2007-01-2233

Transient Clunk Response of a Driveline System: Laboratory Experiment and Analytical Studies

Jaspreet S. Gurm, Wan Joe Chen, Amir Keyvanmanesh, Takeshi Abe Ford Motor Company

Ashley R. Crowther, Rajendra Singh The Ohio State University

Copyright © 2007 SAE International

ABSTRACT

A laboratory experiment is designed to examine the clunk phenomenon. A static torque is applied to a driveline system via the mass of an overhanging torsion bar and electromagnet. Then an applied load may be varied via attached mass and released to simulate the step down (tip-out) response of the system. Shaft torques and torsional and translational accelerations are recorded at pre-defined locations. The static torque closes up the driveline clearances in the pinion/ring (crown wheel) mesh. With release of the applied load the driveline undergoes transient vibration. Further, the ratio of preload to static load is adjusted to lead to either no-impact or impact events. Test A provides a ‘linear’ result where the contact stiffness does not pass into clearance. This test is used for confirming transient response and studying friction and damping. Test B is for mass release with sufficient applied torque to pass into clearance, allowing the study of the clunk. A set of non-linear differential equations describe the experiment and the applicable dry friction coefficients are experimentally found. Various test conditions (corresponding to no impacts, and single-sided or double-sided impacts) are successfully simulated. Numerical and experimental time histories compare well.

INTRODUCTION

Clunk is an impulsive response in the powertrain which is typically initiated by a sharp torque reversal such as from throttle (and engine torque) tip-in or tip-out [1]. The torsional vibration response at the lowest mode, or ‘driveline shuffle or surging’, causes the gears to impact after they pass through the clearance between their backlashes. The oscillation frequency is under 10 Hz and varies with transmission ratio [1-2]. Also, in relation to the size of the torque step there is a corresponding mean change in rigid body motion [2]. Some results (at the vehicle and drivetrain level) are available for clunk experiments [1-3]. Likewise simulations have been used to study clunk, e.g. [4] where the combined effect of

transients in engine torque, braking and road load are considered. In this paper we report a test device reduced in complexity so as to isolate clunk from additional non-linear sources, however friction needed to be considered. Faced with results from any of these experiments, the nature of non-linear response may be difficult to fully understand. Correct diagnosis usually requires numerous tests. Benefits in terms of time and cost reduction could be realized by using analytical studies. We thus apply a non-linear mathematical simulation technique [2] to understand the physics related to the impact event. Any simulation model is limited by its simplifying assumptions so the bench experiment needs to be designed and conducted. In this paper, the steps involved in experiment, rig and model development and findings are discussed. The laboratory experiment helps to refine and then correlate the mathematical model with measured results.

DEVELOPMENT OF AN EXPERIMENT FOR TRANSIENT

The driveline set-up is shown in Figure 1 and includes driveshaft, rear axle and axle shafts. The axle flanges are rigidly attached to the test-bed. The front end of the driveshaft is connected to the torsion bar and supported by a bearing. The torque is measured via a Wheatstone bridge at each end of the driveshaft and at each axle. The length and section of the torsion bar were selected to give a driveline shuffle frequency of around 2-3 Hz. This is similar to 1st gear in a typical vehicle.

The significant clearances in the set-up are in the pinion/ring gear mesh and in the differential gears. The torque is applied to the set-up via masses suspended on an electromagnet attached to the end of the torsion bar (Figure 2). The mass of the torsion bar and magnet provides a mean (static) load, and suspended

masses provide an applied load, , giving total

preload,

sT1

aT1

as TTT 111 += . The clearances in the driveline are closed due to the load. When the masses are

released, the step down of applied torque causes the driveline to undergo transient vibration and it may pass into the clearance. The ratio of applied to static load,

sar TTT 11= , is adjusted to lead to either no-impact (Test A) or impact (Test B) events. Test A forces the system to behave in a ‘linear’ manner (no impacts). It is used for matching system response and studying friction and damping without the complexity of the clearance nonlinearity. Conversely, the larger step down excitation, under Test B, forces the driveline pass into the clearance leading to one or more impact events.

Figure 1. Driveline Experiment Set-Up: With the

Driveshaft Flanges Rigidly Affixed the Driveline is Preloaded and then Excited by Mass Release at the

Electromagnet.

Figure 2. Masses Applied to the Torsion Bar.

Table 1 provides a summary of experiments conducted with a one-piece driveshaft fitted. By adding steel plates on top the torsion bar end the static torque and inertia may be adjusted. Test cases are: Case 1 with steel plates (higher static torque, inertia); Case 2 without steel

plates. For each case the adding of masses (Figure 2) increases the applied load. A higher static torque requires a larger step down to reach clearances in the system. The table also indicates a clunk (Y) or no-clunk (N) result.

Run AddedMass (kg)

1T

(Nm) sT1

(Nm) fT

(Nm)rT Clunk

Event

Case 11 6.67 -158.8 -101 66 0.57 N 2 13.3 -218.5 -101 79 1.16 N 3 16.0 -241.2 -101 96 1.39 N 4 20.0 -275.0 -101 102 1.72 N 5 21.3 -286.2 -101 110 1.83 Y

Case 26 4.0 -90.0 -55 40 0.63 N 7 8.0 -125.0 -55 54 1.27 N 8 12.0 -161.0 -55 67 1.93 Y 9 16.0 -195.5 -55 80 2.55 Y 10 20.0 -228.5 -55 91 3.15 Y 11 24.0 -262.5 -55 104 3.77 Y 12 28.0 -297.0 -55 119 4.40 Y 13 32.6 -334.5 -55 127 5.08 Y

Table 1. Summary of Test Conditions.

TYPICAL EXPERIMENTAL RESULTS NO IMPACT TEST

Figure 3 shows measured torque on the driveshaft (front and rear) and on the axle shafts for Case 1 Run 4 (Table 1) which is a no-impact event. The torque trace can be divided into two distinct regions: First, the masses are released at Point A leading to a sudden change in the torque. From Point A to Point D, the reduction in the amplitude of peaks is almost linear, indicating that this region is dominated by dry friction damping. The frequency of response is around 2.3 Hz. For the region from Point D to standstill, the frequency of response increases to 4.5 Hz. To better understand this behavior, the phase plane plot of torsion bar is plotted in Figure 4. This plot is derived from measured acceleration at the overhanging end of the torsion bar. Moving from the origin the phase plane exhibits non-linear characteristics until Point D, after which the spiral decay of a typical linear system is evident. From this we can conclude that the system exhibits non-linear characteristics even with no impact event(s).

IMPACT TEST

Figure 5 shows the measured torque traces for Case 2 Run 9 (Table 1) for which driveline passes into but not right across, and then out of the clearance, leading to the case of one single-sided impact. Using driveshaft torque we define Point B as the time when the gears pass into clearance and Point C as out of clearance (impact). Figure 6 looks more closely at the impact, showing right axle velocity and torque and angular acceleration at the rear end of the driveshaft. The acceleration time history shows transient impulsive response attributable to clunk and friction discontinuities.

0 0.5 1 1.5 2-250-150-50

Torq

ue (N

m)

0 0.5 1 1.5 2-250-150-50

Torq

ue (N

m)

0 0.5 1 1.5 2-400

-200

Torq

ue (N

m)

0 0.5 1 1.5 2-400

-200

Torq

ue (N

m)

Time (s)

(a)

(b)

(c)

(d)

A D

Figure 3. Measured Driveshaft Torques for No-Impact Test, Case 1, Run 4: (a) Front End of Driveshaft; (b) Rear End of Driveshaft; (c) Right Axle; (d) Left Axle.

-0.2 0 0.2-6

-4

-2

0

2

4

6

8

Velocity (m/s)

Acc

eler

atio

n (m

/s2 )

D

A

Figure 4. Phase Plane Constructed from Measured Acceleration at Torsion Bar End for No-Impact Test,

Case 1, Run 4.

0 0.5 1 1.5 2-200

-100

0

Torq

ue (N

m)

0 0.5 1 1.5 2-200

-100

0

Torq

ue (N

m)

0 0.5 1 1.5 2-300-200-100

0

Torq

ue (N

m)

0 0.5 1 1.5 2-300-200-100

0

Torq

ue (N

m)

Time (s)

C D

A

B (a)

(b)

(c)

(d)

Figure 5. Measured Driveshaft Torques for Impact Test, Case 2, Run 9: (a) Front End of Driveshaft; (b) Rear End

of Driveshaft; (c) Right Axle; (d) Left Axle.

0 0.2 0.4 0.6 0.8 1-0.1

00.1

Velo

city

(rad

/s)

0 0.2 0.4 0.6 0.8 1-200

-100

0

Torq

ue (N

m)

0 0.2 0.4 0.6 0.8 1-100

-500

Acce

lera

tion

(rad

/s2 )

Time (s)

(a)

(b)

(c)

A B

clunk impulse

C

Figure 6. Measured Responses for Impact Test, Case 2, Run 9: (a) Right Axle Velocity; (b) Torque at Rear End of

Driveshaft; (c) Acceleration at Rear End of Driveshaft.

TORSIONAL DRIVELINE MODEL

The lumped parameter torsional model of the driveline is an adaptation of those used for modelling automatic transmission powertrains [2]. As illustrated by Figure 7 it has four degrees of freedom and one geared element. Coordinates of rotation are 1θ , torsion bar, 2θ ,

driveshaft upper flange and joint, 3θ , final drive pinion

gear and 4θ , final drive ring gear (also known as the crown wheel). The system is grounded at the wheel location. Descriptions of system parameters are given in Table 2. With global coordinate vector,

{ }T4321 θθθθ=θ , the equations of motion are:

)()()()()( tttt TθKθCθJ =++ θ , (1)

Where J, C and K are the inertia, damping and stiffness matrices:

[ 4321 JJJJdiag=J ], (2a)

, (2b)

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

+−−+−

−+−−+

=

4342

43443

3443342

32323

23231212

12121

000

000

ccrcrrcrrcrcc

ccccccd

C

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

+−−+−

−+−−

=

4342

43443

3443342

32323

23231212

1212

000

000

kkrkrrkrrkrkk

kkkkkk

K . (2c)

And, T is the torque vector, as given later by Eq. (6).

Tk12

J ,1

1

2J , J ,3

4J ,34c

23k

12c c23

k4 4c

Tfk34

1θ θ 23θ

θ4d1

Figure 7. Driveline Torsional Model.

Symbol Parameter or Dimension J1 Inertia of torsion bar and electromagnet

(k ²)J2 Inertia of driveshaft upper flange and joint (k ²)J3 Inertia of final drive pinion and driveshaft

l fl d j i t (k ²)J4 Inertia of final drive ring gear k12 Torsional stiffness of upper shaft k23 Torsional stiffness of driveshaft k34 Translation stiffness of gear mesh k4 Torsional stiffness of combined axles

(N / d)d1 Viscous damping of torsion bar c12 Viscous damping of upper shaft c23 Viscous damping of driveshaft c34 Viscous damping of gear mesh c4 Viscous damping of combined axles

(N / d)r3 Radius of final drive pinion r4 Radius of final drive ring (crown wheel)

Table 2. Inertia, Stiffness and Damping Elements of the Torsional Model of Figure 7.

Torsional Mode 1 2 3 4 Damped Frequency (Hz) 2.20 186 486 562

Damping Ratio (%) 3.72 2.47 2.48 3.02 Modal Vector

1θ 0.60 0.00 0.00 0.00

2θ 0.58 0.36 0.94 0.67

3θ 0.53 0.89 0.08 0.62

4θ 0.13 0.27 0.33 0.41

Table 3. Damped Natural Frequencies, Damping Ratios and Mode Shapes of the Torsional Model of Figure 7.

Damping parameters are assigned to give reasonable damping ratios, with the inertial (viscous) damping parameter, d1, the effective damper of the lowest mode [2]. Modal properties are given for the model in Table 3. Mode 1 is of main interest and is the global motion of all coordinates at 2.2 Hz (for Case 1) and 2.97 Hz (for Case 2). The difference between the two cases is due to the addition of steel strips to the end of the torsion bar, thereby increasing the inertia of J1.

Figure 8a overlays the left and right axle torques with the torque at the rear end of the driveshaft for Case 1 Run 4. The latter is in terms of torque transferred to one axle, , where is the torque

transferred to axle shaft from driveshaft, is the final

drive ratio and is the measured driveshaft torque. In the absence of friction loss between the input and output torque through the final drive, the torques would balance, i.e. , where and are the torque on right and left axles respectively. Figure 8 illustrates this is clearly not the case, either for initial (static and applied) or final (static) torque loadings or for dynamic torque. There is thus revealed a friction torque loss across the final drive. From the measured data this final drive friction torque (Figure 8b) is calculated using

dsFDds TnT 50.* = dsT*

FDn

dsT

( larads TTT += 50.* ) raT laT

( ) dsFDlaram

f TnTTT −+= . The results also show the friction torque is a function of velocity and of dynamic torque as measured in the shafting; the torque changes sign at the change in velocity as well as decreasing with decreasing shaft torque and visa versa. Given this understanding, the proposed dry friction model (Coulomb) for application to numerical simulations is:

( ) ringkRf TT µθsgn−= , (3)

Where is the simulated dry friction torque, is the simulated torque at the ring gear (which may be taken as the sum of axle torques), is the calculated

velocity of the ring gear and

fT ringT

Rθ

kµ a constant kinetic coefficient of friction, approximated from experimental measurements. Figure 9 provides insight into the latter

parameter with a velocity domain plot of ( ) larafkR TTT +=− µθsgn , using measured values

for friction torque and axle torques.

0 0.5 1 1.5 2-600

-400

-200

0

0 0.5 1 1.5 2-100

0

100

Torq

ue (N

m)

Time (s)

(b)

Torq

ue (N

m)

(a)

Figure 8. Measured and Calculated Torques for No-Impact Test. Case 1, Run 4. a) Right Axle , Left Axle

and Driveshaft . Driveshaft torque is in terms of torque on one axle; b) Final Drive Friction Torque.

Figure 9. Experimentally Determined Friction Co-efficient (Velocity Domain). Case 1, Run 4 (Table 1).

SIMULATIONS WITH CLEARANCE AND FRICTION NONLINEARITIES

The clearance nonlinearity is described with a conditional statement for zero mesh stiffness within the backlash region,

⎩⎨⎧

<≥

=δδ

5.005.0

34

343434 x

xkk , (4)

Where is the relative mesh displacement and is

given by 34x

443334 θθ rrx −= and 34δ is the translational magnitude of pinion/ring backlash. The numerical method (in Matlab) solves for angular accelerations at each time step using angular displacements and velocities from the previous time step. An efficient approach to the programming structure is to use linear algebraic operations where a mesh torque offset is required in the torque vector, T [2]. In terms of mesh force the offset is given by, 34

*F

( ) 34343434* sgn5.0 δkxF = , (5)

Within lash is zero as the mesh stiffness is zero. Considering static and applied torque, final drive friction torque and the above force offset in terms of torque applied to pinion and ring, the torque vector is,

34*F

[ ]Tfas TFrFrTT +−+= 34434311 0 **T , (6)

Where is initially taken from Table 1 as the measured initial friction torque. After mass release

fT

01 =aT and fT is determined via (Equation 3). The transition time (of mass release) is matched to that shown in the experimental results, e.g. for the result of Figure 3, s06.0=rt . The signum function for friction is approximated via a hyperbolic tangent, i.e.

( ) )tanh(500sgn rr θθ ≈ . This type of function smoothens the inherent discontinuity and it can improve the numerical process [5].

Initial conditions, at , are determined via

and given the initially motionless

system, . The initial condition displacements need to be adjusted when applying the clearance algorithm (Equation 4) and force offsets (Equation 5); the pinion (

rtt =)(0)0()0( 1TKθ −=0)0( =θ

3θ ) and other upstream elements ( 1θ , 2θ ) are rotated through an angular displacement equivalent to half the lash, e.g. 33433 5000 rδθθ .)()(* −= .

COMPARISON BETWEEN THEORY AND EXPERIMENTS

NO IMPACT TEST

For the no-impact test we compare experiment and simulation for Case 1, Run 4 in Figures 10 and 11. For this and the other no-impact cases the driveshaft torque, axle torque, final drive friction torque and right axle velocity compare well. In particular, note the close comparisons between the initial torque ramp on mass release, the lowest mode frequency and the reduction in amplitude (due to friction damping) over several cycles.

The close duplication of the non-linear friction torque illustrates that the simplified friction model is effective. Some parameters are fine tuned using the sets of no-impact results to achieve closer correlation (Table 1). The simulations and experimental results are compared only up to the approximate time where the system changes state (Point D); an examination of response after this time is unnecessary with respect to the study of the impact. An explanation for this change in response is that the final drive may lock up at Point D (where the static friction torque exceeds the dynamic torque). For this test (Case 1, Run 4) the simulation did actually show the slightest clunk and the experiment results showed the response almost yielded clunk. A small increase in applied to static torque ratio, to (Case 1 Run 5), is sufficient to now see a clear clunk in the measured data.

831.=rT

0 0.5 1 1.5 2-300

-200

-100

0

Torq

ue (N

m)

0 0.5 1 1.5 2

-400

-200

0

Torq

ue (N

m)

Time (s)

(a)

(b)

A D

Figure 10. No-Impact Test/Simulation Comparison, Case 1, Run 4: (a) Driveshaft Torque; (b) Axle Torque. Key:

Simulated (dotted line) and Measured (solid line).

0 0.5 1 1.5 2-100

0

100

Torq

ue (N

m)

0 0.5 1 1.5 2-0.2

0

0.2

Vel

ocity

(rad

/s)

Time (s)

(a)

(b)

Figure 11. No-Impact Test/Simulation Comparison, Case 1, Run 4: (a) Final Drive Friction Torque; (b) Axle (Ring

Gear) Angular Velocity. Key: Simulated (dotted line) and Measured (solid line).

0 0.5 1 1.5 2-200

-100

0

Torq

ue (N

m)

0 0.5 1 1.5 2-400

-200

0

Torq

ue (N

m)

Time (s)

(a)

(b)

A CB

Figure 12. Impact Test/Simulation Comparison, Case 2,

Run 9: (a) Driveshaft Torque; (b) Axle Torque. Key: Simulated (dotted line) and Measured (solid line).

0 0.5 1 1.5 2-100

0

100

Torq

ue (N

m)

0 0.5 1 1.5-0.2

0

0.2

Vel

ocity

(rad

/s)

Time (s)

(a)

(b)

Figure 13. Impact Test/Simulation Comparison, Case 2, Run 9: (a) Final Drive Friction Torque; (b) Axle (Ring

Gear) Angular Velocity. Key: Simulated (dotted line) and Measured (solid line).

IMPACT TEST

Figures 12 and 13 compare the impact test. Again simulation and experimental correlate well, though after a few cycles they begin to diverge. When the gears pass into clearance (Point B) the torque on the driveshaft and axle is zero, evident by the flat top on what would otherwise be close to a sinusoidal wave form. The period of each cycle is longer than for a no impact test due to this zero torque region. The final drive friction torque (Figure 13a) also has longer time intervals near zero, corresponding to the torque across the ring gear being (in clearance). The test has one single-sided impact (Point C) and the clunk impulse can be seen in the measured acceleration time histories (Figure 6c). Without simulation this clunk signature is difficult to discern from the measured accelerations. The model and simulation reveal that the

0=ringT

other impulse like transients in the time history are due to the friction non-linearity. In the running vehicle case, with a non-zero drivetrain mean speed the friction effects would be different.

EFFECT OF BACKLASH MAGNITUDE ON SIMULATED IMPACTS

Simulations were performed by varying the size of lash for different impact cases. Here run numbers 8 to 13 (of Table 1) are simulated with lashes: 2, 4, 6, 8 and 10 degrees. Based on the lash and the load applied, the system may have either a single (S) or a double (D) sided impact. Double-sided is the case where the meshing elements pass into clearance and then impact on the opposite side, then come back through clearance to impact a second time on the original side of the mesh. Also, there can be multiple single or double-sided impacts. Table 4 shows the solution types for the simulation map: S(1) refers to one single-sided impact, S(2) two single-sided impacts, D(1) one double side- impact and D(2) two double-sided impacts, and so on.

Lash (Degrees) Run 1T

(Nm) 2 4 6 8 10 8 -161 S(1) S(1) S(1) S(1) S(1)9 -196 S(2) S(2) S(2) S(2) S(2)

10 -229 D(1)+S(1) S(2) S(2) S(2) S(2)11 -263 D(1)+S(2) S(3) S(3) S(3) S(3)12 -297 D(1)+S(2) D(1)+S(2) S(3) S(3) S(3)13 -335 D(2)+S(1) D(1)+S(2) D(1)+S(2) S(3) S(3)

Table 4. Solution Types for Case 2 Runs 8-13. Key: S – Single-Sided Impact, D – Double-Sided Impact, (1) –

One Impact Event, (2) – Two Impact Events, (3) – Three Impact Events.

For low load conditions (Runs 8 and 9) there are only single-sided impacts. In this case the gears do not cross over the whole the clearance so increasing the size of lash has no change on the nature of solutions. With increasing load (Runs 10-13) the number of impacts increases in combinations such as one double-sided impact followed by two single-side impacts (e.g. Run 11, 2° Lash). As the lash is increased there is less likelihood of cross over the whole clearance (thus less double-sided impacts), and the solutions move towards S(3). Again, once reaching S(3) increasing the lash brings no change in solutions. Response will be identical for S(n) only solutions, at certain T1, till the lash is varied across some limit value, but any solutions including a D(n) will have the first impact (opposite side of clearance) occurring sooner with smaller lash and visa versa. This will then change the flow of solution following the time of this impact and effect the magnitude of later impacts; even if the solution type remains fixed (e.g. Run 12, 2° and 4° Lash – see also Table 5).

Table 5 shows the simulated pinion peak-to-peak acceleration, as measured immediately after each impact. For example, for solution type S(2) the accelerations at impact S1 and impact S2 are given. For the double-sided impacts the pair of values gives first impact (opposite side of clearance) and second impact (side of clearance for initial separation). As mentioned, with no change of a given S(n) only solution type the accelerations are identical versus varying lash (such as Run 8). For this test scenario, if a single-sided impact is preceded by a double, then the peak-to-peak response for the single-sided is lower as compared to the case of a single-sided impact alone. For example, consider Run 10. When the lash is 2 degrees, the solution type is D(1) and S(1) whereas when the lash is 4 degrees or higher, the type is S(2). The second impact in D1 is similar in magnitude to the first single-side impact in S2. To consider the effect of the load alone we look at the results for 10 degrees lash. This is convenient as no double-sided impacts are found and we can examine the trend for singular clunk events. Plots of the peak-to-peak accelerations for S1 and S2 impacts in Figure 14 show a fairly linear trend with Tr; refer to Table 1 for the values.

Run Lash (o)

S1 (rad/s2)

S2 (rad/s2)

S3 (rad/s2)

D1 (rad/s2)

D2 (rad/s2)

8 2-10 319 - - - - - - 9 2-10 535 - - - - - -

2 382 - - 475 716 - - 104-10 740 378 - - - -

2 507 147 - 620 897 - - 114-10 900 527 158 - - - -

2 631 326 - 940 1100 - - 4 660 304 - 486 1071 - - 12

6-10 1025 695 310 - - - - 2 480 - - 1273 1209 399 7804 832 395 - 856 1218 - - 6 777 405 - 429 1172 - -

13

8-10 1192 808 405 - - - -

Table 5. Peak-to-Peak Acceleration of Pinion Coordinate on Each Impact, S1, S2, S3, D1 and D2 as Simulated for

various Magnitudes of Lash.

300500700900

1100

8 9 10 11 12 13

Run

Pea

k to

Pea

k A

ccel

erat

ion

(rad/

s²)

Figure 14. Peak-to-Peak Acceleration of Pinion

Coordinate on Impact S1 (dotted) and S2 (solid) vs. Run with 10 Degrees Lash

The multiple clunks of Table 5 can be examined as clunk severity versus load and lash magnitude, however a suitable quantifier combining the impacts to one value needs to be applied. In this instance we find an effective average peak-peak value by taking the square root of the sum of squared values for all the impacts. This approach allows a preliminary analysis though studies on how the driver perceives multiple clunks may be warranted. Figure 15 provides these effective peak-peak values as a surface plot against run number and lash magnitude. The figure shows that overall severity of combined multiple clunk events increases with Tr (increasing run number), as quantified with this metric. As the lash is decreased more impacts are occurring and notably the effective averaged peak-peak values are also higher. For this particular system the dynamics therefore yield a case where reduced lash is leading to a more severe case of clunk(s). This example is thereby important to consider in efforts to improve clunk, given the contradiction to the often quoted finding of Krenz [1], ‘clunk severity increases with driveline lash, but not necessarily in a linear relationship’. Krenz did however consider the tip-in, rather than tip-out scenario. Also of interest is the correlation between Krenz’s finding, ‘large changes in the amount of lash may be required to produce a significant change in response’ with our finding that the nature of solution will not change in the case of single- sided impacts until a limiting value of lash is reached.

24

68

10

89

1011

1213

0

1000

2000

3000

4000

Lash (degrees)Run Number

Effe

ctiv

e P

eak-

Pea

k V

alue

s (ra

d/s²

)

Figure 15. Effective Peak-Peak Values Combining Multiple Impacts versus Run Number and Lash

Magnitude.

CONCLUSION

A driveline test rig was designed and utilized to study clunk under several conditions. The rig effectively simulates an engine torque tip-out event in a rear-wheel drive vehicle. A four DOF lumped parameter torsional model for clunk was developed and validated using the

experiment. The model considered the significant inertias and stiffness in the driveline with one lash included at the final drive as a piecewise clearance function. Friction in the final drive was approximated with a hyperbolic tan function. Drivetrain torque measurements allowed calculation of friction torque losses across the final drive. In turn a friction coefficient was approximated for simulation. Tests were conducted with increased applied load (comparable to engine torque tip-out magnitude) for two mean load cases. The no-impact and impact case presented are a sample of a generally good correlation between simulation and experiment across the test matrix. The no-impact case occurs with less applied load (smaller tip-out) and was used to confirm ‘linear’ system response with respect to clunk. The parameters can then be tuned to improve the model; sensitive parameters are effective engine inertia, J1, inertial (viscous) drag, d1 and axle stiffness, k4, which control the lowest mode.

The validated model was used in a simple parametric study of lash and applied load and such results can be related to the vehicle. Taking some examples from the various tests; the driver may hear a single clunk (S(1) solution type), two spaced at around 0.5s (S(2) solution type) or two spaced at around 0.15s (D(1) solution type). In this study, increased engine tip-out magnitude yields a generally linear trend with peak-to-peak accelerations as measured immediately after impact. Nonetheless, some clunk conditions induce a bifurcation at a certain limiting magnitude, at which the timing and number of impacts change. Further, when the number of clunks varies, development of a comparative metric for clunk severity is difficult. Crowther et al. [6] have studied some of these quantifiers and related aspects.

ACKNOWLEDGEMENTS

The authors wish to acknowledge the support of Ford technologist Chris Nouhan, Dave McCredie and Stan Berlinski in their help to set up the test rig.

REFERENCES

1. R. A. Krenz, "Vehicle Response to Throttle Tip-In/Tip-Out", SAE Paper No. 850967.

2. A.R. Crowther, R. Singh, N. Zhang, C. Chapman, "Impulsive Response of an Automatic Transmission System with Multiple Clearances: Formulation, Simulation and Experiment", accepted for publication in the Journal of Sound and Vibration, September 2006.

3. S. Johansson, E. Langjord and S. Pettersson, "Objective Evaluation of Shunt and Shuffle in Vehicle Powertrains", 7th International Symposium on Advanced Vehicle Control, Arnhem, The Netherlands, August 2004.

4. A.R. Crowther, N. Zhang, R. Singh, ‘Development of Clunk Simulation Model for a Rear Wheel Drive Vehicle With Automatic Transmission’, SAE Noise and Vibration Conference, Traverse City, Michigan, May 2005, SAE Paper No. 2005-01-2292

5. T. C. Kim, T.E. Rook, R. Singh, "Effect of Smoothening Functions on the Frequency Response of an Oscillator with Clearance Non-Linearity", Journal of Sound and Vibration, 2003, Vol. 263 (3), pp. 665-678.

6. A.R. Crowther, C. Janello, R. Singh, “Quantification of Impulsive Phenomena in Torsional Systems with Clearances“, accepted for publication in the Journal of Sound and Vibration, October 2006.

CONTACT

A.R. Crowther and R. Singh, Acoustics and Dynamics Laboratory, Mechanical Engineering Department, The Ohio State University, Columbus, OH 43210, USA, Phone: +1 614-292-4147, Fax: +1 614-292-3163, crowther.5@osu.edu, singh.3@osu.edu.