5 angular frequency is indicated. Oscillations. Harmonic vibrations. Characteristics of oscillations: amplitude, period, frequency, cyclic frequency, phase. Logarithmic damping decrement

08.01.2024

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

For a pendulum, this is the maximum distance that 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, and 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(picture 1).

Picture 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 (picture 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.

VIBRATION FREQUENCY, number of oscillations in 1 s. Denoted by .

PERIOD OF oscillation, the shortest period of time after which the oscillating system returns to the same state in which it was at the initial moment, chosen arbitrarily. Period is the reciprocal of the oscillation frequency. The concept of “period” is applicable, for example, in the case of harmonic oscillations, but is often used for weakly damped oscillations.

Circular or cyclic frequencyω

When the argument of the cosine or sine changes by 2π, these functions return to their previous value. Let us find the time period T during which the phase of the harmonic function changes by 2π.

ω(t + T) + α = ωt + α + 2π, or ωT = 2π.

The time T for one complete oscillation is called the oscillation period. Frequency ν is the reciprocal of the period

The unit of frequency is hertz (Hz), 1 Hz = 1 s -1.

The circular or cyclic frequency ω is 2π times greater than the oscillation frequency ν.

Circular frequency is the rate of change of phase over time. Really:

AMPLITUDE (from the Latin amplitudo - value), the greatest deviation from the equilibrium value of a quantity that fluctuates according to a certain, including harmonic, law; see alsoHarmonic oscillations.

PHASE OF OSCILLATIONS argument of the function cos (ωt + φ), describing the harmonic oscillatory process (ω - circular frequency, t - time, φ - initial phase of oscillations, i.e. phase of oscillations at the initial moment of time t = 0)

Displacement, speed, acceleration of an oscillating system of particles.

Energy of harmonic vibrations.

Harmonic vibrations

An important special case of periodic oscillations are harmonic oscillations, i.e. such changes in a physical quantity that follow the law

Where . From a mathematics course we know that a function of type (1) varies from A to -A, and that it has the smallest positive period.

Therefore, a harmonic oscillation of type (1) occurs with amplitude A and period.

Do not confuse cyclic frequency with oscillation frequency.

There is a simple connection between them. Since, ah, then. The quantity is called the phase of oscillation. At t=0 the phase is equal, therefore it is called the initial phase. and work in the future with the last type of recording of this vibration.

It can be shown that vibrations of the form:

where and can be of any sign, using simple trigonometric transformations is always reduced to the form (1), and, and is not equal, generally speaking. Thus, oscillations of type (2) are harmonic with amplitude and cyclic frequency.

Without giving a general proof, we will illustrate this with a specific example.

Let it be required to show that the oscillation

will be harmonic and find the amplitude, cyclic frequency, period and initial phase. ,.

Really,

We see that the fluctuation of the value of S was written down in the form (1). Wherein

Try to see for yourself that Naturally, the recording of harmonic oscillations in form (2) is no worse than the recording in form (1), and in a specific task there is usually no need to switch from recording in this form to recording in another form. ,You just need to be able to immediately find the amplitude, cyclic frequency and period, having in front of you any form of recording of a harmonic vibration.

Sometimes it is useful to know the nature of the change in the first and second time derivatives of the quantity S, which performs harmonic oscillations (oscillates according to the harmonic law). If ,, then differentiating S with respect to time t gives .

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.

Rotation frequency

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

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:

The pulse frequency is measured. They are represented by electronic counting and capacitor types.
The frequency of spectral components is determined. There are heterodyne and resonant types.
Perform spectrum analysis.
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.
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 subcontractive), then infrasound will be created. This type, which in most cases is not audible to us, people can still feel tactilely. When the limit of 20 kilohertz is exceeded, oscillations 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.

Oscillations are a process of changing the states of a system around the equilibrium point that is repeated to varying degrees over time.

Harmonic oscillation - oscillations in which a physical (or any other) quantity changes over time according to a sinusoidal or cosine law. The kinematic equation of harmonic oscillations has the form

where x is the displacement (deviation) of the oscillating point from the equilibrium position at time t; A is the amplitude of oscillations, this is the value that determines the maximum deviation of the oscillating point from the equilibrium position; ω - cyclic frequency, a value indicating the number of complete oscillations occurring within 2π seconds - the full phase of oscillations, 0 - the initial phase of oscillations.

Amplitude is the maximum value of displacement or change of a variable from the average value during oscillatory or wave motion.

The amplitude and initial phase of oscillations are determined by the initial conditions of movement, i.e. position and speed of the material point at the moment t=0.

Generalized harmonic oscillation in differential form

the amplitude of sound waves and audio signals usually refers to the amplitude of the air pressure in the wave, but is sometimes described as the amplitude of the displacement relative to equilibrium (the air or the speaker's diaphragm)

Frequency is a physical quantity, a characteristic of a periodic process, equal to the number of complete cycles of the process completed per unit of time. The frequency of vibration in sound waves is determined by the frequency of vibration of the source. High frequency oscillations decay faster than low frequency ones.

The reciprocal of the oscillation frequency is called period T.

The period of oscillation is the duration of one complete cycle of oscillation.

In the coordinate system, from point 0 we draw a vector A̅, the projection of which on the OX axis is equal to Аcosϕ. If the vector A̅ rotates uniformly with an angular velocity ω˳ counterclockwise, then ϕ=ω˳t +ϕ˳, where ϕ˳ is the initial value of ϕ (oscillation phase), then the amplitude of the oscillations is the modulus of the uniformly rotating vector A̅, the oscillation phase (ϕ ) is the angle between the vector A̅ and the OX axis, the initial phase (ϕ˳) is the initial value of this angle, the angular frequency of oscillations (ω) is the angular velocity of rotation of the vector A̅..

2. Characteristics of wave processes: wave front, beam, wave speed, wave length. Longitudinal and transverse waves; examples.

The surface separating at a given moment in time the medium already covered and not yet covered by oscillations is called the wave front. At all points of such a surface, after the wave front leaves, oscillations are established that are identical in phase.

The beam is perpendicular to the wave front. Acoustic rays, like light rays, are rectilinear in a homogeneous medium. They are reflected and refracted at the interface between 2 media.

Wavelength is the distance between two points closest to each other, oscillating in the same phases, usually the wavelength is denoted by the Greek letter. By analogy with waves generated in water by a thrown stone, the wavelength is the distance between two adjacent wave crests. One of the main characteristics of vibrations. Measured in distance units (meters, centimeters, etc.)

longitudinal waves (compression waves, P-waves) - particles of the medium vibrate parallel(along) the direction of wave propagation (as, for example, in the case of sound propagation);
transverse waves (shear waves, S-waves) - particles of the medium vibrate perpendicular direction of wave propagation (electromagnetic waves, waves on separation surfaces);

The angular frequency of oscillations (ω) is the angular velocity of rotation of the vector A̅(V), the displacement x of the oscillating point is the projection of the vector A onto the OX axis.

V=dx/dt=-Aω˳sin(ω˳t+ϕ˳)=-Vmsin(ω˳t+ϕ˳), where Vm=Аω˳ is the maximum speed (velocity amplitude)

3. Free and forced vibrations. Natural frequency of oscillations of the system. The phenomenon of resonance. Examples .

Free (natural) vibrations are called those that occur without external influences due to the energy initially obtained by heat. Characteristic models of such mechanical oscillations are a material point on a spring (spring pendulum) and a material point on an inextensible thread (mathematical pendulum).

In these examples, oscillations arise either due to initial energy (deviation of a material point from the position of equilibrium and motion without initial speed), or due to kinetic (the body is imparted speed in the initial equilibrium position), or due to both energy (imparting speed to the body deviated from the equilibrium position).

Consider a spring pendulum. In the equilibrium position, the elastic force F1

balances the force of gravity mg. If you pull the spring a distance x, then a large elastic force will act on the material point. The change in the value of the elastic force (F), according to Hooke's law, is proportional to the change in the length of the spring or the displacement x of the point: F= - rx

Another example. The mathematical pendulum of deviation from the equilibrium position is such a small angle α that the trajectory of a material point can be considered a straight line coinciding with the OX axis. In this case, the approximate equality is satisfied: α ≈sin α≈ tanα ≈x/L

Undamped oscillations. Let us consider a model in which the resistance force is neglected.
The amplitude and initial phase of oscillations are determined by the initial conditions of movement, i.e. position and speed of the material point moment t=0.
Among the various types of vibrations, harmonic vibration is the simplest form.

Thus, a material point suspended on a spring or thread performs harmonic oscillations, if resistance forces are not taken into account.

The period of oscillation can be found from the formula: T=1/v=2П/ω0

Damped oscillations. In a real case, resistance (friction) forces act on an oscillating body, the nature of the movement changes, and the oscillation becomes damped.

In relation to one-dimensional motion, we give the last formula the following form: Fc = - r * dx/dt

The rate at which the oscillation amplitude decreases is determined by the damping coefficient: the stronger the braking effect of the medium, the greater ß and the faster the amplitude decreases. In practice, however, the degree of damping is often characterized by a logarithmic damping decrement, meaning by this a value equal to the natural logarithm of the ratio of two successive amplitudes separated by a time interval equal to the oscillation period; therefore, the damping coefficient and the logarithmic damping decrement are related by a fairly simple relationship: λ=ßT

With strong damping, it is clear from the formula that the period of oscillation is an imaginary quantity. The movement in this case will no longer be periodic and is called aperiodic.

Forced vibrations. Forced oscillations are called oscillations that occur in a system with the participation of an external force that changes according to a periodic law.

Let us assume that the material point, in addition to the elastic force and the friction force, is acted upon by an external driving force F=F0 cos ωt

The amplitude of the forced vibration is directly proportional to the amplitude of the driving force and has a complex dependence on the damping coefficient of the medium and the circular frequencies of natural and forced vibrations. If ω0 and ß are given for the system, then the amplitude of forced oscillations has a maximum value at some specific frequency of the driving force, called resonant The phenomenon itself—the achievement of the maximum amplitude of forced oscillations for given ω0 and ß—is called resonance.

The resonant circular frequency can be found from the condition of the minimum denominator in: ωres=√ωₒ- 2ß

Mechanical resonance can be both beneficial and harmful. The harmful effects are mainly due to the destruction it can cause. Thus, in technology, taking into account various vibrations, it is necessary to provide for the possible occurrence of resonant conditions, otherwise there may be destruction and disasters. Bodies usually have several natural vibration frequencies and, accordingly, several resonant frequencies.

Resonance phenomena under the action of external mechanical vibrations occur in internal organs. This is apparently one of the reasons for the negative impact of infrasonic vibrations and vibrations on the human body.

6.Sound research methods in medicine: percussion, auscultation. Phonocardiography.

Sound can be a source of information about the state of a person’s internal organs, which is why methods for studying the patient’s condition such as auscultation, percussion and phonocardiography are widely used in medicine.

Auscultation

For auscultation, a stethoscope or phonendoscope is used. A phonendoscope consists of a hollow capsule with a sound-transmitting membrane applied to the patient's body, from which rubber tubes go to the doctor's ear. A resonance of the air column occurs in the capsule, resulting in increased sound and improved auscultation. When auscultating the lungs, breathing sounds and various wheezing characteristic of diseases are heard. You can also listen to the heart, intestines and stomach.

Percussion

In this method, the sound of individual parts of the body is listened to by tapping them. Let's imagine a closed cavity inside some body, filled with air. If you induce sound vibrations in this body, then at a certain frequency of sound, the air in the cavity will begin to resonate, releasing and amplifying a tone corresponding to the size and position of the cavity. The human body can be represented as a collection of gas-filled (lungs), liquid (internal organs) and solid (bones) volumes. When hitting the surface of a body, vibrations occur, the frequencies of which have a wide range. From this range, some vibrations will fade out quite quickly, while others, coinciding with the natural vibrations of the voids, will intensify and, due to resonance, will be audible.

Phonocardiography

Used to diagnose cardiac conditions. The method consists of graphically recording heart sounds and murmurs and their diagnostic interpretation. A phonocardiograph consists of a microphone, an amplifier, a system of frequency filters and a recording device.

9. Ultrasound research methods (ultrasound) in medical diagnostics.

1) Diagnostic and research methods

These include location methods using mainly pulsed radiation. This is echoencephalography - detection of tumors and edema of the brain. Ultrasound cardiography – measurement of heart size in dynamics; in ophthalmology - ultrasonic location to determine the size of the ocular media.

2)Methods of influence

Ultrasound physiotherapy – mechanical and thermal effects on tissue.

11. Shock wave. Production and use of shock waves in medicine.
Shock wave – a discontinuity surface that moves relative to the gas and upon crossing which the pressure, density, temperature and speed experience a jump.
Under large disturbances (explosion, supersonic movement of bodies, powerful electric discharge, etc.), the speed of oscillating particles of the medium can become comparable to the speed of sound , a shock wave occurs.

The shock wave can have significant energy Thus, during a nuclear explosion, about 50% of the explosion energy is spent on the formation of a shock wave in the environment. Therefore, a shock wave, reaching biological and technical objects, can cause death, injury and destruction.

Shock waves are used in medical technology, representing an extremely short, powerful pressure pulse with high pressure amplitudes and a small stretch component. They are generated outside the patient’s body and transmitted deep into the body, producing a therapeutic effect provided for by the specialization of the equipment model: crushing urinary stones, treating pain areas and the consequences of injuries to the musculoskeletal system, stimulating the recovery of the heart muscle after myocardial infarction, smoothing cellulite formations, etc.