Effective utilizing axial nonlinear characteristics of diaphragm spring and waveform plate to enhance breakaway performances of a clutch

Mechanical Systems and Signal Processing xxx (2018) xxx–xxx

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Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/ymssp

Effective utilizing axial nonlinear characteristics of diaphragm spring and waveform plate to enhance breakaway performances of a clutch Wen-Bin Shangguan a,⇑, Xue-Lai Liu a, Subhash Rakheja a,⇑,1, Qiufeng Hou b a b

School of Mechanical and Automotive Engineering, South China University of Technology, Guangzhou 510641, China Ningbo Hongxie Clutch Co., Ltd., Ningbo 315807, China

a r t i c l e

i n f o

Article history: Received 27 September 2017 Received in revised form 20 April 2018 Accepted 29 May 2018 Available online xxxx Keywords: Clutch Breakaway of a clutch Clutch pedal ride Deformation of diaphragm spring and waveform plate

a b s t r a c t For general-purpose passenger cars with a manual transmission, breakaway performances of a clutch have great effects on ride of clutch operation system. Breakaway performances of a clutch is defined as force versus displacement applied at clutch fingers, and it is nonlinear decided by nonlinear characteristics of diaphragm spring and waveform plate. The relationship between characteristics of clutch breakaway and the ride of clutch operation system is investigated in this paper. A finite element (FE) model is proposed and used to estimate characteristics of clutch breakaway, and the calculated results are compared with experiments and these obtained by simple analytical method that deformation of a diaphragm spring is ignored. In the FE model, the nonlinear characteristics of diaphragm spring and waveform plate at axis direction are included. The influence of nonlinear characteristics of waveform plate and diaphragm spring on clutch breakaway is analyzed using the proposed model and the method for how to utilize nonlinear characteristics of diaphragm spring and waveform plate to enhance breakaway performances of a clutch is proposed. An example is given to demonstrate the effectiveness of the developed in the paper. Ó 2018 Elsevier Ltd. All rights reserved.

1. Introduction In this section, a brief introduction to the structure of a clutch operation system and its performance requirements are presented firstly, then literature review regarding the research status of a clutch operating system and objectives of this research are given; thirdly, factures influencing ride performances of clutch pedal are analyzed and finally motivations of this paper are presented. 1.1. Clutch operation system and performance requirements For general-purpose passenger cars, clutch assembly and pedal system collaborate to achieve clutch disengagement or engagement as shown in Fig. 1. The force applied to the clutch pedal (6) is transferred to the clutch release bearing (7) through the pedal lever (5) and two hydraulic cylinder (4). The pedal system including clutch pedal (6), pedal lever (5) ⇑ Corresponding authors. 1

E-mail address: [email protected] (W.-B. Shangguan). Mechanical & Industrial Engineering, Concordia University, Montreal, Canada (on leave).

https://doi.org/10.1016/j.ymssp.2018.05.060 0888-3270/Ó 2018 Elsevier Ltd. All rights reserved.

Please cite this article in press as: W.-B. Shangguan et al., Effective utilizing axial nonlinear characteristics of diaphragm spring and waveform plate to enhance breakaway performances of a clutch, Mech. Syst. Signal Process. (2018), https://doi.org/10.1016/j. ymssp.2018.05.060

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7

9 8

1

2

3

4

6

5

Fig. 1. Scheme of clutch assembly and pedal system. 1- waveform plate, 2- diaphragm spring, 3-release lever, 4-hydraulic cylinder, 5-pedal lever, 6-clutch pedal, 7-clutch release bearing, 8-hydraulic tubing, 9-assistant spring.

and assistant spring (9). The clutch lever is used to increase the force applied at the clutch pedal, and the force is then transferred to clutch release bearing (7) and the resulted deformation of diaphragm spring makes the pressure plate contacting with the large end of the diaphragm spring moves rightwards or leftwards. The movement of pressure plate rightwards or leftwards makes a clutch to be disengaged or engaged. The ride performance of a clutch pedal is determined by the relation between the force applied at clutch pedal and the resulted displacement. Usually, the shape and configuration of pedal lever, hydraulic cylinder and release lever are hard to be changed since the limited space in engine compartment or car cabin. Therefore, the best and possible way to adjust ride performance of a clutch pedal is to optimize clutch breakaway property. A clutch breakaway property is defined as a relation between the force applied at the release finger and the resulted displacement. The release finger is the small end of a diaphragm spring, or the contacting face between release bearing and diaphragm spring as shown in Fig. 1. It is seen from Fig. 1 that clutch release property is determined by the deformation of diaphragm spring (2) and waveform plate (1). The relation of force versus displacement for diaphragm spring (2) and waveform plate (1) is nonlinear, and usually is hard to adjust to meet the required pedal ride performance. Fig. 2 shows the measured force versus displacement curve for a generic clutch pedal and is used for evaluating clutch pedal ride performances. It is seen that hysteresis exists between the separation and engagement process, and this is generated by the loss of the hydraulic system and the dry friction between the pressure plate and diagram spring or waveform

125 T

Force [N]

100 M

75

Do

t

en

m

e ag

Q

ng

se Di

50

25 P 0

Engagement

25

50

75

100

125

150

Travel [mm] Fig. 2. Measured force versus displacement at the clutch pedal of a passenger car.

Please cite this article in press as: W.-B. Shangguan et al., Effective utilizing axial nonlinear characteristics of diaphragm spring and waveform plate to enhance breakaway performances of a clutch, Mech. Syst. Signal Process. (2018), https://doi.org/10.1016/j. ymssp.2018.05.060

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plate [1–6]. For a passenger car, requirements for ride of a clutch pedal are listed as follows. The data in the following requirements is only used as examples and not as general: (1) The range of maximum pedal force at point M is usually set to 90 N–120 N. (2) During disengagement of a clutch, the force difference range of pedal force between the peak point M and valley point Q is usually set to 10 N–30 N to guarantee the sense of ‘‘inhalation” for a driver. If it is too small, a driver has no feeling of inhalation at the process of depressing the pedal. (3) The range of the maximum pedal displacement (from point P to point T) should be within 120 mm to 140 mm to meet human ergonomics. The relation of force versus displacement for a clutch pedal of a passenger shown in Fig. 2 is determined by the clutch release property, pedal level, hydraulic assisted system and assistant spring of the clutch operation system. The characteristic of force versus displacement of a clutch pedal is an important index for evaluating operating comfort of a clutch. For example, too large pedal force at point M shown in Fig. 2 causes the driver to be tired in the multiple repeating operations, but little pedal force makes a driver has little or no feelings if a driver pedals the clutch pedal. The nonlinearities in clutch pedal system (also called operation system) influencing clutch breakaway performance and clutch pedal ride are the nonlinearities of a diaphragm spring of the large end and a waveform plate of the axial direction. These nonlinearity properties and the combination way have great influence to the clutch breakaway and clutch ride performances. To enhance clutch pedal performance, the working point at the nonlinear force versus displacement of diaphragm spring and the adjacent stiffness of working point, and the curvature of force versus displacement for waveform plate should be optimized to meet the requirements as discussed in Fig. 2.

1.2. Literature review and objectives Published papers investigating clutch and its operation system are divided into three categories: study of diaphragm spring, waveform plate and clutch operation system. For the study of diaphragm spring, relationship between the transmitted torque of a clutch and the axial stiffness of diaphragm spring, working temperature and wear of friction disk are measured using a test bench in Refs. [7–9]. Assuming deformation of diaphragm spring and waveform plate is linear, clutch transmitted torque with properties of diaphragm spring and waveform plate is investigated in Ref. [10]. A method for modeling clutch thermal expansion is presented in Ref. [11], and the effects of thermal expansion on clutch torque is investigated using simulation and experiment. Since waveform plate influences clutch torque transmission from engine to driveline and the force versus displacement of a waveform plate in axial direction depends on its shape and sizes, a waveform plate is discretized with several parameters and sensitivities of each parameter on force versus displacement is investigated using FE analysis in Refs. [12,13]. Also the effects of clutch working temperature on force versus displacement of a waveform plate at conventional clutch and ceramic clutch is investigated in Refs. [14,15], and it is concluded repetitive clutch engagements led to an increase in the temperature of the clutch facings. This temperature rise then results in an increase in the temperature increase of the waveform plate, and lead to change of material properties of a waveform plate and further change load versus deflection curve of a waveform plate. The relations between clutch torque transmissibility and sizes of friction pads, nonlinear force versus displacement property of waveform plate and slip-speed-dependent coefficient of friction are studied in Ref. [16]. The main purpose of this study is to build a model for AMT (Automatic Manual Transmission) control strategy, but deformation of diaphragm spring is ignored in this study. For the study of ride of clutch operation system, the effects of cylinder diameter and the displacement of hydraulic cylinder on ride of clutch operating system are studied in Ref. [16], but the influence of nonlinear properties of the diaphragm spring and the waveform plate of a clutch on clutch pedal ride are not considered in Ref. [16]. The clutch and the clutch actuation system are two important sub-systems for clutch operation system. A dynamic model of hysteresis loop of one clutch actuation system is proposed in Ref. [17,18], and hydraulic system is only considered in the system to simulate hysteresis loop. The influence of the hysteresis loops in the clutch actuation system on the clutch pedal ride in investigated. The influences of friction between a piston and hydraulic cylinder on clutch pedal ride is studied, and a method for estimating the friction coefficient versus cylinder pressure is proposed in Ref. [19]. One way to change clutch pedal ride is to optimize assistant spring (part 9 in Fig. 1), and the methods for estimating optimized stiffness of the spring can be found in Ref. [20,21]. Based on the above reviewing of the existing references, it is concluded that the characteristics of clutch breakaway depend on the characteristics of diaphragm spring and waveform plate and have great influences on clutch pedal ride. The clutch pedal ride also depends on clutch actuation system, where the properties of two hydraulic cylinders (Master cylinder and slave cylinder) and the friction between a piston and hydraulic cylinder are two important factors to influence clutch pedal ride, but they are difficult to adjust or control. The easiest way to adjust clutch pedal ride is to change the breakaway performance depending on nonlinear characteristics of diaphragm spring and waveform plate if configuration of pedal level and stiffness of assistant spring are assumed. But how to design nonlinear characteristics of diaphragm spring and waveform plate to enhance clutch breakaway performance is seldom studied and the investigation of influence elastic deformation of diagram spring on clutch breakaway is not found in the published papers to the best of author’s knowledge. Please cite this article in press as: W.-B. Shangguan et al., Effective utilizing axial nonlinear characteristics of diaphragm spring and waveform plate to enhance breakaway performances of a clutch, Mech. Syst. Signal Process. (2018), https://doi.org/10.1016/j. ymssp.2018.05.060

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Therefore, the objectives of this paper are to the study of relationship between clutch component nonlinearities and elastic deformation and clutch breakaway characteristics, and to develop a method for utilizing nonlinear breakaway property of a clutch to enhance clutch pedal ride performance. 1.3. Factors influencing ride performances of clutch pedal It is seen from Figs. 1 and 2 that relation of force versus displacement of clutch pedal is determined by clutch breakaway performance and the lever ratio of pedal lever and hydraulic cylinder. Due to the limitation of install space at car cabin or engine compartment, the adjustment of pedal lever and hydraulic cylinder is limited. Therefore, the frequently used way to change pedal force versus displacement curve is to adjust clutch breakaway performance and nonlinear force versus displacement relation of waveform plate. The measured force versus displacement curves for the diaphragm spring at large end and wave plate of a generic clutch shown that the force versus displacement curves have nonlinear characteristics. Fig. 3 shows the measured clutch breakaway property of a clutch using the measurement equipment and the methods presented in Section 3. Since the dry friction damping existed, the hysteresis is generated between the disengagement process and the engagement process. The clutch breakaway property is determined by requirements of clutch pedal rid as mentioned in Section 1.1. The point A and the point B in Fig. 3 are defined as the disengagement point with maximum disengagement force and the disengagement point with the maximum disengagement displacement, respectively. The point C is defined as the disengagement points with minimum disengagement force. The breakaway characteristics of a clutch have a great effect on ride performance of a clutch pedal. The maximum separation force, the force at point A in Fig. 3, determines the peak force value of the clutch pedal force. The force difference between the force at peak and valley point at Fig. 3, point A and point C, determines the force difference of a clutch pedal, and the maximum value of disengagement displacement of a clutch determines the maximum value of pedal displacement. For clutch supplies, the possible way to improve ride characteristics of clutch pedal is to adjust clutch breakaway property. The breakaway property of a clutch is determined by axial nonlinear characteristics of the diaphragm spring and waveform plate at one clutch, so it is necessary to study the relationship between these nonlinear characteristics and clutch breakaway characteristic. 1.4. Motivations From the above statements, it is concluded that breakaway property of a clutch has great effects on the ride performance of a clutch pedal, and the breakaway property of a clutch is nonlinear and it is determined by the nonlinear characteristics of the diaphragm spring and waveform plate. Therefore, the motivations of this paper are to study how to utilize effectively nonlinear breakaway property of a clutch to enhance clutch ride performance, and to study relationship between nonlinear property of a clutch breakaway and the nonlinear characteristics of the diaphragm spring and waveform plate.

2. Analysis of clutch breakaway property During the process of clutch disengagement and engagement, the variation of the clutch release force depends on the axial force versus displacement characteristics of the diaphragm spring and waveform plate, as well as clutch traveling distance. Fig. 4(a) shows sketch of a clutch. A driven disk of a clutch is a component assembled by riveting two friction pads with waveform plate. In addition, there are some linear springs inside a driven disk to provide torsional stiffness for a clutch.

A

1200 Disengagementprocess Disengagement process

C

Force(N)

1000

B

800 Engagementprocess Engagement process

600 400 Maximumdisengagementdisplacement Maximum disengagement displacement

200 0 0

2

4 Displacement(mm)

6

8

Fig. 3. Measured release force versus displacement of a clutch at the release finger (small end).

Please cite this article in press as: W.-B. Shangguan et al., Effective utilizing axial nonlinear characteristics of diaphragm spring and waveform plate to enhance breakaway performances of a clutch, Mech. Syst. Signal Process. (2018), https://doi.org/10.1016/j. ymssp.2018.05.060

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6

5

7

l1

FT

4 3

l2 10

2

FA 1 9

x0

8

x1

(a) clutch sketch

FB x2

FT l1

Δ Pivot point

Elastic deformation

FB

(b) Zoom of waveform plate, driven disk and pressure plate

x0' ' x'0

l2

FA

x0 (c) displacement and force of diaphragm spring

Fig. 4. Scheme of engagement and separation for a clutch. 1-crankshaft, 2-flywheel, 3-friction pad, 4-waveform plate, 5-clutch housing, 6-pressure plate, 7diaphragm spring, 8-primary shaft of a transmission, 9-clutch hub, 10-clutch release bearing.

The x0 and x1 in Fig. 4(c) are the displacements of clutch releasing bearing and the displacement at the large end of a diaphragm spring, respectively. The traveling distance of a pressure plate equals to x1 since the pressure plate always contact with the large end of a diagrammatic spring. The D and x2 in Fig. 4(b) are displacement between the pressure plate and friction pad, and the deformation of a waveform plate, respectively. The deformation of a waveform plate is defined as the thickness difference between the thickness of the plate during clutch releasing process and the thickness at status of clutch assembled. The F A is the force applied to the release finger of a diaphragm spring from release bearing, and the F B is a reaction force of the force applied to waveform plate from pressure plate. If elastic deformation of a diaphragm spring is not considered, the resultant force applied to the contact point between the pressure plate and the larger end of a diaphragm can be calculated using

F T ¼ ðF A l2 þ F B l1 Þ=l1

ð1Þ

where l1 and l2 are distances between the pivot point and the larger end and the finger of a diaphragm spring, respectively. Please cite this article in press as: W.-B. Shangguan et al., Effective utilizing axial nonlinear characteristics of diaphragm spring and waveform plate to enhance breakaway performances of a clutch, Mech. Syst. Signal Process. (2018), https://doi.org/10.1016/j. ymssp.2018.05.060

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The traveling distance of the finger (x0 ) in Fig. 4(c) consists of two parts: x00 and x000 . The x00 is the rigid-body displacement of a diaphragm spring and is estimated using

x00 ¼ ðl2 =l1 Þx1

ð2Þ

where x000 is the elastic deformation of a diaphragm spring and can be calculated using FEA. When a driver wants to disengage a clutch, a force F A is applied to the release finger from release bearing, then the pressure plate will move rightwards and the deformation of x2 generates. If the D in Fig. 4(b) is greater than zero, then a clutch is disengaged and no torque is transmitted from engine to transmission. The ride of a clutch pedal operation system mainly depends on the clutch breakaway properties if the location of clutch pedal, torsional stiffness of assistant spring and hydraulic system are fixed. The clutch breakaway property is defined as the relationship between the force applied the release finger of a diaphragm spring (F A in Fig. 4(c)) and the corresponding displacement (x0 in Fig. 4(c)). Fig. 5 shows the breakaway characteristic curve of a clutch with and without waveform plate. It is seen from Fig. 5(a) that if a clutch has no waveform plate and after disengagement (point Z), the force applied at release finger drops dramatically. On the contrary, if a clutch has a waveform plate, the force changes smoothly. This means that during clutch engagement, the maximum force applied at the pedal is greater if a clutch has no waveform plate. As an example shown in Fig. 5(a), the maximum release force is reduced by about 350 N when a waveform plate existed in the clutch. The relationship between the displacement at the larger end of a diaphragm spring (X1) and the displacement at the release finger (X0) of a diaphragm spring is shown in Fig. 5(b). It is demonstrated that no displacement occurs at the release finger before clutch disengagement if a clutch has no waveform plate, while the displacement at the release finger increase smoothly if a waveform plate existed. The pressure plate moves rightwards only if the X1 is greater than zero. In the design of a clutch, the relation of force versus displacement at the large end of a diaphragm spring can be calculated using A-L formulae [10], and the deformation of a waveform plate can be estimated using FEA. To determine relation of the force applied at a clutch pedal and the generated displacement at the pedal, the relation of force versus displacement at the release finger of a diaphragm spring must be known. Since the elastic deformation of diaphragm spring, the force and displacement at the release finger of a diaphragm spring cannot be estimated using lever ratio with l1 and l2 as shown in Fig. 4 (c). The force versus displacement relation at the large end of a diaphragm spring, FT versus X1, and the relation of the deformation and the force applied at the waveform plate, X1 and FB, are shown in Fig. 6. At the status of clutch assembly, there is a pre-force (FB) applied at the waveform plate and the deformation of the plate is assumed to be zero. In the process of clutch disengagement, the resultant force applied at the larger end of a diaphragm spring equals to (FT  FB) as shown in Figs. 4(c) and 6. In determination of clutch breakaway properties, the resultant force (FT  FB) versus X2 must be known. The X2 equal to X1 if the clutch is not totally disengaged. It is seen from Fig. 6 that the relation of (FT  FB) versus X2 is nonlinear and depends on axial characteristics of waveform plate and characteristics of diaphragm spring. 3. Methods for measuring clutch breakaway performance, waveform plate deformation and force versus displacement of clutch pedal The scheme of the equipment and spot for measuring clutch breakaway performance are shown in Fig. 7. A clutch with or without waveform plate is installed firstly at the measurement equipment with the support ring is fixed on the equipment. An actuator with force and displacement sensors (1) is applied to the release finger of the diaphragm spring (2), and a

1400

Z

without waveform plate with waveform plate

1.5

1200 X1/mm

FA/N

1000 800 600

0.5

400

without waveform plate with waveform plate

200 0

1.0

0.0 0

1

2

3

4

5

6

X0/mm

(a) Clutch breakaway

7

8

Z

0

1

2

3

4

5

6

7

8

X0/mm

(b) Pressure plate displacement

Fig. 5. Clutch release force versus release displacement and pressure plate displacement curve for a clutch with/without waveform plate.

Please cite this article in press as: W.-B. Shangguan et al., Effective utilizing axial nonlinear characteristics of diaphragm spring and waveform plate to enhance breakaway performances of a clutch, Mech. Syst. Signal Process. (2018), https://doi.org/10.1016/j. ymssp.2018.05.060

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0

X2/mm 1

7

2

8000

FT FB

6000

FT/N

FT

FB

Work point

4000

2000

0 0

1

2

3 4 X1/mm

5

6

Fig. 6. Relationship between force and displacement at the large end of a diaphragm spring.

Pressing force 1 2 9 3 8 4 5

7

6

(a) Scheme of the equipment

(b) Measurement spot

Fig. 7. Measurement of clutch breakaway with and without waveform plate. 1-Actuator with force and displacement sensor, 2-diaphragm spring, 3-displacement sensor, 4-pressure plate, 5-waveform plate, 6-fixture end, 7-force sensor, 8-friction pad, 9-pivot ring.

displacement sensor (3) and a force sensor (7) are used to measure the displacement and force at the large end of the diaphragm spring (2), respectively. In measuring breakaway characteristic of a clutch with waveform plate, the friction pad (8) and waveform plate (5) are installed between the pressure plate (4) and the fixture end (6). A force sensor (7) is connected with the fixed end to measure the force applied at the large end of diaphragm spring. With the equipment shown in Fig. 7, the breakaway performance (force versus displacement at the release finger of a clutch) and the force versus displacement at the large end of a clutch are obtained. The scheme of the equipment and the test spot for measuring clutch pedal ride (clutch force versus displacement) are shown in Fig. 8. The motor (1) substitutes for an engine and its rotational speed can be adjusted. In the equipment, the clutch (2) is connected with the gearbox (4), and the gearbox (4) is then connected with a brake (5). The purpose of using gearbox and brake is to measure clutch pedal ride at different gears and different drag torques. Fig. 9 shows the locations of force and displacement sensors for measuring the force versus displacement of clutch pedal on the measurement spot. During the measurement, a force is applied to the force sensor installed on the clutch pedal, and the clutch pedal is then moved forwards and backwards. The applied force to clutch pedal and the resultant displacement are then acquiszed. 4. Methods for calculating clutch breakaway performance In order to study the influence of the elastic deformation of diaphragm spring on clutch breakaway performance, two methods are proposed to calculate clutch breakaway. The first one is Finite Element Analysis (FEA) method. A FEA model for estimating clutch breakaway is proposed regarding diaphragm spring and waveform plate as flexible body, and the model is validated by comparing simulated and measured clutch breakaway performance. The second one is an analytical method regarding diaphragm spring as a rigid body, and it is simple and easy to be used for calculating. Please cite this article in press as: W.-B. Shangguan et al., Effective utilizing axial nonlinear characteristics of diaphragm spring and waveform plate to enhance breakaway performances of a clutch, Mech. Syst. Signal Process. (2018), https://doi.org/10.1016/j. ymssp.2018.05.060

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1

2

3

4

5

6

(a) Scheme of the equipment

(b) Test bench Fig. 8. Structure of the equipment for measuring the force versus displacement of clutch pedal. 1-motor, 2-clutch, 3-Clutch pedal system, 4-gearbox, 5-braker, 6-Installation platform.

Displacement sensor

Force sensor

Fig. 9. Test bench for measuring pedal force versus displacement relations.

In the proposed FEA model, the diaphragm spring and waveform plate in a clutch are regarded as flexible bodies. Both the diaphragm spring and waveform plate have been meshed with shell elements according to its thin thickness (less than 3 mm). Comparing with the solid element, shell elements takes up less computing resources and the process of calculation is easier to converge. The pressure plate and friction pad are regard as rigid bodies. The material constants for diaphragm spring and waveform plate are shown in Table 1. The boundary conditions of the proposed FEA model are shown in Fig. 10. It is seen that the friction pad (1) and pivot ring (4) are regarded as fixed constraint. The friction ties are assumed between those contacting daces: friction pad and waveform

Please cite this article in press as: W.-B. Shangguan et al., Effective utilizing axial nonlinear characteristics of diaphragm spring and waveform plate to enhance breakaway performances of a clutch, Mech. Syst. Signal Process. (2018), https://doi.org/10.1016/j. ymssp.2018.05.060

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W.-B. Shangguan et al. / Mechanical Systems and Signal Processing xxx (2018) xxx–xxx Table 1 Material constants. Poisson ration

Density ðkg=m3 Þ

Diaphragm spring

5

2:06  10

0.290

7:74  103

Waveform plate

2:11  105

0.288

7:82  103

Components

Elastic modulus (MPa)

6

5

1 4

3 2

Fig. 10. Boundary conditions of FEA model for estimating breakaway of a clutch. 1, 4-fix constraint, 2, 3, 5, 6-friction ties.

plate (6), friction pad and pressure plate (2), pressure plate and diaphragm spring (5), diaphragm spring and release bearing (3). The coefficient of friction is equal to 0.3. The loads applied at the FEA model are shown in Fig. 11. The loads at different status of a clutch are given as follows. (1) Load at the installing status of a clutch: A pre-install force FB or displacement is applied to the friction disk as shown in Fig. 11(a), and its makes friction disk consisting of two friction pads and a waveform plate contacts with pressure plate and pressure contacts with large end of the diaphragm spring. The pre-install force is determined by making the diaphragm spring to its working point. The working point is defined as a point located at the force versus displacement at the large end of a diaphragm spring if a clutch is assembled. (2) Load at the disengagement of a clutch: A force (FA) or displacement is applied gradually at the release bearing for making a clutch to be disengaged. The force makes the diaphragm spring to be deformed, the pressure plate to move rightwards and the waveform plate to its un-deformed status. (3) Load at the engagement of a clutch: A force FC opposing to FA or displacement is applied gradually at the release bearing for making a clutch to be engaged. The force makes the deformed diaphragm spring return to the un-deformed status, the pressure plate and the waveform plate to move leftwards. The contact force between the release bearing and diaphragm spring, friction pad and wave plate can be obtained from the FAE model. The calculated deformation of the diaphragm by FEA is shown in Fig. 12, and the deformations of diaphragm at different status can be easily measured using the FEA model. If deformation of the diaphragm spring of a clutch is ignored (x000 in Fig. 4 equal to 0), the breakaway property can be estimated using the following simple analytical method. The traveling distance of the finger (x0 ) can be calculated

x0 ¼ x00 ¼

l2 x1 l1

ð3Þ

The force applied to the release finger (F A ) can be calculated

F A ¼ ðF T  F B Þl1 =l2

ð4Þ

Please cite this article in press as: W.-B. Shangguan et al., Effective utilizing axial nonlinear characteristics of diaphragm spring and waveform plate to enhance breakaway performances of a clutch, Mech. Syst. Signal Process. (2018), https://doi.org/10.1016/j. ymssp.2018.05.060

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FB

FA (a) Load for making diaphragm spring to its working point

(b) Load for making clutch to be disengaged

Fc (c) Load for making clutch to be engaged

Fig. 11. Loads applied for making diaphragm spring to its working point, clutch disengagement and engagement (dotted line: initial status, solid line: finial status).

(a) free status

(b) engagement status

(c) disengagement status Fig. 12. Deformation of the diaphragm.

The clutch breakaway property obtained by measurement and calculation using FEA and simple analytical method is shown in Fig. 13, and some key parameters estimated from the breakaway property are given in Table 2 and the relative tolerance results are given in Table 3. It is see that the property obtained by FEA method is much more close to the measurement, which validates the FEA mode and calculation methods. It is also shown that there is a bigger difference between the measurement and calculation if the deformation of diaphragm spring is not considered in estimating clutch break property. 5. Influence of the diaphragm spring and the waveform plate on breakaway performance of a clutch The diaphragm spring and waveform plate are two important components in a clutch to change its breakaway property. Since a diaphragm spring has nonlinear characteristics of displacement versus force at the larger end of the diaphragm spring, it can keep the clamp force applied to pressure plate to be constant after the friction plate is worn [9]. The relation between force and displacement of a waveform plate at clutch axis direction is also nonlinear, and this nonlinear characteristic can reduce vehicle shock during clutch engagement. Both the diaphragm spring and the waveform plate of a clutch have great influence to clutch breakaway property. Please cite this article in press as: W.-B. Shangguan et al., Effective utilizing axial nonlinear characteristics of diaphragm spring and waveform plate to enhance breakaway performances of a clutch, Mech. Syst. Signal Process. (2018), https://doi.org/10.1016/j. ymssp.2018.05.060

W.-B. Shangguan et al. / Mechanical Systems and Signal Processing xxx (2018) xxx–xxx

11

1200 1000

FA/N

800 600 400

Mearsured Simple analytical method FEA

200 0 0

2

4 X0/mm

6

8

Fig. 13. Clutch breakaway property.

Table 2 The measured and estimated clutch breakaway characteristics. Method

Maximum separation force (N)

Force difference between peak and valley point (N)

Maximum separation displacement(mm)

Measured Simple analytical method FEA method

1140 1264 1097

85 94 87

7.1 6.2 7.3

Table 3 The relative tolerances of the parameters of characterizing clutch breakaway. Parameters

Maximum separation force Force difference between peak and valley point Maximum separation

Relative tolerances compared with experiments (%) By Simple analytical method

By FEA

10.8% 10.6% 12.7%

3.8% 2.4% 2.8%

5.1. Working point of diaphragm spring The working point of diaphragm spring is defined as a point at a curve of force versus displacement at the large end of diaphragm spring of when a clutch is assembled. The working point determines the clamp force applied to pressure plate from diaphragm spring and the maximum torque the clutch transmitted. Fig. 14 shows force versus displacement at the large end of a diaphragm, where B2 is the baseline working point, and B1 and B3 are the working points with small and large deformations of diaphragm spring, respectively. The displacements and forces for working point B1, B2 and B3 are (3.8 mm, 5700 N), (4.0 mm, 5450 N), and (4.2 mm, 5050 N), respectively. The influence of working point of diaphragm spring on clutch breakaway characteristic is study using the proposed FEA model in Section 4. Fig. 15 shows the calculated breakaway characteristics of a clutch if diaphragm spring works at different working point. The calculated maximum disengagement force, the force difference at peak and valley point and the maximum disengagement displacement are listed at Table 4. It is concluded from Fig. 15 and Table 4 that a working point with smaller displacement generates the large maximum disengagement force and disengagement displacement, and increases force difference at the peak and the valley point. 5.2. Axis deformation of waveform plate The force versus displacement of a waveform plate has nonlinear characteristics and to study the influence trend of the nonlinear characteristics of waveform plate on the clutch breakaway, the diaphragm spring is assumed as a rigid body in this section. The nonlinear force (FB) versus displacement (X2) of a waveform plate in Fig. 4 is simplified as piece-wise linear with three stages as shown in Fig. 16. The stiffness (k1, k2, k3) and end stops (a, b, c) are determined using lease squares methods with the measured nonlinear force versus displacement of a waveform plate. The nonlinear force (FT) versus displacement (X1) of a diaphragm spring at its large end and at its working range is also simplified as piecewise linear with three stages as Please cite this article in press as: W.-B. Shangguan et al., Effective utilizing axial nonlinear characteristics of diaphragm spring and waveform plate to enhance breakaway performances of a clutch, Mech. Syst. Signal Process. (2018), https://doi.org/10.1016/j. ymssp.2018.05.060

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8000

B1

7000

B2

6000

B3

FT/N

5000 4000 3000 2000 1000 0 0

1

2

3 X1/mm

4

5

6

Fig. 14. Force versus displacement at the large end of a diaphragm spring.

A

1200 1000

B

FA/N

800 600

B3 B2 B1

400 200 0 0

2

4 X0/mm

6

8

Fig. 15. Breakaway characteristic of a clutch under different working point of the diaphragm spring.

Table 4 Influence of working position on clutch breakaway characteristic. Working point

Maximum disengagement force (N)

Force difference at peak and valley point (N)

Maximum disengagement displacement (mm)

B1 B2 B3

1168 1097 1046

156 85 34

8.1 7.3 6.4

Force and displacement curve Load-deflection curve

k3

FB /N

Nonlinear force versus displacement fitted piecewise linear at three stages

k2

k1

a X2 /mm

b

c

Fig. 16. Force versus displacement of a waveform plate: nonlinear and fitted piece wise linear at three stages.

Please cite this article in press as: W.-B. Shangguan et al., Effective utilizing axial nonlinear characteristics of diaphragm spring and waveform plate to enhance breakaway performances of a clutch, Mech. Syst. Signal Process. (2018), https://doi.org/10.1016/j. ymssp.2018.05.060

W.-B. Shangguan et al. / Mechanical Systems and Signal Processing xxx (2018) xxx–xxx 0

0

0

13

0

shown in Fig. 17. The stiffness (k1 , k2 , k3 ) and end stops (a0 , b , c0 ) are determined using lease squares methods with the measured nonlinear force versus displacement of a diaphragm plate. The origin of coordinate is the working point of diaphragm spring. So the X1 and the X2 start at the same point. The relationship between FB and x2 in Fig. 16 is:

8 0 < x2 < a > < k1 x2 FB ¼ k2 x2 þ ðk1  k2 Þa a < x2 < b > : k3 x2 þ ðk1  k2 Þa þ ðk2  k3 Þb b < x2 < c

ð5Þ

The relationship between FT and x1 at working range in Fig. 17 is:

8 0 > < k1 x1 FT ¼ k02 x1 þ ðk01  k02 Þa0 > : 0 0 0 0 0 0 k3 x1 þ ðk1  k2 Þa0 þ ðk2  k3 Þb

0 < x1 < a0 0

a0 < x1 < b

ð6Þ

0

b < x1 < c0

If the deformation of a diaphragm spring is ignored, the clutch breakaway property, the relationship between the FA and x0 in Fig. 4, is calculated by:

 FA ¼

l1 ðFT  FB Þ=l2

if

x0 ¼ l1 x2 =l2

l1 FT =l2

if

x0 ¼ l1 ðx2 þ DÞ=l2

ð7Þ

5.2.1. Influence of maximum axis deformation of a waveform plate The maximum axis deformation of a waveform plate has great influence to the engagement and disengagement process of a clutch. If the maximum axis deformation is too small, the engagement or the disengagement process of a clutch is too quick, and the fast process may cause vehicle judder or shuffle [17]. Fig. 18 shows three different axial forces versus displacement relations for a waveform plate with different shapes. It is seen that the maximum axis deformations of the waveform plate for curves C1, C2 and C3 are 0.68 mm, 0.78 mm and 0.88 mm respectively. The fitted piecewise stiffness and end stops for curves of C1, C2 and C3 are listed at Table 5. It is seen that the stiffness of k1 and k2 the end stops of a and b for curves of C1, C2 and C3 are the same, and the only difference for the fitted piecewise stiffness is k3. The fitted piecewise stiffness and end stops for the force versus displacement curve of a diaphragm spring at its large end at its working displacement range shown in Fig. 17 are given in Table 6. The calculated clutch breakaway curves using FEA simple analytical method are shown in Fig. 19, and some key parameters, such as maximum disengagement force, force difference at peak and valley point and the and the maximum disengagement displacement, are then estimated from the curves at Fig. 19 and are listed at Table 7.

Fig. 17. Force versus displacement of a diaphragm spring at its large end: Nonlinear and the simplified piecewise linear with three stages at its working range.

Please cite this article in press as: W.-B. Shangguan et al., Effective utilizing axial nonlinear characteristics of diaphragm spring and waveform plate to enhance breakaway performances of a clutch, Mech. Syst. Signal Process. (2018), https://doi.org/10.1016/j. ymssp.2018.05.060

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7000

C1 C2 C3

6000

FB/N

5000 4000 3000 2000 1000 0 0.0

0.2

0.4

0.6

0.8

1.0

X2/mm Fig. 18. Force versus displacement of a waveform plate with different maximum axial compression.

Table 5 The fitted piecewise stiffness and end stops for curves of C1, C2 and C3. Curve

k1 (N/mm)

k2 (N/mm)

K3 (N/mm)

a (mm)

b (mm)

c (mm)

C1 C2 C3

2096 2096 2096

4252 4252 4252

26,865 15,548 11,106

0.30 0.30 0.30

0.62 0.62 0.62

0.68 0.78 0.88

Table 6 The stiffness and end stops for the fitted piecewise line of a diaphragm spring. Parameters

Values

Parameters

Values

0

1770

a0 (mm)

0.7

k2 (N/mm)

0

524

k3 (N/mm)

0

210

b (mm) c0 (mm)

k1 (N/mm)

0

1.2 1.7

1000

1000

800

800

FA/N

1200

FA/N

1200

600

600

C1 C2 C3

400 200

200

0 0

2

4 X0/mm

6

C1 C2 C3

400

8

0

0

2

4

6

X0/mm

(a) FEA

(b) Simple analytical method

Fig. 19. The calculated breakaway property of a clutch.

It is concluded from Fig. 19 and Table 7 that the maximum disengagement force and the force difference of peak point and valley point are decreased with the increasing of the maximum axial deformation of the waveform plate, but maximum disengagement displacement of a clutch do not change with maximum axis deformation of a waveform plate. 5.2.2. Influence of curvature of axial force versus displacement curve Curvature of an axial force versus displacement curve is also an important feature of a waveform plate. It determines the disengagement point and engagement point of clutch pedal. Please cite this article in press as: W.-B. Shangguan et al., Effective utilizing axial nonlinear characteristics of diaphragm spring and waveform plate to enhance breakaway performances of a clutch, Mech. Syst. Signal Process. (2018), https://doi.org/10.1016/j. ymssp.2018.05.060

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W.-B. Shangguan et al. / Mechanical Systems and Signal Processing xxx (2018) xxx–xxx Table 7 Parameters of clutch breakaway property estimated by FEA and analytical methods for the different waveform plate. Parameters

Maximum disengagement force (N) Force difference of peak and valley point (N) Maximum disengagement displacement (mm)

Waveform plate (C1)

Waveform plate (C2)

Waveform plate (C2)

FEA

Analytical

FEA

Analytical

FEA

Analytical

1130 124 7.3

1262 262 6.1

1095 88 7.3

1236 236 6.1

1056 49 7.3

1205 205 6.1

7000

C4 C5 C6

6000 5000

FB/N

4000 3000 2000 1000 0 0.0

0.2

0.4

0.6

0.8

X2/mm Fig. 20. Force versus displacement curves with different curvature for a waveform plate.

Three curves of force versus displacement with different curvature of waveform plate (see Fig. 20) are used to analyze the influence of curvature of a curve on clutch breakaway property. The C4, C5 and C6 are used to denote three curves with small, middle and large curvatures, respectively. The fitted stiffness and end stops of the piecewise line curves for curve of C4, C5 and C6 are listed at Table 8. It is seen that the end stops of c for curves of C4, C5 and C6 are the same, but the end stops of a, b, and the stiffness, k1, k2 and k3, for the fitted piecewise line are different. The FEA model and simple analytical mode are used to obtain clutch breakaway performance, and the calculate clutch breakaway and some key parameters are shown in Fig. 21 and Table 9, respectively. Table 8 The fitted piecewise stiffness and end stops for curves of C1, C2 and C3. Curve

k1 (N/mm)

k2 (N/mm)

k3 (N/mm)

a (mm)

b (mm)

c (mm)

C4 C5 C6

3480 1453 1265

6797 4213 6450

12,000 14,500 26,200

0.44 0.30 0.40

0.64 0.62 0.68

0.78 0.78 0.78

1000

1000

800

800

FA/N

1200

FA/N

1200

600

600

400

C6 C5 C4

200

400 200

0 0

2

4

6 X0/mm

(a)

C6 C5 C4

8

0

0

2

X0/mm

4

6

(b)

Fig. 21. Influence of curvature of force versus displacement on clutch breakaway (a) FEA (b) Simple analytical method.

Please cite this article in press as: W.-B. Shangguan et al., Effective utilizing axial nonlinear characteristics of diaphragm spring and waveform plate to enhance breakaway performances of a clutch, Mech. Syst. Signal Process. (2018), https://doi.org/10.1016/j. ymssp.2018.05.060

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Table 9 Key parameters of clutch breakaway property estimated by FEA and analytical methods for the different waveform plate. Parameters

Maximum disengagement force (N) Force difference of peak and valley point (N) Maximum disengagement displacement (mm)

Waveform plate (C4)

Waveform plate (C5)

Waveform plate (C6)

FEA

Analytical

FEA

Analytical

FEA

Analytical

1094 83 7.3

1236 236 6.1

1095 83 7.3

1236 236 6.1

1095 85 7.3

1238 238 6.1

It is shown that both force difference at peak point and valley point and maximum disengagement are increased with the increasing of the curvature of axial stiffness curve. However, this tendency is not obviously compare with that of the maximum axial deformation presented at Section 5.2.1. Besides, the maximum disengagement displacement is not change with the increasing of curvature. 5.3. Summary Based on the above calculation, it is concluded that increasing the displacement of working position of diaphragm spring and the axial compression of waveform plate can decreases difference between peak and valley force and maximum disengagement force; the curvature of axial stiffness curve has little effect on it. The maximum disengagement displacement increased with the decreasing of the working position of diaphragm spring, but maximum disengagement displacement cannot change by adjusting the waveform plate. 6. Applications Fig. 22 shows the measured force versus displacement curve for a generic clutch pedal. The measures for evaluating clutch pedal performance are the maximum pedal force, the force difference between peak and valley point and the maximum pedal displacement, and they equal to 83 N, 3.6 N and 102 mm, respectively. As discussed in Section 1.1, so it is concluded that the above all measures are lower than the required data. Since the configuration of pedal system and locations of hydraulic cylinder are fixed, the only way to change measures is to optimize the nonlinear characteristics of diaphragm spring and waveform plate to make the measures to meet requirements. Based on the analysis results in Sections 5, the working point of diaphragm spring B2 in Fig. 14 and the k3 and the c at waveform plate force versus displacement curve in Fig. 16 have significant effects on the clutch breakaway characteristic. As shown in Fig. 14, working point also determines the piecewise stiffness and end stops of diaphragm spring. As shown in Fig. 18 and Table 4, it is seen than stiffness of k1 and k2, and end stops of a and b are the same even if end stop of c and stiffness of k3 are changed. If the c is changed, the k3 changes with it, so the c and the k3 is simplified as one variable. An optimization model is proposed by taking the weighted function of maximum disengagement force, force difference of peak and valley point and maximum disengagement displacement as the object function, and assuming the maximum displacement of waveform plate (c) and the displacement of the working point (B2) as design variables. The optimization model is:

8 > < Max ww ðPw ; cÞ s:t: 3:8 mm 6 Pw 6 4:0 mm > : 0:6 mm 6 c 6 0:8 mm

ð8Þ

100

A

Pedal Force(N)

80

B

60 40 20

Maximum pedal displacement

0 0

20

40

60

80

100

120

Pedal Displacement(mm) Fig. 22. Measured force versus displacement at pedal of a passenger car.

Please cite this article in press as: W.-B. Shangguan et al., Effective utilizing axial nonlinear characteristics of diaphragm spring and waveform plate to enhance breakaway performances of a clutch, Mech. Syst. Signal Process. (2018), https://doi.org/10.1016/j. ymssp.2018.05.060

W.-B. Shangguan et al. / Mechanical Systems and Signal Processing xxx (2018) xxx–xxx

17

where P w represent displacement of the working point. The objective function, ww , is defined as

ww ¼

F w Dw D w þ þ A1 A2 A3

ð9Þ

where F w , Dw and Dw represent maximum disengagement force, force difference of peak and valley point and maximum disengagement displacement respectively. The A1 , A2 and A3 are the maximum value of F w , Dw and Dw , respectively. The simple analytical method described in Section 4 is used for estimating maximum disengagement force, force difference of peak and valley point and maximum disengagement displacement of a clutch, and sequential quadratic programming (SQP) method is used for obtaining optimized c and P w in Eq. (8). The optimized design variables are shown in Table 10. Using the optimized end stop ðcÞ and displacement of working point, the clutch breakaway property is estimated using the proposed FEA model in Section 4, and the calculated clutch breakaway property is given in Fig. 22. It can be estimated from Fig. 23 that the maximum disengagement force, the force difference between peak point and valley point and the maximum displacement are increased from 1067 N to 1190 N, from 79 N to 167 N and from 7.1 mm to 8.4 mm, respectively. Using the baseline clutch and the optimized clutch, the pedal force versus displacement is measured using the method presented in Section 3 and the measured data are shown in Fig. 23. It is seen from Fig. 24 that the maximum disengagement

Table 10 The optimized design variables. Design variables

Baseline

Optimized

Working point P w ðmmÞ End stop c ðmmÞ

4.0 0.7

3.8 0.8

1200 1000

FA/N

800 600 400

Baseline Optimized

200 0 0

2

4 X0/mm

6

8

Fig. 23. Clutch breakaway characteristic.

100

Pedal Force/N

80 60 40 Basedline

20

Optimized 0 0

20

40 60 80 100 Pedal Displacement/mm

120

140

Fig. 24. Pedal force versus displacement characteristics comparison (Test bench).

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Table 11 Measures for evaluating clutch pedal performance. Parameters

Using baseline clutch

Using the optimized clutch

Required

Maximum pedal force (N) Force difference at Peak and valley point (N) Maximum pedal displacement (mm)

83.3 4.6 101

92.1 13.4 123

90–120 10–30 120–140

force and force difference between peak and valley point are increased by 8.8 N, and the disengagement displacement is increased by 22 mm if the optimized clutch is installed at the clutch operation system. The measures for evaluating clutch pedal performance obtained from Fig. 24 are listed in Table 11. It is seen that all measures are located at the required ranges after using the optimized clutch, which validates the proposed analytical and optimization methods developed in this paper. 7. Conclusion The relationship between nonlinear characteristics of clutch breakaway and the ride of clutch operation system is investigated in this paper. A finite element (FE) model is proposed and used to estimate characteristics of clutch breakaway, and the nonlinear characteristics of diaphragm spring and waveform plate at axis direction are included in the model. The method for how to utilize nonlinear characteristics of diaphragm spring and waveform plate to enhance breakaway performances of a clutch is proposed. The following conclusions are obtained: (1) The breakaway characteristic curve depends on the nonlinear characteristic of axial stiffness of diaphragm spring and waveform plate and the elastic deformation of diaphragm spring. Increasing the working position of diaphragm spring and the axial compression of waveform plate can decreases peak-valley difference force and maximum disengagement force effectively. On the contrary, the curvature of axial stiffness curve has little effect on it. Besides, the maximum disengagement displacement increased with the decreasing of the working position of diaphragm spring. (2) The FE model and measure method proposed in the paper are effective for research the clutch breakaway performances. The test benches for measuring clutch breakaway and force versus displacement of clutch pedal are designed and proposed in this paper for validating the FEA model and calculation methods. Axial nonlinear characteristics for diaphragm spring and waveform plate determine the clutch breakaway performances and have great influence on clutch pedal performance. The elastic deformation of diaphragm spring cannot be ignored for study the clutch breakaway performances. (3) An optimization model is proposed by reducing the working point of diaphragm and increasing the maximum axis deformation of a waveform plate, and all the measures are located at the required ranges after using the optimized clutch.

Acknowledgements Authors would like to acknowledge the support from the Natural Science Foundation of China (Project No.11472107). References [1] A.E. Balau, C.F. Caruntu, C. Lazar, Simulation and control of an electro-hydraulic actuated clutch, Mech. Syst. Signal Process 25 (2011) 1911–1922. [2] R. Nouailletas, E. Mendes, D. Koenig, Hybrid modeling and identification of dry friction systems, application to a clutch actuator, Control Eng. Pract. 18 (2010) 904–917. [3] Tse-Chang Li, Yu-Wen Huang, Jen-Fin Lin, Studies on centrifugal clutch judder behavior and the design of frictional lining materials, Mech. Syst. Sig. Process. 66–67 (2016) 811–828. [4] Shoaib Iqbal, Farid Al-Bender, Agusmian P. Ompusunggu, Bert Pluymers, Wim Desmet, Modeling and analysis of wet friction clutch engagement dynamics, Mech. Syst. Sig. Process. 60–61 (2015) 420–436. [5] Grzegorz Kudra, Jan Awrejcewicz, Michał Szewc, Modeling and simulations of the clutch dynamics using approximations of the resulting friction forces, Appl. Math. Model. 46 (2017) 707–715. [6] Guoqiang Li, Daniel Görges, Optimal control of thegear shifting process for shift smoothness in dual-clutch transmissions, Mech. Syst. Sig. Process. 103 (2018) 23–38. [7] F. Amisano, G. Ercole, G. Mattiazzo, et al, Diaphragm spring clutch dynamic characteristics test bench, SAE Technical Paper Series (1999), No. 1999-010737. [8] G. Ercole, G. Mattiazzo, S. Mauro, et al, Experimental methodologies to diaphragm spring clutch characteristics, SAE Tech. Paper Ser. (2000), No. 200001-1151. [9] G. Mattiazzo, S. Mauro, M. Velardocchia, et al, Measurement of torque transmissibility in diaphragm spring clutch, SAE Technical Paper Series (2002), No. 2002-01-0934. [10] X.J. Li, in: Study on Simulation and Computational Method of the Torque of the Vehicle Diaphragm Spring Clutch, Beijing Jiaotong University, Beijing, 2010, p. 6. [11] M. Hoic, Z. Herold, N. Kranjcetive, et al, Experimental characterization and modeling of dry dual clutch thermal expansion effects, SAE Tech. Paper Ser. (2013), No. 2013-01-0818.

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W.-B. Shangguan et al. / Mechanical Systems and Signal Processing xxx (2018) xxx–xxx

19

[12] N. Cappetti, M. Pisaturo, A. Senatore, Load–deflection curve sensitivity to main shape parameters in a dry clutch cushion spring, in: 7th international conference on engineering computational technology, 2010, pp. 1–14. [13] S. Sfarni, E. Bellenger, J. Fortin, et al, Finite element analysis of automotive cushion discs, Thin-Walls Structure. 47 (2009) 474–483. [14] N. Cappetti, M. Pisaturo, A. Senatore, Modeling the cushion spring characteristic to enhance the automated dry-clutch performance: the temperature effect, IMechE Part D: J. Automobile Eng. 226 (11) (2012) 1472–1482. [15] B. Czel, K. Varadi, A. Albers, M. Mitariu, Fe thermal analysis of a ceramic clutch, Tribol. Int. 42 (5) (2009) 714–723. [16] F. Vasca, L. Iannelli, A. Senatore, et al, Torque transmissibility assessment for automotive dry-clutch engagement, IEEE/ASME Transactions on Mechatronics. 16 (3) (2011) 564–573. [17] Y. Hong, H.W. Park, Simulation of clutch actuation system for commercial vehicle, Int. J. Precis. Eng. Manuf. 11 (6) (2010) 839–843. [18] Y. Hong, H.W. Park, S.J. Yu, et al, A study on clutch actuation system program development using hysteresis method, SAE Technical Paper (2008), 200801-1685. [19] B.S. Lee, Friction characteristics of a hydraulic cylinder for an automotive manual clutch, Psychiatry Res. 140 (4) (2006) 658–666. [20] W.B. Zhu, X.X. Zhao, Y. Gan, et al, Parameter optimization of torsional spring in clutch operating mechanisms, China Mech. Eng. (In Chinese). 27 (23) (2016) 3149–3156. [21] W.B. Shangguan, T. Sun, R.Y. Zheng, et al, The influence of driven disc for a friction clutch on vehicle judder during starting, J. Vib. Eng. (In Chinese). 29 (3) (2016) 488–497.

Please cite this article in press as: W.-B. Shangguan et al., Effective utilizing axial nonlinear characteristics of diaphragm spring and waveform plate to enhance breakaway performances of a clutch, Mech. Syst. Signal Process. (2018), https://doi.org/10.1016/j. ymssp.2018.05.060