Dnamic Contact Loads of Spur Gear Pairs with ddendum Modifications V. tanasiu, I. Doroftei Theor of Mechanisms and Robotics Department, Gh. sachi Technical Universit of Iasi B-dul D. Mangeron, 6-63, Iasi - 75, Romania mail: vatanasi@mail.tuiasi.ro, idorofte@mail.tuiasi.ro bstract The paper presents a dnamic tooth load analsis of spur gears with addendum modifications. The analtical model is developed to simulate the load sharing characteristics through a mesh ccle. The model takes into account the main internal factors of dnamic load as time-varing mesh stiffness and composite tooth profile errors. The specific phenomenon of contact tooth pairs alternation during mesh ccle is integrated in this dnamic load modeling. comparative stud is included, which shows the effects of the factors with an important role in the wa of the dnamic load variation. Kewords: spur gears, dnamic loads, mesh stiffness, profile error, addendum modification.. Introduction The dnamic characteristics of spur gear pairs are significant for the design and motion control of these mechanisms [], []. The position accurac in a motion control sstem is affected b vibration due to nonlinear effects such mesh stiffness or tooth profile errors [], [3], [4]. complete analtical methodolog that covers all influence factors with high accurac is not current available. The time-varing mesh stiffness represents the main cause of undesired vibrations in the case of gear transmissions with high manufacturing precision. Because the tooth pair stiffness varies during the mesh ccle, different evaluation models for gear mesh stiffness have been developed [3-6]. The fluctuation of the dnamic loads of meshing gears represents a relevant parameter in dnamic analsis. In this work, the dnamic model accounts the non-linear time varing mesh stiffness, variable tooth profile errors, and tooth profile modifications in order to obtain reliable data for the prediction of gear dnamic loads. n investigation of the two group of the evaluation methods of the mesh stiffness used in the dnamic analsis of spur gear pairs is included in the paper. The effects of the operation frequenc and load condition od spur gears are considered in the numerical analsis of the dnamic shared loads.. Dnamic Model During the engagement ccle, the contact load does not remain constant. The load variation is mainl caused b the following factors: (i) the alternating engagement of single and double pairs of teeth; (ii) the variation of the mesh stiffness along the line of action; (iii) the deviation of the tooth profile from the theoretical involute profile. In order to build an accurate analtical model of the dnamic tooth load sharing, the parameters used in the model need to be estimated correctl. The mechanical model for a gear pair in mesh is shown in Figure, where the teeth are considered as springs and the gear blanks as inertia masses [4], [6]. In developing this model, the dnamic process is studied in the rotating plane of the gears and the differential equations of motion are developed b using the theoretical line of action. In Figure, for a pair of contacting teeth i, the time-varing mesh stiffness k i (t) and the composite tooth profile error e i (t) act as parameter excitations.
J rb F n F di(t) T T k i(t) e (t) i Figure : Dnamic model of meshing gears The differential equations of motion can be expressed as F n r b J Jθ& + Fd rb = T, () J θ & + Fd rb = T, () where: θ, θ are the rotation angle of the pinion and the driven gear, respectivel; J and J are the mass moments of inertia of the gears; T and T denote the external torques applied on the gear sstem; and r b, r b are the base circle radii of the gears. The dnamic load is expressed as where () = N F F t d di i =, (3) F = θ θ + θ& θ& di ki() t rb rb c( rb rb ) (4) with k i representing the mesh stiffness of the gear pair and c the damping coefficient. B introducing the composite coordinate x = r θ r θ d b b (4) qs. () and () ield an equation of motion in the following form N + & d d + di () = n i = mx&& cx F t F (5) where Fn is the static load and N represents the number of simultaneous tooth pairs in mesh, and () = () + () Fdi t ki t xd ei t (6) with x d being the dnamic displacement and e i (t) the equivalent error of teeth profile. The damping coefficient is calculated b c = ξ mekm, (7) where ξ represents the damping ratio factor and k m is the average mesh stiffness of the gear pair. The meshing resonance frequenc of the gear pair is determined as follows: f n km =, (8) π m where k m is the average mesh stiffness of the gear pair. 3. Parameters for Dnamic Model 3. Mesh Stiffness The gear tooth is modeled to be a nonuniform cantilever beam supported b a flexible fillet region and foundation [7] as shown in Figure. s f s p F F n α' F z p Fig. : Gear tooth deflection model The individual tooth mesh stiffness is defined in the normal direction to the contact surface as k F j = n f j z (9) where F n is the normal tooth load per unit length. The approach methodologies of the tooth mesh stiffness are based on analtical or with finite element methods [], [3], [5]. Two groups of the evaluation methods of the mesh stiffness of spur gear pairs is presented in the paper. The finite element contact modelling is computationall ver difficult to model for
dnamic analsis. Kuang and Yang [5] developed an alternative method and introduced a semi-empirical equation for the single tooth stiffness with and without modification. In this approach, the analtical expression is proposed b using the curve fitting technique on the data drown from a quadratic isoparametric finite element method j r ( ) = ( + ) + ( + 3 ) ( + ) k r x x r w x m () where r, r w, x and z are radius at loading point, pitch radius, addendum modification coefficient and number of teeth and the coefficients,, 3 are computed as functions of the number of teeth z. The q.() is applicable for the following 6. < x < 6. and < z < for solid steel gears. The effect of bending, shear and Hertzian contact deformation is taking into account in the analtical method to calculate the tooth deformation. In this calculus procedure, the total deflection f j of a pair of meshing teeth is expressed as j = bj + fj + H j= j= f f f f, () where: f b is the deflection due to bending, shear and axial deformation of the tooth corresponding to the involutes profile; f f is the deflection due to the flexibilit of the tooth foundation and fillet; f H represents the local compliance of the Hertzian contact. The tooth deflection f b is analticall derived b using the method of the potential energ of deformation [4], [7], and can be expressed in the integral form as follows p ' 4( ) () ' α p ' Fn cos +ν + α = M. tan i fz + d d I where: fz = fb + f f (3) Mi ' = p cosα ' s sin α ' f (4) ( ) The tooth parameter sf = f() and integrands, I and, are formulated in terms of variable with great complexit [7]. The equations of the tooth profile coordinates are established from the geometrical conditions b using the propert of the involute profile and trochoidal curve corresponding to the fillet profile. rb T ρ h ρ O O P T h Fig. 3: Contact of spur gear teeth For a pair of meshing teeth, the contact teeth deflection is given b f H ( ν) 4 F n hh ν = ln π l c bh ( ν) rb (5) where b H is the half-width of surface of contact and h, h represent the distance between contact point and tooth centreline in the normal direction on tooth profile of the pinion and gear, respectivel, and ν is the Poisson s coefficient. (Fig. 3). The teeth pairs in contact act like parallel springs. Therefore, the total mesh stiffness k t during each engagement ccle can be written as a function of the position of contact point on the action line t s s k = k + k, for double-tooth contact k t I s = k, for simple - tooth contact (6) where I and II are the mating points of the teeth pairs. The numerical results obtained from analtical calculus method of mesh stiffness are plotted and compared with the values computed b the equation introduced b Kuang and Yang [5]. Figures 4 and 5 show the variation of the total mesh stiffness from the starting to ending of the contact ccle in relation to the geomatrical specificatuion pf the gears. The numerical values 3
obtained from q.() b using the analtical calculus procedure are represetated with the thin line of the curve. These examples illustrate the effect of the change of the number of teeth and addendum modification coefficients on the amount and variation of the total mesh stiffness. k t N mm μm 3 5 5 5 B D G Figure 4: Numerical results of mesh stiffness for the gear pair with following specifications z = 3 ; z = 3 ; x = ; x = The time-varing mesh stiffness is mainl caused b the following factors: (i) the variation of the single mesh stiffness along the equivalent line of action; (ii) the fluctuation of the total number of total pairs in contact during the engagement ccle. 3. Tooth Profile rror The tooth profile error is defined as the distance between the theoretical involutes profile and the real tooth profile in the normal direction. The profile error function e(t) i due to manufacturing can be defined as ( ) ( ) e t = sin ω t +α (7) i i z where ω z is the mesh frequenc and α is the phase angle. The composite error e s is the sum of tooth errors of the pinion and gears. Tooth profile errors are added to the theoretical profile in normal directions. k t N mm μm 7 4 8 5 9 6 3 B D G 4. Static Load Sharing The total displacement must be the same for each pair of teeth in the region of double tooth contact to maintain tooth contact. From this condition, the static load factor c si can be expressed [7] as ( ) k s ks csi = es es (8) F n ks + ks Figure 5: Numerical results of mesh stiffness for the gear pair with following specifications z = 8, z = 4, x =+ 8,. x = 8. Numerical results presented in Figs. 4 and 5 show that the variation of the mesh stiffness is sensitive to the calculus method used especiall for the case of spur gear with addendum modification coefficients. Referring to Figures 4 and 5, the following mesh points were used to represent the successive positions of contact point of a tooth as it passes through the zone of loading: the initial point of engagement, ; the lowest point of single-tooth contact, B; the highest point of single-tooth contact, D; and the final point of engagement,. Section B and D are double - tooth contact zone and section BD is the single tooth - contact zone. where: e s, e s represent the composite profile deviations at the mating point points of the teeth pairs and k s, k s are the single-tooth-pairstiffness [ N / μ m]. The static factor c si of the single tooth pair I is defined the ratio of the single static load F I to the static load F n. If the effect of tooth errors is neglected in q. (8), the tooth load sharing ratio c si depends on the mesh stiffness onl. In such a case, the sharing loads do not depend on the magnitude of the transmitting load. 5. Dnamic Load Simulation The design parameters of the analzed gear pairs are chosen as: material - steel/steel; number of teeth of the pinion and gear, z =8; z = 4; tooth module, m = 3 [mm]; tooth facewidth, b = 5 [mm], center distance, a = 9 [mm]. 4
Specific characteristics of these gear pairs are shown in Table, where x and x are the addendum modification coefficients of the pinion and gear, and ε α represents the contact ratio. Gear pair x x ε α k m [N/μm] f n [Hz] G.63 546.5 3889 G + -.4 48. 3398 Table. Characteristics of the analzed gears In the analsis of dnamic loads, the transmitting load is defined as W = q(f n / b), where q represents the load factor..4 C di G. q =. ξ =.6 ξ =.7 = / 4 e I = Figure 6: Variation of static and dnamic factor includes gear pairs with different combination of the addendum modification coefficients. nominal value F n / b = [N/mm] corresponding to a medium transmitting load is considered in the numerical analsis. The dnamic factor c di of the single tooth pair is defined the ratio of the single dnamic load F di to the static load F n. Static load is the stead state force resulting from driving torque. The effects of damping coefficient and operational speed on the dnamic factor C di are presented in Figures 6 and 7, where the thin line represents the c si factor. The effect of both, the mesh stiffness and the composite profile error on the dnamic load variation is shown in Figure 8. C d I.4.. G q = e I = = / 4 Figure 8: Variation of static and dnamic factor C di.6.4.. G q = ξ =.7 ξ =.6 = / e I = C di.6.4.. G e I q =.5 = / 4 q =.5 Figure 7: Variation of static and dnamic factor Figure 9: Variation of static and dnamic factor computer program was developed for simulating the dnamic characteristics of spur gear pairs. The equations of motion are solved b the fourth-order Runge-Kutta method. Computer analsis of dnamic characteristics 5
C d.8.6.4. G 5 5 [ mm] Figure : Variation of the total dnamic factor C d.8.6.4. G 5 5 mm [ ] Figure : Variation of the total dnamic factor The dnamic load fluctuation under different transmitting loads is shown in Figure 9. The variation of the dnamic factor C d on the path of contact as a function of the addendum modification coefficients are presented in Figures and. In these figures, the line of contact is represented as abscissas and the position of the path of contact on the abscissas is presented according to the distribution of the addendum modification coefficients. The effect of the variation in mesh stiffness on the change on tooth dnamic load is more effective at lower speeds than at higher speeds. The number of load oscillation results as a ratio of the meshing resonance frequenc f n and the meshing frequenc f z of the gear pair. 6. Conclusion n investigation of the dnamic tooth load sharing among meshing teeth of spur gears with addendum modifications is presented in the paper. The values of the time-varing mesh stiffness along the path of contact are compared b using an exact analtical model and a semiempirical equation. The numerical results obtained b using these two different calculus methodologies showed the sensitiveness of the mesh stiffness in relation to the gear ratio and addendum modification coefficients. The dnamic loads are mainl affected b the time varing mesh stiffness, operating frequenc, and composite tooth profile errors. Under a medium or heav load condition, the effect of mesh stiffness on the load sharing ratio was considerabil larger than the composite profile error due to manufacture. cknowledgements This paper is based on the financial support of the National Universit Research Council of Romania, grant ID_96. References [] J. Wang, I. Howard, The Torsional Stiffness of Involute Spur Gears. Proceedind of the Institution of Mechanical ngineering, C: Journal of Mechanical ngineering Science, Vol.8, No., 3-4, 4. []. M. Vaisha, M., Singh, R. Strategies for Modeling Friction in Gear Dnamics. Journal of Mechanical Design,Transaction of the SM, Vol.5, 383-393, 3. [3] J. Lin, J., G.P. Parker, Mesh Stiffness Variation Instabilities in Two-Stage Gear Sstems. Journal of Vibration and coustics, Transactions of the SM, Vol.4, 68-76,. [4] D. Yang, Z.S. Sun, Rotar Model for Spur Gear Dnamics. Journal of Mechanisms, Transmissions, and utomation in Design. Vol. 7, 59-3, 985. [5] J.H. Kuang, Y.T. Yang, n stimate of Mesh Stiffness and Load Sharing Ratio of a Spur Gear Pair. D-Vol.43-, International Power Transmission and Gearing Conference, Volume, SM, -9, 99. [6] O. Sato, H. Shimojimo, T. Kaneko, Positioning Control of a Gear Train Sstem Including Flexible Shafts. JSM International Journal, Vol.3, No. 67, 465-47, 987. [7] tanasiu, V., nalsis of Teeth Forces in Spur Gear Pairs. Proceedings of the ight IFToMM International Smposium on Theor of Machines and Mechanisms, Bucharest, 37-4,. 6