In our case

2.4.2. Determine the base number of stress cycles .

For steels = 4∙ 10⁶.

2.4.3. Let factor K_bE that reduces variable load conditions to the constant load equivalence be

K_bE = 1

2.4.4. Determine the equivalent number of cycles for the pinion and the gear.

;

.

2.4.5. Determine the durability factor for the pinion and the gear if:

N_bE ≥ N_b0 then K_bL=1,

N_bE < N_b0 then ,

where m=3 for toothed wheels with hardness H ≤ 350 BHN and m=9 if H > 350 BHN.

In our case: , then

, then

2.4.6. Determine safety factor S_b for the pinion and for the gear.

- for wheels made of forged blanks (our case) S_b = 1.75;

- for wheels made of cast blanks S_b = 2.3.

2.4.7. Determine the bending allowable stresses for the gear and the pinion

, .

In our case:

For further calculations we assume that the design allowable bending stress has less value of above calculated stresses .

3. STRENGTH CALCULATION OF THE STRAIGHT SPUR GEARS

Initial data: torque at the pinion shaft T^p =74 N× m; torque at the gear shaft T^g =370 N× m; velocity ratio of the gearing u=5; allowable contact stress [σ _H]=515 MPa; allowable bending stress [σ _b]=255 MPa; hardness of the gear material H^g=285 BHN, angular velocity of the gear shaft
ω ^g = 40 rad/sec.

3.1. Determine the centre distance of the straight spur gears

,

where the sign (“+”) is used for gears with external toothing as in our case; u is the velocity ratio of the spur gears; 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 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 1 depending upon disposition of toothed wheels with respect to bearings and factor ψ _bd= b^g/d^p. Since b^g and d^p are not determined we find this factor by the following formula

,

where gear face width factor ψ _ba is taken from table 3.1 2 depending upon the position of the gear relative to bearings, remembering that the value of ψ _ba should correspond to the standard. The greater ψ _ba the less overall dimensions of the gearing. That is why we select the greater value of ψ _ba.

In our case the gear is located symmetrically relative to support. That is why we take ψ _ba = 0.5, , K_Hβ =1.05.

Obtained magnitude of a_w is rounded up according to the series given in table 3.3. We assume a_w=180 mm.

Table 3.1

Approximate values of K_Hβ

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	1.08 1.22	1.17 1.44	1.28 -	- -	- -	- -
On cantilevers, roller bearings	up to 350 over 350	1.06 1.11	1.12 1.25	1.19 1.45	1.27 -	- -	- -
Symmetrical	up to 350 over 350	1.01 1.01	1.02 1.02	1.03 1.04	1.04 1.07	1.07 1.16	1.11 1.26
Non-symmetrical	up to 350 over 350	1.03 1.06	1.05 1.12	1.07 1.20	1.12 1.29	1.19 1.48	1.28 -

Recommended values of the gear face width factor ψ _ba

Table 3.2

Recommended values of the gear face width factor ψ _ba

Approximate values of K_Hβ

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	1.08 1.22	1.17 1.44	1.28 -	- -	- -	- -
On cantilevers, roller bearings	up to 350 over 350	1.06 1.11	1.12 1.25	1.19 1.45	1.27 -	- -	- -
Symmetrical	up to 350 over 350	1.01 1.01	1.02 1.02	1.03 1.04	1.04 1.07	1.07 1.16	1.11 1.26
Non-symmetrical	up to 350 over 350	1.03 1.06	1.05 1.12	1.07 1.20	1.12 1.29	1.19 1.48	1.28 -