XÂY DỰNG MÔ HÌNH LY HỢP TRÊN CƠ SỞ MÔ HÌNH MA SÁT ĐỘNG LỰC HỌC

While that of the clutch modeled by the LuGre model occurs when the normal force is zero (about 3s), which does not capture the reality behavior of the clutch.. Figure 5 [r]

MODELLING OF AN AUTOMOTIVE CLUTCH BASED

ON DYNAMIC FRICTION MODEL

Tran Van Nhu *

University of Transport and Communication

ABSTRACT

The clutch dynamic model is important in dynamic research and gearshift control for automated manual transmission and dual clutch transmission The dynamic model must describe correctly the behavior of the clutch in the transitional period for simulation and the controller design ensures fast and smooth synchronization This article presents the clutch dynamic model based on the friction dynamic model The developed model was simulated using Matlab/Simulink, the numerical simulation results were compared with the literature models to show the effective of the developed model

Keywords: Clutch; Clutch dynamic model; Dynamic friction model; Powertrain; Stribeck.

The friction clutch is an element important in

the automotive powertrain, its function is to

transmit torque from the engine to the

transmission system by friction torque The

friction clutch is present in the various

powertrain such as the Manual Transmission

(MT), the Automated Manual Transmission

(AMT) and the Dual Clutch Transmission

(DCT) The AMTs and the DCTs have more

transmission and the automatic one It attracts

many researchers for modeling, simulation

and developing the control law to manage the

clutch/dual clutch during gear shifting and

take-off phase In the literature, the friction

clutch is usually modeled by the Coulomb

friction model The Coulomb friction model

does not describe well the Stick-Slip

transition and Stribeck phenomenon In the

clutch control research to enhance the smooth

driving, the transition phase is very important

In this paper the author develops a dynamic

model of friction clutch based on the bristle

models

In the literature, the dynamic friction model is

presented in some works The Dalh’s model

introduced in [1] was developed for the

simulation of control systems This model is

*

Tel: 0972 020094, Email: vannhu.tran@utc.edu.vn

only a function of the relative displacement and the sign of velocity Therefore, it neither captures the Stribeck effect, which is a rate dependent phenomenon, nor does it capture the stick-slip transition The Dalh’s model can

be used to simulate the systems with hysteresis [2] The Bristle model was developed in [3] by Haessig and Friedland This model captures the behavior of the microscopical contact point between two surfaces Each contact point is thought of as a bond between flexible bristles The Bristles model captures the nature of friction, the stick-slip behavior and can be made velocity dependent However, the model is complexity due to the large number of bristles Motion in sticking may be oscillatory since there is no damping of the bristles in the model Canudas

de Wit C et al introduced a dynamic friction model called Lugre [4] Lugre friction is modeled as the average deflection force of elastic springs The average bristles deflection

is a velocity dependent function, which can capture the Stribeck effect One of the disadvantages of the LuGre friction model is that it does not possess the non-drift property

M Aberger and M Otter applied this dynamic friction model for modeling a clutch model [5] However, this clutch model does not capture the behavior of the clutch in the process opened by the non-drift property [6]

In [7] the authors introduced a static friction

model Pacejka with the “Magic” formula,

which capture the Stribeck effect and it

applied for modelling the tyre friction This

model is not continuous and does not capture

the stick-slip transition

In this paper the author introduces a dynamic

clutch model based on the Bristles model The

developed model is simulated and compared

with the literature models to show the effectof

the developed model

CLUTCH MODEL BASED ON THE

STATIC FRICTION MODEL

In the literature, the author majority use

Coulomb friction to model the clutch friction

This model does not capture the Stribeck

effect In [8], the author introduced a clutch

model based on the static friction model with

the friction coefficient depending on the

sliding velocity to capture Stribeck effect

The clutch friction torque in the sliding phase

is determined by equations:

( )sign( )

T     F (1)

where: c is the clutch geometry constant;

n

F is the normal force; r is the sliding

angular velocity,   ( r) is the friction

coefficient depending on the sliding angular

velocity r:

( ) ( ) r s s



 

       (2)

coefficient;s is thestatic friction coefficient;

s

 is the Stribeck angular velocity; s is the

Stribeck exponent

When the clutch is locked, the torque

transmitting through the clutch is the static

friction torque, which is determined

depending on the state of the system (see

equation (21))

CLUTCH MODEL BASED ON THE

LUGRE FRICTION MODEL

The clutch dynamics model based on the

Lugre model developed in [5], with the

relative angular velocity r between the

clutch friction disc and the pressure plate The LuGre clutch model is described in the standard form of a first-order nonlinear differential equation:

- The average deflection of the bristlesz:

| | ( )

r r

r

dz

z







  (3)

where g ( r) is a function depending on the relative angular velocity r, which captures the Stribeck effect and can be depicted as:

 

0

1

g     e  





with 0 is the stiffness of the bristles

- The clutch friction torque is determined by following equation:

dz

dt

where 1 is the damping coefficient, 2 is the linear viscous friction coefficient, c is the clutch geometry constant, Fn is the normal force

THE NEW DYNAMIC MODEL OF THE FRICTION CLUTCH

The surfaces of the clutch disc and pressure plate are very irregular at the microscopic level We visualize this contact as two bodies that make contact through elastic bristles (see Figure 1) [4] For simplicity the bristles on one part are shown as being rigid When a torque applies the bristles is deflected which give rise to the friction torque If the torque applied is sufficiently large, the bristles deflect so much that they will slip

Figure 1 The contact between two surfaces of the

clutch disc and clutch pressure plate

The clutch slipping coefficient is defined as

the ratio of internal slipping angular velocity

   and the angular velocity of the input

shaft 1

1

 









 (6)

where 2 is the angular velocity of the

bristles head (see Figure 1)

In the sliding phase, the bristles are deflected

maximum, they will slip, therefore2  2,

where 2 is the angular velocity of the output

shaft In this phase, the slipping coefficient

(6) becomes

r





The average deflection of the bristles is

denoted asz, the average deflection rate of

the bristles is given:

dz

dt   (8)

dz

dt         

The clutch friction torque is a function

depending on the slipping coefficient  and

the normal force Fn,

T  f  F  g  F (10)

where g ( )  is a function capturing the

Stribeck effect, we use the “Magic” formula [7]

( ) sin tan tan ( )

g  D C  BE B  B

(11)

where:

c

D (12)

1

2

2 sin c

s

  (13)

1

tan / (2 )

tan ( )

s

E





 (14)

where s is the Stribeck slipping

coefficient,BCD is the slope of the line

tangent to the curve g ( )  at the coordinates

of the origin ( 0) (the clutch is locked)

In the locked phase, the clutch torque is independent of the slipping coefficient

(0), it is a linear function of the bristle deflection, Tc z 0, where 0 is the stiffness of the bristles We have

0

c

dt  dt (15)

0

1 dT c dz

dt  dt

  (16) From the equations (9) and (16) we have:

0

1

1 c

r

dT

dt  

    (17)

From the equation (10) we have

( )

( )

( )

( )

n

dF

dg d





(18)

From the equations (17) and (18) we have:

( )

( )

n c

dg d F

d dt dF g dt

 





(19)

The clutch model is described by the equations (10), (11) and (19)

SIMULATION RESULTS Considering a simplified model of powertrain

as shown in Figure 2 In this figure, T in is the engine torque, I1 is the mass moment of inertia of the engine, flywheel, clutch drum and pressure plate, I2 is the mass moment of inertia of the clutch disc, C K, are respectively the stiffness and damping coefficient of the clutch disc and the clutch shaft, I3 is the equivalent mass moment of the transmission system and the vehicle mass

Figure 2 Simplified model of powertrain

The differential equations of the simplified

powertrain model is given [8]

1

1

2

3

1

1

1

; 1, , 3

c

r

I

I

I

i



(20)

where i, i are respectively the angular

velocities and angular displacements of the

engine, the clutch disc and the vehicle, Tr is the

load torque, Tc is the clutch friction torque,

which is modelled in the above sections

For the clutch model based on the static

friction model, it is necessary to determine the

clutch torque in the locked state In the locked

state, we have  1  2 From the first and

second sub-equation of the equation (20)we

can find the clutch torque as following

equation:

2

*

in

c

T I

T

I I

C  K   I





(21) Applying simulation with the parameters of the

simplified powertrain model as following [8]:

I  kgm2;I2  0.1kgm2; I2  2.65kgm2;

16300

parameters of clutch models are [8]: s  0.8;

0.6

c

  ;s  2;s  10rad/s; c 0.28m;

4

0 5.10

  Nm; 1 3Nm; 2  0Nm.s/rad;

3

10

s

   ; B100 The function g ( )  is

shown in Figure 3

Figure 3 Functiong ( ) 

The first simulation is implemented by the time-varying normal force Fn as shown in Figure 4 The engine torque T in and the load torque Tr are constants, the initial slipping angular velocity is r(0) 100 rad/s The simulation results with three clutch models are shown in Figures 4 and Figure 5 At the first stage, the clutch is synchronized (from 0s

to 2s), the normal force Fn increases In this case, the behaviors of the clutch modeled by the three methods are similar Then, at the second stage (from 2s to 3s), the normal force decreases Naturally, the clutch switches from the locked state to the slipping state The state switching of the clutch modelled by the static friction model and the developed model occurs almost at the same time when the normal force decreases to 50daN (at 2.75s, see Figure 4) While that of the clutch modeled by the LuGre model occurs when the normal force is zero (about 3s), which does not capture the reality behavior of the clutch Figure 5 showed that, at the state switching moment (about 2.75s), the clutch torque based

on the static friction model oscillates with a large amplitude The developed model oscillates less and captures the behavior of stick-slip moment

Figure 4 First test - time-varying normal force:

angular velocities

The second test is implemented with a constant normal force (Fn  80daN), the engine torque is time-varying The simulation results shown in Figure 6 In this case, the behavior of the static clutch model and the

developed model are similar, that of Lugre

clutch modelis slightly different from the

other two

Figure 5 First test - time-varying normal force:

clutch torque

According the function of the clutch, the

normal force is variable to synchronize and

disengage the clutch disc In this case, the

developed clutch model captures well the

clutch behavior in the process of

synchronization and disengagement

Figure 6 Second test - time-varying engine

torque: angular velocities

CONCLUSION

In the literature, the clutch model is modelled

based on the static friction model This model

does not capture the stick-slip transition The

clutch model based on the Lugre friction

model capture well the behavior of clutch in the synchronization process However, in the disengagement process, this model does not capture the behavior of the clutch by the non-drift property

The developed model in this paperhas eliminated the disadvantages of the two models above It captures well the Stribeck effect, the stick-slip transition However, this model is not affine in the control inputFn

REFERENCES

1 P Dahl (1968), “A solid friction model”, The Aerospace Corporation, El Segundo, CA, Technical Report TOR-0158H3107–18I-1

2 H Olsson, K J Åström, C C de Wit, M Gäfvert, P Lischinsky (1998), “Friction Models

and Friction Compensation”, Eur J Control, vol

4, No.3, pp 176–195

3 D A Haessig, B Friedland (1991), “On the

modelling and simulation of friction”, J Dyn Syst Meas Control, vol 133(1), pp 354–362

4 C Canudas de Wit, H.Olsson, K.J.Astrom, P.Lischinsky (1995), “A new model for control of

systems with friction”, IEEE Trans Autom Control, vol 40, pp 419–424

5 M Aberger, M Otter (2002), “Modelling Friction in Modelica with the LuGre Friction

model”, trong International Modelica Conference, Proceedings, Oberpfaffenhofen

6 R Nouailletas (2009), “Modélisation hybride, identification, commande et estimation d’états de système soumis à des frottements secs - Application à un embrayage robotisé.”, Grenoble INP, Grenoble

7 H B Pacejka (2006), Tyre and Vehicle Dynamics Butterworth-Heinemann

8 V N Tran (2013), “Vehicle driveability improvement by the powertrain control”, Université de Valenciennes et du Hainaut-Cambrésis, Valenciennes.

TÓM TẮT

XÂY DỰNG MÔ HÌNH LY HỢP TRÊN CƠ SỞ MÔ HÌNH MA SÁT

ĐỘNG LỰC HỌC

Trần Văn Như *

Trường Đại học Giao thông Vận tải

Mô hình động lực học ly hợp quan trọng trong nghiên cứu động lực học và điều khiển quá trình chuyển số trên hệ thống truyền lực tự động hóa AMT và DCT Mô hình động lực học cần mô tả được hành vi của ly hợp trong giai đoạn quá độ để mô phỏng chính xác và thiết kế bộ điều khiển đảm bảo đóng mở ly hợp nhanh và êm dịu Bài báo này tác giả trình bày mô hình động lực học ly hợp xây dựng trên cơ sở mô hình ma sát động lực học Mô hình được mô phỏng bằng phần mềm Matlab/Simulink, kết quả mô phỏng được so sánh với các mô hình trước đây cho thấy sự đáp ứng hành vi của mô hình

Từ khóa: Ly hợp; Mô hình động lực học ly hợp; Mô hình ma sát động lực học; Hệ thống truyền

lực; Stribeck.

Ngày nhận bài: 01/8/2017; Ngày phản biện: 14/8/2017; Ngày duyệt đăng: 30/8/2017

*

Tel: 0972 020094, Email: vannhu.tran@utc.edu.vn