Gears are toothed mechanical parts 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 BC, the famous ancient Greek philosopher Aristotle recorded gears in the literature.Around 250 BC, the mathematician Archimedes also described in the literature a windlass using a turbine worm.Gears from BC are still preserved in the remains of Katsfin in Iraq today.
Gear has a long history in our country.According to historical records, gears were used in ancient China as far back as 400-200 BC.The bronze gear unearthed in Shanxi, my country is the oldest gear that has been discovered so far. As a guide car reflecting the achievements of ancient science and technology, it is a mechanical device with a gear mechanism as the core.During the Italian Renaissance in the second half of the 15th century, the famous all-rounder Leonardo.Da Vinci, not only in culture and art, but also left indelible achievements in the history of gear technologyAfter more than 500 years, the current gear still retains the original sketch at that time.
It was not until the end of the 17th century that people began to study the shape of gear teeth that could correctly transmit motion.In the 18th century, after the European Industrial Revolution, the application of gear transmission became more and more extensive; first, the development of cycloid gears, and then the involute gears, until the beginning of the 20th century, involute gears have been dominant in applications.Later, they developed displacement gears, arc gears, bevel gears, helical gears, and so on.
Modern gear technology has reached:齿轮模数0.004-100毫米；齿轮直径由1毫米-150米；传递功率可达十万千瓦；转速可达十万转/分；最高的圆周速度达300米/秒。
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 categories
There are many types of gears, and the most common classification method is based on the gear axis.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: straight bevel gears, spiral bevel gears, zero-degree bevel gears, etc.
3) Cross-axis gears: There are cross-axis helical gears, worm gears, hypoid gears, etc.
The efficiency listed in the above table is the transmission efficiency and does not include the loss of 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.Cross-axis helical gears and worm gears and other cross-axis gear pairs are driven by relative sliding to generate rotation, so the influence of friction is very large, and the transmission efficiency is reduced compared with other gears.The efficiency of the gear is the transmission efficiency of the gear under normal assembly conditions.If the installation is incorrect, especially when the bevel gear assembly distance is incorrect and the intersection point with the cone has an error, its efficiency will be significantly reduced.
3.1 Gears with parallel shafts
1) Spur gear
The tooth line and the axis line are cylindrical gears in parallel directions.Because it is easy to process, it is the most widely used in power transmission.
2) Rack
A linear rack gear that meshes with a spur gear.It can be regarded as a special case when the pitch circle diameter of the spur gear becomes infinite.
3) Internal gear
A gear that meshes with a spur gear and has gear teeth on the inner side of the ring.Mainly used in applications such as planetary gear transmissions and gear couplings.
4) Helical gear
The tooth line is a cylindrical gear with a spiral line.Because it is stronger than spur gears and runs smoothly, it is widely used.Axial thrust is generated during transmission.
5) Helical rack.
A pinion gear that meshes with a helical gear.This is equivalent to the situation when the pitch diameter of the helical gear becomes infinite.
6) Herringbone gear
The tooth line is a gear composed of two helical gears with left-handed and right-handed rotation.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 cone line.Among bevel gears, they are relatively easy to manufacture.Therefore, it has a wide range of applications as a bevel gear for transmission.
2) Spiral bevel gear
The tooth line is a curved bevel gear with a helix angle.Although it is more difficult to manufacture than straight bevel gears, it is also widely used as a high-strength, low-noise gear.
3) Zero degree bevel gear
Curved bevel gears with a helix angle of zero degrees.Because it has the characteristics of both straight and curved bevel gears, the force on the tooth surface is the same as that of a straight bevel gear.
3.3 Cross-shaft gears
1)Cylindrical worm pair
Cylindrical worm pair is a general term for cylindrical worm and worm gear meshing 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) Cross-axis helical gear
The name of the cylindrical worm pair when it is transmitted between crossed shafts.It can be used in the case of a helical gear pair or a helical gear and a spur gear pair.Although it runs smoothly, it is only suitable for use under light load conditions.
3.4 Other special gears
1) Face gear
A disc-shaped gear that can mesh with a spur gear or a helical gear.Transmission between right-angled shafts and crossed shafts.
2) Drum worm pair
The general term for drum worms and worm gears that mesh with them.Although it is more difficult to manufacture, it can transmit large loads compared to cylindrical worm gears.
3) Hypoid gear
Conical gears that drive between crossed shaftsLarge and small gears are processed eccentrically, similar to spiral gears, and the meshing principle is very complicated.
4. Basic terminology and size calculation of gears
Gears have a lot of specific terms and expressions of gears. In order to enable everyone 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 gear teeth is modulus
m1、m3、m8…被称为模数1、模数3、模数8。Modulus is a universal name in the world. The symbols m (modulus) and numbers (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), symbols (diameter pitches) and numbers (number of gear teeth when the pitch circle diameter is 1 inch) are used to indicate the size of the gear teeth.For example: DP24, DP8, etc.There are also special addressing methods that use symbols (circle section) and numbers (millimeters) to indicate the size of gear teeth, such as CP5 and CP10.
Multiply the modulus by the circumference ratio to get the tooth pitch (p), which is the length between two adjacent teeth.
The formula is:
p=pi ratio x modulus = πm
Comparison of tooth sizes with 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 popular.
The pressure angle is the angle between the radius line and the tangent of the tooth profile at a point (usually a node) on the tooth surface.As shown in the figure, α is the pressure angle.Because α'=α, α'is also the pressure angle.
When the meshing state of tooth A and tooth B looks from the node:
Tooth A pushes point B on the node.The driving force at this time acts on the common normal of tooth A and tooth B.In other words, the common normal is the direction of force and the direction of pressure, and α is the pressure angle.
Modulus (m), pressure angle (α) plus the number of teeth (z) are the three basic parameters of gears,Calculate the size of each part of the gear based on this parameter.
4) Tooth height and tooth thickness
The height of the teeth is determined by the modulus (m).
Total tooth height h=2.25m (=tooth root height+tooth top height)
The tooth tip height (ha) is the height from the tooth tip to the index line. ha=1m.
The tooth root height (hf) is the height from the tooth root to the index line. hf=1.25m.
The basis of 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 index circle diameter (d) of the gear.The pitch, tooth thickness, tooth height, tooth tip height, and tooth root height can be determined based on the index circle.
Index circle diameter d=zm
Addendum circle diameter da=d+2m
Tooth root circle diameter df=d-2.5m
The index circle cannot be seen directly in the actual gear, because the index circle is a circle assumed to determine the size of the gear.
6) Center distance and backlash
When the index circles of a pair of gears are tangentially engaged, 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, backlash is an important factor in order to obtain a smooth meshing effect.Backlash is the gap between the tooth surfaces when a pair of gears mesh.
There is also a gap in the tooth height direction of the gear.This gap is called Clearance.The head clearance (c) is the difference between the tooth root height of the gear and the addendum height of the matching gear.
Head clearance c=1.25m-1m=0.25m
7) Helical gear
A helical gear is a helical gear after the gear teeth of the spur gear are twisted in a spiral shape.Most of the geometry of spur gears can be applied to helical gears.There are two ways for helical gears according to their reference planes:
End face (shaft right angle) datum (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., gears with helical teeth, the spiral direction and coordination are fixed.The spiral 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 coordination of various gears is shown below.
5. The most commonly used gear tooth profile is the involute tooth profile
Just divide the tooth pitch into equal parts on the outer circumference of the friction wheel, install the protrusions, and then mesh and rotate with each other, the following problems will occur:
- The tangent point of the gear tooth slips
- The moving speed of the tangent point is fast and slow
- Produce vibration and noise
The gear transmission needs to be quiet and smooth, and thus the involute curve is born.
1) What is an involute
Wrap a thread with a pencil on one end around the outer circumference of the cylinder, and then gradually release the thread while the thread is taut.At this time, 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, tie 8 pencils to draw 8 involute curves.Then, wind the line in the opposite direction and draw 8 curves in the same way. This is a gear with an involute curve as the tooth profile and 8 teeth.
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 profile, and it is easier to process;
- Because it rolls and meshes on the curve, it can smoothly transmit the rotation movement;
- 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 strong 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 that determines the size of the gear.Base circle and index circle are important geometric dimensions of gears.The 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 index circles of two standard involute gears mesh tangentially at the standard center distance.
When the two wheels are engaged, it looks like two friction wheels (Friction wheels) with index circle diameters d1 and d2 are driving.However, in fact the meshing of the involute gear 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.After this tooth starts to mesh for a period of time, the gear is two-tooth meshing (P1, P3).The meshing continues. When the meshing point moves to the point P2 on the index circle, there is only one meshing gear tooth left.The meshing continues, and when the meshing point moves to the point P3, the next gear tooth starts to mesh at the P1 point, and the state of two teeth meshing is formed again.Just like this, the two-tooth meshing and single-tooth meshing of the gear alternately repeatedly transmit rotational motion.
The common tangent line A-B of the base circle is called the meshing line.The meshing points of the gears are all on this meshing line.
It is represented by a vivid picture, as if the belt is crossed on the outer circumference of the two base circles to transmit power in a rotary motion.
6. Gear displacement is divided into positive displacement and negative displacement
The tooth profiles of the gears we usually use are generally standard involutes. However, there are some situations where the gear teeth need to be modified, such as adjusting the center distance and preventing undercutting of the pinion.
1) Number and shape of gear teeth
The involute tooth profile curve varies with the number of teeth.The more the number of teeth, the more straight the tooth profile curve.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 figure above, for a gear with 10 teeth, part of the involute tooth profile at the root of the tooth has been excavated, causing undercutting.However, if the positive displacement is adopted for the gear with the number of teeth z=10, the diameter of the addendum circle and the tooth thickness of the gear teeth are increased, the same level of gear strength as the gear with 200 teeth can be obtained.
2) Modified gear
The figure below is a schematic diagram of positive displacement gear cutting with the number of teeth z=10.When cutting teeth, the movement of the tool along the radius xm (mm) is called radial displacement (referred to as displacement).
xm=Displacement amount (mm)
x = displacement coefficient
m=Modulus (mm)
The tooth profile changes through positive displacement.The tooth thickness of the gear teeth increases, and the outer diameter (the diameter of the addendum circle) also increases. The gear adopts positive displacement to avoid undercut. The gear displacement can also achieve other purposes, such as changing Center distance, positive displacement can increase center distance, negative displacement can reduce center distance.
Regardless of whether it is a positive displacement or a negative displacement gear, there are restrictions on the displacement amount.
3) Positive and negative displacement
There are positive and negative displacements.Although the tooth height is the same, the tooth thickness is different.The gears with thicker teeth are positively shifted gears, and those with thinner teeth are negatively shifted gears.
When the center distance of the two gears cannot be changed, the small gear is positively modified (to avoid undercutting), and the large gear is negatively modified to make the center distance the same.In this case, the absolute value of the displacement is equal.
4) Meshing of shifting gears
The standard gear is meshed when the index circle of each gear is 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 engagement angle is different from the pressure angle on the index circle (index circle pressure angle).The meshing angle is an important element when designing a displacement gear.
6) The role of gear displacement
It can prevent undercutting due to the small number of teeth during processing;The desired center distance can be obtained by displacement;In the case of a pair of gears with a large gear ratio, the pinion gear that is prone to wear is positively modified to increase the tooth thickness.On the contrary, the large gear is negatively displaced to make the tooth thickness thinner so that the life of the two gears is close.
7. Gear accuracy
Gears are mechanical elements that transmit power and rotation.The performance requirements for gears mainly include:
- Greater power transmission capacity
- Use small gears as much as possible
- low noise
- Correctness
To meet the above 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)The accuracy of the involute tooth profile-the accuracy of the tooth profile
b)The accuracy of the tooth line on the tooth surface-the accuracy of the tooth line
c)Accuracy of tooth/tooth position
- Gear indexing accuracy—single pitch accuracy
- Accuracy of tooth pitch—accumulated tooth pitch accuracy
- The deviation of the probe ball clamped between the two gears in the radial direction—radial runout accuracy
2) Tooth profile error
3) Tooth line error
4) Tooth pitch error
Measure the tooth pitch value on the measuring 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 tooth pitch (Fp) measures the tooth pitch deviation of the all-wheel to make an evaluation.The total amplitude value of the cumulative pitch deviation curve is the total pitch deviation.
5) Radial runout (Fr)
Place the probe (spherical, cylindrical) in the tooth groove one after another, and measure the difference between the maximum and minimum radial distance between the probe and 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, tooth line accuracy, etc. we have described are all methods to evaluate the accuracy of a gear unit.The difference is that there is also a two-tooth surface meshing test method that evaluates the accuracy of the gear after meshing the gear with the measuring 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 center distance.The figure below is the test result of a gear with 30 teeth.There are 30 wavy lines for single tooth radial comprehensive deviation.The total radial deviation is approximately the sum of the radial runout deviation and the single tooth radial deviation.
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 tooth pitch errors is also strong.
8) Conditions for high-precision gears
Calculation of standard spur gears (small gear①, large gear②)
Shift spur gear calculation formula (small gear①, big gear②)
Calculation formula of standard helical tooth (tooth right angle method) (pinion ①, large gear ②)
Calculation formula of shifting helical teeth (tooth right angle method) (pinion ①, large gear ②)
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