Gear. Durability of spur gears

System of nomograms - graphic-analytical calculation of gear transmission for contact strength

In this calculation of the gear for strength, the following condition must be satisfied:   s_H < [s_H],

where   s_H   - contact pressure,  [s_H] - admitted pressure

Hertz formula (compression of cylinders along generatrices):

s_H = Z_E · [w_n/r_пр] < [s_H],                                                  [1]

where

Z_E - the factor which is taking into account mechanical properties of materials spur gear and a wheel;

w_n - normal loading on unit of length of contact lines;

1/r_пр - reduced curvature.

So then,

For steel cogwheels Z_E = 190 МПа^1/2 , Е₁ = Е₂ = 2.1 · 10⁵ МПа, n₁ = n₂ = 0.3.

Total length of contact lines L_S:

L_S ⁼ b_w / Z_e²,                                                                       [5]

where Z_e - coefficient taking into account the total length of contact lines:

Z_e = [(4 - e_a)/3]^0.5,                                                              [6]

where e_a- end overlap coefficient, equal to the ratio of the angle of rotation of the gear - from the position of the engagement of the end profile of its tooth to the disengagement, to the angular step 2·p/z. For gears without displacement of the original profile, the approximate value of the overlap coefficient is calculated by the formula:

e_a= 1.88 - 3.2 · (1/Z₁ ± 1/Z₂),                                               [7]

where Z1 и Z2 - number of gear and wheel teeth.

When the overlap coefficient changes from 1.25 to 1.9 the Z_e coefficient changes from 0.84 to 0.96. For approximate calculations, take the average value Z_e = 0.9 if e_a=1.6.

We express the force normal to the tooth surface through the mathematical ratio of the circumferential component of the force directed along the X axis and the cosine of the angle between the direction of the force and its circumferential component:

F_n = F_t / cos a_t ,                                                        [8]

where  F_t - circumferential force component:

F_t =2 · 10³ T / d ,                                                    [9]

where Т - transmitted torque, N · м; d - pitch diameter, mm.

Taking into account the formulas [5]  and  [7] formula for determining the normal load per unit length of contact lines [3] will take the form:

 w_n = К_Н · F_t · Z_e / [b_w · cos a_t],                                          [10]

Radii of curvature of gear tooth profiles and gear wheel:

r₁=0.5  · d_w1  · sin a_tw,                                                       [11]

r₂=0.5  · d_w2  · sin a_tw,                                                       [12]

Accept:

d_w = d  ·  cos a_t / cos a_tw,

Then the formula for calculating the reduced radius of curvature[4] takes the form:

1/r_пр = [2 ·  (i ±1)]/ [d₁ · i · cos a_t  · tg a_tw].                       [13]

 We get a formula for calculating spur gears without shifting the original contour  (a_tw = a_t = a)  for contact strength: 

s_H = Z_E · Z_e · Z_Н · [(К_Н · F_t / d₁ · b_w)  · (i ± 1)/u]^0.5  < [s_H], [14]

where i - gear ratio, d₁- pitch diameter, Z_Н - coefficient taking into account the shape of the mating surfaces of the teeth.

Z_Н = (1/cos a_t) · (2/tga_tw)^0.5                                                                  [15]

For gears without displacement of the original contour Z_Н =2.5.

In the formula, [13] all components are selected or calculated during the design calculation. Only the load factor remains unknown К_н. So, the load factor used to calculate the contact strength is determined by the formula: 

К_Н = К_НА · К_Нv · К_Нb · К_Нa ,                                               [16]

where

К_НА - external load factor,

К_Нv - internal dynamic load factor,

К_Нb - coefficient of concentration or uneven load along the length of the contact line,

К_Нa - load distribution ratio between the teeth.

Allowed for preliminary calculations К_Н = 1.3 ... 1.5. Smaller values are chosen for more accurate gears and when moving away from supports, larger values for less accurate gears and when located near supports

Consider the listed four coefficients used in the calculation of gears for strength. 

External dynamic load factor К_НА:

- under uniform loading - 1;

- with little unevenness - 1.25;

- with medium unevenness - 1.5;

- with significant unevenness - 1.75.

External dynamic load factor К_Нv depending on the circumferential speed, degree of accuracy and hardness of the teeth:

Note - The numerator is for hardened gear teeth, the denominator is for non-hardened gear teeth. 

The coefficient of concentration or uneven load along the length of the contact line of the gear К_Нb of the gear for teeth with a hardness of less than 350 HB is in the range of 1 ... 1.35, for teeth with a hardness of more than 350 HB is in the range of 1 ... 1.5. A value of 1 is only possible in spur gears with perfectly precise manufacturing and absolutely rigid shafts and bearings. The concentration of the load is the greater, the greater the misalignment of the shafts of the gears, the greater their width and the lower the rigidity of the gears, etc. There are methods that allow obtaining a relatively accurate value of this coefficient, taking into account all of the above factors. However, it is rather difficult to take everything into account in the design calculation, therefore it is proposed to preliminarily take for the first case of reduced hardness К_Нb=1.2 and for the case of increased hardness К_Нb=1.3.

When calculating the gear transmission, the coefficient of load distribution between the teeth К_Нa for the average case is calculated by the formula:

К_Нa = 1 + 0.06 ·  (n_ст - 5),                                    [17]

where n_ст - wheel precision.

When the hardness of the wheel and gear is more than 350HB, they accept К_Нa = 1.

 System of nomograms - graphic-analytical calculation of gear transmission for contact strength

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