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Nonlinear dynamic behaviour and severity of lightly loaded gear rattle under different vibro-impact models and internal excitations

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Abstract

This investigation comprehensively compared the elastic contact and the modified stiff impact models, aiming for the backlash-induced nonlinear vibro-impact gear rattle under lightly loaded conditions. Three meshing force models in backlash are incorporated into the elastic model independently. Unlike the previous uncoupled stiff impact model, the modified impact model employed in this paper considers the coupling between the pinion and gearwheel. Three scenarios were investigated with different components of two internal excitations, static transmission error-induced periodical backlash and the time-varying meshing stiffness. The numerical results show that the free and forced gear motion, nonlinear characteristics and rattle severity are significantly affected by static transmission error rather than time-varying meshing stiffness. Two studied hydrodynamic lubricant models show different damping effects throughout the free vibration response. The forced gear motion and rattle sensitivities show a noticeable difference below 40 rad/s2 and turn to a slight variation above 60rad/s2. The high excitation level taking over the backlash determines the dynamic characteristics more deterministically than the internal excitations. Finally, the hydrodynamic lubricant model containing only the oil squeeze effect will likely match the experimental results obtained from the dedicated rattle test bench under several conditions. The experimentally detected components of the static transmission error on the gearwheel suggest that an acceptable model should be considered in the gear rattle model.

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Data availability statement

The datasets generated and/or analysed during the current study are available from the corresponding author on reasonable request.

Abbreviations

ATol:

Absolute tolerance

DSI:

Double-side impact

EHL:

Elastohydrodynamic lubricant

FFT:

Fast Fourier transform

GDA:

KLINGELNBERG gear deviation analysis

HDL:

Hydrodynamic lubricant

ICE:

Internal combustion engine

ICEFS:

Internal combustion engine function simulation

LOA:

Line of action

MEBDF:

Modified extended backward differentiation formulas

NI:

No impact

R Tol :

Relative tolerance

SSI:

Single-side impact

TVMS:

Time-varying meshing stiffness

a :

Half-length of the oil film

A e :

Excitation level, rad/s2

b:

Nominal half gear backlash

c :

Contact damping

f :

Frequency, Hz

F :

Gear meshing force

h a :

Gear addendum height

h c :

Centre oil film thickness

\({i_g}\) :

Gear ratio

j :

Periodical backlash

J :

Moment of inertia of gear

k :

Stiffness, N/m

l :

Harmonic number of Fourier series

L r :

Rattle severity index, dB

m :

Gear equivalent mass, kg

m n :

Gear normal module, mm

M a :

Alternative torque, Nm

M DT :

Drag torque, Nm

M e :

Dynamic torque, Nm

M m :

Mean torque, Nm

n :

Rotational speed, rpm

r :

Curvature radius at contact point

r c :

Coefficient of restitution

R :

Gear geometry radius

R a :

Roughness

t :

Time

V ent :

Fluid entraining velocity

x :

Relative displacement

z:

Number of teeth

Z:

Common axial face width of a gear pair

\( {\alpha_n}\) :

Gear normal pressure angle, deg

\(\beta \) :

Gear helix angle, deg

\(\xi \) :

Damping ratio

\(\rho \) :

Density

\(\mu \) :

Dynamic viscosity

\(\tilde e\) :

Static transmission error

\(\theta \) :

Rotational displacement

\(\delta \) :

Feigenbaum’s value

bp :

Pinion base circle

bw :

Gearwheel base circle

c :

Gear teeth contact

e :

Excitation

e q :

Equivalent

HRa :

Rahnejat, H. HDL model

g :

Gear meshing

r :

Pinion shaft rotation

s :

Teeth separation

sa :

Sampling

stf :

Modified stiff impact

p :

Pinion

pp :

Pinion pitch circle

pw :

Gearwheel pitch circle

RBr :

Brancati, R. HDL model

w :

Gearwheel

. :

First-time derivation

.. :

Second-time derivation

- :

Mean value

References

  1. Dogan, S.N., Ryborz, J., Bertsche, B.: Design of low-noise manual automotive transmissions. Proc. IMechE, Part K: J. Multi-body Dynamics. 220, 79–95 (2006). https://doi.org/10.1243/14644193JMBD10.

  2. Seaman, R.L., Johnson, C.E., Hamilton, R.F.: Component inertial effects on transmission Design. SAE Technical Paper. 841686 (1984). https://doi.org/10.4271/841686.

  3. Knabe, G.M., Küçükay, F.: Customer orientated approach for evaluation of transmission rattle. SAE Technical Paper. 2012–01–0311 (2012). https://doi.org/10.4271/2012-01-0311.

  4. Singh, R., Xie, H., Comparin, R.J.: Analysis of automotive neutral gear rattle. J. Sound Vib. 131, 177–196 (1989). https://doi.org/10.1016/0022-460X(89)90485-9

    Article  Google Scholar 

  5. Donmez, A., Kahraman, A.: Experimental and theoretical investigation of vibro-impact motions of a gear pair subjected to torque fluctuations to define a rattle noise severity index. J. Vib. Acoust. 144, 041001 (2022). https://doi.org/10.1115/1.4053264

    Article  Google Scholar 

  6. Rigaud, E., Perret-Liaudet, J.: Investigation of gear rattle noise including visualization of vibro-impact regimes. J. Sound Vib. 467, 115026 (2020). https://doi.org/10.1016/j.jsv.2019.115026

    Article  Google Scholar 

  7. Baumann, A., Bertsche, B.: Experimental study on transmission rattle noise behaviour with particular regard to lubricating oil. J. Sound Vib. 341, 195–205 (2015). https://doi.org/10.1016/j.jsv.2014.12.018

    Article  Google Scholar 

  8. Chae, C.-K., Won, K.-M., Kang, K.T., et al.: Measurement of transmission rattle sensitivity and calculation of driveline torsional vibration for gear rattle analysis. SAE Technical Paper. 2005–01–1785 (2005). https://doi.org/10.4271/2005-01-1785.

  9. Wei, Z., Shangguan, W.-B., Liu, X., et al.: Modeling and analysis of friction clutches with three stages stiffness and damping for reducing gear rattles of unloaded gears at transmission. J. Sound Vib. 483, 115469 (2020). https://doi.org/10.1016/j.jsv.2020.115469

    Article  Google Scholar 

  10. Wang, Y., Qin, X., Huang, S., et al.: Design and analysis of a multi-stage torsional stiffness dual mass flywheel based on vibration control. Appl. Acoust. 104, 172–181 (2016). https://doi.org/10.1016/j.apacoust.2015.11.004

    Article  Google Scholar 

  11. Zhou, Y., Shi, X., Rao, W., et al.: Feasibility study of single-mass flywheels with centrifugal pendulum vibration absorbers in vehicles with dual-clutch transmissions. J. Vib. Control 29, 3213–3226 (2023). https://doi.org/10.1177/10775463221093106

    Article  Google Scholar 

  12. Guo, D., Zhou, Y., Zhou, Y., et al.: Numerical and experimental study of gear rattle based on a refined dynamic model. Appl. Acoust. 185, 108407 (2022). https://doi.org/10.1016/j.apacoust.2021.108407

    Article  Google Scholar 

  13. Kamo, M., Yamamoto, H., Koga, H., et al.: Analysis method for the contribution rate of each pair of gears on the driveline gear rattle. SAE Technical Paper. 96072 (1996). https://doi.org/10.4271/960726.

  14. Bozca, M.: Transmission error model-based optimisation of the geometric design parameters of an automotive transmission gearbox to reduce gear-rattle noise. Appl. Acoust. 130, 247–259 (2018). https://doi.org/10.1016/j.apacoust.2017.10.005

    Article  Google Scholar 

  15. Bozca, M., Fietkau, P.: Empirical model based optimization of gearbox geometric design parameters to reduce rattle noise in an automotive transmission. Mech. Mach. Theory 45, 1599–1612 (2010). https://doi.org/10.1016/j.mechmachtheory.2010.06.013

    Article  Google Scholar 

  16. Liu, D., Zhao, G., Fu, Z., et al.: Rattle dynamic model and vibration behaviors of noncircular face gear. J. Comput. Nonlinear Dyn. 17, 061002 (2022). https://doi.org/10.1115/1.4053933

    Article  Google Scholar 

  17. Liu, G., Parker, R.G.: Nonlinear dynamics of idler gear system. Nonlinear Dyn. 53, 345–367 (2008). https://doi.org/10.1007/s11071-007-9317-z

    Article  Google Scholar 

  18. Robinette, D., Beikmann, R., Piorkowski, P., et al.: Characterizing the onset of manual transmission gear rattle part II: analytical results. SAE Int. J. Passenger Cars Mech. Syst. 2, 1365–1376 (2009). https://doi.org/10.4271/2009-01-2069.

  19. Wang, M.Y., Manoj, R., Zhao, M.: Gear rattle modelling and analysis for automotive manual transmissions. Proc. IMechE, Part D: J. Automobile Engineering. 215, 241–258 (2001). https://doi.org/10.1243/0954407011525610.

  20. Karagiannis, K., Pfeiffer, F.: Theoretical and experimental investigations of gear-rattling. Nonlinear Dyn. 2, 367–387 (1991). https://doi.org/10.1007/BF00045670

    Article  Google Scholar 

  21. Kadmiri, Y., Rigaud, E., Perret-Liaudet, J., et al.: Experimental and numerical analysis of automotive gearbox rattle noise. J. Sound Vib. 331, 3144–3157 (2012). https://doi.org/10.1016/j.jsv.2012.02.009

    Article  Google Scholar 

  22. Theodossiades, S., Tangasawi, O., Rahnejat, H.: Gear teeth impacts in hydrodynamic conjunctions promoting idle gear rattle. J. Sound Vib. 303, 632–658 (2007). https://doi.org/10.1016/j.jsv.2007.01.034

    Article  Google Scholar 

  23. Theodossiades, S., De La Cruz, M., H Rahnejat, H.: Prediction of airborne radiated noise from lightly loaded lubricated meshing gear teeth. Appl. Acoustics. 100, 79–86 (2015). https://doi.org/10.1016/j.apacoust.2015.06.014.

  24. Russo, R., Brancati, R., Rocca, E.: Experimental investigations about the influence of oil lubricant between teeth on the gear rattle phenomenon. J. Sound Vib. 321, 647–661 (2009). https://doi.org/10.1016/j.jsv.2008.10.008

    Article  Google Scholar 

  25. Fietkau, P., Bertsche, B.: Influence of tribological and geometrical parameters on lubrication conditions and noise of gear transmissions. Mech. Mach. Theory 69, 303–320 (2013). https://doi.org/10.1016/j.mechmachtheory.2013.06.007

    Article  Google Scholar 

  26. De la Cruz, M., Chong, W.W.F., Teodorescu, M., et al.: Transient mixed thermo-elastohydrodynamic lubrication in multi-speed transmissions. Tribol. Int. 49, 17–29 (2012). https://doi.org/10.1016/j.triboint.2011.12.006

    Article  Google Scholar 

  27. Rahnejat, H.: Computational modelling of problems in contact dynamics. Eng. Anal. 2, 192–197 (1985). https://doi.org/10.1016/0264-682X(85)90031-0

    Article  Google Scholar 

  28. Brancati, R., Rocca, E., Russo, R.: A gear rattle model accounting for oil squeeze between the meshing gear teeth. Proc IMechE Part D J. Auto. Eng. 219, 1075–1083 (2005). https://doi.org/10.1243/095440705X34757

    Article  Google Scholar 

  29. Xia, Y., Pang, J.: Investigation on vibro-impacts of driveline system based on a nonlinear clearance element with time-varying stiffness and oil-squeeze damping. J. Vib. Acoust. 144, 031014 (2022). https://doi.org/10.1115/1.4053477

    Article  Google Scholar 

  30. Jerret-Liaudet, J, Rigaud, E. Some effects of gear eccentricities on automotive rattle noise. In: Proceedings of the ASME 2007 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. Volume 7: 10th International Power Transmission and Gearing Conference. Las Vegas, Nevada, USA. September 4–7, 561–568 (2007). https://doi.org/10.1115/DETC2007-34794.

  31. Ottewill, J.R., Neild, S.A, Wilson, R.E.: Intermittent gear rattle due to interactions between forcing and manufacturing errors. J. Sound Vib. 321, 913–935 (2009). https://doi.org/10.1016/j.jsv.2008.09.050.

  32. Ottewill, J.R., Neild, S.A., Wilson, R.E.: An investigation into the effect of tooth profile errors on gear rattle. J. Sound Vib. 329, 3495–3506 (2010). https://doi.org/10.1016/j.jsv.2010.03.014

    Article  Google Scholar 

  33. Donmez, A., Kahraman, A.: Influence of various manufacturing errors on gear rattle. Mech. Mach. Theory 173, 104868 (2022). https://doi.org/10.1016/j.mechmachtheory.2022.104868

    Article  Google Scholar 

  34. Kahraman, A.: On the response of a preloaded mechanical oscillator with a clearance: Period-doubling and chaos. Nonlinear Dyn. 3, 183–198 (1992). https://doi.org/10.1007/BF00122301

    Article  Google Scholar 

  35. Kahraman, A., Blankenship, G.W.: Experiments on nonlinear dynamic behavior of an oscillator with clearance and periodical time-varying parameters. J. Appl. Mech. 64, 217–226 (1997). https://doi.org/10.1115/1.2787276

    Article  Google Scholar 

  36. Xiao, Z, Zhou, C., Chen, S. et al. Effects of oil film stiffness and damping on spur gear dynamics. Nonlinear Dyn. 96, 145–159 (2019). https://doi.org/10.1007/s11071-019-04780-6.

  37. Li, Z., Zhu, C., Liu, H., et al.: Mesh stiffness and nonlinear dynamic response of a spur gear pair considering tribo-dynamic effect. Mech. Mach. Theory 153, 103989 (2020). https://doi.org/10.1016/j.mechmachtheory.2020.103989

    Article  Google Scholar 

  38. Donmez, A., Kahraman, A.: Characterization of nonlinear rattling behavior of a gear pair through a validated torsional model. J. Comput. Nonlinear Dyn. 17, 041006 (2022). https://doi.org/10.1115/1.4053367

    Article  Google Scholar 

  39. Kahraman, A., Singh, R.: Interactions between time-varying mesh stiffness and clearance non-linearity in a geared system. J. Sound Vib. 146, 135–156 (1991). https://doi.org/10.1016/0022-460X(91)90527-Q

    Article  Google Scholar 

  40. Fernandez-Del-Rincon, A., Diez-Ibarbia, A., Theodossiades, S.: Gear transmission rattle: assessment of meshing forces under hydrodynamic lubrication. Appl. Acoust. 144, 85–95 (2019). https://doi.org/10.1016/j.apacoust.2017.04.001

    Article  Google Scholar 

  41. Diez-Ibarbia, A., Fernandez-del-Rincon, A., Garcia, P., et al.: Gear rattle dynamics under non-stationary conditions: The lubricant role. Mech. Mach. Theory 151, 103929 (2020). https://doi.org/10.1016/j.mechmachtheory.2020.103929

    Article  Google Scholar 

  42. Johnson, O., Hirami, N.: Diagnosis and objective evaluation of gear rattle. SAE Technical Paper. 911082 (1991). https://doi.org/10.4271/911082.

  43. Gill-Jeong, C.: Analysis of the nonlinear behavior of gear pairs considering hydrodynamic lubrication and sliding friction. J. Mech. Sci. Technol. 23, 125–137 (2009). https://doi.org/10.1007/s12206-009-0623-x

    Article  Google Scholar 

  44. Guilbault, R., Lalonde, S., Thomas, M.: Nonlinear damping calculation in cylindrical gear dynamic modeling. J. Sound Vib. 331, 2110–2128 (2012). https://doi.org/10.1016/j.jsv.2011.12.025

    Article  Google Scholar 

  45. Barthod, M., Hayne, B., Tebec, J.-L., et al.: Experimental study of gear rattle excited by a multi-harmonic excitation. Appl. Acoust. 68, 1003–1025 (2007). https://doi.org/10.1016/j.apacoust.2006.04.011

    Article  Google Scholar 

  46. Brancati, R., Rocca, E., Siano, D., et al.: Experimental vibro-acoustic analysis of the gear rattle induced by multi-harmonic excitation. Proc. Inst. Mech. Eng. Part D J. Auto. Eng. 232, 785–796 (2018). https://doi.org/10.1177/0954407017707670

    Article  Google Scholar 

  47. Wan, Z., Cao, H., Zi, Y., et al.: Mesh stiffness calculation using an accumulated integral potential energy method and dynamic analysis of helical gears. Mech. Mach. Theory 92, 447–463 (2019). https://doi.org/10.1016/j.mechmachtheory.2015.06.011

    Article  Google Scholar 

  48. Yi, Y., Huang, K., Xiong, Y., et al.: Nonlinear dynamic modelling and analysis for a spur gear system with time-varying pressure angle and gear backlash. Mech. Syst. Signal Process. 132, 18–34 (2019). https://doi.org/10.1016/j.ymssp.2019.06.013

    Article  Google Scholar 

  49. Rocca, E., Russo, R.: Theoretical and experimental investigation into the influence of the periodical backlash fluctuation on the gear rattle. J. Sound Vib. 330, 4738–4752 (2011). https://doi.org/10.1016/j.jsv.2011.04.008

    Article  Google Scholar 

  50. Cash, J.R., Considine, S.: An MEBDF code for stiff initial value problems. ACM Trans. Math. Softw. 18, 142–155 (1992). https://doi.org/10.1145/146847.146922

    Article  MathSciNet  Google Scholar 

  51. Dion, J.-L., Moyne, S.L., Chevallier, G., et al.: Gear impacts and idle gear noise: experimental study and non-linear dynamic model. Mech. Syst. Signal Process. 23, 2608–2628 (2009). https://doi.org/10.1016/j.ymssp.2009.05.007

    Article  Google Scholar 

  52. Kahnenbley, T., Gravel, G.: Gear standards for reliable measurement of noise-causing tooth flank ripples. Forsch. Ingenieurwes. 83, 537–543 (2019). https://doi.org/10.1007/s10010-019-00366-1

    Article  Google Scholar 

  53. Azar, R.C., Crossley, F.R.E.: Digital simulation of impact phenomenon in spur system. J. Eng. Ind. 99(3), 792–798 (1977). https://doi.org/10.1115/1.3439315

    Article  Google Scholar 

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Acknowledgements

The authors would like to thank MAGNA PT Powertrain (Jiangxi) for the rattle experiments on the test bench and the gear data measurements.

Funding

This project was supported by the National Natural Science Foundation of China (Grant No. 51975080), China Scholarship Council (Grant No. 202106050063) and Jiangxi Province Key Development Project (Grant No. 20202BBE53007).

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Authors and Affiliations

  1. College of Mechanical and Vehicle Engineering, Chongqing University, Chongqing, 400044, China

    Yi Zhou

  2. College of Vehicle Engineering, Chongqing University of Technology, Chongqing, 400054, China

    Xiaohui Shi & Dong Guo

  3. Institute of Machine Components, University of Stuttgart, 70569, Stuttgart, Germany

    Yi Zhou, Martin Dazer & Bernd Bertsche

  4. Product Engineering Department, Magna PT Powertrain (Jiangxi) Co., Ltd, Nanchang, 330013, China

    Ziyuan Mei

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  1. Yi Zhou
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Zhou, Y., Shi, X., Guo, D. et al. Nonlinear dynamic behaviour and severity of lightly loaded gear rattle under different vibro-impact models and internal excitations. Nonlinear Dyn 112, 961–993 (2024). https://doi.org/10.1007/s11071-023-09113-2

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