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LIFE PREDICTION MODEL OF FRICTION MATERIAL FOR TRAVELING WAVE ULTRASONIC MOTOR

Xiu TIAN1,*, Jian-jun QU2, Jian-chun LI1, Yan-li WANG1, Yue-dong LANG1 1 Beijing Aerospace Control Device Institute, Beijing 100039, China

2 School of Mechatronics Engineering, Harbin Institute of Technology, Harbin 150001, China * Corresponding author, E-mail: tianxiu999@163.com; Tel.: 86-10-68388673.

Traveling wave ultrasonic motor (TWUSM) relies on friction force for torque transmission. Wear often exists on friction material (FM) inevitably, and largely affects life of TWUSM. A new life prediction model of FM for TWUSM was proposed. Compared with standing wave USM, wear of TWUSM FM was resulted from scratching of traveling wave crests on stator. Considering motion of a particle in contact interface between stator and rotor, visco-elastic polymer contact model was used to calculate normal pressure distribution and relative sliding speed, which were introduced to traditional wear law of Archard. In the model, special relation between time and distance for traveling wave was used to analyze contact intervals. The change of thickness of FM during service which made decrease of preload was also taken into account. Parameters of common disc-type TWUSM were used to compute life prediction curve, and meanwhile the experimental investigation was carried out. The life prediction result found out to be consistent with the test.

Keywords: Traveling wave ultrasonic motor; Friction material; Wear; Life prediction

1. INTRODUCTION

Ultrasonic motor (USM) relies on frictional force for torque transformation. The wear of FM affects life span of USM seriously. Therefore, research on wear and life prediction of FM of USM is widely studied at present. Ishii proposed a life prediction model for standing wave USM(SWUSM) FM[1]. In the model, a dynamic contact model between stator and rotor for SWUSM was used[2]. However, this model did not fit for TWUSM very well[3]. In several types of USM, TWUSM were paid more attention to practical application because of their advantages, such as continuous contact, good wear resistant, etc. Zheng revealed that there were three stages in the lifetime of TWUSM by investigating the wear course of TWUSM[4]. Then, he established a wear evaluation model, and the evaluation model error was about 15% comparing with test[5].

In this paper, a new life prediction model of FM for TWUSM was proposed. Compared with SWUSM, wear of TWUSM FM was resulted from scratching of traveling wave crests on stator. Considering motion of a particle in contact interface between stator and rotor, visco-elastic polymer contact model was used to calculate normal pressure distribution and relative sliding speed, which were introduced to traditional wear law of Archard. The life prediction curve of TWUSM FM was

calculated and simulated numerical by using MATLAB software. Then, experimental investigation was carried out using a common type TWUSM to verify the model.

2. CAUSE OF WEAR FOR TWUSM FM

The wear of TWUSM FM is generated by the reaction force of stator, as shown in Fig. 1. Stator is regard as a sine wave surface. Its wave crests contact with rotor’s FM. While TWUSM operating, the wave crests push forward, wear of FM is generated by the scratching of traveling wave crests on stator surface.

Figure 1. Wear generation course of TWUSM FM

The wear forms of FM consist of fatigue, plowing and abrasive wear, and are much more complicate. Therefore, to ensure a correct FM wear calculation, we analyze the problem from a microscopic point of view. Q is concerned as a particle on FM interface. When located in the contact region of stator wave crest, Q is subject to a normal dynamic force, and the sliding speed between Q

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and stator is also a variable. When out of contact region, Q is suffered no force, thus no wear created at Q point.

3. CONTACT MODELING OF TWUSM

According to the traditional wear equation of Archard. Wear volume of TWUSM FM V can be expressed by

H

LFkV ⋅= (1)

with k proportional coefficient, F normal dynamic pressure, L sliding distance and H harness of FM. In Eq. (1), the variables of F, L and H can be calculated by the visco-elastic contact model of TWUSM.

Based on Kirchoff plate theory, the neutral plane of stator moves in the way of an ideal traveling wave[6]

)~2cos(),~(~ txAtxw ω

λπ −= (2)

with A amplitude of vibration, λ wavelength, ω exiting frequency and t vibrating time. The spatial fixed coordinate of reference ( zx ~,~ ) is used in Eq. (2). For further simplify equation, a coordinate of reference is fixed on the neutral plane of stator (x, z). The conversion relation between these two coordinates of reference can be expressed as tvxx w ⋅+=~ , with wave propagation velocity vw = λω/2π. Eq. (2) can be simplified as

⎟⎠⎞

⎜⎝⎛=

λxAtxw π2cos),( (3)

The contact state between FM and stator is shown in Fig. 2. Assume the boundary of contact region are -K and L, a is the distance between surface of stator tooth and neutral plane of stator. When Q is in the contact region, its normal displacement, speed and acceleration of friction layer surface can be derived by

⎟⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛=

λλMAxAxunπ2cosπ2cos)( (4)

[ ]

⎟⎠⎞

⎜⎝⎛==

λω xA

dxtxUd

xu ncn

π2sin),()( (5)

[ ]

⎟⎠⎞

⎜⎝⎛−==

λω xA

dxtxUd

xu ncn

π2cos),(

)( 22

2

(6)

Its tangential displacement and speed are denoted by

⎟⎠⎞

⎜⎝⎛=−≈

λλxaA

xdxwdaxut

π2sinπ2)())(()( (7)

⎟⎠⎞

⎜⎝⎛−==

λλω xaAxutxv tt

π2cosπ2)(),( (8)

Figure 2. Contact deformation of FM

3.1. Dynamic pressure in normal direction

The visco-elastic contact model of TWUSM is shown in Fig. 3. kn and kt are the normal and the tangential equivalent stiffness of FM, kn=E·b/h. E, b and h are elastic modulus, width and thickness. cn and ct are normal and tangential equivalent damping coefficient.

Figure 3. Schematic diagram of contact state between

TWUSM stator and rotor

Suppose the length of Q on the FM is dx, in normal direction, using visco-elastic contact model and the application of Newton's dynamics law, the force balance equation of the FM is established as

dxxukdxxucbdxxpdxxum nnnnnn )()()()( −−= (9)

with mn the equivalent quality of micro-unit on the contact layer, p(x) the function of normal distribution pressed on the contact layer. Eq. (4)-(6) is introduced into Eq. (9) to obtain the pressure distribution function

( )

⎥⎦

⎤⎟⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛+

⎢⎣

⎡⎟⎠⎞

⎜⎝⎛−=

λλω

λω

Mkxc

xmkbAxp

nn

nn

π2cosπ2sin

π2cos)( 2

(10)

On the contact boundary, the normal pressure equals to zero, then

0)()( ==− LpKp (11)

The preload F0 is equal to the total value of normal dynamic load pressed on contact layer in a period, then

3

∫∫−−

==L

K

L

L

dxxpnbdxxpnbF )()(0λ

(12)

Therefore, from Eq. (10)-(12), the iteration values of M, mn, K and L can be obtained.

The coordinate is translated back to fixed coordinate ( x~ , t), so from Eq. (10), the normal pressure distribution function becomes as

⎥⎦

⎤⎟⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛ −⋅+

⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −−=

λπω

λπω

ωλπω

Mktxc

txmkbAxP

nn

nn

2cos~2sin

~2cos)()( 2

(13)

The normal dynamic pressure of TWUSM is different from the one of SWUSM. For a TWUSM, the dynamic pressure at of any point on FM is both changed with displacement x~ and time t. Therefore, there are two variables in normal dynamic pressure. For a time t, the total value dynamic pressure on FM is constant. Only its phase transforms with the change of time t. However, for a SWUSM, the dynamic pressure on FM changes with the only variable of time.

3.2. Rotor / stator relative sliding velocity

When Q is in the contact area, and the coordinate is translated to the fixed coordinate ( )tx ,~ , the relative sliding velocity between rotor and stator becomes

RvtxaAxv −⎟⎠⎞

⎜⎝⎛ −−= ω

λλω

~π2cosπ2)~( (14)

3.3. FM hardness

In general, the mechanical properties of materials are affected by temperature. Hardness loss is a function of temperature

THH Δ⋅−= α0 (15)

with H0 hardness of FM at room temperature, N·mm-2, α hardness loss coefficient.

4. LIFE PREDICTION MODEL

For a particle Q, it suffered the normal dynamic pressure dF = p ),~( tx bdx. Within the unit time dt, the relative sliding distance is ttxvL d),~(d ⋅= . By Eq. (1), the amount of wear at point Q in the unit time dt can be obtained by

H

bdxdttxvtxpkdV ),~(),~(= (16)

Therefore, the total amount of wear is the wear volume of entire surface for FM in the running time.

H

dtxdtxvtxPnbkV

∫ ∫=

τ λ

0 0

~),~(),~( (17)

In a period of x and t, stator and rotor of TWUSM are not entirely in contact. On the whole ring surface of rotor, in the total running time of USM, the contact process between stator and rotor is a dynamic contact and separation process. For the traveling wave equation, there exists a relationship, tvx ⋅= w

~ , and x~ corresponds to t. When the point Q moves in space from –K point to L point on the rotor, it moves in time from –K/vw to L/vw. Thus, if ],[~ LKx −∈ , ]/,/[ ww vLvKt −∈ , it can be the value range for the two periodic variables on the FM contact interval.

As worn of FM gradually formed during USM operation, its thickness h changes as a variable. Its thickness becomes thinner, which affects the deformation of rotor, so the normal pressure decreases. Thereby, the contact width of FM decreases, which results in wear rate decrease[7]. In the above model, the change of wear rate is considered by introducing the normal stiffness kn=E·b/h to calculate normal dynamic pressure. Therefore, thickness h affects stiffness of FM, further affecting the contact width and wear rate of FM. Then the wear volume of the material is

ΔhrrV ⋅−⋅= )(4

21

22

π (18)

with Δh the worn thickness of FM, Δh =0.01, 0.02, 0.03…, mm. the integral region of x and t in Eq. (17) are switched to ],0[ λ∈x ， ],0[ Tt ∈ , introduce Eq. (13) and (14) to Eq. (17), and by Eq. (18), using MATLAB programming iteration, the values of FM wear volume correspond to different run-time can be obtained, and the wear and life prediction curve of the FM were drawn.

5. EXPERIMENTAL VERIFICATION

With 60-type TWUSM and PTFE-based FM developed by our group, the validity of life prediction theoretical model was verified by experiments, all calculation parameters related to the model is shown in Table 1.

The flexible rotor with flange structure can provide contact pre-load between stator and rotor. The

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relationship between rotor deformation and pre-pressure is linear. Thus, in the wear process of FM, pre-pressure F is a variable

)2.0( ΔhhkF Z −+⋅= (19)

with h the thickness of FM, h=0.11mm, kZ the stiffness coefficient of rotor, kZ=353N·mm-1. By the above formula, real-time pre-load of FM is calculated.

Table 1. Correlation parameters of TWUSM60

No. Value

Vibration amplitude A(mm) 0.0015 Distance between tooth and neutral plane a(mm) 3. 614 Effective radius of stator tooth r(mm) 27.25 Wavelength λ(mm) 19.02 Number of nodal diameters n 9 Vibration frequency of stator f(kHz) 43.5 Preload force F0(N) 110 Inner diameter of FM r1(mm) 52.5 Outer diameter of FM r2(mm) 56.5 Elastic modulus of FM E(N/mm2) 1500 Dynamic friction coefficient Ct =Cn (N·s/mm2) 0.001 Density ρ(g/mm3) 2.1×10-6 Hardness of FM H(N/mm2) 55

The wear test of USM FM was implemented. In the

experiment, the speed of rotor kept constant, at about 120±10r·min-1. The worn thickness of FM Δh was tested randomly by using dial gauge, the measurement error is in less than 1μm. The wear coefficient is an important parameter for an accurate model. With worn thickness data of FM in the early 900h experiment and this life prediction model, wear coefficient k value of FM is calculated in reverse. It is found that when the running time is greater than 200h, the FM and the stator has entered to a stable worn phase, the value of wear coefficient k is steady at (0.82 ± 0.05) × 10-7, let k = 0.82 × 10-7 to validate the theoretical model.

The comparison chart of wear test result of 60 TWUSM FM and life prediction curve received by the model is shown in Fig. 4. The USM was working continuously during test. The total running time was more than 3000h. It shows that life prediction result for TWUSM FM has good agreement with the experimental result. With running time increases, the wear rate of FM decreases. It is consistent with the analysis results in this paper. Therefore, it is verified that the life prediction model of TWUSM FM is reasonable and effective.

Figure 4. Wear prediction curve of FM for TWUSM

and test value comparison

6. CONCLUSION

A life prediction model of TWUSM FM is established. The model is different from previous models because that it takes the dynamic contact characteristics of TWUSM and the thickness factors of FM into account. The model is simple and practical, so that by the model, we simulated and analyzed life curve of FM for 60-type TWUSM. Compared with the experimental results, it shows that the results calculated by the proposed model are consistent with the experimental results. Therefore, it validated the model reasonable and effective.

REFERENCES

[1] Ishii T, Ueha S, Nakamura K. Wear properties and life prediction of FMs for ultrasonic motors. Japanese Journal of Applied Physics, (34): 2765~2770, 1995.

[2] Nakamura K, Kurosawa M, Ueha S. Design of a hybrid transducer type ultrasonic motor. IEEE trans. on UFFC. 40(4): 395~401, 1993.

[3] Ishii T, Takahashi H, Nakamura K, et al. Wear prediction method of the friction materials used for the ultrasonic motors. J. of Japanese Society of Tribologists, 45(1): 62-71, 2000. (in Japanese)

[4] Zheng W, Zhao CS. Experimental study on life of traveling wave rotary ultrasonic motors. Piezoelectrics & Acoustooptics. 30(1): 90~92, 2008. (in Chinese)

[5] Zheng W, Zhao CS. A wear evaluation of friction materials used for rotary ultrasonic motors. IEEE ultrasonics symp., pp. 1838-1841, 2008.

[6] Storck H, Wallaschek J. The effect of tangential elasticity of the friction layer between stator and rotor in traveling wave ultrasonic motors. International Journal of Non-Linear Mechanics, (38):143-159, 2003.

[7] Wallaschek J. Contact mechanics of piezoelectric ultrasonic motors. Smart Mater. Struct., (7): 369-381,1998