Today, I will talk to you about gears in detail.
1. What is Gear?
A gear is a toothed mechanical part that can mesh with each other. It is extremely widely used in mechanical transmission and the entire mechanical field.
2. The history of gears
As early as 350 B.C., the famous Greek philosopher Aristotle recorded gears in literature. Around 250 BC, the mathematician Archimedes also described the use of a turbine worm in the literature. In the ruins of Katesfin, present-day Iraq, there are still gears from the BC era.
Gears also have a long history on the mainland. According to historical records, as far as 400 ~ 200 B.C. in ancient China has begun to use gears, the bronze gear unearthed in Shanxi, the mainland is the oldest gear that has been found so far, as a reflection of the ancient scientific and technological achievements of the guide car is a mechanical device with the gear mechanism as the core. In the second half of the 15th century, during the Italian Renaissance, the famous all-rounder Leonardo da Vinci not only left an indelible achievement in the history of culture and art, but also in the history of gear technology, after more than 500 years, the current gear still retains the prototype of the sketch at that time.
It wasn't until the end of the 17th century that the shape of the teeth that could correctly transmit motion was studied. In the 18th century, after the industrial revolution in Europe, the application of gear transmission became more and more extensive, first the development of cycloid gears, and then involute gears, until the beginning of the 20th century, involute gears have been used dominantly. Later, displacement gears, arc gears, bevel gears, helical gears and so on were developed.
Modern gear technology has reached: gear modulus 0.004-100 mm, gear diameter from 1 mm to 150 m, transmission power up to 100,000 kilowatts, speed up to 100,000 rpm, maximum circumferential speed up to 300 m/s.
Internationally, power transmission gear devices are developing in the direction of miniaturization, high speed and standardization. The application of special gears, the development of planetary gear devices, and the development of low-vibration and low-noise gear devices are some of the characteristics of gear design.
3. Gears are generally divided into three main categories
There are many types of gears, and the most common way to classify them is according to the axle nature of the gear. Generally, there are three types: parallel axis, intersecting axis and staggered axis.
1) Parallel shaft gears: including spur gears, helical gears, internal gears, racks and helical racks, etc.
2) Intersecting shaft gears: straight bevel gears, arc bevel gears, zero-degree bevel gears, etc.
3) Staggered shaft gears: there are staggered shaft helical gears, worm gears, hypoid gears, etc.
The efficiencies listed in the table above are transmission efficiencies and do not include losses in bearings and stirring lubrication. The meshing of the gear pairs of parallel shafts and intersecting shafts is basically rolling, and the relative sliding is very small, so the efficiency is high. Staggered shaft gear pairs, such as staggered shaft helical gears and worm and worm gears, rotate through relative sliding to achieve power transmission, so the impact of friction is very large, and the transmission efficiency decreases compared with other gears. The efficiency of the gear is the transmission efficiency of the gear under normal assembly conditions. In the event of incorrect installation, especially if the bevel gear is assembled at an incorrect distance and there is an error in the intersection of the cone, its efficiency can be significantly reduced.
3.1 Parallel shaft gears
1) Spur gears
A cylindrical gear with a tooth line parallel to the shaft line. Because of its ease of processing, it is most widely used in power transmission.
2) Rack
A linear rack-and-pinion gear meshed with a spur gear. It can be seen as a special case when the pitch diameter of the spur gear becomes infinite.
3) Internal gears
Meshing with spur gears, a gear with teeth is machined on the inside of the ring. It is mainly used in planetary gear transmission mechanism and gear coupling and other applications.
4) Helical gears
A cylindrical gear with a helical tooth line. It is widely used because of its high specific spur gear strength and smooth operation. Axial thrust is generated during transmission.
5) helical rack and pinion,
A strip gear that engages with a helical gear. This is equivalent to the situation when the pitch diameter of a helical gear becomes infinite.
6) Herringbone gears
The tooth wire is a gear composed of two helical gears that rotate left and right. There is the advantage that it does not produce thrust in the axial direction.
3.2 Intersecting shaft gears
1) Straight bevel gears
Bevel gears with the same tooth wire as the bus bar of the pitch taper wire. Among bevel gears, they are a type that is relatively easy to manufacture. Therefore, it has a wide range of applications as a bevel gear for transmission.
2) Spiral bevel gears
The tooth wire is a curved, bevel gear with a helix angle. Although it is more difficult to make than straight bevel gears, it is also widely used as a high-strength, low-noise gear.
3) Zero-degree bevel gears
Curved bevel gears with a helix angle of zero degrees. Because it has the characteristics of both spur and curved bevel gears, the force situation on the tooth flanks is the same as that of straight bevel gears.
3.3 Staggered shaft gears
1) Cylindrical worm pair
The cylindrical worm pair is a general term for the cylindrical worm and the worm gear that is meshed with it. Smooth operation and large gear ratios can be obtained in a single pair are the biggest characteristics, but they have the disadvantage of low efficiency.
2) Staggered shaft helical gears
The name given to the cylindrical worm pair when it is driven between staggered shafts. It can be used in the case of helical gear pairs or helical gears and spur gear pairs. Although it runs smoothly, it is only suitable for use under light loads.
3.4 Other special gears
1) Face gears
A disc-shaped gear that engages with a spur or helical gear. It is transmitted between orthogonal shafts and staggered shafts.
2) Drum worm pair
A general term for a drum worm and the worm gear that engages with it. Although it is more difficult to manufacture, it can transmit large loads compared to cylindrical worm pairs.
3) Hypoid gears
Bevel-shaped gears that drive between staggered shafts. The large and small gears are eccentrically machined, similar to the arc tooth gears, and the meshing principle is very complex.
4. Basic terminology and sizing of gears
There are many terms and expressions unique to gears, so in order to let everyone know more about gears, here are some basic terms of gears that are often used.
1) The name of each part of the gear
2) The term that indicates the size of the gear teeth is modulus
m1、m3、m8... It is called modulus 1, modulus 3, and modulus 8. Modulus is a common name all over the world, using the symbol m (modulus) and number (mm) to indicate the size of the teeth, the larger the number, the larger the teeth.
In addition, in countries that use imperial units (e.g., the United States), symbols (diameter joints) and numbers (the number of teeth of a gear with a diameter of 1 inch in an indexing circle) are used to indicate the size of the teeth. For example: DP24, DP8, etc. There are also special ways to use symbols (weeks) and numbers (millimeters) to indicate the size of the teeth, such as CP5 and CP10.
Multiply the modulus by pi to give the pitch (p), which is the length between two adjacent teeth.
The formula is:
p=pi x modulus = πm
Comparison of tooth sizes of different modules:
3) Pressure angle
The pressure angle is the parameter that determines the tooth shape of the gear. That is, the inclination of the tooth surface. The pressure angle (α) is generally 20°. Previously, gears with a pressure angle of 14.5° were commonplace.
The pressure angle is the angle between the radius line and the tangent of the tooth shape at a point on the tooth surface (generally referred to as a node). As shown in the figure, the α is the pressure angle. Because α'=α, α' is also the angle of pressure.
When the meshing state of tooth A and tooth B looks like from the node:
A tooth pushes point B on the node. At this time, the driving force acts on the common normal of tooth A and tooth B. That is, the common normal is the direction in which the force is applied and the direction in which the pressure is subjected, and the α is the angle of pressure.
The modulus (m), pressure angle (α) and the number of teeth (z) are the three basic parameters of the gear, and the size of each part of the gear is calculated based on this parameter.
4) Tooth height and tooth thickness
The height of the teeth is determined by the modulus (m).
Full tooth height h=2.25m (= root height + top height)
The top height (ha) is the height from the top of the tooth to the index line. ha=1m。
Root height (HF) is the height from the root of the tooth to the index line. hf=1.25m。
The reference for tooth thickness (s) is half of the tooth pitch. s=πm/2。
5) The diameter of the gear
The parameter that determines the size of the gear is the indexing circle diameter (d) of the gear. Based on the indexing circle, the tooth pitch, tooth thickness, tooth height, tooth top height, and tooth root height can be determined.
The diameter of the indexing circle d=zm
The diameter of the tooth top circle is da=d+2m
齿根圆直径df=d-2.5m
Indexing circles cannot be seen directly in actual gears, because indexing circles are hypothetical circles that determine the size of gears.
6) Center distance and backlash
When the indexing circles of a pair of gears are tangent to each other, the center distance is half of the sum of the diameters of the two indexing circles.
Center distance a=(d1+d2)/2
In the meshing of gears, backlash is an important factor in order to achieve a smooth meshing effect. Backlash is the gap between the tooth surfaces when a pair of gears are meshed.
There is also a gap in the direction of the tooth height of the gear. This void is called the clearance. Backlash (c) is the difference between the root height of the gear and the tip height of the matching gear.
顶隙 c=1.25m-1m=0.25m
7) Helical gears
The gear after the spur gear teeth are twisted in a spiral shape is a helical gear. Most of the spur gear geometry grates can be used for helical gears. There are 2 ways for helical gears according to their datums:
End face (shaft right angle) reference (end face modulus/pressure angle)
Normal (Right Angle) Datum (Normal Modulus/Pressure Angle)
The relationship between the end face modulus mt and the normal modulus mn is mt=mn/cosβ
8) Spiral direction and fit
Helical gears, arc bevel gears, etc., gears with spiral teeth are certain, and the direction and fit of the spiral are certain. The spiral direction means that when the center axis of the gear points up and down, and when viewed from the front, the direction of the teeth pointing to the right is [right-handed], and the upper left is [left-handed]. The fit of the various gears is shown below.
5. The most commonly used gear tooth profile is the involute tooth profile
If you only divide the tooth pitch on the outer circumference of the friction wheel, install protrusions, and then mesh them to rotate with each other, the following problems will occur:
- The tangent point of the tooth produces sliding
- The tangent moves at different times faster and sometimes slowly
- Vibration and noise are generated
The gear tooth transmission is both quiet and sleek, and from this the involute curve is born.
1) What is an involute
Wrap a pencil thread around one end around the outer circumference of the cylinder, and gradually release the thread while the thread is taut. In this case, the curve drawn by the pencil is the involute curve. The outer circumference of the cylinder is known as the base circle.
2) Example of 8-tooth involute gear
After dividing the cylinder into 8 equal parts, tie 8 pencils and draw 8 involute curves. Then, the line is wound in the opposite direction and 8 curves are drawn in the same way, which is a gear with 8 teeth with an involute curve as the tooth shape.
3) Advantages of involute gears
- Even if there is some error in the center distance, it can be meshed correctly;
- It is easier to get the correct tooth shape, and it is easier to process;
- Because it rolls and meshes on a curve, rotational motion can be smoothly transmitted;
- As long as the size of the teeth is the same, one tool can process gears with different numbers of teeth;
- The tooth root is thick and strong.
4) Base circle and indexing circle
The base circle is the base circle that forms the involute tooth shape. The indexing circle is the reference circle that determines the size of the gear. The base circle and the indexing circle are important geometric dimensions of gears. The involute tooth shape is the curve that forms on the outside of the base circle. On the base circle, the angle of pressure is zero degrees.
5) Meshing of involute gears
The indexing circles of the two standard involute gears are tangentially meshed at the standard center distance.
When the two wheels are meshed, it looks like two friction wheels with a diameter of d1 and d2 in the indexing circle. However, in reality, the meshing of involute gears depends on the base circle and not on the indexing circle.
The meshing contact points of the two gear tooth shapes move on the engagement line in the order of P1-P2-P3. Note the yellow teeth in the drive gear. For a period of time after the tooth begins to mesh, the gear is meshed with two teeth (P1, P3). The engagement continues, and when the engagement point is moved to the point P2 on the indexing circle, there is only one engagement tooth left. The meshing continues, and when the meshing point moves to the point P3, the next tooth begins to engage at the P1 point, forming a state of two-tooth meshing again. Just like this, the two-tooth meshing of the gear interacts with the single-tooth meshing to repeatedly transmit the rotational motion.
The common tangent of the base circle A to B is called the meshing line. The meshing points of the gears are all on this meshing line.
It is represented by a figurative diagram, as if the belt is crossed on the outer circumference of the two base circles to make a rotational motion to transmit power.
6. The displacement of gears is divided into positive displacement and negative displacement division
The tooth profile of the gear we usually use is generally a standard involute, but there are some situations where the teeth need to be displaced, such as adjusting the center distance, preventing the pinion from undercutting, etc.
1) The number and shape of the gear
The involute tooth profile curve varies with the number of teeth. The higher the number of teeth, the more straight the tooth curve will be. With the increase of the number of teeth, the tooth deformation of the tooth root becomes thicker, and the strength of the gear teeth increases.
As can be seen from the figure above, the involute tooth shape at the root of the tooth of the gear tooth with 10 teeth is partially excavated, and the undercut phenomenon occurs. However, if the gear with a number of teeth Z=10 is positively displaced, the diameter of the tooth top circle is increased, and the tooth thickness of the gear tooth is increased, the same degree of gear strength can be obtained as that of a gear with a number of 200 teeth.
2) Displacement gears
The figure below is a schematic diagram of the gear positive displacement cutting with the number of teeth Z=10. When cutting teeth, the amount of movement of the tool along the radius direction xm (mm) is called radial displacement (referred to as displacement amount).
xm=displacement (mm)
x = displacement coefficient
m=modulus(mm)
Tooth profile change through positive displacement. The tooth thickness increases, and the outer diameter (tooth top circle diameter) also increases. By adopting a positive displacement of the gear, the occurrence of undercut can be avoided. The displacement of the gears can also achieve other purposes, such as changing the center distance, positive displacement can increase the center distance, and negative displacement can reduce the center distance.
Whether it is a positive or negative displacement gear, there is a limit to the amount of displacement.
3) Positive and negative displacement
There are positive and negative conjugations. Although the tooth height is the same, the tooth thickness is different. The tooth thickness becomes thicker for the positive displacement gear, and the tooth thickness becomes thinner for the negative displacement gear.
When it is not possible to change the center distance of the two gears, the pinion is positively dislocated (to avoid undercut) and the large gear is negatively dislocated so that the center distance is the same. In this case, the absolute value of the displacement is equal.
4) Meshing of displacement gears
Standard gears are meshed in the tangent state of the indexing circle of each gear. The meshing of the dislocated gears, as shown in the figure, is tangential engagement on the meshing joint circle. The angle of pressure on the circle of the meshing joint is called the angle of meshing. The angle of engagement is not the same as the angle of pressure on the indexing circle (indexing circle pressure angle). The engagement angle is an important factor when designing displacement gears.
6) The role of gear displacement
It can prevent the undercut phenomenon caused by the small number of teeth during machining, the desired center distance can be obtained through displacement, and the positive displacement of the pinion that is easy to produce wear is carried out in the case of a large number of teeth ratio of a pair of gears, so that the tooth thickness is thickened. Instead, the large gear is negatively displaced, making the tooth thickness thinner so that the life of the two gears is close.
7. Accuracy of gears
Gears are mechanical elements that transmit power and rotation. The performance requirements for gears are mainly as follows:
- greater power transmission capacity;
- Use gears that are as small as possible;
- Low noise;
- Correctness.
In order to meet the requirements described above, it is necessary to improve the accuracy of gears.
1) Classification of gear accuracy
The accuracy of gears can be broadly divided into three categories:
a) The correctness of the involute tooth shape - the tooth shape accuracy
b) The correctness of the tooth line on the tooth surface - the accuracy of the tooth line
c) Correctness of the teeth/coveolar position
- Indexing accuracy of gear teeth - single pitch accuracy
- Correctness of pitch—Cumulative pitch accuracy
- The deviation of the position of the gauge ball clamped in the radius direction - radial runout accuracy
2) Tooth shape error
3) Tooth line error
4) Tooth pitch error
The pitch value is measured on the measuring circumference centered on the gear shaft.
Single Pitch Deviation (FPT): The difference between the actual pitch and the theoretical pitch.
The cumulative total deviation of the tooth pitch (Fp) is measured and evaluated for the deviation of the tooth pitch of the whole wheel. The total amplitude of the cumulative deviation curve is the total deviation of the pitch.
5) Radial Runout (Fr)
The probes (spherical, cylindrical) are successively placed in the cogging and the difference between the maximum and minimum radial distances from the probe to the gear axis is determined. The eccentricity of the gear shaft is part of the radial runout.
6) Radial Composite Total Deviation (Fi")
So far, the tooth shape, tooth pitch, tooth line accuracy, etc., which we have described, are all methods to evaluate the accuracy of the gear unit. In contrast, there is also a method of meshing the gear with the measuring gear to evaluate the accuracy of the gear. The left and right tooth surfaces of the measured gear are in contact with the measuring gear and rotate for a full circle. Record the change in the distance between the centers. The figure below shows the test results of a gear with a number of 30 teeth. There are a total of 30 wavy lines with a single tooth radial comprehensive deviation. The total radial comprehensive deviation is approximately the sum of the radial runout deviation and the radial comprehensive deviation of a single tooth.
7) The correlation between the various precision of gears
There is a correlation between the accuracy of each part of the gear, generally speaking, the correlation between radial runout and other errors is strong, and the correlation between various pitch errors is also very strong.
8) Conditions for high-precision gears
8. Gear calculation formula
Calculation of standard spur gears (pinion (1), big gear (2))
Calculation formula for shift spur gear (pinion (1), big gear (2))
Calculation formula for standard helical teeth (right-angle tooth method) (pinion (1), large gear (2))
Calculation formula for displacement helical teeth (right-angle tooth method) (pinion (1), large gear (2))