In this article, we'll talk about free fluctuations. Let's consider their examples: mathematical and spring pendulums, and also an oscillatory circuit.

Mechanical oscillations

Oscillatory motion, or mechanical vibrations is the movement of bodies or the state change that occurs over time. Examples in mechanics can be pendulum oscillations of strings and beams hours, the membranes of the loudspeakers, bridges, and other structures.Free oscillations

Periodic is called oscillatory motion, if the values ​​of physical quantities that change during oscillations are repeated at identical intervals of time.

The minimum interval (time) of time through which the position of the body is repeated during the oscillatory motion is called the period of oscillationT. The number of oscillations that a body performs per unit time is called the frequency of oscillationsν .


Among the various vibrational motions, harmonic oscillatory movements are of great importance.

The oscillation is called harmonic, during which the material point deviates from the equilibrium position according to the law of the sine or cosine.

The importance of this movement lies in the fact that many vibrational movements in nature are close to harmonic, and also because complex oscillations can be decomposed into harmonic ones. Let us write down the displacement of a material point with a harmonic motion:

The letter "x "  the deviation of the point that oscillates from the equilibrium position is indicated. The maximum displacement from the equilibrium position is called the amplitude. In our casexmax  = A. Argument(ωt + φ0 )  is called the phase of the oscillation, and the quantityφ0  - the initial phase of the oscillation. The phase allows you to determine the displacement of a point at a certain point in time.

Period of harmonic oscillationT. taking into account the fact that during the period of the oscillation the phase will change to. can be calculated by the formula:

The frequency of free oscillations is:

The speed of a point with harmonic oscillations is found as the first derivative of the time shift:

Acceleration of the point with harmonic oscillations is found as the second derivative of the time shift:

If the body is removed from the equilibrium state in the oscillatory system and released, it will perform so-called free oscillations, which are always damped.

To study the oscillations of a different nature, instruments called "oscilloscopes" are often used. Oscilloscope (from the lat.oscillo  - "hesitate" and the Greek.graph  - "write") - an instrument for observing oscillations and recording them in a graphical form.

The amplitude of oscillations in real systems decreases with time, and the oscillations eventually stop, so free oscillations are always damped.

The period of oscillations does not depend on their amplitude, because in real mechanical systems there are always losses of mechanical energy.

Investigating free oscillations in the "load-spring" system, in the absence of losses of mechanical energy, one can conclude that the period of such oscillations is determined by the formula:

whereω  - cyclic frequency.

The frequency of free oscillations, respectively, is measured by the formula:

Mathematical pendulum

A pendulum considered to be a mathematical pendulum is a limb suspended from an inextensible and weightless thread. A mathematical pendulum is an abstract concept, because, firstly, there are no point bodies in nature, and secondly, there are no absolutely inextensible and weightless threads. However, with a certain approximation, a pendulum suspended on a thread can be considered a mathematical pendulum. When the ball is in a state of equilibrium, then it acts on the force of gravity and the elasticity of the thread, which counterbalance each other, in other words, the resultant of these forces is zero.

The oscillation period of a mathematical pendulum can be calculated from the formula:

where the cyclic frequency of free oscillationsω 2 = l / g,  al  Is the length of the thread.

According to the formula, we can conclude that the period of oscillations of a mathematical pendulum does not depend on the mass of the body, but is determined only by the length of the suspension and by the acceleration of free fall.

Spring pendulum

Another example of harmonic free vibrations are oscillations of a body on a spring. In equilibrium the spring is not yet deformed, the elastic force on the body is not valid. The friction force between the body and the support is also zero. The force of attraction is balanced by the reaction force bearing. If you bring the body out of balance, moving it along the axis OX at a distance x = ± A. and then release, the pendulum will freely oscillate under the force of elasticity, and free oscillations of the pendulum will occur according to the lawx = Asinwt.

The period of free oscillations of the pendulum on the spring is:

where the cyclic frequency of the oscillationsω 2 = k / m, k  - spring stiffness,m  - body mass.

As can be seen from the formula, the period and frequency of oscillations of the spring pendulum do not depend on the acceleration of free fall, but are determined only by the mass of the suspended body and the spring stiffness.

Electrical oscillations in the circuit

An electrical circuit in which free electromagnetic oscillations are possible is called an oscillatory circuit. It consists of a capacitor C, a coil with inductance L and a resistor with resistance R (in the real technical circuit, the resistance of the coil and connecting conductors plays the role of the resistor).

Ohm's law for a closed circuit that does not contain an external current source, and in which free electromagnetic oscillations occur, is written in this form:

whereU = q / C  - voltage on the capacitor, q - charge of the capacitor,J = dq / dt  - current in the circuit.

Free oscillations in the contour are harmonic, therefore they change according to the following law: