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Original Article

Proc IMechE Part J:

J Engineering Tribology
Accelerated life testing of harmonic 0(0) 1–12
! IMechE 2015
driver in space lubrication Reprints and permissions:
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DOI: 10.1177/1350650115586032
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Jun-Yang Li1, Jia-Xu Wang2,3, Guang-Wu Zhou3, Wei Pu3 and

Zhi-Hua Wang4

Abstract
For mechanical moving components in space applications, the accelerated life test for the verification of reliability is very
important. Since traditional methods of the accelerated life test for liquid lubrication have some limitations, a new
method is proposed in this paper, in which the harmonic driver is used as the object of study. First, the life test of a
harmonic driver is performed in vacuum. The results show that the adhesive wear caused by asperity contact at the wave
generator–flexspline interfaces is the crucial factor governing the life of harmonic driver. To clarify the lubricant film
behavior at the wave generator–flexspline interface, a mixed lubrication analysis is conducted. Furthermore, the scheme
of the accelerated life test is discussed. The speed and load are selected as the acceleration stresses of the test, while the
load supported by asperity contact is defined as the characteristic quantity of the life and used to select acceleration
factors. Finally, accelerated life tests under different working conditions are conducted to validate the results of the
theoretical analysis.

Keywords
Harmonic driver, accelerated life test, mixed lubrication analysis, adhesive wear

Date received: 23 August 2014; accepted: 9 March 2015

Introduction and oil viscosity a constant could accelerate the life

Harmonic drives (HD) have been used in many space test under the premise of unchanging lubrication
applications that require high precision, lightweight condition. This is based on the elastohydrodynamic
and compact design components.1 In the past lubrication (EHL) theory – oil film thickness is in
50 years, many new lubricants have been used for proportion to the product of speed and oil viscosity.11
HD with the development of lubrication technology, It has been used by Japan Aerospace Exploration
and the types of lubrication are different according to Agency (JAXA) for the life test of HD in vacuum,12
the working conditions and environments, such as and also by many research institutes for the life test of
liquid lubrication, solid lubrication and solid film/oil liquid lubricated parts in space applications.13–16 In
or grease composite lubrication.2–6 Most published order to adapt to the extreme temperature environ-
reports focused on the failure mechanism of HD by ment in space applications, liquid lubricant usually
conducting life test, and the experimental results show has high viscosity index. Consequently, the tempera-
that the performance degradation and failure of HD ture must be increased greatly to reduce viscosity
may be mainly caused by the wear of flexspline and many times for the purpose of allowing the speed to
circular spline, especially the interface between the
wave generator and flexspline.7–10 1
Post-doctoral Research Station of Control Science and Engineering,
For the space applications, the reliability assess- Chongqing University, Chongqing, People’s Republic of China
2
ment of HD is very important. The life test must be State Key Laboratory of Mechanical Transmission, Chongqing
University, Chongqing, People’s Republic of China
conducted on the ground before a new lubricant is 3
School of Aeronautics and Astronautics, Sichuan University, Chengdu,
used for space mission. Obviously, it may take a Sichuan, People’s Republic of China
long time and cost a great deal of labor and material 4
Lanzhou Institute of Physics, China Academy of Space Technology,
resources when the moving parts are expected to have Lanzhou, Gansu, People’s Republic of China
a long lifetime. However, conducting the accelerated
Corresponding author:
life test can estimate reliability faster and more cost-
Jun-yang Li, Post-doctoral Research Station of Control Science and
effectively. For liquid lubricants, in the 1970s, Hughs Engineering, Chongqing University, No.174 Shazhengjie, Shapingba,
Aircraft of America proposed that raising speed and Chongqing 400044, People’s Republic of China.
reducing viscosity to keep the product of input speed Email: lijunyang1982@sina.com

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2 Proc IMechE Part J: J Engineering Tribology 0(0)

Figure 1. Principle of operation.

Table 1. HD materials.

Part Material

Circular spline 38CrMnSi

Flexspline 40 Cr
Wave generator GCr15
bearing (race and ball)
Wave generator Polytetrafluoroethylene
bearing separator

Table 2. Lubrication.

Part Material
Figure 2. Lubrication of the harmonic driver.
CS/FS Diamond-like carbon films (WC-DLC:H)
and PFPE-based oil (Braycote815Z)
raise many times. In this case, the heat stress may FS/WG PFPE-based oil (Braycote815Z)
arouse new failure mode which is the consequence WG PFPE-based oil (Braycote815Z)
of temperature increase, and that may not be accep- on race and polytetrafluoroethylene
table in accelerated life test. retainer
In the present study, life test and mixed lubrication
analysis of HD with new lubricating materials for
space applications are conducted to simulate the Table 3. The performance parameters of WC-DLC:H.
degeneration of transmission performance and the
Thickness Hardness E-modulus Critical
failure mechanism. Based on the obtained results, a
(lm) (GPa) (GPa) load (mN)
new method of accelerated life test is proposed and
verified by experiments. 1.3 9.2 89.0 92

Specifications and lubrication of HD

For space applications, three critical areas of lubri-
The principle of HD is unique, capable of transmit- cation can be identified for the HD (Figure 2):
ting high torque through an elastically deformable
component. It consists of three concentric elements – The engaged teeth (CS/FS).
(Figure 1)17: – The contact surface between the wave generator
outer ring and the inner surface of the flexspline
1. The circular spline(CS) is a rigid ring with internal (FS/WG).
teeth, engaging the teeth of the flexspline across – The Wave generator bearing.
the major axis of the wave generator.
2. The flexspline(FS) is a non-rigid, thin cylindrical
steel cup with external teeth on a slightly smaller In this research, type HD60 with a reduction ratio of
pitch diameter than the circular spline. 100 is provided for the test, the material characteris-
3. The wave generator (WG) is a thin-raced ball tics and lubrication of the HD are listed in Tables 1
bearing fitted onto an elliptical plug serving as a and 2, and the performance parameters of
high efficiency torque converter. WC-DLC:H is listed in Table 3.

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Li et al. 3

Figure 3. Diagram of performance testing equipment.

Life testing in vacuum

The principle of performance test is showed in (a) 80

Transmission efficiency (%)

Figure 3. For transmission accuracy test, the input 75
angular speed (!i ) and output angular speed (!0 )
are measured by angle encoders. The transmission 70
accuracy is defined as the difference between !0 and 65
!i divided by the reduction ratio. For efficiency test,
the angle encoder at input side is replaced by torque 60
sensor. The working conditions and environment of 55
test are listed as follow:
50
Input speed: 3 r/min 45
Output load torque: 5 Nm 0 2 4 6 8 10
Vacuum quality: <1  103 Pa 6
Total output revolution (10 r)
Room temperature: 20–25  C. (b) 5
Transmission accuracy (' )

Figure 4 shows that the transmission efficiency of 4

the HD has decreased significantly, but
transmission accuracy remains quite constant after 3
the life test of 107 revolutions. Then, the decrease
in efficiency, rather than accuracy, can be used to 2
evaluate the life of HD in the accelerated life test.
After dismantling, no remarkable wear is found on 1
either of the two HD gear pairs (Figure 5),
which means that the tribological performance of 0
the HD has been improved under the condition of 0 2 4 6 8 10
6
solid film-oil composite lubrication. Remarkable Total output revolution (10 r)
wear is observed on the outer ring of the wave gen-
erator (Figure 6) and the inner surface of the flexs- Figure 4. Transmission performance versus life time. (a)
pline (Figure 7). The wear depth of flexspline is Transmission efficiency. (b) Transmission accuracy.
between 10.2 lm and 13.5 lm, which is measured by
a 3D surface profilometer. The results show that sur-
face wear at the interface of FS/WG is the main
reason of efficiency decrease. IR spectrum is used to
analyze the oil and the result shows that the IR
Mixed lubrication analysis
absorption peaks are mainly at 1094 cm1,
1153 cm1, and 1206 cm1 (Figure 8), which are the To clarify the lubricant film behavior at the interface
products of functional groups (–CF3, –CF2–, –CF2– of FS/WG, where the lubrication status changed
O–CF2–) of the oil (Braycote815Z). It means that the depending mainly on the working conditions
major ingredients of oil are not changed, which indi- and environment, a mixed lubrication analysis is
cates no chemical wear occurs, and therefore, lubrica- conducted.
tion severity is mainly the reason of the wear. Besides,
significant adhesive region can be seen from Figures 6
Mixed lubrication model
and 7, which also illustrates that the main wear is
adhesive wear, because the hardness of wave genera- Before the elastic deformation, the diameter of the
tor out ring (43 HRC) is close to that of flexspline (40 bearing outer race of WG is a little smaller than
HRC). that of the inner surface of FS (usually assembled as

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4 Proc IMechE Part J: J Engineering Tribology 0(0)

Figure 5. Conditions of rubbing surface after life test: (a) Circular spline teeth and (b) Flexspline teeth.

Figure 6. Wear condition and surface profile of the bearing outer ring after life test.

Figure 7. Wear condition and surface profile of the inner surface of the flexspline after life test.

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Li et al. 5

a clearance fit: H7/h7). When the elliptical plug is The relationship between normal force at the inter-
assembled and becomes rotating, the inner surface face of FS/WG and output load torque expressed as
of FS and the bearing outer ring of WG always follow
contact at opposite ends of the major axis of the T
ellipse, which can be described by an equivalent roll- FT ¼ 0:378
d1 cos 1
ing contact between a cylinder and a plane (Figure 9).
The effective radius of curvature at the contact can be where T is output load torque, d1 pitch diameter of
expressed as circular spline, and 1 is the pressure angle of gear.

R ¼ R1 R2 =ðR1  R2 Þ
Model formulation
where R1 is the radius of curvature of flexspline at end In the solution domain of a mixed lubrication analy-
of the major axis, R2 is that of the outer ring of wave sis, the pressure is governed by the Reynolds equation
generator at end of the major axis. expressed as follow
The contact rough surfaces have a composite RMS    
roughness, which can be expressed as @  3 @p @  3 @p @ ðhÞ @ ðhÞ

h þ 
h ¼U þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi @x 12 @x @y 12 @y @x @t
¼ R2rms1 þ R2rms2 ð1Þ

where the x-coordinate coincides with the motion

where Rrms1 is RMS roughness of inner surface of direction. The instantaneous lubricant film thickness
flexspline, and Rrms2 is that of the outer ring surface (or the gap between the two rough surfaces) h is cal-
of wave generator bearing. culated by the geometric equation given as follow18

x2
h ¼ h0 þ þ 1 ðx, y, tÞ þ 2 ðx, y, tÞ þ Vðx, y, tÞ:
2Rx
1.1 ð2Þ
1.0
0.9
2855
2924 2358
1689 Here, Rx is the local radius of curvature at the
contact (representing macro-contact geometry), 1
Transmittance (%)

0.8 817
Before testing
0.7 690 and 2 denote the micro-roughness amplitudes of
After testing 627
0.6 1094 two rough surfaces, respectively, and V is the surface
1153
0.5 1206 deformation, which is computed through the follow-
0.4 ing integral.
0.3
Z Z
0.2 2 pð, &Þ
0.1 Vðx, y, tÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffidd&
E0
0.0
ðx  Þ2 þ ð y  &Þ2
4000 3500 3000 2500 2000 1500 1000 500
-1 ð3Þ
Wave number (cm )

Figure 8. Infrared spectrogram of the oil.

Figure 9. Equivalent motion model.

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6 Proc IMechE Part J: J Engineering Tribology 0(0)

An effective viscosity  has been introduced in Table 4. The performance parameters of Braycote815Z oil.
equation (1) to take into account the non-
Parameter Value
Newtonian lubricant properties19
  Viscosity index 350
1 1 0 1 Kinematic viscosity (20  C) 250 (cSt)
¼ sinh ð4Þ
  1 0 Kinematic viscosity (100  C) 100 (cSt)
Pressure–viscosity coefficient 22 (m2/N)
where 0 is a reference shear stress, 1 denotes the Density 1.87 (g/cm3)
shear stress acting on the lower surface, and  is the
Thermal conductivity 0.084 (J/m.s.k)
limiting viscosity at low shear rate, which is assumed
to be a function of pressure and temperature
 
1 1
 ¼ 0 EXP p þ ð
þ pÞð  Þ : ð5Þ
T T0

The density of lubricant is also dependent on pres-

sure, given by
 
0:6  109 p
 ¼ 0 1 þ ð6Þ
1 þ 1:7  109 p

The load balance equation can be expressed as

follow

FT ¼ FC þ FH ð7Þ
Figure 10. 3D surface profiles.
where FC is the load supported by the asperity contact
and FH is the one by the hydrodynamic lubricant film.
When the average elastohydrodynamic film thick-
ness (ha ) is not greater than the composite RMS of 257  257 nodes equally spaced. The corresponding
roughness (), surface asperity contacts would usually dimensionless mesh spacing is sufficiently small.
take place,20 and a certain portion of load is sup- Readers may refer to the following paper for detailed
ported by the asperity contacts. The contact load descriptions. The solution
 convergence
 criterion is set
ratio (Wc ) is the load supported by asperity contacts to be: "p ¼ Pi,j new  Pi,j old  Pi,j new 5 0:00001
divided by the total load. l (l ¼ ha =) is defined as  0:0001
 for the
 solution of pressure and
ratio of average film thickness to the composite RMS "w ¼ wnew  wold  wnew 5 0:00001  0:0001 for the
roughness, which can well describe the global lubrica- load.
tion effectiveness in rough surface contacts. The Figure 11 shows the results of mixed-lubrication
recently developed unified mixed EHL model and analysis, and the areas in light yellow in film thickness
numerical approach have been proven to be capable contours indicate the locations of asperity contact.
of simulating the entire transition from full-film and The contact load ratio Wc is 44.8%, which means
mixed lubrication all the way down to boundary the WG–FS interface is under the condition of
lubrication and dry contact with the contact load mixed lubrication, and further verifies the conclusion
ratio Wc varying from 0 continuously to 1.0.21,22 in Part 2 that the lubrication severity of WG–FS inter-
The detailed numerical procedure will not be repeated face leading to adhesive wear is the crucial factor
in this paper. governing the life of the HD.
In order to guide the selecting of accelerated stress,
mixed-lubrication analysis under different conditions
Results and discussions is conducted. Variation of asperity contact load ratio
The property parameters of the oil (Braycote815Z) due to different input speeds, temperatures and loads
are listed in Table 4. The effective radius of curvature are shown in Figures 12 and 13. As expected, asperity
at the contact is R ¼ 2105 mm. Two machined sur- contact load ratio increases when increasing the tem-
faces with 3D roughness topography are rubbing perature and output load and decreasing the input
against each other, and the Rrms value of each surface speed, as shown in Figure 12. The increased tempera-
is 0.35 lm, which results in a composite RMS rough- ture can cause the oil viscosity and film thickness to
ness of  ¼ 0:5 m. One of the 3D surface profiles is reduce, which will increase the asperity contact area
shown in Figure 10. The solution domain in the pre- and load, and therefore increase the wear and realize
sent study is defined as 3 4 X 4 and 3 4 Y 43. the accelerated life effect. But the oil of Braycote 815Z
The computational grid covering the domain consists has a high viscosity index, and the temperature has a

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Li et al. 7

relatively weak effect on the contact load ratio. When

the speed reaches 80 r/min, it becomes full-film elas-
tohydrodynamic lubrication. Under the condition of
mixed lubrication, the input speed has a significant
influence on the asperity contact load ratio, while
the influence of the output load torque depends on
the current level of asperity contact (see Figure 12).
The increase in asperity contact load ratio can be
resulted from the increased torque input mainly at
low speeds, and this effect would be weakened when
raising the input speed. The results of mixed-lubrica-
tion analysis show that increasing input speed and
load torque at the same time to remain or increase
the level of asperity contact load ratio is feasible for
conducting the accelerated life test.

Principle of accelerated life test

Figure 11. Distributions of oil film thickness and pressure. (a) The accelerated life test of HD should be conducted in
Film thickness and (b) pressure. Speed x ¼ 3 rpm, Torque the way described as follow:
T ¼ 5 N.m, temperature: 20  C.
1. Keeping the failure mechanism unaltered
The failure of the HD is mainly caused by the
adhesive wear of wave generator–flexspline inter-
Asperity contact load ratio Wc (%)

55 o
face under the condition of mixed lubrication in
0C
50 o
the present work. The lubricating condition of
20 C
45 o
wave generator–flexspline interface cannot be
40 C
40 o
changed during the accelerated life test, because
60 C
35 the failure in elastohydrodynamic lubrication is
30 mainly due to fatigue wear instead of adhesive
25 wear.
20 2. Selecting accelerated stress
Mixed EHL
15 In order to shorten testing time, a higher input
10 speed is required. At the same time, the output
5 load must be increased in order to keep the lubri-
0 cation condition unchanged and retain the same
0 20 40 60 80 100 120
level of asperity contact load ratio.
Input speed V (r/min)
3. Establishing accelerated model
The adhesive wear of wave generator–flexspline
Figure 12. Asperity contact load ratio at different input interface is a crucial factor governing the life of
speeds and temperatures for a load torque of 5 N.m. HD, and the degradation of transmission effi-
ciency is directly related to the wear volume. In
the present work, the wear rate model is selected
as the reference accelerated model.
Asperity contact load ratio Wc (%)

3rpm
70 10rpm To describe adhesive wear of lubricated contact,
60 15rpm
20rpm Rowe23 introduced the concept of the fractional film
50 25rpm defect into the Archard theory, which was originally
30rpm
40 50rpm for describing adhesive wear due to dry sliding con-
100rpm tact. The work of Rowe correlates wear with the effec-
30
tiveness of the lubricant as follow
20
10 dV FT
¼ km ð1 þ f2 Þ1=2
ð8Þ
0 dL s
0 5 10 15 20 25 30 35
Load torque T (N.m) where FT is the total normal load, s is the flow pres-
sure under static load, and f is coefficient of friction.
Figure 13. Asperity contact load ratio at different input The parameter km is characteristic of the tendency of
speeds and loads for temperature of 20  C. the couple to wear by adhesion, the parameter
is

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8 Proc IMechE Part J: J Engineering Tribology 0(0)

characteristic of the lubricant for its ability to reduce shear stress in the hydrodynamic film is much lower
the asperity contact area. than the shear stress at asperity contact, the friction
Unfortunately, Rowe’s model does not take into coefficient of asperity contact, fC , is almost equal to
account the fact that the total load is supported by the friction coefficient f. Based on this, equation (8)
the lubricating film and the contacting asperities. In can be modified as follow
mixed lubrication, actually, the total normal load FT
is shared by the hydrodynamic lifting force FH and the dV FC
¼ km ð1 þ f2C Þ1=2
: ð13Þ
asperity interacting force FC . Similarly, the total fric- dL s
tion force is the sum of two components.
According to the criterion of plastic flow for a two-
N Z Z
X dimensional body under combined normal and tan-
Ff ¼ fFT ¼ Ff,C þ Ff,H ¼ ci dAci gential stresses, yielding of the contact asperity will
i¼1
Aci
ð9Þ follow the expression24
Z Z
þ H dAH
m2 þ  2 ¼ s2 ð14Þ
AH
where m is the flow pressure under combined stresses
where Ff,C is the asperity interacting friction force, and is the shear strength.
Ff,H is the hydrodynamic friction force, ci is the From friction theory
shear stress at asperity contact, H is the shear stress
in the hydrodynamic film, and AH is the hydrodyna- Ff,C fC FC
¼ ¼ ¼ fC m ð15Þ
mically lubricated area. Ac Ac
The friction coefficient of a single asperity fci can be
written as where Ff,c is almost equal to the total friction force.
Thus
ci
fci ¼ : ð10Þ
pci s
m ¼ 1=2
: ð16Þ
ð1 þ f2C Þ
Since the coefficient of friction is approximately
constant for all asperity contacts, Ff,C can be Substituting equation (16) into equation (13), one
written as can have

N Z Z
X N Z Z
X dV FC F T Wc
Ff,C ¼ fci pcidAci ¼ fC pcidAci ¼ fC FC ¼ km
¼ km
: ð17Þ
dL m m
i¼1 i¼1
Aci Aci

ð11Þ Since the parameters km ,

, and m are related to
lubricant and flexspline material, FT  Wc can be taken
where fC can easily be determined experimentally. as the characteristic quantity of the life of HD instead
Hence, the coefficient of friction can be obtained of wear rate.
from

Ff,C þ Ff,H fC FC þ Ff,H

Accelerated life test
f¼ ¼ : ð12Þ
FT FT A block diagram of the accelerated life test is showed
in Figure 14. The transmission efficiency is tested, and
In mixed lubrication, although the total normal the total number of output revolutions for the effi-
load is shared by the hydrodynamic film and the ciency dropping down to 50% is defined as the life
asperity contact, the load taken by asperity contact of the HD. The 3D surface profilometer was used to
is the decisive factor of the adhesive wear. As the measure the weight volume of flexspline. The testing

Figure 14. Block diagram of accelerated life test.

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Li et al. 9

Table 5. Environment and conditions of ALT. Table 6. Wear conditions of flexspline inner surface.

Input speed Load torque Vacuum quality Temperature Conditions Wear

of volume Wear
3 r/min 5 Nm ALT (mm3) morphology
15 r/min 5 Nm
15 r/min 10 Nm 100 r/min, 5 N.m 13.5
15 r/min 20 Nm <1  103 Pa 20–25  C
15 r/min 30 Nm
100 r/min 5 Nm
100 r/min 10 Nm

100 r/min, 10 N.m 27.1

80
Transmission efficiency (%)

15 r/min, 5 N.m
75 15 r/min, 10 N.m
15 r/min, 20 N.m
70 15 r/min, 30 N.m
100 r/min, 5 N.m
65 3 r/min, 5 N.m
100 r/min, 10 N.m
60
55
15 r/min, 10 N.m 62.6
50 The end of life

0 2 4 6 8 10 12 14
6
Total output revolution (10 r)

Figure 15. Degradation of the efficiency under different

conditions.

3 r/min, 5 N.m 68. 6

conditions are given in Table 5. The input speed of 3 r/
min and load torque of 5 N.m are used for the normal
life tests, which are typical in engineering reality. In
order to verify the results of the numerical analysis,
the input speed for accelerated life test is raised to
15 r/min and 100 r/min, and the output load torque
is increased accordingly. There are five samples
under each condition, and the average value of three
15 r/min, 20N.m 78.2
samples which are more close to each other among the
five tested results is selected as the life of the HD.
Figure 15 shows the degradation of the efficiency
under different conditions. Under the condition of
x ¼ 100 r/min and T ¼ 5 N.m, when the total number
of output revolutions reaches 11.66  106, the
efficiency is reduced just from 76% to 71%. It seems
that the life of the harmonic drive is increased 15 r/min, 30 N.m 91.8
significantly compared to the case with input speed
of 3 r/min as demonstrated by the corresponding
curves. The influence of the load torque on the rate
of efficiency degradation is negligible when the load
torque increases from 5 Nm to 10 Nm. After disman-
tling, slight wear is found on the inner surface of the
flexspline. The experimental results indicate that the
lubricating condition at the interface between wave
generator and flexspline is changed to nearly full-
film EHL condition when increasing the input speed input speed should be in a certain range or the lubri-
to 100 r/min. Under these conditions, heavier load will cating condition will be changed and the life test of
cause an increase in film pressure without changing harmonic driver cannot be accelerated in this way.
the lubricating condition. The experimental results Under the condition of x ¼ 15 r/min and
and the theoretical analysis reveal that increased T ¼ 5 N.m, the efficiency degradation rate is lower

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10 Proc IMechE Part J: J Engineering Tribology 0(0)

Figure 16 shows the lives of the HD with respect to

12
15 r/min, 5 N.m the loads taken by asperity contact. The life of the HD
Total output revolution (10 r)
6

10 is approximately exponentially decreased with the

15 r/min, 10 N.m
load taken by asperity contact. The HD under the
8 condition of x ¼ 3 r/min and T ¼ 5 N.m and that
under x ¼ 15 r/min and T ¼ 10 N.m have almost the
6 3 r/min, 5 N.m same life and the load taken by asperity contact.
15 r/min, 20 N.m
4 However, the testing time is shortened significantly
15 r/min, 30 N.m
as the input speed is increased. Figure 17 shows the
2 wear rate of flexspline inner surface is nearly propor-
tional to the loads taken by asperity contact, which is
0 agree with the wear calculation formula proposed
0 10 20 30 40 50 60 70 80
Load taken by asperity contact (N) above. The experimental results show the load taken
by asperity contact (FT  Wc ) can be selected as the
characteristic quantity of the HD life during the accel-
Figure 16. Life of HD V.S. the load taken by asperity contact.
erating life tests.

Conclusions
20 15 r/min, 30 N.m When the HD operates with a low speed in vacuum,
the lubricating conditions of all contacting surfaces
Wear rate (10 mm /r)

15 15 r/min, 20 N.m are severe. Solid film-oil composite lubrication sys-

3

tems on gear pairs can effectively prevent wear. The

adhesive wear caused by asperity contacting on the
-6

15 r/min, 10 N.m
10 wave generator–flexspline interface is a crucial factor
governing the life of the HD.
5
15 r/min, 5 N.m 1. The method of accelerated life test of the HD is to
raise both input speed and load without changing
0 the lubrication condition. According to the adhe-
0 10 20 30 40 50 60 70 80 sive wear model, the load taken by asperity con-
Load taken by asperity contact (N) tact (FT  Wc ) is defined as the accelerated factor
and further used to select proper accelerated
Figure 17. Wear rate of flexspline inner surface V.S. the load stress.
taken by asperity contact. 2. Accelerated life tests under different working con-
ditions have been conducted, and the results agree
well with the theoretical mixed lubrication
analysis.
than that under the condition of x ¼ 3 r/min and 3. The scheme of accelerated life test proposed in the
T ¼ 5 N.m. The rate is increased significantly when present paper can be applied to liquid lubricating
the load torque increases from 5 N.m to 30 N.m. interface if adhesive wear is the basic failure mode.
The results are in good agreement with those obtained
from the numerical analysis, i.e. the adhesive wear
Funding
rate is proportional to the load taken by the asperities.
Increasing load torque would increase asperity This work is supported by the Natural Science Foundation
of China (grant nos. 51435001 and 51375506), China
contact load ratio if the wave generator–flexspline
Postdoctoral Science Foundation funded project (grant
interface still operates in mixed lubrication at the
no. 2013M542258) and the Chongqing Postdoctoral
input speed of 15 r/min. Good accelerating effect, Science Foundation funded project (grant no. Xm2014082).
therefore, can be obtained by raising the input speed
and load without changing the lubrication condition. Conflict of interest
After dismantling, no remarkable wear is observed on
None declared.
either gear pairs or wave generator bearing of the HD
under different working conditions, but significant References
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12 Proc IMechE Part J: J Engineering Tribology 0(0)

y, Y coordinate perpendicular to the rolling shear strength (MPa)

direction, Y ¼ y=a (mm) 0 reference shear stress (MPa)
1 shear stress acting on the lower surface
1 pressure angle of gear ( )
(MPa)

lubricant in its ability to reduce the
ci shear stress at asperity contact (MPa)
asperities contact area
H shear stress of hydrodynamic compo-
1 , 2 micro-roughness amplitudes of rough
nent (MPa)
surfaces (lm)
sliding speed of flexspline relative to
 limiting viscosity at low shear rate
wave generator outer ring (rpm)
 effective viscosity
!1 input rotational speed (rpm)
l film thickness ratio (ha =)
!1 rotational speed of flexspline relative to
 composite RMS roughness of two sur-
wave generator outer ring (rpm)
faces (lm)
! rotational speed of flexspline relative to
m flow pressure under combined stresses
wave generator outer ring (rpm)
(MPa)
s flow pressure under static load (MPa)

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