The equation of harmonic oscillations and its significance in the study of the nature of oscillatory processes. Mechanical vibrations Equation of harmonic vibrational motion

§ 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.

The differential equation of damped oscillations , or ,

where r- coefficient of resistance; δ - attenuation coefficient: ; ω 0 - own angular oscillation frequency *

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

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 argument value 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:

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) by setting : . Substituting the expression for the velocity 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 weight; 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

where J- the moment of inertia of the pendulum about the axis of oscillation; t - its weight; 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:

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 =2 cm, s, s, ω = =. 1. Determine the initial phases φ 1 and φ 2 of the components of the

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 ​​of 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

Substitute the values BUT 1 , BUT 2 , φ 1 , φ 2 and perform calculations:

Since the angular frequencies of the added oscillations are the same, the resulting oscillation will have the same frequency ω. This allows us to write the equation of the resulting oscillation in the form , where A\u003d 2.65 cm, rad.

Example 6 A material point participates simultaneously in two mutually perpendicular harmonic oscillations, the equations of which

where a 1 = 1 cm A 2 \u003d 2 cm, . Find the equation for the trajectory of a point. Build a trajectory in compliance with the scale and indicate the direction of movement of the point.

Decision. To find the equation for the trajectory of a point, we eliminate the time t from the given equations (1) and (2). To do this, use

let's use the formula. In this case, therefore

Since according to formula (1) , then the trajectory equation

The resulting expression is the equation of a parabola, the axis of which coincides with the axis Oh. From equations (1) and (2) it follows that the displacement of a point along the coordinate axes is limited and ranges from -1 to +1 cm along the axis Oh and from -2 to +2 cm along the axis OU.

To construct the trajectory, we find by equation (3) the values y, corresponding to a range of values X, satisfying the condition cm, and make a table:

X , CM

Having drawn the coordinate axes and selecting the scale, we will put on the plane hoy found points. Connecting them with a smooth curve, we get the trajectory of a point that oscillates in accordance with the equations of motion (1) and (2) (Fig. 6.5).

In order to indicate the direction of movement of a point, we will follow how its position changes over time. At the initial moment t=0 point coordinates are equal x(0)=1 cm and y(0)=2 cm. At the next time, for example, when t 1 \u003d l s, the coordinates of the points will change and become equal X(1)= -1cm, y( t )=0. Knowing the positions of the points at the initial and subsequent (close) moments of time, it is possible to indicate the direction of movement of the point along the trajectory. On fig. 6.5 this direction of movement is indicated by an arrow (from the point BUT to the origin). After the moment t 2 = 2 s the oscillating point will reach the point D, it will move in the opposite direction.

Tasks

Kinematics of harmonic oscillations

6.1. The point oscillation equation has the form , where ω=π s -1 , τ=0.2 s. Define period T and the initial phase φ of oscillations.

6.2. Define period T, frequency v and initial phase φ of oscillations given by the equation , where ω=2.5π s -1 , τ=0.4 s.

6.3. The point oscillates according to the law , where A x(0)=2 media ; 2) x(0) = cm and ; 3) x(0)=2cm and ; 4) x(0)= and . Build a vector diagram for the moment t=0.

6.4. The point oscillates according to the law , where A\u003d 4 cm. Determine the initial phase φ if: 1) x(0)= 2 media ; 2) x(0)= cm and ; 3) X(0)= cm and ; 4) x(0)=cm and . Build a vector diagram for the moment t=0.

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

Equation (1)

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.

fluctuations 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.

Mechanical vibrations. Oscillation parameters. Harmonic vibrations.

hesitation A process is called exactly or approximately repeating at certain intervals.

A feature of oscillations is the obligatory presence of a stable equilibrium position on the trajectory, in which the sum of all forces acting on the body is equal to zero is called the equilibrium position.

A mathematical pendulum is a material point suspended on a thin, weightless and inextensible thread.

Parameters of oscillatory motion.

1. Offset or coordinate (x) - deviation from the equilibrium position in a given

moment of time.	[x ]=m

2. Amplitude ( xm) is the maximum deviation from the equilibrium position.

[ X m ]=m

3. Oscillation period ( T) is the time it takes for one complete oscillation.

[T ]=c.

0 "style="margin-left:31.0pt;border-collapse:collapse">

Mathematical pendulum

Spring pendulum

m

https://pandia.ru/text/79/117/images/image006_26.gif" width="134" height="57 src="> Frequency (linear) ( n ) – the number of complete oscillations in 1 s.

[n]= Hz

5. Cyclic frequency ( w ) – the number of complete oscillations in 2p seconds, i.e., approximately 6.28 s.

w = 2pn ; [w]=0" style="margin-left:116.0pt;border-collapse:collapse">

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The shadow on the screen fluctuates.

Equation and graph of harmonic oscillations.

Harmonic vibrations - these are oscillations in which the coordinate changes over time according to the law of sine or cosine.

https://pandia.ru/text/79/117/images/image014_7.jpg" width="254" height="430 src="> x=Xmsin(w t+ j 0 )

x=Xmcos(w t+ j 0 )

x - coordinate,

Xm is the oscillation amplitude,

w is the cyclic frequency,

wt+j 0 = j is the oscillation phase,

j 0 is the initial phase of oscillations.

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Graphs are different only amplitude

Graphs differ only in period (frequency)

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If the amplitude of the oscillations does not change over time, the oscillations are called undamped.

Natural vibrations do not take into account friction, the total mechanical energy of the system remains constant: E to + E n = E fur = const.

Natural oscillations are undamped.

With forced oscillations, the energy supplied continuously or periodically from an external source compensates for the losses arising due to the work of the friction force, and the oscillations can be undamped.

The kinetic and potential energy of the body during vibrations pass into each other. When the deviation of the system from the equilibrium position is maximum, the potential energy is maximum, and the kinetic energy is zero. When passing through the equilibrium position, vice versa.

The frequency of free oscillations is determined by the parameters of the oscillatory system.

The frequency of forced oscillations is determined by the frequency of the external force. The amplitude of forced oscillations also depends on the external force.

Resonan c

Resonance called a sharp increase in the amplitude of forced oscillations when the frequency of the action of an external force coincides with the frequency of natural oscillations of the system.

When the frequency w of the change in the force coincides with the natural frequency w0 of the oscillations of the system, the force does positive work throughout the entire period, increasing the amplitude of the body's oscillations. At any other frequency, during one part of the period, the force does positive work, and during the other part of the period, it does negative work.

At resonance, an increase in the oscillation amplitude can lead to the destruction of the system.

In 1905, under the hooves of a squadron of guards cavalry, the Egyptian bridge across the Fontanka River in St. Petersburg collapsed.

Self-oscillations.

Self-oscillations are called undamped oscillations in the system, supported by internal energy sources in the absence of external force change.

Unlike forced oscillations, the frequency and amplitude of self-oscillations are determined by the properties of the oscillatory system itself.

Self-oscillations differ from free oscillations by the independence of the amplitude from time and from the initial short-term impact that excites the process of oscillations. A self-oscillating system can usually be divided into three elements:

1) oscillatory system;

2) energy source;

3) a feedback device that regulates the flow of energy from a source into an oscillatory system.

The energy coming from the source over a period is equal to the energy lost in the oscillatory system over the same time.

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. the smallest period of 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 an argument of a periodic function, which, for a given oscillation amplitude, determines the state of the oscillatory system (displacement, speed, acceleration) of the body at any 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).

Similarly, you can build dependency graphs