Gear. Gear calculation.
Geometrical parameters of cylindrical cogwheels (non corrected)

The standardized parameters characterizing the gear train are:

- tooth module;

- ratio;

- center distance.

The values of these parameters are given in Table 1, as well as links to nomogram systems for graphic-analytical calculation of the main geometric parameters of the gear pair:

- number of teeth of the wheel and gear;

- pitch diameters (initial diameters);

- tooth tip diameters;

- tooth root diameters.  

Table 1. Cylindrical gear - standardized parameters

Gear calculation formulas

Gear ratio

i = Z2 / Z1, 

where  Z2 > Z1  and accordingly  i > 1,  for low-speed gear -  i = 5.6...6.3,  for high-speed gear -  i = 6.3...8. In the gearboxes of machine tools i<4.

Factor of width of cogwheels 

f_a = b / A

Step of a cogwheel

p = 3.14 · m

Initial diameters of cogwheels -  d_w1  и  d_w2. 

District dividing step  p_t

p_t = 3.14 · d / z

Circular module m_t

m_t = d / z

Table 3. Gear parameters

Spur gears with offset initial contour

Consider the correction of spur gears in accordance with the standards that were in effect before 1970.

Correction is recommended for cylindrical and bevel gears, provided that the number of gear teeth is not equal Z₁ and wheels Z₂.

Correction is most appropriate in the following cases:

1) the gear has a small number of teeth (Z₁<17), since in this case the undercut at the root of the tooth is eliminated,

2) at large gear ratios, as it allows to reduce the relative slip of the profiles. 

In the manufacture of corrected wheels, the initial rail is displaced in the direction from the axis of rotation of the wheel - positive displacement, and in the direction of the axis - negative displacement.

Altitude correction

The outer diameter of the gear increases by 2·х·m (positive offset), the outer diameter of the wheel is reduced by 2·х·m (negative offset). Meaning х selected according to the table. With a positive offset, the length of the head of the tooth increases, the length of the stem decreases accordingly. With a negative bias, vice versa. Pitch diameters and spacing remain unchanged.

The amount of displacement of the initial contour of the wheel is indicated in the table on the drawing of the gear

When correcting a gear, the size of the tooth is calculated when measured by a constant chord s_x (tooth thickness at the pitch circle) and tooth height h_x. For positive displacement s_x increases, the tooth thickens at the base and hardens.

The values of the bias coefficients Х₁=-Х₂ are selected based on the materials of the sources of the 1930s. (at profile angle of the original contour 20⁰ and the tooth head height factor equal to 1)

 Corner correction

Recommended subject to: [Z₁+Z₂]<35, and also, if necessary, fit into the required center distance. Center distance and engagement angle during angular correction differ from normal values (uncorrected engagement). 

The calculation is made in the following order:

1. Determining the depth of entry

h_з= A - m · ((Z₁+Z₂)/2 + Xc - 2),

where Хс - is the total shift: Хс = Х₁ + Х₂. Values of the total shift Хс, the shift of the original gear profile Х₁ and the engagement angle  are determined using a table or a nomogram (not shown) depending on the number of teeth of the gear and wheel Z₁, Z₂, and are within Хс - в пределах 0.3...0.94,   Х₁ - 0.274...0.468, engagement angle - 20⁰45'...27⁰12' (values increase as number of teeth decreases).

2. We find the diameter of the circle of the protrusions according to the formula:

- for gear - D_1e = 2·m · (Z₁/2 + X₁ - 1) + 2·h_з.

- for wheel - D_2e = 2·m · (Z₂/2 + X₂ - 1) + 2·h_з.

3. Next, the thickness and height of the tooth is determined - s_x, h_x. The calculation uses tabular data (not given in the text) - the values of the involute (inv) of the pressure angle.

As mentioned above, angular correction is also applied when it is necessary to fit into a certain gear center distance. 

In this case:

1. Determine the actual engagement angle

cos a = 0,93969 ·  m ·  (Z1 + Z2) / 2 · A.

2. According to the table (not given in the text), we find its involute. 

3. We determine the total profile shift by the formula:

Хс= (inv a - 0,014904)  · (Z₁ + Z₂)/0,72974.

4. We calculate the shift of the original gear profile

Х₁ = Х_с · (0,748 - 0,017 · Z₁) / [1,496 - 0,017 · (Z₁ + Z₂)],

and shift of the original wheel profile

Х₂ =Х_с - Х₁.

Next, the above three steps of calculating the angular correction are performed, during which the values of the quantities are determined: h_з, D_1e, D_2e, s_x, h_x.

Consider the correction of spur gears in accordance with the standards in force after 1970.

The original contour correction is applied under the condition: Z₁>11 and the total number - Zс>29.

Corner correction

With a total displacement other than zero, the center distance of the gear train changes by:

y · m = (X1+X2- Dy) · m,

where y- is the coefficient of the perceived (realized) bias, and Dу - equalization bias coefficient determined by the nomogram (not given in the text). 

Recommended values for the shift of the original contour:

1. With an unspecified center distance (in power gears)

- if  9<Z₁<30   -   Х₁ = Х₂ = 0,5,

- if Z₁>30        -   Х₁ = Х₂ = 0.

2. At a given center distance (in power transmissions)

- if  Z₁>20                            -   Х₁ = Х₂ = 0.

- if 13<Z₁<21  и   i > 3,4      -   Х₁ = Х₂ =0.

3. In kinematic pairs:

- if  Z₁>20                               -   Х₁ = Х₂ = 0.

- if 11<Z₁<17  и   Z₂ > 21       -   Х₁ =0,3    Х₂ = -0,3.

In the machine tool industry, the following formula is used to determine the displacement coefficient of the original contour

х = 0,0061 · (100 - Z).

  We present the following table to determine the maximum bias factors - Х₁ и Х₂ under the following conditions:

1) the greatest contact strength;

2) highest bending strength;

3) the greatest wear resistance and the greatest seizing resistance.

Helical spur gears

In developing.

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