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 inductance of the circuit, 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: |
(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 Coriolis force is equal to:
where is point weight,-vectorangular velocity rotating reference system, - the vector of the speed of movement of a point mass in this reference system, square brackets indicate the operation vector product.
Magnitude 
called Coriolis acceleration.
By physical nature
Mechanical(sound,vibration)
Electromagnetic (light,radio waves, thermal)
Mixed type- combinations of the above
By the nature of interaction with the environment
Forced - oscillations occurring in the system under the influence of external periodic influence. Examples: leaves on trees, raising and lowering a hand. During forced oscillations, a phenomenon may occur resonance: a sharp increase in the amplitude of oscillations upon coincidence natural frequencyoscillator and frequency of external influence.
Free (or own)- these are oscillations in a system under the influence of internal forces, after the system is taken out of equilibrium (in real conditions, free oscillations are always fading). The simplest examples of free oscillations are the oscillations of a weight attached to a spring, or a weight suspended on a thread.
Self-oscillations - fluctuations during which the system has a reserve potential energy, spent on oscillations (an example of such a system is mechanical watches). A characteristic difference between self-oscillations and forced oscillations is that their amplitude is determined by the properties of the system itself, and not by the initial conditions.
Parametric - oscillations that occur when any parameter of the oscillatory system changes as a result of external influence.
Random - oscillations in which the external or parametric load is a random process.
Harmonic vibrations
Where XAω
Generalized harmonic oscillation in differential form
(Any non-trivial
Velocity and acceleration during harmonic oscillations.
According to the definition of speed, speed is the derivative of a position with respect to time
Thus, we see that the speed during harmonic oscillatory motion also changes according to the harmonic law, but the speed oscillations are ahead of the phase displacement oscillations by p/2.
Value - maximum speed of oscillatory motion (amplitude of speed fluctuations).
Therefore, for the speed during harmonic oscillation we have: 
,
and for the case of zero initial phase (see graph).
According to the definition of acceleration, acceleration is the derivative of speed with respect to time:
-
second derivative of the coordinate with respect to time. Then: .
Acceleration during harmonic oscillatory motion also changes according to the harmonic law, but acceleration oscillations are ahead of speed oscillations by p/2 and displacement oscillations by p (the oscillations are said to occur in antiphase).
Magnitude
Maximum acceleration (amplitude of acceleration fluctuations). Therefore, for acceleration we have: 
,
and for the case of zero initial phase: 
(see chart).
From the analysis of the process of oscillatory motion, graphs and corresponding mathematical expressions, it is clear that when the oscillating body passes the equilibrium position (the displacement is zero), the acceleration is zero, and the speed of the body is maximum (the body passes the equilibrium position by inertia), and when the amplitude value of the displacement is reached, the speed is equal to zero, and the acceleration is maximum in absolute value (the body changes the direction of its motion).
Harmonic vibrations- 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- displacement (deviation) of the oscillating point from the equilibrium position at time t; A- 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; - full phase of oscillations; - initial phase of oscillations.
Generalized harmonic oscillation in differential form
(Any non-trivial the solution to this differential equation is a harmonic oscillation with a cyclic frequency)
Until now, we have considered natural oscillations, i.e. oscillations that occur in the absence of external influences. External influence was needed only to bring the system out of equilibrium, after which it was left to its own devices. The differential equation of natural oscillations does not contain any traces of external influence on the system: this influence is reflected only in the initial conditions.
Establishment of oscillations. But very often one has to deal with fluctuations that occur with a constantly present external influence. Particularly important and at the same time quite simple to study is the case when the external force is periodic. A common feature of forced oscillations that occur under the influence of a periodic external force is that some time after the onset of the external force, the system completely “forgets” its initial state, the oscillations become stationary and do not depend on the initial conditions. The initial conditions appear only during the period of establishment of oscillations, which is usually called the transition process.
Sinusoidal effect. Let us first consider the simplest case of forced oscillations of an oscillator under the influence of an external force varying according to a sinusoidal law:
Rice. 178. Excitation of forced oscillations of a pendulum
Such external influence on the system can be carried out in various ways. For example, you can take a pendulum in the form of a ball on a long rod and a long spring with low stiffness and attach it to the pendulum rod near the suspension point, as shown in Fig. 178. The other end of a horizontal spring should be made to move according to the law? using a crank mechanism driven by an electric motor. Current
on the pendulum from the spring side, the driving force will be almost sinusoidal if the range of motion of the left end of spring B is much greater than the amplitude of oscillation of the pendulum rod at the point where spring C is attached.
Equation of motion. The equation of motion for this and other similar systems, in which, along with the restoring force and resistance force, a driving external force acting on the oscillator, varying sinusoidally with time, can be written in the form
Here the left side, in accordance with Newton's second law, is the product of mass and acceleration. The first term on the right side represents the restoring force proportional to the displacement from the equilibrium position. For a load suspended on a spring, this is an elastic force, and in all other cases, when its physical nature is different, this force is called quasi-elastic. The second term is the friction force, proportional to the speed, for example, the force of air resistance or the friction force in the axis. The amplitude and frequency of the driving force rocking the system will be considered constant.
Let us divide both sides of equation (2) by mass and introduce the notation
Now equation (2) takes the form
In the absence of a driving force, the right side of equation (4) vanishes and, as one would expect, it reduces to the equation of natural damped oscillations.
Experience shows that in all systems, under the influence of a sinusoidal external force, oscillations are eventually established, which also occur according to a sinusoidal law with the frequency of the driving force co and with a constant amplitude a, but with some phase shift relative to the driving force. Such oscillations are called steady-state forced oscillations.
Steady-state oscillations. Let us first consider the steady-state forced oscillations, and for simplicity we will neglect friction. In this case, equation (4) will not contain a term containing speed:
Let's try to look for a solution corresponding to the steady-state forced oscillations, in the form
Let's calculate the second derivative and substitute it together with in equation (5):
For this equality to be valid at any time, the coefficients for the left and right must be the same. From this condition we find the amplitude of oscillations a:
Let us study the dependence of amplitude a on the frequency of the driving force. The graph of this dependence is shown in Fig. 179. When formula (8) gives Substituting the values here we see that a force constant in time simply shifts the oscillator to a new equilibrium position, shifted from the old one by From (6) it follows that when the displacement
as obviously it should be.
Rice. 179. Dependency graph
Phase relationships. As the frequency of the driving force increases from 0 to steady-state oscillations occur in phase with the driving force and their amplitude constantly increases, slowly at first, and as it approaches to - faster and faster: at the amplitude of the oscillations increases indefinitely
For values of co exceeding the frequency of natural oscillations, formula (8) gives a negative value for a (Fig. 179). From formula (6) it is clear that when the oscillations occur in antiphase with the driving force: when the force acts in one direction, the oscillator is shifted in the opposite direction. With an unlimited increase in the frequency of the driving force, the amplitude of the oscillations tends to zero.
In all cases, it is convenient to consider the amplitude of oscillations to be positive, which is easy to achieve by introducing a phase shift between the forcing
force and displacement:
Here a is still given by formula (8), and the phase shift is equal to zero at and equal to . Graphs of the driving force versus frequency are shown in Fig. 180.
Rice. 180. Amplitude and phase of forced oscillations
Resonance. The dependence of the amplitude of forced oscillations on the frequency of the driving force is non-monotonic. A sharp increase in the amplitude of forced oscillations as the frequency of the driving force approaches the natural frequency of the oscillator is called resonance.
Formula (8) gives an expression for the amplitude of forced oscillations, neglecting friction. It is with this neglect that the amplitude of oscillations turns to infinity when the frequencies coincide exactly. In reality, the amplitude of oscillations, of course, cannot go to infinity.
This means that when describing forced oscillations near resonance, taking friction into account is fundamentally necessary. When friction is taken into account, the amplitude of forced oscillations at resonance turns out to be finite. The greater the friction in the system, the smaller it will be. Far from resonance, formula (8) can be used to find the amplitude of oscillations even in the presence of friction, if it is not too strong, i.e. Moreover, this formula, obtained without taking into account friction, has a physical meaning only when there is still friction . The fact is that the very concept of steady-state forced oscillations is applicable only to systems in which there is friction.
If there were no friction at all, then the process of establishing oscillations would continue indefinitely. In reality, this means that expression (8) obtained without taking into account friction for the amplitude of forced oscillations will correctly describe oscillations in the system only after a sufficiently large period of time after the start of the action of the driving force. The words “a sufficiently long period of time” mean here that the transition process has already ended, the duration of which coincides with the characteristic time of decay of natural oscillations in the system.
At low friction, steady-state forced oscillations occur in phase with the driving force at and in antiphase at both and in the absence of friction. However, near resonance, the phase does not change abruptly, but continuously, and with an exact coincidence of frequencies, the displacement lags in phase behind the driving force by (a quarter of a period). In this case, the speed changes in phase with the driving force, which provides the most favorable conditions for the transfer of energy from the source of the external driving force to the oscillator.
What is the physical meaning of each of the terms in equation (4), which describes the forced oscillations of the oscillator?
What are steady-state forced oscillations?
Under what conditions can formula (8) be used for the amplitude of steady-state forced oscillations, obtained without taking into account friction?
What is resonance? Give examples known to you of the manifestation and use of the phenomenon of resonance.
Describe the phase shift between the driving force and the displacement for different ratios between the frequency co in the driving force and the natural frequency of the oscillator.
What determines the duration of the process of establishing forced oscillations? Give reasons for your answer.
Vector diagrams. You can verify the validity of the above statements if you obtain a solution to equation (4), which describes steady-state forced oscillations in the presence of friction. Since steady-state oscillations occur with the frequency of the driving force c and a certain phase shift, the solution to equation (4), corresponding to such oscillations, should be sought in the form
In this case, speed and acceleration will obviously also change with time according to the harmonic law:
It is convenient to determine the amplitude a of steady-state forced oscillations and the phase shift using vector diagrams. Let us take advantage of the fact that the instantaneous value of any quantity varying according to the harmonic law can be represented as a projection of a vector onto some pre-selected direction, and the vector itself rotates uniformly in the plane with a frequency co, and its constant length is equal to
amplitude value of this oscillating quantity. In accordance with this, we associate each term of equation (4) with a vector rotating with angular velocity, the length of which is equal to the amplitude value of this term.
Since the projection of the sum of several vectors is equal to the sum of the projections of these vectors, equation (4) means that the sum of the vectors associated with the terms on the left side is equal to the vector associated with the value on the right side. To construct these vectors, we write out the instantaneous values of all terms on the left side of equation (4), taking into account the relations
From formulas (13) it is clear that the length vector associated with the quantity is ahead by an angle, the vector associated with the quantity. The length vector associated with the term x is ahead by the length vector, i.e. these vectors are directed in opposite directions.
The relative position of these vectors for an arbitrary moment in time is shown in Fig. 181. The entire system of vectors rotates as a whole with angular velocity c counterclockwise around point O.
Rice. 181. Vector diagram of forced oscillations
Rice. 182. Vector comparable to external force
Instantaneous values of all quantities are obtained by projecting the corresponding vectors onto a pre-selected direction. The vector associated with the right side of equation (4) is equal to the sum of the vectors shown in Fig. 181. This addition is shown in Fig. 182. Applying the Pythagorean theorem, we get
from where we find the amplitude of steady-state forced oscillations a:
The phase shift between the driving force and the displacement as can be seen from the vector diagram in Fig. 182, is negative, since the length vector lags behind the vector Therefore
So, steady-state forced oscillations occur according to the harmonic law (10), where a and are determined by formulas (14) and (15).
Rice. 183. Dependence of the amplitude of forced oscillations on the frequency of the driving force
Resonance curves. The amplitude of steady-state forced oscillations is proportional to the amplitude of the driving force. Let us study the dependence of the amplitude of oscillations on the frequency of the driving force. At low attenuation this dependence has a very sharp character. If then, as co tends to the frequency of free oscillations, the amplitude of forced oscillations a tends to infinity, which coincides with the previously obtained result (8). In the presence of damping, the amplitude of oscillations at resonance no longer goes to infinity, although it significantly exceeds the amplitude of oscillations under the influence of an external force of the same magnitude, but having a frequency far from the resonant one. Resonance curves for different values of the damping constant y are shown in Fig. 183. To find the resonance frequency cutoff, you need to find at which co the radical expression in formula (14) has a minimum. Equating the derivative of this expression with respect to zero (or complementing it to a complete square), we are convinced that the maximum amplitude of forced oscillations occurs at
The resonant frequency turns out to be less than the frequency of free oscillations of the system. At small 7, the resonant frequency practically coincides with As the frequency of the driving force tends to infinity, i.e., at amplitude a, as can be seen from (14), tends to zero. When, that is, under the action of a constant external force, the amplitude If we substitute this here and we get This is a static displacement of the oscillator from the equilibrium position under the action of a constant force and the displacement of the oscillator occurs in antiphase with the driving force. In resonance, as can be seen from (15), the displacement lags behind the external force in phase by The second of formulas (13) shows that in this case the external force changes in phase with the speed, i.e., it always acts in the direction of movement. That this is exactly how it should be is clear from intuitive considerations.
Speed resonance. From formula (13) it is clear that the amplitude of speed oscillations during steady-state forced oscillations is equal to . Using (14) we obtain
Rice. 184. Velocity amplitude during steady forced oscillations
The dependence of the velocity amplitude on the frequency of the external force is shown in Fig. 184. The resonance curve for velocity, although similar to the resonance curve for displacement, differs from it in some respects. So, i.e., under the action of a constant force, the oscillator experiences a static displacement from the equilibrium position and its speed after the transition process ends is zero. From formula (19) it is clear that the velocity amplitude at becomes zero. Velocity resonance occurs when the frequency of the external force exactly coincides with the frequency of free oscillations
How are vector diagrams constructed for steady-state forced oscillations under sinusoidal external influence?
What determines the frequency, amplitude and phase of steady-state forced harmonic oscillations?
Describe the differences between the resonance curves for displacement amplitude and velocity amplitude. What characteristics of the oscillatory system determine the sharpness of the resonance curves?
How is the nature of the resonance curve related to the parameters of the system that determine the damping of its own oscillations?
