What is a gear module? How to Calculate Gear Modulus

What is a gear

A gear is a small wheel with teeth that is attached to a special rotating axis. The surface of the gear in this case can be either conical or cylindrical.

Gear transmissions also have their own classification:

Straight teeth. The most common type of gears, the teeth of which are often located in radial planes.
Beveled. In another way, this type is also called helical, and its use is common in gas and electric tools. In relation to the rotating axis they are at a certain angle.
Worm-shaped. They are also called helical gears and are used primarily for steering a car.
Screw. They often have the shape of a cylinder and are also located along the entire line of the screw. Such gears are located on shafts that are perpendicular to the rotating axis.

These varieties are the most common, but far from the only ones, so the type used directly correlates with what function it will have to perform.

Moreover, each gear has a certain number of teeth, which is determined by its purpose. The difference between the number of teeth used is necessary, since thanks to this factor it becomes possible to regulate the shaft speed and torque. Gears are also divided into driving and driven. The driving gear is the gear to which the torque is supplied from the outside, and the driven gear is the one from which it is removed.

Why is the gear called that?

Technically this is understandable. Initially, a “pinion” is the smallest wheel in a gear train. There are no fewer than six teeth there, even in theory; grip is not ensured. ... In mechanical engineering, the driven wheel of a gearbox is called a wheel.”

Rack

A rack is part of a gear with an infinite pitch circle radius. As a result, its circles are straight parallel lines. The involute profile of the rack also has a rectilinear outline. This property of the involute is the most important in the manufacture of gears. A transmission using a toothed bar (rack) is called a rack and pinion transmission (rack), it is used to convert rotational motion into translational motion and vice versa. The transmission consists of a rack and a spur gear (gear). This type of transmission is used in rack railways.

Characteristics and Application

The gear teeth are in radial planes. The contact line of spur gears is parallel to the axis of rotation.

Depending on the required load characteristics and the accuracy of the transmitted rotation, the module (distance between the centers of the teeth) is selected from 1 to 6.

It is used in moving parts of mechanisms together with a gear rack.

Spur gears are used in all types of automatic gates, heavy-duty conveyor lines, 3D printers, CNC machines and much more.

Gear module parameters

The characteristic under consideration is designated by the letter m, indicating the strength of the gears. The unit is measured in millimeters (the higher the transmission load, the greater the modular value). The following indicators are used in calculating the parameter:

pitch circle diameter;
pitch and number of teeth;
involute (diameter of the main circle);
similar characteristics of dark gear cavities;
tooth height of dark and light wheels.

In the mechanical engineering industry, calculations are carried out using standard values for ease of manufacture and replacement of gears with numbers from 1 to 50.

What is a module in the drawing?

A module is a unified element of any system, consisting of an interchangeable set of mass-produced parts. The module drawing is carried out on the basis of GOST 2.109-73 - a unified system of design documentation (ESKD).

Spur gears [ edit | edit code]

The profile of wheel teeth usually has an involute lateral shape. However, there are gears with a circular tooth profile (Novikov gear with one and two lines of engagement) and with a cycloidal one. In addition, ratchet mechanisms use gears with an asymmetrical tooth profile.

Involute gear parameters:

m
is the wheel module.
The gearing modulus is a linear value π
times smaller than the circumferential pitch P or the ratio of the pitch along any concentric circle of the gear to
π
, that is, the modulus is the number of millimeters of pitch circle diameter per tooth. The dark and light wheels have the same module. The most important parameter, standardized, is determined from the strength calculation of gears. The more loaded the transmission, the higher the module value. All other parameters are expressed through it. The module is measured in millimeters and is calculated using the formula:

Read also: Installation of pressure gauges on GOST pipelines

m = dz = p π >=

>> >

z
- number of wheel teeth
p
- tooth pitch (marked in lilac)
d
- diameter of the pitch circle (marked in yellow)
da
is the diameter of the circle of the vertices of the dark wheel (marked in red)
db
- diameter of the main circle - involute (marked in green)
df
- diameter of the circle of the dark wheel depressions (marked in blue)
haP+hfP
— height of the tooth of the dark wheel,
x+haP+hfP
— height of the tooth of the light wheel

In mechanical engineering, certain values of the gear module m

for the convenience of manufacturing and replacing gears, which are integers or numbers with a decimal fraction:
0.5
;
0.7
;
1
;
1.25
;
1.5
;
1.75
;
2
;
2.5
;
3
;
3.5
;
4
;
4.5
;
5
and so on until
50
. (for more details, see GOST 9563-60 Gear wheels. Modules)

Tooth head height - haP

and the height of the tooth stem -
hfP
- in the case of the so-called.
zero gear (made without offset, a gear with “zero” teeth)
(the offset of the cutting rack cutting the teeth closer or further to the workpiece, and the offset closer to the workpiece is called
negative offset
, and the offset further from the workpiece is called
positive
) are related with modulus
m
as follows:
haP = m;
hfP = 1.25 m , that is:

hf P ha P = 1.25 <>>>=1.25> >

From this we obtain that the tooth height h

(not indicated in the figure):

h = hf P + ha P = 2, 25 m >+>=2,25m> >

In general, it is clear from the figure that the diameter of the circle of the vertices is da

greater than the diameter of the cavities
df
by twice the tooth height
h
.
Based on all this, if you want to practically determine the module m
of a gear without having the necessary data for calculations (except for the number of teeth
z
), then you need to accurately measure its outer diameter
da
and divide the result by the number of teeth
z
plus 2:

m = daz + 2 >> >

Longitudinal line of the tooth [edit | edit code]

Gears are classified depending on the shape of the longitudinal line of the tooth into:

The main geometric dimensions of cylindrical gears are determined from calculations and shown in Fig. 1.

The shape of the gears can be flat - Fig. 3 a) and b) or with a protruding hub - Fig. 3 b). Much less often (usually in single-stage gearboxes) wheels are made with a hub protruding in both directions.

The width S of the ends of the ring gear is taken as .

Chamfers are made at the ends of the ring gear.

Rice. 3. The simplest forms of cylindrical gears manufactured in small-scale production

Rice. 4. Shapes of cylindrical gears produced in mass production

There are two possible designs of gears; integral with the shaft (gear shaft) and separately from it (mounted gear). The quality (rigidity, accuracy, etc.) of the gear shaft is higher, and the manufacturing cost is lower than that of the shaft and pinion gear, therefore all gears of the gearboxes are made integral with the shaft. Mounted gears are used, for example, in cases where, according to operating conditions, the gear must be movable along the axis of the shaft.

Figure 5 shows the designs of the gear shaft: a—high-speed (with a small gear ratio) and b—low-speed (intermediate shaft) stages of a two-stage gearbox. Both designs provide cutting teeth with free tool exit.

Rice. 5. Designs of the gear shaft: a—high-speed (with a small gear ratio) and b—low-speed (intermediate shaft) stages of a two-stage gearbox

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very necessary

How to find the pitch diameter of a gear?

The pitch circle diameter d is one of the main parameters used to calculate the gear: d = m × z, where z is the number of teeth; m – module.

What is the pitch diameter?

The circle that is the initial circle when engaged with the rack is a dividing circle; its diameter is designated d (Fig. 2). For wheels without offset, the pitch circles coincide with the initial ones. The thickness of the tooth along the pitch circle S is equal to the width of the cavity between two teeth e.

Methods for making gears

In the replacement gear manufacturing industry, it is very important to select the most efficient and accurate gear processing method and tooling.

Careful preparation for the hardening stage results in a relatively simple operation of turning a hard part followed by hard machining of the gears. When turning hard parts, a good surface finish is important.

Gear machining will change significantly due to e-mobility, new transmission designs and the need to be both flexible and productive. The focus will be on conventional traditional gear machines, and multi-tasking gear machining will become the norm instead.

The traditional method of producing high volume gears, namely gear milling, requires machines and tools specific to the production of gears and in many cases even to the size of the gear itself.

Gear hobbing is a gear manufacturing process in which gear teeth are formed through a series of cuts using a helical cutting tool. The cutter and gear blank rotate without stopping until all the teeth are cut. Gear hobbing is only possible for external gears.

Advantages:

Reduced overall gear cost compared to high speed steel tools.
High cutting speed.
Extended tool life.
No additional costs.

Cutting

During the manufacturing process, the gear is cut. It is a machining process to create a gear. Gear cutting methods include: broaching, hobbing, milling, forming and grinding.

Broaching is primarily used to make splined or very large gears. The next stage is gear hobbing. For this purpose, special equipment called a milling machine is used. This process produces many types of gears, including spur, spline, worm and helical.

Another gear cutting process is milling, which is not only one of the initial processes but also one of the most important techniques. This uses a router and a router cutter that is run through the gear blank to create the gap between the teeth. One of the main advantages of gear hobbing is that it can produce almost any type of gear.

Finally, there is the finishing and grinding process, where the cut gear is finished by lapping, chipping, polishing, honing or grinding.

Precision cutting produces high-quality gears for agricultural, automotive, manufacturing and other industries.

Rolling

This is a method of creating wheel teeth by plastic deformation, which is carried out using a special tool while rolling it mutually with the workpiece.

Formula for calculating spur gear parameters

To determine the parameters of a spur gear, you will need to perform some preliminary calculations. The length of the initial circle is π×D, where D is its diameter.

Calculation of gear module

The engagement pitch t is the distance between adjacent teeth measured along the starting circle. If we multiply this distance by the number of teeth z, then we should get its length:

After carrying out the transformation, we get:

If we divide the pitch by pi, we get a factor that is constant for a given gear part. This is called the engagement module m.

The dimension of the gear module is millimeters. If you substitute it into the previous expression, you get:

Having performed the transformation, we find:

This implies the physical meaning of the engagement module: it represents the length of the arc of the initial circle corresponding to one tooth of the wheel. The diameter of the protrusion circle De is equal to

where h' is the height of the head.

The height of the head is equal to m:

Carrying out mathematical transformations with substitution, we get:

The diameter of the circle of the depressions Di corresponds to De minus the two heights of the base of the tooth:

where h“ is the height of the tooth stem.

For cylindrical wheels, h“ is equal to a value of 1.25m:

Gear device

Having performed the substitution on the right side of the equality, we have:

which corresponds to the formula:

and if we perform the substitution, we get:

In other words, the head and stem of the tooth have a height ratio of 1:1.25 to each other.

The next important dimension, tooth thickness s, is taken to be approximately equal to:

for cast teeth: 1.53m:
for those made by milling - 1.57m, or 0.5×t

Since the step t is equal to the total thickness of the tooth s and the cavity sв, we obtain formulas for the width of the cavity

for cast teeth: sв=πm-1.53m=1.61m:
for those made by milling - sв = πm-1.57m = 1.57m

The design characteristics of the remaining part of the gear part are determined by the following factors:

forces applied to the part during operation;
configuration of parts interacting with it.

Detailed methods for calculating these parameters are given in such university courses as “Machine Parts” and others. The gear module is widely used in them as one of the main parameters.

To display gears using engineering graphics methods, simplified formulas are used. In engineering reference books and state standards you can find characteristic values calculated for typical gear sizes.

Initial data and measurements

In practice, engineers are often faced with the task of determining the modulus of a real-life gear in order to repair or replace it. At the same time, it also happens that design documentation for this part, as well as for the entire mechanism into which it is included, cannot be found.

The simplest method is the running-in method. Take a gear for which the characteristics are known. Insert it into the teeth of the part being tested and try to roll it around. If the pair engages, then their step coincides. If not, continue the selection. For a helical cutter, choose a cutter suitable for the pitch.

This rule of thumb works well for small gears.

For large ones, weighing tens or even hundreds of kilograms, this method is physically impossible.

Calculation results

Larger ones will require measurements and calculations.

As is known, the module is equal to the diameter of the circle of the protrusions divided by the number of teeth plus two:

m=De/(z+2)

The sequence of actions is as follows:

measure the diameter with a caliper;
count the teeth;
divide the diameter by z+2;
Round the result to the nearest whole number.

Wheel tooth and its parameters

This method is suitable for both straight and helical gears.

What types of gears are there?

Types of gears, gears

Transverse Tooth Profile Typically, gears have a tooth profile with an involute flank shape. ...
Longitudinal tooth line Spur gears...
Internal gears...
Helical gears...
Sector gears...
Gears with circular teeth

Straight and helical wheels

The module and diameter of spur gears is one of the most popular types. The teeth are placed in radial planes, and the contact area of a pair of wheels is parallel to the axis of rotation. The axes of both gears are located in the same way.

Helical wheels are an improved variation of the above modification. The teeth are at a certain angle to the rotational axis. Engagement is smoother and quieter, which allows the elements to be operated in low-noise applications, ensuring the transmission of more torque at high speed. The disadvantages include the increased contact area of the teeth, which provokes increased friction and heating of the parts. This is fraught with reduced power and increased lubricant consumption. In addition, the mechanical action along the gear axis requires the use of thrust bearings to mount the shaft.

Chevron modifications and analogues with internal gearing

Chevron gears overcome mechanical axial force problems. Unlike straight and helical versions, the teeth are made in the form of the letter V. The axial impact of the two halves of the device is compensated by interaction, which makes it possible to avoid the use of thrust bearings on the shaft. This model is independently installed along the axis; one of the working gearboxes is mounted on cylindrical shortened bearings (floating bearings).

The internal gear module is equipped with teeth that are cut inside. Operation of the part involves unilateral rotations of the drive and driven wheels. In this design, less friction is spent, which improves efficiency. Similar devices are used in mechanisms limited in overall dimensions, as well as planetary gears, special pumps and tank turrets.

Screw, circular, sector versions

The helical-type gear module is a cylinder with teeth that are arranged in the helical direction. Such elements are installed on non-intersecting shafts located perpendicular to each other. The alignment angle is 90 degrees.

A sector gear is a part of any gear used in gears where the main element does not need to rotate a full turn. This detail makes it possible to save valuable space in the size of a full-fledged analogue.

Gears in terms of module and number of teeth with a circular arrangement differ in contact contact at one meshing point located parallel to the main axes. The second name of the mechanism is Novikov gear. It provides good driving characteristics, smooth and silent operation, and increased gripping ability. At the same time, the efficiency of such parts is slightly lower than their analogues, and the manufacturing process is much more complicated. These parts have a significantly limited field of operation due to their characteristics.

Bevel gears

Bevel gears have different types, they differ in the shape of the tooth lines, with straight, curved, tangential, and circular teeth. Bevel gears are used in machines to move the mechanism, where it is necessary to transmit rotation from one shaft to another, the axes of which intersect. For example, in automobile differentials, to transmit torque from the engine to the wheels.

Rack

A rack is part of a gear with an infinite pitch circle radius. As a result, its circles are straight parallel lines. The involute profile of the rack also has a rectilinear outline. This property of the involute is the most important in the manufacture of gears. A transmission using a toothed bar (rack) is called a rack and pinion gear (rack), it is used to convert rotational motion into translational motion and vice versa. The transmission consists of a rack and a spur gear (gear). This type of transmission is used in rack railways.

Star

The star gear is the main part of a chain transmission, which is used in conjunction with a flexible element - a chain - to transmit mechanical energy.

Ring gear

A ring gear is a special type of gear whose teeth are located on the side surface. Such a gear usually works in tandem with a spur gear or a drum (lantern wheel) consisting of rods. This type of transmission is used in tower clocks.

Toothed bevel wheels

The design of bevel gears (Fig. 9) differs from the design of cylindrical gears in the relationships in the part of the ring gear.

Rice. 9. Design of bevel gears: a, b, c – turned; c, d – stamped; g, e – cast; e – composite (prefabricated)

What are the Similarities Between Gear and Gear

There are several similarities between a gear and a gear:

Like a gear, a gear can be either a driven or a driving element in the overall system.
The shape of a gear and a gear can be either cylindrical or conical, it all depends on the function that a particular part performs.
With the help of a gear and a gear, you can maneuver based on the speed of the rotating element, either decreasing it or increasing it.
Gears and gears can be used equally effectively on electric and gas-powered tools, but gears are most used because they ensure the stability of the mechanism.
Gears and gears can be used to drive rotational axes.

The external similarities between a gear and a cogwheel are also justified by the fact that often these two elements can perform similar functions and be interchangeable in certain systems and mechanisms.

Let's summarize

Design drawings and diagrams for gears of various configurations are mainly the same for oblique and spur versions. The main differences arise in strength calculations. Graphic displays use characteristics based on typical overall dimensions of gears. Among the assortment presented on the market, it is quite possible to select a gear with the necessary characteristics and strength indicators.

Sources

https://mehmanxona.ru/tehnologii/vidy-shesterenok.html
https://novoe-info.ru/chto-takoe-modul-shesterni/
https://MechPrivod.com/market/zubchataya_shesterenka/shesterenka_zubchataya_cilindricheskaya/
https://novoe-info.ru/kak-nayti-modul-zubchatogo-kolesa/
https://doctordent.su/pulpit/kak-opredelit-modul-zuba-shesterni-po-diametru.html
https://FB.ru/article/429020/modul-shesterni-vidyi-opredelenie-standartnyie-pokazateli

Calculation of bevel gears for strength

Calculation of contact strength

The strength calculation of a bevel gear is based on the assumption that the load-bearing capacity of the teeth of a bevel gear is the same as that of an equivalent cylindrical one (see Fig. 3) with the same tooth length b and a profile corresponding to the average additional cone (the average section of the tooth). However, operating practice has shown that, at the same load level, bevel gears fail faster than cylindrical gears.

Taking into account the transformations and strength conditions, the formula for the verification calculation of steel bevel gears has the form:

σн = 6.7×104√(КнТ1/de13uΘн) ≤ [σ]н,

where T1 – in Nm; d1 – in mm.

For spur bevel gears Θн = Θr = 0.85.

For gears with circular teeth, the values of Θн are taken from reference tables.

The load factor KA for bevel gears can be determined by the formula:

Kn = KA×KNβ×KNv.

Coefficient KA takes into account external dynamic load. It is assigned in the same way as when calculating cylindrical gears.

The KNβ coefficient takes into account the uneven distribution of load along the length of the contact lines of the teeth in mesh. For wheels with circular teeth, this coefficient is determined by the formula:

КНβ = √(КНβ0) provided КНβ ≥ 1.2.

where КНβ0 is a coefficient selected from reference tables depending on the ratio ψbd = b/d1, the hardness of the gears and the transmission scheme.

For most bevel gears, the ratio of the ring gear width (tooth length) to the outer cone distance Kbe = b/Re = 0.285, then:

ψbd = 0.166 √(u2 + 1).

For spur bevel gears, KNβ is selected from reference tables, and KNβ = KNβ0 is taken.

The value of the internal dynamic load coefficient KNv for gears with circular teeth is chosen as for cylindrical helical gears. For bevel spur gears, KHv is also selected according to reference tables, but with a decrease in the degree of accuracy by one.

Design calculation

Having solved the dependence σн = 6.7×104√(КнТ1/de13uΘн) ≤ [σ]н, relative to de1, we obtain the design calculation formula for steel bevel gears:

de1 = 1650 × 3√KnT1/u[σ]Н2 ΘН),

where de1 is the outer pitch diameter of the gear, mm; Т1 – in Nm, [σ]н in N/mm2.

***

Calculation of bevel gear teeth for bending strength

Similar to the calculation of a cylindrical gear transmission, the calculated bending stresses in the teeth of bevel wheels and the condition of their strength are described by the formulas:

σF1 = КFF1YFs1/bmΘF≤ [σ]F1; σF2 = σF1YFs2/ YFs1≤ [σ]F2,

where m or mn is the normal module in the middle section of the bevel gear tooth (reference value); YFs is the coefficient of tooth shape and stress concentration of the equivalent wheel, taking into account the displacement coefficient xe (xn) along zv (zvn); ΘF – coefficient taking into account the influence of the type of bevel wheels on the load-bearing capacity of the transmission; choose according to the recommendations given above.

Load factor KF for bevel gears:

KF = KA×KFβ×KFv,

where KA is a coefficient that takes into account the external dynamic load, depending on the degree of uniformity of loading of the driving and driven transmission links. When specifying the load by a torque cyclogram or a typical loading mode, which takes into account external dynamic loads, KA = 1; KFβ is a coefficient that takes into account the uneven distribution of load along the length of the contact lines of the teeth in mesh.

For spur bevel gears KFβ = K'Fβ;

For wheels with circular teeth:

KFβ = √ K'Fβ provided KFβ ≥ 1.15,

where K'Fβ is determined by the formula KFβ = 0.18 + 0.82KNβ0.

The internal dynamic load coefficient KFv is taken from reference tables. Permissible stresses [σ]F1, [σ]F2 are determined according to the recommendations.

***

Strength calculation of open bevel gears

Open bevel gears are made only with straight teeth and are used at peripheral wheel speeds of less than 2 m/s. The degree of accuracy according to the standards of smoothness and contact is 9th. The dimensions of the gear are determined based on the contact strength of the teeth, followed by testing for bending. When calculating, allowable stresses are taken:

[σ]Н = σНlim/[s]Н; [σ]F = σFlim/[s]F

The internal dynamic load coefficients KHv and KFv are taken from reference tables. The coefficients of uneven load distribution along the length of the contact lines are taken as follows: КНβ = КFβ = 1.

Due to the increased wear of the teeth of open gears, it is recommended to take the engagement modulus value 1.5 times greater than for closed gears of the same size.

***

Planetary gears