In the equation of harmonic vibration, the quantity is standing. Equation of harmonic vibrations. Fundamentals of Maxwell's theory for the electromagnetic field

We examined several physically completely different systems, and made sure that the equations of motion are reduced to the same form

Differences between physical systems appear only in different definition quantities and in various physical sense variable x: this can be a coordinate, angle, charge, current, etc. Note that in this case, as follows from the very structure of equation (1.18), the quantity always has the dimension of inverse time.

Equation (1.18) describes the so-called harmonic vibrations.

The harmonic vibration equation (1.18) is linear differential equation second order (since it contains the second derivative of the variable x). The linearity of the equation means that

if some function x(t) is a solution to this equation, then the function Cx(t) will also be his solution ( C– arbitrary constant);

if functions x 1 (t) And x 2(t) are solutions to this equation, then their sum x 1 (t) + x 2 (t) will also be a solution to the same equation.

A mathematical theorem has also been proven, according to which a second-order equation has two independent solutions. All other solutions, according to the properties of linearity, can be obtained as their linear combinations. It is easy to verify by direct differentiation that the independent functions and satisfy equation (1.18). Means, common decision this equation looks like:

Where C 1,C 2- arbitrary constants. This solution can be presented in another form. Let's enter the value

and determine the angle by the relations:

Then the general solution (1.19) is written as

According to trigonometry formulas, the expression in brackets is equal to

We finally come to general solution of the harmonic vibration equation as:

Non-negative value A called vibration amplitude, - initial phase of oscillation. The entire cosine argument - the combination - is called oscillation phase.

Expressions (1.19) and (1.23) are completely equivalent, so we can use any of them, based on considerations of simplicity. Both solutions are periodic functions time. Indeed, sine and cosine are periodic with a period . Therefore, various states of a system performing harmonic oscillations are repeated after a period of time t*, during which the oscillation phase receives an increment that is a multiple of :

It follows that

Least of these times

called period of oscillation (Fig. 1.8), and - his circular (cyclic) frequency.

Rice. 1.8.

They also use frequency fluctuations

Accordingly, the circular frequency is equal to the number of oscillations per seconds

So, if the system at time t characterized by the value of the variable x(t), then the variable will have the same value after a period of time (Fig. 1.9), that is

The same meaning will naturally be repeated over time 2T, ZT etc.

Rice. 1.9. Oscillation period

The general solution includes two arbitrary constants ( C 1, C 2 or A, a), the values ​​of which must be determined by two initial conditions. Usually (though not necessarily) their role is played by the initial values ​​of the variable x(0) and its derivative.

Let's give an example. Let the solution (1.19) of the equation of harmonic oscillations describe the motion of a spring pendulum. The values ​​of arbitrary constants depend on the way in which we brought the pendulum out of equilibrium. For example, we pulled the spring to a distance and released the ball without initial speed. In this case

Substituting t = 0 in (1.19), we find the value of the constant C 2

The solution thus looks like:

We find the speed of the load by differentiation with respect to time

Substituting here t = 0, find the constant C 1:

Finally

Comparing with (1.23), we find that is the amplitude of the oscillations, and its initial phase is zero: .

Let us now unbalance the pendulum in another way. Let's hit the load so that it acquires an initial speed, but practically does not move during the impact. We then have other initial conditions:

our solution looks like

The speed of the load will change according to the law:

Let's substitute here:

Oscillations These are processes in which a system repeatedly passes through an equilibrium position with greater or lesser periodicity.

Oscillation classification:

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

b) according to form (simple = harmonic; complex, being the sum of simple harmonic vibrations);

V) by degree of frequency = periodic (system characteristics repeat after a strictly defined period of time (period)) and aperiodic;

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

G) on energy – free (one-time input of energy into the system from the outside = one-time external influence); forced (multiple (periodic) input of energy into the system from the outside = periodic external influence); self-oscillations (undamped oscillations that arise due to the system’s ability to regulate the supply of energy from a constant source).

Conditions for the occurrence of oscillations.

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

b) The presence of an external source of energy that is capable of bringing the system out of equilibrium at least once;

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

d) The presence of inertia (inertial element) in the system.

As an illustrative example, consider the movement of a mathematical pendulum. Mathematical pendulum called a small body 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 plumb, the force of gravity is balanced by the tension force of the thread
. When the pendulum deviates from the equilibrium position by a certain angle α a tangential component of gravity appears F=- mg sinα. The minus sign in this formula means that the tangential component is directed in the direction opposite to the deflection of the pendulum. She is a restorative force. At small angles α (about 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 deviates, it rises to a certain height, i.e. he is given a certain reserve of potential energy ( E sweat = mgh). When the pendulum moves to the equilibrium position, potential energy transforms into kinetic energy. At the moment when the pendulum passes the equilibrium position, the potential energy is 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. If there is no friction in the system, the pendulum's oscillations will continue indefinitely.

The equation of harmonic oscillation 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 (A) – amplitude of oscillations, that is, the modulus of maximum displacement,

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

t- time.

The quantity under the cosine sign φ = ω 0 t + φ 0 called phase harmonic vibration. Phase determines the displacement 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 through which certain states of the oscillatory system are repeated is called period of oscillation T.

The physical quantity inverse to the period of oscillation is called oscillation frequency:
. Oscillation frequency ν shows how many oscillations occur per unit time. Frequency unit – hertz (Hz) – one vibration 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 vector diagram method.

The vector diagram method allows you to clearly present all the parameters included in the equation of harmonic oscillations. Indeed, if the amplitude vector A located at an angle φ to the axis X, then its projection onto the axis X will be equal to: x = Acos(φ ) . Corner φ and there is the initial phase. If the vector A bring into rotation with angular velocityω 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) = Acos(ω 0 t+ φ) . The time it takes for the amplitude vector to make one full revolution is equal to the period T harmonic vibrations. The number of vector revolutions per second is equal to the oscillation frequency ν .

Harmonic oscillations are oscillations in which a physical quantity changes over time according to a harmonic (sine, cosine) law. The harmonic vibration 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 - vibration amplitude, dimension A coincides with dimension 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 is a constant approximately equal to 3.14, 2π=6.28

The oscillation period, frequency in hertz and cyclic frequency are related by relations.
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 the oscillations, it is enough to use one of these parameters.

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

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

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

Equation of harmonic vibration

The equation of harmonic oscillation establishes the dependence of the body coordinates on time

The cosine graph at the initial moment has a maximum value, and the sine graph has a zero value at the initial moment. If we begin to examine the oscillation from the equilibrium position, then the oscillation will repeat a sinusoid. If we begin to consider the oscillation from the position of maximum deviation, then the oscillation will be described by a cosine. Or such an oscillation can be described by the sine formula with an initial phase.

Change in speed and acceleration during harmonic oscillation

Not only the coordinate of the body changes over time according to the law of sine or cosine. But quantities such as force, speed and acceleration also change similarly. The force and acceleration are maximum when the oscillating body is at the extreme positions where the displacement is maximum, and are zero when the body passes through the equilibrium position. The speed, on the contrary, in extreme positions is zero, and when the body passes through the equilibrium position, it reaches its maximum value.

If the oscillation is described by the law of cosine

If the oscillation is described according to the sine law

Maximum speed and acceleration values

Having analyzed the equations of dependence v(t) and a(t), we can guess that speed and acceleration take maximum values ​​in the case when the trigonometric factor is equal to 1 or -1. Determined by the formula

« Physics - 11th grade"

Acceleration is the second derivative of a coordinate with respect to time.

The instantaneous speed of a point is the derivative of the point's coordinates with respect to time.
The acceleration of a point is the derivative of its speed with respect to time, or the second derivative of the coordinate with respect to time.
Therefore, the equation of motion of a pendulum can be written as follows:

where x" is the second derivative of the coordinate with respect to time.

For free oscillations, the coordinate X changes with time so that the second derivative of the coordinate with respect to time is directly proportional to the coordinate itself and is opposite in sign.

Harmonic vibrations

From mathematics: the second derivatives of sine and cosine by their argument are proportional to the functions themselves, taken with the opposite sign, and no other functions have this property.
That's why:
The coordinate of a body performing free oscillations changes over time according to the law of sine or cosine.

Periodic changes physical quantity depending on time, occurring according to the law of sine or cosine are called harmonic vibrations.

Oscillation amplitude

Amplitude harmonic oscillations is the modulus of the greatest displacement of a body from its equilibrium position.

The amplitude is determined by the initial conditions, or more precisely by the energy imparted to the body.

The graph of body coordinates versus time is a cosine wave.

x = x m cos ω 0 t

Then the equation of motion describing the free oscillations of the pendulum:

Period and frequency of harmonic oscillations.

When oscillating, the body's movements are periodically repeated.
The time period T during which the system completes one complete cycle of oscillations is called period of oscillation.

Oscillation frequency is the number of oscillations per unit time.
If one oscillation occurs in time T, then the number of oscillations per second

IN International system units (SI) unit of frequency is called hertz(Hz) in honor German physicist G. Hertz.

The number of oscillations in 2π s is equal to:

The quantity ω 0 is the cyclic (or circular) oscillation frequency.
After a period of time equal to one period, the oscillations are repeated.

The frequency of free oscillations is called natural frequency oscillatory system.
Often, for short, the cyclic frequency is simply called the frequency.

Dependence of the frequency and period of free oscillations on the properties of the system.

1.for spring pendulum

The natural frequency of oscillation of a spring pendulum is equal to:

The greater the spring stiffness k, the greater it is, and the less, the greater the body mass m.
A stiff spring imparts greater acceleration to the body, changes the speed of the body faster, and the more massive the body, the slower it changes speed under the influence of force.

The oscillation period is equal to:

The period of oscillation of a spring pendulum does not depend on the amplitude of the oscillations.

2.for thread pendulum

The natural frequency of oscillation of a mathematical pendulum at small angles of deviation of the thread from the vertical depends on the length of the pendulum and the acceleration of gravity:

The period of these oscillations is equal to

The period of oscillation of a thread pendulum at small angles of deflection does not depend on the amplitude of oscillations.

The period of oscillation increases with increasing length of the pendulum. It does not depend on the mass of the pendulum.

The smaller g, the longer the period of oscillation of the pendulum and, therefore, the slower the pendulum clock runs. Thus, a clock with a pendulum in the form of a weight on a rod will fall behind by almost 3 s per day if it is lifted from the basement to the top floor of Moscow University (height 200 m). And this is only due to the decrease in the acceleration of free fall with height.