Natural vibrations are processes that are characterized by a certain repeatability. For example, these include the movement of the pendulum of a clock, a guitar string, the legs of a tuning fork, the activity of the heart.
Mechanical vibrations
Taking into account the physical nature, natural vibrations can be mechanical, electromagnetic, electromechanical. Let's take a closer look at the first process. Natural vibrations occur in cases where there is no additional friction, no external forces. Such movements are characterized by frequency dependence only on the characteristics of the given system.
Harmonic processes
These natural oscillations imply a change in the oscillating quantity according to the cosine (sine) law. Let us analyze the simplest form of an oscillatory system, consisting of a ball suspended on a spring.
In this case, gravity balances the elasticity of the spring. According to Hooke's law, there is a direct relationship between its extension of the spring and the force applied to the body.
Elastic force properties
Own electromagnetic oscillations in the circuit are related to the magnitude of the impact on the system. The elastic force, which is proportional to the displacement of the ball from the equilibrium position, is directed towards the equilibrium state. The movement of the ball under its influence can be described by the law of cosine.
The natural oscillation period will be determined mathematically.
In the case of a spring pendulum, the dependence on its rigidity, as well as on the mass of the load, is revealed. The period of natural oscillations in this case can be calculated by the formula.
Energy at harmonic oscillation
The value is constant if there is no friction force.
As the oscillatory movement occurs, a periodic transformation of kinetic energy into a potential value occurs.
Damped oscillations
Own electromagnetic oscillations can occur when the system is not affected by external forces. Friction contributes to the damping of oscillations, a decrease in their amplitude is observed.
The frequency of natural oscillations in an oscillatory circuit is related to the properties of the system, as well as to the intensity of losses.
With an increase in the attenuation coefficient, an increase in the period of oscillatory motion is observed.
The ratio of amplitudes that are separated by an interval equal to one period is constantvalue throughout the process. This ratio is called the damping decrement.
Natural vibrations in the oscillating circuit are described by the law of sines (cosines).
The oscillation period is an imaginary quantity. The movement is aperiodic. The system, which is removed from the equilibrium position without additional oscillations, returns to its original state. The method of bringing the system to an equilibrium state is determined by its initial conditions.
Resonance
The period of natural oscillations of the circuit is determined by the harmonic law. Forced oscillations appear in the system under the action of a periodically changing force. When compiling the equation of motion, it is taken into account that in addition to the forcing effect, there are also such forces acting during free vibrations: the resistance of the medium, the quasi-elastic force.
Resonance is a sharp increase in the amplitude of forced oscillations when the frequency of the driving force tends to the natural frequency of the body. All vibrations that occur in this case are called resonant.
To reveal the relationship between the amplitude and the external force for forced oscillations, you can use the experimental setup. When the crank handle is rotated slowly, the load on the spring moves up and down similarly to the point of their suspension.
Own electromagnetic oscillations in the oscillatory circuit can be calculated and other physical parameterssystem.
In the case of a faster rotation, the oscillations increase, and when the rotation frequency is equal to the natural one, the maximum amplitude value is reached. With a subsequent increase in the frequency of rotation, the amplitude of the forced oscillations of the analyzed load decreases again.
Resonance characteristic
With a slight movement of the handle, the load almost does not change its position. The reason is the inertia of the spring pendulum, which does not have time under the influence of an external force, so only “trembling in place” is observed.
The natural frequency of oscillations in the circuit will correspond with a sharp increase in the amplitude of the frequency of the external action.
The graph of such a phenomenon is called the resonance curve. It can also be considered for a filament pendulum. If you hang a massive ball on the rail, as well as a number of light pendulums with different thread lengths.
Each of these pendulums has its own oscillation frequency, which can be determined based on the acceleration of free fall, the length of the thread.
If the ball is taken out of equilibrium, leaving the light pendulum without movement, then released, its swings will lead to periodic bending of the rail. This will cause the effect of a periodically changing elastic force on light pendulums, causing them to perform forced oscillations. Gradually, all of them will have an equal amplitude, which will be the resonance.
This phenomenon can also be seen for a metronome, the base of which is connectedthread with the axis of the pendulum. In this case, it will swing with maximum amplitude, then the frequency of the pendulum “pulling” the string corresponds to the frequency of its free oscillations.
Resonance occurs when an external force, acting in time with free vibrations, does work with a positive value. This leads to an increase in the amplitude of the oscillatory motion.
Besides the positive impact, the phenomenon of resonance often performs a negative function. For example, if the tongue of a bell is swinging, it is important for the sound to be produced that the rope act in time with the free oscillating movements of the tongue.
Application of resonance
The operation of the reed frequency meter is based on resonance. The device is presented in the form of elastic plates of different lengths, fixed on one common base.
In case of contact of the frequency meter with an oscillatory system for which it is required to determine the frequency, that plate, the frequency of which is equal to the measured one, will oscillate with the maximum amplitude. After entering platinum into resonance, you can calculate the frequency of the oscillating system.
In the eighteenth century, not far from the French city of Angers, a detachment of soldiers moved in step along a chain bridge, the length of which was 102 meters. The frequency of their steps took on a value equal to the frequency of free vibrations of the bridge, which caused a resonance. This caused the chains to break, the suspension bridge to collapse.
In 1906, for the same reason, the Egyptian bridge in St. Petersburg was destroyed, along which a squadron of cavalrymen moved. To avoid such unpleasant phenomena, now withcrossing the bridge, the military units go at a free pace.
Electromagnetic phenomena
They are interconnected fluctuations of magnetic and electric fields.
Own electromagnetic oscillations in the circuit occur when the system is taken out of equilibrium, for example, when a charge is imparted to a capacitor, a change in the current magnitude in the circuit.
Electromagnetic oscillations appear in different electrical circuits. In this case, the oscillatory movement is performed by the current strength, voltage, charge, electric field strength, magnetic induction, and other electrodynamic quantities.
They can be considered as damped oscillations, since the energy imparted to the system goes to heat.
As forced electromagnetic oscillations are the processes in the circuit, which are caused by a periodically changing external sinusoidal electromotive force.
Such processes are described by the same laws as in the case of mechanical vibrations, but they have a completely different physical nature. Electrical phenomena are a special case of electromagnetic processes that have power, voltage, alternating current.
Oscillatory circuit
It is an electrical circuit that consists of an inductor connected in series, a capacitor with a certain capacitance, a resistance resistor.
When the oscillatory circuit is in a stable equilibrium state, the capacitor has no charge, and no electric current flows through the coil.
Among the main featureselectromagnetic oscillations note the cyclic frequency, which is the second derivative of the charge with respect to time. The phase of electromagnetic oscillations is a harmonic quantity, described by the sine (cosine) law.
The period in the oscillatory circuit is determined by the Thomson formula, depends on the capacitance of the capacitor, as well as the value of the inductance of the coil with current. The current in the circuit changes according to the sine law, so you can determine the phase shift for a certain electromagnetic wave.
Alternating current
In a frame rotating at a constant angular velocity in a uniform magnetic field with a certain value of induction, harmonic EMF is determined. According to Faraday's law for electromagnetic induction, they are determined by the change in magnetic flux, is a sinusoidal value.
When an external EMF source is connected to the oscillatory circuit, forced oscillations occur inside it, occurring with a cyclic frequency ώ, equal in value to the frequency of the source itself. They are undamped movements, since when a charge is made, a potential difference appears, a current arises in the circuit, and other physical quantities. This causes harmonic changes in voltage, current, which are called pulsating physical quantities.
The value of 50 Hz is taken as the industrial frequency of alternating current. To calculate the amount of heat released when passing through an alternating current conductor, the maximum power values \u200b\u200bare not used, since it is reached only in certain periods of time. For such purposes, applyaverage power, which is the ratio of all the energy passing through the circuit during the analyzed period, to its value.
The value of the alternating current corresponds to the constant, which releases the same amount of heat over the period as that of the alternating current.
Transformer
This is a device that increases or decreases voltage without significant loss of electrical energy. This design consists of several plates on which two coils with wire windings are fixed. The primary is connected to an alternating voltage source, and the secondary is attached to devices that consume electrical energy. For such a device, a transformation ratio is distinguished. For a step-up transformer, it is less than one, and for a step-up transformer, it tends to 1.
Auto oscillations
These are called systems that automatically regulate the supply of energy from an external source. The processes that take place in them are considered periodic undamped (self-oscillatory) actions. Such systems include a tube generator of electromagnetic interactions, a bell, a clock.
There are also cases in which different bodies simultaneously participate in oscillations in different directions.
If you add together such movements that have equal amplitudes, you can get a harmonic oscillation with a larger amplitude.
According to the Fourier theorem, a set of simple oscillatory systems, into which a complex process can be decomposed, is considered a harmonic spectrum. It indicates the amplitudes and frequencies of all simple oscillations included insuch a system. Most often, the spectrum is reflected in a graphical form.
Frequencies are marked on the horizontal axis, and the amplitudes of such oscillations are shown along the ordinate axis.
Any oscillatory movements: mechanical, electromagnetic, are characterized by certain physical quantities.
First of all, these parameters include amplitude, period, frequency. There are mathematical expressions for each parameter, which allows you to carry out calculations, quantitatively calculate the desired characteristics.
