Conditions for the occurrence of free fluctuations - Knowledge Hypermarket. Oscillations: mechanical and electromagnetic. Free and forced vibrations. Characteristics Conditions for the existence of mechanical vibrations
Lecture. 1. Oscillations. Shape of vibrations. Types of vibrations. Classification. Characteristics of the oscillatory process. Conditions for the occurrence of mechanical vibrations. Harmonic vibrations.
Oscillations- a process of changing the states of a system around the equilibrium point that is repeated to one degree or another over time. Oscillatory processes are widespread in nature and technology, for example, the swing of a clock pendulum, alternating electric current, etc. The physical nature of oscillations can be different, therefore, mechanical, electromagnetic, etc. oscillations are distinguished. However, different oscillatory processes are described by the same characteristics and the same equations. This implies the expediency of a unified approach to the study of oscillations of various physical natures.
Vibration form may be different.
Oscillations are called periodic if the values of physical quantities that change during the oscillation process are repeated at regular intervals (Fig. 1). (Otherwise the oscillations are called aperiodic). An important special case of harmonic oscillations is identified (Fig. 1).
Oscillations approaching harmonic are called quasi-harmonic.
Fig.1. Types of vibrations
Oscillations of various physical natures have many common patterns and are closely interrelated with waves. The generalized theory of oscillations and waves studies these patterns. The fundamental difference from waves: during oscillations there is no transfer of energy; these are local, “local” energy transformations.
Kinds hesitation. Oscillations vary I am by nature:
mechanical(movement, sound, vibration),
electromagnetic(for example, vibrations in an oscillatory circuit, a cavity resonator , fluctuations in the strength of electric and magnetic fields in radio waves, visible light waves and any other electromagnetic waves),
electromechanical(vibrations of the telephone membrane, piezoquartz or magnetostrictive ultrasound emitter) ;
chemical(fluctuations in the concentration of reacting substances during so-called periodic chemical reactions);
thermodynamic(for example, the so-called singing flame, etc. thermal self-oscillations found in acoustics, as well as in some types of jet engines);
oscillatory processes in space(of great interest in astrophysics are fluctuations in the brightness of Cepheid stars (pulsating variable supergiant stars that change brightness with an amplitude from 0.5 to 2 magnitudes and a period from 1 to 50 days);
Thus, oscillations cover a huge area of physical phenomena and technical processes.
Classification of vibrations according to the nature of interaction with the environment :
free (or own)- these are oscillations in a system under the influence of internal forces, after the system is brought out of equilibrium (in real conditions, free oscillations are almost always damped).
For example, vibrations of a load on a spring, a pendulum, a bridge, a ship on a wave, a string; fluctuations in plasma, density and air pressure during the propagation of elastic (acoustic) waves in it.
For free oscillations to be harmonic, it is necessary that the oscillatory system be linear (described by linear equations of motion), and there is no energy dissipation in it (the latter causes attenuation).
forced- oscillations occurring in the system under the influence of external periodic influence. With forced oscillations, the phenomenon of resonance may occur: a sharp increase in the amplitude of oscillations when the natural frequency of the oscillator coincides with the frequency of the external influence.
self-oscillations- oscillations in which the system has a reserve of potential energy that is spent on oscillations (an example of such a system is a mechanical watch). A characteristic difference between self-oscillations and free oscillations is that their amplitude is determined by the properties of the system itself, and not by the initial conditions.
parametric- oscillations that occur when any parameter of the oscillatory system changes as a result of external influence,
random- oscillations in which the external or parametric load is a random process,
associated vibrations- free vibrations mutually connected systems, consisting of interacting single oscillatory systems. Associated fluctuations have a complex appearance due to the fact that vibrations in one system influence vibrations in another through coupling (generally dissipative and nonlinear)
oscillations in structures with distributed parameters(long lines, resonators),
fluctuation, occurring as a result of the thermal movement of matter.
Conditions for the occurrence of oscillations.
1. For oscillation to occur in a system, it is necessary to remove it from its equilibrium position. For example, for a pendulum, giving it kinetic (impact, push) or potential (deflection of the body) energy.
2. When a body is removed from a stable equilibrium position, a resultant force appears directed towards the equilibrium position.
From an energy point of view, this means that conditions arise for a constant transition (kinetic energy into potential energy, electric field energy into magnetic field energy and vice versa.
3. Energy losses of the system due to the transition to other types of energy (often thermal energy) are small.
Characteristics of the oscillatory process.
Figure 1 shows a graph of periodic changes in the function F(x), which is characterized by the following parameters:
Amplitude - the maximum deviation of a fluctuating quantity from some average value for the system.
Period - the shortest period of time through which any indicators of the state of the system are repeated(the system makes one complete oscillation), T(c).
“Physical and mathematical pendulum” - It is customary to distinguish: Presentation on the topic: “Pendulum”. Mathematical pendulum. Performed by Tatyana Yunchenko. Mathematical pendulum physical pendulum. Pendulum.
“Sound resonance” - The same thing happens with two equally tuned strings. By passing the bow along one string, we will cause vibrations on the other. Having set one tuning fork in vibration, you will notice that the other tuning fork will sound by itself. Concept. Prepared by: Velikaya Yulia Checked by: Sergeeva Elena Evgenievna Municipal Educational Institution “Secondary School No. 36” 2011.
“Oscillating movement” - Extreme left position. Swing. Examples of oscillatory movements. Conditions for the occurrence of oscillations. Amplitude shift. V=max a=0 m/s?. Sewing machine needle. Oscillatory movement. Balance position. Tree branches. V=0 m/s a=max. Far right position. Car springs. Clock pendulum. Feature of oscillatory movement.
“Lesson on mechanical vibrations” - Types of pendulums. Towards a position of equilibrium. Free vibrations. G. Klin, Moscow region 2012. Example: pendulum. Types of oscillatory systems 3. The main property of oscillatory systems 4. Free vibrations. Presentation for a physics lesson. Completed by: physics teacher Lyudmila Antonevna Demashova. 6. An oscillatory system is a system of bodies capable of performing oscillatory movements.
“Pendulum swings” - Cosine. “The world we live in is surprisingly prone to fluctuations” R. Bishop. Types of vibrations. Basic characteristics of the oscillatory process (motion). Mathematical and spring pendulum tests. 7. A weight suspended on a spring was brought out of its equilibrium position and released. Unit of measurement (second s).
“Physics of mechanical vibrations” - Let's talk about vibrations... Parameters of mechanical vibrations. Shows the maximum displacement of the body from the equilibrium position. Oscillatory systems. “There was a merry ball in the castle, the musicians were singing. Period. Video task. Bazhina G.G. – physics teacher at Municipal Educational Institution “GYMNASIA No. 11” in Krasnoyarsk. The breeze in the garden rocked the light swing" Konstantin Balmont.
There are a total of 14 presentations in the topic
2. Moment of inertia and its calculation
According to the definition, the moment of inertia of a body relative to an axis is equal to the sum of the products of the masses of particles by the squares of their distances to the axis of rotation or
However, this formula is not suitable for calculating the moment of inertia; since the mass of a solid body is distributed continuously, the sum should be replaced by an integral. Therefore, to calculate the moment of inertia, the body is divided into infinitesimal volumes dV with mass dm=dV. Then
where R is the distance of the element dV from the axis of rotation.
If the moment of inertia I C about the axis passing through the center of mass is known, then one can easily calculate the moment of inertia about any parallel axis O passing at a distance d from the center of mass or
I O = I C + md 2,
This ratio is called Steiner's theorem: the moment of inertia of a body relative to an arbitrary axis is equal to the sum of the moment of inertia relative to an axis parallel to it and passing through the center of mass and the product of the body mass by the square of the distance between the axes.
3. Kinetic energy of rotation
Kinetic energy of a rigid body rotating around a fixed axis
Differentiating the formula with respect to time, we obtain the law of change in the kinetic energy of a rigid body rotating around a fixed axis:
–
the rate of change of kinetic energy of rotational motion is equal to the power of the moment of force.
dK rotation =M Z Z dt=M Z d K K 2 -K 1 =
those. the change in kinetic energy of rotation is equal to the work done by torque.
4. Flat movement
The motion of a rigid body in which the center of mass moves in a fixed plane, and the axis of its rotation passing through the center of mass remains perpendicular to this plane is called flat movement. This movement can be reduced to a combination of translational movement and rotation around fixed (fixed) axis, since in the C-system the axis of rotation actually remains stationary. Therefore, plane motion is described by a simplified system of two equations of motion:
The kinetic energy of a body performing plane motion will be:
and finally
,
since in this case i " is the rotation speed of the i-th point around a fixed axis.
Oscillations
1. Harmonic oscillator
Oscillations In general, movements that repeat over time are called.
If these repetitions follow at regular intervals, i.e. x(t+T)=x(t), then the oscillations are called periodic. The system that makes
vibrations are called oscillator. The oscillations that a system, left to itself, makes are called natural, and the frequency of oscillations in this case is natural frequency.
Harmonic vibrations vibrations that occur according to the law sin or cos are called. For example,
x(t)=A cos(t+ 0),
where x(t) is the displacement of the particle from the equilibrium position, A is the maximum
offset or amplitude, t+ 0 -- phase oscillations, 0 -- initial phase (at t=0), 
-- cyclic frequency, is simply the oscillation frequency.
A system that performs harmonic oscillations is called a harmonic oscillator. It is important that the amplitude and frequency of harmonic oscillations are constant and independent of each other.
Conditions for the occurrence of harmonic oscillations: a particle (or system of particles) must be acted upon by a force or moment of force proportional to the displacement of the particle from the equilibrium position and
trying to return it to a position of balance. Such a force (or moment of force)
called quasi-elastic; it has the form , where k is called quasi-rigidity.
In particular, it can be simply an elastic force that vibrates a spring pendulum oscillating along the x axis. The equation of motion of such a pendulum has the form:
or 
,
where the designation is introduced.
By direct substitution it is easy to verify that by solving the equation
is a function
x=A cos( 0 t+ 0),
where A and 0 -- constants, to determine which you need to specify two initial conditions: position x(0)=x 0 of the particle and its speed v x (0)=v 0 at the initial (zero) moment of time.
This equation is the dynamic equation of any
harmonic vibrations with natural frequency 0. For the weight on
period of oscillation of a spring pendulum
.
2. Physical and mathematical pendulums
Physical pendulum- is any physical body that performs
oscillations around an axis that does not pass through the center of mass in the field of gravity.
In order for the natural oscillations of the system to be harmonic, it is necessary that the amplitude of these oscillations be small. By the way, the same is true for the spring: F control = -kx only for small deformations of the spring x.
The period of oscillation is determined by the formula:
.
Note that the quasi-elastic moment here is the moment of gravity
M i = - mgd , proportional to the angular deviation .
A special case of a physical pendulum is mathematical pendulum-- a point mass suspended on a weightless inextensible thread of length l. Period small fluctuations mathematical pendulum
3. Damped harmonic oscillations
In a real situation, dissipative forces (viscous friction, environmental resistance) always act on the oscillator from the environment.
, which slow down the movement. The equation of motion then takes the form:
.
Denoting and , we obtain the dynamic equation of natural damped harmonic oscillations:
.
As with undamped oscillations, this is the general form of the equation.
If the medium resistance is not too high
Function 
represents an exponentially decreasing amplitude of oscillations. This decrease in amplitude is called relaxation(weakening) of vibrations, and is called attenuation coefficient hesitation.
Time during which the amplitude of oscillations decreases by e=2.71828 times,
called relaxation time.
In addition to the attenuation coefficient, another characteristic is introduced,
called logarithmic damping decrement-- it's natural
logarithm of the ratio of amplitudes (or displacements) over a period:
.
Frequency of natural damped oscillations
depends not only on the magnitude of the quasi-elastic force and body mass, but also on
environmental resistance.
4. Addition of harmonic vibrations
Let us consider two cases of such addition.
a) The oscillator participates in two mutually perpendicular fluctuations.
In this case, two quasi-elastic forces act along the x and y axes. Then
In order to find the trajectory of the oscillator, time t should be excluded from these equations.
The easiest way to do this is if multiple frequencies:
Where n and m are integers.
In this case, the trajectory of the oscillator will be some closed curve called Lissajous figure.
Example: the oscillation frequencies in x and y are the same ( 1 = 2 =), and the difference in the oscillation phases 
(for simplicity we put 1 =0).
.
From here we find: 
-- the Lissajous figure will be an ellipse.
b) The oscillator oscillates one direction.
Let there be two such oscillations for now; Then
where and 
-- oscillation phases.
It is very inconvenient to add vibrations analytically, especially when they are
not two, but several; therefore geometric is usually used vector diagram method.
5. Forced vibrations
Forced vibrations arise when acting on the oscillator
external periodic force changing according to a harmonic law
with frequency ext: 
.
Dynamic equation of forced oscillations:
For steady state oscillation the solution to the equation is the harmonic function:
where A is the amplitude of forced oscillations, and is the phase lag
from compelling force.
Amplitude of steady-state forced oscillations:
Phase lag of steady-state forced oscillations from external
driving force:
.
\hs So: steady-state forced oscillations occur
with a constant, time-independent amplitude, i.e. don't fade away
despite the resistance of the environment. This is explained by the fact that the work
external force comes to
increase in the mechanical energy of the oscillator and completely compensates
its decrease, occurring due to the action of the dissipative resistance force
6. Resonance
As can be seen from the formula, the amplitude of forced oscillations
And ext depends on the frequency of the external driving force ext. The graph of this relationship is called resonance curve or the amplitude-frequency response of the oscillator.
The value of the frequency of the external force at which the amplitude of oscillations becomes maximum is called resonant frequency res, and a sharp increase in amplitude at in = res -- resonance.
The resonance condition will be the condition of the extremum of the function A( ext):
.
The resonant frequency of the oscillator is determined by the expression:
.
In this case, the resonant value of the amplitude of forced oscillations
The quantity characterizing the resonant response of the system is called quality factor oscillator.
On the contrary, with a sufficiently large resistance 
no resonance will be observed.
Fundamentals of the special theory of relativity. molecular
>> Conditions for the occurrence of free oscillations
§ 19 CONDITIONS FOR THE APPEARANCE OF FREE VIBRATIONS
Let us find out what properties a system must have in order for free oscillations to occur in it. It is most convenient to first consider the vibrations of a ball strung on a smooth horizontal rod under the action of the elastic force of a spring 1.
If you move the ball slightly from the equilibrium position (Fig. 3.3, a) to the right, then the length of the spring will increase by (Fig. 3.3, b), and the elastic force from the spring will begin to act on the ball. This force, according to Hooke's law, is proportional to the deformation of the spring and the direction of the foam to the left. If you release the ball, then under the action of elastic force it will begin to move with acceleration to the left, increasing its speed. In this case, the elastic force will decrease, since the deformation of the spring decreases. At the moment when the ball reaches the equilibrium position, the elastic force of the spring becomes equal to zero. Consequently, according to Newton’s second law, the acceleration of the ball will also become zero.
At this point, the speed of the ball will reach its maximum value. Without stopping in the equilibrium position, it will continue to move to the left by inertia. The spring is compressed. As a result, an elastic force appears, directed to the right and inhibiting the movement of the ball (Fig. 3.3, c). This force, and therefore the acceleration directed to the right, increases in magnitude in direct proportion to the modulus of the displacement x of the ball relative to the equilibrium position.
1 Analysis of the vibrations of a ball suspended on a vertical spring is somewhat more complicated. In this case, the variable elastic force of the spring and the constant force of gravity act simultaneously. But the nature of the oscillations in both cases is completely the same.
The speed will decrease until, in the extreme left position of the ball, it becomes zero. After this, the ball will begin to accelerate to the right. With decreasing displacement modulus x force F control decreases in absolute value and in the equilibrium position again goes to zero. But by this moment the ball has already acquired speed and, therefore, by inertia continues to move to the right. This movement leads to stretching of the spring and the appearance of a force directed to the left. The movement of the ball is slowed down until it comes to a complete stop in the extreme right position, after which the whole process is repeated all over again.
If there were no friction, the movement of the ball would never cease. However, friction and air resistance prevent the ball from moving. The direction of the resistance force both when the ball moves to the right and when it moves to the left is always opposite to the direction of speed. The scope of its oscillations will gradually decrease until the movement stops. With low friction, damping becomes noticeable only after the ball has oscillated a lot. If you observe the movement of the ball over a not very large time interval, then the damping of oscillations can be neglected. In this case, the influence of the resistance force on the voltage can be ignored.
If the resistance force is large, then its action cannot be neglected even over short time intervals.
Place a ball on a spring into a glass with a viscous liquid, for example glycerin (Fig. 3.4). If the spring stiffness is small, then the ball removed from its equilibrium position will not oscillate at all. Under the action of elastic force, it will simply return to its equilibrium position (dashed line in Figure 3.4). Due to the action of the drag force, its speed in the equilibrium position will be practically zero.
In order for free oscillations to occur in a system, two conditions must be met. Firstly, when moving a body from an equilibrium position, a force must arise in the system directed towards the equilibrium position and, therefore, tending to return the body to the equilibrium position. This is exactly how a spring acts in the system we considered (see Fig. 3.3): when the ball moves both to the left and to the right, the elastic force is directed towards the equilibrium position. Secondly, the friction in the system should be quite low. Otherwise, the vibrations will quickly die out. Undamped oscillations are possible only in the absence of friction.
1. What vibrations are called free!
2. Under what conditions do free oscillations occur in the system?
3. What oscillations are called forced! Give examples of forced oscillations.
One of the most interesting topics in physics is oscillations. The study of mechanics is closely connected with them, with how bodies behave when they are affected by certain forces. Thus, when studying oscillations, we can observe pendulums, see the dependence of the amplitude of oscillation on the length of the thread on which the body hangs, on the stiffness of the spring, and the weight of the load. Despite its apparent simplicity, this topic is not as easy for everyone as we would like. Therefore, we decided to collect the most well-known information about vibrations, their types and properties, and compile for you a brief summary on this topic. Perhaps it will be useful to you.
Definition of the concept
Before talking about concepts such as mechanical, electromagnetic, free, forced vibrations, their nature, characteristics and types, conditions of occurrence, it is necessary to define this concept. Thus, in physics, an oscillation is a constantly repeating process of changing state around one point in space. The simplest example is a pendulum. Each time it oscillates, it deviates from a certain vertical point, first in one direction, then in the other. The theory of oscillations and waves studies the phenomenon.
Causes and conditions of occurrence
Like any other phenomenon, oscillations only occur if certain conditions are met. Mechanical forced vibrations, like free ones, arise when the following conditions are met:
1. The presence of a force that removes the body from a state of stable equilibrium. For example, the push of a mathematical pendulum, at which movement begins.
2. The presence of minimal friction force in the system. As you know, friction slows down certain physical processes. The greater the friction force, the less likely it is for vibrations to occur.
3. One of the forces must depend on the coordinates. That is, the body changes its position in a certain coordinate system relative to a certain point.
Types of vibrations
Having understood what an oscillation is, let’s analyze their classification. There are two most well-known classifications - by physical nature and by the nature of interaction with the environment. Thus, according to the first criterion, mechanical and electromagnetic vibrations are distinguished, and according to the second, free and forced vibrations. There are also self-oscillations and damped oscillations. But we will only talk about the first four types. Let's take a closer look at each of them, find out their features, and also give a very brief description of their main characteristics.
Mechanical
It is with mechanical vibrations that the study of vibrations in a school physics course begins. Students begin their acquaintance with them in such a branch of physics as mechanics. Note that these physical processes occur in the environment, and we can observe them with the naked eye. With such oscillations, the body repeatedly makes the same movement, passing a certain position in space. Examples of such oscillations are the same pendulums, the vibration of a tuning fork or guitar string, the movement of leaves and branches on a tree, a swing.
Electromagnetic
After the concept of mechanical vibrations has been firmly grasped, the study of electromagnetic vibrations, which are more complex in structure, begins, since this type occurs in various electrical circuits. During this process, oscillations in electric as well as magnetic fields are observed. Despite the fact that electromagnetic oscillations have a slightly different nature of occurrence, the laws for them are the same as for mechanical ones. With electromagnetic oscillations, not only the strength of the electromagnetic field can change, but also characteristics such as charge and current strength. It is also important to note that there are free and forced electromagnetic oscillations.
Free vibrations
This type of oscillation occurs under the influence of internal forces when the system is removed from a state of stable equilibrium or rest. Free oscillations are always damped, which means their amplitude and frequency decrease over time. A striking example of this type of swing is the movement of a load suspended on a thread and oscillating from one side to the other; a load attached to a spring, either falling down under the influence of gravity, or rising up under the action of the spring. By the way, it is precisely this kind of oscillations that is paid attention to when studying physics. And most of the problems are devoted to free vibrations, and not forced ones.
Forced
Despite the fact that this kind of process is not studied in such detail by schoolchildren, it is forced oscillations that are most often found in nature. A fairly striking example of this physical phenomenon can be the movement of branches on trees in windy weather. Such fluctuations always occur under the influence of external factors and forces, and they arise at any moment.
Oscillation Characteristics
Like any other process, oscillations have their own characteristics. There are six main parameters of the oscillatory process: amplitude, period, frequency, phase, displacement and cyclic frequency. Naturally, each of them has its own designations, as well as units of measurement. Let's look at them in a little more detail, focusing on a brief description. At the same time, we will not describe the formulas that are used to calculate this or that value, so as not to confuse the reader.
Bias
The first of these is displacement. This characteristic shows the deviation of the body from the equilibrium point at a given moment in time. It is measured in meters (m), the generally accepted designation is x.
Oscillation amplitude
This value indicates the greatest displacement of the body from the equilibrium point. In the presence of undamped oscillation, it is a constant value. It is measured in meters, the generally accepted designation is x m.
Oscillation period
Another quantity that indicates the time it takes to complete one complete oscillation. The generally accepted designation is T, measured in seconds (s).
Frequency
The last characteristic we will talk about is the oscillation frequency. This value indicates the number of oscillations in a certain period of time. It is measured in hertz (Hz) and is denoted as ν.
Types of pendulums
So, we have analyzed forced oscillations, talked about free oscillations, which means we should also mention the types of pendulums that are used to create and study free oscillations (in school conditions). Here we can distinguish two types - mathematical and harmonic (spring). The first is a certain body suspended from an inextensible thread, the size of which is equal to l (the main significant quantity). The second is a weight attached to a spring. Here it is important to know the mass of the load (m) and the spring stiffness (k).
conclusions
So, we figured out that there are mechanical and electromagnetic vibrations, gave them a brief description, described the causes and conditions for the occurrence of these types of vibrations. We said a few words about the main characteristics of these physical phenomena. We also figured out that there are forced and free vibrations. We determined how they differ from each other. In addition, we said a few words about pendulums used in the study of mechanical vibrations. We hope this information was useful to you.