4. 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 =59.822 Nm; torque at the gear shaft T^g =209.192 N∙m; velocity ratio of the gearing u=3.55; 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= 28.54 rad/sec.

4.1. 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_Hβ is the load concentration factor; ψ_ba= b^g/a_w is the gear face width factor.

Transformed modulus of elasticity E_tris determined as

,

whereE^pand E^gare 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∙10⁵MPa.

In our case the gear is located symmetrically with respect to supports. That is why ψ_ba=0,4, and K_Hβ=1.04.

We round off obtained magnitude of a_wto the nearest side according to the standard series. We assume a_w=125 mm.

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=0.4∙125=50 mm.

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

,

whereK_mis taken as 6.8 for straight spur gears.

Obtained magnitude of the module should be rounded off to the greater side according to the standard series. 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 assume that m = 2 mm.

4.5. Determine the total number of teeth

4.6. Determine the number of teeth of the pinion

,

wherez_min=17 for straight spur gears.

For our case

.

Round off z^pto the nearest integer numeral. Assumez^p=27.

4.7. Determine the number of teeth of the gear

Upper sign is right for gears with external toothing and down sign is used for gears with internal toothing.

4.8. Specify the velocity ratio of the gearing

u_act= .

The error ε = should be less or equal to 4%.

In our case condition is satisfied, because

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

d^p= m· z^p=2∙27=54 mm,

d^g = 2·a_w d^p=2∙125-54=196 mm.

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

= d^p+ 2·m=54+2∙2=58 mm,

= d^g ∙ 2·m=196+2∙2=200 mm.

4.11. Determine the dedendum circles diameters for the pinion and the gear

= d^p-2.5·m=54-2.5∙2=49 mm,

= d^g 2.5·m=196-2.5∙2=191 mm.

4.12. Determine forces that act in the engagement of straight spur gears:

- turning force ;

- radial force ,

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

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

whereT^p is the torque at the pinion shaft in N∙mm; K_His the design load factor that is determined as K_H=K_H∙K_HV, where K_H is the load concentration factor; K_HV is the dynamic load factor.

The load concentration factor K_His specified depending upon ψ_bd= . That is why K_H=1.04.

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

m/sec,

whereω^g is the angular velocity of the gear and the gearing accuracy of manufacturing.

In our case K_HV=1,24

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≤521.882≤567. Condition is satisfied.

4.14. Determine the maximum bending stress

where K_b is the load concentration factor; K_bv is the dynamic load factor; Y_b is the tooth form factor that is determined depending upon the number of teeth of the gear for the case when the offset factor x=0