Formulas for cyclic rotation frequency. Determination of the shaft speed. HydroMuseum - RPM How RPM is defined in physics
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 quantity 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, 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, there are 2 π radians in the whole circle. 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 revolution period is shown by the following formulas:
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 it is impossible to express exactly π in decimal fractions, 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 with 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 itself provides this speed to it. 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 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 ensure 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.
One of the most common types of movement in nature and technology is rotation. This type of movement of bodies in space is characterized by a set of physical quantities. An important characteristic of any rotation is the frequency. The formula for rotational speed can be found by knowing certain quantities and parameters.
What is rotation?
In physics, it is understood as such a movement of a material point around a certain axis, in which its distance to this axis remains constant. It is called the radius of rotation.
Examples of this movement in nature are the rotation of the planets around the Sun and around their own axis. In technology, rotation is represented by the movement of shafts, gears, the wheels of a car or bicycle, the movement of the blades of windmills.
Physical quantities describing rotation
For the numerical description of rotation in physics, a number of characteristics have been introduced. Let's list them and describe them.
First of all, this is the angle of rotation, denoted θ. Since a full circle is characterized by a central angle of 2 * pi radians, then, knowing the value of θ, by which the rotating body turned in a certain period of time, it is possible to determine the number of revolutions during this time. In addition, the angle θ allows you to calculate the linear path traveled by the body along the curved circle. The corresponding formulas for the number of revolutions n and the distance traveled L are:
Where r is the radius of the circle or the radius of rotation.
The next characteristic of the considered type of motion is the angular velocity. It is usually denoted by the letter ω. It is measured in radians per second, that is, it shows the angle in radians that a rotating body turns in one second. For the angular velocity in the case of uniform rotation, the formula is valid:
Angular frequency, period and angular velocity
It has already been noted above that an important property of any rotational movement is the time it takes to complete one revolution. This time is called the rotation period. It is denoted by the letter T and is measured in seconds. The formula for the period T can be written in terms of the angular velocity ω. The corresponding expression looks like:
The reciprocal of a period is called frequency. It is measured in hertz (Hz). For circular motion, it is convenient to use not the frequency itself, but its angular counterpart. Let's denote it f. The formula for the angular frequency of rotation f is:
Comparing the last two formulas, we arrive at the following equality:
This equality means the following:
- the formulas for the angular frequency and angular velocity coincide, therefore these quantities are numerically equal to each other;
- as well as speed, frequency indicates at what angle in radians a body rotates in one second.
The difference between these quantities is the only one: the angular frequency is a scalar quantity, while the speed is a vector.
Linear rotation speed, frequency and angular frequency
In engineering, for some rotating structures, for example, gears and shafts, their operating frequencies μ and linear speeds v are known. However, each of these characteristics can be used to determine the angular or cyclic frequency.
It was noted above that the frequency μ is measured in hertz. It shows the number of revolutions of a rotating body in one second. The formula for it takes the form:
If we compare this expression with the corresponding equality for f, then the formula for how to find the rotation frequency f through μ describing it will look like:
This formula is intuitive because μ is the number of revolutions per unit of time, while f is the same value, only expressed in radians.
The linear speed v is related to the angular speed ω by the following equation:
Since the modules of f and ω are equal, it is easy to obtain the corresponding formula for the cyclic rotation frequency from the last expression. Let's write it down:
Where r is the radius of rotation. Note that the velocity v increases linearly with increasing radius r, while the ratio of these quantities is a constant. The last conclusion means that if you measure the cyclic rotation frequency at any point in the section of a rotating massive object, then it will be the same everywhere.
The task of determining the cyclic speed of the shaft
Angular speeds contain useful information because they allow you to calculate important physical characteristics such as angular momentum or angular velocity. Let's solve the following problem: it is known that the operating speed of the shaft is 1500 rpm. What is the cyclic frequency for this shaft?
From the units of measurement given in the condition, it is clear that the usual frequency μ is given. Therefore, the formula for the frequency of rotation of the cyclic shaft has the form:
Before using it, you should convert the figure indicated in the condition to standard units of measurement, that is, to reciprocal seconds. Since the shaft makes 1500 revolutions per minute, then in a second it will make 60 times less revolutions, that is, 25. That is, its rotation frequency is 25 Hz. Substituting this number into the formula written above, we obtain the value of the cyclic frequency: f = 157 rad/s.
When designing equipment, it is necessary to know the number of revolutions of the electric motor. To calculate the speed, there are special formulas that are different for AC and DC motors.
Synchronous and asynchronous electric machines
There are three types of AC motors: synchronous, the angular velocity of the rotor of which coincides with the angular frequency of the stator magnetic field; asynchronous - in them, the rotation of the rotor lags behind the rotation of the field; collector, the design and principle of operation of which are similar to DC motors.
Synchronous speed
The rotation speed of an AC electric machine depends on the angular frequency of the stator magnetic field. This speed is called synchronous. In synchronous motors, the shaft rotates at the same speed, which is an advantage of these electric machines.
To do this, in the rotor of high-power machines there is a winding to which a constant voltage is applied, which creates a magnetic field. In low power devices, permanent magnets are inserted into the rotor, or there are pronounced poles.
Slip
In asynchronous machines, the number of revolutions of the shaft is less than the synchronous angular frequency. This difference is called the "S" slip. Due to the slip, an electric current is induced in the rotor, and the shaft rotates. The larger S, the higher the torque and the lower the speed. However, if the slip exceeds a certain value, the electric motor stops, starts to overheat and may fail. The rotational speed of such devices is calculated according to the formula in the figure below, where:
- n is the number of revolutions per minute,
- f - network frequency,
- p is the number of pairs of poles,
- s - slip.
There are two types of such devices:
- With squirrel-cage rotor. The winding in it is cast from aluminum during the manufacturing process;
- With phase rotor. The windings are made of wire and are connected to additional resistances.
Speed control
In the process of work, it becomes necessary to adjust the number of revolutions of electric machines. It is carried out in three ways:
- Increasing the additional resistance in the rotor circuit of electric motors with a phase rotor. If it is necessary to greatly reduce the speed, it is allowed to connect not three, but two resistances;
- Connection of additional resistances in the stator circuit. It is used to start high power electrical machines and to adjust the speed of small electric motors. For example, the number of revolutions of a table fan can be reduced by connecting an incandescent lamp or a capacitor in series with it. The same result gives a decrease in the supply voltage;
- Network frequency change. Suitable for synchronous and asynchronous motors.
Attention! The speed of rotation of collector electric motors operating from the AC network does not depend on the frequency of the network.
DC motors
In addition to AC machines, there are electric motors connected to the DC network. The number of revolutions of such devices is calculated using completely different formulas.
Rated rotation speed
The number of revolutions of the DC machine is calculated using the formula in the figure below, where:
- n is the number of revolutions per minute,
- U - network voltage,
- Rya and Iya - armature resistance and current,
- Ce – motor constant (depends on the type of electric machine),
- F is the magnetic field of the stator.
These data correspond to the nominal values of the parameters of the electric machine, the voltage on the field winding and armature, or the torque on the motor shaft. Changing them allows you to adjust the speed. It is very difficult to determine the magnetic flux in a real motor, therefore, for calculations, the strength of the current flowing through the excitation winding or the armature voltage is used.
The number of revolutions of AC collector motors can be found using the same formula.
Speed control
Adjustment of the speed of an electric motor operating from a DC network is possible over a wide range. It is available in two ranges:
- Up from nominal. To do this, the magnetic flux is reduced with the help of additional resistances or a voltage regulator;
- Down from par. To do this, it is necessary to reduce the voltage at the armature of the electric motor or turn on a resistance in series with it. In addition to reducing the speed, this is done when starting the electric motor.
Knowing what formulas are used to calculate the speed of rotation of the electric motor is necessary when designing and commissioning equipment.
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