Formula for the relationship between cyclic frequency and oscillation period. Frequency, period, cyclic frequency, amplitude, phase of oscillation

(lat. amplitude- magnitude) is the greatest deviation of the oscillating body from the equilibrium position.

For a pendulum, this is the maximum distance by which the ball moves away from its equilibrium position (figure below). For oscillations with small amplitudes, such a distance can be taken as the length of the arc 01 or 02, as well as the lengths of these segments.

The amplitude of oscillations is measured in units of length - meters, centimeters, etc. On the oscillation graph, the amplitude is defined as the maximum (modulo) ordinate of the sinusoidal curve (see figure below).

Oscillation period.

Oscillation period- this is the shortest period of time through which a system oscillating returns again to the same state in which it was at the initial moment of time, chosen arbitrarily.

In other words, the oscillation period ( T) is the time during which one complete oscillation occurs. For example, in the figure below, this is the time it takes for the pendulum bob to move from the rightmost point through the equilibrium point ABOUT to the far left point and back through the point ABOUT again to the far right.

Over a full period of oscillation, the body thus travels a path equal to four amplitudes. The period of oscillation is measured in units of time - seconds, minutes, etc. The period of oscillation can be determined from a well-known graph of oscillations (see figure below).

The concept of “oscillation period”, strictly speaking, is valid only when the values ​​of the oscillating quantity are exactly repeated after a certain period of time, i.e. for harmonic oscillations. However, this concept also applies to cases of approximately repeating quantities, for example, for damped oscillations.

Oscillation frequency.

Oscillation frequency- this is the number of oscillations performed per unit of time, for example, in 1 s.

The SI unit of frequency is named hertz(Hz) in honor of the German physicist G. Hertz (1857-1894). If the oscillation frequency ( v) is equal to 1 Hz, this means that every second there is one oscillation. The frequency and period of oscillations are related by the relations:

In the theory of oscillations they also use the concept cyclical, or circular frequency ω . It is related to the normal frequency v and oscillation period T ratios:

.

Cyclic frequency is the number of oscillations performed per 2π seconds

The time during which one complete change in the emf occurs, that is, one cycle of oscillation or one full revolution of the radius vector, is called period of alternating current oscillation(Figure 1).

Figure 1. Period and amplitude of a sinusoidal oscillation. Period is the time of one oscillation; Amplitude is its greatest instantaneous value.

The period is expressed in seconds and denoted by the letter T.

Smaller units of measurement of period are also used: millisecond (ms) - one thousandth of a second and microsecond (μs) - one millionth of a second.

1 ms = 0.001 sec = 10 -3 sec.

1 μs = 0.001 ms = 0.000001 sec = 10 -6 sec.

1000 µs = 1 ms.

The number of complete changes in the emf or the number of revolutions of the radius vector, that is, in other words, the number of complete cycles of oscillations performed by alternating current within one second, is called AC oscillation frequency.

The frequency is indicated by the letter f and is expressed in cycles per second or hertz.

One thousand hertz is called a kilohertz (kHz), and a million hertz is called a megahertz (MHz). There is also a unit of gigahertz (GHz) equal to one thousand megahertz.

1000 Hz = 10 3 Hz = 1 kHz;

1000 000 Hz = 10 6 Hz = 1000 kHz = 1 MHz;

1000 000 000 Hz = 10 9 Hz = 1000 000 kHz = 1000 MHz = 1 GHz;

The faster the EMF changes, that is, the faster the radius vector rotates, the shorter the oscillation period. The faster the radius vector rotates, the higher the frequency. Thus, the frequency and period of alternating current are quantities inversely proportional to each other. The larger one of them, the smaller the other.

The mathematical relationship between the period and frequency of alternating current and voltage is expressed by the formulas

For example, if the current frequency is 50 Hz, then the period will be equal to:

T = 1/f = 1/50 = 0.02 sec.

And vice versa, if it is known that the period of the current is 0.02 sec, (T = 0.02 sec.), then the frequency will be equal to:

f = 1/T=1/0.02 = 100/2 = 50 Hz

The frequency of alternating current used for lighting and industrial purposes is exactly 50 Hz.

Frequencies between 20 and 20,000 Hz are called audio frequencies. Currents in radio station antennas fluctuate with frequencies up to 1,500,000,000 Hz or, in other words, up to 1,500 MHz or 1.5 GHz. These high frequencies are called radio frequencies or high frequency vibrations.

Finally, currents in the antennas of radar stations, satellite communication stations, and other special systems (for example, GLANASS, GPS) fluctuate with frequencies of up to 40,000 MHz (40 GHz) and higher.

AC current amplitude

The greatest value that the emf or current reaches in one period is called amplitude of emf or alternating current. It is easy to notice that the amplitude on the scale is equal to the length of the radius vector. The amplitudes of current, EMF and voltage are designated by letters respectively Im, Em and Um (Figure 1).

Angular (cyclic) frequency of alternating current.

The rotation speed of the radius vector, i.e. the change in the rotation angle within one second, is called the angular (cyclic) frequency of alternating current and is denoted by the Greek letter ? (omega). The angle of rotation of the radius vector at any given moment relative to its initial position is usually measured not in degrees, but in special units - radians.

A radian is the angular value of an arc of a circle, the length of which is equal to the radius of this circle (Figure 2). The entire circle that makes up 360° is equal to 6.28 radians, that is, 2.

Figure 2.

1rad = 360°/2

Consequently, the end of the radius vector during one period covers a path equal to 6.28 radians (2). Since within one second the radius vector makes a number of revolutions equal to the frequency of the alternating current f, then in one second its end covers a path equal to 6.28*f radian. This expression characterizing the rotation speed of the radius vector will be the angular frequency of the alternating current - ? .

? = 6.28*f = 2f

The angle of rotation of the radius vector at any given instant relative to its initial position is called AC phase. The phase characterizes the magnitude of the EMF (or current) at a given instant or, as they say, the instantaneous value of the EMF, its direction in the circuit and the direction of its change; phase indicates whether the emf is decreasing or increasing.

Figure 3.

A full rotation of the radius vector is 360°. With the beginning of a new revolution of the radius vector, the EMF changes in the same order as during the first revolution. Consequently, all phases of the EMF will be repeated in the same order. For example, the phase of the EMF when the radius vector is rotated by an angle of 370° will be the same as when rotated by 10°. In both of these cases, the radius vector occupies the same position, and, therefore, the instantaneous values ​​of the emf will be the same in phase in both of these cases.

Thus, the total energy of the harmonic oscillation is constant and proportional to the square of the displacement amplitude . This is one of the characteristic properties of harmonic oscillations. Here the constant coefficient k in the case of a spring pendulum means the stiffness of the spring, and for a mathematical pendulum k=mgH. In both cases, the coefficient k is transmitted by the parameters of the oscillatory system.

The total energy of a mechanical oscillatory system consists of kinetic and potential energies and is equal to the maximum value of any of these two components:

Therefore, the total vibration energy is directly proportional to the square of the displacement amplitude or the square of the velocity amplitude.

From the formula:

it is possible to determine the amplitude x m of displacement oscillations:

The displacement amplitude during free oscillations is directly proportional to the square root of the energy imparted to the oscillatory system at the initial moment when the system was brought out of equilibrium.

Kinematics of mechanical free vibrations

1 Displacement, speed, acceleration. To find the kinematic characteristics (displacement, velocity and acceleration) of free oscillations, we will use the law of conservation and transformation of energy, which for an ideal mechanical oscillatory system is written as follows:

Since the time derivative φ " is constant, the angle φ depends linearly on time:

Taking this into account, we can write:

x = x m sin ω 0 t, υ = x m ω 0 cos ω 0 t

Here the value

is the amplitude of the speed change:

υ = υ m cos ω 0 t

Dependence of instantaneous acceleration value a from time t we find as the derivative of speed υ with respect to time:

a = υ " = - ω 0 υ m sin ω 0 t,

a = -a m sin ω 0 t

the “-” sign in the resulting formula indicates that the sign of the projection of the acceleration vector onto the axis along which the oscillations occur is opposite to the sign of the displacement x.

So, we see that with harmonic oscillations, not only the displacement, but also the speed and acceleration change sinusoidally .

2 Cyclic oscillation frequency. The quantity ω 0 is called the cyclic frequency of oscillations. Since the function sin α has a period of 2π in its argument α, and harmonic oscillations have a period of T in time, then

As you study this section, please keep in mind that fluctuations of different physical nature are described from common mathematical positions. Here it is necessary to clearly understand such concepts as harmonic oscillation, phase, phase difference, amplitude, frequency, oscillation period.

It must be borne in mind that in any real oscillatory system there is resistance of the medium, i.e. the oscillations will be damped. To characterize the damping of oscillations, a damping coefficient and a logarithmic damping decrement are introduced.

If oscillations occur under the influence of an external, periodically changing force, then such oscillations are called forced. They will be undamped. The amplitude of forced oscillations depends on the frequency of the driving force. As the frequency of forced oscillations approaches the frequency of natural oscillations, the amplitude of forced oscillations increases sharply. This phenomenon is called resonance.

When moving on to the study of electromagnetic waves, you need to clearly understand thatelectromagnetic waveis an electromagnetic field propagating in space. The simplest system emitting electromagnetic waves is an electric dipole. If a dipole undergoes harmonic oscillations, then it emits a monochromatic wave.

Formula table: oscillations and waves

Physical laws, formulas, variables	Oscillation and wave formulas
Harmonic vibration equation: where x is the displacement (deviation) of the fluctuating quantity from the equilibrium position; A - amplitude; ω - circular (cyclic) frequency; α - initial phase; (ωt+α) - phase.
Relationship between period and circular frequency:
Frequency:
Relationship between circular frequency and frequency:
Periods of natural oscillations 1) spring pendulum: where k is the spring stiffness; 2) mathematical pendulum: where l is the length of the pendulum, g - free fall acceleration; 3) oscillatory circuit: where L is the circuit inductance, C is the capacitance of the capacitor.
Natural frequency:
Addition of oscillations of the same frequency and direction: 1) amplitude of the resulting oscillation where A 1 and A 2 are the amplitudes of the vibration components, α 1 and α 2 - initial phases of the vibration components; 2) the initial phase of the resulting oscillation
Equation of damped oscillations: e = 2.71... - the base of natural logarithms.
Amplitude of damped oscillations: where A 0 is the amplitude at the initial moment of time; β - attenuation coefficient;
Attenuation coefficient: oscillating body where r is the resistance coefficient of the medium, m - body weight; oscillatory circuit where R is active resistance, L is the inductance of the circuit.
Frequency of damped oscillations ω:
Period of damped oscillations T:
Logarithmic damping decrement:

Everything on the planet has its own frequency. According to one version, it even forms the basis of our world. Alas, the theory is too complex to present in one publication, so we will consider exclusively the frequency of oscillations as an independent action. Within the framework of the article, definitions of this physical process, its units of measurement and metrological component will be given. And finally, an example of the importance of ordinary sound in everyday life will be considered. We learn what he is and what his nature is.

What is oscillation frequency called?

By this we mean a physical quantity that is used to characterize a periodic process, which is equal to the number of repetitions or occurrences of certain events in one unit of time. This indicator is calculated as the ratio of the number of these incidents to the period of time during which they occurred. Each element of the world has its own vibration frequency. A body, an atom, a road bridge, a train, an airplane - they all make certain movements, which are called so. Even if these processes are not visible to the eye, they exist. The units of measurement in which oscillation frequency is calculated are hertz. They received their name in honor of the physicist of German origin Heinrich Hertz.

Instantaneous frequency

A periodic signal can be characterized by an instantaneous frequency, which, up to a coefficient, is the rate of phase change. It can be represented as a sum of harmonic spectral components that have their own constant oscillations.

Cyclic frequency

It is convenient to use in theoretical physics, especially in the section on electromagnetism. Cyclic frequency (also called radial, circular, angular) is a physical quantity that is used to indicate the intensity of the origin of oscillatory or rotational motion. The first is expressed in revolutions or oscillations per second. During rotational motion, the frequency is equal to the magnitude of the angular velocity vector.

This indicator is expressed in radians per second. The dimension of cyclic frequency is the reciprocal of time. In numerical terms, it is equal to the number of oscillations or revolutions that occurred in the number of seconds 2π. Its introduction for use makes it possible to significantly simplify the various range of formulas in electronics and theoretical physics. The most popular example of use is calculating the resonant cyclic frequency of an oscillatory LC circuit. Other formulas can become significantly more complex.

Discrete event rate

This value means a value that is equal to the number of discrete events that occur in one unit of time. In theory, the indicator usually used is the second minus the first power. In practice, the hertz is usually used to express the pulse frequency.

Rotational speed

It is understood as a physical quantity that is equal to the number of full revolutions that occur in one unit of time. The indicator used here is also the second minus the first power. To indicate the work done, phrases such as revolutions per minute, hour, day, month, year and others can be used.

Units of measurement

How is oscillation frequency measured? If we take into account the SI system, then the unit of measurement here is hertz. It was originally introduced by the International Electrotechnical Commission back in 1930. And the 11th General Conference on Weights and Measures in 1960 consolidated the use of this indicator as an SI unit. What was put forward as the “ideal”? It was the frequency when one cycle is completed in one second.

But what about production? Arbitrary values ​​were assigned to them: kilocycle, megacycle per second, and so on. Therefore, when you pick up a device that operates at GHz (like a computer processor), you can roughly imagine how many actions it performs. It would seem how slowly time passes for a person. But the technology manages to perform millions and even billions of operations per second during the same period. In one hour, the computer already does so many actions that most people cannot even imagine them in numerical terms.

Metrological aspects

Oscillation frequency has found its application even in metrology. Different devices have many functions:

1. The pulse frequency is measured. They are represented by electronic counting and capacitor types.
2. The frequency of spectral components is determined. There are heterodyne and resonant types.
3. Spectrum analysis is carried out.
4. Reproduce the required frequency with a given accuracy. In this case, various measures can be used: standards, synthesizers, signal generators and other techniques in this direction.
5. The indicators of the obtained oscillations are compared; for this purpose, a comparator or oscilloscope is used.

Example of work: sound

Everything written above can be quite difficult to understand, since we used the dry language of physics. To understand the information provided, you can give an example. Everything will be described in detail, based on an analysis of cases from modern life. To do this, consider the most famous example of vibrations - sound. Its properties, as well as the features of the implementation of mechanical elastic vibrations in the medium, are directly dependent on the frequency.

The human hearing organs can detect vibrations that range from 20 Hz to 20 kHz. Moreover, with age, the upper limit will gradually decrease. If the frequency of sound vibrations drops below 20 Hz (which corresponds to the mi subcontractave), then infrasound will be created. This type, which in most cases is not audible to us, can still be felt tangibly by people. When the limit of 20 kilohertz is exceeded, vibrations are generated, which are called ultrasound. If the frequency exceeds 1 GHz, then in this case we will be dealing with hypersound. If we consider a musical instrument such as a piano, it can create vibrations in the range from 27.5 Hz to 4186 Hz. It should be taken into account that musical sound does not consist only of the fundamental frequency - overtones and harmonics are also mixed into it. All this together determines the timbre.

Conclusion

As you have had the opportunity to learn, vibrational frequency is an extremely important component that allows our world to function. Thanks to her, we can hear, with her assistance computers work and many other useful things are accomplished. But if the oscillation frequency exceeds the optimal limit, then certain destruction may begin. So, if you influence the processor so that its crystal works at twice the performance, it will quickly fail.

A similar thing can be said with human life, when at high frequencies his eardrums burst. Other negative changes will also occur in the body, which will lead to certain problems, even death. Moreover, due to the peculiarities of the physical nature, this process will stretch over a fairly long period of time. By the way, taking this factor into account, the military is considering new opportunities for developing weapons of the future.