Mechanism and Machine Theory 37 (2002) 167±174
www.elsevier.com/locate/mechmt
Determination of power losses in gear transmissions with
rolling and sliding friction incorporated
Y. Michlin *, V. Myunster
Quality Assurance and Reliability, Technion ± Israel Institute of Technology, Haifa 32000, Israel
Received 1 January 2001; accepted 5 September 2001
Abstract
A methodology is described for analysis of gear transmissions with allowance for the power losses due to
both the rolling and sliding friction. The latter friction, incorporated in the traditional methods, does not by
itself account for the wear of the teeth at contact sites adjoining the pitch point, where the rolling losses are
paramount. The approach makes for improved accuracy (esp. in the vicinity of the pitch point) in predicting the losses and the wear already at the design stage. Ó 2002 Elsevier Science Ltd. All rights reserved.
Keywords: Power losses; Rolling and sliding friction; Gear transmissions
1. Introduction
Experience with gear transmissions [1] indicates that the wear of the teeth at contact sites
adjoining the pitch point is often commensurate with, or heavier than, that at other sites. In
evaluating tooth wear, some authors currently resort to coecients associated exclusively with
kinematic parameters (per-unit sliding, accelerated sliding, etc.). This approach is inadequate ± it
fails to explain the wear in the pitch-point zone. There are works in which the kinematic and
dynamic parameters a€ecting the losses are taken into account, but they are experimental in
character [1±3]. Nor do the existing methodologies incorporate the losses due to the rolling e€ect,
and thus fail to describe the losses in the vicinity of the pitch point and to explain the wear in that
zone [4]. We have established that a convenient tool for predicting the wear at the design stage is
the transmission loss coecient (TLC) [5], namely the ratio of the friction loss in the system and
the power input of the driving gear. It is shown how the loss coecient incorporating both the
rolling and sliding frictions depends on the contact stresses; the derived equations and the
*
Corresponding author. Tel.: +972-4-829-4380; fax: +972-4-829-4398.
E-mail address: Ye®[email protected] (Y. Michlin).
0094-114X/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved.
PII: S 0 0 9 4 - 1 1 4 X ( 0 1 ) 0 0 0 7 0 - 2
168
Y. Michlin, V. Myunster / Mechanism and Machine Theory 37 (2002) 167±174
corresponding graphs indicate that this dependence is particularly strong in the vicinity of the
pitch point.
2. The method
During transmission of the load from the driving gear 1 to its driven counterpart 2, the tooth
pro®les undergo deformation, creating a contact band AB (Fig. 1). The load distribution diagrams
under static loading are symmetric about the center of this band, i.e. the resultant of the distributed force passes through the gear contact point K. As the teeth move over each other, the
diagrams are displaced in the opposite direction and the resultant is shifted to point D1 ; this in
turn generates resistance to the motion, or in other words ± rolling friction. It is known that when
the contact point does not coincide with the kinematic pitch point P, the gears undergo sliding
friction, generated by the di€erence in their relative speeds. As a result, the force of their interaction deviates from the normal to the gear surface through the friction angle l. Under the
combined action of the two types of friction, the force is displaced along the line D1 F1 with its
point of application F1 moving away from P in the direction of the driven gear.
The loss can be conveniently represented through the TLC u [5], namely the ratio of the friction
loss in the system and the power input to the driving gear. For the ®rst driving gear, the TLC is
uˆ
H3 H1 H2
ˆ
;
H1
H1
…1†
where H1 is the power input to the driving gear, H2 the power output of the driven gear and H3 is
the friction loss.
H1 is given by
H1 ˆ T1 x1 ;
…2†
where x1 is the angular velocity of the driving gear, T1 is the torque input to the driving gear,
which in turn is given by
T1 ˆ R12 O1 F1 cos…aw ‡ l†;
where R12 is the gear interaction force, aw the pressure angle and l is the friction angle.
Fig. 1. Scheme for determination of friction losses in an external-engagement gear transmission.
…3†
Y. Michlin, V. Myunster / Mechanism and Machine Theory 37 (2002) 167±174
169
The power and torque for the driven gear are obtainable in the same way. Substituting them in
Eq. (1), we have an expression for the TLC
ƒ! ƒƒ!
PO1 F1 O2
ƒ! ƒƒ! :
PO2 F1 O1
u1 ˆ 1
…4†
Here and subsequently the arrows indicate directed segments, for which we adopt the following
sign rule: codirectional segments yield a positive ratio. Positive angles are read o€ counterclockwise from the normal. By this rule, the angle aw is positive and the transmission ratio mG21 ±
negative, as
ƒ!
x2 PO1
…5†
ˆ ƒ! :
mG21 ˆ
x1 PO
2
Considering Eq. (5), Eq. (4) can be rewritten as follows:
ƒ!
PO2
ƒ!
PO1
u1 ˆ 1
ƒ!
PF1
ƒ! mG21 ˆ 1
PF1
ƒ! ƒ!
…PO1 =PF1 † mG21
1 mG21
ˆ
ƒ! ƒ!
ƒ! ƒ! :
…PO1 =PF1 † 1
1 …PO1 =PF1 †
Denoting
ƒ!
PO1
M ˆ ƒ!
PF1
we obtain
u1 ˆ
1 mG21
:
1 M
…6†
As the contact point K moves along the line of action N1 N2 , the projection of the contact band
passes through a position in which the application point of the resultant tooth interaction force
falls on the center line O1 O2 . We call this point S, thereby determining for 1 as the driving gear a
point S2 and for 2 as the driving gear a point S1 .
Thus four segments can be demarcated on N1 N2 : N1 S1 , S1 P ; PS2 and S2 N2 , the segments S1 P and
PS2 forming the pitch-point zone, and ± for the ®rst driving gear ± N1 S1 and S2 N2 forming the
zones ahead of and beyond the pitch point, respectively. For each of these segments, M obeys a
di€erent formula.
Consider the case where the contact point K is on N1 S1 . By the similarity of the triangles N1 PO1
and GPF1
Mˆ
O1 N1
:
F1 G
…7†
By Fig. 1,
F1 G ˆ F1 C ‡ CG:
Since
…8†
170
Y. Michlin, V. Myunster / Mechanism and Machine Theory 37 (2002) 167±174
CG ˆ KD1
…9†
and since in the triangle CD1 F1
F1 C ˆ
…KP ‡ F1 G tan aw † tan l;
…10†
we have, after some fairly simple manipulations,
KD1 KP tan l
:
F1 G ˆ
1 ‡ tan aw tan l
…11†
Introducing the notations, L is the relative displacement of the distributed load resultant
ƒƒ!
KD1
…12†
ƒƒ
ƒ! ˆ L
O1 N1
and n is the relative coordinate of the contact-band center on the line of action
ƒ!
PK
ƒƒ! ˆ n
N1 P
…13†
(the value of n for an involute gear system being numerically equal to the parameter C of the
standard [4]) and substituting them in Eq. (7), we have
Mˆ
1 ‡ tan aw tan l
:
L ‡ n tan aw tan l
…14†
This formula is valid for all points of the line of action. When using it, attention must be paid to
the signs of all the parameters in it, as per Table 1.
The displacement of the resultant, KD1 , is given by
KD1 ˆ k bH ;
…15†
where k is an experimentally determined coecient representing the displacement; bH the semiwidth of the Hertzian contact band AB.
According to Hertz' theory [6]
p
…16†
bH ˆ 4P 0 …k1 ‡ k2 † qR ;
Table 1
Parameter signs in accordance with the position of the contact point on the action line
Driving gear
Parameter
N1 S1
S1 P
PS2
S2 N2
1
aw
l
L
n
+
+
+
M <0
+
+
+
+
+
+
+
+
aw
l
L
n
+
+
+
+
+
M >0
+
+
2
Y. Michlin, V. Myunster / Mechanism and Machine Theory 37 (2002) 167±174
scP
s
p0
;
ˆ
p2 …k1 ‡ k2 † qRP
171
…17†
where scP is the normal contact stresses under contact at the pitch point P; p0 is the load per unit
length of the contact band; qR is the normalized curvature radius of the tooth pro®les at the
contact point, given by [7]
qR ˆ
q1 q2
q1 q2
…18†
with the plus for external and the minus for internal engagement, respectively; qRP is qR under
contact at P; k1 ; k2 are constants, given by
k1 ˆ
1 m21
;
pE2
k2 ˆ
1 m22
;
pE2
…19†
where m1 ; m2 is Poisson's ratio for the driving and driven gear materials, respectively; E1 ; E2 is the
modulus of elasticity in tension and compression, as above.
Eliminating p0 in Eq. (16), we have
p
bH ˆ 2p …k1 ‡ k2 † scP qR qRP :
…20†
By Fig. 1
ƒƒ! ƒ! ƒƒ!
q1 ˆ N1 P ‡ PK ˆ N1 P …1 ‡ n†;
ƒƒ!
ƒƒ!
q2 ˆ N1 N2 q1 ˆ N1 P …mG12 ‡ n†;
where
mG12
ƒƒ!
1
O2 P
ˆ
ˆ ƒƒ! :
mG21 O
1P
At P ; n ˆ 0, hence
ƒ!
ƒƒ! ƒƒ
q1P ˆ N1 P ˆ O1 N1 tan aW ;
ƒƒ
ƒ!
ƒƒ!
q2P ˆ N1 P mG12 ˆ O1 N1 tan aW mG12 :
…21†
…22†
…23†
…24†
…25†
Substituting Eqs. (21), (22), (24) and (25) in Eq. (18), we have
…1 ‡ n†…mG12 ‡ n† ƒƒ
ƒ!
O1 N1 tan aW ;
mG12 1
mG12
ƒƒ
ƒ!
O1 N1 tan aW
ˆ
mG12 1
qR ˆ
…26†
qRP
…27†
and in turn substituting Eqs. (26) and (27) in Eq. (20) yields
s
…1 ‡ n†…mG12 ‡ n†
ƒƒ
ƒ!
bH ˆ 2p …k1 ‡ k2 † scP O1 N1 tan aW
:
…mG12 1†2
…28†
172
Y. Michlin, V. Myunster / Mechanism and Machine Theory 37 (2002) 167±174
Eqs. (12),(15) and (28) yield
s
…1 ‡ n†…mG12 ‡ n†
;
L ˆ 2p k …k1 ‡ k2 † scP tan aW
2
…mG12 1†
…29†
where the sign is plus or minus as per Table 1.
Eqs. (26), (27) and (29) can be used both for external and for internal engagement, insofar as
the signs are automatically accounted for in that of mG12 : negative in the ®rst case, positive in the
second.
The bounds of S1 and S2 are obtained from the condition that the application point of the
resultant should lie in the center line O1 O2 ; this is satis®ed when
jnj ˆ jLj
…30†
with the point S2 determined for 1 as the driving gear, and S1 for 2.
Eqs. (6), (14) and (29) indicate that the farther the contact point K from the pitch point, the
greater the contribution of the sliding friction to the loss coecient. In the pitch-point zone, the
contribution of the rolling friction is greater, and the higher the contact stresses, the higher
the corresponding loss.
Fig. 2 shows, as an example, the change of the loss coecient along the line of action ± for a
pair of gears with parameters Z1 ˆ 23, Z2 ˆ 46, m ˆ 4:5. The contact stresses at the pitch point
were taken as 1760 MPa, the sliding friction coecient as 0.1 (the typical value for a closed gear
system [8]). Curves 1 and 2 refer to the respective patterns u01 without and u1 with allowance for
the rolling friction. Table 2 lists the calculated loss coecients at characteristic points of the action
Fig. 2. Change of TLC along the line of action: 1 ± without allowance for rolling friction; 2 ± with allowance for rolling
friction.
Y. Michlin, V. Myunster / Mechanism and Machine Theory 37 (2002) 167±174
173
Table 2
Loss coecients at characteristic points of the action line
Point
n
u1
u01
u1 =u01
Start of action line …N1 †
Start of active segment of action line (SAP)
Start of zone of single-tooth-pair contact (LPSTC)
Pitch point (P)
End of zone of single-tooth-pair contact (HPSTC)
End of active segment of action line (EAP)
End of action line …N2 †
)1.000
)0.653
)0.153
0
0.098
0.597
2.000
0.0546
0.0399
0.0142
0.0058
0.0106
0.0363
0.0984
0.0546
0.0361
0.0086
0
0.0051
0.0308
0.0984
1.00
1.10
1.64
1
2.08
1.18
1.00
line, for the above example. It shows that at the start and end of the action line the coecient with
the rolling-friction losses u1 taken into account is, respectively, 10 and 18% higher than its
counterpart u01 based on the sliding friction alone. Correspondingly, at the start of the singletooth-pair zone the excess is 64%, and at the end ± more than twofold. As the pitch point is
approached the contribution of the rolling friction increases, and at the point itself it becomes
paramount. Numerical data may di€er from one example to another, but the character of the
curves will be the same.
3. Summary and conclusions
In this way, the proposed methodology for evaluation of gear transmission losses with both the
rolling and sliding friction incorporated ± makes it possible not only to explain the wear in the
pitch-point zone, but also to determine the actual losses along the entire line of action with all
friction factors accounted for. It is shown that the farther the contact point from the pitch point,
the greater the contribution of the sliding friction to the loss coecient. By contrast, in the pitchpoint zone the contribution of the rolling friction is greater, and the higher the contact stresses,
the higher the corresponding loss. The presented formulae make it possible to choose the
transmission parameters at the design stage so as to ensure uniform wear along the tooth pro®le,
since non-uniform wear distorts the transmission ratio in the course of one revolution, generates
additional loads and accelerates the wear process.
Acknowledgements
This research project was supported by the Israel Ministry of Absorption.
References
[1] G.I. Skundin, Mechanical Transmissions of Wheeled and Caterpillar Tractors, Mashinostroenie, Moscow, 1969, 342
pp (in Russian).
[2] G.H. Benedict, B.W. Kelley, Instantaneous coecient of gear tooth friction, ASLE Trans. 4 (1961) 59±70.
[3] Yu.A. Misharin, In¯uence of friction condition on the magnitude of the friction coecient in the case of rolling with
sliding, in: International Conference on Gearing, London, September 1958.
174
Y. Michlin, V. Myunster / Mechanism and Machine Theory 37 (2002) 167±174
[4] AGMA 2001-C95, Fundamental rating factors and calculation methods for involute spur and helical gear teeth.
[5] V.N. Kudryavtsev et al., Calculation and Design of Gear Transmissions (Handbook), Politekhnika Publishing,
St. Peterburg, 1994, 448 pp (in Russian).
[6] S.P. Timoshenko, J.N. Goodier, in: Theory of Elasticity, third ed., McGraw-Hill, New York, 1970, 567 pp.
[7] Gear Design, Manufacturing and Inspection Manual. AE-15, SAE, 1990, 643 pp.
[8] J.E. Shigley, Mechanical Engineering Design, ®fth ed., McGraw-Hill, New York, 1989, 779 pp.
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