Factors KHα, Kbα that take into account non-uniform load distribution between some pairs of teeth

Contact ratio ε _α is found by the following formula

.

The obtained value of σ _H should meet 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 = 1.072;

;

Gear drive accuracy of manufacturing is 9; K_H= 1.072·1.01= 1.083;

K_Hα = 1.13;

;

;

so the strength condition is satisfied.

4.16. Determine the maximum bending stress

,

where K_bβ is the load concentration factor that is determined according to table 3.7; K_bv is the dynamic load factor specified in table 3.8; Y_b is the tooth shape factor that is determined in table 3.9; it depends 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 value of σ _b > [σ _b] it is necessary to increase the module.

In our case: K_bβ = 1.155; K_bv = 1.02; ;

Y_b = 3.61; K_bα = 1.35; ;

;

.

Strength condition is satisfied.

5. STRENGTH CALCULATION OF THE BEVEL GEAR

Initial data: torque at the gear shaft T^g =460 N× m; velocity ratio of the gearing u=3; allowable contact stress [σ _H]=620 MPa; allowable bending stress [σ _b]=168 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_Hβ 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 the 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_Hβ depends upon the hardness of the gear material. If H^g£ 350 BHN, K_Hβ is ranged from1.23 to 1.35. Otherwise (H^g> 350 BHN) K_Hβ is ranges from1.25 to 1.45. It is necessary to note that greater values of K_Hβ are intended for the case when one of tooth wheels is on the cantilever shaft. Let us take K_Hβ = 1.3

The obtained value of should be rounded up 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

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

d₂ = arctg u= arctg 3=71°36ʹ ’,

d₁ = 90°- d₂=90-71.6=18°24ʹ ’.

5.3. Determine the external cone distance

_.

5.4. Determine the face width of the gear

b^g = × R_e=0.285∙ 165.98=47.3 mm.

5.5. Determine the external module

,

where ν _b = 0.85 is the correction factor; K_bβ is the load concentration factor that is determined according to table 3.7 and depends upon ψ _bd factor, where the latter is found as

.

Let us take K_Bβ = 1.32 (for gear arrangement on cantilevers, mounted on roller bearings).

5.6. Determine the number of the gear teeth

and round off to the integer number. In our case .

5.7. Determine the number of the pinion teeth

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

5.8. Specify the velocity ratio of the gearing

u_act= .

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

In this case ε = =

5.9. Specify pitch angles for the pinion and the gear

d₂ = arctg u_act=arctg 3.04=71°48’ʹ,

d₁ = 90°- d₂=18°12’ʹ

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

=103.74+2∙ 3.99∙ cos18°12’=111.32 mm, =315.21+2∙ 3.99∙ cos71°48’=317.70 mm.

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

=103.74–2.4∙ 3.99∙ cos18°12’=94.64 mm, =315.21–2.4∙ 3.99∙ cos71°48’=312.22 mm.

5.13. Specify the external cone distance

.

5.14. Specify the face width of the gear

=0.285∙ 165.92=47.23 mm.

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

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

- turning force

N;

- radial force at the gear

=3108∙ tg20°∙ cos71°48’=353.3 N;

- axial force at the gear

=3108∙ tg20°∙ sin71°48’=1074.4 N.

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

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

K_H=K_Hβ · K_HV.

Load concentration factor K_Hβ is specified according to table 3.2 which depends upon factor ψ _bd= .

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

In this case ψ _bd= , V^g= m/sec. K_H=K_Hβ · 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)· 620=496…682 MPa.

Otherwise it is necessary to change the external pitch diameter and make calculation once more.

5.18. Determine the maximum bending stress

,

where K_bb is the load concentration factor defined according to table 3.7; K_bv is the dynamic load factor given in table 3.8 (for bevel gears we should reduce the degree of accuracy by 1); Y_b is the tooth shape factor that is defined according to table 3.9 depends 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.

6. ANALYSIS AND DESIGN OF SHAFTS

6.1. Find the minimum diameter of speed reducer shafts

,

where T is the torque at the shaft is measured in N·mm; [τ ] is the allowable torsion stress in MPa.

In order to compensate action of bending stresses, the allowable tangential stress is considered as down rated. For steels [τ ] = 15…20 MPa.

The obtained value of d_min is rounded up according to the following standard series: 20, 21, 22, 23, 24, 25, 26, 28, 30, 32, 34, 36, 38, 40, 42, 45, 48, 50, 52, 55, 58, 60, 65, 70, 75, 80, 85, 90, 95, 100, 105, 110, 115, 120, 130, 140, 150.

In general-purpose speed reducers the stepped shafts with solid cross-section are used as a rule.

For the input shaft d_min is the diameter of the shaft cantilever portion where such elements as a half coupling, a pulley, a sprocket or a pinion may be mounted (Fig. 6.1, 6.3). In order to fix the above mentioned elements in the axial direction we use a shoulder which height t₁ may be ranged from 2 to 5 mm depending on the shaft diameter. Recommended values of t₁ are given in table 6.1.

Fig.6.1. Input shaft

The next shaft section of diameter d₂=d₁+2∙ t₁ (the value of d₂ must correspond to standard series) is for seal installation. Seals are used to prevent bearing assemblies from dust and dirt accumulation and lubrication leakage from bearings. For general-purpose speed reducer commercial seals are used more frequently used.

In order to reduce friction at the point of seal contact with the shaft, the corresponding section should be polished. For this purpose this section is additionally surface hardened to 45-50 HRC.

Table 6.1

Recommended values of t ₁ and t ₂