The equation of harmonic oscillations and its importance in the study of the nature of oscillatory processes

All harmonics are mathematical expression.Their properties are characterized by a set of trigonometric equations, the complexity of which is determined by the complexity of the oscillation process, the properties of the system and the environment in which they occur, that is, external factors affecting the oscillation process.

For example, in the mechanics of harmonic oscillation is a movement, which is characterized by:

- straightforward character;

- uneven;

- movement of the physical body, which takes place on a sine or cosine trajectory as a function of time.

Based on these properties, you can reduce the equation of harmonic oscillations, which has the form:

x = A cos ωt or type of x = A sin ωt, where x - the value of the origin, and - the value of the vibration amplitude, ω - ratio.

Such an equation of harmonic oscillations is essential for all the harmonic oscillations, which are discussed in the kinematics and mechanics.

index ωt, which this formula is under the sign of the trigonometric functions, call phase and it determines the location of the vibrating material point at this particular point in time for a given amplitude.When considering the cyclical fluctuations of the index is 2n, it shows the number of mechanical vibrations within a time cycle and is denoted w.In this case, the equation of harmonic oscillations contains it as the measure of cyclic (circular) frequency.

considered by us the equation of harmonic oscillations, as already noted, can take various types, depending on several factors.For example, here is a variant.To consider the differential equation of the free harmonic oscillations, one should consider the fact that they all tend to decay.The different types of vibrations, this phenomenon manifests itself in different ways: stop a moving body, the cessation of radiation in electrical systems.A simple example showing the reduction of the vibrational potential acts of its transformation into thermal energy.

Considered equation is: d²s / dt² + 2β x ds / dt + ω²s = 0. In this formula: s - the value of fluctuating value which characterizes the properties of a system, β - constant, showing the attenuation coefficient, ω- cyclic frequency.

use of such a formula allows approach to the description of oscillatory processes in linear systems with a single point of view, and also to make the design and modeling of oscillating processes in the scientific and experimental level.

For example, it is known that damped oscillations at the final stage of its existence cease to be harmonic, ie the categories of frequency and time for them to become simply meaningless and claims are not recognized.

classic method for studying harmonic vibrations acts harmonic oscillator.In its simplest form it is a system that describes a differential equation of harmonic oscillations: ds / dt + ω²s = 0. However, the variety of oscillatory processes naturally leads to the fact that there are a large number of oscillators.Here they are the main types:

- spring oscillator - normal load, has a certain mass m, which is suspended on an elastic spring.He oscillates harmonic type, which are described by the formula F = - kx.

- physical oscillator (pendulum) - solid, oscillates around a static axis under the influence of a certain strength;

- mathematical pendulum (in nature practically does not occur).It is an ideal model system consisting of an oscillating physical body, which has a certain mass, which is suspended on a rigid weightless thread.