Gearing accuracy of manufacturing

Table 3.6

Dynamic load factor K_HV

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 make calculations once more.

3.14. Determine the maximum bending stress

,

where K_bb is the load concentration factor that is determined according to table 3.7; K_bv is the dynamic load factor determined according to table 3.8; Y_b is the tooth shape factor that is determined by means of table 3.9 depending upon the number of gear teeth when the offset factor x=0.

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

Table 3.7

Approximate values of K_bβ

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 -
Non-symmetrical	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

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

4. STRENGTH CALCULATION OF THE HELICAL SPUR GEARS FOR

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 modulus of elasticity of pinion and gear materials respectively. Since the pinion and the gear are made of steel we may make the conclusion that E_tr = E^p = E^g = 2.1·10⁵ MPa.

Load concentration factor K_Hβ is determined according to table 3.2. This factor depends upon disposition of tooth wheels with respect to bearings and factor ψ _bd= b_g/d_p. Since b_g^g and d_p^p were not determined, we find this factor by the following formula:

,

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

Obtained value of a_w we round up according to the series given in table 3.3.

In this our case: T^g = 464300 N·mm; T^p = 161120 N·mm; u = 4;
[σ _H] = 640 MPa; E_tr = 2.1·10⁵ MPa; [σ _b] = 293.657 MPa.

From table 3.1 we take ψ _ba = 0.5; , and

K_Hβ = 1.073 (for symmetrical gear arrangement and tooth surface hardness up to 350MPa).

Thus

according to table 3.3 we take a_w = 125 mm for the 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. b^g = 0.5·125 = 62.5 mm.

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

,

where K_m is 5.8 for helical spur gears.

The obtained value of the module should be rounded up 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.

, round off up to m_n = 2mm

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 and in our case this condition is not satisfied.

That’s why we takem_n = 2.5mm, then

4.6. Determine the total number of teeth

..

The obtained value of z_Σ should be rounded off to the nearest integer number.

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

.

The value of this angle must be ranged from 8 to 18˚.

4.8. Determine the number of teeth of the pinion

,

where z_min=17· for helical spur gears.

The obtained value of z^p should be rounded off to the nearest integer number. 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 then or equal to 4%. Otherwise the number of teeth z^p, z^g and z_Σ must be rounded down.

In our case the condition is satisfied, as

; .

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

,

d^g = 2· a _w - d^p = 2· 125 – 50.5 = 199.5 mm.

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

,

.

4.13. Determine the dedendum circles 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 developed in the contact zone of teeth

,

where Z_Hβ takes into account rising 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 in table 3.2 and depends upon .

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

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

The dynamic load factor K_HV is specified in 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.

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

Table 4.1