Forced vibrations. Resonance. Forced vibrations Can forced vibrations
In order for the system to perform undamped oscillations, it is necessary to compensate for the loss of oscillation energy due to friction from the outside. In order to ensure that the oscillation energy of the system does not decrease, a force is usually introduced that periodically acts on the system (we will call such a force forcing, and the oscillations are forced).
DEFINITION: forced These are the oscillations that occur in an oscillatory system under the influence of an external periodically changing force.
This force usually plays a dual role:
Firstly, it rocks the system and provides it with a certain amount of energy;
Secondly, it periodically replenishes energy losses (energy consumption) to overcome the forces of resistance and friction.
Let the driving force change over time according to the law:
Let us compose an equation of motion for a system oscillating under the influence of such a force. We assume that the system is also affected by a quasi-elastic force and the resistance force of the medium (which is true under the assumption of small oscillations).
Then the equation of motion of the system will look like:
Or 
.
Having made the substitutions , , - the natural frequency of oscillations of the system, we obtain a non-homogeneous linear differential equation of the 2nd order:
From the theory of differential equations it is known that the general solution of an inhomogeneous equation is equal to the sum of the general solution of a homogeneous equation and a particular solution of an inhomogeneous equation.
The general solution of the homogeneous equation is known:
,
Where 
; a 0 and a- arbitrary const.
.
Using a vector diagram, you can verify that this assumption is true, and also determine the values of “ a" And " j”.
The amplitude of oscillations is determined by the following expression:
.
Meaning " j”, which is the magnitude of the phase lag of the forced oscillation 
from the driving force that determined it, is also determined from the vector diagram and amounts to:
.
Finally, a particular solution to the inhomogeneous equation will take the form:
 |
(8.18) |
This function, combined with
 |
(8.19) |
gives a general solution to an inhomogeneous differential equation that describes the behavior of a system under forced oscillations. The term (8.19) plays a significant role in the initial stage of the process, during the so-called establishment of oscillations (Fig. 8.10).
Over time, due to the exponential factor, the role of the second term (8.19) decreases more and more, and after sufficient time it can be neglected, retaining only the term (8.18) in the solution.
Thus, function (8.18) describes steady-state forced oscillations. They represent harmonic oscillations with a frequency equal to the frequency of the driving force. The amplitude of forced oscillations is proportional to the amplitude of the driving force. For a given oscillatory system (defined by w 0 and b), the amplitude depends on the frequency of the driving force. Forced oscillations lag behind the driving force in phase, and the magnitude of the lag “j” also depends on the frequency of the driving force.
The dependence of the amplitude of forced oscillations on the frequency of the driving force leads to the fact that at a certain frequency determined for a given system, the amplitude of oscillations reaches a maximum value. The oscillatory system turns out to be especially responsive to the action of the driving force at this frequency. This phenomenon is called resonance, and the corresponding frequency is resonant frequency.
DEFINITION: a phenomenon in which a sharp increase in the amplitude of forced oscillations is observed is called resonance.
The resonant frequency is determined from the maximum condition for the amplitude of forced oscillations:
. (8.20)
Then, substituting this value into the expression for the amplitude, we get:
. (8.21)
In the absence of medium resistance, the amplitude of oscillations at resonance would turn to infinity; the resonant frequency under the same conditions (b = 0) coincides with the natural frequency of oscillations.
The dependence of the amplitude of forced oscillations on the frequency of the driving force (or, what is the same, on the oscillation frequency) can be represented graphically (Fig. 8.11). The individual curves correspond to different values of “b”. The smaller “b”, the higher and to the right the maximum of this curve lies (see the expression for w res.). With very large damping, resonance is not observed - with increasing frequency, the amplitude of forced oscillations monotonically decreases (lower curve in Fig. 8.11).
The set of presented graphs corresponding to different values of b is called resonance curves.
Notes regarding resonance curves:
As w®0 tends, all curves come to the same nonzero value, equal to . This value represents the displacement from the equilibrium position that the system receives under the influence of a constant force F 0 .
For w®¥, all curves asymptotically tend to zero, because at high frequencies, the force changes its direction so quickly that the system does not have time to noticeably shift from its equilibrium position.
The smaller b, the more the amplitude near resonance changes with frequency, the “sharper” the maximum.
Examples:
The phenomenon of resonance often turns out to be useful, especially in acoustics and radio engineering.
Losses of mechanical energy in any oscillatory system due to the presence of friction forces are inevitable, therefore, without “pumping” energy from the outside, the oscillations will be damped. There are several fundamentally different ways to create oscillatory systems of continuous oscillations. Let's take a closer look at undamped oscillations under the influence of an external periodic force. Such oscillations are called forced. Let's continue studying the motion of a harmonic pendulum (Fig. 6.9).
In addition to the previously discussed forces of elasticity and viscous friction, the ball is acted upon by an external forcing periodic force varying according to a harmonic law
frequency, which may differ from the natural frequency of the pendulum ω o. The nature of this force in this case is not important to us. Such a force can be created in various ways, for example, by imparting an electric charge to the ball and placing it in an external alternating electric field. The equation of motion of the ball in the case under consideration has the form
Let us divide it by the mass of the ball and use the previous notation for the system parameters. As a result we get forced oscillation equation:
Where f o = F o /m− the ratio of the amplitude value of the external driving force to the mass of the ball. The general solution of equation (3) is quite cumbersome and, of course, depends on the initial conditions. The nature of the motion of the ball, described by equation (3), is clear: under the influence of the driving force, oscillations will arise, the amplitude of which will increase. This transition regime is quite complex and depends on the initial conditions. After a certain period of time, the oscillatory mode will be established and their amplitude will cease to change. Exactly steady state of oscillation, in many cases is of primary interest. We will not consider the transition of the system to a steady state, but will focus on describing and studying the characteristics of this mode. With this formulation of the problem, there is no need to specify initial conditions, since the steady state we are interested in does not depend on the initial conditions, its characteristics are completely determined by the equation itself. We encountered a similar situation when studying the motion of a body under the action of a constant external force and the force of viscous friction
After some time, the body moves at a constant steady speed v = F o /β , which does not depend on the initial conditions and is completely determined by the equation of motion. The initial conditions determine the regime transitional to steady motion. Based on common sense, it is reasonable to assume that in a steady mode of oscillation the ball will oscillate at the frequency of the external driving force. Therefore, the solution to equation (3) should be sought in a harmonic function with the frequency of the driving force. First, let's solve equation (3), neglecting the resistance force
Let's try to find its solution in the form of a harmonic function
To do this, we calculate the dependence of the speed and acceleration of the body on time, as derivatives of the law of motion
and substitute their values into equation (4)
Now you can reduce it by cosωt. Consequently, this expression turns into the correct identity at any time, subject to the fulfillment of the condition
Thus, our assumption about the solution of equation (4) in the form (5) was justified: the steady state of oscillations is described by the function
Note that the coefficient A according to the resulting expression (6) can be either positive (with ω < ω o), and negative (with ω > ω o). The change in sign corresponds to a change in the phase of oscillations by π (the reason for this change will be clarified a little later), therefore the amplitude of the oscillations is the modulus of this coefficient |A|. The amplitude of the steady-state oscillations, as one would expect, is proportional to the magnitude of the driving force. In addition, this amplitude depends in a complex way on the frequency of the driving force. A schematic graph of this relationship is shown in Fig. 6.10
Rice. 6.10 Resonance curve
As follows from formula (6) and is clearly visible on the graph, as the frequency of the driving force approaches the natural frequency of the system, the amplitude increases sharply. The reason for this increase in amplitude is clear: the driving force “during” pushes the ball, when the frequencies completely coincide, the established mode is absent - the amplitude increases to infinity. Of course, in practice it is impossible to observe such an infinite increase: Firstly, this can lead to the destruction of the oscillatory system itself, Secondly, at large amplitudes of oscillations, the resistance forces of the medium cannot be neglected. A sharp increase in the amplitude of forced oscillations as the frequency of the driving force approaches the natural frequency of oscillations of the system is called the phenomenon of resonance. Let us now proceed to the search for a solution to the equation of forced oscillations taking into account the resistance force
Naturally, in this case too, the solution should be sought in the form of a harmonic function with the frequency of the driving force. It is easy to see that searching for a solution in the form (5) in this case will not lead to success. Indeed, equation (8), in contrast to equation (4), contains the particle velocity, which is described by the sine function. Therefore, the time part in equation (8) will not be reduced. Therefore, the solution to equation (8) should be represented in the general form of a harmonic function
in which there are two parameters A o And φ must be found using equation (8). Parameter A o is the amplitude of forced oscillations, φ − phase shift between the changing coordinate and the variable driving force. Using the trigonometric formula for the cosine of the sum, function (9) can be represented in the equivalent form
which also contains two parameters B=A o cosφ And C = −A o sinφ to be determined. Using function (10), we write explicit expressions for the dependences of the speed and acceleration of a particle on time
and substitute into equation (8):
Let us rewrite this expression in the form
In order for equality (13) to be satisfied at any time, it is necessary that the coefficients of the cosine and sine be equal to zero. Based on this condition, we obtain two linear equations for determining the parameters of function (10):
The solution to this system of equations has the form
Based on formula (10), we determine the characteristics of forced oscillations: amplitude
phase shift
At low attenuation, this dependence has a sharp maximum as the driving force frequency approaches ω to the natural frequency of the system ω o. Thus, in this case, resonance may also occur, which is why the plotted dependences are often called a resonance curve. Taking into account weak attenuation shows that the amplitude does not increase to infinity, its maximum value depends on the attenuation coefficient - as the latter increases, the maximum amplitude quickly decreases. The resulting dependence of the oscillation amplitude on the frequency of the driving force (16) contains too many independent parameters ( f o , ω o , γ ) in order to construct a complete family of resonance curves. As in many cases, this relationship can be significantly simplified by moving to “dimensionless” variables. Let us transform formula (16) to the following form
and denote
− relative frequency (the ratio of the frequency of the driving force to the natural frequency of oscillations of the system);
− relative amplitude (the ratio of the oscillation amplitude to the deviation value A o = f/ω o 2 at zero frequency);
− dimensionless parameter that determines the amount of attenuation. Using these notations, function (16) is significantly simplified
since it contains only one parameter − δ . A one-parameter family of resonance curves described by function (16 b) can be constructed, especially easily using a computer. The result of this construction is shown in Fig. 629.
rice. 6.11
Note that the transition to “conventional” units of measurement can be carried out by simply changing the scale of the coordinate axes. It should be noted that the frequency of the driving force, at which the amplitude of the forced oscillations is maximum, also depends on the damping coefficient, decreasing slightly as the latter increases. Finally, we emphasize that an increase in the damping coefficient leads to a significant increase in the width of the resonance curve. The resulting phase shift between the oscillations of the point and the driving force also depends on the frequency of the oscillations and their damping coefficient. We will become more familiar with the role of this phase shift when considering energy conversion in the process of forced oscillations.
the frequency of free undamped oscillations coincides with the natural frequency, the frequency of damped oscillations is slightly less than the natural one, and the frequency of forced oscillations coincides with the frequency of the driving force, and not the natural frequency.
Forced electromagnetic oscillations
Forced These are the oscillations that occur in an oscillatory system under the influence of an external periodic influence.
Fig.6.12. Circuit with forced electrical oscillations
Let us consider the processes occurring in an electric oscillatory circuit ( Fig.6.12), connected to an external source, the emf of which varies according to the harmonic law
,
Where m– amplitude of external EMF,
– cyclic frequency of EMF.
Let us denote by U C voltage across the capacitor, and through i - current strength in the circuit. In this circuit, in addition to the variable EMF (t) the self-induced emf is also active L in the inductor.
The self-induction emf is directly proportional to the rate of change of current in the circuit
.
For withdrawal differential equation of forced oscillations arising in such a circuit, we use Kirchhoff’s second rule
.
Voltage across active resistance R find by Ohm's law
.
The strength of the electric current is equal to the charge flowing per unit time through the cross section of the conductor
.
Hence
.
Voltage U C on the capacitor is directly proportional to the charge on the capacitor plates
.
The self-induction emf can be represented through the second derivative of the charge with respect to time
.
Substituting voltage and EMF into Kirchhoff's second rule
.
Dividing both sides of this expression by L and distributing the terms according to the degree of decreasing order of the derivative, we obtain a second-order differential equation
.
Let us introduce the following notation and obtain
– attenuation coefficient,
– cyclic frequency of natural oscillations of the circuit.
.
(1)
Equation (1) is heterogeneous linear differential equation of the second order. This type of equation describes the behavior of a wide class of oscillatory systems (electrical, mechanical) under the influence of external periodic influence (external emf or external force).
The general solution of equation (1) consists of the general solution q 1 homogeneous differential equation (2)
(2)
and any private solution q 2 heterogeneous equations (1)
.
Type of general solution homogeneous equation (2) depends on the value of the attenuation coefficient . We will be interested in the case of weak attenuation << 0 . При этом общее решение уравнения (2) имеет вид
Where B And 0 – constants specified by the initial conditions.
Solution (3) describes damped oscillations in the circuit. Values included in (3):
– cyclic frequency of damped oscillations;
– amplitude of damped oscillations;
–phase of damped oscillations.
We look for a particular solution to equation (1) in the form of a harmonic oscillation occurring with a frequency equal to the frequency external periodic influence - EMF, and lagging in phase by From him
Where 
– amplitude of forced oscillations, depending on frequency.
Let us substitute (4) into (1) and obtain the identity
To compare the phases of oscillations, we use trigonometric reduction formulas
.
Then our equation will be rewritten as
Let us represent the oscillations on the left side of the resulting identity in the form vector diagram (rice.6.13)..
The third term corresponding to oscillations on the capacitance WITH, having phase (
t
–
) and amplitude 
, we represent it as a horizontal vector directed to the right.
Fig.6.13. Vector diagram
The first term on the left side, corresponding to oscillations in inductance L, will be depicted on the vector diagram as a vector directed horizontally to the left (its amplitude 
).
Second term corresponding to oscillations in resistance R, we represent it as a vector directed vertically upward (its amplitude 
), because its phase is /2 behind the phase of the first term.
Since the sum of three vibrations to the left of the equal sign gives a harmonic vibration 
, then the vector sum on the diagram (diagonal of the rectangle) depicts an oscillation with an amplitude 
and phase
t, which is on
advances the oscillation phase of the third term.
From a right triangle, using the Pythagorean theorem, you can find the amplitude A( )
(5)
And tg as the ratio of the opposite side to the adjacent side.
.
(6)
Consequently, solution (4) taking into account (5) and (6) will take the form
.
(7)
General solution of a differential equation(1) is the sum q 1 and q 2
.
(8)
Formula (8) shows that when a circuit is exposed to a periodic external EMF, oscillations of two frequencies arise in it, i.e. undamped oscillations with the frequency of external EMF
and damped oscillations with frequency 
. Amplitude of damped oscillations 
Over time, it becomes negligibly small, and only forced oscillations remain in the circuit, the amplitude of which does not depend on time. Consequently, steady-state forced oscillations are described by function (4). That is, forced harmonic oscillations occur in the circuit, with a frequency equal to the frequency of the external influence and amplitude 
, depending on this frequency ( rice. 3A) according to law (5). In this case, the phase of the forced oscillation lags behind by
from coercive influence.
Having differentiated expression (4) with respect to time, we find the current strength in the circuit
Where 
– current amplitude.
Let us write this expression for the current strength in the form
,
(9)
Where 
–phase shift between current and external emf.
In accordance with (6) and rice. 2
.
(10)
From this formula it follows that the phase shift between the current and the external emf depends, at constant resistance R, from the relationship between the frequency of the driving EMF and natural frequency of the circuit 0 .
If < 0, then the phase shift between the current and the external EMF < 0. Колебания силы тока опережают колебания ЭДС по фазе на угол .
If > 0 then > 0. Current fluctuations lag behind EMF fluctuations in phase by an angle .
If = 0 (resonant frequency), That = 0, i.e. the current and EMF oscillate in the same phase.
Resonance– this is the phenomenon of a sharp increase in the amplitude of oscillations when the frequency of the external, driving force coincides with the natural frequency of the oscillatory system.
At resonance = 0 and oscillation period
.
Considering that the attenuation coefficient
,
we obtain expressions for the quality factor at resonance T = T 0
,
on the other side
.
The voltage amplitudes across the inductance and capacitance at resonance can be expressed through the quality factor of the circuit
,
(15)
.
(16)
From (15) and (16) it is clear that when
=
0, voltage amplitude across the capacitor and inductance in Q times greater than the amplitude of the external emf. This is a property of sequential RLC circuit is used to isolate a radio signal of a certain frequency 
from the radio frequency spectrum when rebuilding the radio receiver.
On practice RLC circuits are connected to other circuits, measuring instruments or amplifying devices that introduce additional attenuation into RLC circuit. Therefore, the real value of the quality factor of the loaded RLC circuit turns out to be lower than the value of the quality factor, estimated by the formula
.
The real value of the quality factor can be estimated as
Fig.6.14. Determining the quality factor from the resonance curve
,
where f– bandwidth of frequencies in which the amplitude is 0.7 of the maximum value ( rice. 4).
Capacitor voltage U C, on active resistance U R and on the inductor U L reach a maximum at different frequencies, respectively
,
,
.
If the attenuation is low 0 >> , then all these frequencies practically coincide and we can assume that
.
1. Let us find out what energy transformations occur during oscillations of a spring pendulum (see Fig. 80). When a spring is stretched, its potential energy increases and at maximum stretch it has the value E n = .
As the load moves towards the equilibrium position, the potential energy of the spring decreases, and the kinetic energy of the load increases. In the equilibrium position, the kinetic energy of the load is maximum E k = , and the potential energy of the spring is zero.
When a spring is compressed, its potential energy increases and the kinetic energy of the load decreases. At maximum compression, the potential energy of the spring is maximum, and the kinetic energy of the load is zero.
If we neglect the friction force, then at any moment of time the sum of potential and kinetic energies remains unchanged
E = E n + E k = const.
In the presence of a frictional force, energy is spent on doing work against this force, the amplitude of oscillations decreases and the oscillations die out.
Thus, the free oscillations of the pendulum, occurring due to the initial supply of energy, are always fading.
2. The question arises what needs to be done to ensure that the fluctuations do not stop over time. Obviously, to obtain undamped oscillations it is necessary to compensate for energy losses. This can be done in different ways. Let's consider one of them.
You know well that the vibrations of a swing will not die out if you constantly push it, that is, act on it with some force. In this case, the vibrations of the swing are no longer free, they will occur under the influence of an external force. The work of this external force precisely replenishes the energy loss caused by friction.
Let's find out what the external force should be? Let us assume that the magnitude and direction of the force are constant. Obviously, in this case the oscillations will stop, because the body, having passed the equilibrium position, will not return to it. Therefore, the magnitude and direction of the external force must change periodically.
Thus,
forced oscillations are oscillations that occur under the influence of an external, periodically changing force.
Forced vibrations, unlike free ones, can occur at any frequency. The frequency of forced oscillations is equal to the frequency of change of the force acting on the body, in this case it is called forcing.
3. Let's do an experiment. We hang several pendulums of different lengths from a rope fixed in the racks (Fig. 82). Let's deflect the pendulum A from the equilibrium position and leave it to itself. It will oscillate freely, acting with some periodic force on the rope. The rope, in turn, will act on the remaining pendulums. As a result, all pendulums will begin to perform forced oscillations with the frequency of oscillations of the pendulum A.
We will see that all pendulums will begin to oscillate with a frequency equal to the frequency of the pendulum's oscillations A. However, their amplitude of oscillations, except for the pendulum C, will be less than the amplitude of the pendulum oscillations A. The pendulum C, the length of which is equal to the length of the pendulum A, will swing very strongly. Consequently, the pendulum has the greatest oscillation amplitude, the natural frequency of oscillations of which coincides with the frequency of the driving force. In this case they say that it is observed resonance.
Resonance is the phenomenon of a sharp increase in the amplitude of forced oscillations when the frequency of the driving force coincides with the natural frequency of the oscillatory system (pendulum).
Resonance can be observed when the swing oscillates. Now you can explain that the swing will swing more strongly if it is pushed in time with its own vibrations. In this case, the frequency of the external force is equal to the oscillation frequency of the swing. Any push against the movement of the swing will cause a decrease in its amplitude.
4 * . Let's find out what energy transformations occur during resonance.
If the frequency of the driving force differs from the natural frequency of vibration of the body, then the driving force will be directed either in the direction of movement of the body or against it. Accordingly, the work of this force will be either negative or positive. In general, the work of the driving force in this case slightly changes the energy of the system.
Let now the frequency of the external force be equal to the natural frequency of oscillations of the body. In this case, the direction of the driving force coincides with the direction of the body's velocity, and the resistance force is compensated by an external force. The body vibrates only under the influence of internal forces. In other words, the negative work against the resistance force is equal to the positive work of the external force. Therefore, oscillations occur with maximum amplitude.
5. The phenomenon of resonance must be taken into account in practice. In particular, machine tools and machines undergo slight vibrations during operation. If the frequency of these vibrations coincides with the natural frequency of individual parts of the machines, then the amplitude of the vibrations can be very large. The machine or the support on which it stands will collapse.
There are known cases when, due to resonance, an airplane fell apart in the air, the propellers of ships broke, and railway rails collapsed.
Resonance can be prevented by changing either the natural frequency of the system or the frequency of the force causing the oscillations. For this purpose, for example, soldiers crossing a bridge do not walk in step, but at a free pace. Otherwise, the frequency of their steps may coincide with the natural frequency of the bridge and it will collapse. This happened in 1750 in France, when a detachment of soldiers passed across a 102 m long bridge hanging on chains. A similar incident occurred in St. Petersburg in 1906. When a cavalry squadron crossed the Egyptian Bridge over the Fontanka River, the frequency of the horses’ clear step coincided with the vibration frequency of the bridge.
To prevent resonance, trains cross bridges at slow or very fast speeds so that the frequency of wheel impacts on the rail joints is significantly less or significantly greater than the natural frequency of the bridge.
The phenomenon of resonance is not always harmful. Sometimes it can be useful, since it allows you to obtain a large increase in the amplitude of vibrations with the help of even a small force.
The action of a device that allows you to measure the frequency of oscillations is based on the phenomenon of resonance. This device is called frequency meter. His work can be illustrated by the following experiment. A frequency meter model is attached to the centrifugal machine, which consists of a set of plates (tongues) of different lengths (Fig. 83). At the ends of the plates there are tin flags coated with white paint. You can notice that when you change the speed of rotation of the machine handle, different plates begin to vibrate. Those plates whose natural frequency is equal to the rotation frequency begin to vibrate.
Self-test questions
1. What determines the amplitude of free oscillations of a spring pendulum?
2. Is the amplitude of a pendulum's oscillations kept constant in the presence of friction forces?
3. What energy transformations occur when a spring pendulum oscillates?
4. Why are free oscillations damped?
5. What vibrations are called forced? Give examples of forced oscillations.
6. What is resonance?
7. Give examples of harmful manifestations of resonance. What needs to be done to prevent resonance?
8. Give examples of the use of the resonance phenomenon.
Task 26
1. Fill out table 14, writing down what force acts on the oscillatory system if it performs free or forced oscillations; what are the frequency and amplitude of these oscillations; whether they are damped or not.
Table 14
|
Oscillation characteristics |
Type of vibrations |
|
|
Available |
Forced |
|
|
Acting force |
||
|
Frequency |
||
|
Amplitude |
||
|
Attenuation |
||
2 e.Suggest an experiment for observing forced oscillations.
3 e.Study experimentally the phenomenon of resonance using mathematical pendulums you have made.
4. At a certain speed of rotation of the sewing machine wheel, the table on which it stands sometimes sways strongly. Why?
Forced oscillations are those oscillations that occur in a system when an external forcing periodically changing force, called a driving force, acts on it.
The nature (time dependence) of the driving force may be different. This can be a force changing according to a harmonic law. For example, a sound wave, the source of which is a tuning fork, hits the eardrum or microphone membrane. A harmoniously changing force of air pressure begins to act on the membrane.
The driving force can be in the nature of jolts or short impulses. For example, an adult swings a child on a swing, periodically pushing them at the moment when the swing reaches one of its extreme positions.
Our task is to find out how the oscillatory system reacts to the influence of a periodically changing driving force.
§ 1 The driving force changes according to the harmonic law
F resist = - rv x and compelling force F out = F 0 sin wt.
Newton's second law will be written as:
The solution to equation (1) is sought in the form , where is the solution to equation (1) if it did not have the right-hand side. It can be seen that without the right-hand side, the equation turns into the well-known equation of damped oscillations, the solution of which we already know. Over a sufficiently long time, the free oscillations that arise in the system when it is removed from the equilibrium position will practically die out, and only the second term will remain in the solution of the equation. We will look for this solution in the form
Let's group the terms differently:
This equality must be satisfied at any time t, which is only possible if the coefficients of the sine and cosine are equal to zero.
So, a body that is acted upon by a driving force, changing according to a harmonic law, performs oscillatory motion with the frequency of the driving force.
Let us examine in more detail the question of the amplitude of forced oscillations:
1 The amplitude of steady-state forced oscillations does not change over time. (Compare with the amplitude of free damped oscillations).
2 The amplitude of forced oscillations is directly proportional to the amplitude of the driving force.
3 The amplitude depends on the friction in the system (A depends on d, and the damping coefficient d, in turn, depends on the drag coefficient r). The greater the friction in the system, the smaller the amplitude of forced oscillations.
4 The amplitude of forced oscillations depends on the frequency of the driving force w. How? Let us study the function A(w).
At w = 0 (a constant force acts on the oscillatory system), the displacement of the body is constant over time (it must be borne in mind that this refers to a steady state, when the natural oscillations have almost died out).
· When w ® ¥, then, as is easy to see, amplitude A tends to zero.
· It is obvious that at a certain frequency of the driving force, the amplitude of the forced oscillations will take on the greatest value (for a given d). The phenomenon of a sharp increase in the amplitude of forced oscillations at a certain value of the frequency of the driving force is called mechanical resonance.
It is interesting that the quality factor of the oscillatory system in this case shows how many times the resonant amplitude exceeds the displacement of the body from the equilibrium position under the action of a constant force F 0 .
We see that both the resonant frequency and the resonant amplitude depend on the damping coefficient d. As d decreases to zero, the resonant frequency increases and tends to the natural oscillation frequency of the system w 0 . In this case, the resonant amplitude increases and at d = 0 it goes to infinity. Of course, in practice the amplitude of oscillations cannot be infinite, since in real oscillatory systems resistance forces always act. If the system has low attenuation, then we can approximately assume that resonance occurs at the frequency of its own oscillations:
where in the case under consideration is the phase shift between the driving force and the displacement of the body from the equilibrium position.
It is easy to see that the phase shift between force and displacement depends on the friction in the system and the frequency of the external driving force. This dependence is shown in the figure. It is clear that when< тангенс принимает отрицательные значения, а при >- positive.
Knowing the dependence on the angle, one can obtain the dependence on the frequency of the driving force.
At frequencies of the external force that are significantly lower than the natural force, the displacement lags slightly behind the driving force in phase. As the frequency of the external force increases, this phase delay increases. At resonance (if small), the phase shift becomes equal to . When >> the displacement and force oscillations occur in antiphase. This dependence may seem strange at first glance. To understand this fact, let us turn to energy transformations in the process of forced oscillations.
§ 2 Energy transformations
As we already know, the amplitude of oscillations is determined by the total energy of the oscillatory system. It was previously shown that the amplitude of forced oscillations remains unchanged over time. This means that the total mechanical energy of the oscillatory system does not change over time. Why? After all, the system is not closed! Two forces - an external periodically changing force and a resistance force - do work that should change the total energy of the system.
Let's try to figure out what's going on. The power of the external driving force can be found as follows:
We see that the power of the external force feeding the oscillatory system with energy is proportional to the oscillation amplitude.
Due to the work of the resistance force, the energy of the oscillatory system should decrease, turning into internal. Resistance force power:
Obviously, the power of the resistance force is proportional to the square of the amplitude. Let's plot both dependencies on a graph.
In order for the oscillations to be steady (the amplitude does not change over time), the work of the external force during the period must compensate for the energy loss of the system due to the work of the resistance force. The intersection point of the power graphs exactly corresponds to this regime. Let's imagine that for some reason the amplitude of forced oscillations has decreased. This will lead to the fact that the instantaneous power of the external force will be greater than the power of losses. This will lead to an increase in the energy of the oscillatory system, and the amplitude of the oscillations will restore its previous value.
In a similar way, one can be convinced that with a random increase in the amplitude of oscillations, the power losses will exceed the power of the external force, which will lead to a decrease in the energy of the system, and, consequently, to a decrease in the amplitude.
Let's return to the question of the phase shift between the displacement and the driving force at resonance. We have already shown that the displacement lags behind, and therefore the force leads the displacement, by . On the other hand, the velocity projection in the process of harmonic oscillations is always ahead of the coordinate by . This means that during resonance, the external driving force and speed oscillate in the same phase. This means they are co-directed at any given time! The work of the external force in this case is always positive, it all goes to replenish the oscillatory system with energy.
§ 3 Non-sinusoidal periodic influence
Forced oscillations of the oscillator are possible under any periodic external influence, not just sinusoidal. In this case, the established oscillations, generally speaking, will not be sinusoidal, but they will represent a periodic movement with a period equal to the period of the external influence.
An external influence can be, for example, successive shocks (remember how an adult “rocks” a child sitting on a swing). If the period of external shocks coincides with the period of natural oscillations, then resonance may occur in the system. The oscillations will be almost sinusoidal. The energy imparted to the system at each push replenishes the total energy of the system lost due to friction. It is clear that in this case, options are possible: if the energy imparted during a push is equal to or exceeds the friction losses per period, then the oscillations will either be steady or their scope will increase. This is clearly visible in the phase diagram.
It is obvious that resonance is also possible in the case when the period of repetition of shocks is a multiple of the period of natural oscillations. This is impossible with the sinusoidal nature of the external influence.
On the other hand, even if the shock frequency coincides with the natural frequency, resonance may not be observed. If only the friction losses during the period exceed the energy received by the system during the push, then the total energy of the system will decrease and the oscillations will dampen.
§ 4 Parametric resonance
External influence on the oscillatory system can be reduced to periodic changes in the parameters of the oscillatory system itself. The oscillations excited in this way are called parametric, and the mechanism itself is called parametric resonance .
First of all, we will try to answer the question: is it possible to shake up the small oscillations already existing in the system by periodically changing some of its parameters in a certain way.
As an example, consider a person swinging on a swing. By bending and straightening his legs at the “right” moments, he actually changes the length of the pendulum. In extreme positions, a person squats, thereby slightly lowering the center of gravity of the oscillatory system; in the middle position, a person straightens, raising the center of gravity of the system.
To understand why a person swings at the same time, consider an extremely simplified model of a person on a swing - an ordinary small pendulum, that is, a small weight on a light and long thread. To simulate the raising and lowering of the center of gravity, we will pass the upper end of the thread through a small hole and will pull the thread at those moments when the pendulum passes the equilibrium position, and lower the thread the same amount when the pendulum passes the extreme position.
The work of the thread tension force per period (taking into account that the load is lifted and lowered twice per period and that D l << l):
Please note that in brackets there is nothing more than triple the energy of the oscillatory system. By the way, this quantity is positive, therefore, the work of the tension force (our work) is positive, it leads to an increase in the total energy of the system, and therefore to the swing of the pendulum.
Interestingly, the relative change in energy over a period does not depend on whether the pendulum swings weakly or strongly. This is very important, and here's why. If the pendulum is not “pumped up” with energy, then for each period it will lose a certain part of its energy due to the friction force, and the oscillations will die out. And for the range of oscillations to increase, it is necessary that the energy gained exceeds that lost to overcome friction. And this condition, it turns out, is the same - both for a small amplitude and for a large one.
For example, if in one period the energy of free oscillations decreases by 6%, then in order for the oscillations of a pendulum 1 m long not to dampen, it is enough to reduce its length by 1 cm in the middle position, and increase it by the same amount in the extreme position.
Returning to the swing: if you start swinging, then there is no need to squat deeper and deeper - squat the same way all the time, and you will fly higher and higher!
*** Quality again!
As we have already said, for the parametric buildup of oscillations, the condition DE > A of friction per period must be met.
Let's find the work done by the friction force over the period
It can be seen that the relative amount of lifting of the pendulum to swing it is determined by the quality factor of the system.
§ 5 The meaning of resonance
Forced oscillations and resonance are widely used in technology, especially in acoustics, electrical engineering, and radio engineering. Resonance is primarily used when, from a large set of oscillations of different frequencies, one wants to isolate oscillations of a certain frequency. Resonance is also used in the study of very weak periodically repeating quantities.
However, in some cases resonance is an undesirable phenomenon, as it can lead to large deformations and destruction of structures.
§ 6 Examples of problem solving
Problem 1 Forced oscillations of a spring pendulum under the action of an external sinusoidal force.
A load of mass m = 10 g was suspended from a spring with stiffness k = 10 N/m and the system was placed in a viscous medium with a resistance coefficient r = 0.1 kg/s. Compare the natural and resonant frequencies of the system. Determine the amplitude of oscillations of the pendulum at resonance under the action of a sinusoidal force with an amplitude F 0 = 20 mN.
Solution:
1 The natural frequency of an oscillatory system is the frequency of free vibrations in the absence of friction. The natural cyclic frequency is equal to the oscillation frequency.
2 Resonant frequency is the frequency of an external driving force at which the amplitude of forced oscillations increases sharply. The resonant cyclic frequency is equal to , where is the damping coefficient, equal to .
Thus, the resonant frequency is . It is easy to see that the resonant frequency is less than the natural frequency! It is also clear that the lower the friction in the system (r), the closer the resonant frequency is to the natural frequency.
3 The resonant amplitude is
Task 2 Resonance amplitude and quality factor of the oscillatory system
A load of mass m = 100 g was suspended from a spring with stiffness k = 10 N/m and the system was placed in a viscous medium with a resistance coefficient
r = 0.02 kg/s. Determine the quality factor of the oscillatory system and the amplitude of oscillations of the pendulum at resonance under the action of a sinusoidal force with an amplitude F 0 = 10 mN. Find the ratio of the resonant amplitude to the static displacement under the influence of a constant force F 0 = 20 mN and compare this ratio with the quality factor.
Solution:
1 The quality factor of the oscillatory system is equal to , where is the logarithmic damping decrement.
The logarithmic damping decrement is equal to .
Finding the quality factor of the oscillatory system.
2 The resonant amplitude is
3 Static displacement under the action of a constant force F 0 = 10 mN is equal to .
4 The ratio of the resonant amplitude to the static displacement under the action of a constant force F 0 is equal to
It is easy to see that this ratio coincides with the quality factor of the oscillatory system
Problem 3 Resonant vibrations of a beam
Under the influence of the weight of the electric motor, the cantilever tank on which it is installed bent by . At what speed of the motor armature can there be a danger of resonance?
Solution:
1 The motor housing and the beam on which it is installed experience periodic shocks from the rotating armature of the motor and, therefore, perform forced oscillations at the frequency of the shocks.
Resonance will be observed when the frequency of shocks coincides with the natural frequency of vibration of the beam with the motor. It is necessary to find the natural frequency of vibration of the beam-motor system.
2 An analogue of the beam-motor oscillatory system can be a vertical spring pendulum, the mass of which is equal to the mass of the motor. The natural frequency of oscillation of a spring pendulum is equal to . But the spring stiffness and the mass of the motor are not known! What should I do?
3 In the equilibrium position of the spring pendulum, the gravitational force of the load is balanced by the elastic force of the spring
4 Find the rotation of the motor armature, i.e. shock frequency
Problem 4 Forced oscillations of a spring pendulum under the influence of periodic shocks.
A weight of mass m = 0.5 kg is suspended from a spiral spring with stiffness k = 20 N/m. The logarithmic damping decrement of the oscillatory system is equal to . They want to swing the weight with short pushes, acting on the weight with a force F = 100 mN for a time τ = 0.01 s. What should be the frequency of the strokes in order for the amplitude of the weight to be greatest? At what points and in what direction should you push the kettlebell? To what amplitude will it be possible to swing the weight in this way?
Solution:
1 Forced vibrations can occur under any periodic influence. In this case, the steady-state oscillation will occur with the frequency of the external influence. If the period of external shocks coincides with the frequency of natural oscillations, then resonance occurs in the system - the amplitude of oscillations becomes greatest. In our case, for resonance to occur, the period of the shocks must coincide with the period of oscillation of the spring pendulum.
The logarithmic damping decrement is small, therefore, there is little friction in the system, and the period of oscillation of a pendulum in a viscous medium practically coincides with the period of oscillation of a pendulum in a vacuum:
2 Obviously, the direction of the pushes must coincide with the speed of the weight. In this case, the work of the external force replenishing the system with energy will be positive. And the vibrations will sway. Energy received by the system during the impact process
will be greatest when the load passes the equilibrium position, because in this position the speed of the pendulum is maximum.
So, the system will swing most quickly under the action of shocks in the direction of movement of the load as it passes through the equilibrium position.
3 The amplitude of oscillations stops growing when the energy imparted to the system during the impact process is equal to the energy loss due to friction during the period: .
We will find the energy loss over a period through the quality factor of the oscillatory system
where E is the total energy of the oscillatory system, which can be calculated as .
Instead of the loss energy, we substitute the energy received by the system during the impact:
The maximum speed during the oscillation process is . Taking this into account, we get .
§7 Tasks for independent solution
Test "Forced vibrations"
1 What oscillations are called forced?
A) Oscillations occurring under the influence of external periodically changing forces;
B) Oscillations that occur in the system after an external shock;
2 Which of the following oscillations is forced?
A) Oscillation of a load suspended from a spring after its single deviation from the equilibrium position;
B) Oscillation of the loudspeaker cone during operation of the receiver;
B) Oscillation of a load suspended from a spring after a single impact on the load in the equilibrium position;
D) Vibration of the electric motor housing during its operation;
D) Vibrations of the eardrum of a person listening to music.
3 An oscillatory system with its own frequency is acted upon by an external driving force that varies according to the law. The damping coefficient in the oscillatory system is equal to . According to what law does the coordinate of a body change over time?
C) The amplitude of forced oscillations will remain unchanged, since the energy lost by the system due to friction will be compensated for by the energy gain due to the work of the external driving force.
5 The system performs forced oscillations under the action of a sinusoidal force. Specify All factors on which the amplitude of these oscillations depends.
A) From the amplitude of the external driving force;
B) The presence of energy in the oscillatory system at the moment the external force begins to act;
C) Parameters of the oscillatory system itself;
D) Friction in the oscillatory system;
D) The existence of natural oscillations in the system at the moment the external force begins to act;
E) Time of establishment of oscillations;
G) Frequencies of external driving force.
6 A block of mass m performs forced harmonic oscillations along a horizontal plane with period T and amplitude A. Friction coefficient μ. What work is done by the external driving force in a time equal to period T?
A) 4μmgA; B) 2μmgA; B) μmgA; D) 0;
D) It is impossible to give an answer, since the magnitude of the external driving force is not known.
7 Make a correct statement
Resonance is a phenomenon...
A) Coincidence of the frequency of the external force with the natural frequency of the oscillatory system;
B) A sharp increase in the amplitude of forced oscillations.
Resonance is observed under the condition
A) Reducing friction in the oscillatory system;
B) Increasing the amplitude of the external driving force;
C) The coincidence of the frequency of the external force with the natural frequency of the oscillatory system;
D) When the frequency of the external force coincides with the resonant frequency.
8 The phenomenon of resonance can be observed in...
A) In any oscillatory system;
B) In a system that performs free oscillations;
B) In a self-oscillating system;
D) In a system undergoing forced oscillations.
9 The figure shows a graph of the dependence of the amplitude of forced oscillations on the frequency of the driving force. Resonance occurs at a frequency...
10 Three identical pendulums located in different viscous media perform forced oscillations. The figure shows the resonance curves for these pendulums. Which pendulum experiences the greatest resistance from the viscous medium during oscillation?
A) 1; B) 2; AT 3;
D) It is impossible to give an answer, since the amplitude of forced oscillations, in addition to the frequency of the external force, also depends on its amplitude. The condition does not say anything about the amplitude of the external driving force.
11 The period of natural oscillations of the oscillatory system is equal to T 0. What can be the period of the shocks so that the amplitude of the oscillations increases sharply, that is, a resonance arises in the system?
A) T 0; B) T 0, 2 T 0, 3 T 0,…;
C) The swing can be rocked with pushes of any frequency.
12 Your little brother is sitting on a swing, you swing him with short pushes. What should be the period of succession of shocks for the process to occur most efficiently? The period of natural oscillations of the swing T 0.
D) The swing can be rocked with pushes of any frequency.
13 Your little brother is sitting on a swing, you swing him with short pushes. In what position of the swing should the push be made and in what direction should the push be made so that the process occurs most efficiently?
A) Push in the uppermost position of the swing towards the equilibrium position;
B) Push in the uppermost position of the swing in the direction from the equilibrium position;
B) Push in a balanced position in the direction of movement of the swing;
D) You can push in any position, but always in the direction of movement of the swing.
14 It would seem that by shooting from a slingshot at the bridge in time with its own vibrations and making a lot of shots, you can strongly swing it, but this is unlikely to succeed. Why?
A) The mass of the bridge (its inertia) is large compared to the mass of the “bullet” from a slingshot; the bridge will not be able to move under the influence of such impacts;
B) The impact force of a “bullet” from a slingshot is so small that the bridge will not be able to move under the influence of such impacts;
C) The energy imparted to the bridge in one blow is much less than the energy loss due to friction over the period.
15 You are carrying a bucket of water. The water in the bucket swings and splashes out. What can be done to prevent this from happening?
A) Swing the hand in which the bucket is located in rhythm with walking;
B) Change the speed of movement, leaving the length of steps unchanged;
C) Stop periodically and wait for the water vibrations to calm down;
D) Make sure that during the movement the hand with the bucket is positioned strictly vertically.
Tasks
1 The system performs damped oscillations with a frequency of 1000 Hz. Define Frequency v 0 natural vibrations, if the resonant frequency
2 Determine by what value D v resonant frequency differs from natural frequency v 0= 1000 Hz oscillatory system, characterized by a damping coefficient d = 400s -1.
3 A load of mass 100 g, suspended on a spring of stiffness 10 N/m, performs forced oscillations in a viscous medium with a resistance coefficient r = 0.02 kg/s. Determine the damping coefficient, resonant frequency and amplitude. The amplitude value of the driving force is 10 mN.
4 The amplitudes of forced harmonic oscillations at frequencies w 1 = 400 s -1 and w 2 = 600 s -1 are equal. Determine the resonant frequency.
5 Trucks enter a grain warehouse along a dirt road on one side, unload and leave the warehouse at the same speed, but on the other side. Which side of the warehouse has more potholes in the road than the other? How can you determine from which side of the warehouse is the entrance and which is the exit based on the condition of the road? Justify the answer
Forced vibrations vibrations that occur in any system under the influence of a variable external force (for example, vibrations of a telephone membrane under the influence of an alternating magnetic field, vibrations of a mechanical structure under the influence of a variable load, etc.). The nature of a military system is determined both by the nature of the external force and by the properties of the system itself. At the beginning of the action of a periodic external force, the nature of the V. c. changes with time (in particular, V. c. are not periodic), and only after some time periodic V. c. are established in the system with a period equal to the period of the external force (steady-state VC.). The establishment of a voltage in an oscillatory system occurs the faster, the greater the damping of oscillations in this system. In particular, in linear oscillatory systems (See Oscillatory systems), when an external force is turned on, free (or natural) oscillations and oscillations simultaneously arise in the system, and the amplitudes of these oscillations at the initial moment are equal, and the phases are opposite ( rice.
). After the gradual attenuation of free oscillations, only steady-state oscillations remain in the system. The amplitude of the VK is determined by the amplitude of the acting force and the attenuation in the system. If the attenuation is small, then the amplitude of the voltage wave depends significantly on the relationship between the frequency of the acting force and the frequency of natural oscillations of the system. As the frequency of the external force approaches the natural frequency of the system, the amplitude of the VK increases sharply—resonance occurs. In nonlinear systems (See Nonlinear systems), the division into free and VK is not always possible. Lit.: Khaikin S.E., Physical foundations of mechanics, M., 1963.
Great Soviet Encyclopedia. - M.: Soviet Encyclopedia. 1969-1978 .
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Books
- Forced vibrations of shaft torsion when taking into account damping, A.P. Filippov, Reproduced in the original author's spelling of the 1934 edition (publishing house Izvestia of the USSR Academy of Sciences). IN… Category: Mathematics Publisher: YOYO Media, Manufacturer: Yoyo Media,
- Forced transverse vibrations of rods taking into account damping, A.P. Filippov, Reproduced in the original author's spelling of the 1935 edition (publishing house "Izvestia of the USSR Academy of Sciences")... Category: