The number of repetitions of any events or their occurrence in one timer unit is called the frequency. This physical quantity measured in hertz - Hz (Hz). It is denoted by the letters ν, f, F, and is the ratio of the number of recurring events to the period of time during which they occurred.

When an object revolves around its center, we can talk about such a physical quantity as the frequency of rotation, formula:

• N is the number of revolutions around an axis or around a circle,
• t is the time during which they were made.

In the SI system, it is denoted as - s-1 (s-1) and is referred to as revolutions per second (r / s). Other units of rotation are also used. When describing the rotation of the planets around the Sun, they speak of revolutions in hours. Jupiter rotates once every 9.92 hours, while the Earth and Moon rotate in 24 hours.

Rated rotation speed

Before defining this concept, it is necessary to determine what the nominal mode of operation of a device is. This is such an order of operation of the device, in which the greatest efficiency and reliability of the process are achieved over a long period of time. Based on this, the rated rotation speed is the number of revolutions per minute when operating in the nominal mode. The time required for one revolution is 1/v seconds. It is called the rotation period T. So, the relationship between the period of revolution and the frequency has the form:

For your information. The frequency of rotation of the shaft of an asynchronous motor is 3000 rpm, this is the rated speed of rotation of the output shank of the shaft at the rated operating mode of the electric motor.

How to find or find out the rotation frequencies of various mechanisms? For this, a device called a tachometer is used.

Angular velocity

When a body moves in a circle, not all of its points move at the same speed relative to the axis of rotation. If we take the blades of a conventional household fan that rotate around the shaft, then the point located closer to the shaft has a rotation speed greater than the marked point on the edge of the blade. This means that they have different linear speed of rotation. At the same time, the angular velocity of all points is the same.

Angular velocity is the change in angle per unit time, not distance. It is denoted by the letter of the Greek alphabet - ω and has a unit of radians per second (rad / s). In other words, the angular velocity is a vector tied to the axis of rotation of the object.

The formula for calculating the relationship between the angle of rotation and the time interval looks like this:

ω = ∆ϕ/∆t,

• ω is the angular velocity (rad./s);
• ∆ϕ is the change in the angle of deflection during rotation (rad.);
• ∆t is the time spent on the deviation (s).

The designation of the angular velocity is used in the study of the laws of rotation. It is used to describe the motion of all rotating bodies.

Angular velocity in specific cases

In practice, they rarely work with angular velocity values. It is needed in the design development of rotating mechanisms: gearboxes, gearboxes and other things.

You can calculate it using the formula. To do this, use the relationship of the angular velocity and rotational speed.

ω \u003d 2 * π / T \u003d 2 * π * ν,

• π is a number equal to 3.14;
• ν - rotational speed, (rpm).

As an example, the angular velocity and rotational speed of the wheel disk during the movement of the walk-behind tractor can be considered. Often it is necessary to reduce or increase the speed of the mechanism. For this, a device in the form of a gearbox is used, with the help of which the speed of rotation of the wheels is reduced. At top speed movement of 10 km / h, the wheel makes about 60 rpm. After converting minutes to seconds, this value is 1 rpm./s. After substituting the data into the formula, the result will be:

ω \u003d 2 * π * ν \u003d 2 * 3.14 * 1 \u003d 6.28 rad / s.

For your information. Reducing the angular velocity is often required in order to increase the torque or tractive effort of the mechanisms.

How to determine angular velocity

The principle of determining the angular velocity depends on how the movement occurs in a circle. If evenly, then the formula is used:

If not, then you will have to calculate the values ​​​​of the instantaneous or average angular velocity.

The quantity in question is vector, and Maxwell's rule is used to determine its direction. In common parlance - the gimlet rule. The velocity vector has the same direction as the translational movement of the screw having a right-hand thread.

Let's consider an example of how to determine the angular velocity, knowing that the angle of rotation of a disk with a radius of 0.5 m varies according to the law ϕ = 6*t:

ω = ϕ / t = 6 * t / t = 6 s-1

The vector ω changes due to the rotation in space of the axis of rotation and when the value of the modulus of the angular velocity changes.

Angle of rotation and period of revolution

Consider point A on an object rotating around its axis. When turning over a certain period of time, it will change its position on the circle line by a certain angle. This is the angle of rotation. It is measured in radians, because the unit is taken as a segment of a circle equal to the radius. Another measure of the angle of rotation is a degree.

When, as a result of the rotation, point A returns to its original place, it means that it has made a complete revolution. If its movement is repeated n times, then they talk about a certain number of revolutions. Based on this, one can consider 1/2, 1/4 turn and so on. A vivid practical example of this is the path that the cutter makes when milling a part fixed in the center of the machine spindle.

Attention! The angle of rotation has a direction. It is negative when the rotation is clockwise and positive when the rotation is counterclockwise.

If the body moves uniformly along the circle, we can talk about a constant angular velocity during movement, ω = const.

In this case, characteristics such as:

• period of revolution - T, this is the time required for a complete revolution of a point in a circular motion;
• frequency of revolution - ν, this is the total number of revolutions that a point makes along a circular path in a single time interval.

Interesting. According to known data, Jupiter revolves around the Sun in 12 years. When the Earth during this time makes almost 12 revolutions around the Sun. The exact value of the period of revolution of a round giant is 11.86 Earth years.

Cyclic speed (circulation)

Scalar quantity that measures frequency rotary motion, is called the cyclic rotation frequency. This is an angular frequency equal not to the angular velocity vector itself, but to its modulus. It is also called radial or circular frequency.

The cyclic frequency of rotation is the number of revolutions of the body in 2 * π seconds.

For AC motors, this frequency is asynchronous. Their rotor speed lags behind the speed magnetic field stator. The value that determines this lag is called slip - S. In the process of sliding, the shaft rotates, because an electric current appears in the rotor. Slip is permissible up to a certain value, the excess of which leads to overheating of the asynchronous machine, and its windings can burn out.

The device of this type of motor differs from the device of DC machines, where the conductive frame rotates in the field of permanent magnets. A large number of the frame contained the anchor, many electromagnets formed the basis of the stator. In three-phase AC machines, the opposite is true.

When an induction motor is running, the stator has a rotating magnetic field. It always depends on the parameters:

• mains frequency;
• number of pole pairs.

The speed of rotation of the rotor is in direct proportion to the speed of the magnetic field of the stator. The field is created by three windings, which are located at an angle of 120 degrees relative to each other.

Change from angular to linear speed

There is a difference between the linear speed of a point and the angular speed. When comparing the values ​​in the expressions describing the rules of rotation, you can see the commonality between these two concepts. Any point B belonging to a circle with radius R makes a path equal to 2*π*R. In doing so, she makes one turn. Considering that the time required for this is the period T, the modular value of the linear speed of point B is located by the following action:

ν \u003d 2 * π * R / T \u003d 2 * π * R * ν.

Since ω = 2*π*ν, it turns out:

Therefore, the linear speed of point B is greater, the farther from the center of rotation the point is.

For your information. If we consider cities at the latitude of St. Petersburg as such a point, their linear speed is relative to earth's axis is equal to 233 m/s. For objects at the equator - 465 m/s.

The numerical value of the acceleration vector of point B, moving uniformly, is expressed throughRand angular velocity, thus:

a = ν2/ R, substituting here ν = ω* R, we get: a = ν2/ R = ω2* R.

This means that the greater the radius of the circle along which the point B moves, the greater the value of its acceleration modulo. The farther a point of a rigid body is located from the axis of rotation, the greater its acceleration.

Therefore, it is possible to calculate accelerations, modules of velocities of necessary points of bodies and their positions at any moment of time.

Understanding and ability to use calculations and not get confused in definitions will help in practice to calculate linear and angular velocities, as well as freely move from one value to another in calculations.

Video

Sometimes, in relation to cars, questions from mathematics and physics pop up. In particular, one of these issues is the angular velocity. It is related to both the operation of mechanisms and the passage of turns. Let's figure out how to determine this value, what it is measured in and what formulas should be used here.

How to determine the angular velocity: what is this value?

From a physical and mathematical point of view, this value can be defined as follows: these are data that show how fast a certain point rotates around the center of the circle along which it moves.

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This seemingly purely theoretical value is of considerable practical importance in the operation of the car. Here are just a few examples:

• It is necessary to correctly correlate the movements with which the wheels rotate when turning. The angular velocity of the wheel of a car moving along the inner part of the trajectory must be less than that of the outer one.
• It is required to calculate how fast the crankshaft rotates in the car.
• Finally, the car itself, when passing a turn, also has a certain amount of movement parameters - and in practice, the stability of the car on the track and the likelihood of a rollover depend on them.

The formula for the time it takes for a point to rotate around a circle of a given radius

In order to calculate the angular velocity, the following formula is used:

ω = ∆φ /∆t

• ω (read "omega") - actually calculated value.
• ∆φ (pronounced “delta phi”) is the angle of rotation, the difference between the angular position of the point at the first and last moment of the measurement.
• ∆t
  (read "delta te") - the time during which this very shift occurred. More precisely, since "delta" means the difference between the time values ​​at the moment when the measurement was started and when it was finished.

The above formula for angular velocity applies only in general cases. Where we are talking about uniformly rotating objects or about the relationship between the movement of a point on the surface of a part, the radius and time of rotation, it is required to use other relationships and methods. In particular, the rotation frequency formula will already be needed here.

Angular velocity is measured in a variety of units. In theory, rad/s (radian per second) or degree per second is often used. However, this value means little in practice and can only be used in design work. In practice, it is more measured in revolutions per second (or minute, if we are talking about slow processes). In this regard, it is close to the frequency of rotation.

Angle of rotation and period of revolution

Much more common than angle of rotation is rotation frequency, which indicates how many revolutions an object makes in a given period of time. The fact is that the radian used for calculations is the angle in the circle when the length of the arc is equal to the radius. Accordingly, in whole circle there are 2 π radians. The number π is irrational, and it cannot be reduced to either a decimal or a simple fraction. Therefore, in the event that a uniform rotation occurs, it is easier to count it in frequency. It is measured in rpm - revolutions per minute.

If the matter does not concern a long period of time, but only that during which one revolution occurs, then the concept of the period of circulation is used here. It shows how fast one circular motion is made. The unit of measurement here is the second.

The relationship between angular velocity and rotational speed or period of revolution shows following formula:

ω = 2 π / T = 2 π *f,

• ω is the angular velocity in rad/s;
• T is the circulation period;
• f is the rotation frequency.

You can get any of these three values ​​​​from another using the rule of proportions, while not forgetting to translate the dimensions into one format (in minutes or seconds)

What is the angular velocity in specific cases?

Let's give an example of a calculation based on the above formulas. Let's say we have a car. When driving at 100 km / h, its wheel, as practice shows, makes an average of 600 revolutions per minute (f = 600 rpm). Let's calculate the angular velocity.

Since exactly to express π decimals impossible, the result will be approximately equal to 62.83 rad / s.

Relationship between angular and linear velocities

In practice, it is often necessary to check not only the speed at which the angular position of a rotating point changes, but also the speed of it itself in relation to linear motion. In the example above, calculations were made for the wheel - but the wheel moves along the road and either rotates under the influence of the speed of the car, or provides it with this speed itself. This means that each point on the surface of the wheel, in addition to the angular velocity, will also have a linear velocity.

The easiest way to calculate it is through the radius. Since the speed depends on time (which will be the period of revolution) and the distance traveled (which is the circumference), then, given the above formulas, the angular and linear speed will be related as follows:

• V is the linear speed;
• R is the radius.

It is obvious from the formula that the larger the radius, the higher the value of such a speed. With regard to the wheel with the highest speed, a point on the outer surface of the tread will move (R is maximum), but exactly in the center of the hub, the linear speed will be equal to zero.

Acceleration, moment and their connection with mass

In addition to the above quantities, there are several other points associated with rotation. Considering how many rotating parts of different weights are in the car, their practical significance cannot be ignored.

Uniform rotation is an important thing. But there is not a single detail that would spin evenly all the time. The number of revolutions of any rotating assembly, from the crankshaft to the wheel, always eventually rises and then falls. And the value that shows how much the revolutions have increased is called angular acceleration. Since it is a derivative of angular velocity, it is measured in radians per second squared (as linear acceleration is in meters per second squared).

Another aspect is also connected with the movement and its change in time - the angular momentum. If up to this point we could only consider purely mathematical features of the movement, then here it is already necessary to take into account the fact that each part has a mass that is distributed around the axis. It is determined by the ratio of the initial position of the point, taking into account the direction of movement - and the momentum, that is, the product of mass and speed. Knowing the moment of impulse that occurs during rotation, it is possible to determine what load will fall on each part when it interacts with another

Hinge as an example of momentum transfer

A typical example of how all of the above data applies is the constant velocity joint (CV joint). This part is used primarily on front-wheel drive vehicles, where it is important not only to provide a different rate of rotation of the wheels when turning, but also their controllability and the transfer of impulse from the engine to them.

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The design of this node is precisely designed to:

• equalize how fast the wheels spin;
• provide rotation at the moment of rotation;
• guarantee the independence of the rear suspension.

As a result, all the formulas given above are taken into account in the operation of the SHRUS.

Rotation frequency is a physical quantity, a characteristic of a periodic process, equal to the number of complete cycles completed per unit of time. Standard notation in formulas - υ, f , ω or F . The unit of frequency in the International System of Units (SI) is general case is Hertz (Hz, Hz). The reciprocal of the frequency is called the period.

A periodic signal is characterized by an instantaneous frequency, which is the rate of phase change, but the same signal can be represented as a sum of harmonic spectral components that have their own frequencies. The properties of the instantaneous frequency and the frequency of the spectral component are different, you can read more about this, for example, in Fink's book "Signals, Interference, Errors".

In theoretical physics, as well as in some applied electrical and radio engineering calculations, it is convenient to use an additional quantity - cyclic (circular, radial, angular) frequency (denoted ω ). The cyclic frequency is related to the oscillation frequency by the relation ω=2 πf . In a mathematical sense, the cyclic frequency is the first derivative full phase fluctuations in time. The unit of cyclic frequency is radians per second (rad/s, rad/s).

In mechanics, when considering rotational motion, the analogue of cyclic frequency is the angular velocity.

The frequency of discrete events (pulse frequency) is a physical quantity equal to the number of discrete events occurring per unit of time. The unit of frequency of discrete events is a second to the minus first power ( s −1, s−1), but in practice, hertz is usually used to express the pulse frequency.

The rotational speed is a physical quantity equal to the number of full revolutions per unit of time. The unit of rotational speed is a second to the minus first power ( s −1, s−1), revolution per second. Units often used are revolutions per minute, revolutions per hour, etc.

Other quantities related to frequency

• Bandwidth - fmax − fmin
• Frequency interval - log ( fmax / fmin )
• Frequency deviation - Δ f /2
• Period - 1/ f
• Wavelength - υ/ f
• Angular speed (rotation speed) - dφ / dt ; 2πFBP

Metrological aspects

measurements

Frequency meters are used to measure frequency. different types, including: for measuring the frequency of pulses - electronic counting and capacitor, for determining the frequencies of spectral components - resonant and heterodyne frequency meters, as well as spectrum analyzers.

To reproduce the frequency with a given accuracy, various measures are used - frequency standards (high accuracy), frequency synthesizers, signal generators, etc.

Compare frequencies with a frequency comparator or with an oscilloscope using Lissajous figures.

Standards

State primary standard of units of time, frequency and national time scale GET 1-98 - located at VNIIFTRI

Secondary standard of the unit of time and frequency VET 1-10-82 - located in SNIIM (Novosibirsk)

Since the linear speed uniformly changes direction, then the movement along the circle cannot be called uniform, it is uniformly accelerated.

Angular velocity

Pick a point on the circle 1 . Let's build a radius. For a unit of time, the point will move to the point 2 . In this case, the radius describes the angle. The angular velocity is numerically equal to the angle of rotation of the radius per unit time.

Period and frequency

Rotation period T is the time it takes the body to make one revolution.

RPM is the number of revolutions per second.

The frequency and period are related by the relation

Relationship with angular velocity

Line speed

Each point on the circle moves at some speed. This speed is called linear. The direction of the linear velocity vector always coincides with the tangent to the circle. For example, sparks from under a grinder move, repeating the direction of instantaneous speed.

Consider a point on a circle that makes one revolution, the time that is spent - this is the period T. The path traveled by a point is the circumference of a circle.

centripetal acceleration

When moving along a circle, the acceleration vector is always perpendicular to the velocity vector, directed to the center of the circle.

Using the previous formulas, we can derive the following relations

Points lying on the same straight line emanating from the center of the circle (for example, these can be points that lie on the wheel spoke) will have the same angular velocities, period and frequency. That is, they will rotate in the same way, but with different linear speeds. The farther the point is from the center, the faster it will move.

The law of addition of velocities is also valid for rotational motion. If the motion of a body or frame of reference is not uniform, then the law applies to instant speeds. For example, the speed of a person walking along the edge of a rotating carousel is equal to the vector sum of the linear speed of rotation of the edge of the carousel and the speed of the person.

The Earth participates in two main rotational movements: daily (around its axis) and orbital (around the Sun). The period of rotation of the Earth around the Sun is 1 year or 365 days. The Earth rotates around its axis from west to east, the period of this rotation is 1 day or 24 hours. Latitude is the angle between the plane of the equator and the direction from the center of the Earth to a point on its surface.

According to Newton's second law, the cause of any acceleration is a force. If a moving body experiences centripetal acceleration, then the nature of the forces that cause this acceleration may be different. For example, if a body moves in a circle on a rope tied to it, then active force is the elastic force.

If a body lying on a disk rotates along with the disk around its axis, then such a force is the force of friction. If the force ceases to act, then the body will continue to move in a straight line

Consider the movement of a point on a circle from A to B. The linear velocity is equal to v A And v B respectively. Acceleration is the change in speed per unit of time. Let's find the difference of vectors.

Rotary motion is periodic motion.

The period is denoted by the letter T.

To find the period of revolution, you need to divide the rotation time by the number of revolutions:

The rotational speed is indicated by the letter n.

To find the rotational speed, you need to divide the number of revolutions by the time during which these revolutions are completed:

The rotation frequency and the period of revolution are related to each other as reciprocal quantities: The period is measured in seconds: [ T ] = 1 s.

The unit of frequency is a second to the minus first power: [ n ] \u003d 1 s -1.

This unit has its own name - 1 hertz (1 Hz).

Let's draw an analogy between rotational and translational movements.

A progressively moving body changes its position in space relative to other bodies.

Rotating bodies rotate through a certain angle.

If for any equal intervals of time a progressively moving body makes equal movements, the movement is called uniform.

If for any equal intervals of time a rotating body rotates through the same angle, then such rotation is called uniform. The characteristic of uniform translational motion is the speed. The corresponding characteristic of the rotational motion is the angular velocity:

Angular velocity is a physical quantity equal to the ratio the angle of rotation of the body to the time during which this rotation is completed.

Angular velocity indicates the angle through which a body rotates per unit of time.

To get the unit of angular velocity, you need to substitute a unit in its defining formula - 1 radian, and time - 1 s. We get: [ω] = 1

Similarly, you can introduce a characteristic of non-uniform rotation. If the type of non-uniform translational motion is uniformly variable motion, then for rotational motion, the concept of uniformly variable rotation can be introduced.

A characteristic of uniformly variable translational motion is acceleration:

Continuing the analogy further, we write down the equation for displacement during rectilinear uniformly accelerated motion

Since during rotation, the displacement of the body corresponds to the angle of rotation, linear velocity - angular velocity, linear acceleration - angular acceleration, then a similar equation for rotational motion will have the form:

Another equation for translational motion will correspond to the equation for rotational motion:

The method used in this case is called by analogy.

The points of a body that performs rotational motion rotate relative to the axis of rotation by certain angles and move along arcs of circles, passing certain paths. Thus, the characteristics of rotational motion are both angular and linear velocities.

The linear velocity of a point is directed tangentially to the circle along which it moves.

This is evidenced by dirt flying from the wheels of the car or sparks flying from a metal object pressed against the emery wheel.

The further a point is from the axis of rotation, the greater its linear velocity. The angular velocity of points lying on the same radius is the same. Therefore, the linear speed of a point is directly proportional to the radius of the circle along which it rotates.

In time equal to the period, the point travels a path equal to the circumference of the circle. In this case, its linear velocity is equal to The ratio of the angle of rotation to the time of rotation through this angle is equal to the angular velocity

Thus, the linear velocity of a rotating point is related to its angular velocity by the relation:

With uniform rotation, the speed changes in direction, but does not change in magnitude.

Let the rotating body at the initial moment of time be at point A and its speed be tangential. At the next moment of time, the body is at point B. At the same time, its speed changed only in direction and is directed tangentially to the circle.

Let's find the velocity difference vector using the rule of action with vectors. It can be seen from the drawing that the difference vector is directed towards the side close to the center of the circle. How less angle rotation, the closer the velocity vector is directed to the direction to the center of rotation.