I I I I I I I I I I
0 Figure 11 —Concluded,
(a)800 rpm.
friction force is proportional to the nor-
2000
I mal load, and a reversal in sign occurs at
I about the pitch point. The good results
^^ I
for the static case are believed to be
1500 partly due to using instrumentation that
was identical to that used for the static
calibration (i.e., the Wheatstone bridge
circuit). Assessing the accuracy for the
1000
dynamic case is more difficult, since we do
not fully know what to expect. However,
dynamic operation could introduce the fol-
Soo lowing problems:
(1) There could be some change in sen-
sitivity due to the change in signal condi-
tioners (i.e., constant current amplifiers
0
operating through slip rings).
30 28 26 24 22 20 18 16 14 12 10 (2) Resistance variations of the slip
Roll angle on tooth 1, deg rings and other electrical noise can con-
I I I I I I I I I 1 1 taminate dynamic data. This was minimized
30 28 26 24 22 20 18 16 14 12 10 here by the use of synchronous averaging,
Roll angle on tooth 2, deg as described in the test procedure for
(b)2000 rpm. dynamic data.
Figure 11.—Dynamic tooth strains at four speeds and seven torque levels. (3) Other dynamic effects such as gear
Compressive strains are shown as positive for comparison with tensile body vibration can also produce unwanted
data.
8
the summing of the strain magnitudes, as is
Pitch point Pitch point
tooth 1 tooth 2r Normal the case for normal force (see Appendix).
500 force 3 various techniques can be used to minimize
z000 400 --------,-- -r ----'-- ^ ---- errors - synchronous averaging, as carried
300 Tooth 1 Tooth 2 2 out here, and possibly, an adjustment (com-
pensation) of the friction curve to bring
1000 200 1 about zero friction at the pitch point.
100 Frictional The do offset of the strain signal is tri-
0 0 I force o tical. Figure 13 shows the superimposed
curves of normal and friction forces for
-------- -1 four successive revolutions of the gear,
using nonaveraged data. Each curve is
(a) 800 rpm.
based on the corresponding tensile and
5o0 3 compressive strains for that particular
2000 400 --------- revolution. A significant variation in
2 friction estimation is evident from one
300 revolution to the next; this cannot be
1000 200 1 ascribed to the expected small torque fluc-
100 tuations caused by the belt drive.
0 ^ 0 0
Z
Pitch point Pitch point
0U Vo ----------- 2 500 tooth 1 tooth f- 3
2000 force
E (b) 2000 rpm. o0 Z a 400 Tooth 1 2
Z Z 500 3 LL U`o 300 i Tooth 2
2000 400 ---------- 0m 1000 E 200 ____ 2
2 0 8 too rFrictional °
300 Z forte 2
1000 200 - Z 1 0 0 A 0.u2_
100
-1
0 0
-1 30 26 22 18 14 10 -2
(c) 4000 rpm. Roll angle on tooth 1, deg
500 3 30 26 22 18 14 10
2000 Roll angle relative to tooth 2, deg
400 ---------- -- -- -------- -- - \ Figure 13.—Computed dynamic load and friction superimposed for four
2
300 successive revolutions at 4000 rpm and 110-percent torque.
1000
200 1
100 Differences in profile between the

0 0 0 single-tooth calibration gear and the
operating gear result in the tooth contact
-1 point being slightly displaced along the
30 26 22 18 14 10 tooth profile, thereby causing an error in
Roll angle on tooth 1, deg the measured roll angle. This error has
been estimated to be of the same order
30 26 22 18 14 10 (0.250 ) as the error in setting the roll
Roll angle on tooth 2, deg
angle for calibration.
(d) 6000 rpm.
The friction force results obtained
Figure 12.—Computed dynamic load and friction at four speeds and
herein were necessarily qualitative. A
110-percent torque.
logical next step would be to calibrate the
signals. A strain output from the tooth gages with a known friction force. A
fillet gages was observed when the teeth device similar to that of Benedict and
were out of mesh. This is attributed to Kelly [3] (Fig. 14) could be used for this
vibration of the gear body. The effect was purpose. In their application, dynamic
most obvious at higher speeds, appearing at effects prevented Benedict and Kelly from
three times tooth mesh frequency. This obtaining useful results from this device.
frequency component can be seen in the If the device were used only for static
friction force trace at 6000 rpm calibration, this restriction would be
(Fig. 12(d)). removed. Alternatively, with only slight
Using strain outputs to detect fric- modification this setup could be used to
tion requires accurate measurement of apply a known force in the friction force
strain. A 1-percent error in strain meas- direction while the tooth contact position
urement will result in a 10-percent error was held constant.
in force estimation (see Appendix). This
extreme sensivity to measurement error
occurs only with friction force estimation.
It effectively results from using the dif-
ference between the magnitude of the.ten-
sile and compressive strains, rather than