A Contact Force Model Considering Meshing and Collision States for Dynamic Analysisin Helical Gear System
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摘要
The current research on gear system dynamics mainly utilizes linear spring damping model to calculate the contact force between gears. However, this linear model cannot correctly describe the energy transfer process of collision that often occurs in gear system. Focus on the contact-impact events, this paper proposes an improved gear contact force model for dynamic analysis in helical gear transmission system. In this model, a new factor associated with hysteresis damping is developed for contact-impact state, whereas the traditional linear damping factor is utilized for normal meshing state. For determining the selection strategy of these two damping factors, the fundamental contact mechanics of contact-impact event a ected by supporting forces are analyzed. During this analysis, an e ect factor is proposed for evaluating the influence of supporting forces on collision. Meanwhile, a new restitution of coe cient is deduced for calculating hysteresis damping factor, which suitable for both separation and non-separation states at the end of collision. In addition, the time-varying meshing sti ness(TVMS) is obtained based on the potential energy approach and the slice theory. Finally, a dynamic analysis of a helical gear system is carried out to better understand the contact force model proposed in this paper. The analysis results show that the contribution of supporting forces to the dynamic response of contact-impact event within gear pair is important. The supporting forces and dissipative energy are the main reasons for gear system to enter a steady contact state from repeated contact-impact state. This research proposes an improved contact force model which distinguishes meshing and collision states in gear system.
The current research on gear system dynamics mainly utilizes linear spring damping model to calculate the contact force between gears. However, this linear model cannot correctly describe the energy transfer process of collision that often occurs in gear system. Focus on the contact-impact events, this paper proposes an improved gear contact force model for dynamic analysis in helical gear transmission system. In this model, a new factor associated with hysteresis damping is developed for contact-impact state, whereas the traditional linear damping factor is utilized for normal meshing state. For determining the selection strategy of these two damping factors, the fundamental contact mechanics of contact-impact event a ected by supporting forces are analyzed. During this analysis, an e ect factor is proposed for evaluating the influence of supporting forces on collision. Meanwhile, a new restitution of coe cient is deduced for calculating hysteresis damping factor, which suitable for both separation and non-separation states at the end of collision. In addition, the time-varying meshing sti ness(TVMS) is obtained based on the potential energy approach and the slice theory. Finally, a dynamic analysis of a helical gear system is carried out to better understand the contact force model proposed in this paper. The analysis results show that the contribution of supporting forces to the dynamic response of contact-impact event within gear pair is important. The supporting forces and dissipative energy are the main reasons for gear system to enter a steady contact state from repeated contact-impact state. This research proposes an improved contact force model which distinguishes meshing and collision states in gear system.
The current research on gear system dynamics mainly utilizes linear spring damping model to calculate the contact force between gears. However, this linear model cannot correctly describe the energy transfer process of collision that often occurs in gear system. Focus on the contact-impact events, this paper proposes an improved gear contact force model for dynamic analysis in helical gear transmission system. In this model, a new factor associated with hysteresis damping is developed for contact-impact state, whereas the traditional linear damping factor is utilized for normal meshing state. For determining the selection strategy of these two damping factors, the fundamental contact mechanics of contact-impact event a ected by supporting forces are analyzed. During this analysis, an e ect factor is proposed for evaluating the influence of supporting forces on collision. Meanwhile, a new restitution of coe cient is deduced for calculating hysteresis damping factor, which suitable for both separation and non-separation states at the end of collision. In addition, the time-varying meshing sti ness(TVMS) is obtained based on the potential energy approach and the slice theory. Finally, a dynamic analysis of a helical gear system is carried out to better understand the contact force model proposed in this paper. The analysis results show that the contribution of supporting forces to the dynamic response of contact-impact event within gear pair is important. The supporting forces and dissipative energy are the main reasons for gear system to enter a steady contact state from repeated contact-impact state. This research proposes an improved contact force model which distinguishes meshing and collision states in gear system.
引文
[1]P Flores,M Machado,M T Silva,et al.On the continuous contact force models for soft materials in multibody dynamics.Multibody Syst.Dyn.,2011,25(3):357-375.
[2]M Inalpolat,M Handschuh,A Kahraman.Influence of indexing errors on dynamic response of spur gear pairs.Mechanical Systems and Signal Processing,2015,60-61:391-405.
[3]S Zhou,G Song,Z Ren,et al.Nonlinear dynamic analysis of coupled gearrotor-bearing system with the e ect of internal and external excitations.Chinese Journal of Mechanical Engineering,2016,29(2):281-292.
[4]A Fernandez-Del-Rincon,A Diez-Ibarbia,S Theodossiades.Gear transmission rattle:Assessment of meshing forces under hydrodynamic lubrication.Applied Acoustics,2017,144:85-95.
[5]Y A Khulief,A A Shabana.A continuous force model for the impact analysis of flexible multibody systems.Mechanism and Machine Theory,1987,22(3):213-224.
[6]A Kahraman,R Singh.Non-linear dynamics of a spur gear pair.Journal of Sound and Vibration,1990,142(1):49-75.
[7]Y Wen,J Yang,S Wang.Random dynamics of a nonlinear spur gear pair in probabilistic domain.Journal of Sound and Vibration,2014,333(20):5030-5041.
[8]F H Liu,S Theodossiades,L A Bergman,et al.Analytical characterization of damping in gear teeth dynamics under hydrodynamic conditions.Mechanism and Machine Theory,2015,94:141-147.
[9]Z Hu,J Tang,J Zhong,et al.E ects of tooth profile modification on dynamic responses of a high speed gear-rotor-bearing system.Mechanical Systems and Signal Processing,2016,76-77:294-318.
[10]J Zhan,M Fard,R Jazar.A CAD-FEM-QSA integration technique for determining the time-varying meshing sti ness of gear pairs.Measurement,2017,100:139-149.
[11]X H Liang,Z L Liu,J Pan,et al.Spur gear tooth pitting propagation assessment using model-based analysis.Chinese Journal of Mechanical Engineering,2017,30(6):1369-1382.
[12]K H Hunt,F R E Crossley.Coe cient of restitution interpreted as damping in vibroimpact.Journal of Applied Mechanics,1975,42(2):440-445.
[13]T W Lee,A C Wang.On the dynamics of intermittent-motion mechanisms.Part 1:Dynamic model and response.Journal of Mechanisms,Transmissions,and Automation in Design,1983,105(3):534-540.
[14]H M Lankarani,P E Nikravesh.A contact force model with hysteresis damping for impact analysis of multibody systems.Journal of Applied Mechanics,1990,112(3):369-376.
[15]Y Gonthier,J McPhee,C Lange,et al.A regularized contact model with asymmetric damping and dwell-time dependent friction.Multibody System Dynamics,2004,11(3):209-233.
[16]S Hu,X Guo.A dissipative contact force model for impact analysis in multibody dynamics.Multibody System Dynamics,2015,35(2):131-151.
[17]D Xing,Y H Shen,Y Z Wei,et al.A comparative study of the dissipative contact force models for collision under external spring forces.Journal of Computational and Nonlinear Dynamics,2018,13(10):101009.
[18]Y H Shen,D Xing,Y Z Wei,et al.A contact force model considering constant external forces for impact analysis in multibody dynamics.Multibody System Dynamics,2018,44(4):397-419.
[19]H Ma,J Zeng,R Feng,et al.Review on dynamics of cracked gear systems.Engineering Failure Analysis,2015,55:224-245.
[20]F L Litvin,A Fuentes,I Gonzalez-Perez,et al.Modified involute helical gears:computerized design,simulation of meshing and stress analysis.Computer Methods in Applied Mechanics and Engineering,2003,192(33-34)3619-3655.
[21]D C H Yang,J Y Lin.Hertzian damping,tooth friction and bending elasticity in gear impact dynamics.Journal of Mechanisms,Transmissions,and Automation in Design,1987,109(2):189-196.
[22]Q Wang,Y Zhang.A model for analyzing sti ness and stress in a helical gear pair with tooth profile errors.Journal of Vibration and Control,2017,23(2):272-289.
[23]Q Wang,B Zhao,Y Fu,et al.An improved time-varying meshing sti ness model for helical gear pairs considering axial mesh force component.Mechanical Systems and Signal Processing,2018,106:413-429.
[24]H Ma,R Song,X Pang,et al.Time-varying meshing sti ness calculation of cracked spur gears.Engineering Failure Analysis,2014,44:179-194.
[25]P Sainsot,P Velex,O Duverger.Contribution of gear body to tooth deflections-a new bidimensional analytical formula.Journal of Mechanical Design,2004,126(4):748-752.
[26]N Yousfi,B Zghal,A Akrout,et al.Damping models identification of a spur gear pair.Mechanism and Machine Theory,2018,122:371-388.
[27]R Kasuba,J W Evans.An Extended model for determining dynamic loads in spur gearing.Journal of Mechanical Design,1981,103:398-409.
[28]S Chen,J Tang,L Wu.Dynamics analysis of a crowned gear transmission system with impact damping:Based on experimental transmission error.Mechanism&Machine Theory,2014,74:354-369.
[29]J Alves,N Peixinho,M T Da Silva,et al.A comparative study of the viscoelastic constitutive models for frictionless contact interfaces in solids.Mechanism&Machine Theory,2015,85:172-188.
[30]W J Stronge.Rigid body collisions with friction.Proceedings of the Royal Society of London,Series A:Mathematical and Physical Sciences,1990,431(1881):169-181.
[31]R G Parker,X Wu.Vibration modes of planetary gears with unequally spaced planets and an elastic ring gear.Journal of Sound and Vibration,2010,329:2265-2275.
[32]P M T Marquesa,R C Martinsa,J H O Seabrab.Gear dynamics and power loss.Tribology International,2016,97:400-411.
[2]M Inalpolat,M Handschuh,A Kahraman.Influence of indexing errors on dynamic response of spur gear pairs.Mechanical Systems and Signal Processing,2015,60-61:391-405.
[3]S Zhou,G Song,Z Ren,et al.Nonlinear dynamic analysis of coupled gearrotor-bearing system with the e ect of internal and external excitations.Chinese Journal of Mechanical Engineering,2016,29(2):281-292.
[4]A Fernandez-Del-Rincon,A Diez-Ibarbia,S Theodossiades.Gear transmission rattle:Assessment of meshing forces under hydrodynamic lubrication.Applied Acoustics,2017,144:85-95.
[5]Y A Khulief,A A Shabana.A continuous force model for the impact analysis of flexible multibody systems.Mechanism and Machine Theory,1987,22(3):213-224.
[6]A Kahraman,R Singh.Non-linear dynamics of a spur gear pair.Journal of Sound and Vibration,1990,142(1):49-75.
[7]Y Wen,J Yang,S Wang.Random dynamics of a nonlinear spur gear pair in probabilistic domain.Journal of Sound and Vibration,2014,333(20):5030-5041.
[8]F H Liu,S Theodossiades,L A Bergman,et al.Analytical characterization of damping in gear teeth dynamics under hydrodynamic conditions.Mechanism and Machine Theory,2015,94:141-147.
[9]Z Hu,J Tang,J Zhong,et al.E ects of tooth profile modification on dynamic responses of a high speed gear-rotor-bearing system.Mechanical Systems and Signal Processing,2016,76-77:294-318.
[10]J Zhan,M Fard,R Jazar.A CAD-FEM-QSA integration technique for determining the time-varying meshing sti ness of gear pairs.Measurement,2017,100:139-149.
[11]X H Liang,Z L Liu,J Pan,et al.Spur gear tooth pitting propagation assessment using model-based analysis.Chinese Journal of Mechanical Engineering,2017,30(6):1369-1382.
[12]K H Hunt,F R E Crossley.Coe cient of restitution interpreted as damping in vibroimpact.Journal of Applied Mechanics,1975,42(2):440-445.
[13]T W Lee,A C Wang.On the dynamics of intermittent-motion mechanisms.Part 1:Dynamic model and response.Journal of Mechanisms,Transmissions,and Automation in Design,1983,105(3):534-540.
[14]H M Lankarani,P E Nikravesh.A contact force model with hysteresis damping for impact analysis of multibody systems.Journal of Applied Mechanics,1990,112(3):369-376.
[15]Y Gonthier,J McPhee,C Lange,et al.A regularized contact model with asymmetric damping and dwell-time dependent friction.Multibody System Dynamics,2004,11(3):209-233.
[16]S Hu,X Guo.A dissipative contact force model for impact analysis in multibody dynamics.Multibody System Dynamics,2015,35(2):131-151.
[17]D Xing,Y H Shen,Y Z Wei,et al.A comparative study of the dissipative contact force models for collision under external spring forces.Journal of Computational and Nonlinear Dynamics,2018,13(10):101009.
[18]Y H Shen,D Xing,Y Z Wei,et al.A contact force model considering constant external forces for impact analysis in multibody dynamics.Multibody System Dynamics,2018,44(4):397-419.
[19]H Ma,J Zeng,R Feng,et al.Review on dynamics of cracked gear systems.Engineering Failure Analysis,2015,55:224-245.
[20]F L Litvin,A Fuentes,I Gonzalez-Perez,et al.Modified involute helical gears:computerized design,simulation of meshing and stress analysis.Computer Methods in Applied Mechanics and Engineering,2003,192(33-34)3619-3655.
[21]D C H Yang,J Y Lin.Hertzian damping,tooth friction and bending elasticity in gear impact dynamics.Journal of Mechanisms,Transmissions,and Automation in Design,1987,109(2):189-196.
[22]Q Wang,Y Zhang.A model for analyzing sti ness and stress in a helical gear pair with tooth profile errors.Journal of Vibration and Control,2017,23(2):272-289.
[23]Q Wang,B Zhao,Y Fu,et al.An improved time-varying meshing sti ness model for helical gear pairs considering axial mesh force component.Mechanical Systems and Signal Processing,2018,106:413-429.
[24]H Ma,R Song,X Pang,et al.Time-varying meshing sti ness calculation of cracked spur gears.Engineering Failure Analysis,2014,44:179-194.
[25]P Sainsot,P Velex,O Duverger.Contribution of gear body to tooth deflections-a new bidimensional analytical formula.Journal of Mechanical Design,2004,126(4):748-752.
[26]N Yousfi,B Zghal,A Akrout,et al.Damping models identification of a spur gear pair.Mechanism and Machine Theory,2018,122:371-388.
[27]R Kasuba,J W Evans.An Extended model for determining dynamic loads in spur gearing.Journal of Mechanical Design,1981,103:398-409.
[28]S Chen,J Tang,L Wu.Dynamics analysis of a crowned gear transmission system with impact damping:Based on experimental transmission error.Mechanism&Machine Theory,2014,74:354-369.
[29]J Alves,N Peixinho,M T Da Silva,et al.A comparative study of the viscoelastic constitutive models for frictionless contact interfaces in solids.Mechanism&Machine Theory,2015,85:172-188.
[30]W J Stronge.Rigid body collisions with friction.Proceedings of the Royal Society of London,Series A:Mathematical and Physical Sciences,1990,431(1881):169-181.
[31]R G Parker,X Wu.Vibration modes of planetary gears with unequally spaced planets and an elastic ring gear.Journal of Sound and Vibration,2010,329:2265-2275.
[32]P M T Marquesa,R C Martinsa,J H O Seabrab.Gear dynamics and power loss.Tribology International,2016,97:400-411.