fluctuations called movements or processes that are characterized by a certain repetition in time. Fluctuations are widespread in the surrounding world and can have a very different nature. These can be mechanical (pendulum), electromagnetic ( oscillatory circuit) and other types of oscillations.
Free, or own oscillations are called oscillations that occur in a system left to itself, after it has been brought out of equilibrium by an external influence. An example is the oscillation of a ball suspended on a thread.

special role in oscillatory processes has simplest form fluctuations - harmonic vibrations. Harmonic vibrations underlie a unified approach to the study of vibrations different nature, since the oscillations encountered in nature and technology are often close to harmonic, and periodic processes of a different form can be represented as a superposition of harmonic oscillations.

Harmonic vibrations such oscillations are called, in which the oscillating value varies with time according to the law sinus or cosine.

Harmonic vibration equationlooks like:

where A - oscillation amplitude (the value of the greatest deviation of the system from the equilibrium position); -circular (cyclic) frequency. Periodically changing cosine argument - called oscillation phase . The phase of oscillation determines the displacement of the oscillating quantity from the equilibrium position in this moment time t. The constant φ is the value of the phase at time t = 0 and is called the initial phase of the oscillation . The value of the initial phase is determined by the choice of the reference point. The x value can take values ​​ranging from -A to +A.

The time interval T, after which certain states of the oscillatory system are repeated, called the period of oscillation . Cosine is a periodic function with a period of 2π, therefore, over a period of time T, after which the oscillation phase will receive an increment equal to 2π, the state of the system performing harmonic oscillations will repeat. This period of time T is called the period of harmonic oscillations.

The period of harmonic oscillations is : T = 2π/ .

The number of oscillations per unit time is called oscillation frequency ν.
Frequency of harmonic vibrations is equal to: ν = 1/T. Frequency unit hertz(Hz) - one oscillation per second.

Circular frequency = 2π/T = 2πν gives the number of oscillations in 2π seconds.

Graphically, harmonic oscillations can be depicted as a dependence of x on t (Fig. 1.1.A), and rotating amplitude method (vector diagram method)(Fig.1.1.B) .

The rotating amplitude method allows you to visualize all the parameters included in the equation of harmonic oscillations. Indeed, if the amplitude vector BUT located at an angle φ to the x-axis (see Figure 1.1. B), then its projection on the x-axis will be equal to: x = Acos(φ). The angle φ is initial phase. If the vector BUT put into rotation with angular velocity, equal to the circular frequency of oscillations, then the projection of the end of the vector will move along the x-axis and take values ​​ranging from -A to +A, and the coordinate of this projection will change over time according to the law:
.

Thus, the length of the vector is equal to the amplitude of the harmonic oscillation, the direction of the vector in initial moment forms an angle with the x-axis equal to the initial phase of oscillations φ, and the change in the direction angle with time is equal to the phase of harmonic oscillations. The time it takes for the amplitude vector to make one full turn, equal to the period T of harmonic oscillations. The number of revolutions of the vector per second is equal to the oscillation frequency ν.

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

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

The oscillation amplitude 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 smallest period of time after which the system, making oscillations, again returns 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 for which one complete oscillation takes place. For example, in the figure below, this is the time it takes for the weight of the pendulum to move from the rightmost point through the equilibrium point O to the leftmost point and back through the point O again to the far right.

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

The concept of "oscillation period", strictly speaking, is valid only when the values ​​of the fluctuating quantity exactly repeat through certain interval time, i.e. for harmonic oscillations. However, this concept is also applied to cases of approximately repeating quantities, for example, for damped oscillations .

Oscillation frequency.

Oscillation frequency is the number of oscillations 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, then this means that one oscillation is made for every second. The frequency and period of oscillations are related by the relations:

In the theory of oscillations, the concept is also used 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 per 2π seconds.

Changes in time according to a sinusoidal law:

where X- the value of the fluctuating quantity at the moment of time t, BUT- amplitude , ω - circular frequency, φ is the initial phase of oscillations, ( φt + φ ) is the total phase of oscillations . At the same time, the values BUT, ω and φ - permanent.

For mechanical vibrations with an oscillating value X are, in particular, displacement and velocity, for electrical oscillations- voltage and current.

Harmonic vibrations take special place among all types of oscillations, since this is the only type of oscillations, the shape of which is not distorted when passing through any homogeneous environment, i.e., waves propagating from a source of harmonic oscillations will also be harmonic. Any non-harmonic vibration can be represented as a sum (integral) of various harmonic vibrations (in the form of a spectrum of harmonic vibrations).

Energy transformations during harmonic vibrations.

In the process of oscillations, there is a transition of potential energy Wp into kinetic Wk and vice versa. In the position of maximum deviation from the equilibrium position, the potential energy is maximum, the kinetic energy is zero. As you return to the equilibrium position, the speed of the oscillating body increases, and with it, so does kinetic energy, reaching a maximum in the equilibrium position. The potential energy then drops to zero. Further-neck movement occurs with a decrease in speed, which drops to zero when the deflection reaches its second maximum. Potential energy here increases to its initial (maximum) value (in the absence of friction). Thus, fluctuations in kinetic and potential energies occur with a double (compared to the oscillations of the pendulum itself) frequency and are in antiphase (i.e., between them there is a phase shift equal to π ). Total vibration energy W remains unchanged. For a body oscillating under the action of an elastic force, it is equal to:

where v m — maximum speed body (in equilibrium position), x m = BUT- amplitude.

Due to the presence of friction and resistance of the medium free vibrations decay: their energy and amplitude decrease with time. Therefore, in practice, not free, but forced oscillations are used more often.

Harmonic vibrations

Function Graphs f(x) = sin( x) and g(x) = cos( x) on the Cartesian plane.

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

,

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

Generalized harmonic oscillation in differential form

(Any non-trivial solution to this differential equation- there is a harmonic oscillation with a cyclic frequency)

Types of vibrations

Evolution in time of displacement, velocity and acceleration in harmonic motion

• Free vibrations are made under the influence internal forces system after the system has been brought out of equilibrium. For free oscillations to be harmonic, it is necessary that the oscillatory system be linear (described linear equations motion), and there was no dissipation of energy (the latter would cause damping).
• Forced vibrations performed under the influence of an external periodic force. For them to be harmonic, it is sufficient that the oscillatory system be linear (described by linear equations of motion), and external force itself changed over time as a harmonic oscillation (that is, so that the time dependence of this force was sinusoidal).

Application

Harmonic vibrations stand out from all other types of vibrations for the following reasons:

see also

Notes

Literature

• Physics. Elementary textbook Physics / Ed. G. S. Lansberg. - 3rd ed. - M ., 1962. - T. 3.
• Khaykin S. E. Physical foundations mechanics. - M., 1963.
• A. M. Afonin. Physical foundations of mechanics. - Ed. MSTU im. Bauman, 2006.
• Gorelik G.S. Vibrations and waves. Introduction to acoustics, radiophysics and optics. - M .: Fizmatlit, 1959. - 572 p.

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Modern Encyclopedia

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The simplest type of vibrations are harmonic vibrations- fluctuations in which the displacement of the oscillating point from the equilibrium position changes over time according to the sine or cosine law.

So, with a uniform rotation of the ball around the circumference, its projection (shadow in parallel rays of light) performs a harmonic oscillatory motion on a vertical screen (Fig. 1).

The displacement from the equilibrium position during harmonic vibrations is described by the equation (it is called the kinematic law harmonic movement) of the form:

where x - displacement - a value characterizing the position of the oscillating point at time t relative to the equilibrium position and measured by the distance from the equilibrium position to the position of the point at a given time; A - oscillation amplitude - the maximum displacement of the body from the equilibrium position; T - oscillation period - the time of one complete oscillation; those. the shortest amount of time after which values ​​are repeated physical quantities characterizing the oscillation; - initial phase;

The phase of the oscillation at time t. The oscillation phase is the argument periodic function, which at a given oscillation amplitude determines the state of the oscillatory system (displacement, speed, acceleration) of the body at any time.

If at the initial moment of time the oscillating point is maximally displaced from the equilibrium position, then , and the displacement of the point from the equilibrium position changes according to the law

If the oscillating point at is in a position of stable equilibrium, then the displacement of the point from the equilibrium position changes according to the law

The value of V, the reciprocal of the period and equal to the number full oscillations made in 1 s is called the frequency of oscillations:

If in time t the body makes N complete oscillations, then

the value , showing how many oscillations the body makes in s, is called cyclic (circular) frequency.

The kinematic law of harmonic motion can be written as:

Graphically, the dependence of the displacement of an oscillating point on time is represented by a cosine (or sinusoid).

Figure 2, a shows the time dependence of the displacement of the oscillating point from the equilibrium position for the case .

Let us find out how the speed of an oscillating point changes with time. To do this, we find the time derivative of this expression:

where is the amplitude of the velocity projection on the x-axis.

This formula shows that during harmonic oscillations, the projection of the body velocity on the x axis also changes according to the harmonic law with the same frequency, with a different amplitude, and is ahead of the mixing phase by (Fig. 2, b).

To find out the dependence of acceleration, we find the time derivative of the velocity projection:

where is the amplitude of the acceleration projection on the x-axis.

For harmonic oscillations, the acceleration projection leads the phase shift by k (Fig. 2, c).

Similarly, you can build dependency graphs

Considering that , the formula for acceleration can be written

those. for harmonic oscillations, the acceleration projection is directly proportional to the displacement and opposite in sign, i.e. acceleration is directed in the direction opposite to the displacement.

So, the acceleration projection is the second derivative of the displacement, then the resulting ratio can be written as:

The last equality is called equation of harmonic oscillations.

A physical system in which harmonic oscillations can exist is called harmonic oscillator, and the equation of harmonic oscillations - harmonic oscillator equation.