Zilberman A. R. Generator of undamped oscillations // Quantum. - 1990. - No. 9. - P. 44-47.

By special agreement with the editorial board and editors of the journal "Kvant"

Such generators are used in many devices - radios, televisions, tape recorders, computers, electrical organs, etc. - and are very different. Thus, generator frequencies can range from several tens of hertz (low notes in an electric organ) to hundreds of megahertz (television) and even up to several gigahertz (satellite television, radars used by traffic police officers to determine the speed of a car). The power that a generator can deliver to a consumer ranges from several microwatts (a generator in a wristwatch) to tens of watts (a television scan generator), and in some special cases the power can be such that there is no point in writing - you still won’t believe it. The form of oscillations can be as simple as sinusoidal (local oscillator of a radio receiver) or rectangular (computer timer), or very complex - “simulating” the sound of musical instruments (musical synthesizers).

Of course, we will not consider all this diversity, but will limit ourselves to a very simple example - a low-power sinusoidal voltage generator of moderate frequency (hundreds of kilohertz).

As is known, in the simplest oscillatory circuit, consisting of an ideal capacitor and an ideal coil, undamped harmonic oscillations can occur. The equation of the process is easy to obtain by equating (taking into account the signs) the voltages on the capacitor and on the coil - after all, they are connected in parallel (Fig. 1):

\(~\frac qC = -LI"\) .

The current flowing through the coil changes the charge on the capacitor; these quantities are related by the relation

\(~I = q"\) .

Now we can write the equation

\(~q"" + \frac(q)(LC) = 0\) .

The solution to this equation is well known - these are harmonic oscillations. Their frequency is determined by the parameters of the oscillatory circuit\[~\omega = \frac(1)(\sqrt(LC))\] , and the amplitude depends only on the energy that was initially given to the circuit (and which remains constant for an ideal circuit).

What will change if the circuit elements are not ideal, as actually happens in practice (for many years the author has never seen a single ideal coil, although he was very interested in this issue)? Let, for definiteness, all the imperfection of the circuit is due to the fact that the coil, or more precisely, the wire from which it is wound, has an active (ohmic) resistance r(Fig. 2). In fact, of course, the capacitor also has energy losses (although at not very high frequencies you can make a very good capacitor without much difficulty). And the consumer takes energy away from the circuit, which also contributes to the damping of oscillations. In a word, we will assume that r- this is the equivalent value responsible for all energy losses in the circuit. Then Eq. the process takes the form

\(~LI" + rI + \frac(q)(C) = 0\) .

It is clear that it is the second term that prevents us from obtaining the desired equation of undamped oscillations. Therefore, our task is to compensate for this term. Physically, this means that additional energy must be pumped into the circuit, that is, another EMF must be introduced. How can this be done without breaking the chain? The easiest way to use a magnetic field is to create an additional magnetic flux that penetrates the turns of the circuit coil. To do this, not far from this coil, you need to place another coil (Fig. 3) and pass a current through it, the value of which should change according to the desired law, i.e., so that this current creates just such a magnetic field that, penetrating the coil circuit, will create in it such a magnetic flux, which, changing, will induce such an induced emf that will exactly compensate for the term we don’t like in the process equation. This whole long phrase, reminiscent of “the house that Jack built,” is simply a retelling of the Faraday law you know for the phenomenon of electromagnetic induction.

Let's now look at the current that should flow through the additional coil. It is clear that it requires an energy source (to replenish energy losses in the circuit) and a control device that ensures the desired law of current change over time. An ordinary battery can be used as a source, and an electron tube or transistor can be used as a control device.

Transistors come in different types - conventional (they are called bipolar) and field-effect ones, which are further divided into field-effect ones with an insulated gate (they are usually used in digital devices) and with a control p-n-transition. Any field-effect transistor contains a “channel” with two terminals - they are inventively called source and drain, and its conductivity is regulated by applying a control voltage to the third terminal - the gate (Fig. 4). In a field-effect transistor with a control p-n-by a transition - and we will talk about it further - the gate is separated from the channel by just such a transition, for which the gate area is made of the opposite type of conductivity in relation to the channel. For example, if the channel has impurity conductivity of the type p, then the shutter is like n, and vice versa.

When a blocking voltage is applied to the junction U z (Fig. 5), the cross-section of the conducting channel decreases, and at a certain voltage - it is called the cut-off voltage - the channel is completely blocked and the current stops.

Channel current dependence I k from gate voltage U z is shown in Figure 6. This dependence is almost the same as that of an electron tube (triode). It is important to note that the control voltage is a blocking voltage, which means that the current in the control circuit is extremely small (usually it is several nanoamperes), and the control power is correspondingly low, which is very good. At small values ​​of the control voltage, the dependence of the current on voltage can be considered linear and written in the form

\(~I_k = I_0 + SU_z\) ,

Where S- constant value. For a generator, deviations from linearity are also significant, but more on that later.

Figure 7 shows a schematic diagram of a continuous oscillation generator. Here, the control voltage for the field-effect transistor is the voltage on the capacitor of the oscillatory circuit:

\(~U_z = U_C = \frac qC\) ,

and the current through the additional coil is

\(~I_k = I_0 + \frac(Sq)(C)\) .

The additional magnetic flux is proportional to this current, and the additional EMF of the circuit is equal to the derivative of this flux, taken with the opposite sign:

\(~\varepsilon_i = -\Phi" = -(MI_k)" = -\frac(MS)(C) q"\) ,

The minus sign here is quite arbitrary - the coil can be connected to a field-effect transistor either at one end or the other, and the sign of the additional EMF will change to the opposite. In a word, the additional EMF must be such as to compensate for the energy losses in the circuit. Let us write the process equation again:

\(~LI" + rI + \frac(q)(C) - \frac(MS)(C) q" = 0\) .

If you choose the value M such that the fourth term compensates the second, then we get the equation

\(~LI" + \frac(q)(C) = 0\) ,

which corresponds to harmonic undamped oscillations.

How can you influence the size M? It turns out that it will increase if you wind more turns in an additional coil or if this coil is placed closer to the circuit coil. It must be said that the coefficient sufficient for generation M in practice it is quite easy to obtain. It is better to choose this value with some margin - this will result in a circuit not only without losses, but even with pumping energy from an external source (with “negative” losses). When the generator is turned on, the amplitude of the oscillations will initially increase, but after some time it will settle down - the energy entering the circuit in one period will become equal to the energy lost during the same time. Indeed, as the voltage amplitude on the capacitor increases (the control voltage of the field-effect transistor), the transistor begins to amplify worse, since at a large negative voltage the current in the channel circuit stops, and at positive voltages the junction begins to open, which also increases losses in the circuit. As a result, the oscillations are not completely sinusoidal, but if the losses in the circuit are small, the distortion is negligible.

In order to use the resulting oscillations - and this is precisely what the generator is made for - you need to either connect directly to the circuit, or wind another coil. But in both cases, it is necessary to take into account the “leakage” of energy from the circuit and compensate for it, among other losses.

Lecture 12. Mechanical vibrations and waves.

Lecture outline

Harmonic oscillations and their characteristics.

Free undamped mechanical vibrations.

Free damped and forced mechanical vibrations.

Elastic waves.

Harmonic oscillations and their characteristics.

Oscillations processes that are characterized by a certain repeatability over time are called, i.e. fluctuations are periodic changes of any value.

Depending on the physical nature, mechanical and electromagnetic vibrations are distinguished. Depending on the nature of the impact on the oscillating system, free (or natural) oscillations, forced oscillations, self-oscillations and parametric oscillations are distinguished.

Oscillations are called periodic if the values ​​of all physical quantities that change when the system oscillates repeat at equal intervals of time.

Period is the time it takes to complete one complete oscillation:

Where
- number of oscillations per time .

Oscillation frequency- the number of complete oscillations completed per unit of time.

Cyclic or circular frequency - the number of complete oscillations completed in a time of 2 (time units):

.

The simplest type of oscillations are harmonic vibrations, in which the change in value occurs according to the law of sine or cosine (Fig. 1):

,

Where - the value of the changing quantity;

- amplitude of oscillations, maximum value of the changing quantity;

- phase of oscillations at the moment of time (angular time measure);

 0 - initial phase, determines the value at the initial moment of time at
,.

An oscillatory system that performs harmonic oscillations is called harmonic oscillator.

Velocity and acceleration during harmonic vibrations:

Free undamped mechanical vibrations.

Free or own are called the oscillations that a system makes around an equilibrium position after it has somehow been removed from a state of stable equilibrium and presented to itself.

As soon as a body (or system) is removed from an equilibrium position, a force immediately appears that tends to return the body to an equilibrium position. This force is called returning, it is always directed towards the equilibrium position, its origin is different:

a) for a spring pendulum - elastic force;

b) for a mathematical pendulum - the component force of gravity.

Free or natural vibrations are vibrations that occur under the influence of a restoring force.

If there are no friction forces in the system, the oscillations continue indefinitely with a constant amplitude and are called natural undamped oscillations.

Spring pendulum- material point with mass m, suspended on an absolutely elastic weightless spring and oscillating under the action of an elastic force.

Let us consider the dynamics of the natural undamped oscillations of a spring pendulum.

According to Newton's II law,

according to Hooke's law,

Where k– rigidity,
;

or
.

Let's denote cyclic frequency of natural oscillations.

-differential equation of free undamped oscillations.

The solution to this equation is the expression: .

period of oscillation of a spring pendulum.

During harmonic oscillations, the total energy of the system remains constant, a continuous transition occurs V and vice versa.

Math pendulum- a material point suspended on a weightless inextensible thread (Fig. 2).

It can be proven that in this case

Spring and mathematical pendulums are harmonic oscillators (like an oscillatory circuit). A harmonic oscillator is a system described by the equation:

.

Oscillations of a harmonic oscillator are an important example of periodic motion and serve as an approximate model in many problems in classical and quantum physics.

Free damped and forced mechanical vibrations.

In any real system that performs mechanical oscillations, there are always certain resistance forces acting (friction at the point of suspension, environmental resistance, etc.), to overcome which the system expends energy, as a result of which real free mechanical oscillations are always damped.

Damped oscillations- These are oscillations whose amplitude decreases with time.

Let's find the law of amplitude change.

For a spring pendulum of mass m, performing small oscillations under the action of an elastic force
The friction force is proportional to the speed:

where r is the resistance coefficient of the medium; the minus sign means that
always directed opposite to the speed.

According to Newton's II law, the equation of motion of a pendulum has the form:

Let's denote:

differential equation of free damped oscillations.

The solution to this equation is the expression:

,

Where cyclic frequency of free damped oscillations,

 0 - cyclic frequency of free undamped oscillations,

 - attenuation coefficient,

A 0 - amplitude at the initial moment of time (t=0).

- law of decreasing amplitude.

Over time, the amplitude decreases exponentially (Fig. 3).

Relaxation time is the time during which the amplitude decreases in once.

.

Thus, is the reciprocal of the relaxation time.

The most important characteristic of damped oscillations is the logarithmic damping decrement .

Logarithmic damping decrement is the natural logarithm of the ratio of two amplitudes that differ from each other in time by a period:

.

Let's find out its physical meaning.

Z and the relaxation time the system will have time to complete N oscillations:

those. is the reciprocal of the number of oscillations during which the amplitude decreases by a factor of e.

To characterize an oscillatory system, the concept of quality factor is used:

.

Quality factor- physical quantity proportional to the number of oscillations during which the amplitude decreases by e times (Fig. 4,
).

Forced are called oscillations that occur in a system under the influence of a periodically changing external force.

Let the external force change according to the harmonic law:

In addition to the external force, the oscillating system is acted upon by a restoring force and a resistance force, proportional to the oscillation speed:

Forced vibrations occur with a frequency equal to the frequency of the driving force. It has been experimentally established that the displacement lags behind the compelling force in its change. It can be proven that

Where - amplitude of forced oscillations,

- oscillation phase difference And
,

;
.

Graphically forced oscillations are presented in Fig. 5.

E If the driving force changes according to a harmonic law, then the vibrations themselves will be harmonic. Their frequency is equal to the frequency of the driving force, and their amplitude is proportional to the amplitude of the driving force.

Dependence of amplitude on driving force frequency leads to the fact that at a certain frequency determined for a given system, the amplitude reaches a maximum.

The phenomenon of a sharp increase in the amplitude of forced oscillations as the frequency of the driving force approaches the natural frequency of the system (the resonant frequency) is called resonance(Fig. 6).

Elastic waves.

Any elastic body consists of a large number of particles (atoms, molecules) interacting with each other. Interaction forces appear when the distance between particles changes (attraction occurs when stretching, and repulsion when compressed) and are of an electromagnetic nature. If any particle is removed from its equilibrium position by an external influence, then it will pull another particle along with it in the same direction, this second will pull a third, and the disturbance will propagate from particle to particle in the medium at a certain speed, depending on the properties of the medium. If the particle was shifted upward, then under the action of the upper particles, repulsive, and the lower, attractive, it will begin to move down, pass the equilibrium position, move down by inertia, etc., i.e. will perform harmonic oscillatory motion, forcing a neighboring particle to oscillate, etc. Therefore, when a disturbance propagates in a medium, all particles oscillate with the same frequency, each near its equilibrium position.

The process of propagation of mechanical vibrations in an elastic medium is called an elastic wave. This process is periodic in time and space. When a wave propagates, the particles of the medium do not move with the wave, but oscillate around their equilibrium positions. Together with the wave, only the state of oscillatory motion and its energy are transferred from particle to particle of the medium. Therefore, the main property of all waves is the transfer of energy without transfer of matter.

There are longitudinal and transverse elastic waves.

An elastic wave is called longitudinal if the particles of the medium oscillate along the direction of propagation of the wave (Fig. 7).

The relative position of oscillating points is characterized by condensation and rarefaction.

When such a wave propagates through the medium, condensations and rarefaction occur. Longitudinal waves arise in solid, liquid and gaseous bodies in which elastic deformations occur during compression or tension.

An elastic wave is called transverse if the particles of the medium oscillate perpendicular to the direction of propagation of the wave (Fig. 8).

P When a transverse wave propagates in an elastic medium, crests and troughs are formed. A transverse wave is possible in a medium where shear deformation causes elastic forces, i.e. in solids. At the interface between 2 liquids or a liquid and a gas, waves appear on the surface of the liquid; they are caused either by tension forces or gravity forces.

Thus, only longitudinal waves arise inside liquids and gases, while longitudinal and transverse waves occur in solids.

The speed of wave propagation depends on the elastic properties of the medium and its density. The speed of propagation of longitudinal waves is 1.5 times greater than the speed of transverse waves.

Propagating from one source, both waves arrive at the receiver at different times. By measuring the difference in the propagation times of longitudinal and transverse waves, it is possible to determine the location of the source of the waves (atomic explosion, earthquake epicenter, etc.).

On the other hand, the speed of propagation of waves in the earth's crust depends on the rocks lying between the source and receiver of the waves. This is the basis of geophysical methods for studying the composition of the earth's crust and searching for minerals.

Longitudinal waves propagating in gases, liquids and solids and perceived by humans are called sound waves. Their frequency ranges from 16 to 20,000 Hz, below 16 Hz is infrasound, above 20,000 Hz is ultrasound.

Sokolov S.Ya., corresponding member of the USSR Academy of Sciences, in 1927-28. discovered the ability of ultrasonic waves to penetrate metals and developed a technique for ultrasonic flaw detection, constructing the first ultrasonic generator at 10 9 Hz. In 1945, he was the first to develop a method for converting mechanical waves into visible light and created an ultrasonic microscope.

The wave, spreading from the source of vibrations, covers more and more new areas of space.

The geometric location of the points to which the oscillations have propagated at a given time t is called wave front.

The geometric location of points oscillating in the same phase is called wave surface.

There are an infinite number of wave surfaces that can be drawn, but their appearance is the same for a given wave. A wave front represents a wave surface at a given time.

In principle, wave surfaces can be of any shape, and in the simplest case they are a set of parallel planes or concentric spheres (Fig. 9).

The wave is called flat, if its front is a plane.

IN the wave is called spherical, if its front is the surface of a sphere.

IN Waves propagating in a homogeneous isotropic medium from point sources are spherical. At a large distance from the source, a spherical wave can be considered as a plane wave.

Huygens' principle: each point of the wave front (i.e., each oscillating particle of the medium) is a source of secondary spherical waves. The new position of the wave front is represented by the envelope of these secondary waves.

This statement was made in 1690 by the Dutch scientist Huygens. Its validity can be illustrated with the help of waves on the surface of water, which imitate spherical waves arising in the volume of an elastic medium.

and 1 in 1 - front at moment t 1,

and 2 in 2 - front at moment t 2.

Having blocked the surface of the water with an obstacle with a small hole and directed a plane wave at the obstacle, we are convinced that behind the obstacle is a spherical wave (Fig. 10).

Running are called waves that transfer energy in space.

Let us obtain the equation of a traveling plane wave, assuming that the oscillations are harmonic in nature and the Y-axis coincides with the direction of wave propagation.

The wave equation determines the dependence of the displacement of an oscillating particle of the medium on coordinates and time.

Let some particle of the medium IN(Fig. 11) is located at a distance at from the vibration source located at the point ABOUT. At the point ABOUT the displacement of a particle of the medium from the equilibrium position occurs according to a harmonic law,

Where t- time counted from the beginning of oscillations.

At the point CWhere
- time during which the wave leaves the point O gets to the point C, - wave propagation speed.

-plane traveling wave equation.

This equation determines the amount of displacement X oscillating point characterized by coordinate at, at any time t.

If a plane wave does not propagate in the positive direction of the Y axis, but in the opposite direction, then

Because the wave equation can be written as

The distance between nearby points oscillating in the same phase is called the wavelength.

Wavelength- the distance over which the wave propagates during the period of oscillation of the particles of the medium, i.e.

.

Because

where is the wave number.

In general
.

Harmonic vibrations.

Oscillations are processes that differ in varying degrees of repeatability. Oscillatory motion and the waves it causes are very common in nature and technology. Bridges vibrate under the influence of trains passing over them, the eardrum of the ear vibrates, parts of buildings vibrate, and the heart muscle contracts rhythmically.

Depending on the physical nature of the repeating process, vibrations are distinguished: mechanical, electromagnetic, etc. We will consider mechanical vibrations.

Let's consider the simplest mechanical system, consisting of a body (ball) of some mass m, strung on a rod, and a spring with stiffness k, connecting it to a fixed wall. Let us direct the OX axis along the rod, and the origin of coordinates is compatible with the center of the ball, provided that the spring is in an undeformed state. Let's move the ball to a distance X 0 from the equilibrium position (see Fig. 1). Then, from the side of the spring, an elastic force F=-kX 0 (1) will act on the body. This force, as can be seen from equation (1), is proportional to the displacement and is directed in the direction opposite to the displacement. It is called the restoring force. In addition, the system will have a reserve of potential energy
. If you release the load, then under the action of elastic force it will begin to move to the equilibrium position, while its potential energy will decrease, turning into kinetic energy
, the restoring force will decrease and in the equilibrium position will become equal to zero, but the body will not stop in the equilibrium position, but will continue to move by inertia. Its kinetic energy will transform into potential energy, the restoring force will begin to increase, but its direction will change to the opposite. Oscillations will occur in the system. In oscillatory motion, the position of the body at any given moment in time is characterized by the distance from the equilibrium position, which is called displacement. Among the various types of vibrations, the simplest form is harmonic vibration, i.e. one in which the oscillating quantity changes depending on time according to the law of sine or cosine.

1. Undamped harmonic oscillations.

Let a body of mass m be acted upon by a force that tends to return it to the equilibrium position (restoring force) and is proportional to the displacement from the equilibrium position, i.e. elastic force F UPR = -kX. If there is no friction, then the equation of Newton's second law for the body is:

;
or
.

Let's denote
, we get
. (1)

Equation (1) is a linear homogeneous differential equation of the 2nd order, with constant coefficients. The solution to equation (1) will be the law of free or natural undamped oscillations:

,

where A is the value of the largest deviation from the equilibrium position, which is called amplitude (amplitude is a constant, positive value);
- oscillation phase; - initial phase.

G Graphically undamped oscillations are shown in Fig. 2:

T – period of oscillation (time interval of one complete oscillation);
, Where - circular or cyclic frequency,
, ν is called the oscillation frequency.

To find the speed of a material point during harmonic oscillation, you need to take the derivative of the expression for the displacement:

Where
- maximum speed (speed amplitude). Differentiating this expression, we find the acceleration:

Where
- maximum acceleration.

1. Damped harmonic oscillations.

In real conditions, in addition to the restoring force in the oscillating system, there will be a friction force (medium resistance force), which at low speeds is proportional to the speed of the body:
, where r is the resistance coefficient. If we limit ourselves to taking into account the restoring force and the friction force, then the equation of motion will take the form:
or
, dividing by m, we get:
, denoting
,
, we get:
. This equation is called a second-order linear homogeneous differential equation with constant coefficients. The solution to this equation will be the law of free damped oscillations, and will have the following form: .

From the equation it is clear that the amplitude
is not constant, but depends on time and decreases according to an exponential law. As for undamped oscillations, the value ω is called the circular frequency:
, Where
- attenuation coefficient;

-initial phase.

Graphically damped oscillations are presented in Fig. 3.

ABOUT let's limit the oscillation period
or
, which shows that oscillations in the system can only occur if the resistance is insignificant
. The oscillation period is almost equal
.

With increasing damping coefficient, the oscillation period increases and at
turns to infinity. The movement ceases to be periodic. A system removed from an equilibrium position returns to an equilibrium state without oscillating. This kind of motion is called aperiodic.

Figure 4 shows one of the cases of the system returning to the equilibrium position during aperiodic motion. In accordance with the indicated curve, the charge on the membranes of human nerve fibers decreases.

To characterize the rate of attenuation of oscillations, the concept of attenuation coefficient is introduced
. Let us find the time τ during which the amplitude of oscillations will decrease by a factor of ve:

, i.e.

from where βτ=1, therefore . The attenuation coefficient is the inverse in magnitude of the time period during which the amplitude will decrease by a factor of ve. The ratio of amplitude values ​​corresponding to moments of time differing by a period is equal to
is called the damping decrement, and its logarithm is called the logarithmic damping decrement:

.

Free vibrations always damp due to energy losses (friction, environmental resistance, resistance of electric current conductors, etc.). Meanwhile, both in technology and in physical experiments, undamped oscillations are urgently needed, the periodicity of which remains the same as long as the system oscillates at all. How are such oscillations obtained? We know that forced oscillations, in which energy losses are replenished by the work of a periodic external force, are undamped. But where does the external periodic force come from? After all, it, in turn, requires a source of some kind of undamped oscillations.

Undamped oscillations are created by devices that themselves can maintain their oscillations due to some constant source of energy. Such devices are called self-oscillating systems.

In Fig. 55 shows an example of an electromechanical device of this kind. The weight hangs on a spring, the lower end of which is immersed in a cup of mercury when this spring pendulum oscillates. One pole of the battery is connected to the spring at the top, and the other to the cup of mercury. When the load is lowered, the electrical circuit is closed and current flows through the spring. Thanks to the magnetic field of the current, the coils of the spring begin to attract each other, the spring is compressed, and the load receives an upward push. Then the contact is broken, the coils stop tightening, the load falls down again, and the whole process is repeated again.

Thus, the oscillation of the spring pendulum, which would die out on its own, is maintained by periodic shocks caused by the oscillation of the pendulum itself. With each push, the battery releases a portion of energy, part of which is used to lift the load. The system itself controls the force acting on it and regulates the flow of energy from the source - the battery. The oscillations do not die out precisely because during each period exactly as much energy is taken from the battery as is spent during the same time on friction and other losses. As for the period of these undamped oscillations, it practically coincides with the period of natural oscillations of the load on the spring, i.e., it is determined by the stiffness of the spring and the mass of the load.

Rice. 55. Self-oscillations of a load on a spring

In the same way, undamped oscillations of a hammer occur in an electric bell, with the only difference being that in it periodic shocks are created by a separate electromagnet that attracts an armature mounted on the hammer. In a similar way, it is possible to obtain self-oscillations with sound frequencies, for example, to excite undamped oscillations of a tuning fork (Fig. 56). When the legs of the tuning fork move apart, contact 1 closes; current passes through the winding of electromagnet 2, and the electromagnet tightens the legs of the tuning fork. In this case, the contact opens, and then the entire cycle is repeated.

Rice. 56. Self-oscillations of a tuning fork

The phase difference between the oscillation and the force that it regulates is extremely important for the occurrence of oscillations. Let's move contact 1 from the outside of the tuning fork leg to the inside. The closure now occurs not when the legs diverge, but when the legs come closer, i.e., the moment the electromagnet is turned on is advanced by half a period compared to the previous experiment. It is easy to see that in this case the tuning fork will be compressed all the time by a continuously switched on electromagnet, i.e., oscillations will not occur at all.

Electromechanical self-oscillating systems are used very widely in technology, but purely mechanical self-oscillating devices are no less common and important. It is enough to point to any clock mechanism. The undamped oscillations of a pendulum or a clock balancer are supported by the potential energy of a raised weight or by the elastic energy of a wound spring.

Figure 57 illustrates the principle of operation of the Galileo-Huygens pendulum clock (§ 11). This figure shows the so-called anchor passage. A wheel with oblique teeth 1 (running wheel) is rigidly attached to a toothed drum, through which a chain with a weight 2 is thrown. A crossbar 4 (anchor) is attached to the pendulum 3, at the ends of which pallets 5 are fixed - plates curved in a circle with the center on the axis of the pendulum 6. The anchor does not allow the running wheel to rotate freely, but gives it the opportunity to rotate only one tooth for every half-period of the pendulum. But the running wheel also acts on the pendulum, namely, while the tooth of the running wheel is in contact with the curved surface of the left or right pallet, the pendulum does not receive a push and is only slightly slowed down due to friction. But in those moments when the tooth of the running wheel “strikes” along the end of the pallet, the pendulum receives a push in the direction of its movement. Thus, the pendulum makes undamped oscillations, because in certain positions it itself allows the running wheel to push itself in the desired direction. These shocks replenish the energy spent on friction. The period of oscillations in this case almost coincides with the period of natural oscillations of the pendulum, i.e., depends on its length.

Rice. 57. Clock mechanism diagram

Self-oscillations are also vibrations of a string under the action of a bow (in contrast to the free vibrations of a string on a piano, harp, guitar and other non-bowed string instruments, excited by a single push or jerk); self-oscillations include the sound of wind musical instruments, the movement of the piston of a steam engine and many other periodic processes.

A characteristic feature of self-oscillations is that their amplitude is determined by the properties of the system itself, and not by the initial deflection or push, as in free oscillations. If, for example, the pendulum of a clock is deflected too much, then the friction losses will be greater than the energy input from the winding mechanism, and the amplitude will decrease. On the contrary, if the amplitude is reduced, then the excess energy imparted to the pendulum by the running wheel will cause the amplitude to increase. The amplitude at which energy consumption and supply are balanced will be automatically established.

1. 
   Elements of biomechanics 5
2. 
   Mechanical vibrations 14
3. 
   Biophysics of hearing. Sound. Ultrasound 17
4. 
   Biophysics of blood circulation 21
5. 
   Electrical properties of tissues and organs 28
6. 
   Electrocardiography. Rheography 33
7. 
   Basics of Electrotherapy 36
8. 
   Biophysics of vision. Optical instruments 40

2.0 X-rays 49

2.1 Elements of radiation physics. Fundamentals of Dosimetry 54

3. Diadynamic is one of the most famous electrotherapy devices that uses the analgesic and antispasmodic effects of low-frequency currents for medicinal purposes, for example, to improve blood circulation in the body. The procedure is prescribed exclusively by a doctor, duration is 3-6 minutes (for acute conditions daily, for chronic diseases 3 times a week 5-6 minutes).

Indications: diseases of the musculoskeletal system, especially joint pain and

Spine

Electrosleep is a method of electrotherapy that uses pulsed currents of low or sound frequency (1-130 Hz), rectangular in shape, low strength (up to 2-3 mA) and voltage (up to 50 V), causing drowsiness, drowsiness, and then sleep of varying depth and duration.
Indications: diseases of internal organs (chronic ischemic heart disease, hypertension, hypotension, rheumatism, peptic ulcer of the stomach and duodenum, hypothyroidism, gout), diseases of the nervous system (atherosclerosis of cerebral vessels in the initial stage, traumatic cerebropathy, hypothalamic syndrome, migraine , neurasthenia, asthenic syndrome, manic-depressive psychosis, schizophrenia).

Amplipulse therapy is one of the methods of electrotherapy based on the use of sinusoidal modulated currents for therapeutic, prophylactic and rehabilitation purposes.

Undamped harmonic oscillations

Harmonic vibrations occur under the action of elastic or quasi-elastic (similar to elastic) forces described by Hooke’s law:

Where ^F– elastic force;

X – bias;

k– coefficient of elasticity or rigidity.

According to Newton's second law
, Where A– acceleration, A =
.

(2).

Equation (2) – differential equation of undamped harmonic oscillations.

Its solution looks like: or .
^ Characteristics of undamped harmonic oscillations:

X– displacement; A– amplitude; T– period; – frequency; – cyclic frequency, - speed; – acceleration, – phase; 0 – initial phase, E – full energy.

Formulas:

– number of oscillations, – time during which N oscillations occur;

, ; or ;

or ;

– phase of undamped harmonic oscillations;

– total energy of harmonic vibrations.

Damped harmonic oscillations

In real systems involved in oscillatory motion, friction (resistance) forces are always present:

, – resistance coefficient;
- speed.

.

Then we write Newton’s second law:

We write equation (2) in the form:

(3)

His solution is where

– amplitude of oscillations at the initial moment of time;

– cyclic frequency of damped oscillations.

The amplitude of oscillations changes according to an exponential law:

.

Rice. 11. Schedule x= f(t)

Rice. 12. Schedule A t = f(t)

1)
– period of damped oscillations; 2) – frequency of damped oscillations; – natural frequency of the oscillatory system;

3) logarithmic attenuation decrement (characterizes the rate of decrease in amplitude):
.

^ Forced vibrations

To obtain undamped oscillations, the action of an external force is necessary, the work of which would compensate for the decrease in the energy of the oscillating system caused by resistance forces. Such oscillations are called forced.

Law of change of external force:
, Where – amplitude of external force.

We write Newton's second law in the form

Let us introduce the notation
.

The equation of forced oscillations has the form:

The solution to this equation in steady state is:

,

Where

(4)

– frequency of forced oscillations.

From formula (4), when
, the amplitude reaches its maximum value. This phenomenon is called resonance.

^ 1.3 Biophysics of hearing. Sound. Ultrasound.

Wave is the process of propagation of vibrations in an elastic medium.

Wave equation expresses the dependence of the displacement of an oscillating point participating in the wave process on the coordinate of its equilibrium position and time: S = f (x ; t).

If S and X are directed along the same straight line, then the wave longitudinal, if they are mutually perpendicular, then the wave transverse

The equation at point "0" looks like
. The wave front will reach point "x" with a delay in time
.

Wave equation looks like
.

Wave Characteristics:

S– displacement, A– amplitude, – frequency, T– period, – cyclic frequency, - speed.

– wave phase, – wavelength.

Wavelength is the distance between two points whose phases at the same moment of time differ by
.

^ Wave Front– a set of points that simultaneously have the same phase.

Energy flow is equal to the ratio of the energy transferred by waves through a certain surface to the time during which this energy is transferred:

,
.

Intensity:
, – square,
.

The intensity vector showing the direction of propagation of waves and equal to the flux of wave energy through a unit area perpendicular to this direction is called Umov vector.

– density of the substance.
Sound waves

Sound is a mechanical wave whose frequency lies within the range,
– infrasound,
– ultrasound.

There are musical tones (this is a monochromatic wave with one frequency or consisting of simple waves with a discrete set of frequencies - a complex tone).

^ Noise is a mechanical wave with a continuous spectrum and chaotically varying amplitudes and frequencies.

It has
, wherein
.

. 1 decibel (dB) or 1 background = 0.1 B.

The dependence of loudness on frequency is taken into account using equal loudness curves obtained experimentally and is used to assess hearing defects. The method for measuring hearing acuity is called audiometry. A device for measuring loudness is called sound level meter. The sound volume level should be 40 – 60 dB.