Studies on centrifugal clutch judder behavior and the design of frictional lining materials

Mechanical Systems and Signal Processing 66-67 (2016) 811–828

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Studies on centrifugal clutch judder behavior and the design of frictional lining materials Tse-Chang Li a, Yu-Wen Huang a, Jen-Fin Lin a,b,n a b

Department of Mechanical Engineering, National Cheng Kung University, Tainan 701, Taiwan Center for Micro/Nano Science and Technology, National Cheng Kung University, Tainan 701, Taiwan

a r t i c l e i n f o

abstract

Article history: Received 7 April 2014 Received in revised form 28 May 2015 Accepted 11 June 2015 Available online 3 July 2015

This study examines the judder behavior of a centrifugal clutch from the start of hot spots in the conformal contact, then the repeated developments of thermoelastic instability, and finally the formation of cyclic undulations in the vibrations, friction coefficient and torque. This behavior is proved to be consistent with the testing results. Using the Taguchi method, 18 kinds of frictional lining specimens were prepared in order to investigate their performance in judder resistance and establish a relationship between judder behavior and the Ts/Td (Ts: static torque; Td: dynamic torque) and dμ/dVx (μ: friction coefficient; Vx: relative sliding velocity of frictional lining and clutch drum) parameters. These specimens are also provided to examine the effects and profitability with regard to the centrifugal clutch, and find the relative importance of the various control factors. Theoretical models for the friction coefficient (μ), the critical sliding velocity (Vc) with clutch judder, and the contact pressure ratio pn =p (pn: pressure undulation w.r.t. p; p: mean contact pressure) and temperature corresponding to judder behavior are developed. The parameters of the contact pressure ratio and temperature are shown to be helpful to explain the occurrence of judder. The frictional torque and the rotational speeds of the driveline, clutch, and clutch drum as functions of engagement time for 100 clutch cycles are obtained experimentally to evaluate dμ/dVx and Ts/Td. A sharp rise in the maximum pn =p occurred when the relative sliding velocity reached the critical velocity, Vc. An increase in the maximum pn =p generally led to an increase of the (initially negative) dμ/dVx value, and thus the severity of judder. The fluctuation intensity of dμ/dVx becomes a governing factor of the growth of dμ/dVx itself in the engagement process. The mean values of dμ/dVx and Ts/Td for the clutching tests with 100 cycles can be roughly divided into three groups dependent on the fluctuation intensities of these two parameters, for each of which there is a linear relationship. & 2015 Elsevier Ltd. All rights reserved.

Keywords: Clutch judder Frictional lining materials Taguchi method Judder resistance

1. Introduction When a vehicle starts rolling the clutch engagement sometimes generates judder, which may damage the drivetrain components. Judder often manifests itself in the form of noisy torsional vibrations of the drivetrain or a violent surging of the vehicle. Transient torsional oscillations are related to judder. A special relationship between the friction coefficient and sliding

n

Corresponding author at: 1 University Road, Tainan City 701, Taiwan (R.O.C.). Tel.: þ886 6 2757575x62155. E-mail address: [email protected] (J.-F. Lin).

http://dx.doi.org/10.1016/j.ymssp.2015.06.010 0888-3270/& 2015 Elsevier Ltd. All rights reserved.

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velocity becomes the most important source of judder. The torsional vibrations of the clutch system that lead to judder are induced by stick-slip processes at the interface of the clutch friction disc and the flywheel and at pressure plate interfacial contacts. The friction discs of clutches are subjected to considerable thermal loading. High temperatures lead to thermal stress failures, such as surface cracks and permanent distortion [1,2]. Distortions often occur in multidisc clutches, and are in the form of a conical deformation or disc waviness. The temperatures and stresses in the friction discs of a multidisc wet clutch have been studied [3]. In general, the initial distribution of normal pressures on the friction surface can be nonuniform, although it can also be nonuniform due to thermal deformations. This thermoelastic transition is present in many sliding contact systems [4]. This process of pressure changes is unstable and is called thermoelastic instability [5]. In the model proposed by Kennedy and Ling [6] the influences of thermal deformation and wear on the normal contact pressure were taken into account. In contrast, Zagrodzki [7] presented a model of transient thermomechanical phenomena occurring in a multidisc wet clutch that considers the thermoelastic instability effect. Bostwick and Szadkowski [8] studied the self-excited vibrations generated by an increase in the coefficient of friction on a clutch facing due to a decrease in the slip speed on friction surfaces. The amplitude of self-excited vibration depends on the system parameters alone. In Crowther and Zhang [9] a six-degree-of-freedom dynamic model for clutch engagement was used to investigate the effect of low-frequency transients on clutch engagement stick-slip behavior in powertrains. A commercially available sintered friction pad was coupled with a standard gray cast iron pressure plate and tested in a clutch dynamometer to determine the engagement characteristics for the prediction of useful life [10]. The microscopic features of worn sintered friction pads indicated that silica particles provided wear resistance for the pads. Commercial paper-based friction plates with standard steel reaction plates were tested with four stiffness and inertia combinations of a wet clutch system [11]. A low-inertia system showed faster degradation and shorter clutch life for high torsional oscillation. The system was also more shudder sensitive for lower natural frequencies in a less stiff system. An analytical procedure for determining pure stick to stick-slip motions was developed based on linear system analysis [12]. Stick-slip behavior was clearly observed to be a result of engine torque irregularity and nonlinear friction characteristics. Disc inertia significantly affects system dynamics. A study on a wet centrifugal clutch [13] simulated vehicle judder using a clutch unit tester, and the results were utilized as an evaluation index for judder. Based on this, the effects of the components of a friction material on judder were clarified. In Gregori [14] a bench test was developed to measure the judder sensitivity of a friction material used on clutch discs. A methodology was developed for the characterization of clutch facing sensitivity for judder. Stick-slip, present to some degree in almost all actuators and mechanisms with frictional contact, often leads to self-excited vibration. During the engagement process, some of the energy transmitted through the driveline is transformed into other forms of energy by positive damping effects. If for some reason the damping becomes negative, some of the energy transmitted by the clutch can induce self-excited torsional vibrations of the driveline, which can induce judder. Centea et al. [15] showed that the gradient of friction coefficient (μ) and slip velocity (Vx) is a negative value, if dμ/dVx Z  C/Fn (C: damping coefficient; Fn: normal load), the damping is positive and the system is stable. If dμ/dVx o  C/Fn, the damping coefficient of the driveline becomes negative and the vibration system becomes unstable. Karnopp [16] analyzed various friction models in the stickslip region in terms of the variation of the friction force Ft with the relative interfacial contact velocity Vx. Centea et al. [17] investigated the torsional vibration mode of the clutch system. It was proved that judder is related to the type of friction lining material. A transient finite element analysis method was used to analyze the fully coupled thermoelastic instability problem for a brake system [18]. The reliabilities of the analysis technique and simulation model were verified. The cause of judder in clutches has seldom been studied, and thus the present work aims to establish a theoretical model to interpret this behavior, and use the vibrations varying with time and their spectrum analysis to verify its validity. The severity of judder is governed by the rising rate of torque expressed by Ts/Td (Ts: static torque; Td: dynamic torque) and the intensity of torque fluctuations often denoted by dμ/dVx (μ: friction coefficient; Vx: sliding velocity of friction lining). The map of these two governing parameters satisfying the small intensity demand of judders can thus be determined, and so the effects of the fabrication conditions and materials using in friction lining on judder prevention can then be evaluated. The relationships between Ts/Td and dμ/dVx for the specimens were also established in association with the rising rate of friction torque and the intensity of torque fluctuation. 2. Development of theoretical models The present study analyzes the friction coefficient (μ) arising at the contact surface of the friction lining and the clutch drum of a centrifugal-type clutch. A centrifugal-type clutch generally consists of a spider as the input member, three sets of shoes and friction linings as the centrifugal members, and a round-type clutch drum as the output member. One end of the partial-arc shoe is fixed by a pivot (the central point of hinge pin, noted by A1 in Fig. 1) as the input member; a point of this shoe is connected by spring 1 to the pivot side of the adjacent shoe. When the rotational speed of the input member (drive) is sufficiently high, the friction shoe is forced to move outward due to the centrifugal force, and the friction lining rotates to have an engagement with the clutch drum until the friction lining is tightly pressed against the inside surface of the clutch drum without relative sliding velocity. Fig. 1(a) shows the mechanical diagram of one friction lining before starting the engagement with the clutch drum; the subscript i denotes the initial state at this moment. Fig. 1(b) shows the diagram after finishing the full engagement. Δs is defined as the increase in spring length when the friction shoe is subjected to a centrifugal force. The distance of the pivot center (A1) from the clutch center (O) is a. Point C1 and point C2 (not shown in Fig. 1) are the two fixed ends of the

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Fig. 1. Mechanical diagrams of one friction lining (a) before; and (b) after the engagement with the clutch drum.

spring in shoe 1. B is defined as the termination point of the friction lining before the application of the centrifugal force, and B0 is defined as the new termination point of the lining subject to a centrifugal force. Define x as the distance between the tangential line of point B and the inner surface of clutch drum. The rotation angle (φ) of the friction shoe due to the application of a centrifugal force is obtain as [19] ðOB þ xÞ2 OB sin θC2 2 φ ¼ cos  1 a2 þ OB  2a UOB cos θC2 þ a2  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  sin  1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 2 a þ OB  2a UOB cos θC2  a a þ OB  2a U OB cos θC2

ð1Þ

where θc2 denotes the angle between OB and OA1 . Point C1 moves to C 01 and point C2 moves to C 02 when friction shoe 1 moves outward due to a centrifugal force. The angle ∠C 01 A1 C 02 is expressed as [19] 0 1 A2 C 2 sin ð∠A1 A2 C 2  φÞ B C ∠C 1' A1 C 02 ¼ ∠A2 A1 C 2 þ ∠C 1 A1 C 2 þ φ  sin  1 @qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2Þ A pffiffiffi 2 2 3a þ A2 C 2  2 3a U A2 C 2 cos ð∠A1 A2 C 2  φÞ pffiffiffi pffiffiffi 2 where ∠A1 A2 C 2 ¼ cos  1 ððð 3aÞ2 þA2 C 2  A1 C 2 Þ=2 3a U A2 C 2 Þ. The distance between point C 01 and point C 02 is expressed as [19] qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 C 01 C 02 ¼ A1 C 1 þ A1 C 02 2 U A1 C 1 U A1 C 02 U cos ð∠C 01 A1 C 02 Þ ð3Þ where A1 C 02 can be written as [19] qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 2 A1 C 02 ¼ 3a2 þ A2 C 2  2 U 3a U A2 C 2 U cos ð∠A1 A2 C 2  φÞ The elongation length (Δs) of the spring due to the application of a centrifugal force is thus expressed as [19]

Δs ¼ C 01 C 02  C 1 C 2

ð4Þ

A single friction shoe rotating clockwise is subject to several forces, including the centrifugal force Fclu, spring forces Fspr1 and Fspr2, friction forces created in the rubber gasket between the shoe and the drive plate and in the gasket between the pivot and the drive plate, and the normal and tangential (frictional) forces generated at the contact area of the friction lining and clutch drum. Apart from force balance, the moment balance for the pivot is also needed in order to attain the dynamically equilibrium state of the clutch system. The distance from the mass center of the frictional shoe to the center of the clutch is rcm, the mass of the frictional shoe is m, and the rotational speed of the drive in the clutch is ω1. The centrifugal force acting on the frictional shoe is expressed as F clu ¼ mω21 r cm

ð5Þ

If the spring constant is k and the pretension length of the spring is Δs0 before applying any tension force to the spring, then the force Fspr acting on the spring after full contact between the frictional lining and the clutch drum is expressed as F spr ¼ kðΔs0 þ ΔsÞ

ð6Þ

Eq. (6) must be satisfied for each of the two springs in the shoes. Here, the contact pressure (p) formed between the lining and the clutch drum will be obtained later. b is defined as the width of the partial-arc friction lining. The normal force

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acting on an element area in the contact region is denoted as dN. The torques created at the pivot center due to the normal forces and frictional forces formed in the contact area of frictional lining are defined as Mn and Mf, respectively. These are expressed as Z Z θn dNa sin θc ¼ abr p sin θc dθc ð7:aÞ Mn ¼ θ

Z

n

μdNðr  a sin θc Þ ¼ μbr

Mf ¼

Z θn θ

n

pðr  a sin θc Þdθc

ð7:bÞ

where θn ¼an/R, and an and R are the semi-length of the contact area and the effective radius of curvature, respectively. They will be defined later, and can be determined directly from the worn-surface morphology of the frictional lining. The torquemoment balance at the pivot center is expressed as F clu C  F spr1 r s1  F spr2 r s2  F rbr d  Mn þ M f ¼ 0

ð9Þ

As Fig. 1 shows, Fspr1 denotes the spring force evaluated at spring 1, and Fspr2 denotes the spring force evaluated at spring 2 (on the pivot side). The distances from these two springs to the pivot center are rs1 and rs2, respectively. A self-energizing effect is considered in the present study because FcluC  Fspr1rs1  Fspr2rs2 Frbrd40. If the number of friction linings in a clutch is n, then the torque applied to the clutch system can be written as Z θn Z pdθc ð9Þ T ¼ n μrdN ¼ nμbr 2 θ

n

The friction coefficient (μ) is then given as a function of T [19]:

μ¼

nbr 2

T R θn

θ

n

pdθc

ð10Þ

There are four classes of boundary conditions within the contact region. For the class of two solids in sliding contact with their known profiles, the normal displacement due to thermal distortion can be specified. The normal traction (pressure), p, in an elementary area and its tangential traction, q, are related by q ¼ 7 μp, where μ is a constant coefficient of friction. The boundary conditions in the contacts of friction lining and clutch drum fall into this class. The contact parameters, including contact pressure and temperature, are developed for the repeated contacts of the friction lining and the clutch drum to investigate the clutch judders in their engagement. The tangential traction (q) satisfying the boundary conditions of this class [20] is expressed as  n  μEn a þ x γ n2 q¼ ða  x2 Þ1=2 ð11Þ n 2 2Rð1 þ β μ2 Þ1=2 a  x where the x coordinate is defined with its origin at the central point symmetrical with respect to the two ends of the contact area. The reduced modulus En is expressed as 1 1  ν21 1  ν22 ¼ þ E1 E2 En where Gi (i¼1, 2) denotes the solid shear modulus, and νi (i¼ 1, 2) denotes the Poisson's ratio. The subscripts, 1 and 2, of these two material properties are defined for the friction lining and clutch drum, respectively. μ is obtained from the solution of Eq. (10). β is given as [21]   1 ½ð1  2ν1 Þ=G1 ½ð1 2ν2 Þ=G2 β¼ 2 ½ð1  2ν1 Þ=G1 þ½ð1 2ν2 Þ=G2 The exponent

γ on the basis of this kind of boundary conditions is obtained as

1

γ ¼  tan  1 ðβμÞ ffi  βμ=π π provided that βμ is small. an is the semi-length of the contact area in the friction lining, and this is obtained as [20] ðan Þ2 ¼

1 4PR U 1  4γ 2 E n

where P is the normal contact load and R is the effective radius of curvature of the friction lining and the clutch drum. R in the conformal contact is expressed as 1 1 1 ¼  R RFL RCD where RFL (¼r) and RCD are the radii of curvature of the friction lining and the clutch drum, respectively. If the contact area is no longer symmetrical with the two ends of friction lining, its contact center is displaced from the axis of symmetry by a

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distance x0 [20]: x0 ¼ 2γ an

ð12Þ

The contact pressure p is obtained as pðxÞ ¼ ¼

qðxÞ

μ

 n  a þx γ n2 ða x2 Þ1=2 n 2 2Rð1 þ β μ2 Þ1=2 a x En

ð13Þ

The contact load P in the an expression is thus expressed as Z θn bpðxÞ cos θrdθ P¼ θ

ð14Þ

n

Here, θ ¼ x=r and θ ¼ an =r, and  θ r θ r θ . Eq. (14) can then be rewritten as Z an pðxÞ cos ðx=rÞdx P¼b n

n

n

 an

ð15Þ

It can be expected that the stationary clutch drum surface develops “hot spots” where the temperature is significantly in excess of its expected mean value, and this phenomenon was investigated by Barber [5]. Small departures from perfect conformity concentrate the pressure and hence the frictional heating into particular regions of the interface of the friction lining and clutch drum. These regions expand above the level of their surrounding surface and reduce the area of real contact [4]. This process thereby concentrates the contact and elevates the local temperature further, thus leading to thermoelastic instability. If sliding continues at the expanded spots, then the contact pressure is concentrated. These spots are worn down until new contacts occur elsewhere. These new contact spots then proceed to heat, expand and carry the load, while the old ones are relieved of load, and then cool, contract and separate in sequence. This cyclic process has been found in the sliding contact of two conforming surfaces. Since the scale of the hot spots is large compared to the scale of surface roughness, the time period of the cycle is long compared with the time of asperity interactions. Two semi-infinite sliding solid nominally flat surfaces are pressed into contact with a mean pressure p. To avoid the transient nature of microcontacts of the moving surface, the moving surface is taken to be perfectly flat, but conducting behavior is still considered. The stationary solid has a distortivity cn related to the surface heat liberated by friction, and its surface has a small initial undulation of amplitude Δ and wavelength λ. The isothermal pressure required to flatten the waviness is [20] p0 ¼ ðπ En Δ=λÞ cos ð2π x=λÞ

ð16Þ

The mechanism of thermoelastic instability in the contact spots introduces thermal distortion of the surface. The steady thermal distortion (uz ) is given by [20] 2

d uz ^ ¼ cn μV x pðxÞ dx2

ð17Þ

^ where pðxÞ is the pressure developed in the thermal distortion. Vx denotes the relative sliding velocity of the moving surfaces, which is evaluated as Vx ¼r(ωc  ωd), where r denotes the outer radius of the friction lining, and ωc and ωd represent the angular velocities of the clutch and the clutch drum, respectively. The initial sinusoidal undulation of ^ wavelength λ can result in a fluctuation of pressure at the same wavelength. The solution of pðxÞ in Eq. (17) is written as [20] ^ pðxÞ ¼ p þ pn cos ð2π x=λÞ

ð18Þ

The solution of Eq. (17) gives the thermal distortion of the surface as [20] uz ¼  ðcn μV x pn λ =4π 2 Þ cos ð2π x=λÞ 2

ð19Þ 0

The thermal pressure p″ðxÞ required to press the wave flat is added to p ðxÞ in Eq. (16) to obtain pn as [20] pn ¼

π En 2 ðΔ þ cn μV x p0 λ =4π 2 Þ λ

ð20Þ

The distortivity cn in Eq. (17) is expressed as [20] cn ¼ ð1 þ γ Þα=k where 1 ¼ k

1  ν1 =2 1  ν2 =2 þ G2 G1 1  ν1 1  ν2 G1 þ G2 1  ν1

α ¼ 1 G1ν1 G1

ν2 1 G2 ν2 þ1 G2

ð21Þ

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Eq. (20) can be rewritten as pn π En Δ=ðλpÞ ¼ p 1  cn μV x En λ=ð4π Þ

ð22Þ

As the sliding velocity Vx approaches the critical value Vc, given by V c ¼ 4π =ðcn μEn λÞ

ð23Þ

the fluctuation in pressure (pn) given by Eq. (22) theoretically increases to an extremely large value. These sharp rises result in clutch judders. The thermoelastic instability in the contact spots introduces thermal distortion of the surface, as shown in Eq. (19), in the form of sinusoidal undulations. The contact pressure at a hot spot induced by the instability is also a sinusoidal undulation with a variable amplitude in the cyclic form. According to the expression shown in Eq. (9), the value of the frictional torque in the contacting process is dependent upon the friction coefficient and the contact pressure integration over the contact area. As the sliding velocity reaches the Vc in Eq. (23), the intensities of the contact pressure undulation and its integration are expected to be instantly enhanced. The frictional torque in the contacting process is thus also in the form of cyclic undulations, with its amplitude increasing with time. Temperature rise analyses are made in the present study for rectangular frictional lining in sliding contact with the clutch drum. The frictional lining is moving in the x direction with a sliding velocity Vx relative to the clutch drum. The temperature rise T (¼ θ  θambient, θ: specimen temperature; θambient: ambient temperature) distribution in the workpiece due to the generation of the frictional heat arising at the contact zone between the frictional lining and the clutch drum satisfies the unsteady energy equation:

ζC P

∂T ∂T  ∇ UðK ΔTÞ þ ζ C P V x ¼ 0 ∂t ∂x

ð24Þ

where ζ denotes the friction lining density, C P denotes the specific heat and K denotes the thermal conductivity of the friction lining. αn is defined as the thermal diffusivity of the friction lining; then:

αn  K=ζ C P At the initial time, the frictional heat (units: J) is generated as the point heat source θpt. If the contact width of the sliding contact zone is B and the contact length is l, the contact area is Bl. q0 (units: J/s) is defined to be the rate of internal energy change, which is related to the transfer of frictional heat. In practical applications, only some frictional heat is transferred to the friction lining. Let X  x V x t. The y axis is in the direction parallel to the width of the friction lining and the z axis is in the direction normal to the contact surface (toward the inside of the friction lining). If the energy partition corresponding to this part is Rw, the three-dimensional temperature rise distribution on the sliding contact surface for a moving surface heat source with a velocity Vx is expressed as [22] Rw q0 V x T¼ 8K αn π 3=2 ðlBÞ

Z

l=2  l=2

2 4

Z

0

B=2

e  B=2



ðX  X i ÞV x 2αn

@

Z

V2 xt 4α

0

1 3 u2 i e  w  4w A 5 dw dyi dX i w3=2

ð25Þ

h i1=2  2 where ui ¼ ðð ðX  X i Þ2 þ y yi þz2 V x Þ=2αn Þ, and q0 ¼ μV x p. According to theories developed in previous studies [23–25], it is necessary to know the fraction (Rw) of the total energy conducted as heat to the workpiece (friction lining in this study) in order to estimate the workpiece temperature rise distribution. The parameters with subscript “D” are associated with the clutch drum and those with subscript “M” are associated with the friction lining. The apparent area A ( ¼Bl) is defined as the sliding contact area of the friction lining and the clutch drum, and the real contact area (considering asperities) of these two devices is AR. The fraction of the total energy, Rw, in Eq. (25) can be written as [25] RM ¼

1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðkρC ÞD V D AR 1 þ kρC ð ÞM V M A

ð26Þ

where VD denotes the tangential velocity of the clutch drum, and VM denotes the tangential velocity of the friction lining. These two velocities can be determined if the rotational speeds of these two parts are available in the tests. It should be mentioned that the contact pressure and temperature rise during the engagement of sliding contact can be predicted using theoretical models. However, it is very difficult to detect them during practical testing. Nevertheless, their theoretical solutions are provided to establish their relationships with the judder parameters. These theoretical models are presented to evaluate the friction coefficient variations in the engagement process first as the frictional torque is available from a CVT dynamometer (AVL, Japan). This friction coefficient satisfying the torquemoment balance (Eq. (19)) is then applied to determine the pn =p and T values. The friction coefficient is also used to define its gradient with respect to Vx as one of the governing factors of judder.

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3. Specimen preparations and experimental details Examining the economics (i.e., cost savings) of reducing variation is another main purpose of this study. The Taguchi technique uses a different cost model for product characteristics than is typically used, which places more emphasis on reducing variation, particularly when the total variation is within the specification limits for the product. The above characteristic is suitable for the present study, and this technique is thus adopted in the preparations of the friction lining specimens. In this work the contents (wt%) of copper fiber and methyl cellulose, molding pressure and temperature, and molding time are used as the five controlling factors in the evaluation of judder behavior; and three levels (factor values) within the specification limits are arranged for each factor. These five controlling factors are chosen to be of importance to the judder arising in the engagement of friction lining and clutch drum, partly based on the experience of the friction lining manufacturer. It should be stressed that this technique is applied to the design of the orthogonal array (Lʹ18(35)) for the present 18 specimens, and the analysis of variance (ANOVA) is made excluding the optimization from the study. The orthogonal array for these 18 specimens is shown in Table 1. The composite material of the friction lining is made of asbestos-free organic fiber, aramid fiber, phenolic resin, nitrile-butadiene rubber, cashew friction dust, barites, artificial graphite power, coke, and copper powder. The friction lining specimens were made in the shape of a rectangular plate with the dimensions of 37 mm (length)  22 mm (width)  4 mm (thickness). Three pieces of friction lining were then hot-pressed and adhered tightly along the lateral surface of the clutch weight. The geometries of the clutch and some operating conditions are shown in Table 2. The present study used a continuously variable transmission (CVT) system, composed of a drive plate assembly, drive belt, clutch assembly, and driven assembly. Judder tests and analyses of the specimens were carried out on a CVT dynamometer (AVL, Japan). The testing system shown in Fig. 2(a) consists of a drive motor, load motor, drive tachometer, load torque meter, load tachometer, and infrared thermometer. The signals of vertical and horizontal vibrations, frictional torque, and the revolutional speeds of the drive plate, clutch, and clutch drum were collected using a data acquisition system (Prowave Orion, Taiwan) for digital signal processing. The data acquisition system consists of a recorder with eight channels, two NI-9234 acquisition cards (NI, USA), and PCB-352-C68 and PCB-352-C22 accelerometers (USA), as shown in Fig. 2(b), for detecting vibrations in the vertical and horizontal directions, respectively. The synchronous sampling rate was 51.2 kS/s at 102 dB. A three-dimensional confocal microscope (NanoFocus μSurf, Germany) was used to measure the radius of curvature of the partial-arc friction lining with a resolution of 720 nm. In order to obtain Young's modulus of the friction lining specimens, the samples were prepared in the form shown in Fig. 3 for tests using a Micro/Nano Tensile Tester (DDS32 Kammrath & Weiss GmbH, Germany). The experimental results of Young's modulus for the 18 specimens are shown in Table 3. Before the tensile test, specimen topography was measured using a surface roughness measurement instrument (Surfcorder ET-4000 Kosaka, Japan). The average surface roughness and λ values in Eq. (22) for the 18 friction lining specimens are shown in Table 4. The λ values are the mean values of the sums of the five largest peaks and five largest valleys in a sampling length of ls. Table 5 shows the material and thermal properties of the SPCC clutch drum and frictional lining. The thermal effects on these properties were not included in the evaluations of the parameters in this study. The test process from the idle speed to the drive speed up to 500 rpm is defined as one cycle. The sliding velocity Vx with its maximum value is obtained before having the engagement of friction lining and clutch drum. When the sliding velocity reaches zero value the engagement is then completed. Before starting the test, 10 cycles were set up in the computer software as a unit, and 10 units with 100 cycles were given for every judder test. The sampling rates of all signals, including vibrations, are fixed at 5120 Hz. In order to avoid an excessively large amount of data being created in the testing period, the

Table 1 5-factorial, 3-level orthogonal array for the 18 specimens used in this study. Specimen

Copper fiber (wt%)

Methyl cellulose fiber (wt%)

Molding pressure (kgf/cm2)

Molding temperature (1C)

Molding time (h)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

0 0 0 3 3 3 6 6 6 0 0 0 3 3 3 6 6 6

3 6 9 3 6 9 3 6 9 3 6 9 3 6 9 3 6 9

50 80 110 50 80 110 80 110 50 110 50 80 80 110 50 110 50 80

80 100 120 100 120 80 80 100 120 120 80 100 120 80 100 100 120 80

0.5 1 2 1 2 0.5 2 0.5 1 1 2 0.5 0.5 1 2 2 0.5 1

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signals of the first cycle and then every four cycles after that for any two adjacent samples were collected. A total of 26 sets of signals for torque and revolutional speeds are thus available from each test. 4. Results and discussion In the present study the rotational speeds of the drive, clutch, and clutch drum and the signals of frictional torque were acquired as functions of engagement time for the 18 specimens. Fig. 4 shows a schematic diagram of the four parameters for a time period of about 11 s. In the idle state, the drive was rotating at about 1650 rpm. Its rotational speed started to rise rapidly in order to create the centrifugal force and push in the shoe moving outward. The beginning of engagement was at the initial time of having nonzero rotational speed of the clutch drum; while the full engagement was attained when the clutch drum had a rotational speed equal to that of the clutch. In a very short time after the beginning of engagement the clutch torque had risen sharply to about 4 N-m. During the process of engagement, the clutch torque was first found to have fluctuations with a small amplitude, and while the average magnitude gradually reduced the fluctuation amplitude increased significantly in the second half of the engagement process. When the full engagement was finished, the clutch Table 2 Geometries and dimensions of the clutch used in this study. Clutch geometry

Clutch parameter

Vertical distance from clutch to hinge pin, a (m) Width of friction lining, b (m) Vertical distance from centrifugal force to hinge pin, c (m) Distance from the mass center of weight to the center of clutch, rcm (m) Inner radius of clutch drum, r (m) Vertical distance from the spring force of spring 1 to hinge pin, rs1 (m) Vertical distance from the spring force of spring 2 to hinge pin, rs2 (m) Start angle of clutch lining, θc1 (deg.) Finish angle of clutch lining, θc2 (deg.) Mass of the clutch weight, m (kg) Vertical distance from the friction force of rubber damper to the hinge pin, d (m) Spring force, Fspr (N) Friction force of rubber damper, Frbr (N)

0.042 0.024 0.03466 0.0437 0.065 0.0444 0 58.6 91.2 0.223 0.05917 122.1 9

Fig. 2. (a) The CVT dynamometer for clutch tests; (b) the installations of accelerometers for the vibrations in the vertical and horizontal directions.

Fig. 3. Dimensions of friction lining sample for tensile tests.

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torque had a significant drop to about 2.2 N-m. The fluctuations in the second half of the process were found to be cyclic undulations, rather than having a random form. This behavior is ascribed to the judders, which will be verified in a later section. In this study “before contact” refers to the one-second time period before the maximum (relative) sliding velocity, while “after contact” refers to the one-second time period after the maximum sliding velocity. Fig. 5 shows the experimental results of the four parameters for the code-1 specimen in the first cycle after contact. The rotational speed of the clutch drum is slightly increased to about 260 rpm in 1 s. The curve of frictional torque is oscillating due to the judders appearing in the cycle. The low-frequency zero-order-hold torque line is attributed to the combined effect of the signal sampling method mentioned previously and the great magnification of the time scale in the abscissa. The vibration signals of the code-1 specimen at the first cycle before and after contact of the friction lining and the clutch drum are shown in Fig. 6(a) and (b), respectively, in the horizontal and vertical directions. The vibrational profile shown in Table 3 Young's moduli of the 18 specimens prepared in this study. Specimen code

Young's modulus (GPa)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

10.40 15.10 11.40 4.89 6.80 11.30 10.80 4.92 4.58 5.29 8.83 9.49 7.04 6.76 5.16 5.24 4.21 5.35

Table 4 Values of Δ and λ parameters shown in Eq. (24) for the 18 specimens used in this study. Specimen code

Δ (μm)

λ (μm)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

106 179 164 94 93 110 91 78 119 90 98 164 90 111 117 140 128 115

16.56 17.77 19.57 21.83 13.36 11.10 7.92 12.54 13.60 25.70 11.08 23.01 9.42 16.59 11.06 19.70 30.78 18.30

Fig. 6(a) in the vertical direction has a frequency of about 67 Hz related to the rotational speed (4020 rpm) of the drive. The vibrational amplitudes corresponding to the signals after contact are much greater than those before contact. As Fig. 7(a) shows, the judders are cyclic vibrations with a period of about 0.0070 s. Fig. 7(b) shows the vibration magnitudefrequency spectrum corresponding to the vibrational profiles shown in Fig. 6(b). The peak symbolized by “” has a clutch frequency of 27 Hz, which is equal to the rotational speed of the clutch (about 1600 rpm) divided by 60. The drive frequency is slightly elevated to 72 Hz after contact. The peak at a frequency of 144 Hz (symbolized by “  ”) was set as the amplitude modulation frequency, which was twice as large as the drive frequency. The component flaws mainly from the friction lining generate specific defect frequencies in the clutch engagement process. The defect frequencies (fdefect) are obtained as

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Table 5 Density and thermal properties of friction lining and clutch drum.

Clutch drum (SPCC) Friction lining

Density, ρ (kg/m3)

Specific heat, Cp (J/kg  K)

Thermal conductivity, K (W/m K)

Thermal diffusivity, α (m2/s)

7872 2300

481 130.6

60.2 8.5

1.59  10  5 2.83  10  5

Fig. 4. Variations of drive, clutch, and clutch drum rotational speeds and clutch torque with the time of engagement.

fdefect ¼ 7mfdrive þnfdefect, fl (m, n¼1, 2, 3…), where fdrive (¼72 Hz) denotes the drive frequency, and fdefect, fl is the defect frequency of the friction lining. In the present results, m¼2, n¼ 1 and fdefect, fl ¼67 Hz are determined. 137 Hz and 150 Hz (symbolized by “▲”) are then found to be the defect frequencies arising in the process of engagement. Fig. 8(a) shows the variations of the sliding velocity of the friction lining with respect to the clutch drum and the frictional torque with testing time. They were obtained from the first cycle in the code-1 specimen. The sliding velocity slightly increased to the maximum value, and then linearly decreased after this. The frictional torque curve in this time period increased to a level with its mean value gradually close to an asymptotic value (about 8 N-m). Cyclic torque fluctuations then became stronger in the latter part of this time period. The occurrence of the gradually-increasing cyclic torque in the rear part of this time interval is due to the transition from smooth contact to patch configuration contact. This may lead to a small departure (perturbation) in a sinusoidal form from uniform pressure [4]. The pressure perturbation will also give rise to a heating perturbation in a sinusoidal form with the same frequency. The heating perturbation is expressed as a function of the sum of the pressure perturbation and the pressure component (Pth) related to the increase in displacement from the heating. Pth is found to be proportional to (Vx  Vn)  1, where Vn is the critical sliding speed for transition [4]. As Vx increase with time, so Pth in the sinusoidal form also increases its amplitude. This is the so-called thermoelastic instability which can cause the torque to oscillate with an increasing amplitude. In the sliding contact of _ of h_ ¼ μV x p. This process is repeated nominally conformal surfaces, heat is liberated by friction in the interface at a rate (h) for the gradually-increasing new contact spots. These behaviors occur in a cyclic form, with the oscillation amplitude increasing with testing time. When the friction coefficient in Eq. (23) reach the critical velocity Vc, local contact with a judder occurs. Since the λ and En were more or less varied, clutch judders formed intermittently during the engagement process. Fig. 8(b) shows the variations of sliding velocity and frictional torque at the 100th cycle in the code-1 specimen. There is significant difference in the frictional torque compared to that shown in Fig. 8(a). The frictional torques for the 100th cycle in a time period of 1 s after contact can be roughly regressed by a straight line with a positive slope, rather than zero value. Furthermore, the torque fluctuations in the latter time period have amplitudes much greater than those of the 1st cycle; the judder intensity dependent on the frictional torque is thus higher than that of the 1st cycle. Eq. (22) can be used to evaluate pn =p as a function of time in one cycle. Fig. 9(a) and (b) shows the pn =p variations of the code-1 specimen at the 1st and 100th cycles, respectively. In each of these two figures, there exist two pn =p peaks, one at the beginning of the cycle and another at the time with the critical velocity Vc. Between these two peaks the pn =p value is fairly close to 1. The maximum pn =p value occurred at the time of the critical velocity Vc, rather than the maximum friction torque. At the 1st cycle, the maximum pn =p (46.17) occurred at a time of 6.143 s. For the 100th cycle, the maximum pn =p (82.36) occurred at 5.833 s. The maximum pn =p value thus increased with increasing the number of cycles. The maximum values of pn =p for the 18 specimens are shown in Fig. 10(a). The number of cycles with the maximum value of pn =p are shown in Fig. 10(b). The maximum pn =p values of the code-12 and code-17 specimens are the lowest and second lowest, respectively, and that of the code-2 specimen is the highest. In general, the value of maximum pn =p indicates judder severity and the number of cycles with the maximum pn =p reveals the growth rate per cycle to obtain the maximum pn =p value. A

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Fig. 5. Experimental data of drive, clutch, and clutch drum rotational speeds and clutch torque arising at one-second after the maximum sliding velocity for the code-1 specimen operating in the 1st cycle of clutch test.

Fig. 6. Horizontal and vertical vibrational signals created in one second (a) before contact; and (b) after contact for the code-1 specimen in the 1st cycle.

trend is presented that a smaller number of cycles is created frequently together with a larger value of the maximum pn =p, although this feature is not always valid for all of the 18 specimens.

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Fig. 7. (a) Magnification of the vertical vibrational signals created after contact for the code-1 specimen operating in the 1st cycle, and (b) frequency spectrum of vertical vibration signals created after contact for the code-1 specimen operating in the 1st cycle.

Fig. 8. Sliding velocity and frictional torque profiles measured at (a) the 1st cycle, and (b) the 100th cycle in the code-1 specimen.

The temperature rise was evaluated excluding the cumulative thermal effects of sequential cycles before a given cycle. Fig. 11 shows the variations in temperature rise (with the ambient temperature, about 26 1C, as the reference temperature) with time for the 100th cycle for the code-1 specimen. The temperature rise significantly increases before 4.5 s and becomes asymptotic to a constant temperature of 263.1 1C when the maximum value of pn =p ( ¼82.36) is reached. This constant temperature is called the maximum temperature rise. The validity of this prediction cannot be tested directly due to the practical difficulty in measuring the contact temperature. Fig. 12(a) shows the values of the maximum temperature rise for the 18 specimens. All temperatures are evaluated above 200 1C, and strongly depend on the frictional heat input to the friction lining side (Rwq0 in Eq. (25)). The lowest and second lowest values of the maximum temperature rise were obtained for the code-7 and code-2 specimens, respectively. Of note, the highest and second highest values of the maximum pn =p were also obtained for these two specimens. Fig. 12(b) indicates that the data are distributed over a wide range of the maximum temperature rises and the maximum pn =p values, and it does not seem to be possible to obtain a precise regression between these two parameters. Nevertheless, a rough trend is observed in that a specimen with a larger value of max. pn =p is likely to have a smaller value for the maximum temperature rise. This behavior can be explained because a larger separation of two contact surfaces is often formed periodically by a larger value of the maximum pn =p, resulting in a lower value of the maximum temperature rise. Eq. (10) can be used to obtain the variations of the friction coefficient (μ) with time, and thus the sliding velocity at a given clutch cycle. Fig. 13(a) and (b) shows the variations of the friction coefficient with the relative sliding velocity of the friction lining and the clutch drum at the 1st and 100th cycles, respectively. With the modulating behavior demonstrated in Fig. 8 for torque and the definition of μ in Eq. (10), the oscillatory variations in the friction coefficient with sliding velocity can thus be attributed to the thermoelastic instability. The combined effect of friction torque and sliding velocity shows the

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Fig. 9. pn =p solutions for the code-1 specimen in (1) the 1st cycle and (b) the 100th cycle.

Fig. 10. Solutions of the maximum pn =p and the number of clutch cycles with maximum pn =p for 18 specimens.

friction coefficient oscillating with a large amplitude when the engagement was operating at a small sliding velocity. The regression of friction coefficients in the range of low sliding velocities by a straight line is thus roughly acceptable in the determination of dμ/dVx. At the 1st cycle, the friction coefficient fluctuates with respect to its mean value. The mean friction coefficient (μ) at this cycle is quite stable (i.e., the amplitude of μ fluctuations is small). At the 100th cycle, the mean value of the friction coefficient decreases with increasing sliding velocity. These two curves are provided to evaluate the friction coefficient (μ) gradient with respect to the sliding velocity (dμ/dVx). This parameter is one of two that is often used to evaluate the judder resistance of a clutch system. Instead of the local dμ/dVx with either a positive or a negative value, the friction coefficients in the one cycle can be regressed roughly by a straight line with a negative slope. The two straight lines in Fig. 13(a) and (b) shows that the absolute value of dμ/dVx increased with the number of cycles. Fig. 14(a) shows the variations of dμ/dVx with the number of cycles for the code-1, code-2, and code-3 specimens. Each curve exhibits fluctuations. The specimen with greater fluctuations in dμ/dVx has a higher rate of increase of dμ/dVx per cycle. The mean values of the dμ/dVx values for the 100 cycles were determined for these 18 specimens, as shown in Fig. 14 (b). The mean (negative) dμ/dVx values corresponding to the code-7, code-2, and code-10 specimens are in sequence the three highest of the 18 specimens. The values are negative and Z  0.05. A larger negative value of mean dμ/dVx means that a higher friction coefficient is created if the sliding velocity in the process of engagement is reduced at a smaller rate. More noticeable judders are created in specimens with a larger mean dμ/dVx. A comparison of the maximum pn =p and mean dμ/ dVx values of the code-2, code-7 and code-10 specimens indicates that a larger (negative) value of dμ/dVx is favorable for the specimen with a larger value of the maximum pn =p, and thus a stronger judder, except for the code-12 specimen. Td is now defined as the dynamic torque produced at the initial time when the drive revolution had a constant rotational speed. Since the initial time of a constant rotational speed is somewhat hard to identify, it is thus obtained to be the average value of the torques arising at the time period of 1 s after reaching 9.5 m/s as the sliding velocity. Ts is defined as the static

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Maximum temperature rise = 263.1 K

Fig. 11. Solutions of temperature rise in the 100th cycle for the code-1 specimen.

Fig. 12. (a) Solutions of the maximum temperature rise for the 18 specimens, and (b) the data for the maximum pn =p and maximum temperature rise.

torque, which is the maximum value of those torques before completing the full engagement. (Ts-Td) with a positive value can then be used as a parameter to reflect the torque gradient, and thus the probability of severe judder with large cyclic undulations of torque. The experimental data of Ts/Td have been obtained for the 18 specimens. These data show the characteristic that a smaller mean value of Ts/Td was obtained in the specimen with relatively larger fluctuations of Ts/Td. Fig. 15 shows the mean values of Ts/Td for the 18 specimens. The relationship between the parameters of mean dμ/dVx and mean Ts/Td for the 18 specimens is shown in Fig. 16. These 18 specimens can be roughly divided into three classes, and the data in each class can be regressed by straight lines with different slopes. The specimens indicated by “■” are class-A, for which the absolute values of the mean Ts/Td are lower than 1.275. While few studies have discussed what threshold of Ts/Td is acceptable for the clutch design, Ts/Td ¼ 1.2 was adopted in Miyachi et al. [13]. In practice, judders always exist with different intensities, and so acceptable criteria for these two parameters are difficult to set objectively. Ts/Td ¼1.275 is thus adopted in this work to relax the requirement slightly. The code-17 specimen has the smallest values for the mean dμ/dVx and the mean Ts/Td. The data in this class can be regressed by a straight line with a small positive value. The specimens whose Ts/Td value is greater than 1.275 fall into class-B and class-C. The code-2, code-4, and code-7 specimens in class-B can be regressed by a straight line with the largest slope of these three classes; the code-1, code-10, code-12, and code-15 specimens in class-C can be regressed by a straight line with a relatively mild slope, which is larger than that of class-A. The severity of judder arising in a clutch system strongly depends on the mean values of dμ/dVx and Ts/Td. For the specimens in class-B, judder is more sensitive to an increase in mean dμ/dVx than to an increase in mean Ts/Td. For the specimens in class-C, judder is more sensitive to an increase in mean Ts/Td than to an increase in mean dμ/dVx. To eliminate judder, both dμ/dVx and Ts/Td must be sufficiently low. Fluctuations in dμ/dVx associated with the code-2, code-7, and code-10 specimens were found to be the top three strongest. The mean dμ/dVx values of these three specimens are also the top three of the 18 specimens. The fluctuations in

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The 1st cycle

The 100th cycle x: Sliding velocity y: Coefficient of friction

825

x: Sliding velocity y: Coefficient of friction

y= -0.0366x + 0.645

y= 0.000737x + 0.337

Fig. 13. Variations of friction coefficient with relative sliding velocity and their rough regression by a straight line in (a) the 1st cycle and (b) the 100th cycle for the code-1 specimen.

Fig. 14. (a) Variations of dμ/dVx with number of clutch cycles for code-1 to code-3 specimens. (b) Mean dμ/dVx values of the 18 specimens for 100 cycles.

Ts/Td associated with the class-B and class-C specimens are found to be stronger compared to those of class-A specimens. The following conclusions can thus be made: (1) the fluctuation intensities of these two judder parameters have a strong influence on their growth rate per clutch cycle. A specimen with a high fluctuation intensity is apt to have a large mean value of this parameter. (2) The specimens in class-B have a regression line slope different from that of those in class-C. In class-C specimens there is only a high fluctuation intensity in the mean Ts/Td parameter, while in class-B specimens there are high fluctuation intensities for both the mean Ts/Td and dμ/dVx parameters. These results imply that the mean dμ/dVx and mean Ts/Td are two independent parameters. The magnitude of these two parameters is strongly dependent upon their cyclic fluctuation intensity. Analysis of variance (ANOVA) was applied to evaluate the profitabilities of the five control factors: copper fiber and methyl cellulose fiber content levels (wt%), molding pressure (kgf/cm2), temperature (1C), and duration (h). The three parameters mean Ts/Td, mean dμ/dVx, and mean μ are adopted as the objective functions. The signal to noise (S/N) ratios (ηs) for these three objective functions for the 18 specimens were evaluated using the-lower-the-better (LB) characteristic. Normalization was then applied to each η value in order to obtain the normalized S/N ratio (ηn) in a range of 0 o ηn o1. The evaluation of the ηs values in the Lʹ18 orthogonal array was conducted for each of these objective functions, the mean ηs value of each factor was obtained from the ηs values evaluated for a given level. This work is done repeatedly for these three objective functions. In order to take all three objective functions into consideration, the new mean ηs value at every level was obtained from the average of the ηs values established for the objective functions. Table 6 lists the mean ηs values for the five factors at three levels. The effect (E) is defined here as the range between the maximum and minimum ηs values of a control factor in the table. It is often taken as an indication of a control factor's importance with regard to the objective functions. The effective level data show the following sequence: ðEÞMolding pressure 4 ðEÞMolding duration 4ðEÞMethyl cellulose f iber 4 ðEÞCopper f iber 4 ðEÞMolding temperature . This indicates that the molding pressure is the most important factor for the behavior

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demonstrated in these three objective functions. The profitability (Pf) values for these five control factors are shown in Table 7. The highest Pf value was obtained for the molding pressure of the frictional lining, and its profitability is much higher than those of the other factors. 5. Conclusion 1. The judder behavior is theoretically interpreted by starting the hot spots in the conformal contact first, and then developing the thermoelastic instability repeatedly, and finally forming cyclic undulations in the vibrations and frictional torque. The results are found to be consistent with those of the experiment. Theoretical models were developed to

Fig. 15. Mean Ts/Td values for 18 specimens.

Fig. 16. Mean dμ/dVx vs. mean Ts/Td for 18 specimens. Table 6 Mean ηs (S/N ratio) values for the five control factors at three levels. Factor Level, effect and rank Copper fiber (wt%) Methyl cellulose fiber (wt%) Molding pressure (kgf/cm2) Molding temperature (1C) Molding time (h) Level 1 Level 2 Level 3 Effect (Max. Min.) Rank

3.55 3.88 3.82 0.33 4

3.50 3.90 3.84 0.40 3

4.10 3.40 3.75 0.70 1

3.81 3.56 3.89 0.33 5

3.97 3.56 3.71 0.41 2

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Table 7 Profitability (Pf) values for the five control factors. Factor

Degree of freedom f

Sum of square SS

Variance Var

Ratio of variances F

Net sum of squares SS'

Profitability Pf

Copper fiber (wt%) Methyl cellulose fiber (wt%) Molding pressure (kgf/cm2) Molding temperature (1C) Molding time (h) Error Total

2 2 2 2 2 7 17

0.37 0.67 1.89 0.35 0.57 1.14 4.99

0.19 0.33 0.94 0.18 0.29 0.16

1.14 2.04 5.77 1.09 1.74

0.05 0.34 1.56 0.03 0.24 2.78 4.99

0.91 6.79 31.24 0.56 4.87 55.62 100

2.

3.

4.

5.

evaluate the friction coefficient, the relative sliding velocity of the frictional lining and clutch drum and the critical value with regard to producing judders, the contact thermal pressure ratio pn =p, and temperature rise during the engagement of the frictional lining and the clutch drum. These parameters can be applied to predict judder occurrence and behaviors in relation to the Ts/Td and dμ/dVx parameters. Clutch tests were conducted to evaluate the maximum value of pn =p and the maximum temperature rise for the specimens. A specimen with a large maximum pn =p often had a small maximum temperature rise. Most specimens show the characteristic that a larger dμ/dVx is generally advantageous to create a larger maximum pn =p, and thus stronger judder. Mean values of dμ/dVx and Ts/Td for the 100 cycles of the cyclic tests can be used to examine whether specimens are working under thermally stable conditions. In class-A specimens, a small mean dμ/dVx is created along with a small mean Ts/Td. Judder can be attenuated effectively when the two parameters are sufficiently low. In class-B specimens, high fluctuation intensities exist in the parameters of Ts/Td and dμ/dVx; in class-C specimens, high fluctuation intensity only exists in the mean Ts/Td parameter. The fluctuation intensity of the dμ/dVx and Ts/Td parameters influences their growth rate per clutch cycle. A specimen with a high fluctuation intensity is apt to have a large mean value of this parameter. The magnitude of the slope of the dμ/dVx vs. Ts/Td regression line depends on the fluctuation intensities of these two parameters. Small values of dμ/dVx and Ts/Td can be obtained when the fluctuation intensities of these parameters are both small. ANOVA analysis indicates the molding pressure of the frictional lining has the highest effect level for the three objective functions. The highest profitability is also associated with the molding pressure, and its value is much larger than those of the other four control factors.

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