An equation describing harmonic oscillations. oscillatory motion

Changes in time according to a sinusoidal law:

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

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

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

Energy transformations during harmonic vibrations.

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

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

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

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 manifest themselves only in different definitions of the quantity and in a different physical sense of the 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 of harmonic oscillations (1.18) is a second-order linear differential equation (since 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

and define the angle as:

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

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:

Movements that have some degree of repetition are called fluctuations.

If the values ​​of physical quantities that change in the process of movement are repeated at regular intervals, then such a movement is called periodic. Depending on the physical nature of the oscillatory process, mechanical and electromagnetic oscillations are distinguished. According to the method of excitation, vibrations are divided into: free(intrinsic) occurring in the system presented to itself near the equilibrium position after some initial impact; forced- occurring under periodic external influence.

Conditions for the occurrence of free oscillations: a) when the body is removed from the equilibrium position, a force must arise in the system, seeking to return it to the equilibrium position; b) the friction forces in the system must be sufficiently small.

BUT amplitude A is the module of the maximum deviation of the oscillating point from the equilibrium position.

Oscillations of a point occurring with a constant amplitude are called undamped, and fluctuations with gradually decreasing amplitude – fading.

The time it takes for a complete oscillation to take place is called period(T).

Frequency periodic oscillations is the number of complete oscillations per unit of time:

Oscillation frequency unit - hertz(Hz). Hertz is the frequency of oscillations, the period of which is equal to 1 s: 1 Hz = 1 s -1 .

cyclicor circular frequency periodic oscillations is the number of complete oscillations that occur in a time 2p with: . \u003d rad / s.

Harmonic- these are oscillations that are described by the periodic law:

or (1)

where is a periodically changing quantity (displacement, speed, force, etc.), A is the amplitude.

A system whose law of motion has the form (1) is called harmonic oscillator . Sine or cosine argument called oscillation phase. The phase of the oscillation determines the displacement at time t. The initial phase determines the displacement of the body at the time of the beginning of the countdown.

Consider the offset x oscillating body about the equilibrium position. Harmonic oscillation equation:

The first derivative of with respect to time gives an expression for the speed of the body: ; (2)

The speed reaches its maximum value at the time when =1: . The offset of the point at this moment is early to zero = 0 (Fig. 17.1, b).

The acceleration also changes with time according to the harmonic law:

where is the maximum acceleration value. The minus sign means that the acceleration is directed in the direction opposite to the displacement, i.e. acceleration and displacement change in antiphase (Fig. 17.1 in). It can be seen that the speed reaches its maximum value when the oscillating point passes the equilibrium position. At this point, the displacement and acceleration are zero.

1.18. HARMONIC OSCILLATIONS AND THEIR CHARACTERISTICS

Definition of harmonic vibrations. Characteristics of harmonic oscillations: displacement from the equilibrium position, amplitude of oscillations, phase of oscillations, frequency and period of oscillations. Velocity and acceleration of an oscillating point. Energy of the harmonic oscillator. Examples of harmonic oscillators: mathematical, spring, torsional and physical pendulums.

Acoustics, radio engineering, optics and other branches of science and technology are based on the doctrine of oscillations and waves. An important role is played by the theory of vibrations in mechanics, especially in calculations of the strength of aircraft, bridges, certain types of machines and assemblies.

fluctuations are processes that repeat at regular intervals (however, not all repeating processes are fluctuations!). Depending on the physical nature of the repeating process, mechanical, electromagnetic, electromechanical, etc. oscillations are distinguished. During mechanical vibrations, the positions and coordinates of bodies periodically change.

Restoring force - the force under the action of which the oscillatory process occurs. This force tends to return the body or material point deviated from the rest position to its original position.

Depending on the nature of the impact on an oscillating body, free (or natural) vibrations and forced vibrations are distinguished.

Depending on the nature of the impact on an oscillating system, free oscillations, forced oscillations, self-oscillations and parametric oscillations are distinguished.

free (own) oscillations are called such oscillations that occur in a system left to itself after it has been given a push, or it has been taken out of equilibrium, i.e. when only the restoring force acts on the oscillating body. An example is the vibrations of a ball suspended on a thread. In order to cause vibrations, you must either push the ball, or, moving it aside, release it. In the event that no energy dissipation occurs, free oscillations are undamped. However, real oscillatory processes are damped, because an oscillating body is affected by forces of resistance to movement (mainly friction forces).

· compelled such vibrations are called, during which the oscillating system is exposed to an external periodically changing force (for example, vibrations of a bridge that occur when people walking in step pass over it). In many cases, systems perform oscillations that can be considered harmonic.

· Self-oscillations , as well as forced oscillations, they are accompanied by external forces acting on the oscillating system, however, the moments of time when these effects are carried out are set by the oscillating system itself. That is, the system itself controls the external influence. An example of a self-oscillatory system is a clock in which the pendulum receives shocks due to the energy of a raised weight or a twisted spring, and these shocks occur at the moments of the pendulum passing through the middle position.

· Parametric oscillations are carried out with a periodic change in the parameters of the oscillating system (a person swinging on a swing periodically raises and lowers his center of gravity, thereby changing the parameters of the system). Under certain conditions, the system becomes unstable - a random deviation from the equilibrium position leads to the emergence and growth of oscillations. This phenomenon is called parametric excitation of oscillations (i.e., oscillations are excited by changing the parameters of the system), and the oscillations themselves are called parametric.

Despite the different physical nature, the oscillations are characterized by the same regularities, which are studied by general methods. An important kinematic characteristic is the form of vibrations. It is determined by the form of the function of time, which describes the change of one or another physical quantity during oscillations. The most important are those fluctuations in which the fluctuating value changes with time according to the law of sine or cosine . They're called harmonic .

Harmonic vibrations oscillations are called, in which the oscillating physical quantity changes according to the sine (or cosine) law.

This type of oscillation is especially important for the following reasons. First, oscillations in nature and technology often have a character very close to harmonic. Secondly, periodic processes of a different form (with a different time dependence) can be represented as an overlay, or superposition, of harmonic oscillations.

Harmonic oscillator equation

Harmonic oscillation is described by the periodic law:

Rice. 18.1. harmonic oscillation

W

here
- characterizes change any physical quantity during oscillations (displacement of the position of the pendulum from the equilibrium position; voltage on the capacitor in the oscillatory circuit, etc.), A - oscillation amplitude ,
- oscillation phase , - initial phase ,
- cyclic frequency ; value
also called own oscillation frequency. This name emphasizes that this frequency is determined by the parameters of the oscillatory system. A system whose law of motion has the form (18.1) is called one-dimensional harmonic oscillator . In addition to the above quantities, the following concepts are introduced to characterize oscillations: period , i.e. time of one oscillation.

(A period of oscillation T called the smallest period of time after which the states of the oscillating system are repeated (one complete oscillation is performed) and the phase of the oscillation receives an increment 2p).

and frequencies
, which determines the number of oscillations per unit time. The unit of frequency is the frequency of such an oscillation, the period of which is 1 s. This unit is called hertz (Hz ).

Oscillation frequencyn called the reciprocal of the period of oscillation - the number of complete oscillations per unit time.

Amplitude- the maximum value of the displacement or change of a variable during oscillatory or wave motion.

Oscillation phase- argument of a periodic function or describing a harmonic oscillatory process (ω - angular frequency, t- time, - the initial phase of oscillations, that is, the phase of oscillations at the initial moment of time t = 0).

The first and second time derivatives of a harmonically oscillating quantity also perform harmonic oscillations of the same frequency:

In this case, the equation of harmonic oscillations, written according to the cosine law, is taken as a basis. In this case, the first of the equations (18.2) describes the law according to which the speed of an oscillating material point (body) changes, the second equation describes the law according to which the acceleration of an oscillating point (body) changes.

Amplitudes
and
equal respectively
and
. hesitation
ahead of
in phase to ; and hesitation
ahead of
on the . Values A and can be determined from given initial conditions
and
:

,
. (18.3)

Oscillator oscillation energy

P

Rice. 18.2. Spring pendulum

Let's now see what happens to vibration energy . As an example of harmonic oscillations, consider one-dimensional oscillations performed by a body of mass m Under the influence elastic strength
(for example, a spring pendulum, see fig. 18.2). Forces of a different nature than elastic, but in which the condition F = -kx is satisfied, are called quasi-elastic. Under the influence of these forces, bodies also make harmonic oscillations. Let be:

bias:
speed:
acceleration:

Those. the equation for such oscillations has the form (18.1) with natural frequency
. The quasi-elastic force is conservative . Therefore, the total energy of such harmonic oscillations must remain constant. In the process of oscillations, the transformation of kinetic energy occurs E to into a potential E P and vice versa, moreover, at the moments of the greatest deviation from the equilibrium position, the total energy is equal to the maximum value of the potential energy, and when the system passes through the equilibrium position, the total energy is equal to the maximum value of the kinetic energy. Let's find out how the kinetic and potential energy changes with time:

Kinetic energy:

Potential energy:

(18.5)

Considering that i.e. , the last expression can be written as:

Thus, the total energy of the harmonic oscillation turns out to be constant. It also follows from relations (18.4) and (18.5) that the average values ​​of the kinetic and potential energies are equal to each other and half of the total energy, since the average values
and
for the period are 0.5. Using trigonometric formulas, it can be obtained that the kinetic and potential energy change with frequency
, i.e. with a frequency twice the harmonic frequency.

Examples of a harmonic oscillator are spring pendulums, physical pendulums, mathematical pendulums, and torsional pendulums.

1. Spring pendulum- this is a load of mass m, which is suspended on an absolutely elastic spring and performs harmonic oscillations under the action of an elastic force F = -kx, where k is the stiffness of the spring. The equation of motion of the pendulum has the form or (18.8) From formula (18.8) it follows that the spring pendulum performs harmonic oscillations according to the law x \u003d Acos (ω 0 t + φ) with a cyclic frequency

(18.9) and period

(18.10) Formula (18.10) is true for elastic oscillations within the limits in which Hooke's law is fulfilled, i.e. if the mass of the spring is small compared to the mass of the body. The potential energy of a spring pendulum, using (18.9) and the potential energy formula of the previous section, is (see 18.5)

2. physical pendulum- this is a rigid body that oscillates under the action of gravity around a fixed horizontal axis that passes through the point O, which does not coincide with the center of mass C of the body (Fig. 1).

Fig.18.3 Physical pendulum

If the pendulum is deflected from the equilibrium position by a certain angle α, then, using the equation of dynamics of the rotational motion of a rigid body, the moment M of the restoring force (18.11) where J is the moment of inertia of the pendulum about the axis that passes through the suspension point O, l is the distance between the axis and the center of mass of the pendulum, F τ ≈ –mgsinα ≈ –mgα is the restoring force (the minus sign indicates that the directions F τ and α are always opposite; sinα ≈ α since the oscillations of the pendulum are considered small, i.e. the pendulum deviates from the equilibrium position by small angles). We write equation (18.11) as

Or Taking (18.12) we get the equation

Identical to (18.8), whose solution we find and write as:

(18.13) From formula (18.13) it follows that for small oscillations the physical pendulum performs harmonic oscillations with a cyclic frequency ω 0 and a period

(18.14) where the value L=J/(m l) - . The point O" on the continuation of the straight line OS, which is separated from the point O of the suspension of the pendulum at a distance of the reduced length L, is called swing center physical pendulum (Fig. 18.3). Applying the Steiner theorem for the moment of inertia of the axis, we find

That is, OO "is always greater than OS. The suspension point O of the pendulum and the swing center O" have interchangeability property: if the suspension point is moved to the swing center, then the old suspension point O will be the new swing center, and the oscillation period of the physical pendulum will not change.

3. Mathematical pendulum is an idealized system consisting of a material point of mass m, which is suspended on an inextensible weightless thread, and which oscillates under the action of gravity. A good approximation of a mathematical pendulum is a small, heavy ball that is suspended from a long, thin thread. Moment of inertia of a mathematical pendulum

(8) where l is the length of the pendulum.

Since a mathematical pendulum is a special case of a physical pendulum, if we assume that all of its mass is concentrated at one point - the center of mass, then, substituting (8) into (7), we find an expression for the period of small oscillations of a mathematical pendulum (18.15) Comparing formulas (18.13 ) and (18.15), we see that if the reduced length L of the physical pendulum is equal to the length l a mathematical pendulum, then the periods of oscillation of these pendulums are the same. Means, reduced length of a physical pendulum is the length of such a mathematical pendulum, in which the period of oscillation coincides with the period of oscillation of a given physical pendulum. For a mathematical pendulum (material point with mass m suspended on a weightless inextensible thread of length l in the field of gravity with free fall acceleration equal to g) at small angles of deviation (not exceeding 5-10 angular degrees) from the equilibrium position, natural oscillation frequency:
.

4. A body suspended on an elastic thread or other elastic element that oscillates in a horizontal plane is torsion pendulum.

This is a mechanical oscillatory system that uses the forces of elastic deformations. On fig. 18.4 shows the angular analogue of a linear harmonic oscillator that performs torsional vibrations. A horizontally located disk hangs on an elastic thread fixed in its center of mass. When the disk rotates through an angle θ, a moment of forces arises M elastic torsion strain:

where I = IC is the moment of inertia of the disk about the axis passing through the center of mass, ε is the angular acceleration.

By analogy with the load on the spring, you can get.

General information about vibrations

Chapter 6 Vibrational Motion

fluctuations Processes that differ in varying degrees of repetition are called.

Such a property of repeatability is possessed, for example, by the swinging of the pendulum of a clock, the vibrations of a string or the legs of a tuning fork, the voltage between the capacitor plates in the radio receiver circuit, etc.

Depending on the physical nature of the repeating process, oscillations are distinguished:

– mechanical;

– electromagnetic;

– electromechanical, etc.

Depending on the nature of the impact on the oscillating system, there are:

– free (or own);

- forced;

– self-oscillations;

are parametric oscillations.

free or own are called such oscillations that occur in a system left to itself after a push has been given to it or it has been brought out of equilibrium. An example is the oscillation of a ball suspended on a thread (pendulum).

compelled such vibrations are called, during which the oscillating system is exposed to an external periodically changing force.

Self-oscillations are accompanied by the influence of external forces on the oscillating system, however, the moments of time when these influences are carried out are set by the oscillating system itself - the system itself controls the external influence. An example of a self-oscillatory system is a clock in which the pendulum receives shocks due to the energy of a raised weight or a twisted spring, and these shocks occur at the moments of the pendulum passing through the middle position.

At parametric oscillations due to external influences, there is a periodic change in some parameter of the system, for example, the length of a pendulum thread.

The simplest are harmonic vibrations, i.e., such oscillations in which the oscillating quantity (for example, the deviation of the pendulum) changes with time according to the law of sine or cosine.

The most important among the oscillatory motions is the so-called simple or harmonic oscillatory motion.

The nature of such motion is best revealed with the help of the following kinematic model. Let us assume that the geometric point M rotates uniformly around a circle of radius a with a constant angular velocity (Fig. 6.1). Her projection N per diameter, e.g. per axle X, will oscillate from an extreme position to another extreme position and back. Such a fluctuation of the point N called a simple or harmonic oscillation.

To describe it, you need to find the coordinate x points N as a function of time t. Let us assume that at the initial moment of time the radius OM formed with the axis X injection . After time t, this angle will increment and become equal to . From fig. 6.1. it's clear that

. (6.1)

This formula describes analytically the harmonic oscillatory motion of a point N along the diameter.

Value a gives the maximum deviation of the oscillating point from the equilibrium position. It is called amplitude fluctuations. The value 0 is called cyclic frequency. The value is called phase fluctuations, and its value at , i.e., the value - primary phase. After the time has passed

the phase is incremented, and the oscillating point returns to its original position while maintaining the initial direction of motion. Time T called the period of oscillation.

The speed of the oscillating point can be found by differentiating expression (6.1) with respect to time. This gives

Differentiating a second time, we obtain the acceleration

or, using (6.1),

The force acting on a material point during harmonic oscillation is equal to

. (6.6)

It is proportional to the deviation x and has the opposite direction. It is always directed towards the equilibrium position.

Consider the harmonic oscillations of a load on a spring, one end of which is fixed, and a body of mass is suspended from the other m(Fig. 6.2). Let be the length of the undeformed spring. If the spring is stretched or compressed to a length l, then there is a force F tending to return the body to a position of equilibrium. For small tensions, Hooke's law- the force is proportional to the stretching of the spring: . Under these conditions, the equation of motion of the body has the form

Constant k called coefficient elasticity or stiffness of the spring. The minus sign means that the force F directed in the direction opposite to the displacement x, i.e. to the equilibrium position.

When deriving equation (6.7), it was assumed that no other forces act on the body. Let us show that the motion of a body suspended on a spring in a uniform gravitational field obeys the same equation. Let us denote in this case by the letter X elongation of the spring, i.e. difference . The spring pulls the load up with a force, the force of gravity - down. The equation of motion has the form

Let denote the lengthening of the spring in the equilibrium position. Then . Eliminating the weight, we get . We keep the designation , then the equation of motion will take the previous form (6.7). The value of x still means the displacement of the load from the equilibrium position. However, the equilibrium position is shifted by gravity. In addition, in the presence of gravity, the meaning of the quantity changes. Now it means the resultant of the tension forces of the spring and the weight of the load. But all this does not affect the mathematical side of the process. Therefore, one can argue as if there were no gravity at all. This is how we will do it.

The resulting force has the same form as the force in expression (6.6). If we put , then equation (6.7) becomes

. (6.8)

This equation coincides with equation (6.5). Function (6.1) is a solution to such an equation for any values ​​of the constants a and a. This is the general solution. It follows from the foregoing that the load on the spring will perform harmonic oscillations with a circular frequency

and period

. (6.10)

The oscillations described by equation (6.8) are free(or own).

The potential and kinetic energies of the body are given by the expressions

. (6.11)

Each of them changes over time. However, their sum E must remain constant over time:

(6.12)

Everything stated here is applicable to harmonic vibrations of any mechanical systems with one degree of freedom. The instantaneous position of a mechanical system with one degree of freedom can be determined using any one quantity q, called the generalized coordinate, for example, the angle of rotation, displacement along a certain line, etc. The derivative of the generalized coordinate with respect to time is called the generalized speed. When considering the oscillations of mechanical systems with one degree of freedom, it is more convenient to take as the initial one not the Newtonian equation of motion, but the energy equation. Let us assume that a mechanical system is such that its potential and kinetic energies are expressed by formulas of the form

, (6.14)

where d and b are positive constants (system parameters). Then the law of conservation of energy leads to the equation

. (6.15)

It differs from equation (6.12) only in terms of notation, which does not matter in mathematical consideration. From the mathematical identity of equations (6.12) and (6.15) it follows that their general solutions are the same. Therefore, if the energy equation is reduced to the form (6.15), then

, (6.16)

i.e. generalized coordinate q performs a harmonic oscillation with a circular frequency