Nonlinear Dynamic Modeling of Double Helical Gear System

Volume 8 Number 5, October 2014 ISSN 1995-6665 JJMIE Pages 289- 296 Jordan Journal of Mechanical and Industrial Engineering

Chang Qinglin, Hou Li *, Sun Zhijun, Wang Wei, You Yunxia

School of Manufacturing Science and Engineering, Sichuan University, No.24 South Section 1, Yihuan Road, 610065 Chengdu , China.

Received 3 April 2014 Accepted 3 Sep 2014

Abstract

Double helical gear transmission is a very important transmission component. A lot of dynamic models of spur and helical gears for studying dynamic characteristics have been put forward, but the studies on double helical gears are scarce. To study the nonlinear dynamic characteristics of the double helical gear system, the nonlinear dynamic model of double helical gear system was established. Then the dimensionless dynamic model was summarized. Through numerical computation, the nonlinear dynamic characteristic was revealed in some aspect. The vibration amplitude of gear is smaller than the vibration amplitude of pinion at the same direction and the vibration amplitude of the same component at direction is larger than the amplitude of gear at direction. Through the numerical method, we find that there is a plenty of nonlinear dynamic characteristics in the double helical gear system, and thus more research is needed.

Keywords: : Time-Varying Meshing Stiffness; Meshing Damping; Double Helical Gear; Nonlinear Characteristics.

helical gear system, and researched backlash nonlinear 1. Introduction characteristics under different modal parameters. IMAMURA.Y and SATO.S [11] studied the distribution Gear transmission is the most common transmission of dynamic stress in gear systems. Walha L. [12] and his system and is widely used in the production of national partners studied the nonlinear dynamic response of helical industries. It is also a particularly important key gear system with mass eccentricity. Wei [13-14] studied component. It has three basic forms: spur, helical and the friction nonlinear characteristics in-depth in the spur double helical. Because of alternate meshing of single and gear system. Ma Hui [15] analyzed the influence of double spur gear teeth in the transmission, the meshing eccentric load in helical gear system in detail and studied stiffness changed with large amplitude when the the dynamic characteristic in eccentric helical gear system coincidence degree is not an integer, motivating large by the method of modal analysis and coupling analysis. vibration and noise [1-3]. There is not alternate meshing of Wang Qing [16] established the coupling dynamic model single and double gear teeth in the helical gear system and and system differential equations of cylindrical helical the stiffness mutation, so it’s meshing smoother. But there gear transmission considering time-varying mesh stiffness, are some shortcomings in the helical gear system, for backlash and transmission error. N. Leiba [17] studied the example: the dynamic axial load, time-varying mesh wire vibro-impact phenomenon by experimental and numerical length, time-varying frictional force and time-varying method. Song Xiaoguang [18] studied the nonlinear frictional torque [4-9]. The dynamic axial load of double characteristics of the flexible shaft helical gear system helical gear was eliminated though the time-varying considering backlash, bearing radial clearance and gear frictional force and time-varying frictional torque exist. unbalanced force. Tang Jinyuan and Chen Siyu [19-20] Because of its high carrying capacity, smooth proposed an improved nonlinear dynamics model of the transmission, and no axial load, the double helical gear has spur gear. On the basis of the model, the gap nonlinear been widely used. As we all known, many nonlinear characteristics were analyzed. Wang Feng and Wang factors such as time-varying mesh stiffness, time-varying Cheng [21-23] established a dynamic model of mesh damping, meshing impact, backlash, time-varying herringbone gears, then the herringbone gears’ dynamic frictional force and friction torque in double helical gear characteristics were analyzed, but the calculation and system will generate complex nonlinear vibration. analysis of time-varying parameters and nonlinear factors However, studies on system dynamics of double helical were not concerned. Li Wenliang [24] analyzed the gear is seldom, the research on double helical gear system friction force in helical gear and the bending effect. Wei dynamics is very necessary. Jing [25] discussed the backlash nonlinear characteristic in Wesley Blankenship G. and Singh R. [10] studied the helical gear system. Guo Jiashun [26] analyzed dynamic dynamic meshing force, stiffness and transfer matrix in profile modification with herringbone gear tooth based on

* Corresponding author. e-mail: [email protected]. 290 © 2014 Jordan Journal of Mechanical and Industrial Engineering. All rights reserved - Volume 8, Number 5 (ISSN 1995-6665) analytical solutions. Bu Zhonghong [27-28] studied the k inherent characteristic and free vibration characteristics of grz herringbone planetary gear set. Feng Zhigang [29] studied c the meshing static-dynamic vibration characteristics of grz kgry herringbone gear. Yang Xiaofang [30] analyzed the kglrz cgry Ogr distribution of power flow of the three branches of Ogl c herringbone gear drive system by simulation. Ma Shangjun glrz k [31] analyzed the steady-state response of plate glz convergence herringbone gears. Wu Xinyue [32, 33] Tpr cglz c k established a simplified kinetic model of herringbone gear kgly gly pry k plrz cpry and got preliminary solution. Xie Zuiwei [34] analyzed the Opr Tpl stress on herringbone gear tooth surface by using the cplrz contact stress finite element method. Ahmad Al-Shhyab Opl and Ahmet Kahraman [35] developed a non-linear time- k varying dynamic model to study torsional dynamic ply cply behavior for a multi-mesh gear system. Referring to the modeling process of helical gear system and combining its own characteristics, bending- Figure 2. 12 degrees bend-twist-axis coupling vibration model of torsion-axis coupled nonlinear vibration model with 16 double helical gear pair degrees of freedom was established. Nonlinear factors As shown in Figure 2, to establish the nonlinear considered in the process of modeling include time- dynamic model of double-helical gear system of 12 varying mesh stiffness, meshing impact and backlash. degrees, the tooth surface friction and the friction torque Dimensionless operation was adopted to process the are not considered and the generalized displacement system dynamics equations. To solve the dimensionless matrix of 12 degrees can be expressed as: differential equations, the 4-5 Runge-Kutta method was T δ= (,,;yz θ yz ,,;,,; θθθ yz yz ,,) (1) used. Then the nonlinear characteristic of time history pl pl pl pr pr pr gl gl gl gr gr gr diagram, phase diagrams, Lyapunov exponent, Lyapunov Where, yij, z ij ,θ ij ( i= pg , ; j = lr ,)is the translational dimension, Poincare map and global bifurcation graphs in and rotational vibration displacement of center of pinion the double helical gear system was studied. and gear O( i= pg , ; j = lr ,) at yz, direction, ij respectively. 2. Nonlinear Dynamic Model of the Double Helical PP, Gear System Vibration displacement of meshing point lrand GG, lrcan be expressed as: 2.1. The Nonlinear Dynamic Model (2a) yypi =pi +θ pi r The double helical gear is wildly used in industry, such =−β =−+ θβ (2b) as truck, train and other overload machines. A pair of zpi zypi pi tan zpi ( y pi pi r ) tan double helical gear is shown in the Figure.1. From the yy= −θ R (2c) Figure1, we find that the double helical gear could regard gi gi gi (2d) as two helical gear with opposite helical angle. Then the zgi =− zygi gi tanβ =−− zgi ( y gi θβ gi R ) tan 12 degrees bend-twist-axis coupling vibration model of Where, i= lr, . double helical gear pair is established just as Figure2 Taking the double helical gear, seen as two helical shown. gears with opposite helical gears direction, and assuming the two helical gears have the same parameters except the helix angle, normal meshing stiffness of unilateral can be expressed as: ∞

Kmn () t= Kmn ++∑ { an11 cos[ξωh ()] tt bn sin[ξωh ()]} tt ξ =1 (3a) =KK + cos(ω t ) mn mna h

Where: Km is the normal average mesh stiffness of double

helical gear pair; ωh ()t is the meshing frequency of

double helical gear pair; an and bn are the Fourier coefficients in Eq. (3a). Mesh stiffness at the tangential and axial directions can be expressed as: K()= tK () t= K ()cos t β mry mly mn (3b)

KtKtKtmrz ()=mlz () = mn ()sinβ Figure 1. A pair of double helical gear (3c) Where, i= lr, . © 2014 Jordan Journal of Mechanical and Industrial Engineering. All rights reserved - Volume 8, Number 5 (ISSN 1995-6665) 291

The normal damping is expressed as: Through the Newton's second law and considering the dynamic meshing force and backlash in double helical gear Cmn () t= 2ζ v K mn () tm e (4a) pair shown in Figures 1 and 2, the kinetic equation of gear m Where: e is the equivalent mass of double helical gear system shown in Figure 2 can be established as:  •• • JJpg ζ m y+ c y + k y =−+ F() t mg pair, m = ； v is the relative damping of  plpl plypl ply pl yl pl e 22+ JRg b Jr pb  •• •• mzplpl + c plrz ( zpl −+ z pr ) kplrz ( z pl −=− z pr ) Ft zl () (7a) double helical gear pair, and ζ v =0.070 。  ••⋅ Mesh damping at the tangential and axial can be Jθ pl =−+ F() tr T  pl yl pl expressed as:  CtCtCt()= () = ()cosβ •• • mry mly mn (4b)  mpr ypr + cpry ypl + k pry y pl =−+ F yr() t mg pr CtCtCt()= () = ()sinβ  mrz mlz mn (4c)  •• •• − −− −=− Where, i= lr, . mzprpr c plrz ( zpl z pr ) kplrz ( z pl z pr ) Ft zr () (7b)  The tangential and axial dynamic meshing force at left •• Jθ pr =−+ F() tr T and right ends of double helical gear pair can be expressed  pr yr pr as:  •• • m y++=+ c y k y F() t mg Fyl () t= Kmly () tfy (pl −− y gl ely )  glgl glygl gly gl yl gl •• • •• ••• (5a)  + −− mzglgl+++ c glz z gl kz glz gl c glrz () zgl − z gr (7c) Cmly ( ty )( pl y gl ely )   +kglrz ( z gl −= z gr ) Ft zl () Fzl () t= Kmlz () tfz (pl −− z gl elz )  •• ••• (5b)  θ = − Jgl gl Fyl () tR Tgl +Cmlz ( tz )(pl −− z gl elz )  •• • F() t= K () tfy ( −− y e) ++=+ yr mry pr gr ry mgr ygr c gry ygr k gry y gr F yr() t mg gr ••• (5c)  •• • •• +Cmry ( ty )( pr −− y gr ery ) mzgr++− c z gr kz c() zgl − z gr  gr grz grz gr glrz (7d) F() t= K () tfz (pr −− z gr e ) zr mrz ry  −kglrz ( z gl −= z gr ) Ft zr () (5d) •••  •• +Cmrz ( tz )( pr −− z gr ery )  θ = − Jgr gr Fyr () tR Tgr Where: eij ( t ) ( i= lr , ; j = yz , ) are the tangential and Where, mij ( i= pg , ; j = lr ,)are the mass at left and right axial meshing error at left and right ends of double helical ends of pinion and gear, respectively; gear pair, respectively; they can be then expressed in sine function forms Jij ( i= pg , ; j = lr ,) are the moment of inertia at left and ~ rR, as: eij () t=+ eij eaij sin[ωφ h () tt +== eij ] ( i lr , ; j yz , ). right ends of pinion and gear respectively; are reference radius of pinion and gear, respectively. From Eq. (5a) to Eq. (5d), f( ji )( i= lr , ; j = yz , ) is the backlash nonlinear function, and also the tangential 2.2. Dimensionless of Kinetic Equations and axial relative displacement at left and right ends of double helical gear pair, respectively. Assume that the In order to obtain these kinetic equations backlash at left and right ends of double helical gear pair is dimensionless, define the system dimensionless time and , then the tangential equal, and the normal backlash is 2bn dimensionless excitation frequency, respectively as: and axial backlash is 2bb= 2 cos b and tω=t ⋅ tn n (8a) = b 2bban 2 sin , respectively, where β is the helical ωωω= / hn (8b) angle of the double helical gear system. Where: ω is the natural frequency of the system, n

ybit−> () y i b t and ωn= Km mn/ e ; me is the equivalent mass of gear  fy(i )=  0 ( yit≤ b ) (6a) pair; Kmn is the normal average mesh stiffness of gear  pair. ybyit+ () i <− b t Take b as the nominal dimension to take the Eq. (7) −> n zbiai () z b a dimensionless. The dimensionless displacement of double  ： fz(i )=  0 ( zia≤ b ) (6b) helical gear system can be expressed as  zbiai+ () z <− b a Where, i= lr, .

p123= ypl/, bp n = zpl /, bp n = rθ pl / b n ηijk (i= pg , ; j = lrk ,; = yz ,)is the dimensionless support stiffness at bearing at left and right ends of pinion p4= ypr/, bp n 56 = zpr /, bp n = rθ pr / b n and gear at yz, direction, respectively; p7= ygl/, bp n 89 = zgl /, bp n = Rθgl / b n (9) ξ(τ ) (i= lr , ; j = yz , ) is the dimensionless meshing p= y/, bp = z /, bp = Rθ / b mij 10 gr n 11 gr n 12 gr n damping at left and right ends of gear at yz, direction, p1,1 = ybpln/,1,2 = zb ln / respectively; p= y/, bp = z / b η(τ ) (i= lr , ; j = yz , ) is the dimensionless meshing 2,1 rn2,2 rn mij Then the dimensionless kinetic equation of gear system stiffness at left and right ends of gear at yz, direction, can be expressed as: respectively; η ξilrj and ilrj (i= pg ,; j = yz ,) are dimensionless •• • +ξ ++ η ητ internal damping and internal stiffness at left and right  p112ply p ply p1 mlyp () fp (1,1 ) ends of pinion and gear, respectively;  • ~ ~  += 2ξmlyp ()τpF1,1 1 Fii (= 1,12) is the dimensionless external excitation. •• ••  The expressions of dimensionless parameter in Eq.  p+2(ξη pp −+ ) ( pp − ) (10a) 2plrz 25 plrz 25 (10a) to Eq. (10d) are:  • ~  +η ()τfp ( ) += 2ξ ()τ p F cply k ply Ktmly () mlzp 1,2 mlzp 1,2 2 ξ ， η ， ηt  ply = ply = 2 mlyp ( )= 2 •• • 2m ω m ω m ω  ~ pl n pl n pl n +ητ +=ξτ  p3 2[mlyp () fp (1,1 ) 2mlyp () p1,1 ] F3  Ctmly () cplrz k plrz ξt( )= ， ξ = ， η = •• • mlyp 2m ω plrz 2m ω plrz m ω 2 +ξ ++ η ητ pl n pl n pl n  p442pry p pry p4 mryp () fp (2,1 ) Kt() Ct() c  ⋅ ηt( )= mlz ， ξt( )= mlz ， ξ = pry  • ~ mlzp 2 mlzp pry +=ξτ mplω n 2mplω n 2mprω n  2mryp ()pF2,1 4 •• •• k Kt() Ct()  η = pry ， ηt( )= mry ， ξt( )= mry  p5−2(ξηplrz pp 25 −− )plrz ( pp25 − ) (10b) pry 2 mryp 2 mryp mprω n mprω n 2mprω n  • ~  +η ()τfp ( ) += 2ξ ()τp F Kt() Ct() c mrzp 2,2 mrzp 2,2 5 ηt mrz ξt mrz gly  mrzp ( )= 2 ， mrzp ( )= ， ξgly = •• • ~ m ω 2m ω 2m ω  +ητ +=ξτ pr n pr n gl n  p6 2[mryp () fp (2,1 ) 2mryp ()p2,1 ] F6 k c k  η gly ξ glz η glz gly = 2 glz = ， glz = 2 •• • ω ω ω  mgl n 2mgl n mgl n +ξ +− η ητ  p772gly p gly p7 mlyg () fp (1,1 ) Ctmlz () Ktmry () Ctmry ()  • ~ ξt ηt ξt mlzg ( )= ， mryg ( )= 2 ， mryg ( )=  −= 2ξmlyg ()τpF1,1 7 2mglω n mgrω n 2mgrω n  •• • •• Kt() Ct() ~ 2T +ξη + + ξ −+ η mrz mrz pl  p8 2glz p8 glz p88 2(glrz pp8 11 )glrz ( p(10c) ηtmrzg ( )= ， ξtmrzg ( )= ， F = m ω 2 2m ω 3 bm ω 2  • ~ gr n gr n n pl n  −−ητ −ξτ = p11 )mlzg () fp (1,2 ) 2mlzg ()p1,2 F8 ~~~ ~ g ~~~ ~  FFFF=== = ， FFFF= = = =0 ， •• • ~ 1 4 7 10 ω 2 2 5 8 11 −ητ +=ξτ bnn  p9 2[mlyg () fp (1,1 ) 2mlyg () p1,1 ] F9  ~ 2T ~ 2T ~ 2T F = pr ， F =- gl ， F =- gr 。 •• • 6 2 9 2 12 2 bmn prω n bmn glω n bmn grω n  p10 +2ξgry p10 +− η gry p10 η mryg ()τ fp (2,1 )  • ~ The dimensionless backlash nonlinear function can be

 −= 2ξmryg ()τpF2,1 10 expressed as:   •• • •• bbkk  ppij,,−> (ij )  p11 +2ξηgrz p11 + grz p11 −2( ξglrz pp8 −− 11 ) ηglrz ( p8 (10d) bb   nn • ~  b  = ≤ k −−p11 )ηmrzg ()τ fp (2,2 ) − 2ξmrzg ()τp2,2 = F11 fp(ij,, ) 0 (pij ) (11)   bn ⋅ •• •  ~ bbkk   ppij,,+ ()ij <− p12 −2[ηmryg ()τ fp (2,1 ) += 2ξmryg ()τp2,1 ] F12   bbnn Where: i=1, 2; j = 1, 2, k = ta , ξ (i= pg , ; j = lrk ,; = yz ,) Where, . ijk is the dimensionless The dimensionless transmission error can be support damping at bearing at left and right ends of pinion expressed as: and gear at yz, direction, respectively; ~ eij eaij e(τ )=+ sin(ωτ +==φ ) (i lr , ; j yz , ) (12) ij bb eij nn © 2014 Jordan Journal of Mechanical and Industrial Engineering. All rights reserved - Volume 8, Number 5 (ISSN 1995-6665) 293

3. Numerical Results and Discussion 9 ， kply= k pry = k gly = k gry = 6.15 × 10Nm / . ξξ= = × 6 ， glz= grz 0.008 , kglz= k grz 3.03 10Nm / Take the basic parameters as: ξξξξply= pry = gly = gry = 0.008 ， z p = 30 ， zg = 90 ， m = 3 ， α =20 ° ， β =10 ° ， 10 kplrz= k glrz =6.02× 10 Nm / , eij (τ )=0 . g= 9.8 N / kg ， b= 0.1 mm ， B= 60 mm ， n Then Figure 3 to Figure 5 can be obtained by 4-5

Tp =300 Nm⋅ ， Tg =900 Nm ⋅ , fr= 5000 / min , Runge-Kutta method, in which the nonlinear characteristics of double helical gear system can be = = = × 8 ， mp 5 kg , mg 45 kg , Kmna 2 10 Nm / revealed. 8 Kmn =6 × 10 Nm / ,ξξplrz= glrz =0.1,

(a)Time history (b) Time history partial enlarged view

(c)Speed-displacement phase diagram (d) Poincare section

Figure 3. The numerical result of pinion at y direction

(a) Time history (b) Time history partial enlarged view

In Figures 3 to 5, some nonlinear characteristics are amplitude of pinion at the same direction and the vibration revealed. From Figure (3a), Figure (4a) and Figure (5a), amplitude of the same component at z direction is larger we get that the vibration is in the double helical gear than the amplitude of gear at y direction. system. From Figure (3a), Figure (4a), Figure (3b) and From Figures 3 to 5, we find that there are abundant Figure (4b), we find that the amplitude of gear nonlinear characteristics existing in the double helical gear y at direction is smaller than the amplitude of pinion at system. According to their nonlinear characteristics, we direction. From Figure (4a), Figure (5a), Figure (4b) and should take the dynamic performance into consideration Figure (5b), we find that the amplitude of gear when designing the double helical gear system. At the at z direction is larger than the amplitude of gear same time, we could use the vibration signal to monitor the at y direction. To a certain extent, we could obtain that the operation of double helical gear system. vibration amplitude of gear is smaller than the vibration © 2014 Jordan Journal of Mechanical and Industrial Engineering. All rights reserved - Volume 8, Number 5 (ISSN 1995-6665) 295

(a)Time history (b) Time history partial enlarged view

Figure 5. The numerical result of gear at z direction

4. Conclusions Acknowledgments

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