Investigations On Gear Tooth Surface And Bulk Temperatures Using ANSYS P R Thyla PSG College of Technology, Coimbatore, INDIA R Rudramoorthy PSG College of Technology, Coimbatore, INDIA Abstract In gears, the temperature at the conjunction zone between the meshing tooth faces governs the imminent scuffing failure. Failure due to high tooth temperatures can be prevented with the knowledge of temperature distribution in gear teeth under operation. In this work, the prediction of bulk and surface temperatures of the gear tooth is carried out using finite element method, using ANSYS. The investigations are carried out on the tooth of the worm wheel of different sizes of single reduction worm gear boxes under various operating conditions Introduction Conventional gear designs are based on the bending and surface fatigue failure criteria. Failure due to bending is avoided by ensuring in the design that the maximum bending stress in the tooth is lesser than the bending strength of the material. Limiting the Hertzian contact stress induced in the gear tooth to be below the contact compressive strength of the gear material ensures safety in respect of surface fatigue. Dynamic load factors are included in the design to account for effects due to dynamic loading. But these designs do not consider the thermal behaviour of the gears. But in gears that are operated at high speeds and loads, temperature can be of concern and may be a limiting factor. Also, in worm gear drives, due to the large amount of heat generated due sliding friction, temperatures may exceed safe limits during operation. Hence, for gear drives that operate at high speeds and under heavy loads, and in worm gear drives in which the amount of heat generated due to sliding friction is more, it is essential to know the thermal behaviour of the drives at the design stage itself so as to know safe limits of load and speed. Procedure In this work, finite element thermal analysis is carried out on a tooth of the worm wheel to predict the temperature distribution in the tooth. The problem of determining the bulk temperature is considered as a steady-state heat transfer problem in which a state of thermal equilibrium is reached after many cycles of revolution. In order to predict the bulk and the surface temperatures of the gear tooth, a single gear tooth of the worm gear is modeled using UNWINS method of involute profile generation and the solid model generated by coordinate transformation. The reason for modeling a single gear tooth is that the gear tooth is symmetrical in shape and identical heat generation occurs in all the teeth of the gear. The solid model of the tooth of the worm wheel is discretised using 3D, 1-noded tetrahedral elements, ie., SOLID87 element. The finite element mesh of the gear tooth is shown in Figure 1. A convergence test was carried out, by varying the element size so as to ensure correctness of mesh density. The results of the test is depicted in Figure 2.
Figure 1. Finite element mesh of the gear tooth 8 Temperature 6 4 2 2 3 4 5 6 7 Elements Figure 2. Convergence test Boundary Conditions In the steady state thermal analysis of the gear tooth, the frictional heat generated and the convective heat transfer from the different surfaces of the tooth form the boundary conditions. The frictional heat input to the gear tooth-working surface is calculated using the following analysis. The instantaneous heat generated per unit area, per unit time due to sliding of the two gear teeth is given by Heat generation rate Q = V s W f
with the sliding velocity Vs given by, V s = V w / cos λ The frictional force W f is given by W f = µ W, W being the normal load. The coefficient of friction at the mesh is dependent on the sliding velocity and the viscosity of the lubricant. It is estimated as µ = 1.6/(υ.15 * V e.15 * Vs.35 * Rr.5 ) with the entraining velocity V e and the rolling radius R r determined as d D n υ V s V e R r λ φ V e R r = pitch circle diameter of the worm, m = pitch circle diameter of the worm wheel, m = speed of the worm, rpm = viscosity of lubricant, centistokes = Sliding velocity in ft/min = entraining velocity in ft/min = relative radius of curvature on the normal section, inches = lead angle of worm, degrees = pressure angle, degrees =.524*d*n*sin (λ) sin (φ) =.5*D*[(1/cos (λ)) 2 ]* sin (φ), where, The distribution of the heat generated to the two mating teeth depends on the velocity and material properties of the gears. A partition factor is used to estimate the proportion of the heat that goes into the worm and that which goes into the wheel. In a gear tooth, different portions of the tooth have different cooling rates. The heat transfer coefficients for sides, top land and flanks of the gear are estimated as follows. The heat transfer coefficient (h s ) on the gear hub surface is established from the principle of rotating disks as where, Nu = Nusselt number, ω = angular velocity of the rotating disk, rad/s K = thermal conductivity, W/m 2 K ν = kinematic viscosity, m s /s h s = Nu * K * (ω/ν).5 In a gearbox with splash type of lubrication, the wheel rotates in air. Hence, a Nusselts number of.5, which corresponds to air, is adopted. There may be a slight error in this assumption since the cooling of the gear surfaces due to the fling-off of the lubricant is not accounted. Because of the uneven gear tooth surface, it is difficult to estimate the heat transfer coefficient (h t ) on the flank and lands of the tooth. Hence, an approximate estimate is based from the principle of flow across cylinders, which is given by h t = Nu *K/D
where, Re = Reynolds number Pr = Prandtl number C,m = constants Nu = C * Re m Pr.333 Analysis Using the Unwin s method, gear teeth were modeled for four different sizes of single reduction worm gear boxes and the finite element mesh created. The relevant boundary conditions were estimated and applied on the finite element model and steady state thermal analyses were carried out. The investigations were made with variations in load, speed and viscosity grade of lubricant. Analysis Results & Discussion Temperature Distribution in Tooth The temperature distribution in a tooth of the worm wheel of an 8-inch worm gearbox is shown in Figure 4. The figure shows the temperature profile at steady state. From the figure, it can be seen that a large volume of the tooth is at the same temperature, which is much lesser than the highest temperature. The hottest zone is confined to a thin skin of the gear tooth and prevails over a small region at the mid-section of the tooth face The peak temperature occurs at the middle of the face width, at a point below the pitch line. The pattern of temperature distribution is more or less the same in wheels of different sizes, and also under different operating conditions. The temperature distributions at various sections along the face width of the gear tooth are shown in Figure 5. The temperature at the root is more than the temperature at the tip in gears of all sizes, and under all the different operating conditions. The difference in temperature between the tip and the root differs with the applied load and input speed conditions. The difference increases with increase in load. Figure 3. Temperature profile in gear tooth of 3-inch gearbox
Figure 4. Isotherms at different sections of the gear tooth
Temerature rise, C 3 2 2 Output torqu, Nm Bulk Surface Figure 5. Variation of tooth temperature rise with applied load 3 Temperature rise, C 2 7 9 1 13 Input speed, rpm Bulk Surface Figure 6. Variation of tooth temperature rise with input speed Temperature rise, C 2 2 3 4 5 Kinematic Viscosity of oil, cst Bulk Surface Figure 7. Variation of tooth temperature rise with oil viscosity
Influence of operating parameters on tooth temperature The gear tooth temperature is dependent on a large number of parameters. In this work, the effect of the three operating parameter, namely, applied load, input speed and the viscosity of the lubricating oil used are studied. The variations of the bulk and the surface temperatures of the tooth of the gear in one of the test gearboxes for different values of applied load, input speed and the viscosity of the lubricating oil used are shown in Figure 5 to Figure 7. It can be seen that both the bulk temperature rise and the surface temperature rise increase linearly with increase in the applied load. They also increase with increase in input speed and decrease with an increase in the viscosity of the oil used, but the degree of variation in these two cases is very less compared to that due to change in load Conclusion In this work, a methodology has been developed to predict the bulk and the surface temperatures in gear teeth using finite element analysis using ANSYS. The tooth frictional losses and different convective heat transfer coefficients for different portions of the tooth form the input to the model. Steady state thermal analysis is carried out on gear teeth of different sizes of commercial gearboxes under different operating conditions to predict the temperature distribution. The methodology developed can be used for evaluating the thermal characteristics of gearboxes at the design stage itself, eliminating the need for physical experimentation. These virtual experiments aid in the rating of gearboxes for their thermal capacity and to design cooling requirements. References 1) Alastair Cameron, 1981 Basic Lubrication Theory, 3 rd Edition, Ellis Hardwood 2) Patir, N., and Cheng, H. S., Prediction of the Bulk Temperature in Spur Gears Based on Finite Element Temperature Analysis, ASLE Transactions, Vol. 22.pp. 25-36, 1979 3) Townsend, D. P., and Akin, L. S., Analytical and Experimental Spur Gear Tooth Temperature as Affected by Operating Variable, ASME Journal of Mechanical Design, Vol. 13, pp. 216-226, 1981 4) Wang, K.L., and Cheng, H.S., Numerical solution to the dynamic load, film thickness and surface temperature in spur gears Part I, Analysis, ASME Journal of Mechanical Design, vol 13, pp177-187, 1981 5) Wang, K.L., and Cheng, H.S., Numerical solution to the dynamic load, film thickness and surface temperature in spur gears Part II, Results, ASME Journal of Mechanical Design, vol 13, pp188-194, 1981