Gearing accuracy of manufacturing

Types of gear drives	Peripheral speed V, m/sec
under 5	5 - 8	8 – 12.5	over 12.5
Straight spur gears
Helical spur gears
Straight bevel gears			-	-
Spiral bevel gears

Table 3.6

Dynamic load factor K_HV

Gearing accuracy of manufacturing	Tooth surface hardness, BHN	Peripheral speed V, m/sec
	up to 350 over 350	1.04/1.02 1.03/1.00	1.07/1.03 1.05/1.01	1.14/1.05 1.09/1.02	1.21/1.06 1.14/1.03	1.29/1.07 1.19/1.03	1.36/1.08 1.24/1.04
	up to 350 over 350	1.04/1.01 1.03/1.01	1.08/1.02 1.06/1.01	1.16/1.04 1.10/1.02	1.24/1.06 1.16/1.03	1.32/1.07 1.22/1.04	1.40/1.08 1.26/1.05
	up to 350 over 350	1.05/1.01 1.04/1.01	1.10/1.03 1.07/1.01	1.20/1.05 1.13/1.02	1.30/1.07 1.20/1.03	1.40/1.09 1.26/1.04	1.50/1.12 1.32/1.05

Note: The figures in the numerators refer to straight spur gears and those in the denominators, to helical spur gears.

Obtained value of σ _H should correspond to the following condition:

σ _H = (0.8…1.1)∙[σ _H ].

Otherwise it is necessary to change the center distance a _w and recalculate the gearing.

In our case 412,4≤488,78≤567. Condition is satisfied.

3.14. Determine the maximum bending stress

where K_{b b} is the load concentration factor that is determined by table 3.7; K_bv is the dynamic load factor determined from table 3.8; Y_b is the tooth form factor that is determined by means of table 3.9 depending upon the number of teeth of the gear for the case when the offset factor x =0

If obtained magnitude of σ _b >[σ _b ] it is necessary to increase the module.

Table 3.7

Approximate values of K_{b β}

Gear arrangement with respect to bearings	Tooth surface hardness, BHN
0.2	0.4	0.6	0.8	1.2	1.6
On cantilevers, ball bearings	up to 350 over 350	1.16 1.33	1.37 1.70	1.64 -	- -	- -	- -
On cantilevers, roller bearings	up to 350 over 350	1.10 1.20	1.22 1.44	1.38 1.71	1.57 -	- -	- -
Symmetrical	up to 350 over 350	1.01 1.02	1.03 1.04	1.05 1.08	1.07 1.14	1.14 1.30	1.26 -
Nonsymmetrical	up to 350 over 350	1.05 1.09	1.10 1.18	1.17 1.30	1.25 1.43	1.42 1.73	1.61 -

Table 3.8

Dynamic load factor K_bV

Gearing accuracy of manufacturing	Tooth surface hardness, BHN	Peripheral speed V, m/sec
	up to 350 over 350	1.08/1.03 1.03/1.01	1.16/1.06 1.05/1.02	1.33/1.11 1.09/1.03	1.50/1.16 1.13/1.05	1.62/1.22 1.17/1.07	1.80/1.27 1.22/1.08
	up to 350 over 350	1.10/1.03 1.04/1.01	1.20/1.06 1.06/1.02	1.38/1.11 1.12/1.03	1.58/1.17 1.16/1.05	1.78/1.23 1.21/1.05	1.96/1.29 1.26/1.08
	up to 350 over 350	1.13/1.04 1.04/1.01	1.28/1.07 1.07/1.02	1.50/1.14 1.14/1.04	1.72/1.21 1.21/1.06	1.98/1.28 1.27/1.08	1.25/1.35 1.34/1.09

Note: The figures in the numerators refer to straight spur gears and those in the denominators, to helical spur gears.

Table 3.9

Tooth form factor Y_b

z or zv
Y b	4.27	4.07	3.98	3.92	3.88	3.81	3.8	3.75	3.7	3.66	3.65	3.62	3.61	3.6

4. ANALYSIS OF THE HELICAL SPUR GEARS FOR STRENGTH

4.1. Determine the center distance of the helical spur gears

,

where u is the velocity ratio of the gearing; T^g is the torque at the gear shaft in N·mm; [ σ_H ] is the allowable contact stress in MPa; E_tr is the transformed modulus of elasticity in MPa; K_Hβ is the load concentration factor; ψ_ba= b^g/a_w is the gear face width factor.

Transformed modulus of elasticity E_tr is determined as

,

where E^p and E^g are correspondingly modules of elasticity of pinion and gear materials. Since the pinion and the gear are made of steel we can make the conclusion that E_tr = E^p = E^g = 2.1·10⁵ MPa.

Load concentration factor K_Hβ is determined by means of table 3.2 depending on disposition of toothed wheels with respect to bearings and factor ψ_bd= b_g/d_p. Since b_g and d_p were not determined we find this factor by the following formula

,

where gear face width factor ψ_ba is determined from table 3.1 depending on the disposition of the gear relative to bearings taking into account that the value of this factor should correspond to standard. The greater ψ_ba the less overall dimensions of the gearing. That is why we select the greater magnitude of ψ_ba.

Obtained magnitude of a_w should be rounded off to the nearest greater side according to the series given in table 3.3.

In our case: T^g = 464.3 N·m; T^p = 119.67 N·m; u = 4;
[σ_H] = 515,5 MPa;E_tr = 2.1·10⁵ MPa;[σ_b] = 255.6 MPa, symmetrical disposition of gears with respect to supports.

From table 3.1 we take ψ_ba = 0.5; then ,

From table 3.2 we take K_Hβ = 1.07 (for symmetrical gear arrangement and tooth surface hardness up to 350).

Than

Round off obtained magnitude of to the nearest greater side according to table 3.3. That is why we take a_w = 140 mm for further calculations.

4.2. Determine the nominal pitch circle diameter of the gear

.

4.3. Determine the face width of the gear

b^g = ψ_ba·a_w = 0.5·140 = 70 mm.

4.4. Determine the normal module according to the strength condition for bending

,

where K_m is taken as 5.8 for helical spur gears.

Obtained magnitude of the module should be rounded off to the greater side according to the standard series given in table 3.4. It is necessary to note that for general-purpose speed reducers the minimum value of the module is m_min = 2 mm, in our case

,

we take m_n = 2 mm for further calculations.

4.5. Determine the helix angle

.

For helical spur gears this angle should be ranged from 8 to 18˚. Otherwise, it is necessary to change the normal module m_n. As in our case that is less than , that’s why we take m_n= 3mm, than

4.6. Determine the total number of teeth

.

Obtained value of z_Σ should be rounded off to the nearest integer numeral, assume z_Σ = 92.

4.7. Specify the helix angle according to the integer number of z_Σ

Obtained value of this angle is ranged from 8 to 18˚. Condition is satisfied.

4.8. Determine the number of teeth of the pinion

,

where for helical spur gears z_min=17· .

Obtained value of z^p should be rounded off to the nearest integer numeral. If z^p <17· it is necessary to decrease the module or to use nonstandard toothed wheels.

In our case

4.9. Determine the number of teeth of the gear

.

4.10. Specify the velocity ratio of the gearing

u_act= .

The error ε = should be less or equal to 4%. Otherwise the number of teeth z^p, z^g and z_Σ must be rounded off to the other side.

In our case condition is satisfied, since

;

4.11. Determine the nominal pitch circle diameters for the pinion and the gear

,

d^g = 2· a _w - d^p = 2·140 – 54.78 = 225.22 mm.

4.12. Determine the addendum circle diameters for the pinion and the gear

,

.

4.13. Determine the dedendum circle diameters for the pinion and the gear

,

.

4.14. Determine forces that act in the engagement of the helical spur gears:

- turning force ;

- radial force ;

- axial force

where α_w=20˚ is the pressure angle for the pitch circle.

4.15. Determine the maximum contact stress that develops in the contact zone of teeth

,

where Z_Hβ takes into account rising the contact strength of the helical spur gears in comparison with the straight spur gears; T^p is the torque at the pinion shaft in N·mm; K_H is the design load factor that is determined as

,

where K_Hβ is the load concentration factor; K_HV is the dynamic load factor.

The load concentration factor K_Hβ is specified by table 3.2 depending upon .

In order to determine K_HV it is necessary to find the peripheral speed V^g of the gear

where ω^g is the angular velocity of the gear and the gearing accuracy of manufacturing (table 3.5),.

The dynamic load factor K_HV is determined by table 3.6.

Factor Z_Hβ is determined in the following way

,

where K_Hα takes into account non-uniform load distribution between several pairs of teeth; ε_α is the contact ratio.

Table 4.1

Factors K_Hα, K_bα that take into account non-uniform load distribution between some pairs

Peripheral speed V, m/sec	Accuracy of manufacturing	KHα	Kbα
To 5		1.03	1.07
		1.07	1.22
		1.13	1.35
From 5 to 10		1.05	1.2
		1.10	1.3
From 10 to 15		1.08	1.25
		1.15	1.40

K_Hα depends upon the accuracy of manufacturing and the peripheral speed and is determined according to table 4.1.

Contact ratio ε_α is found by the following formula

.

Obtained value of σ_H should correspond to the following condition:

σ_H = (0.8…1.1)·[σ_H ].

Otherwise it is necessary to change the center distance a_w and recalculate the gearing.

In our case: ; K_Hβ = 1.07;

;

Accuracy of manufacturing gear drive is 9; K_H = 1.07·1.01= 1.0807;

; K_Hα = 1.13;

;

;

Obtained value of σ _H should correspond to the following condition:

σ _H = (0.8…1.1)∙[σ _H ].

Otherwise it is necessary to change the center distance a _w and recalculate the gearing.

In our case 412,4≤483,287≤567. Condition is satisfied.

4.16. Determine the maximum bending stress

,

where K_bβ is the load concentration factor that is determined by table 3.7; K_bv is the dynamic load factor determined from table 3.8; Y_b is the tooth form factor that is determined by means of table 3.9 depending on the number of teeth of the equivalent straight spur gear for the case when the shift factor x=0.

Factor Z_bβ is the analogy of Z_Hβ and is determined as

,

where K_bα is chosen from table 4.1; is the correction factor.

If obtained magnitude of σ_b > [σ_b] it is necessary to increase the module.

In our case:K_bβ = 1.14; K_bv = 1.07; ;

Y_b = 3.61; K_bα = 1.35; ; ;

.

Strength condition is satisfied.

5. ANALYSIS OF THE BEVEL GEARS FOR STRENGTH

Let us analyze the bevel gears for strength if torque at the pinion shaft T^p =460 N×m; torque at the gear shaft T^g =153 N×m velocity ratio of the gearing u =3.15; allowable contact stress [σ _H ]=640 MPa; allowable bending stress [σ _b ]=293 MPa, hardness of the gear material H^g =285 BHN.

5.1. Determine the external pitch diameter of the gear

,

where T^g is the torque at the gear shaft in N·mm; E_tr is the transformed modulus of elasticity; K_{H β} is the load concentration factor; u is the velocity ratio; ν _H = 0.85 is the correction factor that takes into account reducing bevel gears strength in comparison with spur gears; [σ _H ] is the allowable contact stress; ψ _bR = b^g/R_e is the gear face width factor that determines proportions of the face width of the gear with respect to the external cone distance. Factor ψ _bR must be less than 0.3. Recommended value of ψ _bR = 0.285.

Since both pinion and gear are made of steel, the transformed modulus of elasticity E_tr =2.1·10⁵ MPa.

Load concentration factor K_{H β} depends upon the hardness of the gear material. If H^g £350 BHN K_{H β} is ranged from1.23 to 1.35. Otherwise (H^g >350 BHN) K_{H β} is ranged from 1.25 to 1.45. It is necessary to note that greater values of K_{H β} are assumed for the case when one of toothed wheels is on the cantilever shaft.

After calculation the obtained magnitude of should be rounded off to the greater side according to standard series given in table 5.1. In our case we assume =315 mm.

Table 5.1

Standard values of the external pitch diameter

Series 1

Series 2

-

-

Note. Series 1 should be preferred to Series 2.

5.2. Determine pitch angles for the pinion and for the gear.

d₂ = arctg u = arctg 3.15=72.38°, d₁ = 90°- d₂=90-72.38=17.62°.

5.3. Determine the external cone distance

_.

5.4. Determine the face width of the gear

b^g = × R_e =0.285∙165.25=47.1 mm.

5.5. Determine the external module

,

where ν _b = 0.85 is the correction factor; K_{b β} is the load concentration factor that is determined according to table 3.7 depending upon ψ _bd factor, where the latter is found as

.

5.6. Determine the number of teeth of the gear

and round off to the integer numeral. Assume .

5.7. Determine the number of teeth of the pinion

and round off to the integer numeral too. In our case

5.8. Specify the velocity ratio of the gearing

u_act = .

The error ε = should be less or equal to 4%. Otherwise, we should round off values of z^p and z^g to the other side.

In our case ε = =

5.9. Specify pitch angles for the pinion and the gear

d₂ = arctg u_act =arctg 3.12=72.23°, d₁ = 90°- d₂=17.77°

5.10. Determine external pitch diameters of the pinion and the gear.

, .

5.11. Determine diameters of addendum circles at the outer section for the pinion and the gear

=101.05+2∙2.35∙cos17.77°=105.53 mm, =314.91+2∙2.35∙cos72.23°=316.33 mm.

5.12. Determine diameters of dedendum circles in the outer section for the pinion and the gear.

=101.05–2.4∙2.35∙cos17.77°=95.68 mm, =314.9–2.4∙2.35∙cos72.23°=313.18 mm.

5.13. Specify the external cone distance

.

5.14. Specify the face width of the gear

=0.285∙165.36=47.28 mm.

5.15. Determine mean pitch diameters for the pinion and for the gear

, .

5.16. Determine forces that act in the engagement of the bevel gears

- turning force ;

- radial force at the gear

=3407.03∙tg20°∙cos72.23°=378.46 N;

- axial force at the gear

=3407.03∙tg20°∙sin72.23°=1180.89 N.

5.17. Determine the maximum contact stress that develops in the contact zone of teeth:

where T^p is in N·mm; K_H is the design load factor determine as

K_H=K_{H β}· K_HV.

Load concentration factor K_Hβ is specified by means of table 3.2 depending upon factor ψ_bd= .

Dynamic load factor K_HV is determined according to table 3.6 depending upon the peripheral speed of the gear (V^g = ) and the accuracy of manufacturing (table 3.5). In order to use table 3.6 for bevel gears we should reduce the degree of accuracy by 1.

In our case ψ _bd = , V ^g= m/sec. K_H=K_{H β}· K_HV =1.16∙1.11=1.29.

Obtained value of σ _H should correspond to the following condition

σ _H =(0.8…1.1)·[σ _H ]=(0.8…1.1)·640=512…704 MPa.

Otherwise it is necessary to change the external pitch diameter and recalculate the gearing. In our case strength condition is satisfied.

5.18. Determine the maximum bending stress

where K_{b b} is the load concentration factor that is determined by table 3.7; K_bV is the dynamic load factor determined from table 3.8 (for bevel gears we should reduce the accuracy of manufacturing by 1); Y_b is the tooth form factor that is determined by means of table 3.9 depending upon the number of teeth of the equivalent straight spur gear for the case when the offset factor x =0;
ν _b = 0.85 is the correction factor; m_m = is the mean module.