1. What is a gear?
Gears are toothed mechanical parts that can mesh with each other. It is widely used in mechanical transmission and the entire mechanical field.
2. The history of gears
As early as 350 BC, the famous ancient Greek philosopher Aristotle recorded gears in literature. Around 250 BC, the mathematician Archimedes also described in the literature a windlass using a turbine worm. Gears dating from BC are still preserved in the remains of Kaisfern in present-day Iraq.
Gear has a long history in China. According to historical records, gears have been used in ancient China as far back as 400-200 BC. The bronze gears unearthed in Shanxi, China are the oldest gears that have been discovered so far. As a guide car reflecting the achievements of ancient science and technology, the gear mechanism is used. The core mechanism. During the Italian Renaissance in the second half of the 15th century, the famous all-rounder Leonardo da Vinci left indelible achievements not only in culture and art, but also in the history of gear technology. After more than 500 years, the current The gears still retain the prototypes sketched at the time.
It wasn't until the end of the 17th century that people began to study the shape of the gear teeth that would transmit motion correctly. In the 18th century, after the European Industrial Revolution, the application of gear transmission became more and more extensive; first, cycloid gears were developed, and then involute gears were developed. Until the beginning of the 20th century, involute gears had an advantage in application. Since then, variable gears, arc gears, bevel gears, helical gears, etc. have been developed.
Modern gear technology has reached: gear module 0.004-100 mm; gear diameter from 1 mm to 150 meters; transmission power up to 100,000 kilowatts; rotational speed up to 100,000 rpm; the highest peripheral speed up to 300 m/s.
Internationally, the power transmission gear device is developing in the direction of miniaturization, high speed and standardization. The application of special gears, the development of planetary gears, and the development of low-vibration and low-noise gears are some of the characteristics of gear design.
3. Gears are generally divided into three categories
There are many kinds of gears, and the most common classification method is according to the gear shaft. Generally divided into 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: there are straight bevel gears, spiral bevel gears, zero-degree bevel gears, etc.
3) Staggered shaft gears: there are staggered shaft helical gears, worm gears, hypoid gears, etc.
The efficiency listed in the table above is the transmission efficiency, excluding the loss of bearing and stirring lubrication. The meshing of the gear pairs of the parallel shaft and the intersecting shaft 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 gears, because they rotate through relative sliding to achieve power transmission, the impact of friction is very large, and the transmission efficiency is reduced compared with other gears. The efficiency of a gear is the transmission efficiency of the gear under normal assembly conditions. If there is an incorrect installation, especially if the bevel gear is not assembled at the correct distance, resulting in an error in the intersection of the same cone, its efficiency will drop significantly.
3.1 Gears with parallel shafts
1) Spur gear
The tooth line and the axis line are parallel to the cylindrical gear. Because it is easy to process, it is most widely used in power transmission.
2) Rack
A linear rack-shaped gear that meshes with a spur gear. It can be regarded as a special case when the pitch diameter of the spur gear becomes infinite.
3) Internal gear
A gear with gear teeth machined on the inner side of the ring to mesh with the spur gear. Mainly used in applications such as planetary gear transmissions and gear couplings.
4) Helical gears
The tooth line is a helical cylindrical gear. It is widely used because of its higher strength and smoother operation than spur gears. Axial thrust is generated during transmission.
5) Helical gear rack,
A bar gear that meshes with a helical gear. It corresponds to the situation when the pitch diameter of the helical gear becomes infinite.
6) Herringbone gear
The tooth line is a gear formed by the combination of two left-handed and right-handed helical gears. It has the advantage of not generating thrust in the axial direction.
3.2 Intersecting shaft gears
1) Straight bevel gear
A bevel gear whose tooth line is the same as the generatrix of the pitch bevel line. Among the bevel gears, it is the type that is relatively easy to manufacture. Therefore, it has a wide range of applications as bevel gears for transmission.
2) Spiral bevel gear
The tooth line is a curved, bevel gear with a helical angle. Although it is more difficult to manufacture than straight bevel gears, it is also widely used as a gear with high strength and low noise.
3) Zero-degree bevel gear
Curved bevel gear with zero helix angle. Because it has the characteristics of both straight and curved bevel gears, the force on the tooth surface is the same as that of straight bevel gears.
3.3 Staggered shaft gears
1) Cylindrical worm pair
Cylindrical worm pair is a general term for a cylindrical worm and a worm gear that meshes with it. Quiet operation and a single pair can obtain a large transmission ratio as its biggest feature, but it has the disadvantage of low efficiency.
2) Staggered shaft helical gear
The name of the cylindrical worm pair when it is transmitted between staggered shafts. Can be used with helical gear pairs or helical and spur gear pairs. Although the operation is smooth, it is only suitable for use under light loads.
3.4 Other special gears
1) Face gear
Disc-shaped gears that can mesh with spur or helical gears. Transmission between orthogonal and staggered axes.
2) Drum-shaped worm pair
The general term for the drum-shaped worm and the worm gear that meshes with it. Although it is more difficult to manufacture, it can transmit a large load compared to a cylindrical worm pair.
3) Hypoid gear
Conical gears that drive between staggered shafts. The large and small gears are eccentrically processed, similar to the spiral gears, and the meshing principle is very complicated.
4. Basic terminology and dimension calculation of gears
There are many gear-specific terms and expression methods for gears. In order to enable you to understand gears more, here are some basic gear terms that are often used.
1) The name of each part of the gear
2) The term for the size of the gear teeth is the modulus
m1, m3, m8... are called modulo 1, modulo 3, modulo 8. The modulus is a common name all over the world. The symbol m (modulus) and the number (mm) are used to indicate the size of the gear teeth. The larger the number, the larger the gear teeth.
In addition, in countries that use imperial units (such as the United States), the size of the gear teeth is indicated by a symbol (diameter pitch) and a number (the number of teeth of a gear when the diameter of the index circle is 1 inch). For example: DP24, DP8, etc. There are also special calling methods that use symbols (weeks) and numbers (millimeters) to indicate the size of the gear teeth, such as CP5, CP10.
The pitch (p) can be obtained by multiplying the modulus by the pi, and the pitch is the length between two adjacent teeth.
The formula is:
p=pi x modulus = πm
Size comparison of gear teeth of different modules:
3) Pressure angle
The pressure angle is a parameter that determines the gear tooth profile. That is, the inclination of the gear tooth surface. The pressure angle (α) is generally 20°. In the past, gears with a pressure angle of 14.5° were common.
The pressure angle is the angle formed between the radius line and the tangent of the tooth shape at a point (generally a node) on the tooth surface. As shown, α is the pressure angle. Since α'=α, α' is also the pressure angle.
When the meshing state of tooth A and tooth B is viewed from the node:
The A tooth pushes the B point on the node. At this time, the driving force acts on the common normal of tooth A and tooth B. That is to say, the common normal is the acting direction of the force and the direction of the pressure, and α is the pressure angle.
The modulus (m), the pressure angle (α) and the number of teeth (z) are the three basic parameters of the gear, and the dimensions of each part of the gear are calculated based on these parameters.
4) Tooth height and tooth thickness
The height of the gear teeth is determined by the modulus (m).
Total tooth height h=2.25m (= tooth root height + tooth tip height)
The addendum height (ha) is the height from the addendum to the index line. ha=1m.
Root height (hf) is the height from the root to the index line. hf=1.25m.
The reference for tooth thickness (s) is half the tooth pitch. s=πm/2.
5) Diameter of gear
The parameter that determines the size of the gear is the index circle diameter (d) of the gear. Based on the index circle, the tooth pitch, tooth thickness, tooth height, tooth tip height and tooth root height can be determined.
Index circle diameter d=zm
Addendum diameter da=d+2m
Root circle diameter df=d-2.5m
The index circle is not directly visible in the actual gear, because the index circle is a hypothetical circle for determining the size of the gear.
6) Center distance and backlash
When the index circles of a pair of gears mesh tangentially, the center distance is half of the sum of the diameters of the two index circles.
Center distance a=(d1+d2)/2
In the meshing of gears, the backlash is an important factor in order to obtain a smooth meshing effect. Backlash is the gap between the tooth surfaces of a pair of gears when they mesh.
There is also a gap in the tooth height direction of the gear. This gap is called Clearance. Top clearance (c) is the difference between the tooth root height of the gear and the tooth top height of the mating gear.
Head clearance c=1.25m-1m=0.25m
7) Helical gears
A gear obtained by twisting the teeth of a spur gear helically is a helical gear. Most of the spur gear geometries are applicable to helical gears. There are 2 types of helical gears according to their datum planes:
End face (shaft right angle) reference (end face modulus/pressure angle>
Normal surface (tooth right angle) datum (normal modulus/pressure angle)
The relationship between the end face modulus mt and the normal modulus mn mt=mn/cosβ
8) Spiral direction and fit
Helical gears, spiral bevel gears, etc., the gear teeth are helical, and the helical direction and coordination are certain. The helical direction means that when the central axis of the gear points up and down, when viewed from the front, the direction of the gear teeth points to the upper right is [right rotation], and the upper left is [left rotation]. The fit of the various gears is shown below.
5. The most commonly used gear profile is the involute profile
If only the outer circumference of the friction wheel is divided into equal pitches, the protrusions are installed, and then they mesh with each other and rotate, the following problems will occur:
The tangent point of the gear teeth produces slip
The movement speed of the tangent point is sometimes fast and sometimes slow
Vibration and noise
The gear teeth are both quiet and smooth, which is why the involute curve is born.
1) What is an involute
Wrap a thread with a pencil attached at one end around the outer circumference of the cylinder, and gradually release the thread while the thread is taut. At this point, the curve drawn by the pencil is the involute curve. The outer circumference of the cylinder is called the base circle.
2) Example of 8-tooth involute gear
After dividing the cylinder into 8 equal parts, attach 8 pencils and draw 8 involute curves. Then, wind the wire in the opposite direction, and draw 8 curves in the same way. This is the gear with the involute curve as the tooth shape and the number of teeth is 8.
3) Advantages of involute gears
Even if the center distance is somewhat wrong, it can be meshed correctly;
It is easier to get the correct tooth shape, and it is easier to process;
Because of the rolling engagement on the curve, the rotational motion can be smoothly transmitted;
As long as the size of the gear teeth is the same, one tool can machine gears with different numbers of teeth;
The roots are thick and strong.
4) Base circle and index circle
The base circle is the base circle that forms the involute tooth shape. The index circle is the reference circle for determining the size of the gear. The base circle and the index circle are the important geometric dimensions of the gear. An involute tooth profile is a curve formed on the outside of the base circle. The pressure angle is zero degrees on the base circle.
5) Meshing of involute gears
The reference circles of two standard involute gears mesh tangentially at standard center-to-center distances.
When the two wheels mesh, it looks like two friction wheels (Friction wheels) with a diameter of d1 and d2 are driving. However, the meshing of involute gears actually depends on the base circle rather than the index circle.
The meshing contact points of the two gear tooth profiles move on the meshing line in the order of P1-P2-P3. Note the yellow teeth in the drive gear. For a period of time after this tooth starts to mesh, the gear is in two-tooth meshing (P1, P3). The meshing continues, and when the meshing point moves to point P2 on the index circle, there is only one meshing tooth left. The meshing continues, and when the meshing point moves to the point P3, the next gear tooth starts meshing at the point P1, and the state of two-tooth meshing is formed again. Just like this, the two-tooth meshing of the gear interacts with the single-tooth meshing to repeatedly transmit rotational motion.
The common tangent A-B of the base circle is called the line of engagement. The meshing points of the gears are all on this meshing line.
It is represented by an image, as if the belts are crossed over the outer circumferences of the two base circles to perform a rotational motion to transmit power.
6. The displacement of the gear is divided into positive displacement and negative displacement
The tooth profiles of the gears we usually use are generally standard involutes. However, there are also some cases where the gear teeth need to be shifted, such as adjusting the center distance and preventing the undercut of the pinion.
1) Number of teeth and shape of the gear
The involute profile curve varies with the number of teeth. The greater the number of teeth, the more straight line the tooth profile curve is. As the number of teeth increases, the tooth profile of the tooth root becomes thicker, and the strength of the gear teeth increases.
As can be seen from the above figure, for a gear with 10 teeth, part of the involute tooth profile at the root of the gear teeth is dug out, resulting in undercutting. However, if positive displacement is used for a gear with z=10 teeth, the diameter of the addendum circle is increased, and the tooth thickness of the gear teeth is increased, a gear strength equivalent to that of a gear with 200 teeth can be obtained.
2) Displacement gear
The figure below is a schematic diagram of the positive displacement of the gear with the number of teeth z=10. When cutting teeth, the movement of the tool along the radial direction xm (mm) is called radial displacement (referred to as displacement).
xm = displacement (mm)
x = displacement coefficient
m=modulus (mm)
Tooth profile change through positive displacement. The tooth thickness of the gear teeth increases, and the outer diameter (the diameter of the tip circle) also increases. By adopting a positive displacement of the gear, the occurrence of undercut (Undercut) can be avoided. The displacement of the gear can also achieve other purposes, such as changing Center distance, positive displacement can increase center distance, negative displacement can reduce center distance.
Whether it is a positive displacement gear or a negative displacement gear, there is a limit to the displacement amount.
3) Positive displacement and negative displacement
There are positive and negative displacements. Although the tooth height is the same, the tooth thickness is different. A gear with thicker teeth is a positive displacement gear, and a gear with a thinner tooth thickness is a negative displacement gear.
When the center distance of the two gears cannot be changed, positive displacement of the pinion gear (avoid undercutting), and negative displacement of the large gear, so that the center distance is the same. In this case, the absolute values of the displacement amounts are equal.
4) Meshing of the displacement gear
Standard gears are meshed in a state where the index circles of each gear are tangent. The meshing of the shifted gears, as shown in the figure, is tangential meshing on the meshing pitch circle. The pressure angle on the meshing pitch circle is called the meshing angle. The meshing angle is different from the pressure angle on the index circle (the index circle pressure angle). The meshing angle is an important factor when designing a displacement gear.
6) The role of gear displacement
It can prevent the undercut phenomenon caused by the small number of teeth during processing; the desired center distance can be obtained by displacement; when the gear ratio of a pair of gears is large, the pinion that is prone to wear can be positively displaced, Make the teeth thicker. Conversely, a negative displacement is performed on the large gear to make the tooth thickness thinner so that the lifespan of the two gears is similar.
7. Accuracy of gears
Gears are mechanical elements that transmit power and rotation. The performance requirements for gears mainly include:
Greater power transmission capacity;
Use the smallest possible gears;
low noise;
correctness.
In order to meet the above-mentioned requirements, improving the accuracy of gears will become a problem that must be solved.
1) Classification of gear accuracy
The accuracy of gears can be roughly divided into three categories:
a) Correctness of involute tooth profile - tooth profile accuracy
b) The correctness of the tooth line on the tooth surface - the tooth line accuracy
c) Correctness of tooth/gap position
Indexing Accuracy of Gear Teeth—Single Pitch Accuracy
Accuracy of Pitch - Cumulative Pitch Accuracy
The deviation of the position of the ball clamped between the two gears in the radial direction—radial runout accuracy
2) Tooth profile error
3) Tooth line error
4) Pitch error
The pitch value is measured on a measurement circle 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 pitch (Fp) is determined by measuring the pitch deviation of the whole wheel to make an evaluation. The total amplitude value of the pitch cumulative deviation curve is the total pitch deviation.
5) Radial runout (Fr)
Place the probes (spherical, cylindrical) one after the other in the tooth slot, and measure the difference between the maximum and minimum radial distances from the probe to the gear axis. The eccentricity of the gear shaft is part of the radial runout.
6) Total radial deviation (Fi")
So far, the tooth profile, pitch, and tooth line accuracy that we have described are all methods of evaluating the accuracy of a single gear. In contrast to this, there is also a method of a two-tooth surface meshing test for evaluating gear accuracy after meshing a gear with a measurement gear. The left and right tooth surfaces of the measured gear are in contact with the measuring gear and rotate for a full circle. Changes in center distance are recorded. The figure below shows the test results of a gear with 30 teeth. There are a total of 30 wavy lines for the radial comprehensive deviation of a single tooth. The total radial deviation value is approximately the sum of the radial runout deviation and the radial comprehensive deviation of a single tooth.
7) Correlation between various precisions of gears
The accuracy of each part of the gear is related. Generally speaking, the radial runout has a strong correlation with other errors, and the correlation between various pitch errors is also very strong.
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