Harmonic oscillation is a phenomenon of periodic change of some quantity, in which the dependence on the argument has the character of a sine or cosine function. For example, a quantity that varies in time as follows harmonically fluctuates:

where x is the value of the changing quantity, t is time, the remaining parameters are constant: A is the amplitude of the oscillations, ω is the cyclic frequency of the oscillations, is the full phase of the oscillations, is the initial phase of the oscillations.

Generalized harmonic oscillation in differential form

(Any non-trivial solution of this differential equation is a harmonic oscillation with a cyclic frequency)

Types of vibrations

Free oscillations are performed under the action of the internal forces of the system after the system has been taken out of equilibrium. For free oscillations to be harmonic, it is necessary that the oscillatory system be linear (described by linear equations of motion), and there should be no energy dissipation in it (the latter would cause damping).

Forced oscillations are performed under the influence of an external periodic force. For them to be harmonic, it is sufficient that the oscillatory system be linear (described by linear equations of motion), and the external force itself changes over time as a harmonic oscillation (that is, that the time dependence of this force is sinusoidal).

Harmonic vibration equation

gives the dependence of the fluctuating value S on time t; this is the equation of free harmonic oscillations in explicit form. However, the equation of oscillations is usually understood as a different record of this equation, in differential form. For definiteness, we take equation (1) in the form

Differentiate it twice with respect to time:

It can be seen that the following relation holds:

which is called the equation of free harmonic oscillations (in differential form). Equation (1) is a solution to differential equation (2). Since equation (2) is a second-order differential equation, two initial conditions are necessary to obtain a complete solution (that is, to determine the constants A and   included in equation (1); for example, the position and speed of an oscillatory system at t = 0.

A mathematical pendulum is an oscillator, which is a mechanical system consisting of a material point located on a weightless inextensible thread or on a weightless rod in a uniform field of gravitational forces. The period of small eigenoscillations of a mathematical pendulum of length l, motionlessly suspended in a uniform gravitational field with free fall acceleration g, is equal to

and does not depend on the amplitude and mass of the pendulum.

A physical pendulum is an oscillator, which is a rigid body that oscillates in the field of any forces about a point that is not the center of mass of this body, or a fixed axis perpendicular to the direction of the forces and not passing through the center of mass of this body.

Harmonic oscillations are oscillations in which a physical quantity changes over time according to a harmonic (sinusoidal, cosine) law. The harmonic oscillation equation can be written as follows:
X(t) = A∙cos(ω t+φ )
or
X(t) = A∙sin(ω t+φ )

X - deviation from the equilibrium position at time t
A - oscillation amplitude, the dimension of A is the same as the dimension of X
ω - cyclic frequency, rad/s (radians per second)
φ - initial phase, rad
t - time, s
T - oscillation period, s
f - oscillation frequency, Hz (Hertz)
π - constant approximately equal to 3.14, 2π=6.28

The oscillation period, frequency in hertz and cyclic frequency are related by relationships.
ω=2πf , T=2π/ω , f=1/T , f=ω/2π
To remember these relationships, you need to understand the following.
Each of the parameters ω, f, T uniquely determines the others. To describe oscillations, it is sufficient to use one of these parameters.

Period T is the time of one fluctuation, it is convenient to use it for plotting fluctuation graphs.
Cyclic frequency ω - used to write the equations of oscillations, allows you to carry out mathematical calculations.
Frequency f - the number of oscillations per unit of time, is used everywhere. In hertz, we measure the frequency to which radios are tuned, as well as the range of mobile phones. The frequency of vibrations of strings is measured in hertz when tuning musical instruments.

The expression (ωt+φ) is called the oscillation phase, and the value of φ is called the initial phase, since it is equal to the oscillation phase at the time t=0.

The sine and cosine functions describe the ratios of the sides in a right triangle. Therefore, many do not understand how these functions are related to harmonic oscillations. This relationship is demonstrated by a uniformly rotating vector. The projection of a uniformly rotating vector makes harmonic oscillations.
The picture below shows an example of three harmonic oscillations. Equal in frequency, but different in phase and amplitude.

fluctuations are called movements or processes that are characterized by a certain repetition in time. Oscillatory processes are widespread in nature and technology, for example, the swing of a clock pendulum, alternating electric current, etc. When the pendulum oscillates, the coordinate of its center of mass changes, in the case of alternating current, the voltage and current in the circuit fluctuate. The physical nature of oscillations can be different, therefore, mechanical, electromagnetic, etc. oscillations are distinguished. However, various oscillatory processes are described by the same characteristics and the same equations. From this comes the feasibility unified approach to the study of vibrations different physical nature.

The fluctuations are called free, if they are made only under the influence of internal forces acting between the elements of the system, after the system is taken out of equilibrium by external forces and left to itself. Free vibrations always damped oscillations because energy losses are inevitable in real systems. In the idealized case of a system without energy loss, free oscillations (continuing as long as desired) are called own.

The simplest type of free undamped oscillations are harmonic oscillations - fluctuations in which the fluctuating value changes with time according to the sine (cosine) law. Oscillations encountered in nature and technology often have a character close to harmonic.

Harmonic vibrations are described by an equation called the equation of harmonic vibrations:

where BUT- amplitude of fluctuations, the maximum value of the fluctuating value X; - circular (cyclic) frequency of natural oscillations; - the initial phase of the oscillation at a moment of time t= 0; - the phase of the oscillation at the moment of time t. The phase of the oscillation determines the value of the oscillating quantity at a given time. Since the cosine varies from +1 to -1, then X can take values ​​from + A before - BUT.

Time T, for which the system completes one complete oscillation, is called period of oscillation. During T oscillation phase is incremented by 2 π , i.e.

Where . (14.2)

The reciprocal of the oscillation period

i.e., the number of complete oscillations per unit time is called the oscillation frequency. Comparing (14.2) and (14.3) we obtain

The unit of frequency is hertz (Hz): 1 Hz is the frequency at which one complete oscillation takes place in 1 s.

Systems in which free vibrations can occur are called oscillators . What properties must a system have in order for free oscillations to occur in it? The mechanical system must have position of stable equilibrium, upon exiting which appears restoring force towards equilibrium. This position corresponds, as is known, to the minimum of the potential energy of the system. Let us consider several oscillatory systems that satisfy the listed properties.

Changes in time according to a sinusoidal law:

where X- the value of the fluctuating quantity at the moment of time t, BUT- amplitude , ω - circular frequency, φ is the initial phase of oscillations, ( φt + φ ) is the total phase of oscillations . At the same time, the values BUT, ω and φ - permanent.

For mechanical vibrations with an oscillating value X are, in particular, displacement and speed, for electrical oscillations - voltage and current strength.

Harmonic vibrations occupy a special place among all types of vibrations, since this is the only type of vibration whose shape is not distorted when passing through any homogeneous medium, i.e., waves propagating from a source of harmonic vibrations will also be harmonic. Any non-harmonic vibration can be represented as a sum (integral) of various harmonic vibrations (in the form of a spectrum of harmonic vibrations).

Energy transformations during harmonic vibrations.

In the process of oscillations, there is a transition of potential energy Wp into kinetic Wk and vice versa. In the position of maximum deviation from the equilibrium position, the potential energy is maximum, the kinetic energy is zero. As we return to the equilibrium position, the speed of the oscillating body increases, and with it the kinetic energy also increases, reaching a maximum in the equilibrium position. The potential energy then drops to zero. Further-neck movement occurs with a decrease in speed, which drops to zero when the deflection reaches its second maximum. Potential energy here increases to its initial (maximum) value (in the absence of friction). Thus, the oscillations of the kinetic and potential energies occur with a double (compared to the oscillations of the pendulum itself) frequency and are in antiphase (i.e., there is a phase shift between them equal to π ). Total vibration energy W remains unchanged. For a body oscillating under the action of an elastic force, it is equal to:

where v m- the maximum speed of the body (in the equilibrium position), x m = BUT- amplitude.

Due to the presence of friction and resistance of the medium, free oscillations damp out: their energy and amplitude decrease with time. Therefore, in practice, not free, but forced oscillations are used more often.

fluctuations called such processes in which the system repeatedly passes through the equilibrium position with a greater or lesser frequency.

Oscillation classification:

a) by nature (mechanical, electromagnetic, fluctuations in concentration, temperature, etc.);

b) in form (simple = harmonic; complex, which are the sum of simple harmonic vibrations);

in) according to the degree of periodicity = periodic (characteristics of the system are repeated after a strictly defined period of time (period)) and aperiodic;

G) in relation to time (undamped = constant amplitude; damped = decreasing amplitude);

G) energy – free (single input of energy into the system from outside = single external action); forced (multiple (periodic) supply of energy to the system from the outside = periodic external influence); self-oscillations (undamped oscillations arising due to the system's ability to regulate the flow of energy from a constant source).

Conditions for the occurrence of oscillations.

a) The presence of an oscillatory system (a pendulum on a suspension, a spring pendulum, an oscillatory circuit, etc.);

b) The presence of an external source of energy that is able to bring the system out of equilibrium at least once;

c) The emergence of a quasi-elastic restoring force in the system (i.e., a force proportional to the displacement);

d) Presence of inertia (inertial element) in the system.

As an illustrative example, consider the movement of a mathematical pendulum. Mathematical pendulum called a body of small size, suspended on a thin inextensible thread, the mass of which is negligible compared to the mass of the body. In the equilibrium position, when the pendulum hangs on a plumb line, the force of gravity is balanced by the force of the thread tension
. When the pendulum deviates from the equilibrium position by a certain angle α there is a tangential component of gravity F=- mg sinα. The minus sign in this formula means that the tangential component is directed in the direction opposite to the pendulum deflection. She is a restoring force. At small angles α (of the order of 15-20 o), this force is proportional to the displacement of the pendulum, i.e. is quasi-elastic, and the oscillations of the pendulum are harmonic.

When the pendulum is deflected, it rises to a certain height, i.e. he is given a certain amount of potential energy ( E sweat = mgh). When the pendulum moves to the equilibrium position, the transition of potential energy into kinetic energy occurs. At the moment when the pendulum passes the equilibrium position, the potential energy is equal to zero, and the kinetic energy is maximum. Due to the presence of mass m(mass is a physical quantity that determines the inertial and gravitational properties of matter) the pendulum passes the equilibrium position and deviates in the opposite direction. In the absence of friction in the system, the pendulum will continue to oscillate indefinitely.

The harmonic oscillation equation has the form:

x(t) = x m cos (ω 0 t+φ 0 ),

where X- displacement of the body from the equilibrium position;

x m (BUT) is the oscillation amplitude, that is, the maximum displacement modulus,

ω 0 - cyclic (or circular) frequency of oscillations,

t- time.

The value under the cosine sign φ = ω 0 t + φ 0 called phase harmonic vibration. Phase determines the offset at a given time t. The phase is expressed in angular units (radians).

At t= 0 φ = φ 0 , That's why φ 0 called initial phase.

The period of time after which certain states of the oscillatory system are repeated is called period of oscillation T.

The physical quantity reciprocal to the period of oscillation is called oscillation frequency:
. Oscillation frequency ν shows how many oscillations are made per unit of time. Frequency unit - hertz (Hz) - unicycle per second.

Oscillation frequency ν related to cyclic frequency ω and oscillation period T ratios:
.

That is, the circular frequency is the number of complete oscillations that occur in 2π units of time.

Graphically, harmonic oscillations can be represented as a dependence X from t and the method of vector diagrams.

The method of vector diagrams allows you to visualize all the parameters included in the equation of harmonic oscillations. Indeed, if the amplitude vector BUT placed at an angle φ to the axis X, then its projection onto the axis X will be equal to: x = Acos(φ ) . Injection φ and there is an initial phase. If the vector BUT put into rotation with an angular velocity ω 0 equal to the circular frequency of oscillations, then the projection of the end of the vector will move along the axis X and take values ​​ranging from -A before +A, and the coordinate of this projection will change over time according to the law: x(t) = BUTcos(ω 0 t+ φ) . The time it takes for the amplitude vector to make one complete revolution is equal to the period T harmonic vibrations. The number of revolutions of the vector per second is equal to the oscillation frequency ν .