COMPARISON OF SINUSOIDAL AND INVOLUTE SPUR GEARS BY MESHING CHARACTERISTICS

P. Tkach, PhD, Assoc. Prof., P. Nosko, DSc, Prof., O. Bashta, PhD, Assoc. Prof., Yu. Tsybrii, PhD, O. Revyakina, PhD, Assoc. Prof., G. Boyko, PhD, Assoc. Prof. 1 E.O. Paton Electric Welding Institute of the National Academy of Sciences of Ukraine, 11 Kazymyr Malevych Str., Kyiv, Ukraine, 03150; e-mail: pavlotkach78@gmail.com 2 National Aviation University, 1 Kosmonavta Komarova Ave., Kyiv, Ukraine, 03058 3 Taras Shevchenko Lugans'k National University, 1 Gogol Square, Starobilsk, Ukraine, 92703 4 Volodymyr Dahl East Ukrainian National University, 59-а Tsentralny Ave., Severodonets'k, Ukraine, 93400


Materials and Methods.
For the qualitative estimation of gearing serviceability, a number of geometric-kinematic indicators can be used, which can be obtained for the sinusoidal gearing based on the results of study [13].
1. Sliding velocity: where 1 ω is an angular velocity of pinion; u is a gear ratio; h is a height of sinusoidal reference profile in fractions of gearing module m. It equals to a radius of the circle which creates the sinusoid (Fig. 1); λ is a parameter for the sinusoid, this angle λ varies from 0 to p λ for the working part of profile p h , from p λ to / 2 π for the root fillet ( Fig. 1). It should be noted that h and λ clearly determine the basic rack in the system of coordinates related to the reference profile sin r x h = λ , / 2 r y = λ . 2. The velocity of contact point in the direction normal to the contact line on the pinion's and gear's surfaces: 2 2 2 sin 4 cos 1 4 cos 2 1 where і ω is an angular velocity of pinion ( 1 і = ) or gear ( 2 і = ), rad/s; і R is a radius of pinion's ( 1 і = ) or gear's ( 2 і = ) pitch circle in fractions of m; 3. The total velocity of contact points: 4. The specific sliding on the pinion's ( 1 і = ) and gear's ( 2 і = ) teeth surface: 2 2 ( 1) sin (4 cos 2 1) sin (4 cos 2 1) In the equation (2) and (4) the sign "+" is taken for the pinion, and "-" for the gear. For a more complete estimation of gearing quality the complex meshing characteristics depending on the geometric-kinematic indicators (1) -(5) may be applied.
1. The surface strength is evaluated based on the assumption that in the result of gearing conformability under the load the surface stresses have constant values. This assumption is usually used in calculations of traditional gearing, because it is consistent well with the experimental results. It was used in study [13] for the determination the relative characteristic of surface strength. If the expression [13] is converted for spur gears it will be obtained: 2  This characteristic is actually the load factor for the surface stresses determine. It contains the geometric parameters of tooth and affects the critically allowable load for the condition of surface strength: where w b is a width of gearing; k is a proportionality factor; determined from the Hertz equation: where n q is a force acting on the unit of length of contact line (directed normally to the working surfaces of the teeth); red E is a reduced modulus of elasticity of gears' materials. Obviously, the critical load corresponds to the limit value of max k for [ ] H H σ =σ . Then, taking into account (8) and (9), the maximum specific force: 2. The scuffing is usually estimated by the values of teeth working surfaces temperature which is determined by the Block's theory [14]. However, for the comparative evaluation of various gearings the instantaneous flash temperature in the zone of contact [15] based on the Block theory may be used: The coefficient 1.84 in this equation contains the physical characteristics of gears and oil. Since for comparison we will consider gears made of identical materials and the same oil, we will use a relative characteristic: In the equation (11) f is the coefficient of sliding friction in the teeth contact zone. It is defined according to [16] where n q is in kgf/cm; red E is in kg/sm 2 ; red χ is in 1/cm; V Σ and S V is in cm/s; HB is a Brinell's hardness number of harder gear's teeth, kg/cm 2 ; Ra is a surface roughness of harder gear's teeth, cm; ν is an oil viscosity, cSt. The equation (12) where 4 3 ( , ) 0.47 0.13 10 0.4 10 The parameters' measurement units included in (12) and (13) are given in accordance with [15,16]. The specific load n q in the equations (11), (12) and (13) taking into account the equations (7) -(9) equals: 3. Wear of teeth active surfaces may be estimated by the relative value of wear layer thickness [16]: where Ω is a coefficient depending on the lubricant properties, it doesn't take into account the geometry of the teeth; I is an intensity of surfaces wear, which for the run-in surfaces is determined in accordance with the recommendations of I.V. Kragelsky [16] by the equation: where K is a coefficient depending on the elastic properties and hardness of gear's material; f t is a parameter of frictional fatigue curve. The equation (15) taking into account (9), (14) and (16) may be represented as: The relative value of wear: may be used as for a comparative evaluation of gearings with different teeth profiles but made of identical materials and works in identical conditions under the same load 4. The oil film's thickness is determined on the basis of dependence shown in the study [17]: where 0 µ is a dynamic viscosity of oil at atmospheric pressure; α is an oil viscosity's piezo coefficient, which depends on the temperature and surface stress in the contact zone of the teeth.
The equation (18), taking into account (14), may be represented as: The relative value of oil film's thickness obtained from (19) after excluding parameters that don't depend on the teeth geometry: can be used for the comparative evaluation of gearing with different teeth profiles. 5. The specific work of frictional forces on the contact line area of unit length is determined by the equation [15]: According to the results of work [13], the gearing power loss is an integral characteristic, based on the geometric-kinematic (1) and complex (6) characteristic as well as specific load (14). It is defined as: where 10 ϕ , 20 ϕ is the pinion's turning angles, that correspond to the starting and end of the gearing. Correspondence of pinion's turning angle with the position of contact line on the profile is determined from the equation [18]: The bending strength of sinusoidal tooth in the first approximation was evaluated in [19]. Its results are in good agreement with the assumption of Yu.V. Anikin [20] about the sinusoidal tooth's advantages. However, a more detailed estimation of stress state of sinusoidal teeth is quite huge and complex task, so it should be extract in an independent study.
So, the geometric-kinematic (1) -(5), complex (6), (11), (17), (20), (21), as well as integral (22) characteristics was applied for comparison of sinusoidal gearing with conventional one. The geometrickinematic meshing characteristics of conventional gearing were determined using the formulas of [21]. It may be applied because quasi-involute arc or crowded gear have the involute spur gearing in a midsection. The dependence of ϕ on the contact line position for the conventional gearing is also determined by [21]. For comparison, we choose gearings with the parameters presented in the Table 1. Results and Discussion. Geometric-kinematic characteristics of both gearings are presented in Fig. 2 -5 and complex characteristics are presented in Fig. 6 -12. The solid and dashed lines correspond to sinusoidal and conventional gearing respectively. The coordinates r x which correspond to the starting and end points of gearing was defined with accordance to recommendations of [18] and [21] for sinusoidal and conventional gearing respectively. Analysis of Table 2 shows that both transmissions have the same values in the near-pitch zone. This fact is due to two reasons. Firstly, the curvature of the sinusoidal reference profile on the pitch line according to the results of [18,20] equals zero. It means that at the pitch point the sinusoidal gearing has meshing characteristics of the involute one. Secondly, the profile angle of the sinusoidal reference profile on the pitch line is 20° ( Table 1).
The geometric-kinematic characteristics of sinusoidal gear at the end point of gearing are 1.18…10.65 times better. Complex characteristics are 6.57…36.81 times better.
The most of sinusoidal gear's characteristics at the beginning of the gearing are much better. This fact is due to the dependence of complex characteristics on reduced curvature and specific sliding. The beginning of the gearing takes place at the lowest point of the pinion's tooth profile where the values of reduced curvature and specific slides of involute gears are usually high. On the contrary, both of characteristics of sinusoidal gearing have very low values at the beginning of gearing.
The gearing power loss, defined by formula (22) for sinusoidal gearing. It should be noted that the value of gearing power loss calculated taking into account the value of V Σ by the recommendations of [15] equals 0.011. Consequently, the formula (22) is in good correlation with formula of [15]. So, sinusoidal gearing has an advantage of 2.13 times by an integral characteristic.