Analysis of Friction-Induced Vibration Leading to Eek Noise in a Dry Friction Clutch. Abstract

Transcription

1 The 22 International Congress and Exposition on Noise Control Engineering Dearborn, MI, USA. August 19-21, 22 Analysis o Friction-Induced Vibration Leading to Eek Noise in a Dry Friction Clutch P. Wickramarachi * and R. Singh Acoustics and Dynamics Laboratory, Center or Automotive Research The Ohio State University, Columbus, Ohio , USA G. Bailey LuK, Inc., 341 Old Airport Road - Wooster Ohio 44691, USA Abstract This study deals with a severe noise problem (Eek) that arises during the engagement o a dry riction clutch in a vehicle with manual transmission. Measurements have shown that near the ull engagement the pressure plate suddenly starts vibrating (rigid body wobbling mode) with a requency close to the irst natural requency o the rotational sub-system. This sel-excited vibration problem exhibits typical signs o a dynamic instability associated with a constant riction coeicient. The transient event results in the generation o high transient noise levels and oten consumers will change clutches prematurely in an eort to eliminate this noise. Our work ocuses on developing a linearized lumped-parameter model o the clutch. The eect o key parameters on the system stability is examined by calculating complex eigensolutions. Results o this analytical study are in agreement with experimental observations. It is seen that the instability o the rigid body wobbling mode is controlled by the riction orces. This mode may, however, be also aected by the irst bending mode o the pressure plate. Thereore, a stier plate could lead to a design with a reduced tendency to Eek. 1. Introduction A strong squealing noise is produced during the engagement o a dry riction clutch in a manual transmission. This is known as Eek noise and it dictates the overall perception o vehicle quality. Despite many experimental studies, this riction-induced phenomenon remains poorly understood. One hypothesis is that Eek is triggered by a dynamic instability o the system composed o the pressure plate and the disc cushion. It has been experimentally observed that, during the Eek event, the pressure plate vibrates in a rigid body wobbling motion at the natural requency o the rotational system (out o plane). This is correlated with * Currently with Data Physics Corporation, <wickramarachi@dataphysics.com>

2 a Single-Degree-O-Freedom model (SDOF) that includes the pressure plate inertia, the diaphragm spring stiness, and the cushion stiness only. In addition, experimental studies show that the modal properties o the pressure plate itsel, due to the abrication process, seem to play a signiicant role in the Eek mechanism(s). To clariy some o these issues, we develop an analytical model that could be used to identiy key parameters aecting the system stability. Based on experimental modal analyses (though not presented here), the complex eigenvalue method is applied to a linearized lumped-parameter model. It is used to demonstrate the potential coupling between the pressure plate-wobbling mode and the irst elastic deormation mode o the pressure plate. Then stability maps are developed in terms o the riction coeicient and the pressure plate geometry and structural stiness. 2. Problem Formulation Two conditions are typically necessary to generate the transient Eek noise (Figure 1). First, the engine speed should be within the RPM range. Conversely, the transmission speed does not seem to have a signiicant eect. Since the slip speed is relatively high, one may assume a constant coeicient o riction µ at the threshold o Eek. This shows that the Eek noise may not ollow the symptoms o classical stick-slip phenomena [1]. Second, the clutch pedal motion (bearing travel) must be such that the clutch is close to ull engagement. Observe this particular position (where Eek begins to occur) in Figure 1 and note that the pressure plate and the disc cushion are already in contact. Measurements o noise spectra exhibit a dominant requency w, which is related to the bearing travel, around 45 Hz (Figure 2.1). This requency has been correlated to the wobbling rigid body motion o the pressure plate (both out-o-plane rotations) rom dynamometer tests and the SDOF model. Thereore the pressure plate and the disc cushion are assumed to be the main components that control the Eek phenomenon. This observation is reinorced by many experiments as elements external to the clutch (gears, release system, engine pulsations, etc.) and internal to the clutch (diaphragm spring and cover) have not signiicantly inluenced the tendency to Eek [2]. FULL Eek threshold Dominant Eek requency (4< w < 55 Hz) Figure 1: Eek noise measurement (in-vehicle) Figure 2.1: Time-requency diagram

3 However, a change in the pressure plate material rom gray-iron (Plate G) to vermicular iron (Plate V) seems to reduce the occurrence o Eek. Such a change could alter the modal properties o the pressure plate, as reported in some brake-squeal studies [4]. Hence, we need to investigate the inluence o the pressure plate elastic deormations. From experimental modal analyses, the second requency b ( 9 Hz) o Figure 2.1 could be linked to the irst bending mode o the plate. Observe the presence o this mode ( b ) at t = 2.8 s in Figure 2.2, which is quickly replaced by the wobbling mode ( w and harmonics) at t = 3.1 s in Figure 2.3. Our study will then attempt to develop a preliminary simulation model o the clutch. At the threshold o Eek, vibration amplitudes are considered suiciently small to build a linearized model that could reproduce the trends o observed dynamic instabilities. Here, the lumped parameter approach is preerred over the FE method. Our study examines a sel-excited system, without any external torque excitations. Moreover, our model will demonstrate a possible coupling between the rigid body-wobbling mode and the irst bending mode o the pressure plate. The pressure plate is assigned an initial angle in order to inspect the amplitude growth with time and demonstrate the existence o dynamic instability..7 Spectrum at 2.8 sec.7 Spectrum at 3.1 sec.6 w 2 w 3 w.6 w 2 w 3 w.5.5 Magnitude (V).4.3 b Magnitude (V) b Frequency (Hz) Frequency (Hz) Figure 2.2: Spectrum at t = 2.8 sec (mode at b ) Figure 2.3: Spectrum at t = 3.1 sec (mode at w ) 3. Lumped Parameter Model The lumped parameter model o Figure 3 incorporates the inertia o a deormable pressure plate along with the cushion stiness k c that is split into our springs o stinesses k x, k, k y and k. In order to consider the nonlinear characteristic o the cushion, the stinesses k y and k are respectively divided and multiplied by a ratio γ, as explained in Figure 4 and the ollowing. Assume that the bearing travel is such that the Eek threshold point (point P) is reached. This would correspond to a plate lito υ = υ eek. Now suppose that the plate is already vibrating about the x-axis with amplitude θ xo. This will change the normal spring orces at locations A and B and consequently the stinesses k y and k by assuming a constant stiness slope around point P, as shown by Equation 1.

4 θ y θ x (a) (b) Figure 3: Lumped-parameter model, (a) 2 rotational DOFs θ x and θ y, (b) 4 translational DOFs in z direction k k y kc = 4γ kc = γ 4 where γ =.9 (1) The riction orces rom the cushion are also included by assuming a constant µ due to high slip speeds (>7 RPM). In terms o the Degrees-O-Freedom (DOFs), this model is composed o two DOFs corresponding to the rigid body out-o-plane rotations (wobbling mode) and our additional translational DOFs (normal to the pressure plate) to incorporate the irst 2-nodal diameter mode o the pressure plate. By choosing appropriate lumped bending stinesses k, we estimate the requency o this bending mode to be close to b. The value m corresponds to a quarter o the total plate mass. k y k Figure 4: Modeling o cushion nonlinear eect Figure 5: Free-body diagrams Next, the ree-body diagrams are drawn in Figure 5. Here, even though the plate is represented as a rigid body or the sake o clarity, it can also elastically deorm as explained beore. The normal spring orces and the corresponding riction orces are expressed as:

5 N N N N x y = k = k = k = k x y ( z rθ ( z + rθ y ( z + rθ y x ) ) ) ( z rθ ) x F F F F x y = µ N = µ N = µ N = µ N x y (2) The resulting governing equations or the linear, undamped and unorced system are obtained in terms o the mass [M] and stiness [K] matrices and the generalized coordinate vector {X}. [ M ]{ X & } + [ K]{ X} = {} (3) where T T = [ θ x θ y zx z z y z, M = diag([ I x I y m m m m ]) X ] r µ y K = rk y rk 2( k ) ( ) ( ) ( ) y µ lr kx Gx G r 2k y r k 2k 2 lr( k ) r ( k ) r( k 2k ) r( 2k ) G G x rk rk x x 2k x 2k 2k y y 2k, G ij = (µ, r, l, k, k ij ), or i={x,y}, j={,r} 4. Stability Studies and Design Maps The asymmetric nature o [K], as controlled by µ and γ, yields complex eigenvalues λ k = σ k ± ω k, where the imaginary part ω k is the requency o mode k (in rad/s) and the real part σ k is an index o stability. When σ k is positive, the amplitude o vibration grows exponentially in time. The eect o µ on λ k is examined in Figure 6. Observe two distinct wobbling modes with requencies 1 and 2 (in Hz), where the requency gap is ound to be related to the ratio γ or low values o µ. Nevertheless, as µ is increased, so is the coupling between those two modes. It results in stable and unstable modes occurring at the same requency c. This would occur when µ =.35, which is typical or disc linings. Next, the eect o the pressure plate structural stiness k is investigated in Figure 7, which maps the requencies and real parts o the wobbling modes. As k is increased, one passes rom an unstable region (or Plate G) to a stable region (or Plate V). This observation is in agreement with measurements and experimental modal analyses that show that Plate V is essentially stier than Plate G. Thereore, the Young s modulus does have a signiicant impact on the system stability and on Eek noise.

6 Stable wobbling mode Unstable wobbling mode Figure 6: Predicted eigenvalues with varying µ Figure 7: Predicted eigenvalues with varying k Next, an animation o complex wobbling modes is developed to visualize phase dierences (such as Figure 8). As the pressure plate wobbles, a circle indicates the location along the edge that has the maximum positive amplitude (z-direction). At µ =.35, this circle actually moves along the edge in the direction o the riction orces or the unstable mode (represented by the eigenvector ψ 1 ) and in the opposite direction or a stable mode (ψ 2 ). This means that the axis o rotation (dash line) is actually rotating as a unction o time and in a dierent direction as shown in this igure. Hence, the riction orces may enhance energy o an unstable mode, while they may dissipate energy o a stable mode. At lower µ values, the axis o rotation remains stationary, which shows the absence o any coupling between θ x and θ y. Figure 8: Animation o complex wobbling modes or µ =.35, showing the directions o the traveling waves induced by ψ 1 and ψ 2 (moving axis o rotation) Stability maps are presented next as they reveal the eects o pressure plate geometry (inner and outer radii r i and r o, and thickness 2l), pressure plate structural stiness (k ) and disc cushion characteristics (µ and k c ). First, a map in terms o r i and r o is displayed in Figure 9. Deining δ =.5(r o - r i ), the radii are varied rom their baseline value minus δ to their baseline value plus δ. The increasing levels o instability are represented with a color code. Two straight lines indicate the baseline values. It is ound that larger radii would result in an

7 increased stability. However, r o seems to have a greater inluence than r i. For practical purposes, it would be more suitable to maintain the same contact area between the pressure plate and the disc. A smaller surace is not recommended since it would produce more heat and reduce the clutch lie. Note that the heat capacity o the clutch is also aected by the pressure plate mass. From these considerations, it seems that r o is the parameter that should be increased, since a 15% increase would theoretically result in a stable system. In addition, this change would increase the torque capacity o the clutch. STABLE STABLE Plate V Plate G Figure 9: Stability map with varying r i and r o Figure 1: Stability map with varying l and k Second, the stability map o Figure 1 shows that a thicker pressure plate would result in a decrease o stability. This is conceivable since l controls the torque generated by the riction orces (recall Figure 5). This torque is responsible or the wobbling mode, because it couples θ x with θ y. Interestingly a 1% decrease in l would make the system stable. As with the previous design change proposed or r o, one needs to consider the consequences o a thinner plate in terms o clutch lie and perormance. For example, a change in l would aect k, which is related to the bending mode o the pressure plate. The next map (Figure 11) compares the inluence o µ on plates G and V. Note that plate V is stable or µ <.45. As or plate G, decreasing µ by 1% (.35 to.31) would make the system stable. Once again, this change may cause problems in terms o the clutch perormance, since the transmitted torque would be also reduced. To compensate, the clamp load (normal load at ull engagement) would have to be increased accordingly. Finally, a stability map is obtained in terms o k and k c, in Figure 12. The latter is varied up to its maximum value at ull engagement. In parallel, the eigenvalues are computed via an iterative process to get the corresponding values or k and k y at each point on the plot. Figure 9 shows the loss o stability as the clutch is being engaged. Here it occurs around k c 26 MN/m or plate G, whereas the instability region is avoided with plate V at the same value. Such predictions match experimental observations airly well.

8 STABLE kc at point E Plate V Plate G Plate V Plate G STABLE Figure 11: Stability map with varying µ and k Figure 12: Stability map simulating a clutch engagement with varying k 5. Conclusion The simulation study has given us some insights into the mechanism(s) o Eek noise in clutches. Speciically, the riction-induced vibration o the pressure plate-disc cushion system has been correlated with a loss o dynamic stability. Our models describe a 6-DOF lumpedparameter system, and we include both rigid body wobbling mode and irst elastic deormation mode o the plate. The complex eigenvalue method is then utilized to map the system stability in terms o the riction coeicient, pressure plate geometry and structural stiness o the pressure plate. The latter is ound to be a key parameter. By controlling the requency o the irst elastic deormation mode, we can alter the coupling with the wobbling mode. This instability mechanism and its eect are correlated with experimental observations. Future work should ocus on developing a more analytical basis or the Eek phenomena. Acknowledgements We thank Dr. Todd Rook o BFGoodrich and the engineering team at LuK, Inc, or their valuable help in solving parts o the Eek mystery. Reerences 1. A. J. McMillan, A Non-linear Friction Model or Sel-excited Vibrations, Journal o Sound and Vibration 25(3), , Discussions with LuK, Inc. engineers, S. W. Kung, K. B. Dunlap and R. S. Ballinger, Complex Eigenvalue Analysis or Reducing Low Frequency Brake Squeal, SAE paper # D. J. Feld and D. J. Fehr, Complex Eigenvalue Analysis Applied to an Aircrat Brake Vibration Problem, ASME Design Engineering Technical Conerences, DE-Vol 84-1, Volume 3 Part A, 1995.