We considered 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: it 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 equation harmonic vibrations(1.18) is linear differential equation second order (because it contains the second derivative of the variable x). The linearity of the equation means that
if any function x(t) is a solution to this equation, then the function Cx(t) will also be his solution ( C is an arbitrary constant);
if functions x 1 (t) and x 2 (t) are solutions of this equation, then their sum x 1 (t) + x 2 (t) will also be a solution to the same equation.
A mathematical theorem is also proved, 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 check by direct differentiation that the independent functions and satisfy equation (1.18). So the general solution to this equation is:
where C1,C2 are arbitrary constants. This solution can also be presented in another form. We introduce the quantity
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and define the angle as:
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Then the general solution (1.19) is written as
According to the trigonometry formulas, the expression in brackets is
We finally arrive at general solution of the equation of harmonic oscillations as:
Non-negative value A called oscillation amplitude, - the initial phase of the oscillation. The whole cosine argument - the combination - is called oscillation phase.
Expressions (1.19) and (1.23) are perfectly equivalent, so we can use either of them for reasons of simplicity. Both solutions are periodic functions of time. Indeed, the sine and cosine are periodic with a period . Therefore, various states of a system that performs harmonic oscillations are repeated after a period of time t*, for which the oscillation phase receives an increment that is a multiple of :
Hence it follows that
The least of these times
called period of oscillation (Fig. 1.8), a - his circular (cyclic) frequency.
Rice. 1.8.
They also use frequency hesitation
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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 same value, the variable will have after a period of time (Fig. 1.9), that is
The same value, of course, will be repeated after a while. 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 should 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 take 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 velocity. In this case
Substituting t = 0 in (1.19), we find the value of the constant From 2
The solution thus looks like:
The speed of the load is found by differentiation with respect to time
Substituting here t = 0, find the constant From 1:
Finally
Comparing with (1.23), we find that is the oscillation amplitude, and its initial phase is equal to zero: .
We now bring the pendulum out of equilibrium 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 put it here:
The simplest type of vibrations are harmonic vibrations- fluctuations in which the displacement of the oscillating point from the equilibrium position changes over time according to the sine or cosine law.
So, with a uniform rotation of the ball around the circumference, its projection (shadow in parallel rays of light) performs a harmonic oscillatory motion on a vertical screen (Fig. 1).
The displacement from the equilibrium position during harmonic vibrations is described by an equation (it is called the kinematic law of harmonic motion) of the form:
where x - displacement - a value characterizing the position of the oscillating point at time t relative to the equilibrium position and measured by the distance from the equilibrium position to the position of the point at a given time; A - oscillation amplitude - the maximum displacement of the body from the equilibrium position; T - oscillation period - the time of one complete oscillation; those. smallest span time after which the values of physical quantities characterizing the oscillation are repeated; - initial phase;
The phase of the oscillation at time t. The oscillation phase is the argument periodic function, which, for a given oscillation amplitude, determines the state oscillatory system(displacement, velocity, acceleration) of the body at any given time.
If at the initial moment of time the oscillating point is maximally displaced from the equilibrium position, then , and the displacement of the point from the equilibrium position changes according to the law
If the oscillating point at is in a position of stable equilibrium, then the displacement of the point from the equilibrium position changes according to the law
The value V, the reciprocal of the period and equal to the number of complete oscillations performed in 1 s, is called the oscillation frequency:
If in time t the body makes N complete oscillations, then
the value 
, showing how many oscillations the body makes in s, is called cyclic (circular) frequency.
The kinematic law of harmonic motion can be written as:
Graphically, the dependence of the displacement of an oscillating point on time is represented by a cosine (or sinusoid).
Figure 2, a shows the time dependence of the displacement of the oscillating point from the equilibrium position for the case .
Let us find out how the speed of an oscillating point changes with time. To do this, we find the time derivative of this expression:
where is the amplitude of the velocity projection on the x-axis.
This formula shows that during harmonic oscillations, the projection of the body velocity on the x axis also changes according to the harmonic law with the same frequency, with a different amplitude, and is ahead of the mixing phase by (Fig. 2, b).
To find out the dependence of acceleration, we find the time derivative of the velocity projection:
where is the amplitude of the acceleration projection on the x-axis.
For harmonic oscillations, the acceleration projection leads the phase shift by k (Fig. 2, c).
§ 6. MECHANICAL OSCILLATIONSBasic formulas
Harmonic vibration equation
where X - displacement of the oscillating point from the equilibrium position; t- time; BUT,ω, φ- respectively amplitude, angular frequency, initial phase of oscillations; - phase of oscillations at the moment t.
Angular oscillation frequency
where ν and T are the frequency and period of oscillations.
The speed of a point making harmonic oscillations,
Harmonic acceleration
Amplitude BUT the resulting oscillation obtained by adding two oscillations with the same frequencies occurring along one straight line is determined by the formula
where a 1 and BUT 2 - amplitudes of oscillation components; φ 1 and φ 2 - their initial phases.
The initial phase φ of the resulting oscillation can be found from the formula
The frequency of beats arising from the addition of two oscillations occurring along the same straight line with different, but close in value, frequencies ν 1 and ν 2,
The equation of the trajectory of a point participating in two mutually perpendicular oscillations with amplitudes A 1 and A 2 and initial phases φ 1 and φ 2,
If the initial phases φ 1 and φ 2 of the oscillation components are the same, then the trajectory equation takes the form
i.e., the point moves in a straight line.
In the event that the phase difference , the equation takes the form
i.e., the point moves along an ellipse.
Differential equation of harmonic vibrations of a material point
, or 
, where m is the mass of the point; k-
coefficient of quasi-elastic force ( k=tω 2).
The total energy of a material point making harmonic oscillations,
The period of oscillation of a body suspended on a spring (spring pendulum),
where m- body mass; k- spring stiffness. The formula is valid for elastic vibrations within the limits in which Hooke's law is fulfilled (with a small mass of the spring in comparison with the mass of the body).
The period of oscillation of a mathematical pendulum
where l- pendulum length; g- acceleration of gravity. Oscillation period of a physical pendulum
where J- the moment of inertia of the oscillating body about the axis
fluctuations; a- distance of the center of mass of the pendulum from the axis of oscillation;
Reduced length of a physical pendulum.
The above formulas are exact for the case of infinitely small amplitudes. For finite amplitudes, these formulas give only approximate results. At amplitudes no greater than the error in the value of the period does not exceed 1%.
The period of torsional vibrations of a body suspended on an elastic thread,
where J- the moment of inertia of the body about the axis coinciding with the elastic thread; k- the stiffness of an elastic thread, equal to the ratio of the elastic moment that occurs when the thread is twisted to the angle by which the thread is twisted.
Differential equation of damped oscillations 
, or ,
where r- coefficient of resistance; δ - damping coefficient: ;ω 0 - natural angular frequency of vibrations *
Damped oscillation equation
where A(t)- amplitude of damped oscillations at the moment t;ω is their angular frequency.
Angular frequency of damped oscillations
О Dependence of the amplitude of damped oscillations on time
I
where BUT 0 - amplitude of oscillations at the moment t=0.
Logarithmic oscillation decrement
where A(t) and A(t+T)- the amplitudes of two successive oscillations separated in time from each other by a period.
Differential equation of forced vibrations
where is an external periodic force acting on an oscillating material point and causing forced oscillations; F 0 - its amplitude value;
Amplitude of forced vibrations
Resonant frequency and resonant amplitude 
and
Examples of problem solving
Example 1 The point oscillates according to the law x(t)=
,
where A=2 see Determine initial phase φ if
x(0)=cm and X , (0)<0. Построить векторную диаграмму для мо- мента t=0.
Solution. We use the equation of motion and express the displacement at the moment t=0 through initial phase:
From here we find the initial phase:
* In the previously given formulas for harmonic oscillations, the same value was simply denoted by ω (without the index 0).
Substitute the given values into this expression x(0) and BUT:φ=
= 
. The value of the argument is satisfied by two angle values:
In order to decide which of these values of the angle φ also satisfies the condition , we first find:
Substituting into this expression the value t=0 and alternately the values of the initial phases and, we find
T 
ok as always A>0 and ω>0, then only the first value of the initial phase satisfies the condition. Thus, the desired initial phase
Based on the found value of φ, we will construct a vector diagram (Fig. 6.1). Example 2 Material point with mass t\u003d 5 g performs harmonic oscillations with a frequency ν =0.5 Hz. Oscillation amplitude A=3 cm. Determine: 1) speed υ points at the time when the offset x== 1.5 cm; 2) the maximum force F max acting on the point; 3) Fig. 6.1 total energy E oscillating point.
and we obtain the velocity formula by taking the first time derivative of the displacement:
To express the speed in terms of displacement, time must be excluded from formulas (1) and (2). To do this, we square both equations, divide the first by BUT 2 , the second on A 2 ω 2 and add:
, or 
Solving the last equation for υ , find
Having performed calculations according to this formula, we obtain
The plus sign corresponds to the case when the direction of the velocity coincides with the positive direction of the axis X, minus sign - when the direction of speed coincides with the negative direction of the axis X.
Displacement during harmonic oscillation, in addition to equation (1), can also be determined by the equation
Repeating the same solution with this equation, we get the same answer.
2. The force acting on a point, we find according to Newton's second law:
where a - acceleration of a point, which we get by taking the time derivative of the speed:
Substituting the acceleration expression into formula (3), we obtain
Hence the maximum value of the force
Substituting into this equation the values of π, ν, t and A, find
3. The total energy of an oscillating point is the sum of the kinetic and potential energies calculated for any moment of time.
The easiest way to calculate the total energy is at the moment when the kinetic energy reaches its maximum value. At this point, the potential energy is zero. So the total energy E oscillating point is equal to the maximum kinetic energy
We determine the maximum speed from formula (2), setting: 
. Substituting the speed expression into formula (4), we find
Substituting the values of the quantities into this formula and performing calculations, we obtain
or mcJ.
Example 3 At the ends of a thin rod l= 1 m and weight m 3 =400 g small balls are reinforced with masses m 1=200 g and m 2 =300g. The rod oscillates about the horizontal axis, perpendicular to
dicular rod and passing through its middle (point O in Fig. 6.2). Define period T vibrations made by the rod.
Solution. The oscillation period of a physical pendulum, which is a rod with balls, is determined by the relation
where J- t - its mass; l FROM - distance from the center of mass of the pendulum to the axis.
The moment of inertia of this pendulum is equal to the sum moments of inertia of the balls J 1 and J 2 and rod J 3:
Taking the balls as material points, we express the moments of their inertia:
Since the axis passes through the middle of the rod, then its moment of inertia about this axis J 3 = =. Substituting the resulting expressions J 1 , J 2 and J 3 into formula (2), we find the total moment of inertia of the physical pendulum:
Performing calculations using this formula, we find
Rice. 6.2 The mass of the pendulum consists of the masses of the balls and the mass of the rod:
Distance l FROM we find the center of mass of the pendulum from the axis of oscillation, based on the following considerations. If the axis X direct along the rod and align the origin with the point O, then the desired distance l is equal to the coordinate of the center of mass of the pendulum, i.e.
Substituting the values of quantities m 1 , m 2 , m, l and performing calculations, we find
Having made calculations according to formula (1), we obtain the oscillation period of a physical pendulum:
Example 4 The physical pendulum is a rod with a length l= 1 m and weight 3 t 1 With attached to one of its ends by a hoop with a diameter and mass t 1 . Horizontal axis Oz
pendulum passes through the middle of the rod perpendicular to it (Fig. 6.3). Define period T oscillations of such a pendulum.
Solution. The oscillation period of a physical pendulum is determined by the formula
(1)
where J- the moment of inertia of the pendulum about the axis of oscillation; t - its mass; l C - the distance from the center of mass of the pendulum to the axis of oscillation.
The moment of inertia of the pendulum is equal to the sum of the moments of inertia of the rod J 1 and hoop J 2:
(2).
The moment of inertia of the rod relative to the axis perpendicular to the rod and passing through its center of mass is determined by the formula 
. In this case t= 3t 1 and
We find the moment of inertia of the hoop using the Steiner theorem 
,where J-
moment of inertia about an arbitrary axis; J 0
-
moment of inertia about the axis passing through the center of mass parallel to the given axis; a - the distance between the specified axes. Applying this formula to the hoop, we get
Substituting expressions J 1 and J 2 into formula (2), we find the moment of inertia of the pendulum about the axis of rotation:
Distance l FROM from the axis of the pendulum to its center of mass is
Substituting into formula (1) the expressions J, l c and the mass of the pendulum , we find the period of its oscillation:
After calculating by this formula, we get T\u003d 2.17 s.
Example 5 Two oscillations of the same direction are added, expressed by the equations ; X 2 = =, where BUT 1 = 1 cm, A 2 \u003d 2 cm, s, s, ω \u003d \u003d. 1. Determine the initial phases φ 1 and φ 2 of the components of the oscillation
bani. 2. Find the amplitude BUT and the initial phase φ of the resulting oscillation. Write the equation for the resulting oscillation.
Solution. 1. The equation of harmonic oscillation has the form
Let's transform the equations given in the condition of the problem to the same form:
From the comparison of expressions (2) with equality (1), we find the initial phases of the first and second oscillations:
Glad and 
glad.
2. To determine the amplitude BUT of the resulting fluctuation, it is convenient to use the vector diagram presented in rice. 6.4. According to the cosine theorem, we get
where is the phase difference of the oscillation components. Since 
, then, substituting the found values φ 2 and φ 1 we get rad.
Substitute the values BUT 1 , BUT 2 and into formula (3) and perform the calculations:
A= 2.65 cm.
The tangent of the initial phase φ of the resulting oscillation can be determined directly from Figs. 6.4: 
, whence the initial phase
Harmonic vibrations are vibrations in which 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 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. 
Generalized harmonic oscillation in differential form:
In order for free vibrations to occur according to the harmonic law, it is necessary that the force tending to return the body to the equilibrium position is proportional to the displacement of the body from the equilibrium position and is directed in the direction opposite to the displacement:
where is the mass of the oscillating body.
A physical system in which harmonic oscillations can exist is called harmonic oscillator, and the equation of harmonic oscillations is harmonic oscillator equation.
1.2. Addition of vibrations
It is not uncommon for a system to simultaneously participate in two or more independent oscillations. In these cases, a complex oscillatory motion is formed, which is created by superimposing (adding) vibrations to each other. Obviously, the cases of summation of oscillations can be very diverse. They depend not only on the number of added oscillations, but also on the oscillation parameters, on their frequencies, phases, amplitudes, directions. It is not possible to review all the possible variety of cases of summation of oscillations, therefore we will confine ourselves to considering only individual examples.
Addition of harmonic oscillations directed along one straight line
Consider the addition of equally directed oscillations of the same period, but differing in the initial phase and amplitude. The equations of the added oscillations are given in the following form:
where and are displacements; and are the amplitudes; and are the initial phases of the added oscillations.
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It is convenient to determine the amplitude of the resulting oscillation using a vector diagram (Fig. 2), on which the vectors of amplitudes and summed oscillations are plotted at angles and to the axis, and the amplitude vector of the total oscillation is obtained by the parallelogram rule.
If we uniformly rotate the system of vectors (parallelogram) and project the vectors onto the axis , then their projections will make harmonic oscillations in accordance with given equations. The mutual arrangement of the vectors , and at the same time remains unchanged, so the oscillatory motion of the projection of the resulting vector will also be harmonic.
This implies the conclusion that the total movement is a harmonic oscillation having a given cyclic frequency. We define the amplitude modulus BUT resulting fluctuation. Into an angle (from the equality of opposite angles of a parallelogram).
Consequently,
from here: .
According to the cosine theorem,
The initial phase of the resulting oscillation is determined from:
Relationships for phase and amplitude make it possible to find the amplitude and initial phase of the resulting motion and to compose its equation: .
beats
Let us consider the case when the frequencies of two added oscillations differ little from each other , and let the amplitudes be the same and the initial phases , i.e.
We add these equations analytically:
Let's transform
| Rice. 3. |
Beats can be observed when two tuning forks sound, if the frequencies and vibrations are close to each other.
Addition of mutually perpendicular vibrations
Let material point simultaneously participates in two harmonic oscillations occurring with the same periods in two mutually perpendicular directions. These directions can be associated rectangular system coordinates , placing the origin at the point's equilibrium position. Let us denote the displacement of the point C along the axes and , respectively, through and . (Fig. 4).
Let's consider several special cases.
1). The initial phases of oscillations are the same
Let us choose the moment of the beginning of the countdown in such a way that the initial phases of both oscillations are equal to zero. Then the displacements along the axes and can be expressed by the equations:
Dividing these equalities term by term, we obtain the equations for the trajectory of point C:
or .
Consequently, as a result of the addition of two mutually perpendicular oscillations, point C oscillates along a straight line segment passing through the origin (Fig. 4).
| Rice. four. |
The oscillation equations in this case have the form:
Point trajectory equation:
Consequently, point C oscillates along a straight line segment passing through the origin, but lying in other quadrants than in the first case. Amplitude BUT resulting fluctuations in both considered cases is equal to:
3). The initial phase difference is .
The oscillation equations have the form:
Divide the first equation by and the second by:
We square both equalities and add them. We obtain the following equation for the trajectory of the resulting movement of the oscillating point:
The oscillating point C moves along an ellipse with semi-axes and . With equal amplitudes, the trajectory of the total motion will be a circle. In the general case, for , but a multiple, i.e., , when adding mutually perpendicular oscillations, the oscillating point moves along curves called Lissajous figures.
Lissajous figures
Figures of Lissajous- closed trajectories drawn by a point that simultaneously performs two harmonic oscillations in two mutually perpendicular directions.
First studied by the French scientist Jules Antoine Lissajous. The shape of the figures depends on the relationship between the periods (frequencies), phases and amplitudes of both oscillations(Fig. 5).
| Fig.5. |
In the simplest case of equality of both periods, the figures are ellipses, which, with a phase difference or degenerate into line segments, and with a phase difference and equality of amplitudes turn into a circle. If the periods of both oscillations do not exactly coincide, then the phase difference changes all the time, as a result of which the ellipse is deformed all the time. When significantly different periods Lissajous figures are not observed. However, if the periods are related as integers, then after a time interval equal to the least multiple of both periods, the moving point returns to the same position again - Lissajous figures of a more complex form are obtained.
The Lissajous figures fit into a rectangle whose center coincides with the origin of coordinates, and the sides are parallel to the coordinate axes and located on both sides of them at distances equal to the oscillation amplitudes (Fig. 6).
