Damped oscillations

Vibrational processes surround a person everywhere. This phenomenon is due to the fact that, firstly, in nature there are many environments (physical, chemical, organic, etc.) within which oscillations, including damped oscillations, occur. Secondly, in the surrounding reality 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 current flow in wires, light phenomena, propagation of radio waves and much more. In the end, the person himself, or rather the human body, is an oscillatory system, whose life is provided by various types of vibration - heartbeat, respiratory process, blood circulation, limb movement.

Therefore, they are studied by various sciences, including interdisciplinary ones. The simplest and basic in this study are free oscillations. They are characterized by the exhaustion of the energy of the vibrational pulse, so they eventually cease, and therefore such fluctuations are determined by the concept of damped oscillations.

In the oscillatory systems, the process of energy loss objectively occurs (in mechanical systems - due to friction, in electrical systems - due to the presence of electrical resistance). That is why such damped oscillations can not be classified as harmonic. Given this initial statement, we can mathematically express the events that occur, for example, in mechanically damped oscillations: F = -rV = -r dx / dt. In this formula, r is the coefficient of resistance, a constant value. From the formula, we can conclude that the value of the speed (V) for a given system is proportional to the value of the resistance. But the presence of the sign "-" means that the force vector (F) and velocity have a multidirectional character.

Applying the equation of Newton's second law, and taking into account the influence of the resistance forces, the equation characterizing the damped oscillations of the process of motion takes the following form: in the presence of resistance forces it looks like: 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 analogous equation can be obtained for an electric circuit with allowance for damping, adding to the left-hand side of the equation the value of the voltage drop across the resistance UR. 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 coefficient of friction r is replaced by the electrical resistance of the circuit R; 2 β = R / L, where: К - resistance of the circuit, L - length of the circuit.

If we construct the corresponding graphs on the basis of these formulas, we can see that the graph of damped oscillations is very similar to that of harmonic oscillations, but the amplitude of the oscillation gradually decreases exponentially.

Taking into account the fact that oscillations can occur by various oscillatory systems and occur in different environments, it is necessary to mention which system we are considering in each particular case. From this condition depends not only the features of the flow of oscillatory processes, but the reverse effect - 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 during the performance the elasticity of the spring does not change, the force of gravity acting on the load, and in electrical systems, the resistance versus speed or the acceleration of the oscillating quantity remain unchanged. Such oscillatory systems are called linear.