This article presents the clutch dynamic model based on the friction dynamic model

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.

INTRODUCTION^*

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 advantages compared to the manual 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( )

c c r r n

T 

   

F

(1) where: _c is the clutch geometry constant;

Fn is the normal force;



_r is the sliding angular velocity,

  (

_r

)

is the friction coefficient depending on the sliding angular velocity



_r:

( _r) _c ( _{s c})e ^{r s s}

  

      ^

(2)

where



_c is the Coulomb friction 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

dt g

 

   (3) where g

( 

_r

)

is a function depending on the relative angular velocity



_r, which captures the Stribeck effect and can be depicted as:

 



²



0

( _r) 1 _c ( _{s c}) ^{r s}

g     e ^{ }



    (4)

with



₀ is the stiffness of the bristles.

- The clutch friction torque is determined by following equation:

0 1 2

c r c n

T z dz F

  dt   

 

    (5) where



₁ is the damping coefficient,



₂ is the linear viscous friction coefficient, _c is the clutch geometry constant, F_n 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

1 2

 

  and the angular velocity of the input shaft



₁

1 2

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

1 2 1 2

1 1 1

    r

   

  

   (7) The average deflection of the bristles is denoted asz, the average deflection rate of the bristles is given:

2 2

dz

dt   (8)

1 2 1 2 1

( ) ( ) _r

dz

dt         

     (9)

The clutch friction torque is a function depending on the slipping coefficient  and the normal force F_n,



^,



^{( )}

c n c n

T  f  F  g  F (10) where g

( ) 

is a function capturing the Stribeck effect, we use the “Magic” formula [7]

 

 



^{1 1}



( ) sin tan tan ( )

g  D C ^ BE B ^ B (11) where:

D



c (12) 2 1

2 sin ^c

s

C 

 

  

   

  (13)

 

1

tan / (2 ) tan ( )

s

s s

B C

E B B

 

 ^ 

 

 (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, T_cz



₀, where



₀ is the stiffness of the bristles. We have

0

dTc dz

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

( ) ( )

( ) ( )

c n

c n

n

c n

dT dg dF

F g

dt dt dt

dg d dF

F g

d dt dt

  

 

 



 

   

 

   

(18)

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

 

0 1

( )

( )

c n r

n c

dg d F d dt

g dF 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 1 2 1 2

2

3 1 2 1 2

3

1 ( )

1 ( ) ( )

1 ( ) ( )

; 1,..., 3

in c

c

r

i i

T T I

T C K

I

C K T

I i



    

    

 

 

    

    

 

(20)

where _i, _i are respectively the angular velocities and angular displacements of the engine, the clutch disc and the vehicle, T_r is the load torque, T_c 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:



1 2 1 2



1

2

*

1 2

( ) ( )

in c

T T I

I I

C  K   I

 

 

 

(21) Applying simulation with the parameters of the simplified powertrain model as following [8]:

1 2.7

I  kgm²;I₂ 

0.1

kgm²; I₂ 

2.65

kgm²; 16300

C Nm/rad; K 60Nm.s/rad. The parameters of clutch models are [8]:



_s 

0.8

;

c

0.6



 ;



_s 

2

;



_s 

10

rad/s;



_c 0.28m;

4

0 5.10

  Nm;



₁

3

Nm;



₂ 

0

Nm.s/rad;

10 3

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 F_n as shown in Figure 4. The engine torque T_in and the load torque T_r 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 F_n 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 (F_n 

80

daN), 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 inputF_n.

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