Equation of harmonic oscillations and its significance in the study of the nature of oscillatory processes

All harmonic oscillations have a mathematical expression. Their properties characterize a set of trigonometric equations, the complexity of which is determined by the complexity of the oscillatory process itself, the properties of the system and the environment in which they occur, ie, external factors affecting the oscillatory process.

For example, in mechanics, a harmonic oscillation is a movement that is characteristic of:

- rectilinear nature;

- unevenness;

- movement of the physical body, which occurs on a sinusoidal or cosine trajectory, and depending on time.

Based on these properties, we can give the equation of harmonic oscillations, which has the form:

X = A cos ωt or the form x = A sin ωt, where x is the coordinate value, A is the amplitude of the oscillation, and ω is the coefficient.

Such an equation of harmonic oscillations is fundamental for all harmonic oscillations, which are considered in kinematics and mechanics.

The index ωt, which in this formula stands under the sign of the trigonometric function, is called the phase, and it determines the location of the oscillating material point at a given particular time instant at a given amplitude. When considering cyclic oscillations, this index is 2n, it shows the number of mechanical oscillations within the time cycle and is denoted by w. In this case, the harmonic oscillation equation contains it as an indicator of the value of the cyclic (circular) frequency.

The equation of harmonic oscillations considered by us, as already noted, can assume various forms, depending on a number of factors. For example, here's an option. To consider the differential equation of free harmonic oscillations, one must take into account the fact that they all have attenuation. In various types of oscillations, this phenomenon manifests itself in different ways: stopping a moving body, stopping radiation in electrical systems. The simplest example, showing a decrease in the vibrational potential, is its transformation into thermal energy.

The equation under consideration has the form: d²s / dt² + 2β x ds / dt + ω²s = 0. In this formula: s is the value of the oscillating quantity that characterizes the properties of this or that system; β is a constant showing the attenuation coefficient; ω is the cyclic frequency.

The use of such a formula allows one to approach the description of oscillatory processes in linear systems from a single point of view, and also to design and model oscillatory processes at the scientific and experimental level.

For example, it is known that damped oscillations at the final stage of their manifestation are no longer harmonious, that is, the categories of frequency and period for them become simply meaningless and are not reflected in the formula.

A classical method of studying harmonic oscillations is a harmonic oscillator. In its simplest form, it represents a system that describes such a differential equation of harmonic oscillations: ds / dt + ω²s = 0. But the variety of oscillatory processes naturally leads to the existence of a large number of oscillators. We list their main types:

- Spring oscillator - a conventional load, which has a certain mass m, which is suspended on an elastic spring. He performs oscillatory movements of a harmonic type, which are described by the formula F = - kx.

- a physical oscillator (pendulum) - a solid body oscillating around a static axis under the influence of a certain force;

- mathematical pendulum (in nature, almost never occurs). It is an ideal model of a system that includes a vibrating physical body that has a certain mass that is suspended on a rigid, weightless thread.