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Development of High Fidelity Dynamic Model with Thermal Response for Single Plate Dry Clutch.

INTRODUCTION

In automotive transmission, clutch plays vital role during gear shift. Quality of shift depends on clutch actuation and synchronization in synchromesh gearbox. The single plate diaphragm spring dry clutch is most commonly used in MT and AMT. It has several advantages with respect to ergonomics and operation. It reduces the pedal effort and improves clutch engagement, wear life, reliability etc. discussed in [1].

The mathematical model of clutch has several applications in clutch control design and development. For instance, in case of AMT development, [2] clutch control is optimized and proved with the high fidelity mathematical model. The mathematical model of clutch can also be used to come up with optimised clutch design. [1] builds virtual design system platform for clutch which saves the time, manpower and efforts in product development cycle. The proposed platform is based on numerical approach.

In order to study the effects of various parameters, which affect torque carrying capacity of the clutch, requires a model which includes all the required phenomena. [4,5] Includes dependence of the torque transmissibility on wear, slip speed, friction pads geometry. The influence of the diaphragm spring, flat/cushion spring and the torsional damper springs is also taken into account to model the system. This model also finds the dependence of the transmitted torque on contact friction and pressure distribution (uniform pressure or uniform wear). Moreover, transient engagement dynamics is developed in [5] correlating the parameters like pressure plate lift and release bearing travel characteristics, clutch pedal kinematics during engagement and vehicle driveline dynamics during vehicle startup.

The coulomb friction model is used in [6] for rotational system of clutch. The diaphragm spring characteristic have been modelled as a combination of both linear and nonlinear system. While model in [7] follows approach for real time application, a fixed time step clutch model has been developed. The thermal characteristics influence the torque carrying capacity. Therefore [8,9] develop torque model considering thermal expansion of parts. This has been studied by simulating the torque transmission in a heavy-duty truck dry clutch at slipping conditions [8]. [1,10] include the wear of clutch which is responsible to reduce the thickness of friction pads. This modelling gives the flexibility to observe the temperature during slippage and after engagement.

This paper focuses on developing a high fidelity dynamic clutch model. The experiment based data is used to model nonlinearity of diaphragm and cushion spring. A different approach is undertaken to model the thermal effect of individual friction surfaces on torque carrying capacity.

CLUTCH ENGAGEMENT DYNAMICS

Figure 1 shows the simplified schematic diagram of single plate dry friction clutch in disengaged condition. The schematic gives a brief idea about clutch parts and their positions in assembly. Flywheel is mounted on input shaft and it forms one friction pair with clutch disc. The clutch is coupled to output shaft by means of spline. The clutch disc is free to move on output shaft within confined space. The clutch disc has two friction pads on either side and cushion spring is sandwiched between the friction pads by means of rivets. The intention of using cushion spring is to allow smooth engagement. The torsion damping springs are fitted on the central plane of the clutch disc in a circular fashion to damp the torsional oscillations. This entire subsystem of friction pads, cushion spring and torsion damping springs forms a clutch disc assembly. Pressure plate along with another friction pad of clutch disc forms second friction pair. The pressure plate, diaphragm spring and cover forms clutch cover assembly. Clutch cover assembly is usually bolted to flywheel and both together acts as a single unit. The diaphragm spring is a steel disc having hole at center, disc is slotted radially. This forms actuating release lever fingers and are operated by CRB. The outer ends of the slots are provided with enlarged blunting holes. These blunt holes help to distribute the concentrated pressure created during clutch actuation i.e. deflection of the fingers by CRB. The engagement and disengagement phenomena is governed by CRB travel. Figure 2 Shows, two clutch conditions (a) engaged condition and (b) disengaged condition. The clutch is said to be engaged when the cushion spring is compressed to flat disc by means of force applied by pressure plate. The compression force defines the torque carrying capacity of clutch. [x.sub.bs] defines the CRB travel and [x.sub.cs] cushion spring travel. The CRB travel and cushion spring are dependent such that [x.sub.cs] = f([x.sub.bs]).

The mathematical model of clutch has been built by considering the physical systems involved and their dynamic interactions in multi domain environment. To simplify and to increase fidelity level, rotational and translational dynamics are modelled separately. Each sub system defines dynamic actions occurring at their subsystem level and their details are discussed in further subsections.

The considerations and assumptions made for clutch modelling are as follows

* Nonlinearity of both the diaphragm and cushion spring is considered.

* Variation of friction coefficient with respect to temperature is taken into account.

* Frictional torque generation on either side of clutch disc has been modelled separately.

* The temperature rises of friction pad due to frictional energy has been modelled.

* Effect of torsional damping spring has been ignored.

* There is no friction in translational motion of clutch disc.

Translational Model

The translational system includes the interaction of clutch parts which exhibit the translational degree of freedom and plays role in transmitting the force applied at CRB to generate the friction torque. Let us assume subscript fp, cp and bs represent friction plate, clutch pressure plate and bearing sleeve respectively. The symbols m, C, k, x indicate mass, damping constant, stiffness and displacement respectively.

The force applied at CRB is responsible for deflection of diaphragm, cushion spring and acceleration of friction pad, CRB. Figure 3 shows, the free body diagram of translational system.

The mass of friction pad is very small so its effect on the system is neglected. The ease of modelling of complex system is obtained by simplifying system without much effect on its principle operation. The simplified FBD for translational system can be drawn as follows,

In figure 4, F is the force applied at CRB for clutch actuation and [F.sub.fc] is the force responsible for friction torque generation. The stationary member which is left to the cushion spring represents flywheel and right is the clutch pressure plate. The [F.sub.fc] applied at both these members is responsible for total friction torque generation.

The equation of motion of simplified translational system by applying D Alembert's principle can be written as

"[mathematical expression not reproducible]"

Rearranging above equation

[[m.sub.bs][x.sub.ds] + [k.sub.ds][x.sub.ds] - [k.sub.ds][x.sub.cs] = F] (1)

[[m.sub.cp] [x.sub.cs] + ([k.sub.cs] + [k.sub.ds])[x.sub.cs] - [k.sub.ds][x.sub.ds] = 0] (2)

In above equations x and x represents linear velocity and acceleration. The diaphragm and cushion springs are nonlinear springs. The stiffness characteristic depends on their deflection i.e. [k.sub.ds] = f([x.sub.ds]) & [k.sub.cs] = f([x.sub.cs]).

Above equations can be written in matrix form

"[mathematical expression not reproducible]"

Friction torque is function of cushion spring deflection [x.sub.cs]. Cushion spring deflection is a function of diaphragm spring and diaphragm spring deflection is function of force applied at CRB.

Rotational Model

The rotational system includes the systems of clutch which exhibits rotational degree of freedom and plays role in clutch engagement. Let us assume the subscript e, f, c and v represent engine, flywheel, clutch disc and vehicle respectively. The symbol J, T, [omega] & [theta] represent inertia, torque, angular speed and position.

In figure 5, [T.sub.fc] is the total friction torque generated due to compression of cushion spring and is the sum of friction torque generated between flywheel-friction pad and pressure plate-friction pad. The equation of motion of rotational system can be written as

[([J.sub.e] + [J.sub.f])[[omega].sub.f] = [[T.sub.e] - [T.sub.fc]]] (3)

[([J.sub.c] + [J.sub.v])[[omega].sub.c] = [[T.sub.fc] - [T.sub.v]]] (4)

In above equation [T.sub.v] and [J.sub.v] define vehicle torque due to road loads and vehicle effective inertia at clutch output respectively. The vehicle torque is function of vehicle mass 'w', coefficient of rolling resistance '[gaamma]', road grade '[theta]', coefficient of drag '[C.sub.d]', air density '[rho]', vehicle frontal area 'a', vehicle velocity 'v', tire rolling radius '[r.sub.r]', gearbox ratio '[N.sub.gb]', differential ratio '[N.sub.df] and gravitational constant 'g'. It is given by

[[T.sub.v] = [[r.sub.]/[N.sub.gb][N.sub.df] (mg sin [theta] + [gamma]mg cos [theta] + 1/2[C.sub.d][rho]a[v.sup.2])]]

The equations (3) and (4) hold good when there is slip i.e. ([[omega].sub.f] - [[omega].sub.c]) [not equal to] 0. But Lockup condition of the clutch can be defined when as [[omega].sub.f] = [[omega].sub.c] i.e. slip is zero. Therefore, equation (3) and {4} can be equated and written as

"[mathematical expression not reproducible]" (5)

Above equation can be written for [T.sub.fc]

"[mathematical expression not reproducible]" (6)

From above equation we can define lock up condition of clutch as,

([[omega].sub.f] - [[omega].sub.c]) = 0 &

(Friction torque due to applied force F) [greater than or equal to] [T.sub.fc]

Friction Torque

The single plate dry clutch has two pair of friction surfaces. First between flywheel-clutch disc and second is clutch disc-pressure plate. Both these pair of friction surfaces are responsible for generating total friction torque. Therefore, total friction torque can be given as

[[T.sub.fc] = [[[mu].sub.fcd][F.sub.fc] [R.sub.eff] + [[mu].sub.cdp] [F.sub.fc] [R.sub.eff]]] (7)

In above equation [R.sub.eff] is effective dynamic contact radius and is function of cushion spring compression. Let us assume i, o represents inner and outer physical quantity.

"[mathematical expression not reproducible]" (8)

[R.sub.eff] is maximum when [x.sub.cs] is maximum (i. e. when cushion spring is completely compressed) and it will be minimum when [x.sub.cs] = 0. ft represents friction coefficient and |[[mu].sub.fcd], [[mu].sub.cdp] are friction coefficient between flywheel- clutch disc, clutch disc-pressure plate respectively. Both the friction coefficients are function of temperature (T).

[[mu] = f (T)] (9)

Thermal Model

The friction torque is a function of coefficient of friction and coefficient of friction is function of temperature. Therefore, the temperature effect has been studied in detail taking account of engine and frictional heat.

Frictional Heat Model

The instantaneous heat generation ([H.sub.fc]) because of the friction between rotating bodies (flywheel-clutch disc or clutch disc-pressure plate) due to the application of normal friction force ([F.sub.fc] is given by

[[H.sub.fc] = [[mu][F.sub.fC][R.sub.eff]([[omega].sub.f] - [[omega].sub.c])]] (10)

The part of generated [H.sub.fc] is responsible for rise of the temperature of friction pad, flywheel and pressure plate. Following generalized equation can be used to model this phenomenon for flywheel, friction pad and clutch disc. Let [H.sub.fc1] define the amount of heat used to raise the temperature of body and m, [C.sub.p], [DELTA]T are mass, specific heat and temperature difference between initial and final temperature.

[H.sub.fc1] = m[C.sub.p][[DELTA]T] (11)

The remaining part of the heat is conducted through the body. Figure 6 shows the model of heat flow for frictional heat. It is assumed that the temperature of entire contact surface of both the pair of friction surface is constant. Let [H.sub.fc2] define the amount of heat transferred through conduction. Applying one dimensional Fourier's conduction equation for friction pad, flywheel and pressure plate the generalized governing equation can be written as,

"[mathematical expression not reproducible]" (12)

In equation (12) K, A, [DELTA]T & [DELTA]x represent thermal conductivity, cross sectional area (perpendicular to heat flow), Temperature different between two ends and width (along the direction of heat flow) respectively. Convective heat (Q) transfer occurs from outer side of flywheel and clutch which are exposed to atmosphere, generalized equation can be given as

[Q = -hA'[DELTA]T] (13)

In equation (13) h, A'& [delta]T represent convective heat transfer coefficient, surface area (exposed to atmosphere) and temperature difference between body and atmosphere respectively. These three equations (11), (12) and (13) define the heat flow in the system.

Engine Heat Flow Model

In automotive application, clutch disc forms one pair of friction surfaces with flywheel. Therefore, the temperature of flywheel plays important role in defining torque carrying capacity. The rise in temperature depends on heat rejected by engine and part of heat conducted to flywheel via crankshaft. The remaining heat is carried away by oil by means of convective heat transfer, as it is submerged in oil. Figure 7 shows simple block diagram of engine heat flow model. The arrows indicate direction of heat flow from crank shaft to flywheel.

The rise in temperature of flywheel depends on multiple phenomenon. Therefore, it is difficult to get the accurate value of temperature. The data from engine test bed has been collected and it has been found that maximum rise in temperature of flywheel is about 70 to 80[degrees]C. This temperature data is used to simulate the effect on torque carrying capacity.

EXPERIMENTAL DATA AND SIMULATION

The nonlinear characteristics of diaphragm and cushion springs are obtained based on experimental data for small car clutch application. The experimental data has been used to formulize the fourth order equation which represents its characteristic behavior. Figure 8 shows the release load characteristics of clutch at tip of diaphragm spring (F verses [x.sub.ds]).

In a similar way the nonlinear characteristics of cushion spring is fitted using third order polynomial based on experimental data shown in figure 9. The graph shows clamping load verses cushion spring defiection([F.sub.fc], verses [x.sub.cs]). The clamping load i. e. friction force ([F.sub.fc]) defines the toque carrying capability of clutch.

Figure 10 shows, the thermal characteristics of friction coefficient of clutch disc friction material. It can be seen that the friction coefficient decreases quadratically with the temperature. The friction characteristics shown in figure 10 is included in clutch simulation model for studying the effect of frictional and engine heat on torque carrying capacity.

Based on rotational and translational dynamics of system, model has been built in Simulink and Simscape simulation environment. Model is simulated along with the engine and vehicle dynamics to get the results for real time vehicle condition. Table 1 shows vehicle and single plate dry clutch parameters used for simulation. Clutch engagement is done in such a way that it accelerates the vehicle without engine stall. Figure 11 shows the clutch engagement process of normal driving condition. The slip during engagement is responsible to generate the frictional heat. Figure 12 and 13 shows the simulation result for model with considering the effect of engine heat say case I and results shown in figure 14 and 15 are without engine heat say case II. Figure 12 and 14 shows the temperature verses time for both the pair of friction surfaces for case I and II. Initial temperature of pair one is higher in case I because of engine heat. As engagement starts the temperature of both friction pairs starts increasing. The rise in temperature and rate of rise in temperature of friction pair 2 (friction disc-clutch pressure plate) is higher than friction pair 1 (flywheel-clutch pressure plate) for same normal engagement force in both the case. The frictional heat generated between pair one and pair two is used to raise the temperature of its adjacent masses and to conduct the heat through them. The mass of flywheel is higher as compared to pressure plate, therefore the major part of the heat generated between the pair one is used to raise the temperature of flywheel only. Figure 13 shows, the frictional torque generated during engagement verses time with effect of engine temperature. The initial phase of engagement shows the gradual torque addition from engine to vehicle to avoid the engine stall during engagement, once the slip is zero the clutch is engaged completely. The difference in torque carrying capacity of pair 1 and 2 of friction surface is less in case I than case II, can be seen in figure 13 and 15. This is because the temperature difference between both the pair of friction surface is higher in case II than case I. The percentage decrease in torque carrying capacity for With the same parameters shown in table 1 the model has been simulated for starting the vehicle on grade of 30% from stand still condition. Figure 16, 17 and 18 shows performance characteristics for consecutive clutch engagements. First clutch engagement is to move the vehicle from stand still. As soon as vehicle starts moving, brake is applied to bring the vehicle back to stationary condition and second clutch engagement is done after 66 seconds to move the vehicle again. Intention of this simulation is to know, the capability of multiple consecutive stand still starts on gradient without clutch failure. Figure 17 shows, the temperature profile of the friction surfaces. For first engagement the rise in the temperature of both the surfaces is under tolerable limit but during second consecutive engagement the impact of first engagement along with more slip causes severe effect on clutch temperature and coefficient of friction. To observe the temperature profile during and post engagement the simulation is extended up to 250 seconds. The peak temperature of second friction pair is for short duration. Cooling of these heated surfaces occurs such that second friction surface comes in equilibrium with first and then they cool together to environment temperature. This phenomenon is shown in figure 17. The possibility of the second friction pair burn off is high in second engagement as temperature of surface reaches to 765 K.

CONCLUSIONS

A unique approach of system disintegration has been followed to develop single plate dry clutch. The model has been built by understanding detailed functionality of subsystems and their participation during engagement. The separate models of translational and rotational systems have helped to understand detailed dynamics and interaction within the subsystems. The individual friction model between flywheel-clutch disc and clutch disc-clutch pressure plate along with engine and frictional heat have helped in understanding the variation in torque carrying capacity. The simulation for normal and worst driving conditions are performed. The decrease in torque carrying capacity is tolerable in normal driving condition but for 30% gradient multiple starts cause the clutch to overheat.

REFERENCES

[1.] (John) Willyard, J., "Heavy Duty, Large Single Plate Diaphragm Spring, Dry Clutches," SAE Technical Paper 892476, 1989, doi:10.4271/892476.

[2.] Huang, H., Di, D., Chu, Y., and Guehmann, C., "Model-Based Optimization for an AMT Clutch Control during the Vehicle Starting," SAE Int. J. Passeng. Cars - Electron. Electr. Syst. 8(1):90-98, 2015, doi:10.4271/2015-01-0161.

[3.] Gong Y. P. and Bian X. J., "Virtual Simulation Application to Clutch Design System," 2010 Third International Conference on Information and Computing, Wuxi, Jiang Su, 2010, pp. 162-165, doi:10.1109/ICIC.2010.225

[4.] Vasca F., Iannelli L., Senatore A. and Reale G., "Torque Transmissibility Assessment for Automotive Dry-Clutch Engagement," IEEE/ASME Transactions on Mechatronics, vol. 16, no. 3, pp. 564-573, June 2011, doi:10.1109/TMECH.2010.2047509

[5.] Dutta Trinoy and Baruah Lopamudra, "Engagement Model of Dry Friction Clutch with Diaphragm Spring," International Journal of Engineering Research, Volume No.3, Issue No. 11, pp: 704-710, Nov. 2014, ISSN:2319-6890,2347-5013

[6.] Zhang Hailiang, Yu Zhuoping, Zhong Zaimin and Wu Bonian, et al. "Modelica based modeling of automotive transmission," Transportation Electrification Asia-Pacific (ITEC Asia-Pacific), 2014 IEEE Conference and Expo, Beijing, 2014, pp. 1-5.doi:10.1109/ITEC-AP2014.6940686

[7.] Bachinger M., Stolz M. and Horn M., "Fixed step clutch modeling and simulation for automotive real-time applications," 2014 American Control Conference, Portland, OR, 2014, pp. 2593-2599.doi: 10.1109/ACC.2014.6858933

[8.] Myklebust A. and Eriksson L., "Torque model with fast and slow temperature dynamics of a slipping dry clutch," 2012 IEEE Vehicle Power and Propulsion Conference, Seoul, 2012, pp. 851-856, doi:10.1109/VPPC.2012.6422728

[9.] Hoic, M., Herold, Z., Kranjcevic, N., Deur, J. et al., "Experimental Characterization and Modeling of Dry Dual Clutch Thermal Expansion Effects," SAE Int. J. Passeng. Cars - Mech. Syst. 6(2):775-785, 2013, doi:10.4271/2013-01-0818.

[10.] Myklebust A. and Eriksson L., "Modeling, Observability, and Estimation of Thermal Effects and Aging on Transmitted Torque in a Heavy Duty Truck with a Dry Clutch," IEEE/ASME Transactions on Mechatronics, vol. 20, no. 1, pp. 61-72, Feb. 2015, doi:10.1109/TMECH.2014.2303859

ACKNOWLEDGMENTS

I would like to express my deep and sincere gratitude to TATA Motors, Pune for giving me this opportunity. I would also like to thank Mr. Prasanta Sarkar for giving me opportunity to work on this project. Special thanks to Mr. Anurag Mishra for his support. I am extremely grateful to Advanced Engineering team for their support during entire work.

DEFINITIONS/ABBREVIATIONS

MT - Manual transmission

AMT - Automated manual Transmission.

CRB - Clutch release bearing

I/P & O/P - Input & output

Amar Penta, Prasad Warule, Sanjay Patel, and Lohit Dhamija

Tata Motors, Ltd.

CONTACT INFORMATION

Amar Penta

Manager

Advanced Engineering, ERC, Tata Motors, Pune

APP664576@tatamotors.com

pentaamar@gmail. com

+91-9823267451
Table 1. Vehicle and clutch parameters for simulation.

m           1000 kg              [gamma]      0.01256
[theta]     8[degrees]           [C.sub.d]    0.33
[r.sub.r]   0.238 m              [rho]        1.225 kg/[m.sup.3]
[N.sub.gb]  2.44                 a            2.7 [m.sup.2]
[N.sub.df]  4.9                  g            9.81 m/[s.sup.2]
[J.sub.e]   0.02151 kg[m.sup.2]  [R.sub.i]    49 mm
[I.sub.f]   0.03962 kg[m.sup.2]  [R.sub.o]    77 mm
[J.sub.c]   0.00093 kg[m.sup.2]  [m.sub.cp]   0.892 kg
[J.sub.v]   0.45 kg[m.sup.2]     [m.sub.bs]   0.198 kg
Thermal conductivity of flywheel, pressure    63,60 & 0.34
plate & friction pad                          W/mk
Convective heat transfer coefficient of air   5 W/[m.sup.2]k
Thickness of friction pad, pressure plate &   2.8,9 & 35 mm
flywheel
Ambient temperature of casing environment     40 [degrees]C
Mass of flywheel, pressure plate and          5.67, 0.91 &
friction pad                                  0.32 kg
Specific heat of flywheel, pressure plate     110, 134 & 1632
and friction pad                              J / kgK
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Author:Penta, Amar; Warule, Prasad; Patel, Sanjay; Dhamija, Lohit
Publication:SAE International Journal of Passenger Cars - Mechanical Systems
Article Type:Report
Date:Apr 1, 2017
Words:3789
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