frictional torque analyzing and testing of gimbaled-mirror

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

http://ieeexplore.ieee.org/search/searchresult.jsp?searchWithin=%22Authors%22:.QT.Hilkert,%20J.M..QT.&newsearch=true

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)

2

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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.

3



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).

4



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)

5

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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.

6



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).

7



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.

8

Fig.6 Schematic diagram of axial and radial coupling thermal deformation of gimbal. Except for radial deformation, axial



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.

9



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.

10

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.



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.

11

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.



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.

12



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).

13



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.

14



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



References

Aboul-Seoud, A. L. and Moharam, H. M., A generalized viscosity correlation for undefined petroleum fractions,

Chemical Engineering Journal, Vol.72, No.3(1999) , pp.253-256

Cheock, K. C., Hu, H. and Loh, N. K., Modeling and Identification of a Class of Servomechanism Systems with Stick-

Slip Friction, Journal of Dynamic Systems, Measurement, and Control, Vol.110, No.3(1988) , pp.324-328

Cousseau, T., Graça, B., Campos, A. and Seabra, J., Friction torque in grease lubricated thrust ball bearings, Tribology

International, Vol.44, No.5(2011) , pp.523-531

Cousseau, T., Graça, B., Campos, A. and Seabra, J., Influence of grease rheology on thrust ball bearings friction torque,

Tribology International, Vol.46, No.1(2012) , pp.106-113

Cubalchini, R., Technology and design needs for precision pointing systems, Proceedings of the 34th conference on

design & control(1995), pp.762~767

Du, F., Zhang, J.and Wen, H., Analysis, testing and control of telescope’s high-precision drive system in low-temperature

environment, Proc. SPIE 9151, Advances in Optical and Mechanical Technologies for Telescopes and

Instrumentation, 91513B (2014)

DuPont company, DuPont™ Krytox® Aerospace Grade Oils and Greases, http://www2.dupont.com/Lubricants /en_US

/assets / downloads.

lubricated with wind turbine gear oils at constant temperature, Tribology International, Vol.67 (2013) , pp.194-202

Gerstenberger, J. and Poll G., Rolling bearing lubrication with grease at low temperatures,Tribology Series, Vol.39

(2001) , pp.303-312

Harris, T.A., Rolling bearing analysis, 3rd ed. 2001. p. 193–7, 446–76, 506–8.

Hilkert, J.M. Inertially stabilized platform technology Concepts and principles, Control Systems, IEEE, Vol.28,

No.1(2008) , pp.26-46

Jiang Shaona，Chen Xiaoyang，Gu Jiaming，et al．Analysis of friction torque of sensitive ball bearing under low

speed．China Mechanical Engineering, No.5(2010),pp.510-514(in chinese)

Kobzova, R. I., Klimov, K. I. and Shulzhenko, I. V., Influence of Temperature on Static Frictional Torque of Grease-

lubricated Bearings, Chemistry and Technology of Fuels and Oils, Vol.10, No.3(1974) , pp.227-229

Lagunowich, C. A., Sobek, R., Mcever, M. and Danyo, G. D., Friction effects on large gimbaled EO directors, Proc.

SPIE, Acquisition, Tracking, Pointing, and Laser Systems Technologies XXI, Vol. 6569 ,656908 (2007)

Levin, J. and Ioannou P. A., Adaptive control with neuro-adaptive disturbance rejection,17th Mediterranean Conference

on Control and Automation(2009), pp.570–575

Lin, C. and Hsiao, Y., Adaptive feedforward control for disturbance torque rejection in seeker stabilizing loop, IEEE

Transactions on Control Systems Technology, Vol.9, No.1 (2001), pp. 108~112

Olsson, H. and Astrom, K. J., Friction Generated Limit Cycles. IEEE Transactions on Control Systems Technology,

Vol.9, No.4 (2001),pp. 629~636

Ostensen, J.O., Astrom, H. and Hoglund, E., Analysis of a Grease-Lubricated Roller Bearing under Arctic Conditions,

Journal of Engineering Tribology, Vol.209, No.3 (1995), pp. 213~220

Palmgren, A., Ball and Roller Bearing Engineering(1959),p.34-41, Burbank

Snare, B., Rolling resistance in lightly loaded bearings, The Ball Bearing Journal, Vol.153 (1968), pp.19-24

16

Fernandes, C. M.C.G., Amaro P. M.P., Martins, R. C. and Seabra, J. H. O.,Torque loss in cylindrical roller thrust bearings

http://ieeexplore.ieee.org/search/searchresult.jsp?searchWithin=%22Authors%22:.QT.Hilkert,%20J.M..QT.&newsearch=true

http://pij.sagepub.com/search?author1=H+%C3%85str%C3%B6m&sortspec=date&submit=Submit



Stockum, L., Profeta, J. and Ballou, L., Precision stabilization system design to reduce the effects of friction, Proc. SPIE,

Acquisition, Tracking, and Pointing II, Vol.887 (1988) , pp.159-167

Wikström, V. and Höglund, E., Starting and Steady-state Friction Torque of Grease Lubricated Rolling Element Bearings

at Low Temperatures-Part I : a Parameter Study, Tribology Transactions, Vol.39, No.3(1996a) , pp.517-526

Wikström, V. and Höglund, E., Starting and Steady-state Friction Torque of Grease Lubricated Rolling Element Bearings

at Low Temperatures-Part Ⅱ: Correlation with less-Complex Text Methods, Tribology Transactions, Vol.39,

No.3(1996b) , pp.684-690

Wikström, V., Larsson, R. and Höglund E., A Theoretical Analysis of Shear Stresses and Roller Slip in Rolling Bearings

During Low-Temperature Starting, Tribology Series, Vol.32 (1997) , pp.423-444

Yang, J.,Plummer, A. and Xue,Y., Dynamic friction modelling without drift and its application in the simulation of a

valve controlled hydraulic cylinder system, Journal of Advanced Mechanical Design, Systems, and Manufacturing,

Vol.8, No.6(2014) , pp.1-6

Yu, S. and Zhao, Y., Simulation study on a friction compensation method for the inertial platform based on the

disturbance observer, Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace

Engineering, Vol.222, No.3(2008) , pp.341-346

17

frictional torque analyzing and testing of gimbaled-mirror

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