Modeling Parametric Vibration of Multistage Gear Systems as a Tool for Design Optimization
This work presents a numerical model developed to
simulate the dynamics and vibrations of a multistage tractor gearbox.
The effect of time varying mesh stiffness, time varying frictional
torque on the gear teeth, lateral and torsional flexibility of the shafts
and flexibility of the bearings were included in the model. The model
was developed by using the Lagrangian method, and it was applied to
study the effect of three design variables on the vibration and stress
levels on the gears. The first design variable, module, had little effect
on the vibration levels but a higher module resulted to higher bending
stress levels. The second design variable, pressure angle, had little
effect on the vibration levels, but had a strong effect on the stress
levels on the pinion of a high reduction ratio gear pair. A pressure
angle of 25o resulted to lower stress levels for a pinion with 14 teeth
than a pressure angle of 20o. The third design variable, contact ratio,
had a very strong effect on both the vibration levels and bending
stress levels. Increasing the contact ratio to 2.0 reduced both the
vibration levels and bending stress levels significantly. For the gear
train design used in this study, a module of 2.5 and contact ratio of
2.0 for the various meshes was found to yield the best combination
of low vibration levels and low bending stresses. The model can
therefore be used as a tool for obtaining the optimum gear design
parameters for a given multistage spur gear train.
[1] Tamminana, V. K., Kahraman, A. and Vijayakar, S., "A study of the
Relationship Between the dynamic factor and the dynamic transsmission
error of spur gear pairs," in ASME 2005 International Design
Engineering Technical Conferences and Computers and Information in
Engineering conference, September 24-28 2005.
[2] M. Vaishya and R. Singh, "Strategies for Modeling Friction in Gear
Dynamics," Journal of Mechanical Design, vol. 125, pp. 383-393, 2003.
[3] J. H. Kuang and A. D. Lin, "The Effect of Tooth Wear on the Vibration
Spectrum of a Spur Gear Pair," Journal of Vibrations and Acoustics,
vol. 123, pp. 311-317, 2001.
[4] M. Vaishya and R. Singh, "Analysis of Periodically Varying Gear Mesh
Systems with Coulomb Friction Using Floquet Theory," Journal of
Sound and Vibration, vol. 243(3), pp. 525-545, 2001.
[5] R. G. Parker, S. M. Vijayakar and T. Imajo, "Non-linear Dynamic Response
of a Spur Gear Pair: Modelling and Experimental Comparisons,"
Journal of Sound and Vibrations, vol. 237(3), pp. 435-455, 2000.
[6] L. Gelman, V. Giurgiutiu and A. Bayoumi, "Statistical Analysis of the
Dynamic Mean Excitation for a Spur Gear," Journal of Vibrations and
Acoustics, vol. 127, pp. 204-207, 2005.
[7] G. Bonori, A. O. Andrisano and F. Pellicano, "Stiffness Evaluation
and Vibration in a Tractor Gear," ASME International Mechanical
Engineering Congress and Exposition, 2004.
[8] M. A. Faith and O. Milosav, "Gear Vibration in Supercritical Mesh-
Frequency Range," Faculty of Mechanical Engineering (FME), Belgrade,
vol. 32, pp. 87-94, 2004.
[9] R. G. Timothy, "Computer- Aided Design Software for Torsional Analysis,"
Master-s thesis, Virginia Polytechnic Institute and State University,
1998.
[10] J. Lin and R. G. Parker, "Sensitivity of Planetory Gear Natural Frequencies
and Vibration Modes to Model Parameters," Journal of Sound and
Vibration, vol. 228(1), pp. 109-128, 1999.
[11] Shengxiang Jia, Ian Howard and Jiande Wang, "The Dynamic Modeling
of Multiple Pairs of Spur Gears in Mesh, Including Friction and
Geometric Errors," International Journal of Rotating Machinery, vol. 9,
pp. 437-442, 2003.
[12] T. L. Krantz and M. Rashidi, "Vibration Analysis of a Split Path
Gearbox," Tech. Rep., Army Research Laboratory, NASA, 1995.
[13] Jian Lin and R. G. Parker, "Mesh Stiffness Variation Instabilities in a
Two stage Gear System," Transactions of ASME, Journal of Vibration
and Acoustics, vol. 124, pp. 68-76, 2002.
[14] F. K. Choy, Y. K. Tu and D. P. Townsend, "Vibration Signature Analysis
of a Multistage Gear Transmission," Tech Rep., NASA Lewis Research
Center, 1989.
[15] H. J. Sutherland and D. P. Burwinkle, "The Spectral Content of the
Torque Loads on a Turbine Gear Tooth," Wind Energy, ASME, vol. 16,
pp. 91-97, 1995.
[16] M. L. James, G. M. Smith and P. W. Whaley, Vibration of Mechanical
and Structural systems. Harper Collins College Publishers, 2nd ed.,
1994.
[17] Hsiang-Hsi Lin, Computer - Aided Design and Analysis of Spur Gear
Dynamics. PhD Thesis, University of Cincinnati, 1985.
[18] K. L. Johnson, Contact Mechanics. Cambridge University Press, 1985.
[19] H. Xu and A. Kahraman, "Prediction of Mechanical Efficiency of
Parallel - Axis Gear Pairs," Transactions of ASME, vol. 129, pp. 58-68,
2007.
[20] D. W. Dudley, GEAR HANDBOOK: The Design, Manufacture and
Application of Gears. McGraw-Hill Publishing Company, Newyork,
1st ed., 1962.
[1] Tamminana, V. K., Kahraman, A. and Vijayakar, S., "A study of the
Relationship Between the dynamic factor and the dynamic transsmission
error of spur gear pairs," in ASME 2005 International Design
Engineering Technical Conferences and Computers and Information in
Engineering conference, September 24-28 2005.
[2] M. Vaishya and R. Singh, "Strategies for Modeling Friction in Gear
Dynamics," Journal of Mechanical Design, vol. 125, pp. 383-393, 2003.
[3] J. H. Kuang and A. D. Lin, "The Effect of Tooth Wear on the Vibration
Spectrum of a Spur Gear Pair," Journal of Vibrations and Acoustics,
vol. 123, pp. 311-317, 2001.
[4] M. Vaishya and R. Singh, "Analysis of Periodically Varying Gear Mesh
Systems with Coulomb Friction Using Floquet Theory," Journal of
Sound and Vibration, vol. 243(3), pp. 525-545, 2001.
[5] R. G. Parker, S. M. Vijayakar and T. Imajo, "Non-linear Dynamic Response
of a Spur Gear Pair: Modelling and Experimental Comparisons,"
Journal of Sound and Vibrations, vol. 237(3), pp. 435-455, 2000.
[6] L. Gelman, V. Giurgiutiu and A. Bayoumi, "Statistical Analysis of the
Dynamic Mean Excitation for a Spur Gear," Journal of Vibrations and
Acoustics, vol. 127, pp. 204-207, 2005.
[7] G. Bonori, A. O. Andrisano and F. Pellicano, "Stiffness Evaluation
and Vibration in a Tractor Gear," ASME International Mechanical
Engineering Congress and Exposition, 2004.
[8] M. A. Faith and O. Milosav, "Gear Vibration in Supercritical Mesh-
Frequency Range," Faculty of Mechanical Engineering (FME), Belgrade,
vol. 32, pp. 87-94, 2004.
[9] R. G. Timothy, "Computer- Aided Design Software for Torsional Analysis,"
Master-s thesis, Virginia Polytechnic Institute and State University,
1998.
[10] J. Lin and R. G. Parker, "Sensitivity of Planetory Gear Natural Frequencies
and Vibration Modes to Model Parameters," Journal of Sound and
Vibration, vol. 228(1), pp. 109-128, 1999.
[11] Shengxiang Jia, Ian Howard and Jiande Wang, "The Dynamic Modeling
of Multiple Pairs of Spur Gears in Mesh, Including Friction and
Geometric Errors," International Journal of Rotating Machinery, vol. 9,
pp. 437-442, 2003.
[12] T. L. Krantz and M. Rashidi, "Vibration Analysis of a Split Path
Gearbox," Tech. Rep., Army Research Laboratory, NASA, 1995.
[13] Jian Lin and R. G. Parker, "Mesh Stiffness Variation Instabilities in a
Two stage Gear System," Transactions of ASME, Journal of Vibration
and Acoustics, vol. 124, pp. 68-76, 2002.
[14] F. K. Choy, Y. K. Tu and D. P. Townsend, "Vibration Signature Analysis
of a Multistage Gear Transmission," Tech Rep., NASA Lewis Research
Center, 1989.
[15] H. J. Sutherland and D. P. Burwinkle, "The Spectral Content of the
Torque Loads on a Turbine Gear Tooth," Wind Energy, ASME, vol. 16,
pp. 91-97, 1995.
[16] M. L. James, G. M. Smith and P. W. Whaley, Vibration of Mechanical
and Structural systems. Harper Collins College Publishers, 2nd ed.,
1994.
[17] Hsiang-Hsi Lin, Computer - Aided Design and Analysis of Spur Gear
Dynamics. PhD Thesis, University of Cincinnati, 1985.
[18] K. L. Johnson, Contact Mechanics. Cambridge University Press, 1985.
[19] H. Xu and A. Kahraman, "Prediction of Mechanical Efficiency of
Parallel - Axis Gear Pairs," Transactions of ASME, vol. 129, pp. 58-68,
2007.
[20] D. W. Dudley, GEAR HANDBOOK: The Design, Manufacture and
Application of Gears. McGraw-Hill Publishing Company, Newyork,
1st ed., 1962.
@article{"International Journal of Mechanical, Industrial and Aerospace Sciences:50398", author = "James Kuria and John Kihiu", title = "Modeling Parametric Vibration of Multistage Gear Systems as a Tool for Design Optimization", abstract = "This work presents a numerical model developed to
simulate the dynamics and vibrations of a multistage tractor gearbox.
The effect of time varying mesh stiffness, time varying frictional
torque on the gear teeth, lateral and torsional flexibility of the shafts
and flexibility of the bearings were included in the model. The model
was developed by using the Lagrangian method, and it was applied to
study the effect of three design variables on the vibration and stress
levels on the gears. The first design variable, module, had little effect
on the vibration levels but a higher module resulted to higher bending
stress levels. The second design variable, pressure angle, had little
effect on the vibration levels, but had a strong effect on the stress
levels on the pinion of a high reduction ratio gear pair. A pressure
angle of 25o resulted to lower stress levels for a pinion with 14 teeth
than a pressure angle of 20o. The third design variable, contact ratio,
had a very strong effect on both the vibration levels and bending
stress levels. Increasing the contact ratio to 2.0 reduced both the
vibration levels and bending stress levels significantly. For the gear
train design used in this study, a module of 2.5 and contact ratio of
2.0 for the various meshes was found to yield the best combination
of low vibration levels and low bending stresses. The model can
therefore be used as a tool for obtaining the optimum gear design
parameters for a given multistage spur gear train.", keywords = "bending stress levels, frictional torque, gear designparameters, mesh stiffness, multistage gear train, vibration levels.", volume = "2", number = "2", pages = "151-8", }