16 Dynamic factor, K K
vv16.1 Purpose of 16.1 Purpose of K K vv
The dynamic factor accounts for load fluctuations arising from contact conditions at the gear mesh. The The dynamic factor accounts for load fluctuations arising from contact conditions at the gear mesh. The main influences are:
main influences are:
a) gear tooth accuracy;
a) gear tooth accuracy;
b) the tooth contact frequency divided by the natural frequency of torsional oscillations due to pinion and b) the tooth contact frequency divided by the natural frequency of torsional oscillations due to pinion and wheel inertias acting against the mesh stiffness.
wheel inertias acting against the mesh stiffness.
The portion of a typical gear load accounted for by the dynamic factor is illustrated in Figure 9.
The portion of a typical gear load accounted for by the dynamic factor is illustrated in Figure 9.
The values included in this standard are appropriate.
The values included in this standard are appropriate.
NO
NOTETE At lAt low-ow-loaload the vd the valalue ofue of K K vv may be higher than given by the standard, but the stress will not exceed the stress at the may be higher than given by the standard, but the stress will not exceed the stress at the maximum rating of the gear on which the values of
maximum rating of the gear on which the values of K K vv are based. If a gear pair is operating at or near resonance speed or at multiples are based. If a gear pair is operating at or near resonance speed or at multiples or sub-multiples of resonance speed (particularly the second and third harmonics and sub-harmonics, respectively) then a thorough or sub-multiples of resonance speed (particularly the second and third harmonics and sub-harmonics, respectively) then a thorough dynamic analysis is recommended. This is beyond the scope of this standard.
dynamic analysis is recommended. This is beyond the scope of this standard.
16.2 Calculation of 16.2 Calculation of K K vv The value of
The value of K K vv is calculated from the equation: is calculated from the equation:
For helical gears of overlap ratios greater than or equal to unity, For helical gears of overlap ratios greater than or equal to unity,
K
K v350v350 == K K v350v350¶¶
where
where K K v350v350¶¶ is obtained from Figure 10. is obtained from Figure 10.
For spur gears, For spur gears,
K
K v350v350 == K K v350v350 ! ! where
where K K v350v350 ! ! is obtained from Figure 11. is obtained from Figure 11.
For helical gears of overlap ratio less than unity:
For helical gears of overlap ratio less than unity:
The value of
The value of B B is calculated from the equation:is calculated from the equation:
where
where X X is obtained from Table 7. is obtained from Table 7.
If the value of
If the value of F F tt K K A A //bb is less is less thanthan 100, then 100, then useuse F F tt K K A A //bb == 101000..
If the procedure is being used to calculate a maximum rating, then estimate the value of
If the procedure is being used to calculate a maximum rating, then estimate the value of F F tt K K A A //bb from the from the equation:
equation:
where
where Ö Ö HPHP is the lesser of is the lesser of Ö Ö HP1HP1 and andÖ Ö HP2HP2..
It is advisable to check the accuracy of this estim
It is advisable to check the accuracy of this estim ate when the rating has been calculated anate when the rating has been calculated an d, if necessary,d, if necessary, re-calculate
re-calculate K K vv using the new value of using the new value of F F tt..
(20)
K v350v350 ¶ ¶ is obtained from Figure 10; is obtained from Figure 10;
K
K v350v350! ! is obtained from Figure 11. is obtained from Figure 11.
(22)
Table 7 — Values of Table 7 — Values of X X
16.3 Values of auxiliary parameter 16.3 Values of auxiliary parameter Q Q vv 16.3.1
16.3.1 Q Q vv takes account of the shift in the resonance frequency of the gear pair when both or either of the takes account of the shift in the resonance frequency of the gear pair when both or either of the pinion and wheel are not solid gears. The general equation for
pinion and wheel are not solid gears. The general equation for Q Q vv is: is:
where
where M M red 0red 0 is the value of is the value of M M redred for the particular example of a solid pinion matin for the particular example of a solid pinion matin g with a solid wheel and:g with a solid wheel and:
where where
16.3.2
16.3.2 Q Q vv and and M M redred for some common gear arrangements when the pinion and wheel material are of the for some common gear arrangements when the pinion and wheel material are of the same density are given in a) to d) below. If the density of the pinion and wheel materials are different, same density are given in a) to d) below. If the density of the pinion and wheel materials are different, Q Q vv is calculated from equations (24) to (27).
is calculated from equations (24) to (27).
a) For a solid pinion meshing with a solid wheel:
a) For a solid pinion meshing with a solid wheel:
b) For a solid pinion meshing with a wheel with a fabricated rim:
b) For a solid pinion meshing with a wheel with a fabricated rim:
where
M 11 is the moment of inertia of the pinion and is the moment of inertia of the pinion and
(26) (26) M
M 22is the moment of inertia of the wheel, andis the moment of inertia of the wheel, and
(27)
BS
BS 43 436- 6-3:1 3:1986 986
c) Planetary gears c) Planetary gears
1) for sun pinion with planet gear wheel:
1) for sun pinion with planet gear wheel:
where where
2) planet gear with annulus rigidly connected to the gear case.
2) planet gear with annulus rigidly connected to the gear case.
In this and other cases where the mass of the stationary annulus is sufficiently large to be assumed In this and other cases where the mass of the stationary annulus is sufficiently large to be assumed infinite:
infinite:
3) planet gear with rotating annulus:
3) planet gear with rotating annulus:
where where
d
do2o2is the outside diameter of the annulus.is the outside diameter of the annulus.
(d) Idler gears (d) Idler gears
where where M
M ll,, M M idlidl and and M M 22 are the reduced masses of the small gear (pinion), idler gear and large gear are the reduced masses of the small gear (pinion), idler gear and large gear (wheel) respectively.
(wheel) respectively.
(31) (31)
n
npp lala is the number of planets meshing with the sun;is the number of planets meshing with the sun;
M
M plapla is the moment of inertia of the planet gear and is calculated from equation (27);is the moment of inertia of the planet gear and is calculated from equation (27);
M
M sunsun is the moment of inertia of the sun pinion and is calculated from equation (26).is the moment of inertia of the sun pinion and is calculated from equation (26).
(32)
Figure 10 — K v350 ¶ for helical gears, && ¶ ¶WW1
Figure 11 — K v350! for spur gears
d C o p y : U n i v e r s i t y o f M a n c h e s t e r L i b r a r y , T h e U n i v e r s i t y o f M a n c h e s t e r L i b r a r y , 0 4 / 0 2 / 2 0 1 3 1 6 : 3 7 , U n c o n t r o l l e d C o p y , ( c ) T h e B r i t i s h S t a n d a r d s I
BS 436-3:1986
17 Load distribution factors, K
H!and K
H ¶17.1 Purpose of the face load factor for contact stress K H ¶
K H ¶ is the maximum specific load divided by the mean specific load.
It accounts for the increase in local load due to mal-distribution of load across the face of the gear caused by deflections, alignment tolerances and helix modifications.
These include:
pinionshaft bending deflections pinionshaft torsional deflections
misalignment because of manufacturing tolerances end relief 4)
helix correction4) crowning4)
17.2 Calculation of K H ¶
17.2.1 The calculation of the face load factor involves the following.
a) determination of the mean load intensity wm;
b) determination of the mesh misalignment due to deflections and manufacturing tolerance modified by the effect of running-in and helix modifications F ¶y;
c) determination of mesh stiffness, c * d) calculation of the face load factor, K H ¶
17.2.2 Calculate the value of beff from the equation:
17.2.3 Calculate the value of wm·
If the tangential load on the gears is known and the procedure is being used to calculate a safety factor, then wm is calculated from the equation:
If the calculated value of wm is less than 100 N/mm then use wm = 100 N/mm.
If the procedure is being used to determine the maximum rating of the gear pair, then an estimated value of F m is calculated from the equation:
where Ö HP is the minimum of Ö HP1 andÖ HP2 then:
wm est is then used in place of wm in the following analysis.
If higher accuracy is required, then the procedure can be used iteratively by calculating a value of K H ¶ using wm est, then re-calculating using wm est/ K H ¶. Three such iterations will normally converge to give a constant value of K H ¶.
17.2.4 Calculate the value of F ¶y . If:
a) the gears are helix corrected, or
b) the gear layout does not conform to Figure 12, or
c) substantial forces other than pure shaft torque are to be applied (e.g. pulley loads), or d) wheel shaft deflections are significant,
then F ¶y is calculated by a thorough analysis of all contritutions to the mesh misalignment (bearing clearances, case and shaft deflections, manufacturing tolerances, etc.).
4)Recommendations on the design of end relief, helix correction and crowning are given in Appendix C.
beff = b – lc/2 (35)
wm= F t K A K v/beff (36)
(37)
wm est= F m est/beff (38)
L i c e n s e d C o p y : U n i v e r s i t y o f M a n c h e s t e r L i b r a r y , T h e U n i v e r s i t y o f M a n c h e s t e r L i b r a r y , 0 4 / 0 2 / 2 0 1 3 1 6 : 3 7 , U n c o n t r o l l e d C o p y , ( c ) T h e B r i t i s h S t a n d a r d s I n s t i t u t i o n 2 0 1 2
Otherwise F ¶y is calculated as follows.
1) Calculate the value of f sh.
Values of K , l and s are defined in Figure 12 and A is taken from Table 8. If the value of s is greater than 0.3l then use
Then for spur and single helical gears:
or for double helical gears
The value of s for double helical gears is the distance to the centre of the helix which is nearer to the torqued end of the gear.
Table 8 — Auxiliary value, A
2) Calculate f ma. The value of fma depends on the manufacturing tolerance of the gears, the case and bearings and the bearing clearances.
For gears without helix modifications and without any adjustment on assembly, use:
where f H ¶ is the larger of the tooth alignment tolerances of the pinion and wheel given in Table 5 of BS 436-2:1970.
For gears with crowning or gears where the contact is adjusted on assembly
provided that this assumption is verified by inspection of the contact marking under light load.
For gears with suitable end relief
3) Calculate F ¶x from the equation:
The negative sign is to be used only if the gears are adjusted on assembly and if the contact pattern is inspected to justify the assumption.
The value of F ¶x is to be the maximum of i) the value from equation (45);
ii) 0.005 wm;
s = 0.3 l (39)
(40)
(41)
Gear pairs without crowning
or end relief
Gears with suitably chosen
crowning
Gear pairs with suitably chosen
end relief
mm· 4m/N mm·4m/N mm·4m/N
0.023 0.012 0.016
f ma= f H ¶ (42)
f ma= 0.5 f H ¶ (43)
f ma= 0.7 f H ¶ (44)
F ¶x= |1.33 f sh ± f ma| (45)
d C o p y : U n i v e r s i t y o f M a n c h e s t e r L i b r a r y , T h e U n i v e r s i t y o f M a n c h e s t e r L i b r a r y , 0 4 / 0 2 / 2 0 1 3 1 6 : 3 7 , U n c o n t r o l l e d C o p y , ( c ) T h e B r i t i s h S t a n d a r d s I
BS 436-3:1986
4) The value of qy is obtained from Figure 13. If the pinion and wheel material are different then:
5) Calculate the value of F ¶y from the equation:
17.2.5 For gears conforming to the basic rack profile specified in BS 436-1 or BS 436-2 and with 1.2u(!u1.9 average values of c * are:
c * = 20 N/(mm· 4m) for a steel/steel gear pair;
c * = 18.2 N/(mm· 4m) for a steel/SG cast iron gear pair;
c * = 16.8 N/(mm· 4m) for an SG cast iron/SG cast iron gear pair;
c * = 14.8 N/(mm· 4m) for a steel/grey cast iron gear pair;
c * = 11 N/(mm· 4m) for a grey cast iron/grey cast iron gear pair.
qy= (qy1+ qy2)/2 (46)
F ¶y= qyF ¶x (47)
Factor K
Shrink fit Key fit
a)
0.0 (s = 0)
0.0 (s = 0)
b)
0.0 (s = 0)
0.0 (s = 0)
c) 0.48 0.8
d) – 0.48 – 0.8
e) 1.33 1.33
f) – 0.36 – 0.6
g) – 0.6 – 1.0
Figure 12 — Constant K for calculation of f sh
L i c e n s e d C o p y : U n i v e r s i t y o f M a n c h e s t e r L i b r a r y , T h e U n i v e r s i t y o f M a n c h e s t e r L i b r a r y , 0 4 / 0 2 / 2 0 1 3 1 6 : 3 7 , U n c o n t r o l l e d C o p y , ( c ) T h e B r i t i s h S t a n d a r d s I n s t i t u t i o n 2 0 1 2
If greater accuracy is required use the procedure in Appendix E.
17.2.6 Calculate the value of K H ¶ from either:
17.3 Purpose of the transverse load factor for contact stress K H!
The transverse load factor for contact stress accounts for the mal-distribution of load down the tooth f lank due to profile and pitch deviations and tooth modifications.
17.4 Calculation of K H! For gears with
º
*< 2where f pe is the single pitch tolerance given in Table 3 of BS 436-2:1970.
For gears with
º
* W 2(48)
(49)
Figure 13 — Values of qy
(50)
d (51)
C o p y : U n i v e r s i t y o f M a n c h e s t e r L i b r a r y , T h e U n i v e r s i t y o f M a n c h e s t e r L i b r a r y , 0 4 / 0 2 / 2 0 1 3 1 6 : 3 7 , U n c o n t r o l l e d C o p y , ( c ) T h e B r i t i s h S t a n d a r d s I
BS 436-3:1986
Table 9 — Minimum and maximum values of K H!
17.5 Running-in allowance y!
For through hardened steels and cast steels:
when
For cast iron and bronze
when
For surface hardened steels
subject to y!u3 at any speed.
If the pinion and wheel are of different materials then