Can forced oscillations occur in an oscillatory system? Conversion of energy during oscillatory motion

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 approximately we can assume that 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 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 oscillator oscillations 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 in a certain way any of its parameters.

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 with a mass of 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 of 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; B) 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 oscillations are those that occur in an oscillatory system under the influence of an external periodically changing force. This force, as a rule, performs a dual role: firstly, it rocks the system and provides it with a certain supply 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:

Having made 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:

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:

The value “j”, which represents the magnitude of the phase lag of the forced oscillation from the driving force that caused it, is also determined from the vector diagram and is:

Finally, a particular solution to the inhomogeneous equation will take the form:

This function in total gives the general solution to the inhomogeneous differential equation that describes the behavior of the system under forced oscillations. Term (2) plays a significant role in the initial stage of the process, during the so-called establishment of oscillations (Fig. 1). Over time, due to the exponential factor, the role of the second term (2) decreases more and more, and after sufficient time has passed, it can be neglected, retaining only term (1) in the solution.

Fig 1.

Thus, function (1) 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. Detlaf A.A., Yavorsky B.M. Physics course: textbook for colleges. - 4th ed., rev. - M.: Higher. school, 2012. - 428 p.

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 called resonant frequency.

In a number of cases, the oscillatory system oscillates under the influence of an external force, the work of which periodically compensates for the loss of energy due to friction and other resistance. The frequency of such oscillations does not depend on the properties of the oscillating system itself, but on the frequency of changes in the periodic force under the influence of which the system makes its oscillations. In this case, we are dealing with forced oscillations, that is, with oscillations imposed on our system by the action of external forces.

The sources of disturbing forces, and therefore forced oscillations, are very diverse.

Let us dwell on the nature of disturbing forces found in nature and technology. As already indicated, electric machines, steam or gas turbines, high-speed flywheels, etc. due to the imbalance of the rotating masses, they cause vibrations of rotors, floors of building foundations, etc. Piston machines, which include internal combustion engines and steam engines, are a source of periodic disturbing forces due to the reciprocating movement of some parts (for example, a piston), the exhaust of gases or steam.

Typically, disturbing forces increase with increasing machine speed, so the fight against vibrations in high-speed machines becomes extremely important. It is often carried out by creating a special elastic foundation or installing an elastic suspension of the machine. If the machine is rigidly fixed to the foundation, then the disturbing forces acting on the machine are almost entirely transmitted to the foundation and then through the ground to the building in which the machine is installed, as well as to nearby structures.

In order to reduce the effect of unbalanced forces on the base, it is necessary that the natural frequency of vibration of the machine on the elastic base (gasket) be significantly lower than the frequency of the disturbing forces, determined by the number of revolutions of the machine.

The reason for the forced oscillations of the ship, the rolling of ships, are waves that periodically impinge on a floating ship. In addition to the rocking of the ship as a whole under the influence of rough water, forced oscillations (vibration) of individual parts of the ship's hull are also observed. The cause of such vibrations is the imbalance of the ship's main engine, which rotates the propeller, as well as auxiliary mechanisms (pumps, dynamos, etc.). During the operation of ship mechanisms, inertial forces of unbalanced masses arise, the repetition frequency of which depends on the number of revolutions of the machine. In addition, forced vibrations of the ship can be caused by the periodic impact of the propeller blades on the ship's hull. Sommerfeld A., Mechanics. Ї Izhevsk: Scientific Research Center “Regular and Chaotic Dynamics”, 2001. Ї168 p.

Forced vibrations of the bridge can be caused by a group of people walking along it in step. Oscillations of a railway bridge can occur under the action of couplers connecting the drive wheels of a passing locomotive. The reasons that cause forced vibrations of rolling stock (electric locomotive, steam locomotive or diesel locomotive, and cars) include periodically repeated impacts of wheels on rail joints. Forced vibrations of cars are caused by repeated impacts of wheels on uneven road surfaces. Forced vibrations of elevators and lifting cages of mines occur due to uneven operation of the lifting machine, due to the irregular shape of the drums on which the ropes are wound, etc. The reasons that cause forced vibrations of power lines, tall buildings, masts and chimneys can be gusts of wind.

Of particular interest are forced vibrations of aircraft, which can be caused by various reasons. Here, first of all, one should keep in mind the vibration of the aircraft caused by the operation of the propeller group. Due to the imbalance of the crank mechanism, running engines and rotating propellers, periodic shocks occur that support forced vibrations.

Along with the oscillations caused by the action of the external periodic forces discussed above, external influences of a different nature are also observed in airplanes. In particular, vibrations arise due to poor streamlining of the front part of the aircraft. Poor flow around the superstructures on the wing or a non-smooth connection between the wing and the fuselage (body) of the aircraft leads to vortex formations. The air vortices, breaking away, create a pulsating flow that hits the tail and causes it to shake. Such shaking of the aircraft occurs under certain flight conditions and manifests itself in the form of shocks that do not occur quite regularly, every 0.5-1 second.

This kind of vibration, associated mainly with the vibration of parts of the aircraft due to turbulence in the flow around the wing and other front parts of the aircraft, is called “buffing”. The phenomenon of buffing, caused by the disruption of flows from the wing, is especially dangerous when the period of impacts on the tail of the aircraft is close to the period of free vibrations of the tail or fuselage of the aircraft. In this case, buffeting-type fluctuations increase sharply.

Very interesting cases of buffing were observed when dropping troops from the wing of an aircraft. The appearance of people on the wing led to vortex formations, causing vibrations in the aircraft. Another case of empennage buffeting on a two-seater aircraft was caused by the fact that a passenger was sitting in the rear cockpit and his protruding head contributed to the formation of vortices in the air flow. In the absence of a passenger in the rear cabin, no vibrations were observed.

Bending vibrations of the propeller caused by disturbing forces of an aerodynamic nature are also important. These forces arise due to the fact that the propeller, when rotating, passes the leading edge of the wing twice during each revolution. The air flow velocities in the immediate vicinity of the wing and at some distance from it are different, and therefore the aerodynamic forces acting on the propeller must periodically change twice for each revolution of the propeller. This circumstance is the reason for the excitation of transverse vibrations of the propeller blades.

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  compelling 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 influence 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 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 a changing coordinate and a 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 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
).

The 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 strength 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.

In 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

.

Unlike 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 replenishes the losses spent on overcoming, and therefore the total no still remains unchanged.

Forced oscillations, 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 vibrations of a children's swing, vibrations of a needle in a sewing machine, a piston in a car engine cylinder, 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 as the frequency of the driving force approaches the natural frequency of the oscillatory system.

Resonance arises due to the fact that when an 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 an external force acts “out of step,” then this force alternately performs either negative or positive work and, as a result, the energy of the system changes slightly.

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 in the room began to shake when a heavy truck passed along the street, this means that the natural frequency of vibration of the glass is equal to the frequency of vibration of the car. If the sea waves resonate with the period of the ship, the motion 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 may be destroyed. However, resonance can also be beneficial. The phenomenon of resonance is used when tuning a radio receiver to a specific broadcast frequency, as well as in many other cases.

Examples of problem solving

EXAMPLE 1

Exercise	The end of the spring of a horizontal pendulum, the load of which has a mass of 1 kg, is acted upon by a variable force whose oscillation frequency is 16 Hz. Will resonance be observed if the spring stiffness is 400 N/m?
Solution	Let us determine the natural frequency of the oscillatory system using the formula: Hz Since the frequency of the external force is not equal to the natural frequency of the system, the phenomenon of resonance will not be observed.
Answer	The phenomenon of resonance will not be observed.

EXAMPLE 2

Exercise	A small ball is suspended on a 1 m long thread from the ceiling of a carriage. At what speed of the car will the ball vibrate especially strongly under the influence of the wheels hitting the rail joints? Rail length 12.5 m.
Solution	The ball performs forced oscillations with a frequency equal to the frequency of impacts of the wheels on the rail joints: If the dimensions of the ball are small compared to the length of the thread, then the system can be considered to have a natural frequency of oscillations: the amplitude of forced undamped oscillations is maximum in the case of resonance, i.e. When . Thus we can write:

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 V. k.). 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), division into free and free-flowing systems 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 oscillations” are in other dictionaries:

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Oscillations arising in the cosmic l. system under the influence of alternating ext. influences (for example, fluctuations in voltage and current in an electrical circuit caused by an alternating emf; vibrations of a mechanical system caused by an alternating load). The character of V. K. is determined by... ... Big Encyclopedic Polytechnic Dictionary

They arise in the system under the influence of periodic external influences (for example, forced oscillations of a pendulum under the influence of a periodic force, forced oscillations in an oscillatory circuit under the influence of a periodic electromotive force). If… … Big Encyclopedic Dictionary

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They arise in a system under the influence of periodic external influences (for example, forced oscillations of a pendulum under the influence of a periodic force, forced oscillations in an oscillatory circuit under the influence of a periodic emf). If the frequency... ... Encyclopedic Dictionary

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: