Formula for calculating cyclic frequency. Basic formulas in physics - vibrations and waves

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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. Oscillations high frequency fade faster than low frequencies.

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 onto 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 dividing at the moment 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 created 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 (immunization of 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 oscillation 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 oscillations. 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 different vibrations, it is necessary to provide for the possible occurrence of resonant conditions, in 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; therefore, methods for studying the patient’s condition such as auscultation, percussion and phonocardiography are well 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. The phonocardiograph consists of a microphone, an amplifier, and a system frequency filters and 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.

In the world around us, there are many phenomena and processes that, by and large, are invisible not because they do not exist, but because we simply do not notice them. They are always present and are the same imperceptible and obligatory essence of things, without which it is difficult to imagine our life. Everyone, for example, knows what an oscillation is: in its most general form, it is a deviation from a state of equilibrium. Well, okay, the top of the Ostankino tower has deviated by 5 m, but what next? Will it freeze like that? Nothing of the kind, it will begin to return back, will slip past the state of equilibrium and will deviate in the other direction, and so on forever, as long as it exists. Tell me, how many people actually saw these quite serious vibrations of such a huge structure? Everyone knows, it fluctuates, here and there, here and there, day and night, winter and summer, but somehow... it’s not noticeable. The reasons for the oscillatory process are another question, but its presence is an inseparable feature of all things.

Everything around oscillates: buildings, structures, clock pendulums, leaves on trees, violin strings, the surface of the ocean, the legs of a tuning fork... Among the oscillations, there are chaotic ones, which do not have strict repeatability, and cyclic ones, in which the oscillating body passes through the time period T full set its changes, and then this cycle repeats exactly, generally speaking, for an infinitely long time. Usually these changes imply a sequential search of spatial coordinates, as can be observed in the example of oscillations of a pendulum or the same tower.

The number of oscillations per unit time is called frequency F = 1/T. Frequency unit - Hz = 1/sec. It is clear that the cyclic frequency is a parameter of oscillations of the same name of any type. However, in practice, it is customary to refer this concept, with some additions, primarily to vibrations of a rotational nature. It just so happens in technology that it is the basis of most machines, mechanisms, and devices. For such oscillations, one cycle is one revolution, and then it is more convenient to use the angular parameters of movement. Based on this, rotational movement is measured in angular units, i.e. one revolution is equal to 2π radians, and the cyclic frequency ῳ = 2π / T. From this expression the connection with frequency F is easily visible: ῳ = 2πF. This allows us to say that the cyclic frequency is the number of oscillations (full revolutions) in 2π seconds.

It would seem, not in the forehead, so... Not quite so. The 2π and 2πF factors are used in many equations of electronics, mathematical and theoretical physics in sections where oscillatory processes are studied using the concept of cyclic frequency. The formula for resonant frequency, for example, is reduced by two factors. If the unit “rev/sec” is used in calculations, the angular, cyclic, frequency ῳ numerically coincides with the value of frequency F.

Vibrations, as the essence and form of existence of matter, and its material embodiment - the objects of our existence, are of great importance in human life. Knowledge of the laws of oscillations has made it possible to create modern electronics, electrical engineering, and many modern machines. Unfortunately, fluctuations do not always bring a positive effect; sometimes they bring grief and destruction. Unaccounted for vibrations, the cause of many accidents, cause materials, and the cyclic frequency of resonant vibrations of bridges, dams, and machine parts leads to their premature failure. The study of oscillatory processes, the ability to predict the behavior of natural and technical objects in order to prevent their destruction or failure to operate is the main task of many engineering applications, and the inspection of industrial facilities and mechanisms for vibration resistance is a mandatory element of operational maintenance.

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:

Harmonic oscillations are oscillations performed according to the laws of sine and cosine. The following figure shows a graph of changes in the coordinates of a point over time according to the cosine law.

picture

Oscillation amplitude

The amplitude of a harmonic vibration is the greatest value of the displacement of a body from its equilibrium position. The amplitude can take on different values. It will depend on how much we displace the body at the initial moment of time from the equilibrium position.

The amplitude is determined by the initial conditions, that is, the energy imparted to the body at the initial moment of time. Since sine and cosine can take values in the range from -1 to 1, the equation must contain a factor Xm, expressing the amplitude of the oscillations. Equation of motion for harmonic vibrations:

x = Xm*cos(ω0*t).

Oscillation period

The period of oscillation is the time it takes to complete one complete oscillation. The period of oscillation is designated by the letter T. The units of measurement of the period correspond to the units of time. That is, in SI these are seconds.

Oscillation frequency is the number of oscillations performed per unit of time. The oscillation frequency is designated by the letter ν. The oscillation frequency can be expressed in terms of the oscillation period.

ν = 1/T.

Frequency units are in SI 1/sec. This unit of measurement is called Hertz. The number of oscillations in a time of 2*pi seconds will be equal to:

ω0 = 2*pi* ν = 2*pi/T.

Oscillation frequency

This quantity is called the cyclic frequency of oscillations. In some literature the name circular frequency appears. The natural frequency of an oscillatory system is the frequency of free oscillations.

The frequency of natural oscillations is calculated using the formula:

The frequency of natural vibrations depends on the properties of the material and the mass of the load. The greater the spring stiffness, the greater the frequency of its own vibrations. The greater the mass of the cargo, the lower frequency own vibrations.

These two conclusions are obvious. The stiffer the spring, the greater the acceleration it will impart to the body when the system is thrown out of balance. The greater the mass of a body, the slower the speed of this body will change.

Free oscillation period:

T = 2*pi/ ω0 = 2*pi*√(m/k)

It is noteworthy that at small angles of deflection the period of oscillation of the body on the spring and the period of oscillation of the pendulum will not depend on the amplitude of the oscillations.

Let's write down the formulas for the period and frequency of free oscillations for a mathematical pendulum.

then the period will be equal

T = 2*pi*√(l/g).

This formula will be valid only for small deflection angles. From the formula we see that the period of oscillation increases with increasing length of the pendulum thread. The longer the length, the slower the body will vibrate.

The period of oscillation does not depend at all on the mass of the load. But it depends on the acceleration of free fall. As g decreases, the oscillation period will increase. This property is widely used in practice. For example, to measure the exact value of free acceleration.

Angular frequency is expressed in radians per second, its dimension is the inverse of the dimension of time (radians are dimensionless). The angular frequency is the time derivative of the oscillation phase:

Angular frequency in radians per second is expressed in terms of frequency f(expressed in revolutions per second or vibrations per second), as

If we use degrees per second as the unit of angular frequency, the relationship to ordinary frequency is as follows:

Finally, when using revolutions per second, the angular frequency is the same as the rotational speed:

The introduction of cyclic frequency (in its main dimension - radians per second) allows us to simplify many formulas in theoretical physics and electronics. Thus, the resonant cyclic frequency of an oscillatory LC circuit is equal to whereas the usual resonant frequency is . At the same time, a number of other formulas become more complicated. The decisive consideration in favor of cyclic frequency was that the factors and , which appear in many formulas when using radians to measure angles and phases, disappear when cyclic frequency is introduced.

Formula for calculating cyclic frequency. Basic formulas in physics - vibrations and waves

Formula table: oscillations and waves

Oscillation amplitude

Oscillation period

Oscillation frequency

See also

See what “Cyclic frequency” is in other dictionaries:

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