Oscillatory processes surround a person everywhere. This phenomenon is due to the fact that, firstly, in nature there are many media (physical, chemical, organic, etc.), within which vibrations occur, including damped vibrations. Secondly, in the reality surrounding us there is a huge variety of oscillatory systems, the very existence of which is associated exclusively with oscillatory processes. These processes surround us everywhere, they characterize the flow of current in wires, light phenomena, the propagation of radio waves and much more. In the end, the man himself, or rather the human body, is an oscillatory system, whose life is provided by various types of oscillations - heartbeat, respiratory process, blood circulation, limb movement.
Therefore, they are studied by various sciences, including interdisciplinary. The simplest and initial in this study are free oscillations. They are characterized by the exhaustion of the energy of the vibrational impulse, therefore they, in the end, cease, and therefore such oscillations are defined by the concept of damped oscillations.
In oscillatory systems, the process of energy loss occurs objectively (in mechanical systems due to friction, in electrical systems due to the presence of electrical resistance). That is why such damped oscillations cannot be classified as harmonic. Given this initial statement, it is possible to mathematically express the damped oscillations occurring, for example, in mechanics, as expressed by the formula: F = - rV = -r dx / dt. In this formula, r is the resistance coefficient, a constant. According to the formula, we can conclude that the value of speed (V) for this system is proportional to the value of resistance. But the presence of the “-” sign means that the force (F) and velocity vectors are multidirectional.
Applying the equation of Newton’s second law, and taking into account the influence of resistance forces, the equation characterizing the damped oscillations of the motion process takes the following form: in the presence of resistance forces it has the form: d ^ 2x / dt2 + 2β dt / dt + ω2 x = 0. In this formula β is the attenuation coefficient, which shows the intensity of this phase of the oscillatory process.
A completely similar equation can be obtained for the electric circuit taking into account the attenuation by adding the value of the voltage drop across the resistance UR to the left side of the equation. Only in this case, the differential equation is written not for the time displacement (t), but for the charge on the capacitor q (t); the friction coefficient r is replaced by the electrical resistance of the circuit R; wherein 2 β = R / L, where: K is the circuit resistance, L is the chain length.
If, on the basis of these formulas, we construct the corresponding graphs, then we can see that the graph of damped oscillations is very similar to the graphs of harmonic oscillations, but the amplitude of oscillations gradually decreases exponentially.
Given the fact that oscillations can occur by various oscillatory systems and occur in different environments, we should make a reservation about which system we are considering in each particular case. Not only features of the course of oscillatory processes depend on this condition, but the opposite effect occurs - the nature of the oscillations determines the system itself and its classification place. We, in this case, considered one in which the properties of the system itself remain unchanged in the study of the oscillatory process. For example, we assume that in the process of making the spring’s elasticity, the force of gravity acting on the load does not change, and in electrical systems the dependences of resistance on the speed or acceleration of an oscillating quantity remain unchanged. Such oscillatory systems are called linear.