INTRODUCTIONAll powertrain or driveline systems have clearances due to design, manufacturing error, and wear. The presence of backlashes, gaps, or multi-staged elements generates transient phenomena depending on the mean and dynamic loads. The entire family of such events can be classified into three cases as shown in Figure 1, where θ is relative angular motion and T is transmitted torque through a component. The first is the no impact case (Figure 1(a)), which occurs when the range of motion does not cross a clearance transition (±Θb). The second is the rattle case (Figure 1(b)), which occurs when pulsating motion about some mean displacement crosses one (single-sided impact) or two (double-sided impact) clearance transitions. The third is the clunk case (Figure 1(c)), which occurs when the mean operating point suddenly changes and the resulting motion crosses one or two clearance transitions. The severity of clunk depends on the mean operating points, transition rate, and dissipation mechanisms [1,2].

Figure 1.

Figure 1. (cont.) Clearance-induced impact regimes: (a) no impact or constant contact, (b) rattle case with single and double-sided impacts, and (c) clunk case with single and double-sided impacts where θ is relative angular displacement, T is transmitted torque, and Θb is a clearance transition. Key: (

) - transmitted torque, ( ) - mean displacement point, and ( ) - dynamic displacement range.

Development of a Non-Linear Clutch Damper Experiment Exhibiting Transient Dynamics

Michael Krak, Jason Dreyer, and Rajendra SinghOhio State Univ.

ABSTRACTMany powertrain structural sub-systems are often tested under steady state conditions on a dynamometer or in a full vehicle. This process (while necessary) is costly and time intensive, especially when evaluating the effect of component properties on transient phenomena, such as driveline clunk. This paper proposes a laboratory experiment that provides the following: 1) a bench experiment that demonstrates transient behavior of a non-linear clutch damper under non-rotating conditions, 2) a process to efficiently evaluate multiple non-linear clutch dampers, and 3) generates benchmark time domain data for validation of non-linear driveline simulation codes. The design of this experiment is based on a previous experimental work on clunk. A commercially available non-linear clutch damper is selected and the experiment is sized accordingly. The stiffness and hysteresis properties of the clutch damper are assumed from the measured quasi-static torque curve provided by the manufacturer. Dissipation sources within the experiment are estimated as well. Instrumentation and time domain signal processing issues are discussed and typical measurements are compared to a simplified model for verification.

CITATION: Krak, M., Dreyer, J., and Singh, R., "Development of a Non-Linear Clutch Damper Experiment Exhibiting Transient Dynamics," SAE Int. J. Passeng. Cars - Mech. Syst. 8(2):2015, doi:10.4271/2015-01-2189.

2015-01-2189Published 06/15/2015

Copyright © 2015 SAE Internationaldoi:10.4271/2015-01-2189saepcmech.saejournals.org

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Non-linear clutch dampers may be designed to mitigate clearance induced impacts in a powertrain system. This component serves to transmit torque and reduce vibration from the engine to the transmission [3]. A typical quasi-static performance curve of a three-stage, symmetric clutch damper is shown in Figure 2, where θ is the relative angular motion between the clutch damper and shaft, and TD is transmitted torque. Positive angular displacement is referred to as “drive-side” and negative angular displacement as “coast-side.” Clutch dampers may include several non-linear features (intentionally or otherwise), such as clearances (±Θb), multi-staged stiffness (KI, KII, KIII), pre-load (TP), and frictional hysteresis (HI, HII) [3].

Figure 2. Typical quasi-static performance curve of a three-stage, symmetric clutch damper with relative angular displacement θ and transmitted torque TD. Key: Θ - stage or clearance transition, K - torsional stiffness, and H - frictional hysteresis. Stages are denoted by subscripts I, II, and III (stopper), pre-load by subscript P, and clearances by subscript b.

Selection of non-linear clutch damper features could be more efficiently facilitated by experimentally validated non-linear driveline simulation codes. Powertrain structural sub-systems are often tested on dynamometers or in full vehicles under steady-state conditions. This process, while necessary, is costly in terms of time, expertise, and instrumentation, especially when evaluating the effects of component properties on transient phenomena. To address this, the current paper proposes a laboratory experiment that intends to achieve the following objectives:

1. Demonstrate the transient (clunk-like) behavior of a non-linear clutch damper under non-rotating conditions; and

2. Generate benchmark time domain data for validation of non-linear driveline simulation codes.

NEW NON-LINEAR CLUTCH DAMPER EXPERIMENTThe clunk phenomenon has been investigated by Gurm et al. [1] and Crowther et al. [2], who proposed non-linear vibratory (non-rotating) experiments that are excited by an external step-like torque. To ensure that these systems are definite, at least one component is grounded, such as a wheel rotor [1] or gear [2]. A torsion arm is attached to another component, which is free to vibrate, and a

step-like force is applied to the arm via a variable mass drop [1,2]. During the system response, impacts occur at clearance locations, such as gear meshes in transmissions or differentials [1,2].

A physical illustration of the new non-linear clutch damper experiment is proposed in Figure 3. The new experiment is a non-rotating system that is excited by an external step-like torque, similar to previous work [1,2]. For simplicity, a flywheel is chosen as the ground and a clutch assembly is bolted directly to it. The clutch assembly houses a greased sleeve bearing and two parallel non-linear clutch dampers. The clutch is engaged so that clutch dampers are compressed against the flywheel; a no-slip condition is assumed between the flywheel and clutch dampers. A shaft contacts the two clutch dampers through a spline, and is radially supported by the sleeve bearing and a ball bearing that is exterior to the clutch assembly. A torsion arm is attached to the shaft and a step-like force is applied to its free end. Due to the very high torque capacity of the clutch dampers, application of the force through a mass drop [1,2] would be impractical and unsafe. Therefore, two pneumatic cylinders with a quick-release mechanism are utilized. All components are supported by steel structures which are grounded to a bed plate.

Figure 3. Physical illustration of the new non-linear clutch damper experiment. Key: A - flywheel (ground), B - clutch assembly, C - clutch shaft with spline, D - ball bearing, E - torsion arm, F - pneumatic cylinders with quick release mechanism, G - steel bed plate (ground), and H - structural support (ground).

Figure 4. Non-linear model of the new clutch damper experiment. Key: A - lumped torsional inertia of torsion arm, shaft, and spline, B - lumped bearing interface, D - non-linear clutch dampers, and E - flywheel (ground).

For design purposes, the experiment may be simply described as a non-linear, single degree of freedom (SDOF) system shown in Figure 4. The equation of motion is the following

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(1)

It is assumed that the torsion arm, clutch shaft, and spline are a single rigid body with torsional inertia J and angular motion θ(t). The interface between the shaft and the two bearings is modeled as a torsional viscous damping C and sliding friction . Multi-staged stiffness KD(θ) and frictional hysteresis are provided by the clutch dampers which are assumed to be identical, symmetric, synchronous, and have negligible torsional inertia. Stiffness and hysteresis properties are assumed from the quasi-static performance curves provided by the manufacturer. The non-linear function

is the torque transmitted through the clutch dampers in series with the spline clearance, as illustrated in Figure 2. The step-like external torque is denoted by T(t).

SIZING AND EXCITATION CALCULATIONSFor sizing purposes, the operating range of the experiment is assumed to be within the drive-side stage II of the clutch dampers. The corresponding SDOF linear system has inertia J and stiffness KII with a natural frequency of ωn = (KII / J)0.5 (rad/s). The target natural frequency is assumed to be approximately equal to the typical driveline surge mode (20π rad/s = 10 Hz) [2]. Since the clutch shaft and spline are production components with known inertia JB, the inertia of the torsion arm (JA) must satisfy the following

(2)

The external step-like torque T(t) is approximated by a ramp function with initiation at t = 0 of duration of te, initial torque T0, and final torque Tf, as illustrated in Figure 5(a). It is defined by the following where

(3)

To ensure a truly non-linear response, the initial and final operating points of the system are designed to lie on different stages of , as illustrated in Figure 5(b). The initial operating point (θ0,T0) lies on drive-side stage II and the final point (θf,Tf) can reside on drive-side stage I or pre-load. Achieving both (θ0,T0) and (θf,Tf) would require proper sizing of the pneumatic actuation system and torsion arm.

Figure 5.

Figure 5. (cont.) Conceptual illustration of the external torque T(t): (a) time-domain representation and (b) operating points on the clutch damper. Key: (θ0,T0) - initial operating point, and (θf,Tf) - final operating point.

INSTRUMENTATION AND TYPICAL MEASUREMENTSA laser vibrometer [4] and data acquisition system [5] are chosen to directly measure the velocity of the torsion arm and shaft, as illustrated in Figure 6. The translational velocity of the laser point along the horizon is denoted by , the radial distance from the shaft axis to the laser point by r(t), and the angular position of the arm by θ(t) (w.r.t. the clutch damper datum) and ϕ(t) (w.r.t. the horizon). A reflective surface for the laser point is provided by a square finished block, which is rigidly attached to the torsion arm such that the measurement surface is in-plane with the shaft axis. Prior to excitation, ϕ0 and r0 are measured with a digital level and ruler, respectively; initial translational displacement u0 is calculated from u0 = r0 sin(ϕ0). The system excitation is then applied and signal

is measured with a sampling frequency of 1.28 kHz; triggering procedures are not discussed in this paper. The translational displacement u(t) is then estimated by numerically integrating [6]. Angular displacement θ(t) is calculated using the following where θf is determined from the static balance of Tf

(4)

(5)

Angular velocity , acceleration , and jerk are estimated through numerical differentiation [6], and all signals are synchronized so that the maximum value of occurs at t = 0.

Figure 6. Schematic of the instrumentation system. Key: A - torsion arm, B - clutch shaft, C - block with finished surfaces, D - laser, and E - laser vibrometer.

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Typical measured responses are shown in Figure 7. All motions are normalized by the clearance transition Θb and natural frequency ωn. The responses have three distinct non-linear regimes: double-sided impact, single-sided impact, and no impact. The double-sided impact regime (DS) is characterized by contact and contact loss with the drive and coast-side pre-loads (at ); there are significant positive and negative peaks. The single-sided impact regime (SS) is characterized by contact and contact loss with the drive-side pre-load (at ) and by significant negative peaks only. In the no impact regime (NI), there is constant contact with the drive-side pre-load (

) and the response is nearly sinusoidal. In addition to non-linear regimes, the responses exhibit a distinct time-varying period, say with respect to peak values of .

Figure 7. Typical measured responses. Key: ( ) - measured motion, ( ) - stage transition, ( ) - response regime transition, A - DS to SS transition, and B - SS to NI transition time.

ESTIMATION OF DISSIPATION SOURCES AT BEARINGA rotating, sub-experiment is developed to estimate the viscous damping C and sliding friction . The clutch dampers are removed from experiment E1, and the torsion arm is replaced with an air motor and slender shaft, as illustrated in Figure 8. Torsional strain in the slender shaft is measured by strain gages in a full bridge configuration, and the gages are calibrated through static loading. An optical tachometer is used to measure the angular velocity of the clutch shaft.

Figure 8. Physical illustration of the bearing sub-experiment. Key: A - air motor, D - coupling, E - slender shaft, G - strain gage and transmitter, I - receiver, L - coupling, N - clutch shaft, P - ball bearing, Q - sleeve bearing, and R - optical tachometer.

The sub-experiment is described as a two degree of freedom, rotating system, as illustrated in Figure 9. The air motor is modeled as torsional inertia element JM, the clutch shaft as torsional inertia element JB, and the slender shaft as torsional stiffness KMB. It is assumed that the velocities and are constant, and the corresponding equation of motion for JB is the following where the torque measured by the strain gages is TMB = KMB (θM - θB)

(6)

Figure 9. Non-linear model of the bearing sub-experiment. Key: A - torsional inertia of air motor, D - torsional stiffness of slender shaft, E - torsional inertia of clutch shaft, and G - viscous damping and sliding friction of the bearing interface.

First, the air motor is turned slowly by hand and TMB is measured. It is assumed that viscous damping is negligible at low velocity (

), and hence the amplitude of is estimated by F ≈ -TMB. Second, the air motor drives the shafts at a relatively high velocity. However, to overcome issues related to system resonance, the velocity is limited by the following

(7)

Velocity and torque TMB are then measured, and viscous damping C is estimated by the following

(8)

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SIMULATION OF THE NON-LINEAR EXPERIMENTThe non-linear clutch damper experiment is described by the SDOF non-linear model given in Figure 2 and equation (1), and is simulated using a commercial multi-domain system simulation software [7]. The model is built within the code using an assembly of graphical components that represent mathematical models of mechanical elements, as illustrated in Figure 10. External torque T(t) is provided by a time-dependent piecewise function generator, J by a torsional inertia element, and C by a torsional viscous damper. Friction is provided by a sliding friction element and described by the following where σ is an empirical regularizing factor (say )

(9)

Torque is assumed to be a sum of elastic and hysteretic torques

(10)

The elastic torque TK(θ) and multi-staged stiffness KD(θ) are provided by sets of parallel torsional springs and can be described by the following

(11)

(12)

The hysteretic torque and multi-staged hysteresis HD(θ) are provided by sliding friction elements and can be described by the following where it is assumed that the transition between HI and HII in the pre-load stage is linear

(13)

(14)

Figure 10. Graphical representation of the new non-linear clutch damper experiment in a commercial, one dimensional simulation software. Key for components: A - external torque, B - torsion arm and shaft inertia, C - bearing interface, D - stage I hysteresis, E - spline backlash and stage I stiffness, F - stage II hysteresis, G - pre-load and stage II stiffness, and H - stage III stiffness.

The simulation is numerically integrated using an algorithm similar to Gear's method [7]. The maximum allowable time step and resolution of the resulting time array is set equal to the sampling period of the measured response (0.781 ms). Two simulation cases are considered, denoted S1 and S2. The method of estimating parameters for each case is listed in Table 1; all parameters, except viscous damping C, are similar across S1 and S2. For simulation S1, C is estimated using the bearing sub-experiment (see Figures 8 and

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9). However for S2, C is estimated by using a typical damping ratio of the driveline torsional surge mode (C = 2ζ(2KIIJ)0.5 where ζ = 0.04) [2]. The value of C for S2 is significantly higher than that for S1 (CS2 ≈ 1000CS1).

Table 1. Parameter estimation methods for simulations cases S1 and S2.

A comparison of simulation with typical measurements is shown in Figures 11 and 12 for cases S1 and S2 respectively. Similar to the measurements, the predictions have three distinct response regimes (double-sided, single-sided, and no impact) with respect to stage transition ±ΘI. However, the regime transition times as predicted by S1 occur much later than those in the measurements and predictions from S2. Also, the predicted peak-to-peak by S2 is more accurate than S1. This is illustrated in Figure 13 where the peak-to-peak is labeled for the 2nd, 4th, and 6th response periods. It is evident that simulation S2 offers a better prediction than S1, and that there is improved agreement between prediction and measurement as

.

Figure 11. Comparison of predictions from simulation S1 with typical measurements. Key: ( ) - measured motion, ( ) - predicted response, ( ) - regime transition for the measured response, ( ) - response regime transition for the predicted response, ( ) - stage transitions, A - DS to SS transition, and B - SS to NI transition time.

Figure 12. Comparison of predictions from simulation S2 with typical measurements. Key: ( ) - measured motion, ( ) - predicted response, ( ) - regime transition for the measured response, ( ) - response regime transition for the predicted response, ( ) - stage transitions, A - DS to SS transition, and B - SS to NI transition time.

Figure 13. Comparison of peak-to-peak angular acceleration for measured and predicted responses: (a) measurement, (b) prediction from simulation S1, and (c) prediction from simulation S2. Key: ( ) - angular acceleration, ( ) - peak-to-peak angular acceleration, and (2nd, 4th, 6th) - response period.

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CONCLUSIONA new non-rotating experiment that exhibits transient behavior of non-linear clutch dampers is proposed in this paper. The experiment contains production components, such as a clutch assembly with two parallel clutch dampers. As previously stated, it is assumed that the clutch dampers are identical, synchronous, and ideally indexed, and that stiffness and hysteresis properties can be estimated from measured quasi-static performance curves. The excitation for the experiment is an external step-like torque, which is consistent with prior work on vehicle clunk [1,2]. Typical measured responses show three distinct clearance-induced impact regimes: double-sided impact, single-sided impact, and no impact. Predictions of a single degree of freedom non-linear model exhibit similar response regimes, regime transition times, and peak-to-peak angular acceleration. Therefore, the objectives of the experiment are achieved as the experimental measurements are successfully verified. Further refinements to the non-linear model are discussed in the companion paper [8].

REFERENCES1. Gurm, J., Chen, W., Keyvanmanesh, A., Abe, T. et al., “Transient Clunk

Response of a Driveline System: Laboratory Experiment and Analytical Studies,” SAE Technical Paper 2007-01-2233, 2007, doi:10.4271/2007-01-2233.

2. Crowther, A., Singh, R., Zhang, N., and Chapman, C., “Impulsive Response of an Automatic Transmission System with Multiple Clearances: Formulation, Simulation and Experiment,” Journal of Sound and Vibration, 306, 444-466, 2007.

3. Shaver, R., “Manual Transmission Clutch Systems,” (Warrendale, Society of Automotive Engineers, Inc., 1997), ISBN 978-1560919841.

4. PSV-400 Scanning Vibrometer, Polytec Inc., http://www.polytec.com (Accessed January 2015).

5. PSV-400-M4 Data acquisition system, Polytec Inc., http://www.polytec.com (Accessed January 2015).

6. MATLAB version 2014a, Natick Massachusetts: The MathWorks Inc, 2014.

7. LMS Imagine.Lab AMESim version 11.2.0: LMS Imagine SA, 2012.8. Saleh, A., Krak, M., Dreyer, J., and Singh, R., “Development of Refined

Clutch-Damper Subsystem Dynamic Models Suitable for Time Domain Studies,” SAE Technical Paper 2015-01-2180, 2015, doi:10.4271/2015-01-2180.

CONTACT INFORMATIONProfessor Rajendra SinghAcoustics and Dynamics LaboratoryNSF I/UCRC Smart Vehicle Concepts CenterDept. of Mechanical and Aerospace EngineeringThe Ohio State Universitysingh.3@osu.edu

Phone: 614-292-9044www.AutoNVH.orghttp://svc.engineering.osu.edu/

ACKNOWLEDGMENTSThe authors acknowledge Eaton Corporation (Clutch Division) for supporting this research. We would like to thank Luiz Pereira and Brian Franke for their assistance with experimental studies. Siemens

Corporation is thanked for granting access to LMS Imagine Lab AMESim software package. Further, we acknowledge the member organizations of the Smart Vehicle Concepts Center (www.SmartVehicleCenter.org) and the National Science Foundation Industry/University Cooperative Research Centers program (www.nsf.gov/eng/iip/iucrc) for partially supporting this basic research.

DEFINITIONS

SymbolsC - torsional viscous damping

f - frequency (Hz)

F - sliding friction

H - frictional hysteresis

J - torsional inertia

K - torsional stiffness

r - radial distance from shaft axis to laser point

t - time

T - torque

u, - translational displacement and velocity (laser vibrometer)

θ, , , - angular displacement, velocity, acceleration, and jerk (w.r.t. to the clutch damper datum)

Θ - stage or clearance transition (angular)

γ - frictional hysteresis regularizing factor

ρ - external torque regularizing factor

σ - velocity regularizing factor for friction and hysteresis

ϕ - angular displacement (w.r.t. horizon)

ω - frequency (rad/s)

Subscripts0 - initial value

I,II,III - clutch damper stage index

A - torsion arm

B - clutch shaft and spline

b - clearance (backlash)

D - clutch damper

f - final value

M - air motor

n - natural

P - pre-load

s - sampling

S1,S2 - simulation index

Superscripts- - normalized

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AbbreviationsDS - double-sided impact regime

NI - no impact regime

SDOF - single degree of freedom

SS - single-sided impact

S1, S2 - simulation index

w.r.t. - with respect to

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