ADVANCED ENGINEERING
2(2008)1, ISSN
1846-5900
COMPARISON OF ISO AND MAX 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 ISO standard and proposed MAX 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, MAX method, ISO
1 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 ISO standard [1] defines form factor YFa, stress correction factor YSa and the corresponding tip factor
Sa Fa FS Y Y
Y = . (1)
All of them are defined as functions of bottom tooth thickness sFn in critical sec-tion and bending moment arm relevant to load applicasec-tion at the tooth tip hFa (Fig. 1).
Fig. 1. Critical section at tooth-root fillet by 30° tangent - ISO [1]
The precise definition of tooth-root thickness sFn 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. ISO [1] defines critical section at tooth-root fillet by the ϕ =30° tangent. Proposed MAX method [2] defines critical section in
sFn 30° 30° Fb hFa αFan ρF
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 MAX method. To compute needed geometry of helical gears both methods can be used but first equivalent spur gear with the number of teeth zn should be defined: ) cos /(cos2 b n z β β z = , (2)
where zis the number of teeth of pinion (or wheel), β is the helix angle and βb is the base helix angle.
This procedure is suggested in the ISO standard [1]. 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 [1], [2] and [3]. The comparison of results is made and presented as follows.
2 NOMINAL TOOTH-ROOT STRESS AT TOOTH-ROOT
Nominal tooth-root stress σF0 is maximum local tensile stress produced at the tooth-root when an error-free gear pair is loaded by the static nominal torque. ISO 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 ISO method C and MAX method has been made for the given test gear parameters.
Nominal tooth-root stress by ISO method C is defined with following equation: β εY Y Y Y bm F Sa Fa n t F0 = σ , (3)
where Ft is the nominal load tangential to reference cylinder, b is the facewidth and mn is normal module.
The form factor YFa 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 YSa 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 YFS =YFaYSa defined by (1) accounts all influences covered by YFa and YSa. The helix factor Yβ takes in account that the bending moment intensity at the tooth-root 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 Yε 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.
3 CRITICAL SECTION AT TOOTH-ROOT FILLET
Transverse component of nominal load Fb is needed to calculate nominal tooth-root stress σF0. 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
n 2 n Fn Fan n Fa Fa cos ) / ( cos ) / ( 6 α α m s m h Y = , (4) ) / 3 , 2 21 , 1 /( 1 ) / 13 , 0 2 , 1 ( Fa Fn S Fa Fn Sa s h q h s Y = + + . (5)
Meanings of hFa, sFn andαFan are shown on Fig.1. The critical section at tooth-root fillet is defined by the ϕ = 30° tangent. MAX method defines critical section in that point of tooth-root fillet where tip factor has maximum value according to
max Sa Fa FS( ) (Y Y )
Y ϕ = . (6) The value of tooth-root fillet tangent ϕ is varying depending on gear parameters when applying MAX method.
4 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, ISO method C and proposed MAX method. Gear parameters needed for calculation of tooth-root stress are taken for test gears according to Tab. 1. All needed expressions and formulas are taken from [1], [2], and [3]. Complete list of equations and code listings are not presented in this article due to its length.(a) (b)
Computer program is developed in no licence cost programming software package „Microsoft Visual Basic 2005 Express Edition V8.0.5“ with use of the Microsoft.NET Framework technologies version 3.0. Fig. 2 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 αn, helix angle β, normal module mn, number of teeth of a pinion z, tool addendum factor
n m h h* a0/ 0 a = , (7)
tool tip radius factor
n m / 0 a * 0 a ρ ρ = (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.
5 TEST GEAR PARAMETERS
Comparison of ISO method C and MAX method has been made for the given test gear parameters. Table 1 shows gear parameters for test gear 1 and 2. In this case value of addendum modification coefficient is 0,5. Table 2 shows parameters for test gear 3 and 4. Value of addendum modification coefficient is 0,0. Table 3 shows parameters for test gear 5 and 6. Value of addendum modification coefficient is -0,5.
Test gear 1. Test gear 2.
Normal pressure angle αn 20° 15°
Tool addendum ha0 = 1,25 mn 12,5 mm 3,75 mm Tip radius of the tool ρa0 = 0,25 mn 2,5 mm 0,75 mm
Normal module mn 10 mm 3 mm
Number of teeth of pinion (or wheel) z 25 22
Helix angle β 0° 0°
Addendum modification coefficient x 0,500 0,500 Tab. 1. Test gear 1 and 2 parameters
Test gear 3. Test gear 4.
Normal pressure angle αn 20° 20°
Tool addendum ha0 = 1,25 mn 25 mm 20 mm Tip radius of the tool ρa0 = 0,25 mn 5 mm 4 mm
Normal module mn 20 mm 16 mm
Number of teeth of pinion (or wheel) z 40 55
Helix angle β 0° 0°
Addendum modification coefficient x 0,000 0,000 Tab. 2. Test gear 3 and 4 parameters
Test gear 5. Test gear 6.
Normal pressure angle αn 20° 15°
Tool addendum ha0 = 1,25 mn 10 mm 12,5 mm Tip radius of the tool ρa0 = 0,25 mn 2 mm 2,5 mm
Normal module mn 8 mm 10 mm
Number of teeth of pinion (or wheel) z 20 25
Helix angle β 0° 0°
Addendum modification coefficient x -0,500 -0,500 Tab. 3. Test gear 5 and 6 parameters
6 ISO AND MAX METHOD COMPARISON
On Fig. 3 are shown results for test gear 1 obtained by ISO method C and proposed MAX method. Fig. 4.shows results for test gear 2 .
Fig. 3. Comparison of ISO and proposed MAX method for test gear 1
Fig. 4. Comparison of ISO and proposed MAX method for test gear 2 YFS YSa YFa ISO (30°) MAX (33,78°) YFS YSa YFa 0,2 0,6 1 1,4 ϕ/r 5 4 3 2 1 YFS YSa YFa ISO (30°) MAX (33,82°) 0,2 0,6 1 1,4 ϕ/ rad YFS YSa YFa 5 4 3 2 1
Fig. 5. Comparison of ISO and proposed MAX method for test gear 3
2 4 5
Fig. 6. Comparison of ISO and proposed MAX method for test gear 4
Tip factor value for different test gears calculated with proposed MAX method is in all cases up to 3% different than the one obtained by ISO standard. When the value of addendum modification coefficient is 0,5 and -0,5 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 (MAX method) are given in Tab. 4.
0,2 0,6 1 1,4 ϕ/ rad YFS YSa YFa 5 4 3 2 1 0,2 0,6 1 1,4 ϕ/ rad YFS YSa YFa 5 4 3 2 1 YSa YFa YFS ISO (30°) MAX (42,9°) YFS YSa YFa ISO (30°) MAX (42,97°)
Fig. 7. Comparison of ISO and proposed MAX method for test gear 5
Fig. 8. Comparison of ISO and proposed MAX method for test gear 6 1 2 3 4 5 6 7 0,2 0,6 1 ϕ/rad 1,4 ISO (30°) MAX (22,92°) YFS YFa YSa YFS YSa YFa 1 2 3 4 5 6 7 0,2 0,6 1 ϕ/rad 1,4 ISO (30°) MAX (26,35°) YFS YFa YSa YFS YSa YFa
Test gear ϕ YFS (ISO) YFS (MAX) Difference in % N1 33,78° 4,3 4,32 0,5 % N2 33,82° 4,83 4,84 0,2 % N3 42,9° 4,3 4,4 2,3 % N4 42,97° 4,26 4,39 3 % N5 22,92° 5,79 5,8 0,17 % N6 26,35° 6,22 6,2 0,32 %
Tab. 4. ISO and MAX method comparison
7 CONCLUSION
The calculation of tip factor using different methods [1] and [2] gave similar results with small deviations that confirm the use of MAX method as alternative to common ISO method. Several experimental researches [3] justified the use of MAX method, too. Further research would include the use of MAX method in analysis of tip factor behaviour by varying other gear parameters (i.e. tool tip radius, tool addendum, etc.). References:
[1] International standard - ISO 6336-1, Part 1 and part 3, 1. Edition, International Organiza-tion for standardizaOrganiza-tion, Switzerland, 1996;
[2] 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 1981;
[3] KISSsoft Manual - Calculation - Programs for machine design, www.KISSsoft.ch;
[5] MAAG-ZAHNRADER AG: MAAG – Taschenbuch, 2te erweiterte und erganzte Auflage, Zurich/Schweiz, 1985;
[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, 1993;
[7] Obsieger, B., Prijenosi sa zupčanicima, Zigo, Rijeka, 2003; [8] Oberšmit, E., Ozubljenja i zupčanici, SNL, Zagreb, 1982;
[9] Obsieger, B., Analitički prikaz profila zuba zupčanika dobivenih odvaljivanjem
proizvolj-nog matematički definiranog osnovnog profila, Tehnički Fakultet Rijeka, Rijeka, 1977;
Authors: Glažar, Vladimir, PhD. student, University of Rijeka, Faculty of Engineering, [email protected];
Obsieger, Boris D. Sc. Prof, University of Rijeka, Faculty of Engineering, [email protected]; Gregov, Goran, PhD. student, University of Rijeka, Faculty of Engineering,
