Abstract
A dynamic lumped-parameter gear model incorporating the effects of a time-varying and asymmetric mesh stiffness and a backlash nonlinearity is formulated to analyze the spur gear rattle response under the idling condition. The proposed theory assumes a rectangular time-varying mesh stiffness function. The phase shift between the mesh stiffness for forward and backward contacts is examined. Numerical studies are employed to examine the effects of engine torque fluctuations and tooth surface friction on the gear rattle response and the corresponding tooth impact behavior. Comparisons between the results from the time-invariant mesh stiffness model and the proposed time-varying mesh stiffness model reveal differences in the gear responses, especially when the mean rotational speed of the fluctuating gear pair is non-zero. The analysis reveals significant effects on the high frequency response components. However, the idling gear dynamics are relatively insensitive to tooth surface friction.
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Abbreviations
- T p /T g :
-
external and braking torques
- Ω p /Ω g :
-
nominal operational speeds of pinion and gear
- θ a :
-
angular position of the tooth of pinion as it just moves into meshing zone
- θ c :
-
angular position of another tooth of pinion corresponding to θ a
- θ d :
-
angular position of the tooth of pinion when it just left meshing
- θ b :
-
angular position of another tooth of pinion corresponding to θ d
- θ p :
-
angular position of the tooth of pinion when gear teeth contacts at pitch point
- θ A /θ C /θ C /θ B /θ p :
-
only for the backward tooth contact in Figure 1(b)
- θ a /θ c /θ d /θ b /θ p :
-
only for forward tooth contact in Figure 1(a)
- LOA:
-
line of action
- OLOA:
-
off line of action
- K m /C m :
-
meshing stiffness and damping coefficient
- K d /K s :
-
mesh stiffness for double-tooth and single-tooth engagement respectively
- ξ(t):
-
dynamic transmission error
- η :
-
one-half of the backlash of the gear teeth flank
- e(t):
-
transmission error function
- R ij :
-
radius, subscript i − p, g for pinion and gear respectively, j-a, p, b for addendum, pitch and base circles respectively
- z p /z g :
-
number of teeth of pinion/gear
- J p /J g :
-
inertial moment of pinion/gear
- m p /m g :
-
mass of pinion/gear
- λ :
-
distance of the two adjacent teeth of pinion along the LOA
- θ p (t):
-
rotating angle of pinion normalized by λ divided by R pb
- α :
-
pressure angle on pitch line
- θ 0 :
-
illustrated in Figure 1(a)
- F k /F c :
-
elastic force/dissipative force
- ɛ :
-
manufacturing errors on the tooth
- Δ:
-
contact-type coefficient
- \({{y_p ,\dot y_p ,\ddot y_p } \mathord{\left/ {\vphantom {{y_p ,\dot y_p ,\ddot y_p } {y_{g,} \dot y_{g,} \ddot y_g }}} \right. \kern-\nulldelimiterspace} {y_{g,} \dot y_{g,} \ddot y_g }}\) :
-
lateral displacement, velocity, acceleration of pinion/gear along LOA
- \({{x_p ,\dot x_p ,\ddot x_p } \mathord{\left/ {\vphantom {{x_p ,\dot x_p ,\ddot x_p } {x_{g,} \dot x_{g,} \ddot x_g }}} \right. \kern-\nulldelimiterspace} {x_{g,} \dot x_{g,} \ddot x_g }}\) :
-
lateral displacement, velocity, acceleration of pinion/gear along OLOA
- K pBx /K gBx :
-
stiffness of the supporting bearing of pinion/gear along OLOA
- C pBx /K gBx :
-
damping of the supporting bearing of pinion/gear along OLOA
- K pBx /K gBx :
-
stiffness of the supporting bearing of pinion/gear along LOA
- C pBx /K gBx :
-
damping of the supporting bearing of pinion/gear along LOA
- θ p /θ g :
-
angular displacement of pinion/gear
- \({{\dot \theta _p } \mathord{\left/ {\vphantom {{\dot \theta _p } {\dot \theta _g }}} \right. \kern-\nulldelimiterspace} {\dot \theta _g }}\) :
-
angular velocity of pinion/gear
- \({{\ddot \theta _p } \mathord{\left/ {\vphantom {{\ddot \theta _p } {\ddot \theta _g }}} \right. \kern-\nulldelimiterspace} {\ddot \theta _g }}\) :
-
angular acceleration of pinion/gear
- F f /M f :
-
friction force/moment
- N :
-
contact force
- C g :
-
damping between gear and shaft due to lubrication
- Γ:
-
profile contact ratio of gears
- A :
-
amplitude of angular acceleration
- f e :
-
frequency of angular acceleration
- u :
-
friction coefficient
- ν :
-
rotational speed of pinion
- σ :
-
contact ratio
References
Brancati, R., Rocca, E. and Russo, R. (2005). A gear rattle model accounting for oil squeeze between the meshing gear teeth. Proc. Institution of Mechanical Engineers—Part D: J. Automobile Engineering, 219, 1075–1083.
Brauer, J. (2005). Transmission error in anti-backlash conical involute gear transmissions: A global-local FE approach. Finite Elements in Analysis and Design, 41, 431–457.
Cai, Y. (1995). Simulation on the rotational vibration of helical gears in consideration of the tooth separation phenomenon (A new stiffness function of helical involute tooth pair). Trans. ASME, J. Mechanical Design, 117, 460–469.
Chaari, F., Fakhfakh, T. and Haddar, M. (2009). Analytical modeling of spur gear tooth crack and influence on gear mesh stiffness. European J. Mechanics A/Solids, 28, 461–468.
Cornell, R. W. (1981). Compliance and stress sensitivity of spur gear teeth. ASME J. Mech. Des., 103, 447–459.
Crowther, A. R., Singh, R., Zhang, N. and Chapman, C. (2007). Impulsive response of an automatic transmission system with multiple clearances: Formulation, simulation and experiment. J. Sound and Vibration, 306, 444–466.
Dion, J.-L., Moyne, S. L., Chevallier, G. and Sebbah, H. (2009). Gear impacts and idle gear noise: Experimental study and non-linear dynamic model. Mechanical Systems and Signal Processing, 23, 2608–2628.
Dogan, S. N. (1999). Loose part vibration in vehicle transmissions — gear rattle. Trans. J. Engineering and Environmental Science, 23, 439–454.
Fujimoto, T. and Kizuka, T. (2001). An improvement of the prediction method of the idling rattle in manual transmission—in the case of the manual transmission with backlash eliminator. SAE Paper No. 2001-01-1164.
Fujimoto, T., Chikatani, Y. and Kojima, J. (1987). Reduction of idling rattle in manual transmission. SAE Paper No. 870395.
Han, B. K., Cho, M. K., Kim, C., Lim, C. H. and Kim, J. J. (2009). Prediction of vibrating forces on meshing gears for a gear rattle using a new multi-body dynamic model. Int. J. Automotive Technology 10,4, 469–474.
He, S., Cho, S. and Singh, R. (2008). Prediction of dynamic friction forces in spur gears using alternate sliding friction formulations. J. Sound and Vibration, 309, 843–851.
Heinrichs, R. and Bodden, M. (1999). Perceptual and Instrumental Description of the gear rattle phenomenon for diesel vehicles. 6th Int. Cong. Sound and Vibration, 99, Technical University of Demark, Lyngby.
Houser, D. R. and Harianto, J. (1998). Load Distribution Program Manual. The Ohio State University. Columbus. Ohio.
Houser, D. R., Harianto, J. and Ueda, Y. (2004). Determining the source of gear whine noise. Gear Solutions, 16–23.
Howard, I., Jia, S. and Wang, J. (2001). The dynamic modelling of a spur gear in mesh including friction and a crack. Mechanical Systems and Signal Processing 15,5, 831–853.
Kim, C., Han, B., Cho, M. and Lim, C. (2008). Topology optimization of transmission housings for minimizing the gear rattling noise. 8th World Cong. Computational Mechanics (WCCM8), 5th. European Cong. Computational Methods in Applied Sciences and Engineering (ECCOMAS 2008), Venice, Italy.
Kim, T. C. and Singh, R. (2001). Dynamic interactions between loaded and unloaded gear pairs under rattle conditions. SAE Paper No. 2001-01-1553.
Luo, A. C. J. and O’Connor, D. (2011). Periodic and chaotic motions in a gear-pair transmission system with impacts. Nonlinear Science and Complexity, Part 1, 13–24.
Padmanabhan, C., Rook, T. and Singh, R. (1995). Modelling of automobile gear rattle phenomenon: State of the art. SAE Paper No. 951316.
Pimsarn, M. and Kazerounian, K. (2002). Efficient evaluation of spur gear tooth mesh load using pseudointerference stiffness estimation method. Mech. Mach. Th., 37, 769–786.
Rebbechi, B., Oswald, F. and Townsend, D. (1996). Measurement of gear tooth dynamic friction. ASME, DE-, 88, Proc. 7th Power Transmission and Gearing Conf., 355–363.
Russo, R., Brancati, R. and Rocca, E. (2009). Experimental investigations about the influence of oil lubricant between teeth on the gear rattle phenomenon. J. Sound and Vibration, 321, 647–661.
Sarkar, N., Ellis, R. E. and Moore, T. N. (1997). Backlash detection in geared mechanisms: Modelling, simulation and experimentation. Mechanical Systems and Signal Processing, 11, 391–408.
Shih, S., Yruma, J. and Kittredge, P. (2001). Drivetrain noise and vibration troubleshooting. SAE Paper No. 2001-01-2809.
Shim, Y., Kauh, S. K. and Ha, K.-P. (2011). Evaluation of idle stability through in-situ torque measurement in automatic transmission vehicles. Int. J. Automotive Technology 12,3, 315–320.
Sirichai, S. (1999). Torsional Properties of Spur Gears in Mesh Using Nonlinear Finite Element Analysis. Ph.D. Dissertation. Curtin University of Technology.
Tangasawi, O., Theodossiades, S. and Rahnejat, H. (2007). Lightly loaded lubricated impacts: Idle gear rattle. J. Sound and Vibration, 308, 418–430.
Theodossiades, S., Tangasawi, O. and Rahnejat, H. (2007). Gear teeth impacts in hydrodynamic conjunctions promoting idle gear rattle. J. Sound and Vibration, 303, 632–658.
The Ohio State University (1993). MULTILDP Version 8.22. Computer Program, Gear Dynamics and Gear Noise Research Laboratory. The Ohio State University. Columbus. Ohio.
Umezawa, K., Suzuki, T. and Sato, T. (1986). Vibration of power transmission helical gears (approximate equation of tooth stiffness). Bulletin of JSME, 29, 1605–1611.
Vaishya, M. and Singh, R. (2001). Analysis of periodically varying gear mesh systems with coulomb friction using floquet theory. J. Sound and Vibration 243,3, 525–545.
Walha, L., Fakhfakh, T. and Haddar, M. (2009). Nonlinear dynamics of a two-stage gear system with mesh stiffness uctuation, bearing exibility and backlash. Mechanism and Machine Theory, 44, 1058–1069.
Wang, J. (2003). Numerical and Experimental Analysis of Spur Gears in Mesh. Ph.D. Dissertation. Curtin University of Technology.
Wang, M., Manoj, R. and Zhao, W. (2001). Gear rattle modeling and analysis for automotive manual transmissions. Proc. Institution of Mechanical Engineers—Part D: J. Automobile Engineering, 215, 241–258.
Wang, M., Zhao, W., Manoj, R. (2002). Numerical modelling and analysis of automotive transmission rattle. J. Vibration and Control, 8, 921–943.
Weber, C. (1949). The Deformation of Loaded Gears and the Effect on Their Load-carrying Capacity. Sponsored Research. British Dept. Sci. and Ind. Res. Report No.3. Germany.
Wu, S., Zuo, M. J. and Parey, A. (2008). Simulation of spur gear dynamics and estimation of fault growth. J. Sound and Vibration, 317, 608–624.
Yakoub, R. Y., Corrado, M., Forcelli, A., Pappalardo, T. and Dutre, S. (2004). Prediction of system-level gear rattle using multibody and vibro-acoustic techniques. SAE Paper No. 2004-32-0063.
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Chen, Z.G., Shao, Y.M. & Lim, T.C. Non-linear dynamic simulation of gear response under the idling condition. Int.J Automot. Technol. 13, 541–552 (2012). https://doi.org/10.1007/s12239-012-0052-1
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DOI: https://doi.org/10.1007/s12239-012-0052-1