Contact and bending limits of endurance

Heat treatment	Tooth hardness	Gear material
case	core and root
Normalizing, martempering	Brinell 180 to 350	Carbon and alloy steels, such as 45 (0.45C), 40X (0.40C-Cr), 40XH (0.40C-Cr-Ni), 50XH (0.50C-Cr-Ni), and 35XM (0.35C-Cr-Mo)
Full hardening	Rockwell C, 40 to 55	Carbon and alloy steels, such as 45 (0.45C), 40X (0.40C-Cr), 40XH (0.40C-Cr-Ni), and 35XM (0.35C-Cr-Mo)
Surface hardening	Rockwell C, 40 to 58	Rockwell C, 25 to 35	Alloy steels, such as 40X (0.40C-Cr), 40XH (0.40C-Cr-Ni), 50XH (0.50C-Cr-Ni), and 35XM (0.35C-Cr-Mo)
Case hardening	Rockwell C, 54 to 64	Rockwell C, 30 to 45	Alloy steels, such as 20XH2M (0.20C-Cr-2Ni-Mo)
Nitriding	Rockwell C, 50 to 60	Rockwell C, 24 to 40	Alloy steels, such as 40XH2MA (0.40C-Cr-2Ni-Mo, quality)		(of tooth core)

where n^p and n^g are correspondingly rotational speeds of the pinion and the gear; c is the number of gears meshing with the gear being analyzed.

In our case c = 1, n^p = 967 rpm; n^g = 241.75 rpm.

.

2.3.5. Determine the durability factor for the pinion and for the gear.

if N_Hi ≥ N_HO then K_HL=1,

if N_Hi < N_HO then .

In our case

, , ,

consequently

, , ,

consequently

2.3.6. Determine the safety factor S_H for the pinion and for the gear.

- for homogeneous structure of the material

(heat treatment is normalizing, martempering

and full hardening) S_H = 1.1;

- for heterogeneous structure of the material

(heat treatment is surface hardening,

case hardening, nitriding) S_H = 1.2.

2.3.7. Determine the contact allowable stresses for the gear and for the pinion

.

In our case:

.

If H^p-H^g 70BHN we assume as the design allowable contact stress the less magnitude of above calculated stresses, where H^p and H^g are correspondingly hardness of the pinion and gear materials.

Otherwise, the design allowable contact stress is determined by the following formula

.

In our case H^p-H^g < 70BHN. That is why for further calculations we take as the design allowable contact stress .

2.4. Determine the allowable bending stresses of the pinion and for the gear.

2.4.1. Determine the limits of the bending endurance for the pinion and for the gear . For this purpose we use table 2.1.

.

2.4.2. Determine the base number of stress cycles .

For steels = 4∙10⁶.

2.4.3. Determine the design number of stress cycles for the pinion and the gear.

;

.

2.4.4. Determine the durability factor for the pinion and for the gear.

if N_bi ≥ N_b0 then K_bL=1,

if N_bi < N_b0 then ,

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

In our case: , consequently

, consequently

2.4.5. Determine the 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.

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

, .

In our case:

For further calculations we assume as the design allowable bending stress the less magnitude of above calculated stresses .

3. ANALYSIS OF THE STRAIGHT SPUR GEARS FOR STRENGTH

Let us carry out the analysis of the straight spur gears for strength if 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 ]=515,455 MPa; allowable bending stress [σ _b ]=255,6 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 upper sign (“+”) is right for gears with external toothing and down sign (“-”) is right for gears with internal toothing; 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_{H β} 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 moduli 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_{H β} is determined by means of table 3.2 depending upon 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 the gear face width factor ψ _ba is determined from table 3.1 depending upon the disposition of the gear relative to bearings and taking into account that the value of ψ _ba should correspond to standard. The greater ψ _ba the less overall dimensions of the gearing.

In our case the gear is located symmetrically with respect to supports. That is why ψ _ba =0,5, and K_{H β}=1.11.

Obtained magnitude of a_w we round off to the nearest greater side according to the series given in table 3.3. We assume a_w =160 mm.

Table 3.1

Recommended values of the gear face width factor ψ _ba

Gear arrangement with respect to bearings	Tooth hardness	ψ ba
Symmetrical	Any	0.315; 0.4; 0.5
Non-symmetrical	Brinell BHN, up to 350 Rockwell C, 40 upwards	0.315; 0.4 0.25; 0.315
On shaft cantilevers	Brinell BHN, up to 350 Rockwell C, 40 upwards	0.25 0.2
For herringbone gears	Any	0.4; 0.5; 0.63
For internal gears	Any	0.2

Table 3.2

Approximate values of K_{H 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	1.08 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
Nonsymmetrical	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 -