Modelling, Analysis and Simulation of Clutch Engagement Judder and Stick-Slip.
Powertrain systems can be investigated for vibration and noise characteristics through dynamic modelling, analytical and numerical analysis and testing. These investigations lead to a better understanding of the causes and effects of vibration and noise and thus allow improvement in design of trucks, passenger cars and buses.
Clutches in buses, cars, trucks and other vehicles are used to gradually engage the engine to the drivetrain while avoiding unpleasant shocks, jerks and excessive drivetrain wear. A basic clutch (see Figure. 1) has two plates that can be moved together by an actuator that exerts a force on one of the two plates. One is pressure plate that is connected by an axle to the gearbox and remaining part of the powertrain. The other is the friction plate that is connected to the crankshaft. As the clutch engages plates are pushed together by the actuator. When the plates touch, torque is transmitted from the engine to the drivetrain. The vehicle now starts to move. After some amount of time the speeds of two plates become equal. The plates are then sticking and the engine is directly connected to the drivetrain. To achieve a successful engagement, the right input force has to be applied by the actuator. This can be done by the driver through a foot pedal.
Judder is an abnormal vibration in a mechanical system, especially due to grabbing between two friction surfaces. i.e., vibration is produced instead of smooth gradual engagement. Judder in automotive clutches has been attributed to an increasing friction coefficient with decreasing slipping velocity, dubbed as a negative friction gradient, and is known to reduce with system damping (Kani etal., (1992)).
Stick-slip is the surfaces alternating between sticking to each other and sliding over each other. Some examples of stick-slip are music comes from bowed instruments, squeaking sound of balloon when rubbing with fingers, squeaking sound of chalk on a blackboard, squeaky shoes on floor, noise of stopping train, noise of car brake, clutch and tires. Stick-slip in clutch is a phenomenon with dependence on friction characteristics, system dynamics, and external tangential and normal forcing (Martins et al., (2004)). Most studies concentrate on systems with sliding interface between one moving and one stationary surface or between one moving and a substrate moving at a predetermined velocity. A great deal of analytical work has been published for a various systems. In automotive clutches the sliding interface is between two masses with undetermined velocities in a non-linear, non-autonomous system, thus, investigation using numerical methods is appropriate.
Reasons for Clutch Stick-Slip and Judder
Stick-slip motion may be regular or irregular. In the case of frictional sliding, stick-slip can have serious and often undesirable consequences, which resulting in noise, high energy loss (friction), surface damage (wear) and component failure.
Clutch slip is a condition where the full power of engine does not reach the transmission, i.e., the engine speeds up but the vehicle speed does not increase as it should. It occurs when the clutch disc is not gripped firmly and it slips between the flywheel and pressure plate as the member rotates.
Clutch slippage begins as a minor problem. At first, it will occur on initial and hard accelerations. As the vehicle moves on problem will occur on upshifts, downshifts and on any kind of acceleration. If it is left uncorrected, it becomes worse. Eventually, clutch discs becomes so badly worn that there is not enough friction present to move the vehicle.
Some of the reasons that the clutch may slip are overheating of pressure plate, deep grooves and traces of overheating on the pressure plate, facing contaminated on the inner portion, facing contaminated by grease or oil, facing carbonized, damaged diaphragm spring fingers, damaged idle damper, facing material worn down to rivets, facing scored on the flywheel side, wear marks on the release bearing inner bore and gearbox snout worn.
Sometimes problem is not with slipping but with sticking. The common reasons the clutch may stick are broken or stretched clutch cable, misadjusted linkage, air in hydraulic line, mismatched clutch components.
Broken or stretched clutch cable: Cable needs the right amount of tension to push or pull effectively.
Air in hydraulic line: Air affects the hydraulics by taking up space the fluid needs to build pressure.
Misadjusted linkage: When your foot hits the pedal, the linkage transmits wrong amount of force.
Mismatched clutch component: Not all aftermarket parts work with clutch. The right combination of components is essential for proper operations.
The reasons that clutch may judder are loose or worn out clutch facings, loose rivets, misalignment of pressure plate with flywheel, flywheel may be loose on the crankshaft flange and bent splinted clutch shaft.
Rabeih and Crolla (1996) developed a torsional vibration model for a manual transmission truck coupled with a vehicle body longitudinal model and a vertical vibration model. They researched system response both in steady state running and transient running condition during and after clutch engagement. They varied the gradient of coefficient of friction versus stick-slip speed of clutch plates for the transient condition and compared results. For a system with zero or positive gradient of friction they found that the system acted with a transient damped response while for a negative gradient of friction the system acted with an unstable response. The final result they found was that the self-excited vibrations reduced with the high levels of system damping.
Bostwick and Szadkwoski (1998) developed a model of a dry clutch tested with facing dynamometer. Like Centea et al., they also commented that clutch judder also occurs due to misalignments in the drivetrain that may produce fluctuating pressure between sliding surfaces, thermo elastic phenomena on contact surfaces and external torsional vibration in certain resonant conditions. They tested the friction material with negative sloped friction characteristics which produces torsional oscillations and compared the result with numerical model. Stick-slip was found numerically, but not validated with testing; a conclusion was drawn that it was associated with the self-excited vibration. This is reasonable as stick-slip is highly dependent on system dynamics.
Centea et al., (2001) researched dry clutch in manual transmission analytically. Through stability analysis they found that the negative friction slope produce clutch judder and that poor pedal acceleration could produce clutch judder even with a positively sloped friction characteristics. The judder oscillations were higher for more negative friction slopes. Stick-slip occurred in the simulations with the highest amplitude of oscillation before engagement.
In this paper a four degree of freedom torsional system model is used to represent a simplified powertrain system during clutch engagement. This powertrain system can be fitted with manual transmission. Clutch judder and stick-slip are researched qualitatively. First the system is defined, dynamic mathematical model provided and the equations of motion for the system with the clutch sticking and sliding are derived from the free body diagram. Then a stability analysis is performed for the linearized system, which demonstrates through Eigen values that how friction characteristics affect system stability, and therefore, clutch judder.
Stick-slip algorithms are outlined and numerical results for nominal engagements for the research of contributing factors to judder and stick-slip are presented in the form of graphs
THE FOUR DEGREE OF FREEDOM (DOF) POWERTRAIN SYSTEM
The four degree-of-freedom (4DOF) torsional system is shown in figure 2. The free body diagram of 4DOF torsional system is shown in figure 3. The model has four rotating inertias ([J.sub.1], [J.sub.2], [J.sub.3] and [J.sub.4]) connected by shaft elements ([k.sub.1] and [k.sub.2]) or frictional contact (clutch engagement). The first inertia [J.sub.1], represents the engine and flywheel, the second inertia [J.sub.2], represents
the one side of the clutch; they are connected by shaft stiffness [k.sub.1]. The third inertia [J.sub.3], represents the other side of the clutch and transmission gear set. The second and third inertias [J.sub.2] and [J.sub.3], are connected by frictional contact. The fourth inertia [J.sub.4], represents the driveline inertia and vehicle mass; they are connected by shaft stiffness [k.sub.2].
The equations of motion for the system with the clutch ([J.sub.2] and [J.sub.3]), either in stiction (sticking) or sliding are derived from the free body diagram shown in figure 3 and as follows
[J.sub.1][[theta].sub.1] + [k.sub.1]([[theta].sub.1]-[[theta].sub.2]) + [c.sub.1][[theta].sub.1] =[T.sub.1] (1)
[J.sub.2][[theta].sub.2] + [k.sub.1]([[theta].sub.1]-[[theta].sub.2]) + [c.sub.2][[theta].sub.2] =[T.sub.c] (2)
[J.sub.3][[theta].sub.3] + [k.sub.2]([[theta].sub.3]-[[theta].sub.4]) + [c.sub.3][[theta].sub.3] =[T.sub.c] (3)
[J.sub.4][[theta].sub.3] + [k.sub.2]([[theta].sub.3]-[[theta].sub.4]) + [c.sub.4][[theta].sub.4] =[T.sub.4] (4)
[J.sub.1][[theta].sub.1] + [k.sub.1]([[theta].sub.1]-[[theta].sub.2/3]) + [c.sub.1][[theta].sub.1] =[T.sub.1] (5)
([J.sub.2] + [J.sub.3])[[theta].sub.2/3] -[k.sub.1]([[theta].sub.1]-[[theta].sub.2/3]) + [c.sub.2][[theta].sub.2/3] + [k.sub.2]([[theta].sub.2/3] -[[theta].sub.4]) + [c.sub.3][[theta].sub.2/3] = 0 (6)
([J.sub.4][[theta].sub.4] + [k.sub.2])[[theta].sub.2/3] -[[theta].sub.4])+[c.sub.4][[theta].sub.4] =[T.sub.4] (7)
[[theta].sub.2/3] represents both coordinate 2 and 3; they are locked together and have the same solution in this state.
Nominal parameters used for the simplified powertrain system are given in Table 1.
Stability analysis for the slipping clutch can be performed for different friction characteristics. This is achieved by neglecting the non-autonomous or constant parts of the system of equations (1) to (4) and finding the eigen values of its system matrix (Centea et al., (2001)).
In the 4DOF torsional system the engine torque [T.sub.1] the resistance torque, [T.sub.4], are either non-autonomous or constant. The clutch torque [T.sub.c], is autonomous with a constant pressure. The only non-constant parameter in the clutch torque equation when sliding is the friction coefficient ([[micro].sub.s]). The coefficient of friction is dependent on the relative sliding velocity. Refer Figure. 4.
Assuming constant pressure across the surface of the clutch plates, the equation for clutch torque is given by,
[T.sub.c] = [N.sub.s] [R.sub.m][[micro].sub.s] F (8)
[N.sub.s] - Number of friction surfaces
[R.sub.m] - Clutch mean radius
[[micro].sub.s] - Coefficient of sliding friction
F - Clutch actuating force (friction force)
The mean clutch radius is determined from the clutch outside ([r.sub.0]) and inside ([r.sub.i]) radii
[R.sub.m] = 2([r.sub.0.sup.3]-[r.sub.i.sup.3])/3([r.sub.0.sup.2]-[r.sub.0.sup.2]) (9)
The coefficient of friction can be linearized to an initial condition value and gradient, [m.sub.s] Refer figure. 4.
[[micro].sub.s] = [[micro].sub.s0] + [m.sub.s]([[theta].sub.2]-[[theta].sub.3] (10)
Thus Tc can be written as
[T.sub.c] = [[m.sub.s]([[theta].sub.2] -[[theta].sub.3])][N.sub.s][R.sub.m]F + [[micro].sub.s0][N.sub.s][R.sub.m]F (11)
Now the sliding clutch torque T is defined by two separate terms.
[[m.sub.s] ([[theta].sub.2] - [[theta].sub.3])][N.sub.s][R.sub.m]F is dependent on the relative velocity between the two sides of the clutch and [[micro].sub.s0][N.sub.s][R.sub.m] F is constant.
Coulomb friction model is the most commonly used friction model. It is also known as dry friction and it is being used in lubricated and contact boundaries modelling. The coulomb friction model is simplified as
F = [F.sub.c]sign(v)
[F.sub.c] is the coulomb friction sliding force and [nu] = [theta] is the sliding speed (velocity)
Coulomb sliding friction force is defined as
where, N - normal load in the contact
Coulomb friction can cause problems in simulations due to the properties of sign function. The coulomb friction model is shown in the Figure.5. It is noticed that Fc is independent of the sliding velocity between the surfaces under its action (Edson Luciano Duque E.L. et al., (2012)).
A viscous friction model can be formulated as
F = [k.sub.v]v
where [k.sub.v] is the viscous coefficient
The viscous friction model is shown in the Figure.6.
Since the coulomb friction model is problematic as regards both the analysis and simulation of a system behavior, a combination of the viscous friction model and the coulomb friction model could be advantageous. Using the combined Coulomb and viscous friction model eliminates the difficulty in determining the friction force at zero sliding speed both at start up and at direction change. The combined coulomb and viscous friction model is shown in the Figure.7. (Duque et al., (2012)).
The friction force F, value is taken as 6000 N in this work (refer Table 1).
The system, equations (1) to (4) can now be analyzed as the linearized homogeneous system as follows
[mathematical expression not reproducible] (12)
[mathematical expression not reproducible] (13)
[mathematical expression not reproducible] (14)
[J.sub.4][[theta].sub.4]-[k.sub.2]([[theta].sub.3]-[[theta].sub.4]) + [c.sub.4][[theta].sub.4] = 0 (15)
The final equation of motion for the system will have the standard form of
The system matrices are
[mathematical expression not reproducible]
Reduced to first order form
[mathematical expression not reproducible]
The system matrix is
[mathematical expression not reproducible]
where [0.sub.4] is a 4 x 4 zero matrix and [I.sub.D4] is a 4 x 4 unit matrix.
The system matrix is used for standard stability analysis. With zero damping on coordinates, the damping matrix takes damping as zero and only includes the effect of the friction slope. The stability analysis yields two eigen values that are complex conjugate pairs representing the upper (coordinates 1 and 2) and lower (coordinates 3 and 4) system frequencies of torsional vibration and two eigen values that are zero frequency pairs representing the rigid body models of the upper and lower system.
The eigen values and its effect on system when disturbed is given in Table 2.
The eigen values of the system matrix A is obtained by using E = eig (A) in matlab. Three different gradients of the coefficient of friction are used for the stability analysis; zero gradient, positive gradient and negative gradient.
Table 3. shows the complex conjugate pairs and corresponding frequencies using the nominal parameters of the simplified powertrain system given in Table.1 with three different values of the gradient of the coefficient of friction [m.sub.s].
By examining the real part of the complex conjugate paired eigen values, the stability of the torsional vibration modes they represent can be determined. For a zero gradient, the system has two pure imaginary complex conjugate pairs and zero real complex conjugate pairs will respond with undamped oscillations (as shown in Figure. 8) in those modes to any disturbance to equilibrium. For a positive gradient, the system has two stable roots and will respond with damped oscillations (as shown in Figure. 9). For the negative gradient the system has two unstable roots and will exhibit self-excited oscillation (as shown in Figure. 10).
When the clutch engages the system changes state as there is one less degree-of-freedom. In this model 4DOF is used, so when clutch engages the system changes to 3DOF. If there is not enough holding torque TST, the clutch may slip again. In the numerical simulation the algorithm determines the state of the system at every time step, the simulation is piecewise and non-linear. The state is determined by first checking the stick-slip inequality and then if necessary checking the holding torque versus shearing torque inequality. The equations of motion for each state, the sliding clutch (equations (1) to (4)) or sticking clutch (equations (5) to (7)) are then solved as dictated. While sliding the direction of sliding dictates the sign of the clutch torque. If the system commences sliding from sticking, the sign of clutch torque needs to be determined from the torque flow in the system.
An algorithm has been used to model this clutch stick-slip. Tc is a non-linear friction torque, a function of sliding velocity and normal force
[mathematical expression not reproducible] (17)
[[epsilon].sub.tol] - tolerance of zero velocity for numerical simulations
[T.sub.s] - clutch slipping torque dependent on kinetic friction
[T.sub.ST] - clutch holding torque dependent on static friction
The clutch slipping torque is same as the actual clutch torque Tc fequation (8))
[T.sub.S] = [N.sub.m][R.sub.m][[micro].sub.s] F (18)
The clutch holding torque, [T.sub.ST], is the same equation with the sliding coefficient of friction replaced with the static coefficient, [[micro].sub.s0]
[T.sub.ST] = [N.sub.m][R.sub.m][[micro].sub.s]F (19)
The shearing torque, [T.sub.INT], between the clutch plates with zero relative velocity is difficult to calculate exactly, Referring to equations (2) or (3), it can be seen that we now have two unknowns in each equation, angular acceleration and clutch torque, they cannot be solved for [T.sub.c], but by combining the two equations the shearing torque is cancelled: providing the sticking equation (6). Numerically this can be worked around to give a reasonably accurate solution. This is achieved by using the acceleration value from the proceeding time step as an approximation so as to solve equation (2) and (3) for an approximation value of [T.sub.c], in this case actually [T.sub.INT]. Using this method a fine time step is required around the point of interest.
Thus the shearing torque can be determined through the left-hand side of the clutch
[mathematical expression not reproducible] (20)
or the right-hand side of the clutch
[mathematical expression not reproducible] (21)
Equations (19) and (20) can also be combined to
[mathematical expression not reproducible] (22)
In simulations, the flow of equations (20) and (21) can be compared to assess error, as the values for [T.sub.INT] from each equation should be the same. This is critical in the region where the value of [T.sub.INT] determines the state of the system. Reducing the time step so the difference between values is minimal is essential.
Numerical simulations have been programmed in MATLAB for the powertrain system of Figure. 2. and the Simulation steps are given below,
* Solves equation of motion (16) for clutch engagement
* Piecewise on engaged (3DOF) and disengaged (4DOF) states
* Equation of motion reduced to first order for use with MATLAB's runge-kutta method
* Includes stick-slip algorithms which govern non-linear dynamics
* Coordinates 1 and 2 rotated at typical engine speeds for a full throttle shift from second to third; Angular velocity [[theta].sub.1]=[[theta].sub.2]=630 rad/s.
* Coordinates 3 and 4 rotated at the lower speed; Angular velocity [[theta].sub.3]=[[theta].sub.4]= 210 rad/s.
* Same values for initial angular displacement twist in each simulation; [[theta].sub.1] = [[theta].sub.2] = [[theta].sub.3] = [[theta].sub.4] = 0.
* Engine torque, T1, is assumed as a constant value of 200 Nm.
* Vehicle aerodynamic drag, rolling resistance and gradient torques, T4, are combined and set to a constant value of 100Nm.
* Static coefficient [micro]s0 is set to 0.13.
* The tolerance of relative velocity for the stick-slip algorithm is set at 5 rad/s.
* The initial conditions for the simulation provide balanced spring torque.
* Time stepping reduced around stick point.
* Care should be taken when using tolerance of zero velocity.
In the simulations, the influence of the following parameters on stick-slip were examined:
1. Static friction coefficient values.
2. Applied pressure fluctuations.
3. Judder approaching engagement.
4. Externally applied torque fluctuations.
RESULTS AND DISCUSSION
Static Friction Coefficient Values
Simulations were performed using the friction coefficient determined through equation (10) with a [[micro].sub.s0] value of 0.13 and the three different values for [m.sub.s] given in Table. 3. Figure. 11. 12 & 13 presents the velocities of the coordinates (coordinates 1 and 2 in upper system and coordinates 3 and 4 in lower system). Coordinate 3 alone responds with resonance because the value of coordinate 4 is much greater than coordinate 3. Refer Table 1. The results demonstrate the centre (undamped), stable or unstable dynamics predicted through stability analysis for different slopes of the friction gradient. This can be seen in the velocity plots or more clearly seen in the torque plots shown in Figure 14, 15 & 16 and it presents the spring torques in the two shafts over the engagement of coordinated 2 and 3. In Figure 14, 15 & 16.
Shaft 1 torque has more number of frequencies than shaft 2 torque because shaft 1 is rotating faster than the shaft 2. For the positively sloped gradient, torsional oscillations were damped. For the negative sloped gradient, torsional oscillations were self-excited. For zero gradient, torsional oscillations were stable or undamped.
Applied Pressure Fluctuations
The fluctuation of applied pressure was found to have a strong effect on whether the system would stick-slip when the clutch approaches engagement. The pressure was fluctuated at the systems two natural frequencies (35.6 Hz and 88.5 Hz) with the amplitude at 10 percent of the base pressure value, the gradient of the coefficient of friction was set as zero (i.e. constant friction).
Clutch slipping torque,
T = [N.sub.s][R.sub.m][[micro].sub.s]F + 0.1* [N.sub.s][R.sub.m][[micro].sub.s]F * sin(2[pi] * forcingfrequency* time)
Coefficient of friction gradient,
In real powertrain systems, the hydraulic system will not have oscillations at values corresponding so closely to natural frequencies in the system with constant amplitudes, however for this analysis the intention was to obtain maximum response from the pressure fluctuations, thus these values were used.
Figure 17, 18, 19 & 20 shows the pressure fluctuations results of the simulations.
From the Figure 17 we can notice that for the 35.6 Hz simulation the lower system (coordinates 3 and 4) responds with resonance; as a result the torsional vibration is large by the time engagement occurs. From the Figure 18, it is notable that the stick-slip cycle occurs for around 0.15s (i.e. from 0.35s to 0.50 s). Here stick-slip algorithm is included. The torsional oscillations cause the shearing torque, [T.sub.INT], to exceed the holding (sticking) torque, TST, so the engaged clutch slips after a short period of sticking. When holding torque exceeds the shearing torque the disengaged clutch will sticks again. The stick-slip cycle repeats itself until the torsional oscillations are no more enough to cause the sticking system to break away again
[T.sub.ST] < [T.sub.INT]
[T.sub.ST] [greater than or equal to] [T.sub.INT]
From the Figure 19. it can be seen, for the 88.5 Hz simulation the upper system (cords 1 and 2) responds with resonance. Stick-slip occurs again for the same reasons, however the frequency is higher corresponding to the higher frequency torsional oscillations. From the Figure 20, it is notable that the stick-slip occurs for around 0.12s (i.e. from 0.38s to 0.50s). If the system includes damping then the torsional oscillations would not have strong enough to cause stick-slip, depending on the strength of the damping.
The reason why the upper system responds with resonance in Figure 17 and the lower system responds with resonance in Figure 19 is given below
For upper system,
[mathematical expression not reproducible]
[f.sub.1]=[[omega].sub.1]/2[pi] = 88.5Hz
For lower system,
[mathematical expression not reproducible]
[f.sub.1]=[[omega].sub.2]/2[pi] = 35.6Hz
Refer Table 1. for the parameter values. The resonance occurs on the upper and system when the natural frequency comes near or equal to the forcing frequency.
Note, in Figure 18 & 20, in all the shearing torque/holding torque plots, the cyan color line represents the the relative velocity of the engaging clutch plates (coordinates 2 and 3), the blue and green color line represents the holding torque, and the red color line represents the actual torque between clutch plates, either friction torque (during sliding) or shearing torque (during sticking)
Judder was found to increase the likelihood of stick-slip. Figure 21 & 22. Shows one of the friction curves used and the result for this case. For the simulation the static coefficient of friction was increased from 0.13 to 0.16. In Figure 21, the judder is difficult to see in the velocity plot, however, it begins to grow around 0.25s. In this simulation also stick-slip algorithm included. In large plot (Figure 22), the sliding torque can be seen to increase up to engagement, where there is a short period of sticking, until the shearing torque exceeds the holding torque, this is followed by a short period of slipping, after which plates stay locked. It is clearly understand from the plot that even small amount of torsional oscillation would have produced more stick-slip cycles.
Externally Applied Torque Fluctuations
Like judder, the external fluctuations also found to increase the likelihood of stick and slip. Figure 23. presents the results of a simulation with the engine torque fluctuated at 88.5 Hz, the natural frequency of the upper system. The expected resonant condition when slipping can be seen in the velocity plot (Refer Figure 24). The system experiences sticking and slipping, after clutch engagement (Refer Figure. 25). As with judder simulation of Figure 22. it can be seen from the holding torque/shearing torque plot that even a small amount of torsional oscillation would have produce severe stick-slip. The effect of changing the value of static coefficient is straightforward. If the static coefficient of friction is increased from 0.13 to 0.14 or 0.15, then the torsional oscillation would reduce and so stick-slip will be reduced. This is due to the sticking torque being larger (A. Crowther et al., (2004)).
In this work the systematic approach for modelling, analysis and simulation of clutch engagement judder and stick-slip for 4DOF powertrain system using Matlab is presented. Stability analysis was performed for the system showed that if the friction coefficient characteristics were flat, no damping would be present due to engagement. If the characteristics were positive, then positive damping would be present in the system. If the characteristics were negative, then negative damping or instability would be present in the system. Because of more heat dissipation in the positive slope friction coefficient, the upper system engaged with lower system earlier than the flat friction coefficient. For negative slope friction coefficient the upper and lower system never engage. It was found that shaft 1 torque has more number of frequencies than the shaft 2 torque because shaft 1 torque rotated at high speed.
A stick-slip algorithm was developed for the clutch engagement, the algorithm required numerical solutions with a fine time step to allow the use of the previous time steps acceleration data to solve the shearing torque between two masses in contact. The occurrence of slipping state from the sticking state was found to dependent on two broad parameters, the holding torque, which is dependent on clutch pressure, static friction and clutch geometry, and the shearing torque, which is dependent on the system dynamics. It was found that even with small amount of torsional oscillation, the judder and external torque fluctuations were found to increase the likelihood for the system to stick and slip. Reducing the torsional oscillation by damping was found to decrese the likelihood of stick-slip and increasing the static coefficient of friction was also found to decrease the likelihood of stick-slip. The results obtained from analytical and numerical simulations can be extended to a experimental testing program and the details of the test rig and testing program are provided in (Crowther and Zhang (2004)).
1. Kani, H., Miyake, J. and Ninomiya, T., "Analysis of the friction surface on clutch judder," JSA Eng. Rev., Tech Notes, 1992, 12(1), 82-84
2. Rabeih, E. and Crolla, D., "Coupling of Driveline and Body Vibrations in Trucks," SAE Technical Paper 962206, 1996, doi:10.4271/962206.
3. Bostwick, C. and Szadkowski, A., "Self-Excited Vibrations During Engagements of Dry Friction Clutches," SAE Technical Paper 982846. 1998, doi: 10.4271/982846.
4. Martins, J. A. C., Oden, J. T. and Simoes, F. M. F., "Recent advances in engineering science-a study of static and kinetic friction," 2004.
5. Centea, D., Rahnejat, H. and Menday, M., "Non-linear multibody dynamic analysis for the study of clutch torsional vibrations (judder)," Applied Mathematical Modelling, Volume 25, Issue 3, January-February 2001, Pages 177-192, 2004. doi:10.1016/S0307-904X(00)00051-2
6. Centea, D., Rahnejat, H. and Menday, M., "The influence of the interface coefficient of friction upon the propensity to judder in automotive clutches," Proceedings of Institution of Mechanical Engineers, Part D: Journal of Automobile Engineering, 213 (3), pp. 245-258, 1999.
7. Crowther A., Zhang N, Liu D.K., Jeyakumaran J.M., "Analysis and Simulation of Clutch Engagement Judder and Stick-Slip in Automotive Powertrain Systems," Proceedings of the Institution of Mechanical Engineers, Part D: Journal of Automobile Engineering December 1. 2004 vol. 218 no. 12 1427-1446, doi: 10.1243/0954407042707731.
8. Crowther, A. R. and Zhang, N, "Dynamic modelling and applications for passenger car powertrains," Acoustics Australia, Vol. 32 April (2004) No. 123
9. Edson Luciano Duque E. L, Barreto Marco Antonio, Fleury Agenor de Toledo, "Use of different friction models on the automotive clutch energy simulation during vehicle launch", ABCM Symposium Series in Mechatronics - Vol. 5, Page 1376, 2012.
A - system matrix
c,C - damping coefficient (Nms /rad)
F - clutch actuation force (N)
I - Inertia (matrix) (kg /m2)
J - inertia (kg /m2)
k, K - stiffness coefficient (Nm /rad)
[m.sub.s] - gradient of friction coefficient
[N.sub.s] - number of clutch friction surfaces
[r.sub.i] - clutch inside radius (m)
[r.sub.0] - clutch outside radius (m)
[R.sub.m] - clutch mean radius (m)
[T.sub.INT] - clutch shearing torque (in stiction) (Nm)
[T.sub.s] - slipping clutch torque (Nm)
[T.sub.ST] - clutch holding torque (in stiction) (Nm)
[theta] - angular displacement (rad)
[epsilon] - tolerance of zero velocity (rad /s)
[[micro].sub.s] - sliding frictioncoefficient
[[micro].sub.s0] - static friction coefficient
NOTATION SPECIFIC TO A 4 DOF SYSTEM
[T.sub.c] - actual clutch torque(Nm)
SUBSCRIPTS FOR INERTIA ELEMENTS AND ANGULAR COORDINATES
1 - engine and flywheel /torque converter pump
2 - clutch drum
3 - clutch hub
4 - driveline and vehicle
Kumaraguru College of Technology
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Jayabalaji G, M.Tech
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Table 1. Nominal parameters used for simplified powertrain system. Parameter Value Parameter Value J1 0.2 kg/m2 c2 0 Nms/rad J2 0.1 kg/m2 c3 0 Nms/rad J3 0.1 kg/m2 c4 0 Nms/rad J4 3 kg/m2 Ns 10 k1 20600 Nm/rad F 6000 N k2 4840 Nm/rad ro 0.0635 m c1 0 Nms/rad ri 0.051 m Table 2. Eigen value and its effect on system when disturbed Eigen value Effect on system when disturbed Driven away from steady-state Positive real number value Negative real number Driven back to steady-state value Remains at position at which it was Zero disturbed Identical to another eigen value Effects cannot be determined Complex, Positive real number Oscillates around steady-state value with increasing amplitude Complex, Negative real number Oscillates around steady-state value with decreasing amplitude Imaginary Oscillates around steady-state value with constant amplitude Table. 3. Stability analysis results for various gradients of the coefficient of friction Gradient of Real part of Imaginary part of Frequency coefficient of complex conjugate complex (Hz) friction, ms pairs conjugate pairs 0.0000 +5.5588i 88.4707 0.0000 -5.5588i 88.4707 0 0.0000 +2.2364i 35.5929 0.0000 -2.2364i 35.5929 -0.0115 + 5.5587i 88.4690 -0.0115 - 5.5587i 88.4690 0.0001 -0.0167 + 2.2362i 35.5919 -0.0167 - 2.2362i 35.5919 0.0115 + 5.5587i 88.4690 0.0115 -5.5587i 88.4690 -0.0001 0.0167 + 2.2362i 35.5919 0.0167 - 2.2362i 35.5919
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|Author:||Sivanesan, M.; Jayabalaji, G.|
|Publication:||SAE International Journal of Passenger Cars - Mechanical Systems|
|Date:||Apr 1, 2017|
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