In a system that is not oscillatory. Forced oscillation equation and its solution. Resonance. Examples of problem solving
Forced vibrations oscillations that occur in any system under the action 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 action of a variable load, etc.). The nature of a V. to. 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. to. changes with time (in particular, V. to. are not periodic), and only after some time, periodic V. to. with a period equal to the period of the external force (steady-state VK.). The establishment of V. to. in an oscillatory system occurs the faster, the greater the damping of oscillations in this system. In particular, in linear oscillatory systems, when an external force is switched on, free (or natural) oscillations and V. to. 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 vibrations remain in the system. The amplitude of V. to. is determined by the amplitude of the acting force and the attenuation in the system. If the damping is small, then the amplitude of the V. to. essentially depends on the ratio between the frequency of the acting force and the frequency of natural oscillations of the system. When the frequency of the external force approaches the natural frequency of the system, the amplitude of the V. to. increases sharply - Resonance sets in. In non-linear systems (see Nonlinear systems), the division into free and free space is not always possible. Lit.: Khaikin S. E., Physical foundations of mechanics, M., 1963.
Great Soviet Encyclopedia. - M.: Soviet Encyclopedia. 1969-1978 .
See what "Forced vibrations" is in other dictionaries:
Forced vibrations- Forced vibrations. The dependence of their amplitude on the frequency of the external action at different attenuation: 1 weak attenuation; 2 strong attenuation; 3 critical attenuation. FORCED OSCILLATIONS, oscillations that occur in any system in ... ... Illustrated Encyclopedic Dictionary
forced vibrations- Oscillations occurring under the periodic influence of an external generalized force. [Non-destructive testing system. Types (methods) and technology of non-destructive testing. Terms and definitions (reference guide). Moscow 2003] forced ... ... Technical Translator's Handbook
Forced oscillations are oscillations that occur under the influence of external forces that change over time. Self-oscillations differ from forced oscillations in that the latter are caused by periodic external influences and occur with the frequency of this ... Wikipedia
FORCED OSCILLATIONS, oscillations that occur in any system as a result of periodically changing external influences: forces in a mechanical system, voltage or current in an oscillatory circuit. Forced vibrations always occur with ... ... Modern Encyclopedia
Oscillations arising in the c.l. system under the action of periodic ext. forces (e.g. oscillations of the phone membrane under the influence of an alternating magnetic field, oscillations of a mechanical structure under the influence of an alternating load). Har r V. to. is defined as external. by force... Physical Encyclopedia
Oscillations arising in the c.l. system under the influence of AC. ext. influences (for example, fluctuations in voltage and current in an electrical circuit caused by alternating emf; fluctuations in a mechanical system caused by alternating load). The character of V. to. is determined by ... ... Big encyclopedic polytechnic dictionary
They arise in the system under the action of a periodic external influence (for example, forced oscillations of a pendulum under the action of a periodic force, forced oscillations in an oscillatory circuit under the action of a periodic electromotive force). If a… … Big Encyclopedic Dictionary
Forced vibrations- (vibration) - oscillations (vibration) of the system, caused and maintained by force and (or) kinematic excitation. [GOST 24346 80] Forced oscillations - oscillations of systems caused by the action of loads that vary in time. [Industry… … Encyclopedia of terms, definitions and explanations of building materials
- (Constrained vibrations, forced vibrations) body vibrations caused by a periodically acting external force. If the period of forced oscillations coincides with the period of natural oscillations of the body, a resonance phenomenon is obtained. Samoilov K. I. ... ... Marine Dictionary
FORCED VIBRATIONS- (see) arising in any system under the influence of external variable influence; their character is determined both by the properties of the external influence and by the properties of the system itself. As the frequency of the external influence approaches the frequency of its own ... Great Polytechnic Encyclopedia
Occur in the system under the action of a periodic external influence (for example, forced oscillations of a pendulum under the action of a periodic force, forced oscillations in an oscillatory circuit under the action of a periodic emf). If the frequency... ... encyclopedic Dictionary
Books
- Forced vibrations of shaft torsion with attenuation taken into account, A.P. Filippov, Reproduced in the original author's spelling of the 1934 edition (publishing house `Proceedings of the Academy of Sciences of the USSR`). AT… Category: Mathematics Publisher: YoYo Media, Manufacturer: YoYo Media,
- Forced transverse vibrations of rods with damping taken into account, A.P. Filippov, Reproduced in the original author's spelling of the 1935 edition (publishing house "Proceedings of the Academy of Sciences of the USSR") ... Category:
In contrast to free oscillations, when the system receives only once (when the system is removed from ), in the case of forced oscillations, the system absorbs this energy from a source of external periodic force continuously. This energy makes up for the losses spent on overcoming, and therefore the total no remains unchanged.
Forced vibrations, unlike free ones, can occur at any frequency. coincides with the frequency of the external force acting on the oscillatory system. Thus, the frequency of forced oscillations is determined not by the properties of the system itself, but by the frequency of the external influence.
Examples of forced vibrations are the vibrations of a children's swing, the vibrations of a needle in a sewing machine, the vibrations of a piston in an automobile engine cylinder, the vibrations of the springs of a car moving on a rough road, etc.
Resonance
DEFINITION
Resonance- this is the phenomenon of a sharp increase in forced oscillations when the frequency of the driving force approaches the natural frequency of the oscillatory system.
Resonance occurs due to the fact that at , the external force, acting in time with free vibrations, always has the same direction from the oscillating body and does positive work: the energy of the oscillating body increases and becomes large. If the external force acts “not in time”, then this force alternately performs either negative or positive work, and as a result, the energy of the system changes insignificantly.
Figure 1 shows the dependence of the amplitude of forced oscillations on the frequency of the driving force. It can be seen that this amplitude reaches a maximum at a certain frequency value, i.e. at , where is the natural frequency of the oscillatory system. Curves 1 and 2 differ in the magnitude of the friction force. At low friction (curve 1), the resonance curve has a sharp maximum; at a higher friction force (curve 2), there is no such sharp maximum.
We often encounter the phenomenon of resonance in everyday life. If the windows trembled in the room when a heavy truck was passing along the street, this means that the natural frequency of the windows is equal to the frequency of the machine. If sea waves are in resonance with the period of the ship, then the pitching becomes especially strong.
The phenomenon of resonance must be taken into account when designing bridges, buildings and other structures that experience vibration under load, otherwise, under certain conditions, these structures can be destroyed. However, resonance can also be useful. The phenomenon of resonance is used when tuning a radio receiver to a certain frequency of broadcasting, as well as in many other cases.
Examples of problem solving
EXAMPLE 1
| Exercise | At the end of the spring of a horizontal pendulum, the load of which has a mass of 1 kg, a variable force acts, the oscillation frequency of which is 16 Hz. Will resonance be observed if the spring rate is 400 N/m. |
| Decision | Let us determine the natural frequency of the oscillatory system by the formula:
Since the frequency of the external force is not equal to the natural frequency of the system, the resonance phenomenon will not be observed. |
| Answer | The resonance phenomenon will not be observed. |
HzEXAMPLE 2
| Exercise | A small ball is suspended on a thread 1 m long from the ceiling of the car. At what speed of the car will the ball vibrate especially strongly under the impact of the wheels on the rail joints? Rail length 12.5 m. |
| Decision | The ball performs forced vibrations with a frequency equal to the frequency of the wheels hitting the rail joints: If the dimensions of the ball are small compared to the length of the thread, then the system can be considered, the natural frequency of which is:
the amplitude of forced undamped oscillations is maximum in the case of resonance, i.e. when . Thus it is possible to write: |
In this lesson, everyone will be able to study the topic “Transformation of energy during oscillatory motion. damped vibrations. Forced vibrations. In this lesson, we will consider what kind of energy transformation occurs during oscillatory motion. To do this, we will conduct an important experiment with a horizontal spring pendulum system. We will also discuss issues related to damped oscillations and forced oscillations.
The lesson is devoted to the topic "Conversion of energy during oscillatory motion." In addition, we will consider the issue related to damped and forced oscillations.
Let's get to know this question with the next important experiment. A body is attached to the spring, which can oscillate horizontally. Such a system is called a horizontal spring pendulum. In this case, the effect of gravity can be ignored.
Rice. 1. Horizontal spring pendulum
We will assume that in the system of friction forces, there are no resistance forces. When this system is in equilibrium and no oscillation occurs, the velocity of the body is 0 and there is no deformation of the spring. In this case, this pendulum has no energy. But as soon as the body is shifted relative to the equilibrium point to the right or to the left, in this case we will do the work of communicating energy in this oscillatory system. What happens in this case? The following happens: the spring is deformed, its length changes. We give the spring potential energy. If you now release the load, do not hold it, then it will begin to move towards the equilibrium position, the spring will begin to straighten and the deformation of the spring will decrease. The speed of the body will increase, and according to the law of conservation of energy, the potential energy of the spring will be converted into the kinetic energy of the body's motion.
Rice. 2. Stages of oscillation of a spring pendulum
Deformation∆x of the spring is determined as follows: ∆x = x 0 - x. Having considered the deformation, we can say that all the potential energy is stored in the spring: .
During oscillations, the potential energy is constantly converted into the kinetic energy of the bar: .
For example, when the bar passes the equilibrium point x 0 , the deformation of the spring is 0, i.e. ∆x=0, therefore, the potential energy of the spring is 0 and all the potential energy of the spring has turned into the kinetic energy of the bar: E p (at point B) \u003d E k (at point A). Or 
.
As a result of this movement, potential energy is converted into kinetic energy. Then the so-called phenomenon of inertia comes into play. A body that has a certain mass, by inertia, passes the point of equilibrium. The speed of the body begins to decrease, and the deformation, the elongation of the spring increases. It can be concluded that the kinetic energy of the body decreases, and the potential energy of the spring begins to increase again. We can talk about the transformation of kinetic energy into potential.
When the body finally stops, the speed of the body will be equal to 0, and the deformation of the spring will become maximum, in this case we can say that all the kinetic energy of the body has turned into the potential energy of the spring. In the future, everything is repeated from the beginning. If one condition is met, such a process will occur continuously. What is this condition? This condition is the absence of friction. But the force of friction, the force of resistance is present in any system. Therefore, with each subsequent movement of the pendulum, energy losses occur. Work is being done to overcome the force of friction. Friction force to the law of Coulomb - Amonton: F TP \u003d μ.N.
Speaking of oscillations, we must always remember that the force of friction leads to the fact that gradually all the energy stored in a given oscillatory system is converted into internal energy. As a result, the oscillations stop, and once the oscillations stop, then such oscillations are called damped.
damped vibrations - vibrations, the amplitude of which decreases due to the fact that the energy of the oscillatory system is spent on overcoming the forces of resistance and friction forces.
Rice. 3. Graph of damped oscillations
The next type of oscillations that we will consider, the so-called. forced vibrations. Forced vibrations called such vibrations that occur under the action of a periodic, external force acting on a given oscillatory system.
If the pendulum oscillates, then in order for these oscillations not to stop, each time an external force must act on the pendulum. For example, we act on the pendulum with our own hand, make it move, push it. It is imperative to act with some force and make up for the loss of energy. So, forced vibrations are those vibrations that occur under the action of an external driving force. The frequency of such oscillations will coincide with the frequency of the external acting force. When an external force begins to act on the pendulum, the following happens: at first, the oscillations will have a small amplitude, but gradually this amplitude will increase. And when the amplitude acquires a constant value, the oscillation frequency also acquires a constant value, they say that such oscillations have been established. Forced oscillations have been established.
established forced vibrations make up for the loss of energy precisely due to the work of an external driving force.
Resonance
There is a very important phenomenon that is quite often observed in nature and technology. This phenomenon is called resonance. "Resonance" is a Latin word and is translated into Russian as "response". Resonance (from lat.resono - “I respond”) - the phenomenon of an increase in the amplitude of the forced oscillations of the system, which occurs when the frequency of the external action of the force approaches the frequency of the natural oscillation of the pendulum or this oscillatory system.
If there is a pendulum that has its own length, mass or spring stiffness, then this pendulum has its own oscillations, which are characterized by frequency. If an external driving force begins to act on this pendulum and the frequency of this force begins to approach the natural frequency of the pendulum (coincides with it), then a sharp increase in the oscillation amplitude occurs. This is the phenomenon of resonance.
As a result of such a phenomenon, the oscillations can be so large that the body, the oscillatory system itself, will collapse. There is a known case when a line of soldiers walking across the bridge, as a result of such a phenomenon, simply collapsed the bridge. Another case when, as a result of the movement of air masses, sufficiently powerful gusts of wind, a bridge collapsed in the United States. This is also a phenomenon of resonance. The oscillations of the bridge, their own vibrations, coincided with the frequency of gusts of wind, the external driving force. This caused the amplitude to increase so much that the bridge collapsed.
They try to take this phenomenon into account when designing structures and mechanisms. For example, when a train is moving, the following may happen. If a wagon is moving and this wagon begins to sway to the beat of its movement, then the amplitude of oscillations can increase so much that the wagon can derail. There will be a crash. To characterize this phenomenon, curves are used, which are called resonant.
Rice. 4. Resonance curve. Curve peak - maximum amplitude
Of course, resonance is not only fought, but also used. It is mostly used in acoustics. Where there is an auditorium, a theater hall, a concert hall, we must take into account the phenomenon of resonance.
List of additional literature:
Are you familiar with resonance? // Quantum. - 2003. - No. 1. - P. 32-33 Physics: Mechanics. Grade 10: Proc. for in-depth study of physics / M.M. Balashov, A.I. Gomonova, A.B. Dolitsky and others; Ed. G.Ya. Myakishev. - M.: Bustard, 2002. Elementary textbook of physics. Ed. G.S. Landsberg, T. 3. - M., 1974
Forced vibrations are called such vibrations that occur in the system under the action of an external driving periodically changing force, called the driving force.
The nature (dependence on time) of the driving force can be different. It can be a force that changes according to the harmonic law. For example, a sound wave, the source of which is a tuning fork, hits the eardrum or microphone membrane. A harmonically changing force of air pressure begins to act on the membrane.
The driving force can be in the form of shocks or short impulses. For example, an adult swings a child on a swing, periodically pushing them at the moment when the swing comes to one of the extreme positions.
Our task is to find out how the oscillatory system reacts to the action of a periodically changing driving force.
§ 1 The driving force changes according to the harmonic law
F cont = - rv x and driving force F out \u003d F 0 sin wt.
Newton's second law is 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 side. It can be seen that without the right-hand side, the equation turns into the equation of damped oscillations known to us, the solution of which we already know. For a sufficiently long time, free oscillations that arise in the system when it is taken out of equilibrium 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 hold at any time t, which is possible only if the coefficients at the sine and cosine are equal to zero.
So, the body, on which the driving force acts, changing according to the harmonic law, makes an 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 factor 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? We study the function A(w).
When w = 0 (a constant force acts on the oscillatory system), the displacement of the body is unchanged over time (it must be borne in mind that this refers to the steady state, when the natural oscillations have almost died out).
· When w ® ¥, then, as it is easy to see, the amplitude A tends to zero.
· Obviously, at some frequency of the driving force, the amplitude of the forced oscillations will take on the largest 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.
Interestingly, 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 factor d. As d decreases to zero, the resonant frequency increases and tends to the frequency of natural oscillations of the system w 0 . In this case, the resonant amplitude increases and, at d = 0, turns to infinity. Of course, in practice, the amplitude of oscillations cannot be infinite, since resistance forces always act in real oscillatory systems. If the system has low damping, then approximately we can assume that the resonance occurs at the frequency of natural 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 the force and the 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 seen that at< 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 less than its own, 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 . At >>, the displacement and force fluctuations occur in antiphase. Such a dependence may seem strange at first glance. To understand this fact, let us turn to the energy transformations in the process of forced oscillations.
§ 2 Energy transformations
As we already know, the oscillation amplitude is determined by the total energy of the oscillatory system. Previously, it was shown that the amplitude of forced oscillations remains unchanged with 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 the matter. 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 energy. Resistance force power:
Obviously, the power of the drag force is proportional to the square of the amplitude. Let's plot both dependencies on the graph.
In order for the oscillations to be steady (the amplitude does not change over time), the work of the external force over the period must compensate for the energy losses of the system due to the work of the resistance force. The point of intersection of power graphs just corresponds to this mode. 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 oscillation amplitude will restore its previous value.
Similarly, it can be seen that with a random increase in the oscillation amplitude, the loss power 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 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, which means that the force is ahead of the displacement by . On the other hand, the velocity projection in the process of harmonic oscillations always leads the coordinate by . This means that at resonance, the external driving force and speed oscillate in the same phase. So they are co-directed at any moment of time! The work done by the external force is always positive in this case. all goes to replenish the oscillatory system with energy.
§ 3 Non-sinusoidal periodic action
Forced oscillations of an oscillator are possible under any periodic external influence, and not just a sinusoidal one. In this case, the steady-state 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 pushes (remember how an adult “swings” 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. In this case, 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 the push is equal to or exceeds the friction losses for the period, then the oscillations will either be steady-state, or their amplitude will increase. This is clearly seen in the phase diagram.
It is obvious that the resonance is also possible in the case when the repetition period 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 loss per period exceeds the energy received by the system during the push, then the total energy of the system will decrease and the oscillations will be damped.
§ 4 Parametric resonance
An external influence on an oscillatory system can be reduced to a periodic change 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, let's try to answer the question: is it possible to swing the small oscillations already existing in the system by periodically changing some of its parameters in a certain way.
As an example, consider swinging a person on a swing. By bending and straightening his legs at the “necessary” moments, he actually changes the length of the pendulum. In the extreme positions, the person squats, thereby slightly lowering the center of gravity of the oscillatory system; in the middle position, the person straightens up, 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 we will pull the thread at those moments when the pendulum passes the equilibrium position, and lower the thread by the same amount when the pendulum passes the extreme position.
The work of the thread tension force for the period (taking into account the fact that the load is lifted and lowered twice per period and that D l << l):
Please note that in parentheses is nothing but the tripled energy of the oscillatory system. By the way, this value 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 hence to the swinging 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 damp out. And in order for the range of oscillations to increase, it is necessary that the acquired energy exceed the energy lost to overcome friction. And this condition, it turns out, is the same - both at a small amplitude and at 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 to not damp, 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.
Back to the swing: once you start swinging, there is no need to squat deeper and deeper - squat the same way all the time and you will fly higher and higher!
*** Goodness again!
As we have already said, for the parametric buildup of oscillations, it is necessary to fulfill the condition DE > A friction per period.
Find the work of the friction force for the period
It can be seen that the relative value of the lifting of the pendulum for its buildup is determined by the quality factor of the system.
§ 5 Significance of resonance
Forced vibrations and resonance are widely used in engineering, especially in acoustics, electrical engineering, and radio engineering. Resonance, first of all, is used when, from a large set of oscillations of different frequencies, they want to select 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, since it can lead to large deformations and destruction of structures.
§ 6 Examples of problem solving
Task 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 a stiffness k = 10 N/m and the system was placed in a viscous medium with a drag coefficient r = 0.1 kg/s. Compare the natural and resonant frequencies of the system. Determine the amplitude of the pendulum oscillations at resonance under the action of a sinusoidal force with an amplitude F 0 = 20 mN.
Decision:
1 The natural frequency of an oscillating system is the frequency of free oscillations in the absence of friction. Natural cyclic frequency is , oscillation frequency .
2 The resonant frequency is the frequency of the external driving force at which the amplitude of the forced vibrations increases sharply. The resonant cyclic frequency is , where is the attenuation coefficient equal to .
Thus, the resonant frequency is . It is easy to see that the resonant frequency is less than its own! It can also be seen that the lower the friction in the system (r), the closer the resonant frequency to its own.
3 The resonant amplitude is
Task 2 Resonant amplitude and quality factor of an oscillatory system
A load of mass m = 100 g was suspended from a spring with a stiffness k = 10 N/m and the system was placed in a viscous medium with a drag coefficient
r = 0.02 kg/s. Determine the quality factor of the oscillatory system and the amplitude of the pendulum oscillations 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 action of a constant force F 0 = 20 mN and compare this ratio with the quality factor.
Decision:
1 The quality factor of the oscillatory system is , where is the logarithmic damping decrement.
The logarithmic damping decrement is .
We find 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 .
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
Task 3 Resonance 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 number of revolutions of the armature of the motor can there be a danger of resonance?
Decision:
1 The body of the engine and the beam on which it is installed experience periodic shocks from the side of the rotating armature of the motor and, therefore, perform forced oscillations with the frequency of the shocks.
Resonance will be observed when the repetition frequency of shocks coincides with the natural frequency of oscillation of the beam with the motor. It is necessary to find the natural oscillation frequency of the beam-motor system.
2 An analogue of the oscillating system beam - motor can be a vertical spring pendulum, the mass of which is equal to the mass of the motor. The natural frequency of oscillation of the spring pendulum is . But the stiffness of the spring and the mass of the motor are not known! How to be?
3 In the equilibrium position of the spring pendulum, the force of gravity of the load is balanced by the force of elasticity of the spring
4 We find the rotation of the armature of the engine, i.e. jolt frequency
Problem 4 Forced oscillations of a spring pendulum under the action of periodic shocks.
A weight of mass m = 0.5 kg is suspended from a helical spring with stiffness k = 20 N/m. The logarithmic damping decrement of the oscillatory system is . They want to swing the weight with short jerks, acting on the weight with a force F = 100 mN for a time τ = 0.01 s. What should be the frequency of repetition of impacts in order for the amplitude of the kettlebell to be the largest? At what moments and in what direction should the kettlebell be pushed? To what amplitude will it be possible to swing the kettlebell in this way?
Decision:
1 Forced vibrations can occur with any periodic action. In this case, the steady oscillation will occur with the repetition rate of the external action. If the period of external shocks coincides with the frequency of natural oscillations, then resonance occurs in the system - the amplitude of oscillations becomes the largest. In our case, for the onset of resonance, the period of repetition of 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 the pendulum in a viscous medium practically coincides with the period of oscillation of the pendulum in vacuum:
2 Obviously, the direction of the shocks must coincide with the speed of the kettlebell. In this case, the work of the external force replenishing the system with energy will be positive. And the vibrations will sway. The energy received by the system during the impact
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 when it passes the equilibrium position.
3 The oscillation amplitude stops growing when the energy imparted to the system during the impact will be equal to the energy loss due to friction over the period: .
We find the energy loss for the period through the quality factor of the oscillatory system
where E is the total energy of the oscillatory system, which can be calculated as .
We substitute instead of the energy of losses the energy received by the system during the impact:
The maximum speed during oscillation is . With this in mind, we get .
§7 Tasks for independent solution
Test "Forced vibrations"
1 What vibrations are called forced?
A) Oscillations occurring under the action of external periodically changing forces;
B) Oscillations that occur in the system after an external push;
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) Vibration of the loudspeaker diffuser during the operation of the receiver;
C) Oscillation of a load suspended from a spring after a single impact on the load in the equilibrium position;
D) Vibration of the body of the electric motor during its operation;
E) Vibrations of the tympanic membrane of a person listening to music.
3 An oscillatory system with a natural frequency is affected by an external driving force that changes according to the law . The damping coefficient in the oscillatory system is . According to what law does the coordinate of the body change over time?
C) The amplitude of forced oscillations will remain unchanged, since the energy losses of the system due to friction will be compensated 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 an oscillatory system of energy at the moment of the beginning of the action of an external force;
C) Parameters of the oscillatory system itself;
D) Friction in the oscillatory system;
E) The existence of natural oscillations in the system at the moment the external force begins to act;
E) The time of establishment of oscillations;
G) Frequencies of the external driving force.
6 A bar 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 the period T?
A) 4μmgA; B) 2μmgA; C) μmgA; D) 0;
E) It is not possible to give an answer, since the magnitude of the external driving force is not known.
7 Make a correct statement
Resonance is the 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) Reduction of friction in the oscillatory system;
B) Increase in the amplitude of the external driving force;
C) 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;
C) In a self-oscillatory system;
D) In a system that performs 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 in different viscous media perform forced oscillations. The figure shows the resonance curves for these pendulums. Which of the pendulums experiences the greatest resistance from the viscous medium during the oscillation process?
A) 1; B) 2; IN 3;
D) It is not possible 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 says nothing about the amplitude of the external driving force.
11 The period of natural vibrations of the oscillatory system is equal to T 0 . What can be the period of repetition of shocks so that the amplitude of oscillations increases sharply, that is, a resonance occurs in the system?
A) T 0; B) T 0, 2 T 0, 3 T 0,…;
C) You can swing the swing with pushes of any frequency.
12 Your little brother is sitting on a swing, you rock him with short pushes. What should be the period of aftershocks in order for the process to proceed most efficiently? The period of natural oscillations of the swing T 0 .
D) You can swing the swing with pushes of any frequency.
13 Your little brother is sitting on a swing, you rock him with short pushes. In what position of the swing should the push be made and in which direction should the push be made in order for the process to take place most efficiently?
A) Push in the extreme upper position of the swing in the direction of the equilibrium position;
B) Push in the extreme upper position of the swing in the direction from the equilibrium position;
B) Push in a position of balance in the direction of movement of the swing;
D) You can push in any position, but always in the direction 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, it can be strongly shaken, 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 the slingshot, the bridge will not be able to move under the influence of such blows;
B) The impact force of the “bullet” from the 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 sways and splashes out. What can be done to prevent this from happening?
A) Waving the hand in which the bucket is located in time with walking;
B) Change the speed of movement, leaving the length of the steps unchanged;
C) Periodically stop and wait for the vibrations of the water to calm down;
D) Make sure that during the movement the hand with the bucket is located strictly vertically.
Tasks
1 The system performs damped oscillations with a frequency of 1000 Hz. Determine the frequency v0 natural vibrations, if the resonant frequency
2 Determine how much D v the resonant frequency is different from the natural frequency v0= 1000 Hz of an oscillatory system characterized by a damping coefficient d = 400s -1 .
3 A mass of 100 g, suspended on a spring of stiffness 10 N/m, performs forced oscillations in a viscous medium with a drag coefficient r = 0.02 kg/s. Determine the damping factor, resonant frequency and amplitude. The amplitude value of the driving force is 10 mN.
4 Amplitudes of forced harmonic oscillations at frequencies w 1 = 400 s -1 and w 2 = 600 s -1 are equal to each other. Determine the resonant frequency.
5 Trucks enter a grain warehouse on a dirt road from 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 to determine from which side of the warehouse the entrance and which exit are determined by the condition of the road? Justify your answer
In order for the system to perform undamped oscillations, it is necessary to replenish the energy losses of oscillations due to friction from the outside. In order for the energy of the oscillations of the system not to decrease, a force is usually introduced that periodically acts on the system (we will call such a force forcing, and oscillations forced).
DEFINITION: forced called such vibrations that occur in an oscillatory system under the action of an external periodically changing force.
This force, as a rule, performs a dual role:
First, it shakes the system and gives it 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 with 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 the quasi-elastic force and the drag force of the medium (which is valid under the assumption of small oscillations).
Then the equation of motion of the system will look like:
Or 
.
After substituting , , - the natural frequency of oscillations of the system, we obtain a non-homogeneous linear differential equation of the 2nd order:
It is known from the theory of differential equations 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 make sure that such an assumption is true, and also determine the values of “ a" and " j”.
The oscillation amplitude is determined by the following expression:
.
Meaning " j”, which is the magnitude of the phase delay of the forced oscillation 
from the driving force that caused it, is also determined from the vector diagram and is:
.
Finally, a particular solution of the inhomogeneous equation will take the form:
 |
(8.18) |
This function, together with
 |
(8.19) |
gives a general solution to an inhomogeneous differential equation describing the behavior of a system under forced vibrations. 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).
In the course of time, due to the exponential factor, the role of the second term in (8.19) decreases more and more, and after a sufficient time it can be neglected, keeping only the term (8.18) in the solution.
Thus, function (8.18) describes steady forced oscillations. They are 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 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 amount of 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 oscillation amplitude reaches its maximum value. The oscillatory system is 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: the phenomenon in which there is a sharp increase in the amplitude of forced oscillations 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 oscillation frequency.
The dependence of the amplitude of forced oscillations on the frequency of the driving force (or, what is the same, on the frequency of oscillations) can be represented graphically (Fig. 8.11). Separate curves correspond to different values of “b”. The smaller “b”, the higher and to the right lies the maximum of this curve (see the expression for w res.). With very large damping, resonance is not observed - with increasing frequency, the amplitude of forced oscillations decreases monotonically (lower curve in Fig. 8.11).
The set of presented graphs corresponding to different values of b is called resonance curves.
Remarks about resonance curves:
As w®0 tends, all curves come to one non-zero value equal to . This value represents the displacement from the equilibrium position that the system receives under the action of a constant force F 0 .
As w®¥ all curves asymptotically tend to zero, since at a high frequency, the force changes its direction so quickly that the system does not have time to noticeably shift from the equilibrium position.
The smaller b, the stronger the amplitude near the resonance changes with frequency, the "sharper" the maximum.
Examples:
The phenomenon of resonance is often useful, especially in acoustics and radio engineering.