All harmonic vibrations have a mathematical expression. Their properties are characterized by 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, i.e., by external factors affecting the oscillatory process.
For example, in mechanics, harmonic oscillation is a movement that is characterized by:
- straightforward character;
- unevenness;
- the movement of the physical body, which occurs along a sinusoidal or cosine path, and depending on time.
Based on these properties, we can bring 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 oscillation amplitude, ω is the coefficient.
Such an equation of harmonic oscillations is fundamental for all harmonic oscillations, which are considered in kinematics and mechanics.
The exponent ωt, which in this formula stands under the sign of a trigonometric function, is called the phase, and it determines the location of the vibrating material point at a given specific moment in time at a given amplitude. When considering cyclic vibrations, this indicator is equal to 2 liters, it shows the number of mechanical vibrations within the time cycle and is denoted by w. In this case, the equation of harmonic oscillations contains it as an indicator of the magnitude of the cyclic (circular) frequency.
The equation of harmonic oscillations considered by us, as already noted, can take various forms, depending on a number of factors. For example, here is such an option. To consider the differential equation of free harmonic oscillations, it should be borne in mind that they are all characterized by damping. In various types of oscillations, this phenomenon manifests itself in different ways: the stopping of a moving body, the cessation of radiation in electrical systems. The simplest example showing a decrease in the vibrational potential is its conversion 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 a particular system, β is a constant indicating 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, as well as to design and simulate oscillatory processes at the scientific and experimental level.
For example, it is known that the damped oscillations at the final stage of their manifestation already cease to be harmonic, that is, the categories of frequency and period for them become simply meaningless and are not reflected in the formula.
The classic way to study harmonic oscillations is a harmonic oscillator. In its simplest form, it represents a system that is described by such a differential equation of harmonic oscillations: ds / dt + ω²s = 0. But the variety of oscillatory processes naturally leads to the fact that there are a large number of oscillators. We list their main types:
- spring oscillator - an ordinary load with a certain mass m, which is suspended on an elastic spring. He makes oscillatory movements of a harmonic type, which are described by the formula F = - kx.
- physical oscillator (pendulum) - a solid body oscillating around a static axis under the influence of a certain force;
- a mathematical pendulum (almost never occurs in nature). It is an ideal model of a system including an oscillating physical body with a certain mass, which is suspended on a rigid weightless thread.