Natural vibrations are processes that are distinguished by a certain repeatability. For example, they include the movement of the pendulum of a clock, a guitar string, tuning fork legs, and heart activity.
Mechanical vibrations
Given the physical nature, natural vibrations can be mechanical, electromagnetic, electromechanical. Let's consider the first process in more detail. Natural oscillations occur in cases where there is no additional friction, no external forces. Such movements are characterized by a frequency dependence only on the characteristics of a given system.
Harmonic processes
These natural vibrations suggest a change in the oscillating quantity according to the law of cosine (sine). Let us analyze the simplest form of the oscillatory system, consisting of a ball suspended on a spring.
In this case, gravity balances the spring elasticity. According to Hooke's law, there is a direct relationship between its extension of the spring and the force applied to the body.
Elastic properties
Natural electromagnetic oscillations in the circuit are associated with 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 to the equilibrium state. The movement of the ball under its influence can be described by the law of cosine.
The period of natural oscillations will be determined mathematically.
In the case of a spring pendulum, a dependence is revealed on its stiffness, as well as on the mass of the load. The period of natural oscillations in this case can be calculated by the formula.
Harmonic Energy
The value is constant if there is no friction force.
As the oscillatory movement occurs, the kinetic energy is periodically converted into a potential value.
Damped oscillations
Natural electromagnetic oscillations can occur in the case 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 vibrations in the oscillatory circuit is associated with the properties of the system, as well as with the intensity of the 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 a constant throughout the process. A similar relation is called the attenuation decrement.
Natural vibrations in the oscillatory circuit are described by the law of sines (cosines).
The oscillation period is an imaginary quantity. The movement has an aperiodic character. The system, which is removed from the equilibrium position without additional fluctuations, returns to its original state. The method of bringing the system into equilibrium is determined by its initial conditions.
Resonance
The period of natural oscillations of the circuit is determined by harmonic law. Forced oscillations appear in the system under the action of periodically changing forces. When drawing up the equation of motion , it is taken into account that in addition to the coercive effect, there are also such forces acting during free vibrations: resistance of the medium, quasi-elastic force.
Resonance is a sharp increase in the amplitude of forced oscillations as 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 identify 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 down and up similarly to their suspension point.
Natural electromagnetic oscillations in the oscillatory circuit can be calculated and other physical parameters of the system.
In the case of a faster rotation, the oscillations increase, and at a rotation frequency equal to the eigenvalue, the maximum value of the amplitude is reached. With a subsequent increase in the rotation frequency, the amplitude of the forced oscillations of the analyzed load decreases again.
Resonance characteristic
With insignificant movement of the handle, the load almost does not change its position. The reason is the inertia of the spring pendulum, which does not keep up under the influence of external force, so only "jitter 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 external influences.
The graph of this phenomenon is called the resonance curve. It can also be considered for a thread pendulum. If you hang a massive ball on the rail, as well as a certain number of lightweight pendulums with different lengths of thread.
Each of these pendulums has its own oscillation frequency, which can be determined based on the acceleration of gravity, the length of the thread.
If the ball is brought out of equilibrium, leaving the light pendulum without movement, then released, its swings will lead to periodic bending of the rack. This will cause the impact of a periodically changing elastic force on the light pendulums, forcing them to make forced vibrations. Gradually, all of them will have an amplitude of equal magnitude, which will be the resonance.
This phenomenon can also be seen for the metronome, the base of which is connected by a thread with the axis of the pendulum. In this case, it will swing with maximum amplitude, then the frequency of the “pendulum” pulling the string of the pendulum corresponds to the frequency of its free vibrations.
Resonance occurs when an external force, acting in time with free vibrations, performs work with a positive value. This leads to an increase in the amplitude of the oscillatory motion.
In addition to the positive impact, the resonance phenomenon often also performs a negative function. For example, if the tongue of a bell is swinging, to get a sound, it is important that the rope acts in one beat with the free oscillatory movements of the tongue.
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 basis.
In case of contact of the frequency meter with the oscillating system, for which it is required to determine the frequency, the plate with the frequency equal to the measured one will oscillate with maximum amplitude. After platinum enters the resonance, the frequency of the oscillatory system can be calculated.
In the eighteenth century, not far from the French city of Angers, a detachment of soldiers moved in a leg along a chain bridge, whose length was 102 meters. The frequency of their steps took 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 collapsed.
In 1906, for the same reason, the Egyptian bridge in St. Petersburg was destroyed, along which the cavalry squadron moved. To avoid such unpleasant phenomena, now, when crossing the bridge, military units are taking a free step.
Electromagnetic phenomena
They are interconnected oscillations of the magnetic and electric fields.
Natural electromagnetic oscillations in the circuit arise when the system is brought out of equilibrium, for example, when the charge is communicated to the capacitor, changes in the current magnitude in the circuit.
Electromagnetic vibrations appear in different electrical circuits. In this case, the oscillatory motion is carried out by current strength, voltage, charge, electric field strength, magnetic induction, and other electrodynamic quantities.
They can be considered as damped oscillations, since the energy supplied to the system goes to heating.
Processes in a circuit that are caused by a periodically changing external sinusoidal electromotive force act as forced electromagnetic oscillations.
Similar processes are described by the same laws as in the case of mechanical vibrations, but have a completely different physical nature. Electrical phenomena is a special case of electromagnetic processes with power, voltage, alternating current.
Oscillation circuit
It is an electric circuit, which consists of an inductance coil connected in series, a capacitor with a specific capacity, and a resistance resistor.
With a stable equilibrium state of the oscillatory circuit, the capacitor does not have a charge, and no electric current flows through the coil.
Among the main characteristics of electromagnetic oscillations, the cyclic frequency, which is the second time derivative of the charge, is noted. The phase of electromagnetic oscillations is a harmonic quantity, described by the law of sine (cosine).
The period in the oscillatory circuit is determined by the Thomson formula, depends on the capacitance of the capacitor, as well as the magnitude of the inductance of the coil with current. The current in the circuit changes according to the sine law, therefore, it is possible to determine the phase shift for a certain electromagnetic oscillation.
Alternating current
In a frame rotating at a constant angular velocity in a uniform magnetic field with a certain magnitude of induction, the harmonic emf is determined. According to the Faraday law for electromagnetic induction, they are determined by a change in magnetic flux, which 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, because when a charge is made, a potential difference appears, a current and other physical quantities appear in the circuit. 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 the alternating current. In order to calculate the amount of heat released when passing through an alternating current conductor, the maximum power values are not used, since it is achieved only in some time intervals. For such purposes, average power is used, which is the ratio of all energy passing through the circuit for the analyzed period to its value.
The value of alternating current corresponds to a constant, emitting during the period the same amount of heat as that of alternating current.
Transformer
This is such a device that increases or decreases voltage without significant loss of electrical energy. This design consists of several plates on which two coils are fixed, having wire windings. The primary is connected to an AC voltage source, and the secondary is attached to devices that consume electrical energy. For such a device, a transformation coefficient is isolated. For a step-up transformer, it is less than unity, and for a step-up transformer, it tends to 1.
Self-oscillations
These are called systems that automatically regulate the supply of energy from an external source. The processes that occur in them are considered periodic undamped (self-oscillating) actions. These systems include a tube generator of electromagnetic interactions, a bell, a clock.
There are also cases in which in different directions simultaneously different bodies participate in vibrations.
If you add together such movements that have equal amplitudes, you can get harmonic oscillation with a larger amplitude.
By the Fourier theorem, the totality of simple oscillatory systems, which can be decomposed into a complex process, is considered a harmonic spectrum. It indicates the amplitudes and frequencies of all simple oscillations included in such a system. Most often, the spectrum is reflected in 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, such parameters include amplitude, period, frequency. For each parameter, there are mathematical expressions, which allows you to carry out calculations, quantify the desired characteristics.