BULETNUL NSTTUTULU POLTEHNC DN AŞ Publicat de Universitatea Tehnică Gh Asachi, aşi Tomul LV (LV), Fasc 1, 28 Secţia CONSTRUCŢ DE MAŞN DYNAMC ASPECTS OF FRCTON FORCE DSTRBUTON OF SPUR GEARS BY VRGL ATANASU* Abstract The paper evidences an alternative methodology to evaluate friction forces as a function of shared normal load and instantaneous friction coefficient of contacting teeth under dynamic conditions The shared dynamic load is analyzed in relation to the time-varying mesh stiffness The calculus methodology of the instantaneous friction coefficient is developed according to the experimental results Key words: spur gears, tooth friction forces, coefficient of friction 1 ntroduction The geared servomechanisms are used in many applications to transmit power and motion Accurate dynamic modelling of these devices becomes important in order to reduce the undesired vibrations The friction force is considered as a dynamic excitation source in the models of spur gear pairs [4], [ [1-14] Usually, dynamic models of spur gear pairs are based on the assumption that the load is equally shared among the tooth pairs in contact [1], [12] n these models, Coulumb friction is assumed with a constant coefficient of friction μ These assumptions not lead to a realistic dynamic model of gear pairs Both, the average and the instantaneous friction coefficient are difficult to calculate The existing equations for these coefficients [3], [5] cannot be used with good results in the case of dynamic analysis An alternativ calculus methodology for the instantaneous friction coefficient of spur gear pairs is proposed in the paper The frictional forces F f on the gear tooth surfaces are computed by considering instantaneous friction coefficient and the sharing normal load of the meshing cycle in relation to the variable mesh stiffness
38 Virgil Atanasiu Construcţii 2 Friction Force 21 nstantaneous Coefficient of Friction t is generally agreed that the coefficient of friction does not remain constant during a meshing cycle Figure 1 presents the distribution of normal and friction forces among the pairs of teeth in contact The experimental values of the instantaneous coefficient of friction at the meshing teeth were obtained by Radzimovsky [7], and Rebbechi et al [9] On this basis, the curve drawn in Fig 1 is considered in the analytical model for the evaluation of the instantaneous coefficient of friction n this figure, A,B,C denote the beginning of engagement, end of engagement and pitch point of the pinion, respectively This model satisfies the boundary conditions of the sliding friction The pitch point C is taken as origin and the curve corresponding to the recess path can be approximately represented by the following expression proposed by Rao [8], (1) y y μ = a1 + a2 l l 2 where y and l are shown in Fig 2 The constants a 1 and a 2 can be established by considering Eq (1) and the boundary conditions for the equation From these conditions results a1 = 3 μm, a2 = 1 5 μm This methodology is also fully valid for the aproach length, by considering the specific geometrical parameters of this meshing zone F f F f µ A F n B C D p b F n E A C y l dy E Fig1 Normal and friction forces Fig2 Variation of coefficient of analytical spur gear model friction The analytical relation for determining the average coefficient of friction μm in gear teeth are derived from roller and gear tests Thus, in [16], the formula for calculation the coefficient of friction be used especially for spur
de Maşini Bul nst Polit aşi, t LV (LV), f 1, 28 39 gear pair with the pitch point in the middle of the path of contact When la l g it is necessary to consider the specific geometry parameters of the gear pair which is analyzed, where l a = AC and l g = CD n this way, it is better to use the average coefficient of friction μ ma and μ mg which are calculated in the middle length of the approach path and of the recess length, respectively, by using Eq (1) The numerical results obtained by using the Eq (3) are in good agreement with the experimental results of the average coefficient of friction obtained in [16] The friction coefficient μ changes its sign with direction of relative sliding V s, ie (2) μ = μsgn(y) where sgn is the signum function defined as (3) sgn( y ) = 1,if V sgn( y ) = 1,if V s s > < The rolling friction coefficient can be calculated by the method of Crook [6] t is practically negligible compared to the sliding friction coefficient The only point of exception is the pitch point 23 Shared Friction Forces Usually, the distribution factor of the friction force corresponding to a tooth pair is defined as (4) c f n n μmf = F n n Eq(2), F represents the normal force corresponded to a single gear pair, and F n is the total normal force, where Fn = 5Fn for double-tooth contact and F n = Fn for single-tooth contact Under static condition, the distribution factor c fs is proposed as follows
4 Virgil Atanasiu Construcţii (5) fs = μc c where μ represents the instantaneous coefficient of friction and c is the shared factor of the normal load ks (6) c = k + k and k s represents the single mesh stiffness [2] Under dynamic condition, the distribution factor follows (7) c fd = μc d s s c fd is proposed as where μ represents the instantaneous coefficient of friction and dynamic shared factor of the normal load c d is the (8) c d Fd = F n To determine the variation of dynamic contact load as a function of contact position, it is necessary to express the equation of motion in the following form (9) m & x d + cx& d + Fdi () t = Fn where m is the equivalent mass m on the line of action, c represents the damping coefficient, ant t shows the meshing time The individual dynamic load is expressed as N i=1 (1) F ( t) k ( t) [ x e ( t) ] = di i d + i where x d is the dynamic displacement and N represents the number of simultaneous tooth pairs in mesh For a pair of contacting teeth i, the time-
de Maşini Bul nst Polit aşi, t LV (LV), f 1, 28 41 varying mesh stiffness k i ( t ) and the composite tooth profile error ei () t act as parameter excitations The total dynamic load is expressed as (11) Fd Fdi () t = N i= 1 where N represents the number of simultaneous tooth pairs in mesh in the actual position of the gear pairs 3 Case llustrations The specifications of the gear pairs which are selected in these illustrations are shown in Table 1, where z 1, z2 represent the number of teeth of the pinion and gear, respectively; x 1, x 2 are the addendum modification coefficients; ε α is the contact ratio The segments AB, AC, BD, DE are the geometrical parameters of the analyzed gear pairs These characteristics are for spur gear pairs having face-width of gears, b = 2[ mm], tooth module m = 25[ mm] and center distance, a = 9[ mm] Table 1 Specifications of the gear pairs z 1 z 2 x 1 x 2 ε AB AC BD DE α [mm] [mm] [mm] [mm] GA1 18 54 165 479 652 259 392 GA2 18 54 +5-5 153 392 343 346 528 15 μ 1 GA 1 5-5 -1-15 A C E Fig3 Variation of the coefficient of friction 15 μ 1 GA 2 5-5 -1-15 A C E Fig4 Variation of the coefficient of friction
42 Virgil Atanasiu Construcţii c f 12 8 4-4 -8-12 c f s c f d GA 1 c f 12 8 4-4 -8 ω = 1 s -1-12 A B C D E ω = 3 s -1 c f s c f d GA 1 A B C D E Fig5 Variation of the friction Fig6 Variation of the friction factors c fs i and c fd factors c fs i and c fd c f d 8 4-4 GA 2 8 4 c f d -4-8 ω = 1-1 s -8 ω = 3 s -1-12 -12 A C E A C Fig7 Variation of dynamic factor for pinion speed ω = 1s 1 c fd GA 2 Fig8 Variation of dynamic factor c fd for pinion speed ω = 3s 1 E A computer program is developed to compute the numerical values of the instantaneous coefficient of friction along the line action in terms of the addendum modification coefficients and pinion speed n investigating the effects of gear geometry, the influence of addendum modification coefficients is analyzed The nominal specific load is F n / b = 2[ N / mm] The variation of the tooth friction coefficient for a mesh cycle is shown in Figures 3 and 4 Figures 5 and 6 present the variation of the factors c fs and c fd corresponding to friction forces along the engagement line when static and dynamic mesh stiffness are considered Figures 7 and 8 presents the variation of the factor c fd for the mesh cycle by considering the pinion speed ω = 1s 1 and ω = 3s 1, respectively The analysis of these results permits to underline especially the effcts of the geometrical parameters of gears and operating regim on the amount and variation of the gear friction forces
de Maşini Bul nst Polit aşi, t LV (LV), f 1, 28 43 4 Conclusion 1 An alternative methodology to evaluate friction forces as a function of shared normal load and instantaneous friction coefficient of contacting teeth under dynamic conditions The time variable mesh stiffness is included in the dynamic model of gear pairs 2 The comparisons of the variation of the friction forces for a mesh cycle are presented for different calculus methodologies 3 The effects of addendum modification coefficients and pinion speed on the shared friction force are analyzed in order to obtain information for the best design solution Acknowledgements This paper is based on the financial support of the National University Research Council of Romania, CNCSS - PNCD, grant no76/27, Cod D_296 Submitted: Accepted: *Technical University Gh Asachi asi, Department of Mechanisms and Robotics asi,romania, e-mail: vatanasi@mailtuiasiro R E F E R E N C E S 1 A t a n a s i u, V An Analytical nvestigation of the Time-Varying Mesh Stiffness of Helical Gears Buletinul Pasi, Gh Asachi Technical University, Tomul XLV, 1-2, sv, pp7-17, (1998) 2 A t a n a s i u, V, L e o h c h i, D Dynamic Analysis of Spur Gears with Tooth Profile Modifications Proceedings of the Ninth FToMM nternational Symposium on Theory of Machines and Mechanisms, Bucharest, 313-318, (25) 3 B e n e d i c t G H, K e l l e y B W, nstantaneous Coefficients of Gear Tooth Friction ASLE Transaction, 4, 57 7 (1961) 4 G u n d a, R, S i n g h, R Dynamic Analysis of Sliding Friction in a Gear Pair Proceedings of DETC 3, ASME Design Engineering Technical Conferences and Computers and nformation in Engineering Conference, Chicagi, llinois USA, September 2-6,1-8 (23) 5 M a r t i n K F, A Review of Friction Predictions in Gear Tooth Wear, 49, 2, 21 238 (1978) 6 M a r t i n K F, The Efficiency of nvolute Spur Gears Trans of the ASME, Journal of Techanical Design, 13(1981) 7 R a d z i m o v s k y E, M i r a r e f i A, Dynamic Behaviour of Gear Systems and Variation of Coefficient of Friction and Efficiency During the Engagement Cycle ASME Journal of Engineering ndustry, 97, 4, 1247-1281 (1975) 8 R a o A C, Gear Friction Coefficient and Forces Wear, 53, 87-93 (1979) 9 R e b b e c h i, B, O s w a l d, F B, T o w n s e n d, D P Measurement of Gear Tooth Dynamic Friction DE-Vol88, PowerTransmission and Gearing Conference, ASME, 355-363 (1996)
44 Virgil Atanasiu Construcţii 1 V a i s h y a, M, H o u s e r, DR Modeling and Analysis of Sliding Friction in Gear Dynamics ASME 2 Design Engineering Technical Conferences and Computers and nformation in Engineering Conference, Sept1-13,2, Baltimore, Maryland, 61-61 (2) 11 V a i s h y a, M, S i n g h, R Sliding Friction-nduced Non-Linearity and Parametric Effects in Gear Dynamics Journal of Sound and Vibration, 248, 671-694, (21) 12 V a i s h y a, M, S i n g h, R Strategies for Modeling Friction in Gear Dynamics Journal of Mechanical Design, Transaction of the ASME, Vol125, 383-393 (23) 13 V e l e x, P, S a i n s o t, P An analytical study of tooth friction excitations in erroless spur and helical gears Mechanism and Machine Theory, 37, 641-658 (22) 14 W a i b o e r, R, R o n a l d, A Velocity Dependence of Joint Friction in Robotic Manipulators with Gear Transmissions Multibody Dynamics 25, ECCOMAS Thematic Conference, Madrid, Spain, 21-24 June, 1-19, (25) 15 W i n k, H C, S e r p a, A L Performance assessment of solutions methods for load distribution problem of gear teeth Mechanism and Machine Theory, 43, 8-94, (28) 16 * * * Calculation of Load Capacity of Spur and Helical Gears Calculation of Scuffing Load Capacity SO 6336/4, DN 399/4 ANALZA FORŢELOR DNAMCE DE FRECARE LA ANGRENAJE CLNDRCE (Rezumat) n lucrare se prezintă o metodologie de evaluare a forţelor de frecare corespunzător fiecărei perechi de dinţi în angrenare, prin considerarea coeficientului de frecare instanataneu şi a distribuţiei forţei dinamice Modelarea analitică a coeficientului de frecare corespunde rezultatelor experimentale, iar forţele dinamice includ rigiditatea variabilă a danturii în timpul funcţionării