Bulletin of the JSME
Journal of Advanced Mechanical Design, Systems, and ManufacturingVol.11, No.5, 2017
Paper No.17-00392© 2017 The Japan Society of Mechanical Engineers[DOI: 10.1299/jamdsm.2017jamdsm0052]
1. Introduction
The seeker is one of the most important parts of a missile. It’s used to track the target and to provide the inertial
stabilization to the detector’s pointing vector (Yu and Zhao, 2008). A seeker is generally composed of a detector, a
controller for acquisition and tracking and a two-axis orthogonal gimbal system. In the task of long-range precision strike
or target positioning, the stability of seeker’s line-of-sight (LOS) is crucial to hit accuracy and target location accuracy
of a missile. As a highly integrated and high-precision optical-mechatronic system, there might be many factors in a
seeker which might degrade the stabilization precision of LOS, such as disturbance torques, gyro noise and drift,
structural resonance frequency, control bandwidth, etc (Hilkert, 2008). Amongst these factors, nonlinear frictional
moment, which mainly comes from bearings for directly driven gimbal, is normally the major stabilization error source
(Stockum, et al., 1988). The nonlinear behavior caused by friction has an adverse influence on the ultra-low velocity and
high precision position control of servo mechanisms (Yang, et al., 2014), which could lead to stick-slip and limit cycles
during movement of inner and outer gimbals (Olsson and Astrom, 2001, Cheock, et al., 1988). Moreover, friction
degrades stabilization by acting as the transference mechanism for base motion disturbance (Lagunowich, et al., 2007).
With the development of lightweight, miniaturized seekers, the effect of nonlinear friction is becoming more and more
pronounced.
Temperature is a critical factor that could change the value of friction torque and its fluctuation property. Ambient
temperature of a missile would be very different because of the broad geographical deployment and different seasons,
and the environment temperature might change in a wide range of -40℃~+60℃ or much more severe. Especially, in the
course of high-speed flight and high-g maneuver, a missile would go through airspace with different temperature and
generate immense heat because of friction with atmospheric air, so the temperature of gimbal fluctuates sharply during
missile flight. As a consequence, friction torque of gimbal would rise or fall along with the change of ambient temperature.
1
Frictional torque analyzing and testing of Gimbaled-mirror
seeker under changing environment temperature
Naihui YU* and Jianzhong SHANG* *College of Mechatronics Engineering and Automation, National University of Defense Technology
137 Deya Road, Kaifu District, Changsha City, Hunan, China
E-mail: [email protected]
Received: 6 August 2017; Revised: 25 September 2017; Accepted: 4 October 2017
Abstract Nonlinear friction is the main stabilization error source of missile’s seeker. In this paper, the frictional torque of a gimbaled-mirror seeker under changing environment temperature is analyzed. The torque is consisted of rolling friction due to elastic hysteresis, sliding friction due to pivoting and friction torque due to lubricant viscosity. Based on the theory of thermo-elasticity, axial and radial coupling thermal deformation between precision bearing and its installation structure are both included in the analysis model. Coupling deformation dramatically changes the actual contact angle and axial force of bearing, which consequently change the value of frictional torque. Temperature effect on viscosity of lubricant is also studied and improved Walther Equation is used to fit the viscosity- temperature relationship of lubricant. A measure system based on electrical measure method was established and temperature test chamber was used to simulate the temperature changing. Experiment results have shown that this model is accurate for friction torque calculation of gimbaled seeker under changing environment temperature.
Keywords : Frictional torque, Bearing, Environment temperature, Gimbaled-mirror seeker, Coupling thermal deformation, Viscosity of lubricant
2© 2017 The Japan Society of Mechanical Engineers[DOI: 10.1299/jamdsm.2017jamdsm0052]
Yu and Shang, Journal of Advanced Mechanical Design, Systems, and Manufacturing, Vol.11, No.5 (2017)
These would be great challenges for all disturbance compensation methods using in the seeker servo control system and
directly influence the performance of the seeker. Scholars all over the world have conducted a lot of theoretical analysis
and experimental researches for bearing friction torque separately over recent decades. And there are lots of papers about
mathematical models or fitting models of bearing friction torque and its impact factors. Considering the influence of
temperature on bearing friction torque, most researches focus on the viscosity of lubricants or greases (Kobzova, et al.,
1974, Ostensen, et al., 1995, Wikstrom, et al., 1996a,1996b, Wikstrom, et al., 1997,Gerstenberger and Poll,2001).
Thrust bearings lubricated with several different greases were tested on a modified Four-Ball Machine and friction torque
were measured,the results show that the friction torque depends on the viscosity of the grease base oil, on its nature, on
the coefficient of friction in full film conditions, and on the interaction between grease thickener and base oil (Cousseau,
et al., 2011, Cousseau, et al., 2012, Fernandes, et al., 2013). All above researches were conducted for bearings separately
and didn’t take into account the deformation of mount constructions, such as the shaft and bearing house, at different
temperatures.
Considering this problem at a system-level, Du et al. (2014) conducted some analysis and experiments on the
telescope’s drive system at low temperature, because low-temperature environment on Antarctic would cause significant
performance deterioration of telescope. According to the paper, increase of the viscosity of grease and decrease of the
clearance of bearing and gear are two major factors of the increase in load torque of drive system with temperature
dropping. And the analysis also shows that motor’s electromagnetism is little affected by temperature if the suitable
material is selected. However, there are very few researches about the influence of temperature variation on the friction
torque at the system-level, especially for seekers. The influence of environment temperature on the friction torque of
gimbal has two aspects. 1. Different temperature would change the viscosity of bearing lubricant, and then change the
bearing friction torque. 2. Different materials and constructions of inner and outer gimbal frames lead to different thermal
deformations. Besides, coupling thermal deformations would happen between inner ring of bearing and shaft and
similarly between outer ring and bearing housing. All these would change the initial contact angle and preload of bearing,
and then change the bearing friction torque. So the friction torque of gimbal is a system-level problem and not a problem
only related to bearings.
How to reduce the effect of nonlinear friction torque on stabilization accuracy has always been a hot spot in research
areas of high precision position servomechanism, such as seekers of missile, photoelectric stabilized platforms, high-
precision drive system of telescopes and so on (Lin and Hsiao, 2001, Levin and Ioannou, 2009). However, most
researches focused on the field of servo control and friction compensation methods. Small friction torque and its
uniformity in different angular position are the foundation for all compensation methods, especially in an environment
of dynamic changing temperature. Using mechanical method to reduce the disturbance torque is still the first choice. So
we must study the impacting mechanism of design parameters, assembly parameters and environment temperature on the
friction torque of servomechanism. At present, most research studied the effect of some factors alone or only analyzed
the bearing itself. But the change of the friction torque of seekers was a joint influence by multiple factors. In this paper,
we will analyze the coupling effect of assembly interference, thermal deformation, lubrication, tightening moment on the
friction torque of gimbal at different environment temperatures. A thermal cycling chamber was used to simulate the
vary temperature and a measurement system of friction torque was built for a gimbal seeker.
2. Nomenclature
a = half length of the ball-race contact ellipse p = interfacial stress between bearing and its house
b = half width of the ball-race contact ellipse pe = the bearing axial clearance after the change of
temperature
B =total curvature Pd = radial clearance of bearing
b0 = bearing width Qn = ball-raceway normal load
𝑑𝐼 = groove diameter of inner raceway R = radius of the curved Hertzian contact
𝑑𝑂 = groove diameter of outer raceway s0 =dimensionless parameter related to bearing type and
lubrication method (used for calculating ML)
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2© 2017 The Japan Society of Mechanical Engineers[DOI: 10.1299/jamdsm.2017jamdsm0052]
Yu and Shang, Journal of Advanced Mechanical Design, Systems, and Manufacturing, Vol.11, No.5 (2017)
dm
D
= mean bearing diameter
= initial diameter of cylinder
v0 =Kinematic viscosity of lubricant at operating
temperature. In the case of grease, the decisive factor is
the viscosity of the base oil at operating temperature
D1 = the diameter of cylinder when temperature
changes
α0 = initial rolling element-race contact angle
Dm = outside diameter of bearing z = the number of rolling element of bearing
Dw = ball diameter α = actual rolling element-race contact angle
E = elasticity modulus μ = coefficient of sliding friction
fi =curvature coefficient of inner raceway 𝛾 = Poisson ratio
fo =curvature coefficient of outer raceway φ = linear expansion coefficient
l1 = the length of shaft =the axial deformation of bearing under the force Fa
l2 = the distance of bearing house P = the thickness of pre-tightening gasket
LE = elliptic integral of second kind Rd = radius after thermal deformation
LB = initial installation distance of the bearings ∆𝑇 = variation of temperature
n =operating speed ∆ = interference amount
3. Friction torque composition of indirect pointing gimbal
Precision pointing systems can be categorized as direct or indirect pointing systems. Direct pointing systems directly
point the line of sight of a sensor or antenna in the desired direction. Indirect pointing systems orient a mirror in front of
a sensor so the reflected line of sight points in the desired direction (Cubalchini, 1995). Load mass of indirect pointing
systems is very light and the mirror could be driven by torque motor directly. This could significantly improve the
resonance frequency and control bandwidth of servo mechanism, which is important for disturbance torque suppression.
But the friction-torque to inertia ratio (TF/J) is a key design parameter in the stabilization system (Stockum, et al., 1988)
and smaller load makes the servo mechanism more sensitive to nonlinear friction torque. High value of TF/J would cause
great challenges on the stability of LOS. So reducing the friction torque by mechanical method should be the first strategy
employed for a gimbal-mirror system, while the moment of inertia of gimbal axis is very small. This involves an overall
system-level design consideration, including structure of frames, quality of bearings, pre-tighten moment in the process
of assembly, etc.
Outer frame
Inner frame
Torque motor
Mirror
The base
High precision
angle sensor
High precision
angle sensor
torque motor
Bearings are the main way of force transmission for both gimbal axis, and their friction torque is the main
disturbance torque. In order to improve the performance of a seeker, friction torque of bearings must be small as far as
possible within a permissible budget. Temperature variation would change viscosity of grease and cause uneven thermal
Fig. 1 Typical structure of indirect pointing gimbal. A gimbal is composed of azimuth axis, elevation axis, a mirror and the
base. Each axis contains a torque motor for drive and a high precision resolver for angular position feedback. Each
rotating shaft is supported by two angular contact bearings and the bearings are exerted a pre-tightening force
through preload nuts. Gyros are installed on the base for strapdown stabilization.
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2© 2017 The Japan Society of Mechanical Engineers[DOI: 10.1299/jamdsm.2017jamdsm0052]
Yu and Shang, Journal of Advanced Mechanical Design, Systems, and Manufacturing, Vol.11, No.5 (2017)
deformation of frames and bearings if different materials were used, which would change the initial contact angle and
preload of bearings. So models of friction torque including all above factors must be established.
In general, bearing friction torque is derived from the elastic hysteresis of contact materials, lubricant viscosity, ball
spin sliding friction, friction between rolling elements and cage, etc (Harris, 2001).
3.1 Rolling friction due to elastic hysteresis of contact materials
When a ball rolls on raceways, front and back of contact regions have asymmetric pressure distributions because of
elastic hysteresis of contact materials. Resistance moment generated by front contact region is higher than driving torque
generated by back contact region, so a rolling friction is produced. Based on Hertz contact theory, elastic hysteresis
friction force produced between a roller and inner or outer raceway is shown in Eq. (1) and it was derived by Jiang(Jiang,
et al, 2010).
3
16
rSE n
bF Q
R
(1)
Where 𝛼𝑟 is energy loss percentage because of elastic hysteresis and generally, 𝛼𝑟 ≤ 1% for metal.Therefore,
the friction torque of the bearing axis caused by one ball in the inner and outer raceway can be expressed respectively as
Eq. (2).
3( )
2 16 2 2
3( )
2 16 2 2
r I n m w IIEI SEI
I
O r O n m w OEO SEO
O
b Q d D cosdM F
R
d b Q d D cosM F
R
(2)
Where I is the contact angle of the ball with the inner raceway and O is the contact angle with the outer raceway.
They are usually similar and defined as a common angle .
Bearing friction torque caused by hysteresis is the sum of friction torques of all balls and is shown in Eq. (3).
, ,
1 1
z z
R EI j EO j
j j
M M M
(3)
As the load of gimbal-mirror system is just a mirror, the radial load can be neglected. Assuming that the axial pre-
tightening force is Fa, rolling element-race normal load Qn can be shown in Eq. (4).
sin
an
FQ
Z (4)
3.2 Sliding friction due to pivoting on contact ellipse
Spin motion of steel balls would happen in the ball bearing’s running process and sliding friction is produced. Based
on Hertz contact theory similarly, pressure distribution in the contact area of ball with the inner or outer raceway can be
expressed as Eq. (5).
12 2 23
, 12
nQ x yP x y
ab a b
(5)
So sliding friction torque on the elliptical contact is shown in Eq. (6).
1 2
2
1 22
11
2 22 21
2 2
,1
3 31
2 8
a b x an
S j n Ea b x a
Q x yM x y dxdy Q aL
ab a b
(6)
For the ball sliding in the inner or outer raceway is controlled by the maximum contact friction force, so the SM is
the maximum of SIM and SOM .And the total sliding friction torque can be calculated in Eq. (7).
, ,1
max( , )Z
S SI j SO jj
M M M
(7)
The friction torque in the axial direction of the bearing caused by the steel balls sliding is shown as Eq. (8).
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2© 2017 The Japan Society of Mechanical Engineers[DOI: 10.1299/jamdsm.2017jamdsm0052]
Yu and Shang, Journal of Advanced Mechanical Design, Systems, and Manufacturing, Vol.11, No.5 (2017)
sinSSM M (8)
3.3 Friction torque due to lubricant viscosity
For bearings at low speeds, Palmgren (1959) has established experimental equations for friction torque caused by
lubricant viscous, which is shown in Eq. (9).
7 2/3 3
0 0 0
7 3
0 0
10 ( ) 2000
160 10 2000
,
,
m
L
m
s v n d v n
d nsM
v
(9)
3.4 The total friction torque and ball load
The total friction torque of bearing is the sum of all friction torque components and can be show in Eq. (9).
R S LM M M M (10)
As can be seen from Eq. (3), Eq. (6), Eq. (9) and Eq. (10), the factors affecting bearing friction torque include the
geometric parameters of bearing, bearing load or pre-tightening force, lubricant viscous, rotating speed of bearing and so
on. In a varying environment temperature, thermal deformation would happen to bearing and its installation structures
and the lubricant viscosity would also change. In this paper, only elevation axis bearings are considered and the analytical
method of azimuth axis ones are similar.
4. Coupling thermal deformation of bearing and its installation structure
For the seeker under investigation, the main structure material of elevation axis (inner frame) is steel and that of
azimuth axis (outer frame) is casting aluminum alloy for weight reduction. Therefore, coupling thermal deformation
between outer ring of bearing and bearing house would happen as the temperature changing. Similarly, coupling
deformation between inner frame and outer frame of gimbal would happen and extra axial preload acting on bearings
could produce simultaneously. These would change the initial parameters and then change the friction torque of bearings.
Two 26-mm-dia angular contact ball bearings are used in the inner frame of gimbal and its dimension parameters at
normal temperature (or 20 ℃) is shown in Tab. 1.
Table 1 Dimension parameters of angular contact ball bearings.
Dimension parameters The values
Outside diameter[mm] 26
Inside diameter[mm] 17
Width [mm] 5
Ball number 15
Initial contact angle[°] 15
groove diameter of inner raceway[mm] 19.1109
groove diameter of outer raceway[mm] 23.8891
Ball diameter[mm] 2.3810
Curvature coefficient of inner raceway 0.56
Curvature coefficient of outer raceway 0.54
4.1 Radial coupling thermal deformation of bearing
The gimbal shafts rotate in ultra-low angular velocity generally, so the heat generated by friction is limited and
temperature of bearings and shafts would remain basically consistent with the environment temperature. Based on the
theory of thermo-elasticity in constant temperature field, radial thermal deformation of a cylindrical structure under
different constant temperatures can be simply expressed in Eq. (11). Therefore, in a free state, the radial thermal
deformation of bearing house and outer ring of bearing can be shown in Eq. (12).
1 1D D T (11)
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2© 2017 The Japan Society of Mechanical Engineers[DOI: 10.1299/jamdsm.2017jamdsm0052]
Yu and Shang, Journal of Advanced Mechanical Design, Systems, and Manufacturing, Vol.11, No.5 (2017)
1 1
1 1
(1 ), (1 )
(1 ) 2 , (1 )
O
L
O S m m S
I m L AA
d d T D D T
D D T D D T
(12)
Where subscript S and A represent steel and aluminium, respectively.
Based on the compatibility condition of displacement and force, the final deformation of bearing and bearing house
could be expressed in Eq. (13). 2 2
1 1 1
1 2 1 2 2
1 1
2 2
1 1 1
2 2 1 2 2
1 1
2
1 1
3 1 2 2
1 1
2
1 1
0 2 1 2
1 1
2
2
1( ) ( )
2 2
1( ) ( )
2 2
1( )
2 ( )
1( )
2 ( )
A
A
o
I L I
I I
L I
m m O
m m S
S m O
m O
o
S m O
L I
L L
L IA
pD D Dr D D
E D D
pD D dr D D
E D d
pD dr d
E D d
pD Dr D D
E D
d
D
(13)
Solutions for the equations were shown in Eq. (14).
1 1 1 2
2 2 1 1 1 1 2
2
1 1
1 2 2
1 1
2
1 1
2 1 2
1
2
2
1
( ) / ( )
( ) / ( )
2
( )
2
( )
m I
m m I
m O
O
S m O
L I
L L
L I
O
A
p D D k k
D k D k D k k
pD dd
E D d
pD DD D
D D
d
E
(14)
Where
2 2
1 1 1
1 2 2
1 1
( )m m O
S
S m O
D D dk
E D d
and
2 2
1 1 1
2 2 2
1 1
( )I L I
A
A L I
D D Dk
E D D
.
Fig. 2 Schematic diagram of bearing’s coupling deformation at changing temperature. In a free state and at lower
temperature, the outer diameter and inner diameter of bearing house are 𝐷𝐿1 and 𝐷𝐼1 , respectively, while bearing’s
outer diameter and groove diameter of outer raceway are 𝐷𝑚1 and 𝑑𝑜1 , respectively. In a constrained state of
assembly, coupling deformation would happen. Mounting hole of bearing house would become larger under dilation
force and outer ring of bearing would become smaller under compression force.
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2© 2017 The Japan Society of Mechanical Engineers[DOI: 10.1299/jamdsm.2017jamdsm0052]
Yu and Shang, Journal of Advanced Mechanical Design, Systems, and Manufacturing, Vol.11, No.5 (2017)
So the groove diameter of outer raceway1 is
2
1 1 1 1
21 2
1 1 1 2
2
2 ( )
( )( )
m O m I
S m O
O O
D d D D
dd
D kd
E k
, if the temperature changes ∆𝑇
degrees. The groove diameter of inner raceway is 2 1 (1 )II I Sd d d T and there is no coupling deformation between
the bearing and shaft as their materials are almost same. The ball diameter is 2 1 (1 )ww w SD D D T . And radial
clearance of bearing could be calculated by the equation 2 2 2 22d o i wP d d D . The parameters of bearing in Tab. 1 were
used for calculation of coupling thermal deformation. Linear expansion coefficient of steel and aluminium was selected
a typical value as 11.7×10-6 mm/mm/℃ and 21.7×10-6 mm/mm/℃ , respectively. The effect of coupling thermal
deformation on groove diameter and radial clearance was shown in Fig. 3 and Fig. 4, when amount of interference ∆ was
zero and the temperature is 20℃. As can be seen from the figures, coupling thermal deformation greatly decreased the
diameter and clearance of bearing as temperature dropping, which both were important for friction torque calculation. If
coupling deformation was not considered, the clearance would keep almost constant.
Fig. 4 The effect of coupling thermal deformation on radial clearance of bearing. In a free state, the clearance of bearing
is almost invariant because of the consistent deformation of outer and inner diameter of bearing (black). However,
the clearance decreases dramatically with the temperature dropping (red).
Fig. 3 The effect of coupling thermal deformation on groove diameter of bearing outer raceway. In a free state, the diameter
of bearing would linearly shrink with the temperature dropping (black). In the assembly state, the deformation of
bearing house would exert a pressure because of different material and this coupling deformation would cause the
shrink nonlinearity and much more severe than that in a free state (red).
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2© 2017 The Japan Society of Mechanical Engineers[DOI: 10.1299/jamdsm.2017jamdsm0052]
Yu and Shang, Journal of Advanced Mechanical Design, Systems, and Manufacturing, Vol.11, No.5 (2017)
Interference fit is usually used in bearing assembly and different amount of interference would change bearing’s
clearance and contact angle. In order to explore the effect of interference fit on bearing’s contact angle in a varying
temperature environment, different amounts of interference between groove diameter of bearing outer raceway and
bearing house was selected to calculate, which was shown in Fig. 5. The figure shows that larger amount of interference
would decrease the contact angle much more drastically. Especially, the contact angle of bearing would drop from 15°
to zero as the temperature dropping, if ∆=0.008mm or lager amount was selected.
4.2 Axial and radial coupling thermal deformation of gimbal
Supposing the temperature changes ∆T and the axial force is Fa, then the new axial sizes of shaft, bearing and
bearing house show as Eq. (15).
deformation would also happen in a changing temperature. L is effective length of shaft and b0 is actual calculation
width of bearing before temperature changing. The installation distance of bearing house is L+2b0. As the
temperature dropping, these dimensions shrink to l1 and l2, respectively
Fig. 5 The effect of interference fit on contact angle of bearing as temperature changing. Different amounts of interference,
∆=0.004mm (purple), 0.006mm (black), 0.008mm (red), were used to calculate the contact angle of bearing with
temperature changing.
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Fig.6 Schematic diagram of axial and radial coupling thermal deformation of gimbal. Except for radial deformation, axial
2© 2017 The Japan Society of Mechanical Engineers[DOI: 10.1299/jamdsm.2017jamdsm0052]
Yu and Shang, Journal of Advanced Mechanical Design, Systems, and Manufacturing, Vol.11, No.5 (2017)
1 0
2
2 1
0
( 2 )(1 )
(1 )
/ 2
2
a
S
S S
a
A
A
e
P
A
TE A
TE
Fl L b
Fl L
P
l
b
b l
A
b
(15)
eP can be calculated in section 4.1.
Simplification for the Eq. (15) was shown in Eq. (16).
20
1
2( ) e P
a
b P CF
C
(16)
Where0
1
2 21
2
E
L b L
A AC
E
and 2 0( 2 )(1 () 1 )S AC L b T L T .
To solve Fa and δp in Eq. (16), another equation about force and deformation of bearing is needed.
In extremely low speed cases, there is almost no centrifugal force in bearing and contact angles of inner raceway
and outer raceway are equal. Under the action of axial preload Fa, two angular contact bearings of back-to-back or face-
to-face arrangement have same axial displacement and can be expressed in Eq. (17) and Eq. (18).
1.5
0
2
cossin 1
cos
a
w
F
ZD K
(17)
0sin( )
cos
wBD
(18)
Iterative equation of Eq. (17) using Newton-Raphson method is shown in Eq. (19).
1.5
0
2
1.5 0.5
20 002
cossin 1
cos
cos cos1cos 1 1.5 tan 1 cos
cos cos
w
w
a
a
F
ZD K
F
ZD K
(19)
Where Fa and is replaced by Eq. (16) and Eq. (18), and then
0
2
1
2 cos
cos
a waF F BD
C
.
By calculating Eq. (19), the actual contact angle could be solved. Substitute it into Eq. (18), the axial displacement
caused by axial force could be resolved. And substitute to Eq. (16), Fa could be solved.
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2© 2017 The Japan Society of Mechanical Engineers[DOI: 10.1299/jamdsm.2017jamdsm0052]
Yu and Shang, Journal of Advanced Mechanical Design, Systems, and Manufacturing, Vol.11, No.5 (2017)
Figure7 show the axial force changes with the thickness of pre-tightening gasket and the temperature as giving an
interference ∆=0.006mm. Larger amount of pre-tightening gasket thickness causes larger axial force, and different
temperatures cause the axial force firstly decrease and then increase. For different expansion coefficients and different
structures, the bearing would be extruded in radial and released in axial as the temperature decreasing, and it would
turn opposite as the temperature increasing. With the change of temperature, the effect of axial and radial thermal
deformation on the axial force is opposite, due to the different material of outer and inner frame of gimbal. Therefore,
the axial force curve is 'U' shape curve. The radial and axial coupling deformations would lead to varying axial force,
and there would be a relatively low axial force area. In particularly, with the temperature going up, the axial force
would increase sharply as the interference between the bearing house and the outer ring of the bearing reduces to zero
and appears loose contact in radial direction. From -40ºC to 50ºC, the axial force changes in a complex manner. As the
temperature dropping, the shrinkage of aluminum alloy housing is larger than that of steel inner frame. Therefore, the
bearing is released and the axial force should drop. However, the radial thermal deformation of gimbal is also
considered. The complex manner is the coupling effect of axial and radial thermal deformation.
Fig. 7 The effect of pre-tightening amount on axial force of bearing as temperature changing under an interference
∆=0.006mm. Different thickness of pre-tightening gasket, =0mm(black), 0.01mm(red), 0.02mm(purple),
0.03mm (green), were used to calculate the axial force of bearing.
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Fig. 8 The effect of interference fit on axial force of bearing as temperature changing under the pre-tightening amount
mm. Different amounts of interference, ∆=0.004mm (green), 0.006mm (red), 0.008mm (black), were used
to calculate the axial force.
2© 2017 The Japan Society of Mechanical Engineers[DOI: 10.1299/jamdsm.2017jamdsm0052]
Yu and Shang, Journal of Advanced Mechanical Design, Systems, and Manufacturing, Vol.11, No.5 (2017)
Fig.8 shows the interference had little effect on axial force in low temperature as giving a pre-tightening amount
0.03P mm, but larger amount of interference would reduce the axial force in high temperature area. Similarly, when
the bearing house and the outer ring appears loose contact in radial direction, the axial force would increase sharply.
4.3 Friction torque with different effects
With the interference, axial pre-tightening amount and temperature changing, the bearing friction torque would
change dramatically. According to section 3.1 and section 3.2, the bearing friction torque caused by hysteresis and ball
pivoting sliding are calculated. Fig.9 and Fig.10 show the sum of the two kinds of friction torque by giving a factor and
changing other two factors.
Fig.9 shows larger pre-tightening amount causes larger friction torque, and the friction torque in low temperature is
very high. Fig.10 shows in low temperature larger interference causes larger friction torque but it would turn oppositely
in high temperature.
Fig.9 The effect of pre-tightening amount fit on friction torque of bearing as temperature changing under an interference
∆=0.006mm. Different thickness of pre-tightening gasket, =0mm (red),0.01mm (yellow), 0.02mm (black),
0.03mm (blue), were used to calculate the friction torque.
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Fig.10 The effect of interference fit on friction torque of bearing as temperature changing under the pre-tightening amount
mm. Different amounts of interference, ∆=0.004mm (green), 0.006mm (red), 0.008mm (black), were used
to calculate the friction torque.
2© 2017 The Japan Society of Mechanical Engineers[DOI: 10.1299/jamdsm.2017jamdsm0052]
Yu and Shang, Journal of Advanced Mechanical Design, Systems, and Manufacturing, Vol.11, No.5 (2017)
5. Temperature effect on viscosity of lubricant and total friction torque analysis
Wikström (1996a) has proved through a series of experiments that most important to start-up torque was base oil
viscosity for grease -lubricated rolling element bearings at low temperatures. Viscosity is a key performance index of
lubricant and mainly come from the cohesion between the molecules in the fluid. The distance and cohesion between the
liquid molecules would change with variation of temperature, which could cause the change of lubricant viscosity.
Walther Equation was most often used to fit the relationship between viscosity and temperature (Aboul-Seoud and
Moharam, 1999).
1 1 1log log logv c a b T (20)
Where v is kinematic viscosity, T is absolute temperature, a1, b1 and c1 are three constants.
Except for Eq.(20), another equation as Eq.(21) is found high precession to some lubricants in fitting the relationship
between viscosity and temperature.
1 1 1log log log logv c a b T (21)
Choosing the Krytox@143AA lubricant which is produced by DuPont Company as an example, the relationship
between viscosity and temperature fitted by Eq.(20) and Eq.(21) are calculated. According to the experiment data coming
from DuPont Company, the two fitting equations are confirmed accurately as possible in Eq. (22). And the errors are
shown in Table 2.
log log 0.932 9.597 -3.773log
log log 1.08 8.6195- 21.2471log log
v T
v T
(22)
Table 2 Fitting results of The 143AA lubricant viscosity with two equation
t(℃) Experimental viscosity Fitting viscosity
1 Error (%)
Fitting viscosity
2 rror (%)E
-32 12000 11762.07 -1.983 11983.82 -0.135
0 340 348.26 2.431 337.73 -0.668
20 85 87.71 3.188 85.78 0.918
38 35 34.98 -0.042 34.65 -0.993
40 32 32.04 0.123 31.78 -0.688
99 5.3 5.252 -0.897 5.354 1.014
100 5.2 5.140 -1.155 5.240 0.768
204 1.1 1.108 0.760 1.094 -0.540
Table 2 shows that the fitting results are quite close to the experimental results. The errors with Eq. (21) is less than
Eq.(20), basically less than 1%.
In order to calculate the total frictional torque of gimbal, the fitting parameters in Eq. (21) were used for ML
calculation. The interference amount ∆ and the thickness of pre-tightening gasket P were assumed to 0.006mm and
0.03mm, respectively. Two same angular contact ball bearings were used for calculation and their detailed parameters
were shown in Tab. 1. Coupling thermal deformation was calculated by equations in section 4 and the friction torque
were calculated by equations in section 2. As shown in Fig. 11, the total frictional torque remains almost constant in the
interval of 0℃ to +60℃. However, the total frictional torque rise exponentially as the temperature drops from -40℃
and the main factor is the viscosity of lubricant. The maximum value of total frictional torque is 3.17Nmm at -40℃
while ML is 2.30 Nmm and MS+MR is 0.84Nmm. Due to coupling thermal deformation, the value of MS+MR rises from
0.38 Nmm to 0.84Nmm.
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2© 2017 The Japan Society of Mechanical Engineers[DOI: 10.1299/jamdsm.2017jamdsm0052]
Yu and Shang, Journal of Advanced Mechanical Design, Systems, and Manufacturing, Vol.11, No.5 (2017)
6. A test of gimbal friction torque in different environment temperatures
Through the theory analysis in this paper, we have confirmed that the change of gimbal friction torque in different
temperatures is caused by lubricant viscosity and structure thermal deformation. A measurement system was set up to
verify the results of mathematical analysis. And in order to simulate the change of environmental temperature, the seeker
gimbal was placed in a rapid temperature change test chamber (RTCTC). And friction torque of its pitch axis was
measured in different temperatures.
Table 3 Main parameters of rapid temperature change test chamber.
Internal Dimension[mm] 1000×1000×800
Range of Fast Temperature Change[oC] -70 to +150
Indication Resolution[oC] 0.1
Control Accuracy of Temperature[oC] 0.5
Temperature Rising/Falling Velocity
Rising/Falling velocity with linear control :
3~15°C/min, adjustable
Rising/Falling velocity non-linear control :
3~30°C/min, adjustable
There are many measuring methods for servomechanism’s friction torque. However, the alternative method was
limited as the gimbal placed in a closed chamber. In this experiment system, electrical measure method was selected,
because no sensor was needed to add on the gimbal and no direct operation was needed to make on the gimbal. Output
torque Tm of DC moment motor is linearly proportional to its armature current Ia generally. And the scaling factor is
electric torque coefficient KT. So the output torque of the motor could be got by measuring its electric current. In a steady
state, the motor torque balance equation can be expressed as Eq. (23).
0 0m f T fT T T K I I (23)
Where Tf and If are the load torque of motor and its corresponding current respectively, T0 and I0 are the no-load
output torque and its corresponding current of motor respectively, which were provided by the motor manufacturer. If
the other disturbance torques, such as cable interference torque, mass eccentricity and the like, be excluded in the
measuring system, the load torque of motor Tf should be regarded as the friction torque.
In order to obtain the friction torque of gimbal, armature current of moment motor must be measured accurately
and electric torque coefficient should be calibrated carefully. A separate motor calibration device was set up for KT
calibration, including a DC moment motor using in the gimbal, a servo driver for motor drive, a programmable multi-
axes controller (PMAC) for motor control, a micro-range torque sensor for small torque measurement, a Data Acquisition
-40 -30 -20 -10 0 10 20 30 40 50 600
0.5
1
1.5
2
2.5
3
3.5
Temperature (℃)
Fri
ctio
n t
orq
ue
of
gim
bal
(N
mm
)
ML
MS+M
R M
Fig.11 Calculation results of total friction torque M and its components in the changing temperature. ML, MS and MR are
calculated by Eq. (9) , Eq. (8) and Eq. (3), respectively. M is calculated by Eq. (10).
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2© 2017 The Japan Society of Mechanical Engineers[DOI: 10.1299/jamdsm.2017jamdsm0052]
Yu and Shang, Journal of Advanced Mechanical Design, Systems, and Manufacturing, Vol.11, No.5 (2017)
Card (DAC) for torque value recording, etc. the PMAC was used to drive the torque motor from static to starting and
read its current when the motor just began rotating. At the same time, The DAC was used to record the torque value from
the torque sensor. Using these data, a calibrated value of KT could be got. Different weights were used to change the load
of moment motor and we can obtain an accurate value of current-torque relation by taking the average of all measures.
Experiment settings of the friction torque of gimbal are as follows.
1. In order to reduce the effects of interference torque on the measurement results, static balance of the gimbal must
be carried before the measurement and all cables was in a relaxed state.
2. Due to the high requirement on the properties of low speed of servomechanism, the motor speed was set to 1/3
rpm and the speed was kept constant when measuring the dynamic friction torque.
3. In accordance with the requirements of the gimbal, the rotation range is from -5° to +10° on the basis of
mechanical zero. The rotation range was set to- 6 ° to + 11 ° in order to eliminate the data of motor speed-up phase.The
friction torques of clockwise direction and counter-clockwise direction were both measured.
4. The temperatures of RTCTC were set from 40 ℃ to + 60 ℃ and measurement was conducted every 20 ℃.Each
temperature point was kept for 1 h before a measure was carried.
As can be seen from Fig. 12, friction torque of different positions is not constant but fluctuating. This related to the
surface quality of bearings, motor torque ripple or other relevant factors. Moreover, temperature has great influence on
friction torque of gimbal. Through data analysis, it could be found that the average of friction torque of clockwise
direction was 3.19 Nmm at -40 ℃ but the average is just 1.05 Nmm at +60℃. The friction torque of lowest temperature
was three times as that of highest temperature and its fluctuation was much more severe at -40 ℃.Similarly, the average
of counter-clockwise direction was -2.45 Nmm at -40 ℃ and that was-1.34 Nmm at +60℃.
Fig.12 Measurement curve of dynamic friction torque of pitch axis. The friction torque of clockwise direction and
counterclockwise direction were measured using electrical measure method. The temperature was -40℃(purple) and
+60℃(yellow), respectively.
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2© 2017 The Japan Society of Mechanical Engineers[DOI: 10.1299/jamdsm.2017jamdsm0052]
Yu and Shang, Journal of Advanced Mechanical Design, Systems, and Manufacturing, Vol.11, No.5 (2017)
7. Conclusions
A friction torque model of gimbal-mirror seeker under changing environment temperature was proposed. The
coupling thermal deformation of bearing and structures was considered in the analysis model. The relationship of
temperature and lubricant viscosity was fit by improved Walther Equation. Total frictional torque including three
components was calculated.
It was found that radial coupling thermal deformation greatly decreased the diameter ,clearance and contact angle
of bearing as temperature dropping. Especially, the contact angle of bearing would drop from 15°to zero before the
temperature dropped to -40℃, if ∆=0.008mm or lager amount was selected. If coupling deformation was not considered,
the clearance and contact angle would keep almost constant as the inner ring and outer ring of bearing deforming
synchronously. The axial coupling deformation would change the contact angle and axial force of bearing. The thickness
of pre-tightening gasket was also included in the analysis model. The total frictional torque was calculated using this
temperature range. The coupling deformation must be considered if different materials were used in the outer and inner
frame of seeker.
Finally, a measure system of frictional torque using electrical measure method was established for gimbaled seeker
and rapid temperature change test chamber was used to simulate the changing of environment temperature. Besides, a
separate motor calibration device was set up for KT calibration. The results have shown that the calculating result based
on the friction model proposed in this paper was almost consistent with the experimental data.
Acknowledgements
This work is supported by the National Natural Science Foundation of China (No.51675527).
15
As shown in Fig. 13, The calculating result based on the friction model proposed in this paper was almost consistent
with the experimental data, especially in the range of low temperature. This is very important for structural designers and
control engineers of seekers to improve the system design. Besides, though the friction torques of two directions were
almost same at 20 ℃, averages of torque also has bigger difference with the temperature falling down, which was caused
by uneven distribution of grease at lower temperature.
Fig.13 Comparison of experimental results and theoretical analysis. The average of experimental data was obtained at
different temperature points. The mean values of frictional torque in clockwise direction (blue) and counterclockwise
direction (black) are not completely identical. The calculating result (red) was almost consistent with the
experimental data.
comprehensive model. According to the calculation results, lubricant viscosity is the main factor, especially in the low
2© 2017 The Japan Society of Mechanical Engineers[DOI: 10.1299/jamdsm.2017jamdsm0052]
Yu and Shang, Journal of Advanced Mechanical Design, Systems, and Manufacturing, Vol.11, No.5 (2017)
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