COMPARISON OF ISO AND MAX METHOD IN DETERMINING TIP FACTOR OF INVOLUTE GEARS

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1 ADVANCED ENGINEERING (008), ISSN COMPARN OF AND METHOD IN DETERMINING TIP FACTOR OF INVOLUTE GEARS Glažar, V.; Obsieger, B. & Gregov, G. Abstract: Two methods for determining tip factor of external involute gears generated with rack type cutter are compared. The critical section at tooth-root fillet is obtained in accordance with standard and proposed method. The analysis and comparison of results acquired by both methods are given. Calculations are made for several values of gear parameters. The comparison of results has been made followed with the commentary and final conclusions. Keywords: spur gear, tooth-root stress, tip factor, method, INTRODUCTION The fracture of single tooth on a gear can lead to failure of entire assembly. To calculate tooth-root stress limit and permissible bending stress standard [] defines form factor, stress correction factor and the corresponding tip factor = YFaYSa. () All of them are defined as functions of bottom tooth thickness s Fn in critical section and bending moment arm relevant to load application at the tooth tip h Fa (Fig. ). F b α Fan hfa 0 0 ρ F s Fn Fig.. Critical section at tooth-root fillet by 0 tangent - [] The precise definition of tooth-root thickness s Fn in critical section is needed due to more accurate calculation of tooth-root stress limit and permissible bending stress that can lead to the failure of entire assembly. [] defines critical section at toothroot fillet by the ϕ = 0 tangent. Proposed method [] defines critical section in 9

2 that point of tooth-root fillet where tip factor has maximum value. Consequently, the value of tooth-root fillet tangent ϕ is varying depending on gear parameters when applying method. To compute needed geometry of helical gears both methods can be used but first equivalent spur gear with the number of teeth z n should be defined: z n = z /(cos βb cos β ), () where z is the number of teeth of pinion (or wheel), β is the helix angle and β b is the base helix angle. This procedure is suggested in the standard []. The basic assumption is that the tooth profile of helical gear in its normal section is identical or very similar to the tooth profile of equivalent spur gear. Profile of tooth root fillet and its geometrical data of helical gear in its normal section are also considered to be the same as the correspondent equivalent spur gear. All needed geometry is than calculated for equivalent spur gear by equations presented in [], [] and []. The comparison of results is made and presented as follows. NOMINAL TOOTH-ROOT STRESS AT TOOTH-ROOT Nominal tooth-root stress σ F0 is maximum local tensile stress produced at the toothroot when an error-free gear pair is loaded by the static nominal torque. standard defines several methods for determination of the tooth-root stress. Method A allows any appropriate method (e. g. FEM, BEM, etc.). Method B involves the assumption that the determinant tooth-root stress occurs with application of load at the outer point of single pair tooth contact of spur gears. Method C is simplified method of calculation derived from method B. Comparison of method C and method has been made for the given test gear parameters. Nominal tooth-root stress by method C is defined with following equation: σ Ft F0 = Y ε Y β bm, () n where F t is the nominal load tangential to reference cylinder, b is the facewidth and m n is normal module. The form factor takes into account the influence on nominal tooth-root stress of the tooth form, with load applied at the tooth tip. The stress correction factor takes into account the conversion of the nominal bending stress determined for application of load at the tooth tip, to the local tooth-rooth stress. The corresponding tip factor = YFaYSa defined by () accounts all influences covered by and. The helix factor Y β takes in account that the bending moment intensity at the toothroot of helical gears, as a consequence of the oblique lines of contact, is less than the corresponding values for the virtual spur gears used as bases for calculation. The contact ratio factor Y ε takes into account the transformation of the local stress determined for application of load at the tooth-root and approximates the value relevant to application of load at the outer point of single pair tooth contact. By means of this factor, account is taken of the influence of the stress correction factor of the load distribution over several points of contacts and that of the tooth bending moment. 0

3 CRITICAL SECTION AT TOOTH-ROOT FILLET Transverse component of nominal load F b is needed to calculate nominal tooth-root stress σ F0. Influence of radial force is neglected due to its positive behaviour on tensile side. Stress is bigger on non critical compressive side. Single tooth is observed as console and bending moment arm relevant to load application at the tooth tip should be defined. To simplify calculation form factor, stress correction factor and tip factor are considered. Previously mentioned factors are obtained with next definitions 6( hfa / mn ) cosα Fan =, () ( sfn / mn ) cosα n Y (, 0, / ) /(,, Fa / Fn ) Sa = + s h s Fn hfa q + S. () Meanings of h Fa, s Fn and α Fan are shown on Fig.. The critical section at tooth-root fillet is defined by the ϕ = 0 tangent. method defines critical section in that point of tooth-root fillet where tip factor has maximum value according to Y ϕ =. (6) FS ( ) ( YFaYSa ) max The value of tooth-root fillet tangent ϕ is varying depending on gear parameters when applying method. THE ALGORITHM FOR CALCULATION OF TOOTH GEOMETRY Special computer program has been developed for calculation of nominal tooth-root stress of external involute spur gears generated with rack type cutter. Program uses new algorithm for faster and more precise calculation of tooth geometry. Critical section is calculated with two different methods, method C and proposed method. Gear parameters needed for calculation of tooth-root stress are taken for test gears according to Tab.. All needed expressions and formulas are taken from [], [], and []. Complete list of equations and code listings are not presented in this article due to its length. (a) (b) Fig.. Computer interface of developed program

4 Computer program is developed in no licence cost programming software package Microsoft Visual Basic 00 Express Edition V8.0. with use of the Microsoft.NET Framework technologies version.0. Fig. a, b shows interface of developed program for calculation of nominal tooth-root stress of external involute spur gears generated with rack type cutter. Needed input data are as follows: normal pressure angle α n, helix angle β, normal module m n, number of teeth of a pinion z, tool addendum factor h a 0 = ha0 / m n, (7) tool tip radius factor * ρ a0 = ρa0 / m n (8) and addendum modification coefficient x. Calculation starts by choosing appropriate button control. Results of calculation are immediately shown on computer screen and saved in each time different file (.txt). This principle allows later use of output data. Program can be easily upgraded with other procedures and published on web due to no cost.net technology. * TEST GEAR PARAMETERS Comparison of method C and method has been made for the given test gear parameters. Table shows gear parameters for test gear and. In this case value of addendum modification coefficient is 0,. Table shows parameters for test gear and. Value of addendum modification coefficient is 0,0. Table shows parameters for test gear and 6. Value of addendum modification coefficient is -0,. Test gear. Test gear. Normal pressure angle α n 0 Tool addendum h a0 =, m n, mm,7 mm Tip radius of the tool ρ a0 = 0, m n, mm 0,7 mm Normal module m n 0 mm mm Number of teeth of pinion (or wheel) z Helix angle β 0 0 Addendum modification coefficient x 0,00 0,00 Tab.. Test gear and parameters Test gear. Test gear. Normal pressure angle α n 0 0 Tool addendum h a0 =, m n mm 0 mm Tip radius of the tool ρ a0 = 0, m n mm mm Normal module m n 0 mm 6 mm Number of teeth of pinion (or wheel) z 0 Helix angle β 0 0 Addendum modification coefficient x 0,000 0,000 Tab.. Test gear and parameters

5 Test gear. Test gear 6. Normal pressure angle α n 0 Tool addendum h a0 =, m n 0 mm, mm Tip radius of the tool ρ a0 = 0, m n mm, mm Normal module m n 8 mm 0 mm Number of teeth of pinion (or wheel) z 0 Helix angle β 0 0 Addendum modification coefficient x -0,00-0,00 Tab.. Test gear and 6 parameters 6 AND METHOD COMPARN On Fig. are shown results for test gear obtained by method C and proposed method. Fig..shows results for test gear. (0 ) (,78 ) 0, 0,6 ϕ/r, Fig.. Comparison of and proposed method for test gear (0 ) (,8 ) 0, 0,6 ϕ/rad, Fig.. Comparison of and proposed method for test gear

6 (0 ) (,9 ) 0, 0,6 ϕ/rad, Fig.. Comparison of and proposed method for test gear (0 ) (,97 ) 0, 0,6 ϕ/rad, Fig. 6. Comparison of and proposed method for test gear Tip factor value for different test gears calculated with proposed method is in all cases up to % different than the one obtained by standard. When the value of addendum modification coefficient is 0, and -0, results are almost the same in both cases. The biggest differences are in case when there is no change of addendum (value of addendum modification coefficient is 0). Complete recapitulation of results for test gears and values of tooth-root fillet tangent ( method) are given in Tab..

7 7 6 (,9 ) (0 ) 0, 0,6, ϕ /rad Fig. 7. Comparison of and proposed method for test gear 7 6 (6, ) (0 ) 0, 0,6 ϕ /rad, Fig. 8. Comparison of and proposed method for test gear 6

8 Test gear ϕ () () Difference in % N,78,, 0, % N,8,8,8 0, % N,9,,, % N,97,6,9 % N,9,79,8 0,7 % N6 6, 6, 6, 0, % 7 CONCLUSION Tab.. and method comparison The calculation of tip factor using different methods [] and [] gave similar results with small deviations that confirm the use of method as alternative to common method. Several experimental researches [] justified the use of method, too. Further research would include the use of method in analysis of tip factor behaviour by varying other gear parameters (i.e. tool tip radius, tool addendum, etc.). References: [] International standard - 66-, Part and part,. Edition, International Organization for standardization, Switzerland, 996; [] Obsieger, B., Calculation of geometry factors for external helical gears generated with rack type cutters, Znanstveno-stručni skup nauka o konstruiranju i konstruiranje pomoću računala, Zagreb, lipanj 98; [] KISSsoft Manual - Calculation - Programs for machine design, [] MAAG-ZAHNRADER AG: MAAG Taschenbuch, te erweiterte und erganzte Auflage, Zurich/Schweiz, 98; [6] Flašker, J.; Glodež, S. & Pehan, S., The influence of functional contact area on the stress field in a gear tooth root, Journal of mechanical engineering, Ljubljana, 99; [7] Obsieger, B., Prijenosi sa zupčanicima, Zigo, Rijeka, 00; [8] Oberšmit, E., Ozubljenja i zupčanici, SNL, Zagreb, 98; [9] Obsieger, B., Analitički prikaz profila zuba zupčanika dobivenih odvaljivanjem proizvoljnog matematički definiranog osnovnog profila, Tehnički Fakultet Rijeka, Rijeka, 977; Authors: Glažar, Vladimir, PhD. student, University of Rijeka, Faculty of Engineering, vladimir.glazar@riteh.hr; Obsieger, Boris D. Sc. Prof, University of Rijeka, Faculty of Engineering, boob@riteh.hr; Gregov, Goran, PhD. student, University of Rijeka, Faculty of Engineering, goran.gregov@riteh.hr 6