Harmonic vibrations
Function Graphs f(x) = sin( x) and g(x) = cos( x) on the Cartesian plane.
harmonic oscillation- fluctuations in which a physical (or any other) quantity changes over time according to a sinusoidal or cosine law. The kinematic equation of harmonic oscillations has the form
,where X- displacement (deviation) of the oscillating point from the equilibrium position at time t; BUT- oscillation amplitude, this is the value that determines the maximum deviation of the oscillating point from the equilibrium position; ω - cyclic frequency, a value showing the number of complete oscillations occurring within 2π seconds - the full phase of oscillations, - the initial phase of 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
Evolution in time of displacement, velocity and acceleration in harmonic motion
- Free vibrations are made under the action of the internal forces of the system after the system has been brought 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 vibrations 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).
Application
Harmonic vibrations stand out from all other types of vibrations for the following reasons:
see also
Notes
Literature
- Physics. Elementary textbook of physics / Ed. G. S. Lansberg. - 3rd ed. - M ., 1962. - T. 3.
- Khaykin S. E. Physical foundations of mechanics. - M., 1963.
- A. M. Afonin. Physical foundations of mechanics. - Ed. MSTU im. Bauman, 2006.
- Gorelik G.S. Vibrations and waves. Introduction to acoustics, radiophysics and optics. - M .: Fizmatlit, 1959. - 572 p.
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See what "Harmonic vibrations" are in other dictionaries:
Modern Encyclopedia
Harmonic vibrations- HARMONIC OSCILLATIONS, periodic changes in a physical quantity that occur according to the sine law. Graphically, harmonic oscillations are represented by a sinusoid curve. Harmonic oscillations are the simplest type of periodic motion, characterized by ... Illustrated Encyclopedic Dictionary
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HARMONIC OSCILLATIONS, periodic motion such as the movement of a PENDULUM, atomic vibrations, or vibrations in an electrical circuit. A body performs undamped harmonic oscillations when it oscillates along a line, moving by the same ... ... Scientific and technical encyclopedic dictionary
Oscillations, at k ryh physical. (or any other) value changes over time according to a sinusoidal law: x=Asin(wt+j), where x is the value of the oscillating value in the given. moment of time t (for mechanical G. to., for example, displacement or speed, for ... ... Physical Encyclopedia
harmonic vibrations- Mechanical vibrations, in which the generalized coordinate and (or) the generalized speed change in proportion to the sine with an argument linearly dependent on time. [Collection of recommended terms. Issue 106. Mechanical vibrations. Academy of Sciences ... Technical Translator's Handbook
Oscillations, at k ryh physical. (or any other) quantity changes in time according to a sinusoidal law, where x is the value of the oscillating quantity at time t (for mechanical G. to., for example, displacement and speed, for electrical voltage and current strength) ... Physical Encyclopedia
HARMONIC OSCILLATIONS- (see), in which physical. the value changes over time according to the law of sine or cosine (for example, changes (see) and speed during oscillation (see) or changes (see) and current strength with electric G. to.) ... Great Polytechnic Encyclopedia
They are characterized by a change in the oscillating value x (for example, the deviation of the pendulum from the equilibrium position, the voltage in the alternating current circuit, etc.) in time t according to the law: x = Asin (?t + ?), where A is the amplitude of harmonic oscillations, ? corner… … Big Encyclopedic Dictionary
Harmonic vibrations- 19. Harmonic oscillations Oscillations in which the values of the oscillating quantity change in time according to the law Source ... Dictionary-reference book of terms of normative and technical documentation
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Harmonic Wave Equation
The harmonic oscillation equation establishes the dependence of the body coordinate on time
The cosine graph has a maximum value at the initial moment, and the sine graph has a zero value at the initial moment. If we begin to investigate the oscillation from the equilibrium position, then the oscillation will repeat the sinusoid. If we begin to consider the oscillation from the position of the maximum deviation, then the oscillation will describe the 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 with time according to the law of sine or cosine. But such quantities as force, speed and acceleration also change in a similar way. The force and acceleration are maximum when the oscillating body is in the extreme positions where the displacement is maximum, and are equal to zero when the body passes through the equilibrium position. The speed, on the contrary, in the extreme positions is equal to zero, and when the body passes the equilibrium position, it reaches its maximum value.
If the oscillation is described according to the law of cosine
If the oscillation is described according to the sine law
Maximum speed and acceleration values
After analyzing the equations of dependence v(t) and a(t), one can guess that the maximum values of speed and acceleration are taken when the trigonometric factor is equal to 1 or -1. Determined by the formula
HARMONIC VIBRATION MOTION
§1 Kinematics of harmonic oscillation
Processes that repeat over time are called oscillations.
Depending on the nature of the oscillatory process and the excitation mechanism, there are: mechanical oscillations (oscillations of pendulums, strings, buildings, the earth's surface, etc.); electromagnetic oscillations (oscillations of alternating current, oscillations of vectors and in an electromagnetic wave, etc.); electromechanical vibrations (vibrations of the telephone membrane, loudspeaker diffuser, etc.); vibrations of nuclei and molecules as a result of thermal motion in atoms.
Let's consider the segment [OD] (radius-vector) making rotational motion around the point 0. The length of |OD| = A . Rotation occurs at a constant angular velocity ω 0 . Then the angle φ between the radius vector and the axisxchanges over time according to the law
where φ 0 is the angle between [OD] and the axis X at the timet= 0. Projection of the segment [OD] onto the axis X at the timet= 0
and at an arbitrary point in time
(1)
Thus, the projection of the segment [OD] on the x axis oscillates along the axis X, and these fluctuations are described by the cosine law (formula (1)).
Oscillations that are described by the cosine law
or sinus
called harmonic.
Harmonic vibrations are periodical, because the value of x (and y) is repeated at regular intervals.
If the segment [OD] is in the lowest position in the figure, i.e. dot D coincides with the point R, then its projection on the x-axis is zero. Let's call this position of the segment [OD] the position of equilibrium. Then we can say that the value X describes the displacement of an oscillating point from its equilibrium position. The maximum displacement from the equilibrium position is called amplitude fluctuations
Value
which stands under the cosine sign is called the phase. Phase determines the displacement from the equilibrium position at an arbitrary point in timet. Phase at the initial moment of timet = 0 equal to φ 0 is called the initial phase.
T
The period of time during which one complete oscillation takes place is called the period of oscillation. T. The number of oscillations per unit time is called the oscillation frequency ν.
After a period of time equal to the period T, i.e. as the cosine argument increases by ω 0 T, the movement is repeated, and the cosine takes the same value
because cosine period is equal to 2π, then, therefore, ω 0 T= 2π
thus, ω 0 is the number of oscillations of the body in 2π seconds. ω 0 - cyclic or circular frequency.
harmonic wave pattern
BUT- amplitude, T- period, X- offset,t- time.
We find the speed of the oscillating point by differentiating the displacement equation X(t) by time
those. speed vout of phase with offset X on theπ /2.
Acceleration - first derivative of velocity (second derivative of displacement) with respect to time
those. acceleration a differs from the phase shift by π.
Let's build a graph X(
t)
, y(
t)
and a(
t)
in one estimate of coordinates (for simplicity, we take φ 0 = 0 and ω 0 = 1)
Free or own oscillations that occur in a system left to itself after it has been taken out of equilibrium are called.
§ 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 oscillations 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.
Decision. 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.
Decision. The oscillation period of a physical pendulum, which is a rod with balls, is determined by the relation
where J- t - its mass; l With - distance from the center of mass of the pendulum to the axis.
The moment of inertia of this pendulum is equal to the sum of the 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 With 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.
Decision. 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 With 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.
Decision. 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 oscillations - oscillations performed according to the laws of sine and cosine. The following figure shows a graph of the change in the coordinate of a point over time according to the law of cosine.
picture
Oscillation amplitude
The amplitude of a harmonic oscillation is the largest value of the displacement of the body from the equilibrium position. The amplitude can take on different values. It will depend on how much we displace the body at the initial moment of time from the equilibrium position.
The amplitude is determined by the initial conditions, that is, the energy imparted to the body at the initial moment of time. Since the sine and cosine can take values in the range from -1 to 1, the equation must contain the factor Xm, which expresses the amplitude of the oscillations. Equation of motion for harmonic vibrations:
x = Xm*cos(ω0*t).
Oscillation period
The period of oscillation is the time it takes for one complete oscillation. The period of oscillation is denoted by the letter T. The units of the period correspond to the units of time. That is, in SI it is seconds.
Oscillation frequency - the number of oscillations per unit time. The oscillation frequency is denoted by the letter ν. The oscillation frequency can be expressed in terms of the oscillation period.
v = 1/T.
Frequency units in SI 1/sec. This unit of measurement is called Hertz. The number of oscillations in a time of 2 * pi seconds will be equal to:
ω0 = 2*pi* ν = 2*pi/T.
Oscillation frequency
This value is called the cyclic oscillation frequency. In some literature, the name circular frequency is found. The natural frequency of an oscillatory system is the frequency of free oscillations.
The frequency of natural oscillations is calculated by the formula:
The frequency of natural oscillations depends on the properties of the material and the mass of the load. The greater the stiffness of the spring, the greater the frequency of natural oscillations. The greater the mass of the load, the lower the frequency of natural oscillations.
These two conclusions are obvious. The stiffer the spring, the greater the acceleration it will impart to the body when the system is unbalanced. The greater the mass of the body, the slower this speed of this body will change.
Period of free oscillations:
T = 2*pi/ ω0 = 2*pi*√(m/k)
It is noteworthy that at small deflection angles, the period of oscillation of the body on the spring and the period of oscillation of the pendulum will not depend on the amplitude of the oscillations.
Let's write down the formulas for the period and frequency of free oscillations for a mathematical pendulum.
then the period will be
T = 2*pi*√(l/g).
This formula will be valid only for small deflection angles. From the formula we see that the period of oscillation increases with the length of the pendulum thread. The longer the length, the slower the body will oscillate.
The period of oscillation does not depend on the mass of the load. But it depends on the free fall acceleration. As g decreases, the oscillation period will increase. This property is widely used in practice. For example, to measure the exact value of free acceleration.