Nonlinear vibration of crowned gear pairs considering the effect of Hertzian contact stiffness
 77 Downloads
Abstract
This study aims to analyze the influence of lead crowning modification of teeth on the vibration behavior of a spur gear pair. Two dynamic rotational models including an uncrowned and crowned gear are examined. Hertzian mesh stiffness is computed using tooth contact analysis in quasistatic state along a complete mesh cycle of teeth mesh. The dynamic orbits of the system are observed using some useful attractors which expand our understanding about the influence of crown modification on the vibration behavior of the gear pair. Nonlinear impact damper consists of noninteger compliance exponents identify energy dissipation of the system beneath the surface layer. By augmenting tooth crown modification, the surface penetration increases and consequently normal pressure of the contact area becomes noticeable. Finally, the results show modification prevents gear pair to experience period doubling bifurcation as the numerical results proved. Using this new method in dynamic analysis of contact, broaden the new horizon in analyzing of the surface of bodies in contact.
Keywords
Nonlinear vibration Crowned gear Hertzian contact Finite element method1 Introduction
Gear transmission systems are known as discontinues dynamical systems that transmit power and motion between shafts. During the transmission, mating gears have their own dynamic characteristics with one or more common points in local contact zones. Impact and friction complete the power transmission of system [1] which might be accompanied by repeated impacts called Vibroimpact process. Excessive noise and large dynamic load are striking features of Vibroimpact transmission systems which are in conflict with the demand for acoustic comfort especially in automotive industries. Tooth contact analysis (TCA) of gears paves the way of study on the nonlinear vibration of those systems by means of analytical and computational techniques.
A number of researches have been carried out on the influence of crown modification in the quasistatic state of the mesh stiffness, transmission error and deflection of gears and so on such as Refs. [2, 3]. Similarly, some papers studied the influence of geometric modification and fault on the dynamics of gears. An experimental study is presented by Gelman et al. [4] in order to test the performance of the wavelet spectral kurtosis techniques, which is a way to diagnose gear tooth fault early. Chen et al. [5] used experimental data, static transmission error and backlash, to investigate impact behavior of crowned gear in terms of the dynamic load factor. Again, impact damping was the major concern of the study by Lijuan [6] which considered nonlinear damping coefficient with integer and noninteger compliance exponents.
The total displacement of gears in mesh is a summation of some deformations such as deflection due to bending, shear and surface indentation. Sánchez et al. [7] studied the mesh stiffness of spur gear pairs by considering the Hertzian effect and presented mesh stiffness equations approximately. To achieve the mesh stiffness, Sánchez considered some factors such as bending, shear, compressive and contact deflections. The presented equation determined the load sharing ratio and calculated the load at any point of the path of contact for spurs gears. Cheng et al. [8] determined the timevarying mesh stiffness with considering the effect of Hertzian deflection and different loads by means of finite element method. An analytical model is presented to calculate mesh stiffness by Ma et al. [9]. This study investigated the mesh stiffness for gear pairs with tip relief. It led to inhibition about overestimate the mesh stiffness during doubletooth contact and consideration of the tooth flexibility effects.
What makes this paper utterly different from the others is directly considering the effect of the Hertzian deflection on the dynamic gear behavior by entering into the dynamic equation. In order to use Hertizian contact damping and Hertzian contact restoring force accurately in dynamic equation of motion, it is necessary to separate surface penetration from other deflections of mesh in spite of making the complex model.
Any change in surface characteristics of mesh arises some changes in the contact pattern of bodies and consequent change in the vibration behavior of the system. Distinguishing the deflection of elements paves the way of study on contact fatigue [10] in dynamic state when the contacting stresses are repetitive at the surface points. Based on this issue, in the present study, the influence of Hertzian contact pattern on the dynamic transmission error of gear mesh has been located at the center of attention to show the effects of crown modification on the nonlinear behavior of the system, comparing crowned gear with uncrowned one. Thanks to the “HelicalPair” software [11, 12] developed in the Center Intermech MO.RE. (Aster, High Technology Network of the Emilia Romagna Region) which enabled us to generate deformable models with different amplitudes of crowning. It should be mentioned that “Helicalpair” has been widely used in the recent studies and the correctness of the results has been proved in Refs. [12, 13, 14, 15].
Moreover, equivalent stiffness of the present gear pair consists of two terms. The first term possess the bending flexibility of the teeth and also flexibility of the gear web and hub. The second term is the flexibility of the Hertzian contact deformation. Each of these two terms behaves as a spring with time dependent stiffness. In this paper the portion of total stiffness for Hertzian contact deformation analyzed separately and its effectiveness illustrated clearly. Finally, the nonlinear dynamics of the gear pair considering the equivalent stiffness is presented.
2 Physical model
The second part of Eq. 1 obtained from the equilibrium point between springs and dampers. \(\delta_{h}\) corresponds the net displacements of the spring associated with \(k_{h}\) which shows dynamic penetration of tooth surfaces in contact and \(x\) is the dynamic transmission error of the system (displacement of equivalent mass) where \(x\left( t \right) = r_{1} \theta_{1} \left( t \right)  r_{2} \theta_{2} \left( t \right)\). \(\theta_{1} \left( t \right)\) is the angular position of the driver wheel (pinion), \(\theta_{2} \left( t \right)\) is the angular position of the driven wheel (gear); \(r_{1}\) and \(r_{2}\) are based radii; \(b_{c}\) is half of the backlash due to operating center distance modification along the line of action, \(c\) is constant viscous damping and \(c_{h}\) denotes Hertzian damping; \(I_{1}\) and \(I_{2}\) are the rotary inertia of pinion and gear respectively. \(k\left( t \right)\) and \(k_{h} \left( t \right)\) are time varying mesh stiffness and Hertzian stiffness, respectively. Notice that \(k\left( t \right)\) does not cover Hertzian mesh stiffness and \(T_{e} = T_{1} r_{1} /I_{1} + T_{2} r_{2} /I_{2}\). \(T_{1}\) and \(T_{2}\) are applied torques on pinion and gear, respectively.
In addition, in forward motion when \(x \ge b_{c}\), the surface penetration grows by exponent \(n = 3/2\) for crowned teeth and \(n = 1\) for uncrowned teeth [17].
Numerical parameters of the gear pairs
Parameters  Pinion  Gear 

Number of teeth  23  34 
Module (mm)  3  3 
Pressure angle (°)  20  20 
Face width (mm)  20  20 
Module of elasticity (MPa)  206,000  206,000 
Poisson ratio  0.3  0.3 
Density (kg/m^{3})  7850  7850 
Contact ratio  1.559  
Backlash  0.1172  
Viscous damping coefficient  0.015 
In Eq. (4), \(k_{mh}\) is the average value of torsional Hertzian stiffness. Amplitudes \(k_{hj}\) and phases \(\varphi_{j}\) are obtained from the discrete Fourier transform for S = 9 samples, discrete rotational position over a mesh cycle. Notice that, in order to decrease nonlinearity of the Eq. 1, the time varying pattern of \(k\) has been neglected by considering the average value of corresponding stiffness, using Fourier transform. For unmodified tooth \(k = 4.17 \times 10^{5} \,{\text{N/mm}}\) and for crowned pair \(k = 2.79 \times 10^{5} \,{\text{N/mm}}\). The dynamic model employs a number of assumptions. First of all, there is no friction such as sliding friction in the dynamic model as the study of Refs. [19, 20]. The gears are considered perfectly rigid with flexibility of mesh and surface and no mounting errors and misalignment are included. The tooth profile for both models is perfectly involute. The surfaces are continuous and nonconforming [21].
3 Computing Hertzian mesh stiffness of contact
Obviously, the contact area \(A_{c} = \pi ab\) and compression \(\delta\) during this period have the highest amount as TCA proved. Then continues through c1 (the duration of double tooth contact) to reach to the lowest point of single tooth contact (LPSTC), \(c\) is the contact ratio, see Table 1. \(A_{c}\) seems to be lower than the previous segment because of division of the transmitted load between two pair of teeth in contact. Consequently, lower compression appears. Finally, the mating teeth experience the second single tooth contact duration, during the remained time, \(2  \left( {c  \varepsilon } \right)\). \(\varepsilon\) is calculable if the approach and recession length of contact line be accessible.
Comparison of Fig. 6a, b reveals that by lead crowning modification, a small change appears in contact pattern and remarkable change in stiffness which is the consequence of different approach toward penetration (using different exponents).
4 Dynamic analysis of modified and unmodified gears
Parameters of gear system
Model  Crown amplitude \(\beta \,\left( {\text{mm}} \right)\)  Compliance exponent n 

Uncrowned  0  1 
Crowned  0.08  1.5 
Rootmeansquares (the square root of the arithmetic mean of the squares of a set of responses) of both cases reveal that the unmodified gear pair with 149 μm endures much more deflection than the modified gear pair with 85.5 μm, akin to its static deflection of finite element modeling (FEM). Likewise, the surface penetration of the second model is deflected 5.5 times than the first one. Moreover, while the fluctuation of modified gear pair (match with an excitation fluctuation period) is rather substantial and reflects two contact losses, the unmodified gear pair fluctuates in lower amplitude but meets six contact loss along a period of its fluctuation.
There are closed loops for each contact loss or impact. Note that the amounts of dissipated energy are different as the initial velocity of contact for each meeting might be different.
5 Conclusion

This paper defines the factors which produce the spur gear mesh stiffness. The effect of Hertzian mesh stiffness and tooth bending mesh stiffness are presented.

The Hertzian dynamic force and possible teeth separation are presented for the spur gears with crowned and uncrowned teeth.

Surface penetration for the dynamic loads is illustrated. Be means of this result, the portion of dynamic transmission error due to Hertzian surface deformation is specified.
Notes
Acknowledgements
The authors would like to thank the Lab SIMECH/INTERMECH MO.RE. (HIMECH District, Emilia Romagna Region) particularly Prof. Francesco Pellicano and Dr. Marco Barbieri for providing “HelicalPair” software.
Compliance with ethical standards
Conflict of interest
The authors declare that they have no conflict of interest to this work.
References
 1.Luo AC, Guo Y (2012) Vibroimpact dynamics. Wiley, HobokenzbMATHGoogle Scholar
 2.Li S (2015) Effects of misalignment error, tooth modifications and transmitted torque on tooth engagements of a pair of spur gears. Mech Mach Theory 83:125–136CrossRefGoogle Scholar
 3.Gurumani R, Shanmugam S (2011) Modeling and contact analysis of crowned spur gear teeth. Eng Mech 18(1):65–78Google Scholar
 4.Gelman L, Chandra NH, Kurosz R, Pellicano F, Barbieri M, Zippo A (2016) Novel spectral kurtosis technology for adaptive vibration condition monitoring of multistage gearboxes. InsightNonDestr Test Cond Monit 58(8):409–416CrossRefGoogle Scholar
 5.Siyu C, Jinyuan T, Lijuan W (2014) Dynamic analysis of crowned gear transmission systen with impact damping: based on experimental transmission error. Mech Mach Theory 74:254–269Google Scholar
 6.Wu LJ, Tang JY, Chen SY (2014) Effect of nonlinear impact damping with noninteger compliance exponent on gear dynamic characteristics. J Cent South Univ 21:3713–3721CrossRefGoogle Scholar
 7.Sánchez MB, Pleguezuelos M, Pedrero JI (2017) Approximate equations for the meshing stiffness and the load sharing ratio of spur gears including hertzian effects. Mech Mach Theory 109:231–249CrossRefGoogle Scholar
 8.Ma H, Zeng J, Feng R, Pang X, Wen B (2016) An improved analytical method for mesh stiffness calculation of spur gears with tip relief. Mech Mach Theory 98:64–80CrossRefGoogle Scholar
 9.Cheng Q, Shao Y, Liu J, Yin L, Du M, Yang Y (2017) Calculation of timevarying mesh stiffness affected by load based on FEM. In: ASME 2017 international design engineering technical conferences and computers and information in engineering conference, pp V010T11A010–V010T11A010Google Scholar
 10.Glaese W, Shaffer S Contact fatigue, battelle laboratories. ASM Handbook, Fatigue and Fracture 19, 331–336Google Scholar
 11.Barbieri M, Zippo A, Pellicano F (2014) Adaptive gridsize finite element modeling of helical gear pairs. Mech Mach Theory 82:17–32CrossRefGoogle Scholar
 12.Motahar H, Samani F, Molaie M (2015) Nonlinear vibration of the bevel gear with teeth profile modification. Nonlinear Dyn 83(4):1875–1884. https://doi.org/10.1007/s110710152452z CrossRefGoogle Scholar
 13.Faggioni M, Samani FS, Bertacchi G, Pellicano F (2011) Dynamic optimization of spur gears. Mech Mach Theory 46(4):544–557CrossRefGoogle Scholar
 14.Bonori G (2006) Static and dynamic modeling of gear transmission error. Ph.D. Thesis, University of Modena and Reggio EmiliaGoogle Scholar
 15.Bonori G, Barbieri M, Pellicano F (2008) Optimum profile modifications of spur gears by means of genetic algorithms. J Sound Vib 313(3):603–616CrossRefGoogle Scholar
 16.Tellı S, Kopmaz O (2006) Free vibrations of a mass grounded by linear and nonlinear springs in series. J Sound Vib 289(4):689–710CrossRefGoogle Scholar
 17.Lankarani HM, Nikravesh PE (1994) Continuous contact force models for impact analysis in multibody systems. J Sound Vib 5(2):193–207Google Scholar
 18.Hunt KH, Crossley FRE (1975) Coefficient of restitution interpreted as damping in vibroimpact. Int J Appl Mech 42(2):440–445CrossRefGoogle Scholar
 19.Kahraman A, Singh R (1990) Nonlinear dynamics of a spur gear pair. J Sound Vib 142(1):49–75CrossRefGoogle Scholar
 20.Liu G, Robert GP (2009) Impact of tooth friction and its bending effect on gear dynamics. J Sound Vib 320(4):1039–1063CrossRefGoogle Scholar
 21.Tesfahunegn YA, Rosa F, Gorla C (2010) The effects of the shape of tooth profile modifications on the transmission error, bending, and contact stress of spur gears. Proc Inst Mech Eng Part C J Mech Eng Sci 224(8):1749–1758CrossRefGoogle Scholar
 22.Li S (2007) Effects of machining errors, assembly errors and tooth modifications on loading capacity, loadsharing ratio and transmission error of a pair of spur gears. Mech Mach Theory 42(6):698–726CrossRefGoogle Scholar
 23.Tavakoli M, Houser D (1986) Optimum profile modifications for the minimization of static transmission errors of spur gears. J Mech Transm Autom Des 108(1):86–94CrossRefGoogle Scholar
 24.Litvin FL, Fuentes A (2004) Gear geometry and applied theory. Cambridge University Press, CambridgeCrossRefGoogle Scholar
 25.GonzalezPerez I, Iserte JL, Fuentes A (2011) Implementation of Hertz theory and validation of a finite element model for stress analysis of gear drives with localized bearing contact. Mech Mach Theory 46(6):765–783CrossRefGoogle Scholar
 26.Machado M, Moreira P, Flores P, Lankarani HM (2012) Compliant contact force models in multibody dynamics: evolution of the Hertz contact theory. Mech Mach Theory 53:99–121CrossRefGoogle Scholar