(lat. amplitude- value) is the greatest deviation of the oscillating body from the equilibrium position.
For a pendulum, this is the maximum distance the ball moves from its equilibrium position (figure below). For oscillations with small amplitudes, such a distance can be taken as the arc length 01 or 02, and 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 the figure below).
Oscillation period.
Oscillation period - this is the smallest time interval after which the system, performing 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 during which one complete oscillation is completed. For example, in the figure below, this is the time during which the pendulum weight moves from the rightmost point through the equilibrium point ABOUT to the leftmost point and back through the point ABOUT back to the far right.
Thus, for a full period of oscillation, the body travels 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 \u200b\u200bof the oscillating quantity are exactly repeated after a certain period of time, that is, for harmonic oscillations. However, this concept also applies to cases of approximately repeating quantities, for example, for damped oscillations.
Oscillation frequency.
Oscillation frequency Is the number of vibrations per unit of time, for example, 1 s.
The SI unit of frequency is called hertz(Hz) in honor of the German physicist G. Hertz (1857-1894). If the vibration frequency ( v) is equal to 1 Hz, then this means that for every second one oscillation occurs. The frequency and period of oscillations are related by the relations:
In the theory of vibrations, they also use the concept cyclical, or circular frequency ω ... It is related to the usual frequency v and oscillation period T ratios:
.
Cyclic frequency Is the number of oscillations made during 2π seconds.
Until now, we have considered natural oscillations, that is, 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 itself. The differential equation of natural vibrations does not contain traces of external influence on the system at all: this influence is reflected only in the initial conditions.
Establishing hesitation. But very often one has to deal with vibrations that occur with the constantly present external influence. Especially important and at the same time simple enough to study is the case when the external force has a periodic character. A common feature of forced oscillations occurring under the action of a periodic external force is that some time after the start of the external force, the system completely "forgets" its initial state, the oscillations become stationary and do not depend on the initial conditions. Initial conditions appear only during the period of the establishment of oscillations, which is usually called a transient process.
Sinusoidal impact. Let us first consider the simplest case of forced oscillations of an oscillator under the action of an external force that changes according to a sinusoidal law:
Figure: 178. Excitation of forced oscillations of the pendulum
Such an external influence on the system can be carried out different ways... For example, you can take a ball-shaped pendulum 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. Should the other end of a horizontal spring be made to move according to the law? using a crank mechanism driven by an electric motor. The current
on the pendulum from the side of the spring, the driving force will be almost sinusoidal if the range of motion of the left end of the spring B is much greater than the amplitude of oscillations of the pendulum rod at the point of attachment of the spring C.
Equation of motion. The equation of motion for this and other similar systems, in which, along with the restoring force and the resistance force, a compelling external force that changes sinusoidally with time acts on the oscillator, can be written as
Here the left side, in accordance with Newton's second law, is the product of mass and acceleration. The first term on the right is a 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 frictional force proportional to the speed, for example, the force of air resistance or the force of friction in the axis. The amplitude and frequency of the driving force swinging the system will be considered constant.
We divide both sides of Eq. (2) by mass and introduce the notation
Now equation (2) takes the form
In the absence of a driving force, the right-hand side of Eq. (4) vanishes and, as expected, it reduces to the equation of natural damped oscillations.
Experience shows that in all systems, under the action of a sinusoidal external force, oscillations are eventually established, which also occur according to a sinusoidal law with the frequency of the driving force ω and with a constant amplitude a, but with a certain 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 vibrations, and for simplicity we neglect friction. In this case, in equation (4) there will be no term containing the velocity:
Let's try to look for a solution corresponding to the steady forced oscillations, in the form
Let's calculate the second derivative and substitute it together with into equation (5):
For this equality to be true at any time, the coefficients at the left and right must be the same. From this condition we find the amplitude of oscillations a:
Let us investigate the dependence of the amplitude a on the frequency of the driving force. The graph of this dependence is shown in Fig. 179. When formula (8) gives Substituting here the values \u200b\u200bwe see that the 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
as obviously it should be.
Figure: 179. Dependency graph
Phase relationships. As the frequency of the driving force grows from 0 to the steady-state oscillations occur in phase with the driving force, and their amplitude constantly increases, slowly at first, and as co k approaches, faster and faster: at the amplitude of the oscillations increases indefinitely
For values \u200b\u200bof ω exceeding the natural frequency, 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 displaced in the opposite direction. With an unlimited increase in the frequency of the driving force, the amplitude of the oscillations tends to zero.
It is convenient to consider the oscillation amplitude in all cases to be positive, which is easy to achieve by introducing a phase shift between the forcing
strength and displacement:
Here, a is still given by formula (8), and the phase shift is zero at and is equal to at. Graphs of the dependence on the frequency of the driving force are shown in Fig. 180.
Figure: 180. Amplitude and phase of forced oscillations
Resonance. The dependence of the amplitude of the forced oscillations on the frequency of the driving force has a non-monotonic character. A sharp increase in the amplitude of forced oscillations when 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 oscillation amplitude goes to infinity with exact coincidence of frequencies. In reality, the oscillation amplitude, of course, cannot go to infinity.
This means that when describing forced oscillations near resonance, it is fundamentally necessary to take into account friction. When friction is taken into account, the amplitude of forced vibrations at resonance is finite. It will be the less, the more friction in the system. Far from resonance, formula (8) can be used to find the vibration amplitude even in the presence of friction, if it is not too strong, i.e., moreover, this formula, obtained without friction, has physical meaning only when friction is still ... The fact is that the very concept of steady forced oscillations is applicable only to systems in which there is friction.
If there were no friction at all, then the process of establishing vibrations would continue indefinitely. In reality, this means that expression (8) for the amplitude of forced vibrations obtained without taking into account friction will correctly describe the vibrations in the system only after a sufficiently long time interval after the onset of the driving force. The words "a sufficiently long period of time" mean here that the transient process has already ended, the duration of which coincides with the characteristic decay time of natural oscillations in the system.
With low friction, steady forced oscillations occur in phase with a driving force at and in antiphase at, as in the absence of friction. However, near resonance, the phase does not change abruptly, but continuously, and with exact coincidence of frequencies, the displacement lags behind in phase from the driving force by (by a quarter of the 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 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 vibrations obtained without taking into account friction?
What is resonance? Give examples of the manifestation and use of the resonance phenomenon known to you.
Describe the phase shift between the driving force and the displacement at different ratios between the frequency ω 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 charts. It is possible to verify the validity of the above statements if we obtain a solution to Eq. (4), which describes the steady forced oscillations in the presence of friction. Since the steady-state oscillations occur with the frequency of the driving force с and some phase shift, the solution of equation (4) corresponding to such oscillations should be sought in the form
In this case, the speed and acceleration, obviously, will 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 the projection of the vector onto some preselected direction, and the vector itself rotates uniformly in the plane with frequency ω, and its constant length is
the amplitude value of this oscillating quantity. In accordance with this, we will associate each term of equation (4) with a vector rotating with an 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 members on the left side is equal to the vector compared with the value on the right side. To construct these vectors, we write out the instantaneous values \u200b\u200bof all terms on the left side of equation (4), taking into account the relations
From formulas (13) it can be seen that the vector of length associated with the value is ahead of the vector associated with the value by the angle The vector of length associated with the term x is ahead of the vector of length, i.e., these vectors are directed in opposite directions.
The relative position of these vectors for an arbitrary moment of time is shown in Fig. 181. The entire system of vectors rotates as a whole with angular velocity counterclockwise around the point O.
Figure: 181. Vector diagram of forced oscillations
Figure: 182. Vector comparable to external force
The instantaneous values \u200b\u200bof all quantities are obtained by projecting the corresponding vectors onto a preselected direction.The vector compared to 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 obtain
whence we find the amplitude of steady-state forced oscillations a:
The phase shift between the driving force and the displacement as seen from the vector diagram in Fig. 182 is negative, since the length vector lags behind the vector Therefore
So, the established forced oscillations occur according to the harmonic law (10), where a and are determined by formulas (14) and (15).
Figure: 183. Dependence of the amplitude of forced vibrations 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 investigate the dependence of the amplitude of oscillations on the frequency of the driving force. At low attenuation y, this dependence is very sharp. If then, as ω 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 resonance. The resonance curves for different values \u200b\u200bof the damping constant y are shown in Fig. 183. To find the resonance frequency of the cut, you need to find at which the radical expression in the formula (14) has a minimum. Equating the derivative of this expression with respect to zero (or complementing it to full square), we are convinced that the maximum amplitude of forced oscillations takes place at
The resonant frequency turns out to be less than the frequency of free vibrations of the system. At small 7, the resonance frequency practically coincides with When the frequency of the driving force tends to infinity, i.e., at the amplitude a, as can be seen from (14), tends to zero. When, i.e., under the action of a constant external force, the amplitude If we substitute 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 in phase from the external force by The second of formulas (13) shows that in this case the external force changes in phase with velocity, i.e., it acts all the time in the direction of motion. That this is exactly the way it should be is clear from intuitive reasons.
Resonance of speed. From formula (13) it can be seen that the amplitude of the velocity oscillations at steady forced oscillations is equal to. Using (14), we obtain
Figure: 184. The amplitude of the speed at 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, with, i.e., under the action of a constant force, the oscillator experiences a static displacement from the equilibrium position and its speed after the end of the transient process is equal to zero. It is seen from formula (19) that the velocity amplitude at vanishes. Velocity resonance occurs when the frequency of the external force exactly coincides with the frequency of free oscillations
How are vector diagrams for steady-state forced oscillations constructed under sinusoidal external influences?
What determines the frequency, amplitude and phase of steady-state forced harmonic oscillations?
Describe the differences in 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 natural oscillations?
When studying this section, it should be borne in mind that hesitation of different physical nature are described from unified mathematical positions. Here it is necessary to clearly understand such concepts as harmonic oscillation, phase, phase difference, amplitude, frequency, oscillation period.
It should be borne in mind that in any real oscillatory system there are resistances of the medium, i.e. oscillations will be damped. To characterize the damping of oscillations, the damping coefficient and the logarithmic damping decrement are introduced.
If vibrations are performed under the influence of an external, periodically changing force, then such vibrations are called forced. They will be continuous. The amplitude of the forced vibrations depends on the frequency of the driving force. When the frequency of forced vibrations approaches the frequency of natural vibrations, the amplitude of the forced vibrations increases sharply. This phenomenon is called resonance.
Turning to the study of electromagnetic waves, one must clearly understand thatelectromagnetic wave is an electromagnetic field propagating in space. The simplest system that emits electromagnetic waves is an electric dipole. If the dipole performs harmonic oscillations, then it emits a monochromatic wave.
Formula table: oscillations and waves
|
Physical laws, formulas, variables |
Oscillation and Wave Formulas |
||||||
|
Harmonic equation: where x is the displacement (deviation) of the fluctuating quantity from the equilibrium position; A is the amplitude; ω - circular (cyclic) frequency; α is the initial phase; (ωt + α) - phase. |
|||||||
|
Relationship between period and circular frequency: |
|||||||
|
Frequency: |
|||||||
|
The relationship of circular frequency to frequency: |
|||||||
|
Periods of natural oscillations 1) spring pendulum: where k is the stiffness of the spring; 2) mathematical pendulum: where l is the length of the pendulum, g - acceleration of gravity; 3) oscillatory circuit: where L is the inductance of the circuit, C is the capacitance of the capacitor. |
|
||||||
|
Natural frequency: |
|||||||
|
Addition of vibrations of the same frequency and direction: 1) the amplitude of the resulting fluctuation where A 1 and A 2 are the amplitudes of the vibration components, α 1 and α 2 - initial phases of vibration components; 2) the initial phase of the resulting oscillation |
|
||||||
|
Damped oscillation equation: e \u003d 2.71 ... is the base of natural logarithms. |
|
||||||
|
Damped oscillation amplitude: where A 0 is the amplitude at the initial moment of time; β is the attenuation coefficient; |
|
||||||
|
Attenuation coefficient: oscillating body where r is the coefficient of resistance of the medium, m is body weight; oscillatory circuit where R is active resistance, L is the loop inductance. |
|||||||
|
Damped oscillation frequency ω: |
|
||||||
|
Damped oscillation period T: |
|
||||||
|
Logarithmic damping decrement: |
The Coriolis force is:
where is point weight,-vectorangular velocityrotating frame of reference, is the velocity vector of a point mass in this frame of reference, square brackets denote the operation vector product.
The quantity 
called Coriolis acceleration.
By physical nature
Mechanical(sound,vibration)
Electromagnetic (shine,radio waves, thermal)
Mixed type- combinations of the above
By the nature of interaction with the environment
Forced - vibrations occurring in the system under the influence of external periodic influences. Examples: leaves on trees, raising and lowering an arm. With forced vibrations, the phenomenon may occur resonance: a sharp increase in the amplitude of the oscillations when the natural frequencyoscillatorand the frequency of external influences.
Free (or own)- these are oscillations in the system under the influence of internal forces, after the system is taken out of the equilibrium state (in real conditions, free oscillations are always fading). The simplest examples of free vibrations are the vibrations of a weight attached to a spring or a weight suspended from a string.
Self-oscillations - fluctuations in which the system has a margin potential energyspent on making 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 - vibrations arising from a change in any parameter of the oscillatory system as a result of external influence.
Random - oscillations in which the external or parametric load is a random process.
Harmonic vibrations
where xANDω
Generalized harmonic oscillation in differential form
(Any non-trivial
Harmonic speed and acceleration.
According to the definition of speed, speed is the time derivative of the coordinate
Thus, we see that the speed during harmonic oscillatory motion also changes according to a harmonic law, but the speed oscillations are ahead of the phase displacement oscillations by p / 2.
Value - the maximum speed of the oscillatory motion (the amplitude of the speed oscillations).
Therefore, for the speed with harmonic oscillation, we have: 
,
and for the case of a zero initial phase (see graph).
According to the definition of acceleration, acceleration is the time derivative of speed:
-
the second derivative of the coordinate in time. Then:.
Acceleration during harmonic oscillatory motion also changes according to a harmonic law, but acceleration oscillations are ahead of velocity oscillations by p / 2 and displacement oscillations by p (they say that oscillations occur in antiphase).
The quantity
Maximum acceleration (amplitude of acceleration oscillations). Therefore, for acceleration we have: 
,
and for the case of a zero initial phase: 
(see graph).
From the analysis of the process of oscillatory motion, graphs and the corresponding mathematical expressions, it can be seen that when the oscillating body passes the equilibrium position (displacement is zero), the acceleration is zero, and the body's speed is maximum (the body passes the equilibrium position by inertia), and when the amplitude value of the displacement is reached, the velocity 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; AND- the amplitude of oscillations is a 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; - 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)