Transmissions are used not only to transmit power, but also to provide the ability to adjust the mechanical advantage to the mechanism. As discussed in the introduction to this unit, in some cases the electric motor itself has sufficient power to perform a specific task, but the output characteristics of the electric motor do not meet the requirements. An electric motor that rotates VERY fast, but with very low torque, is not suitable for lifting heavy loads. In such cases, it becomes necessary to use a gear ratio to change the output characteristics and create a balance of torque and speed.

Imagine a bicycle: a cyclist has limited power, and wants to ensure maximum use of this power at any given time.

By changing the mechanical advantage, the speed of movement is changed. Power is the amount of work done per unit of time. The greater the amount of work. the lower the speed of its implementation.

Example 8.1 shows that if the lever moves 1 meter on the input side, the lever moves 4 meters on the output side. The difference is proportional to the ratio between the leverage lengths.

Output Length / Input Length = 8/2 = 4

It is interesting that both distances are covered at the same time. Let's imagine that a shift of the lever at the entrance by 1 meter occurs in 1 second, so that the speed of movement at the entrance is 1 meter per second. At the same time, an output offset of 4 meters also occurs in 1 second, so the speed here is 8 meters per second. Outlet speed MORE inlet speed due to the ratio between the lengths of the levers.

Example 8.2 presents the same system as in example 8.1, but now a force of 4 Newtons acts on the input. What is the resultant output force?

First of all, it is necessary to calculate the applied moment in the center of rotation caused by the input force using the formulas from Block 7:

Torque = Force x Distance from the center of gravity = 4 N x 2 m = 8 Nm

Next, it is necessary to calculate the resultant force at the output:

Force = Torque / Distance = 8 Nm / 8 m = 1 Newton

Looking at these two examples, we see that if the system shifts by 1 meter under the influence of an input force of 4 Newton, then at the output it will shift by 4 meters under the action of a force of 1 Newton. With less force, the lever moves faster!

We can see how a mechanical advantage (expressed in the form of levers) can be used to control the input force in order to obtain the desired output. Transmission works on the same principle.

A spur gear is essentially a series of levers. The larger the diameter of the gear, the longer the lever.

As can be seen from Example 8.3, the result of the torque applied to the first gear is the linear force arising at the tips of its teeth. The same force acts on the tips of the gear teeth, with which the first gear engages, causing the second to rotate by the action of torque. The diameters of the gears become the length of the levers, while the change in torque is equivalent to the ratio of the diameters. If the small gears drive more gears, the torque increases. If the large gears drive the small gears, the torque is reduced.

In Example 8.4, if the input 36-tooth gear rotates a distance of one tooth (d = width of 1 tooth), this means that it rotates 1 / 36th of its full revolution (a1 = 360/36 = 10 degrees). When turning, it sets in motion a 60-tooth gear, forcing the latter to also shift by 1 tooth. However, for a 60-tooth gear, this means an offset of only 1/60 of the full revolution (a2 = 360/60 = 6 degrees).

When the small gear travels a certain distance in a given time interval, the large gear travels a smaller distance. This means that the large gear rotates more slowly than the small. This principle works in both directions. If the small gears drive more gears, the speed decreases. If large gears drive small gears, speed increases.

From examples 8.1 - 8.4 it is seen that the ratio between the sizes of two gears interlocking with each other is proportional to the change in torque and speed between them. This is called the gear ratio.

As discussed above, the number of gear teeth is directly proportional to its diameter, therefore, to calculate the gear ratio, instead of the diameter, you can simply count the teeth.

The gear ratio is expressed as (gear teeth): (gear teeth), therefore the pair of gears presented above can be described as 12:60 (or 36 to 60).

The gear ratio is calculated by the formula (teeth of the driven gear) / (teeth of the pinion gear)

Therefore, gear ratio = gear teeth / gear teeth = 60/36 = 1.67

As discussed above, the gear ratio is expressed as (gear teeth): (gear teeth), so the pair of gears presented above can be expressed as 12:60 (or 12 to 60).

The gear ratio is calculated by the formula (teeth of the driven gear) / (teeth of the pinion gear)

Therefore, the gear ratio = Teeth of the driven gear / Teeth of the pinion gear = 60/12 = 5

Looking at the example presented above.

The maximum overload torque of the second shaft can be calculated by the formula:

Output torque = Input torque x Gear ratio

Output torque = 1.5 Nm x 5 = 7.5 Nm

The free speed of the second shaft can be calculated by the formula:

Output Speed = Input Speed / Gear Ratio = 100 rpm / 5 = 20 rpm

The second shaft, therefore, rotates at a free speed of 20 rpm, while the maximum overload torque is 7.5 Nm. As the speed decreases, the torque increases.

For the second example, calculations can be made in the same way.

Gear ratio = Driven gear teeth / Driven gear teeth = 12/60 = 0.2

Output torque = Input torque x Gear ratio = 1.5 Nm x 0.2 = 0.3 Nm

Output Speed = Input Speed / Gear Ratio = 100 rpm / 0.2 = 500 rpm

The second shaft, therefore, rotates at a free speed of 500 rpm, while the ultimate overload torque is 0.3 Nm. With increasing speed, the torque decreases.

# How to calculate gear ratio

How to calculate the gear ratio of mechanical gears.

In this article I will give an example of calculating the gear ratio of gears of different diameters, with different numbers of teeth. This calculation is used when it is important to determine, for example, the speed of rotation of the gearbox shaft with a known drive speed and tooth characteristics.

Naturally, it is possible to measure the frequency of rotation of the output shaft, however, in some cases, calculation is required. In addition, in theoretical mechanics, when designing various components and mechanisms, it is necessary to calculate gears in order to obtain a given rotation speed.

The term gear ratio is very ambiguous. It echoes the term gear ratio, which is not entirely true. Speaking of the gear ratio, we mean how many revolutions the driven wheel (gear) will make relative to the drive.

For a correct understanding of the processes and the structure of the gear, you should first familiarize yourself with GOST 16530-83.

So, consider an example of calculation using two gears.

To calculate the gear ratio, we must have at least two gears. This is called a gear train. Usually, the first gear is the drive gear and is located on the drive shaft, the second gear is called the driven gear and rotates when it engages with the drive gear. Pi this between them can be many other gears, which are called intermediate. To simplify the calculation, consider a gear with two gears.

In the example, we have two gears: the drive gear (1) and the driven gear (2). The easiest way is to count the number of teeth on the gears. Count the number of teeth on the pinion gear. You can also see the markings on the gear housing.

Imagine that a pinion gear (red) has 40 teeth, and a driven gear (blue) has 60 teeth.

Divide the number of teeth of the driven gear by the number of teeth of the pinion gear to calculate the gear ratio. In our example: 60/40 = 1.5. You can also record the answer as 3/2 or 1.5: 1.

Such a gear ratio means that the red, pinion gear must complete one and a half turns so that the blue, pinion gear makes one revolution.

Now let's complicate the task using more gears. Add another gear with 14 teeth to our gear. Let's make her the lead.

Let's start with the yellow pinion gear and move in the direction of the pinion gear. For each pair of gears, we calculate our gear ratio. We have two pairs: yellow-red, red-blue. In each pair, we consider the first gear as the leading gear, and the second as the driven gear.

In our example, the gear ratios for the idler are 40/14 = 2.9 and 60/40 = 1.5.

We multiply the values of the gear ratios of each pair and get the total gear ratio of the gear transmission: (20/7) × (30/20) = 4.3. That is, to calculate the gear ratio of the entire gear transmission, it is necessary to multiply the gear ratios for the intermediate gears.

We now determine the speed.

Using the gear ratio and knowing the speed of the yellow gear, you can easily calculate the speed of the driven gear. Typically, the speed is measured in revolutions per minute (rpm). Consider an example of a gear train with three gears. Suppose that the speed of the yellow gear is 340 rpm. We calculate the frequency of rotation of the red gear.

We will use the formula: S1 × T1 = S2 × T2,

S1 - frequency of rotation of the yellow (leading) gear,

T1 - the number of teeth of the yellow (leading) gear,

S2 - frequency of rotation of the red gear,

T2 - the number of teeth of the red gear.

In our case, you need to find S2, but by this formula you can find any variable.

340 rpm × 7 = S2 × 40

It turns out that if the leading yellow gear rotates at a frequency of 340 rpm, then the driven, red gear will rotate at a speed of about 60 rpm. In the same way, we calculate the frequency of rotation of the red-blue pair. The result - the speed of rotation of the blue gear - will be the desired speed of rotation of the entire gear transmission.

## General definition

A good example of changing the speed is easiest to observe on a simple bike. Man slowly pedals. The wheel spins much faster. The change in the number of revolutions occurs due to 2 sprockets connected in a chain. When a large one, rotating with the pedals, makes one revolution, a small one, standing on the rear hub, scrolls several times.

### Torque transmission

The mechanisms use several types of gears that change the torque. They have their own characteristics, positive qualities and disadvantages. The most common programs:

Belt transmission is the simplest in execution. It is used to create home-made machines, in machine tools to change the speed of rotation of the working unit, in cars.

The belt is pulled between 2 pulleys and transmits rotation from the master to the follower. Performance is low as the belt slides over a smooth surface. Thanks to this, the belt unit is the safest way to transmit rotation. When overloaded, the belt slips and the driven shaft stops.

The transmitted number of revolutions depends on the diameter of the pulleys and the coefficient of adhesion. The direction of rotation does not change.

The transitional structure is a belt gear.

There are projections on the belt, and teeth on the gear. This type of belt is located under the hood of the car and connects the sprockets on the axles of the crankshaft and carburetor. When overloaded, the belt breaks, as this is the cheapest part.

The chain consists of sprockets and a chain with rollers. The transmitted speed, force and direction of rotation do not change. Chain transmissions are widely used in transport mechanisms, on conveyors.

### Gear characteristic

In a gear transmission, the driving and driven parts interact directly, due to the engagement of the teeth. The basic rule of operation of such a node is that the modules should be the same. Otherwise, the mechanism will jam. It follows that the diameters increase in direct proportion to the number of teeth. Some values can be replaced by others in the calculations.

Module - the size between the same points of two adjacent teeth.

For example, between the axes or points on the involute in the midline The size of the module consists of the width of the tooth and the gap between them. It is better to measure the module at the intersection of the base line and the axis of the tooth. The smaller the radius, the more the gap between the teeth is distorted along the outer diameter, it increases to the top of the nominal size. Ideal forms of involute can practically only be on the rail. Theoretically on a wheel with a maximum infinite radius.

A part with fewer teeth is called a gear. Usually it is driving, transmitting torque from the engine.

The gear wheel has a larger diameter and is driven in pairs. It is connected to the work node. For example, it transfers the rotation with the necessary speed to the wheels of the car, the spindle of the machine.

Typically, gear speed decreases and power increases. If a pair of a part with a larger diameter, the leading one, at the output the gear has a larger number of revolutions, it rotates faster, but the power of the mechanism decreases. Such transfers are called downshifts.

### Why do you need a parasite

When the gear and wheel interact, several values change at once:

- number of revolutions
- power
- direction of rotation.

Only in planetary units with cutting of teeth along the inner diameter of the crown the direction of rotation is preserved. With external gearing, two identical gears are put in a row. Their interaction does not change anything except the direction of movement. In this case, both gear parts are called gears, no wheels. The second, intermediate, was called the “parasite,” since it is not involved in the calculations, it only changes sign.

### Types of gear joints

The gearing may have a different tooth shape on the parts. It depends on the initial load and the location of the axes of the mating parts. There are types of gear movable joints:

The most common and easy to perform spur gearing. The outer surface of the tooth is cylindrical. The arrangement of the axles of the gear and wheel is parallel. The tooth is located at right angles to the end face of the part.

When there is no way to increase the width of the wheel, but it is necessary to transfer a lot of force, the tooth is cut at an angle and due to this, the contact area is increased. The gear ratio calculation does not change. The node is becoming more compact and powerful.

The lack of helical gears in the additional load on the bearings. The force from the pressure of the driving part acts perpendicular to the plane of contact. In addition to radial, axial force appears.

To compensate for the voltage along the axis and to further increase the power allows the chevron connection. The wheel and gear have 2 rows of oblique teeth directed in different directions. The transmission number is calculated similarly to spur gearing according to the ratio of the number of teeth and diameters. Chevron gearing is complicated. It is placed only on mechanisms with a very large load.

In the bevel gear, the axles are angled. The work element is cut in a conical plane. The gear ratio of such pairs can be 1, when you only need to change the plane of action of the force. To increase the power, a semicircular tooth is cut. The transmitted number of revolutions is considered only by tooth, the diameter is mainly used in calculating the dimensions of the node.

The helical gear has a tooth cut at an angle of 45 °. This allows you to arrange the axis of the working elements perpendicularly in different planes.

The worm gear does not have a gear; the worm replaces it. Axes of parts do not intersect. They are perpendicular in space, but in different planes. The gear ratio of the pair is determined by the number of thread starts on the worm.

In addition to the above, they produce other types of transmissions, but they are extremely rare and do not belong to the standard ones.

### Multi-stage gearboxes

How to choose the right gear ratio. The engine usually produces several thousand revolutions per minute. At the exit - the wheels of the car and the spindle of the machine, this rotation speed will lead to an accident. The power of the executing mechanism is not enough for the working tool to cut metal and the wheels to move the car. One pair of gearing will not be able to provide the required lowering or the driven part must be huge.

Создается многоступенчатый узел с несколькими парами зацеплений. Передаточное число редуктора считается как произведение чисел каждой пары.

U_{р} - gear ratio

Before you select the gear ratio of the gearbox, you need to determine the number of pairs, the direction of rotation of the output shaft, and make the calculation in the reverse order, based on the maximum permissible wheel dimensions.

In a multi-stage gearbox, all gear parts located between the pinion gear at the gearbox input and the driven gear ring on the output shaft are called intermediate gears. Each individual pair has its own transmitting number, gear and wheel.

### Gearbox and gearbox

Any gearbox with gearing is a gearbox, but the converse is not true.

The gearbox is a gearbox with a movable shaft on which gears of different sizes are located. Moving along the axis, it includes in the work of one or the other pair of parts. The change occurs due to the alternate connection of various gears and wheels. They differ in diameter and transmitted speed. This makes it possible to change not only speed, but also power.

### Car transmission

In the machine, the translational movement of the piston is converted into a rotational crankshaft. Transmission is a complex mechanism with a large number of different nodes interacting with each other. Its purpose is to transfer rotation from the engine to the wheels and adjust the number of revolutions - the speed and power of the car.

The transmission includes several gearboxes. This is, first of all:

- gearbox - speeds
- differential.

The gearbox in the kinematic scheme immediately stands behind the crankshaft, changes the speed and direction of rotation.

By switching - moving the shaft, the gears on the shaft are connected alternately with different wheels. When the reverse speed is turned on, the direction of rotation changes through the parasitic, the car as a result moves backward.

The differential is a bevel gear with two output shafts located on the same axis opposite each other. They look in different directions. The gear ratio of the differential gear is small, within 2 units. It changes the position of the axis of rotation and direction. Due to the location of the bevel gears opposite each other, when engaged with one gear, they rotate in the same direction relative to the position of the axis of the car, and transmit the torque directly to the wheels. The differential changes the speed and direction of rotation of the driven cones, and behind them the wheels.

## How to calculate gear ratio

The gear and wheel have a different number of teeth with the same module and a proportional size of the diameters. The gear ratio shows how many revolutions the driving part will make to rotate the follower in a full circle. Gears have a rigid connection. The transmitted number of revolutions in them does not change. This negatively affects the operation of the unit in conditions of overload and dust. The tooth cannot slip like a belt over a pulley and breaks.

### Calculation without resistance

In calculating the gear ratio, the number of teeth on each part or their radii is used.

Where u_{12} - gear ratio of gears and wheels,

Z_{2} and Z_{1} - respectively, the number of teeth of the driven wheel and pinion gear.

The sign “+” is set if the direction of rotation does not change. This applies to planetary gears and gears with cutting teeth along the inner diameter of the wheel. In the presence of a parasite - intermediate parts located between the pinion gear and the ring gear, the direction of rotation changes, as with the external connection. In these cases, “-” is put in the formula.

When two parts are externally connected by means of a parasite located between them, the gear ratio is calculated as the ratio of the number of teeth of the wheel and gear with the “+” sign. The parasite does not participate in the calculations, it only changes direction, and accordingly the sign in front of the formula.

Usually, a clockwise direction is considered positive. The sign plays an important role in the calculation of multi-stage gearboxes. The gear ratio of each transmission is determined separately in the order in which they are located in the kinematic chain. The sign immediately shows the direction of rotation of the output shaft and the working unit, without additional charting.

The calculation of the gear ratio of a gearbox with multiple gears - multi-stage, is determined as the product of gear ratios and is calculated by the formula:

The gear ratio calculation method allows you to design a gearbox with predetermined output values of the number of revolutions and theoretically find the gear ratio.

Rigid gearing is rigid. Parts cannot slip relative to each other, as in a belt drive and change the ratio of the number of rotations. Therefore, the output speed does not change, do not depend on overload. The calculation of the angular velocity and the number of revolutions is correct.

### Gear efficiency

For a real gear ratio calculation, additional factors should be considered. The formula is valid for angular velocity, as for the moment of force and power, they are much smaller in a real gearbox. Their value reduces the resistance of the transmission moments:

- friction of adjoining surfaces,
- bending and twisting of parts under the influence of force and resistance to deformation,
- loss on dowels and slots,
- friction in bearings.

Each type of connection, bearing and assembly has its own correction factors. They are included in the formula. Designers do not make calculations for the bending of each key and bearing. The reference book contains all the necessary coefficients. If necessary, they can be calculated. Formulas do not differ in simplicity. They use elements of higher mathematics. The calculations are based on the ability and property of chromium-nickel steels, their ductility, tensile strength, bending, kink, and other parameters, including the dimensions of the part.

As for the bearings, in the technical reference book by which they are selected, all the data for calculating their operating condition are indicated.

When calculating the power, the main indicator of gearing is the contact spot, it is indicated in percent and its size is of great importance. Perfectly shaped and touched throughout the involute can only have painted teeth. In practice, they are manufactured with an error of several hundredths of a millimeter. During operation of the unit under load on the involute, spots appear in the places where the parts act on each other. The larger the area on the tooth surface they occupy, the better the force transmitted during rotation.

All coefficients are combined together, and the result is the value of the efficiency of the gearbox. The efficiency is expressed as a percentage. It is determined by the ratio of power on the input and output shafts. Ches more gears, joints and bearings, the lower the efficiency.

## Gear ratio

The gear ratio is the same as the gear ratio. The magnitude of the angular velocity and the moment of force varies in proportion to the diameter, and accordingly to the number of teeth, but has the opposite value.

The larger the number of teeth, the lower the angular velocity and the impact force - power.

In a schematic representation of the magnitude of the force and displacement, the gear and wheel can be represented as a lever with support at the point of contact of the teeth and sides equal to the diameters of the mating parts. When shifted by 1 tooth, their extreme points pass the same distance. But the angle of rotation and torque on each part is different.

For example, a gear with 10 teeth rotates 36 °. At the same time, the part with 30 teeth moves 12 °. The angular velocity of a part with a smaller diameter is much larger, 3 times. At the same time, the path that the point passes on the outer diameter is inversely proportional. On the gear, the movement of the outer diameter is less. The moment of force increases inversely with the ratio of displacement.

Torque increases with the radius of the part. It is directly proportional to the size of the impact arm - the length of the imaginary lever.

The gear ratio shows how much the moment of force has changed when it was transmitted through gearing. The digital value matches the transmitted speed.

The gear ratio of the gearbox is calculated by the formula:

where u_{12} - gear ratio of the gear relative to the wheel,

ω_{1} and ω_{2} - the angular velocity of the leading and driven element of the connection,

The ratio of angular velocities can be calculated through the number of teeth. In this case, the direction of rotation is not taken into account and all numbers with a positive sign.

The gear transmission has the highest efficiency and the least protection against overload - the element of application of force breaks, you have to make a new expensive part with complicated manufacturing technology.

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