The equation of motion of a harmonic oscillator has the form. §15

The simplest model of the vibrational motion of atoms in a diatomic molecule can be a system of two masses T/ and w?, connected by an elastic spring. The vibration of two atoms relative to the center of mass can be replaced by the vibration of one equivalent

mass relative to the initial zero point R= 0, where

R- distance between masses, R e- position of the equilibrium point.

In the classical consideration, it is assumed that the spring is ideal - the elastic force F is directly proportional to the deformation - the deviation from equilibrium x = R-R e, according to Hooke's law:

Where To- elasticity constant. Thus, the force is directed towards returning to the equilibrium position.

Using Hooke's and Newton's laws together (F-ta), can be written:

(denoting ). The solution to such an equation is known to be

serve harmonic functions

Where xo- amplitude, and

Using the reduced mass /l we get:

A measure of the potential energy of a system V serves work

IN quantum mechanics analysis of oscillatory motion for a simple model of a harmonic oscillator is quite complex. It is based on solving the Schrödinger equation

(y/- vibrational wave function, E - total energy particles) and is beyond the scope of our presentation.

For a quantum oscillator it is only possible discrete series values of energy E and frequencies in accordance with the formula E=hv. In addition, the minimum value of the oscillator energy is not zero. This quantity is called zero energy, it corresponds to the lowest energy level of the oscillator and is equal to , its existence can be explained based on the Heisenberg uncertainty relation.

Thus, in accordance with quantum mechanics the energy of the harmonic oscillator is quantized:

Where v- oscillatory quantum number, which can take the value y=0, 1, 2, 3,....

When an oscillator interacts with quanta electromagnetic radiation three factors should be taken into account: 1) population of levels (probability of finding a molecule at a given energy level); 2) the frequency rule (Bohr), according to which the energy of a quantum must correspond to the difference in the energy of any two levels;

3) selection rule for quantum transitions: transition probability, i.e. the intensity of the lines in the absorption spectrum is determined by the quantity transition dipole moment (see theoretical introduction). In the case of the simplest harmonic oscillator, the selection rule is obtained from considering the wave functions. It states that transitions can only occur between adjacent levels (“one step”): the vibrational quantum number changes by one Av= 1. Since the distances between adjacent levels are the same, the absorption spectrum of a harmonic oscillator should contain only one line with a frequency

Since, in accordance with the Boltzmann distribution at room temperature and more low temperatures the lowest vibrational level is populated, then the most intense transition is from the very low level(d=0), and the frequency of this line coincides with the frequency of weaker transitions from higher levels to the adjacent, higher level.

Graphs of harmonic oscillator wave functions for different meanings energies are shown in Figure 2.3. They represent solutions of the Schrödinger equation for a harmonic oscillator

Where N, - normalizing factor, H 0- Hermite polynomials, x = R-R e- deviation from the equilibrium position.

Transition dipole moment for vibrational transitions, R0(or M„) is equal to:

Where ju - dipole moment molecules; hesitation

solid wave functions of the initial and final states, respectively. From the formula it is clear that the transition is allowed,

if at the equilibrium point - the dipole moment of the molecule

changes near the position of the equilibrium point, (curve ju=f(R) does not pass through the maximum at this point). The integral (the second factor in the formula) must also not be equal to zero. It can be shown that this condition is met if the transition occurs between adjacent levels, hence additional rule selection Ai = 1.

In the case of diatomic molecules, vibrational spectra can be observed only for heteronuclear molecules; for homonuclear molecules there is no dipole moment and does not change during vibrations. The vibrational spectra of CO2 exhibit vibrations (antisymmetric stretching and bending), in which the dipole moment changes, but symmetric vibrations, in which it remains unchanged, do not appear.

Harmonic oscillator

Harmonic oscillator(in classical mechanics) - a system that, when displaced from an equilibrium position, experiences a restoring force F, proportional to the displacement x(according to Hooke's law):

Where k- system rigidity coefficient.

If F is the only force acting on the system, then the system is called simple or conservative harmonic oscillator. Free oscillations of such a system represent periodic movement around the equilibrium position (harmonic oscillations). The frequency and amplitude are constant, and the frequency does not depend on the amplitude.

Mechanical examples of a harmonic oscillator are a mathematical pendulum (with small angles of deflection), a torsion pendulum, and acoustic systems. Among other analogues of the harmonic oscillator, it is worth highlighting the electric harmonic oscillator(see LC circuit).

Free vibrations

Conservative harmonic oscillator

As a model of a conservative harmonic oscillator, we take a mass load m, fixed to the spring by rigidity k .

Let x- displacement of the load relative to the equilibrium position. Then, according to Hooke's law, a restoring force will act on it:

Then total energy has a constant value

Simple harmonic motion - this is the movement of a simple harmonic oscillator, periodic motion that is neither forced nor damped. A body in simple harmonic motion is exposed to a single variable force, which is directly proportional to the displacement in magnitude x from the equilibrium position and is directed in the opposite direction.

This movement is periodic: the body oscillates around the equilibrium position according to a sinusoidal law. Each subsequent oscillation is the same as the previous one, and the period, frequency and amplitude of the oscillations remain constant. If we assume that the equilibrium position is at a point with coordinate, equal to zero, then the offset x body from the equilibrium position at any time is given by the formula:

Where A- amplitude of oscillations, f- frequency, φ - initial phase.

The frequency of movement is determined characteristic properties system (for example, the mass of a moving body), while the amplitude and initial phase are determined by the initial conditions - the displacement and speed of the body at the moment the oscillations begin. The kinetic and potential energies of the system also depend on these properties and conditions.

Simple harmonic motion can be mathematical models various types movements such as the oscillation of a spring. Other cases that can be roughly considered as simple harmonic motion are the motion of a pendulum and the vibration of molecules.

Simple harmonic motion is the basis of some ways of analyzing more complex types of motion. One of these methods is the method based on the Fourier transform, the essence of which boils down to the expansion of more complex type movements into a series of simple harmonic movements.

F- restoring force, x- movement of the load (spring deformation), k- coefficient spring stiffness.

Any system in which simple harmonic motion occurs has two key properties:

When a system is thrown out of equilibrium, there must be a restoring force that tends to return the system to equilibrium.
The restoring force must be exactly or approximately proportional to the displacement.

The load-spring system satisfies both of these conditions.

Once a displaced load is subjected to a restoring force, it accelerates and tends to return to its original position. starting point, that is, to the equilibrium position. As the load approaches the equilibrium position, the restoring force decreases and tends to zero. However, in the situation x = 0 the load has a certain amount of motion (impulse), acquired due to the action of the restoring force. Therefore, the load overshoots the equilibrium position, beginning to deform the spring again (but already in opposite direction). The restoring force will tend to slow it down until the speed becomes zero; and the force will again strive to return the load to its equilibrium position.

As long as there is no energy loss in the system, the load will oscillate as described above; such a movement is called periodic.

Further analysis will show that in the case of a load-spring system, the motion is simple harmonic.

Dynamics of simple harmonic motion

For vibrations in one-dimensional space, taking into account Newton's Second Law ( F= m d² x/d t² ) and Hooke's law ( F = −kx, as described above), we have a second-order linear differential equation:

m- body weight, x- its movement relative to the equilibrium position, k- constant (spring stiffness coefficient).

The solution to this differential equation is sinusoidal; one solution is:

Where A, ω and φ are constant quantities, and the equilibrium position is taken as the initial one. Each of these constants represents an important physical property movements: A is the amplitude, ω = 2π f- circular frequency, and φ - initial phase.

Universal circular motion

Simple harmonic motion can in some cases be considered as a one-dimensional projection of universal circular motion. If an object moves with a constant angular velocity ω along a circle of radius r, the center of which is the origin of the plane x−y, then such a movement along each of coordinate axes is simple harmonic with amplitude r and circular frequency ω.

A weight like a simple pendulum

At small angles the motion simple pendulum is close to simple harmonic. The period of oscillation of such a pendulum attached to a rod of length ℓ with acceleration free fall g is given by the formula

This shows that the period of oscillation does not depend on the amplitude and mass of the pendulum, but depends on the acceleration of gravity g, therefore, with the same length of the pendulum, on the Moon it will swing more slowly, since gravity is weaker there and less value free fall acceleration.

This approximation is correct only for small deflection angles, since the expression for angular acceleration is proportional to the sine of the coordinate:

I- moment of inertia; V in this case I = mℓ 2 .

what does it do angular acceleration directly proportional to the angle θ, and this satisfies the definition of simple harmonic motion.

Damped harmonic oscillator

Taking the same model as a basis, we will add the force of viscous friction to it. The force of viscous friction is directed against the speed of movement of the load relative to the medium and is proportional to this speed. Then full strength, acting on the load, is written as follows:

Carrying out similar actions, we get differential equation, describing a damped oscillator:

Here the designation is introduced: . The coefficient is called the attenuation constant. It also has the dimension of frequency.

The solution breaks down into three cases.

, where is the frequency of free oscillations. , Where

Critical damping is remarkable in that it is at critical damping that the oscillator most quickly tends to the equilibrium position. If the friction is less than critical, it will reach the equilibrium position faster, but will “overshoot” it due to inertia and will oscillate. If the friction is greater than critical, then the oscillator will exponentially tend to the equilibrium position, but the more slowly, the greater the friction.

Therefore, in dial indicators (for example, in ammeters), they usually try to introduce critical attenuation so that its readings can be read as quickly as possible.

The damping of an oscillator is also often characterized by a dimensionless parameter called the quality factor. Quality factor is usually denoted by the letter . By definition, the quality factor is equal to:

The higher the quality factor, the slower the oscillator oscillations decay.

An oscillator with critical damping has a quality factor of 0.5. Accordingly, the quality factor indicates the behavior of the oscillator. If the quality factor is greater than 0.5, then the free movement of the oscillator represents oscillations; Over time, it will cross the equilibrium position an unlimited number of times. A quality factor less than or equal to 0.5 corresponds to non-oscillatory motion of the oscillator; V free movement it will cross the equilibrium position at most once.

The quality factor is sometimes called the gain factor of the oscillator, since with some methods of excitation, when the excitation frequency coincides with the resonant one, the amplitude of oscillations turns out to be approximately times greater than when excited at a low frequency.

Also, the quality factor is approximately equal to the number of oscillatory cycles during which the oscillation amplitude decreases by a factor, multiplied by .

In the case of oscillatory motion, damping is also characterized by such parameters as:

Life time vibrations (aka decay time, it's the same relaxation time) τ - time during which the amplitude of oscillations will decrease in e once.

This time is considered as the time required for the attenuation (cessation) of oscillations (although formally free oscillations continue indefinitely).

Forced vibrations

Oscillator oscillations are called forced when some additional external influence is applied to it. This effect can be produced by various means and by various laws. For example, force excitation is the effect on a load of a force that depends only on time according to a certain law. Kinematic excitation is the effect on the oscillator by the movement of the spring attachment point along given law. It is also possible to be affected by friction, when, for example, the medium with which the load experiences friction moves according to a given law.

Lecture 1

OSCILLATIONS. WAVES. OPTICS

The first scientists to study oscillations were Galileo Galilei and Christiaan Huygens. Galileo established the independence of the oscillation period from the amplitude. Huygens invented the pendulum clock.

Any system that, when slightly disturbed from its equilibrium position, exhibits stable oscillations is called a harmonic oscillator. IN classical physics such systems are a mathematical pendulum within small angles of deflection, a load within small amplitudes of oscillation, electrical circuit, consisting of linear elements capacitance and inductance.

(1.1.1)

Where X A

Speed of an oscillating material point

If a periodically repeating process is described by equations that do not coincide with (1.1.1), it is called anharmonic. A system that performs anharmonic oscillations is called an anharmonic oscillator.

1.1.2 . Free vibrations of systems with one degree of freedom. Complex form submissions harmonic vibrations

In nature, small oscillations that a system makes near its equilibrium position are very common. If a system removed from an equilibrium position is left to itself, that is, no external forces act on it, then such a system will perform free undamped oscillations. Let's consider a system with one degree of freedom.

Where

, (1.1.4)

Expression (1.1.5) coincides with equation (1.1.3) of free harmonic oscillations, provided that

, Where A=Xe-iα

1.1.3 . Examples oscillatory movements various physical nature

Harmonic oscillator. Spring, physical and mathematical pendulums

Harmonic oscillator is called a system that oscillates, described by an equation of the form (140.6);

The oscillations of a harmonic oscillator are important example periodic motion and serve as an exact or approximate model in many problems of classical and quantum physics. Examples of a harmonic oscillator are spring, physical and mathematical pendulums, oscillatory circuit(for currents and voltages so small that the circuit elements could be considered linear).

1. Spring pendulum- is a load of mass T, suspended on a perfectly elastic spring and performing harmonic oscillations under the action elastic force F = – kx, Where k- spring stiffness. Equation of motion of a pendulum

From expressions (142.1) and (140.1) it follows that the spring pendulum performs harmonic oscillations according to the law x=A with s (w 0 t + j) with cyclic frequency

Formula (142.3) is valid for elastic vibrations within the limits within which Hooke’s law is satisfied (see (21.3)), i.e., when the mass of the spring is small compared to the mass of the body. Potential energy spring pendulum, according to (141.5) and (142.2), is equal to

2. Physical pendulum- a rigid body that, under the influence of gravity, oscillates around a stationary horizontal axis, passing through the point ABOUT, not coinciding with the center of mass WITH bodies (Fig. 201).

If the pendulum is tilted from its equilibrium position by a certain angle a, then, in accordance with the equation of dynamics of rotational motion of a rigid body (18.3), the moment M restoring force can be written as

Where J- moment of inertia of the pendulum relative to the axis passing through the suspension point Oh, l – the distance between it and the center of mass of the pendulum, F t = – mg sin a » – mg a. - restoring force (the minus sign is due to the fact that the directions Ft And a always opposite; sin a » a corresponds to small oscillations of the pendulum, i.e. small deviations of the pendulum from the equilibrium position). Equation (142.4) can be written as

identical to (142.1), the solution of which (140.1) is known:

From expression (142.6) it follows that for small oscillations the physical pendulum performs harmonic oscillations with a cyclic frequency w 0 (see (142.5)) and period

Where L=J/(ml) - reduced length physical pendulum.

Dot ABOUT' on the continuation of the straight line OS, distant from the point ABOUT suspension of the pendulum at a distance of the given length L, called swing center physical pendulum (Fig. 201). Applying Steiner's theorem (16.1), we obtain

i.e. OO' always more OS. Suspension point ABOUT pendulum and swing center ABOUT' have property of interchangeability: if the suspension point is moved to the center of swing, then the previous point ABOUT suspension

will become the new center of swing, and the period of oscillation of the physical pendulum will not change.

3. Mathematical pendulum- This idealized system consisting of a material point with mass T, suspended on an inextensible weightless thread, and oscillating under the influence of gravity. Good approximation mathematical pendulum is a small heavy ball suspended on a thin long thread. Moment of inertia of a mathematical pendulum

Where l- length of the pendulum.

Since a mathematical pendulum can be represented as special case physical pendulum, assuming that all its mass is concentrated at one point - the center of mass, then, substituting expression (142.8) into formula (1417), we obtain an expression for the period of small oscillations of a mathematical pendulum

Comparing formulas (142.7) and (142.9), we see that if the reduced length L physical pendulum is equal to the length l mathematical pendulum, then the periods of oscillation of these pendulums are the same. Hence, reduced length of a physical pendulum- this is the length of such a mathematical pendulum, the period of oscillation of which coincides with the period of oscillation of a given physical pendulum.

Ideal harmonic oscillator. The ideal oscillator equation and its solution. Amplitude, frequency and phase of oscillations

OSCILLATIONS

HARMONIC VIBRATIONS

Ideal harmonic oscillator. The ideal oscillator equation and its solution. Amplitude, frequency and phase of oscillations

Oscillation is one of the most common processes in nature and technology. Oscillations are processes that repeat over time. Hesitate high rise buildings and high-voltage wires under the influence of wind, the pendulum of a wound clock and a car on springs while driving, river level throughout the year and temperature human body in case of illness. Sound is fluctuations in air pressure, radio waves are periodic changes electrical tension and magnetic field, light is also electromagnetic vibrations. Earthquakes - soil vibrations, ebbs and flows - changes in sea and ocean levels caused by the attraction of the moon, etc.

Oscillations can be mechanical, electromagnetic, chemical, thermodynamic, etc. Despite such diversity, all oscillations are described by the same differential equations.

A harmonic oscillator can be considered linear if the displacement from the equilibrium position is directly proportional to the disturbing force. The oscillation frequency of a harmonic oscillator does not depend on the amplitude. For an oscillator, the principle of superposition is satisfied - if several disturbing forces act, then the effect of their total action can be obtained as a result of adding the effects from active forces separately.

Harmonic vibrations are described by the equation (Fig. 1.1.1)

(1.1.1)

Where X-displacement of the oscillating quantity from the equilibrium position, A– amplitude of oscillations, equal to the value maximum displacement, - the phase of oscillations, which determines the displacement at the moment of time, - the initial phase, which determines the magnitude of the displacement at the initial moment of time, - the cyclic frequency of oscillations.

The time of one complete oscillation is called the period, , where is the number of oscillations completed during the time.

The oscillation frequency determines the number of oscillations performed per unit time; it is related to the cyclic frequency by the relation , then the period.

Thus, the speed and acceleration of the harmonic oscillator also varies according to harmonic law with amplitudes and respectively. In this case, the velocity is ahead of the displacement in phase by , and the acceleration by (Fig. 1.1.2).

From a comparison of the equations of motion of a harmonic oscillator (1.1.1) and (1.1.2) it follows that , or

This second order differential equation is called the harmonic oscillator equation. Its solution contains two constants A and , which are determined by the task initial conditions

Stable balance corresponds to a position of the system in which it potential energy has a minimum ( q– generalized coordinate of the system). The deviation of the system from the equilibrium position leads to the emergence of a force that tends to return the system back. The value of the generalized coordinate corresponding to the equilibrium position is denoted by , then the deviation from the equilibrium position

We will count the potential energy from minimum value. Let us accept the resulting function, expand it into a Maclaurin series and leave the first term of the expansion, we have: o

Where . Then, taking into account the introduced notations:

, (1.1.4)

Taking into account expression (1.1.4) for the force acting on the system, we obtain:

According to Newton's second law, the equation of motion of the system has the form: ,

and has two independent solutions: and , so general solution:

From formula (1.1.6) it follows that the frequency is determined only own properties mechanical system and does not depend on the amplitude and initial conditions of motion.

The dependence of the coordinates of an oscillating system on time can be determined in the form of the real part complex expression , Where A=Xe-iα– complex amplitude, its module coincides with the usual amplitude, and its argument coincides with the initial phase.

Chemist's Handbook 21

Chemistry and chemical technology

Harmonic law of motion

Mechanical, in which rotational motion is converted into oscillatory motion (mainly eccentric and cam mechanisms). The law of motion of the driven link can be close to harmonic. These exciters are used in some types of screens, vibrating centrifuges, and worm mixers.

IN classical mechanics to find the law of motion of a system of points (coordinates qi as functions of time), it is necessary to solve the system of Newton’s equations. With an arbitrarily chosen coordinate system, the general solution of these equations with potential (VII, 7) does not lead to the harmonic form of q (t). However, it is easy to show that with the help of linear combinations of the coordinates q, - it is possible to construct new coordinates, each of which changes according to a harmonic law with a certain frequency (c. Such coordinates

Indeed, the vibrations of two atoms connected by a bond are similar to the vibrations of a pair of spheres held together by a spring. For small shifts, the restoring force is proportional to the displacement, and if such a system is set in motion, the oscillations will be described by the law of simple harmonic motion.

The best operating conditions for the regenerator would be created if the piston did not move harmoniously, but stopped at the end of each stroke. However, a fairly high efficiency can be obtained by using, due to its simplicity, the harmonic law of piston motion.

When hesitating working environment in a pipeline or in any other pressure channel, the distribution of flow velocities over the flow cross section differs from the law that describes this distribution in the case of steady motion of the medium. Thus, when the laminar flow of liquid oscillates in a round cylindrical pipe, the parabolic distribution of velocities is disrupted, which, as is known from hydraulics, is characteristic of laminar steady motion of liquid in a pipe. At harmonic change pressure gradient along the pipe, the velocity distribution can be found using formula (9.42). To do this, instead of (s), you should substitute the Laplace image of the harmonic law of pressure gradient change in the formula and then perform inverse conversion. The function (t, r) obtained in this way is given in the work.

It is clear that there is no need to implement a cycle with intermittent movement of pistons in the designs of industrial machines. For any law of piston motion, in particular for a harmonic one (for a crank drive), the thermodynamic efficiency of an ideal Stirling machine is equal to unity.

In these installations, a simplified, close to harmonic, law of motion of the rods was adopted - the articulated four-bar linkage of the pumping machine was replaced by crank mechanisms. This assumption is generally accepted and, as experiments have shown, is completely justified for the conditions of the experiments.

Internal state diatomic molecule defined if its state is specified electron shell, as well as characteristics of the rotational motion of the molecule as a whole and the vibrational motion of the nuclei. Rotation and vibrations are considered to a first approximation to be independent of the electronic state of the molecule. The simplest model for describing the rotational and vibrational motions of a diatomic molecule is the rigid rotator - harmonic oscillator model, according to which the rotation of the molecule as a rigid rotator and vibrations of nuclei according to the harmonic law are considered independently. Classic description for this model, see chap. IV., 5. Let us write in the same approximation the expression for the energy of a diatomic molecule, using quantum mechanical formulas (VII.19), (VII.20) and (UP.22)

A change in the amplitude of vibrations, as well as a transition from harmonic to shock mode of vibration, is achieved by installing replaceable eccentrics, the profile of which is determined by the law of motion of the pusher with the working table and a block of coaxial cylinders mounted on it.

In section e it was noted that if the energy of molecules is expressed by the sum of a certain number of terms that are quadratic either with respect to spatial coordinates () or with respect to momenta (/z), then the form of the distribution law does not depend on exactly how many terms are included in the expression for kinetic and how much - in the expression for potential energy. However, the derivation of the law is simplified if we consider same number terms expressing potential kinetic energy. Physically, this corresponds to the assumption that the total motion of molecules is represented by the number of 5 independent harmonic oscillators. The energy of the molecule in this case can be written as follows:

In spectrometers with constant acceleration relative speed The motion of the source and absorber periodically changes according to a linear or harmonic law, which makes it possible to record the spectrum under study in a given speed interval. Typically, in such spectrometers, information is recorded in the memory of a multichannel analyzer operating in a time mode, when the memory channels are opened synchronously with the speed cycle.

One of the expressions quantum laws is the discreteness of the energy levels of the body performing periodic movements. Consider, as an example, the harmonic oscillation of an oscillator. The energy of a classical harmonic oscillator can vary continuously. This energy is equal to yA 2 ( highest value potential energy at x = A). Elastic constant

Forced vibrations. Let's consider longitudinal vibrations linear elastic system with one degree of freedom under the action of a driving force P if), changing according to a harmonic law. Initially, we accept the assumption that there are no inelastic resistance forces. The equation of motion in this case (Fig. 3.7, a) has the form tx = -Py + P (/), which after substitutions P = cx, dm = social and P (/) = Po sin (oi) gives

If we were dealing with classical system, then, under certain initial conditions, in principle, it would be possible to excite a movement in which only one of the normal coordinates would change. Then, when this normal coordinate changes, changes in all bond lengths, bond angles, etc., proportional to this coordinate with coefficients would be observed If normal coordinates would change according to a harmonic law, then everything geometric parameters the molecules would also change according to a harmonic law, and all geometric parameters would pass through their equilibrium values in the same phase. An example of normal vibrations for a water-type XY2 molecule is shown in Fig. 8 2

If the electrons of a substance are slightly displaced from their equilibrium positions, then they are subject to the action of a restorative action, the magnitude of which is assumed to be proportional to the displacement. In this case, the movement of electrons turns out to be a simple harmonic oscillation. The passage of light through a system containing a number of such electrical oscillators is equivalent to the appearance of an additional electric force, which, according to Maxwell's theory, turns out to be one of the components of electromagnetic oscillations of light. When light passes through, the electric field changes with a corresponding frequency and affects the movement of the oscillating electron according to the law of conservation of energy. Speed (and therefore kinetic energy) the propagation of light in matter is less than in vacuum, therefore, the kinetic energy of electrons interacting with light increases. Thus, light tends to change the movement of electrons in a molecule and acts in the opposite direction to the force tending to keep the electron in its original position.

This measurement option can also be implemented during torsional vibrations of a tubular sample, if the outer cylinder is installed motionless, the inner cylinder is mounted on a torsion bar and the torque acting on it is set according to the harmonic law. If we now measure the phase difference between the torque and the angle of rotation of the cylinder, as well as the amplitude of the twist angle, then the calculation scheme for determining O will be reduced to the above-mentioned formulas (VI. 15) and (VI. 16). However, if we measure the ratio of torque to the angular velocity of the cylinder, then this corresponds problem about,b determining the system impedance.

In conclusion, we note that from the point of view of the complete and physically reasonable quantitative description dynamics of fluids, all considered models are only a first approximation for describing diffusion and oscillations in water, since a number of simplifications were used in their construction. Only in the limit of long sedentary life times (this can occur at low temperatures) or with strong electrostriction of water molecules in the hydration shell of ions is the harmonic approximation and simple model hopping diffusion [equation (4-5) table. 4] are legal. At high temperatures and in solutions in which the bonds between water molecules are weakened by ions, the vibrations become sharply anharmonic, slowed down by relaxation and diffusion movements. In this case, the behavior of the liquid is more consistent with the behavior of the system free particles[Equation(37)]. The assumption that there is no correlation between diffusion and oscillatory motions is also controversial issue. Recently, Raman et al.

In the next section. 11.3 row will be disassembled simple examples, allowing one to estimate the contributions to the heat capacity of individual decomposed degrees of freedom. In this case, more attention will be paid to a system consisting of particles with two possible energy states, and a harmonic oscillator, since using their example it is possible to relatively simply and at the same time fairly fully analyze the relationship between molecular motion and the heat capacity of the system. For more complex systems It is often easy to estimate the heat capacity at average temperatures based on classical law uniform distribution by degrees of freedom.

The laws of motion of microparticles in quantum mechanics differ significantly from classical ones. On the one hand, they behave (for example, during collisions) like particles with indivisible charges and mass, on the other hand, like waves with a certain frequency (wavelength) and characterized by wave functionа1з - property, otral See pages where the term Harmonic Law of Motion is mentioned Notaries in Novoalekseevka Free ads in the Notaries section in Novoalekseevka. There are no announcements yet, be the first! The predecessors of modern notaries could be found in ancient Egypt, […]

OSCILLATIONS. WAVES. OPTICS

OSCILLATIONS

Lecture 1

HARMONIC VIBRATIONS

Ideal harmonic oscillator. The ideal oscillator equation and its solution. Amplitude, frequency and phase of oscillations

Oscillation is one of the most common processes in nature and technology. Oscillations are processes that repeat over time. High-rise buildings and high-voltage wires oscillate under the influence of the wind, the pendulum of a wound clock and a car on springs while driving, the river level throughout the year and the temperature of the human body during illness. Sound is fluctuations in air pressure, radio waves are periodic changes in the strength of the electric and magnetic field, light is also electromagnetic fluctuations. Earthquakes - soil vibrations, ebbs and flows - changes in sea and ocean levels caused by the attraction of the moon, etc.

Oscillations can be mechanical, electromagnetic, chemical, thermodynamic, etc. Despite such diversity, all oscillations are described by the same differential equations.

Any system that, when slightly disturbed from its equilibrium position, exhibits stable oscillations is called a harmonic oscillator. In classical physics, such systems are a mathematical pendulum within small angles of deflection, a load within small amplitudes of oscillation, and an electrical circuit consisting of linear elements of capacitance and inductance.

Harmonic vibrations are described by the equation (Fig. 1.1.1)

(1.1.1)

Where X-displacement of the oscillating quantity from the equilibrium position, A– the amplitude of oscillations, equal to the value of the maximum displacement, - the phase of oscillations, which determines the displacement at the moment of time, - the initial phase, which determines the value of the displacement at the initial moment of time, - the cyclic frequency of oscillations.

The time of one complete oscillation is called the period, , where is the number of oscillations completed during the time.

The oscillation frequency determines the number of oscillations performed per unit time; it is related to the cyclic frequency by the relation , then the period.

Speed of an oscillating material point

acceleration

Thus, the speed and acceleration of the harmonic oscillator also change according to the harmonic law with amplitudes and respectively. In this case, the velocity is ahead of the displacement in phase by , and the acceleration by (Fig. 1.1.2).

From a comparison of the equations of motion of a harmonic oscillator (1.1.1) and (1.1.2) it follows that , or

This second order differential equation is called the harmonic oscillator equation. Its solution contains two constants A and , which are determined by setting the initial conditions

1.1.2 . Free vibrations of systems with one degree of freedom. Complex form of representation of harmonic vibrations

In nature, small oscillations that a system makes near its equilibrium position are very common. If a system removed from an equilibrium position is left to itself, that is, no external forces act on it, then such a system will perform free, undamped oscillations. Let's consider a system with one degree of freedom.

Stable equilibrium corresponds to a position of the system in which its potential energy has a minimum ( q– generalized coordinate of the system). The deviation of the system from the equilibrium position leads to the emergence of a force that tends to return the system back. The value of the generalized coordinate corresponding to the equilibrium position is denoted by , then the deviation from the equilibrium position

We will count the potential energy from the minimum value. Let us accept the resulting function, expand it into a Maclaurin series and leave the first term of the expansion, we have: o

Where . Then, taking into account the introduced notations:

, (1.1.4)

Taking into account expression (1.1.4) for the force acting on the system, we obtain:

According to Newton's second law, the equation of motion of the system has the form: ,

Expression (1.1.5) coincides with equation (1.1.3) of free harmonic oscillations, provided that

and has two independent solutions: and , so the general solution is:

From formula (1.1.6) it follows that the frequency is determined only by the intrinsic properties of the mechanical system and does not depend on the amplitude and on the initial conditions of motion.

The dependence of the coordinates of an oscillating system on time can be determined in the form of the real part of the complex expression , Where A=Xe-iα– complex amplitude, its module coincides with the usual amplitude, and its argument coincides with the initial phase.

1.1.3 . Examples of oscillatory movements of various physical natures

Oscillations of a load on a spring

Let us consider the oscillations of a load on a spring, provided that the spring is not deformed beyond its elasticity limits. Let us show that such a load will perform harmonic oscillations relative to the equilibrium position (Fig. 1.1.3). Indeed, according to Hooke's law, a compressed or stretched spring creates a harmonic force:

Where – spring stiffness coefficient, – coordinate of the equilibrium position, X– coordinate of the load (material point) at the moment of time, – displacement from the equilibrium position.

Let us place the origin of the coordinate at the equilibrium position of the system. In this case.

If the spring is stretched by an amount X, then release at the moment of time t=0, then the equation of motion of the load according to Newton’s second law will take the form -kx=ma, or , And

(1.1.6)

This equation coincides in form with the equation of motion (1.1.3) of a system performing harmonic oscillations; we will seek its solution in the form:

. (1.1.7)

Substituting (1.17) into (1.1.6), we have: that is, expression (1.1.7) is a solution to equation (1.1.6) provided that

If at the initial moment of time the position of the load was arbitrary, then the equation of motion will take the form:

Let us consider how the energy of a load undergoing harmonic oscillations changes in the absence external forces(Fig. 1.14). If at the moment t=0 tell the load the displacement x=A, then its total energy will become equal to the potential energy of the deformed spring, the kinetic energy is zero (point 1).

A force acts on the load F= -kx, tending to return it to the equilibrium position, so the load moves with acceleration and increases its speed, and, consequently, kinetic energy. This force reduces the displacement of the load X, the potential energy of the load decreases, turning into kinetic energy. The load-spring system is closed, so its total energy is conserved, that is:

. (1.1.8)

At the moment of time, the load is in the equilibrium position (point 2), its potential energy is zero, and its kinetic energy is maximum. Maximum speed we find the load from the law of conservation of energy (1.1.8):

Due to the reserve of kinetic energy, the load does work against the elastic force – and the equilibrium position passes. Kinetic energy gradually turns into potential energy. When the load has a maximum negative displacement – A, kinetic energy Wk=0, the load stops and begins to move to the equilibrium position under the action of an elastic force F= -kx. Further movement occurs in a similar way.

Pendulums

By pendulum we mean solid, which oscillates around under the influence of gravity fixed point or axles. There are physical and mathematical pendulums.

A mathematical pendulum is an idealized system consisting of a weightless inextensible thread on which a mass is suspended, concentrated at one material point.

A mathematical pendulum, for example, is a ball on a long thin thread.

The deviation of the pendulum from the equilibrium position is characterized by the angle φ , which forms a thread with a vertical (Fig. 1.15). When the pendulum deviates from the equilibrium position, a moment of external forces (gravity) occurs: , Where m- weight, – pendulum length

This moment tends to return the pendulum to the equilibrium position (similar to the quasi-elastic force) and is directed opposite to the displacement φ , so there is a minus sign in the formula.

The equation for the dynamics of rotational motion for a pendulum has the form: Iε=,

We will consider the case of small oscillations, therefore sin φ ≈φ, denote ,

we have: , or , and finally

This is the equation of harmonic vibrations, its solution:

The oscillation frequency of a mathematical pendulum is determined only by its length and the acceleration of gravity, and does not depend on the mass of the pendulum. The period is:

If a oscillating body cannot be imagined as material point, then the pendulum is called physical (Fig. 1.1.6). We write the equation of its motion in the form:

In case of small fluctuations , or =0 , where . This is the equation of motion of a body performing harmonic oscillations. The frequency of oscillation of a physical pendulum depends on its mass, length and moment of inertia relative to the axis passing through the suspension point.

Let's denote . Magnitude is called the reduced length of a physical pendulum. This is the length of a mathematical pendulum whose period of oscillation coincides with the period of a given physical pendulum. A point on a straight line connecting the point of suspension with the center of mass, lying at a distance of a reduced length from the axis of rotation, is called the center of swing of a physical pendulum ( ABOUT'). If the pendulum is suspended at the center of swing, then the reduced length and period of oscillation will be the same as at the point ABOUT. Thus, the suspension point and the swing center have the properties of reciprocity: when the suspension point is transferred to the swing center, the previous suspension point becomes the new swing center.

A mathematical pendulum that swings with the same period as the physical one under consideration is called isochronous to this physical pendulum.

1.1.4. Addition of oscillations (beats, Lissajous figures). Vector description of the addition of oscillations

The addition of identically directed oscillations can be done using the method vector diagrams. Any harmonic oscillation can be represented as a vector as follows. Let's select an axis X with the starting point at the point ABOUT(Fig.1.1.7)

From the point ABOUT let's construct a vector that makes an angle with axle X. Let this vector rotate with angular velocity. Projection of a vector onto an axis X is equal to:

that is, it performs harmonic oscillations with an amplitude A.

Consider two harmonic oscillations of the same direction and the same cyclic small, given by vectors And . Axis offsets X are equal:

the resulting vector has a projection and represents the resulting oscillation (Fig. 1.1.8), according to the cosine theorem. Thus, the addition of harmonic oscillations is carried out by the addition of vectors.

Let us perform the addition of mutually perpendicular oscillations. Let a material point do two things mutually perpendicular vibrations frequency:

The material point itself will move along a certain curvilinear trajectory.

From the equation of motion it follows: ,

. (1.1.9)

From equation (1.1.9) we can obtain the equation of the ellipse (Fig. 1.1.9):

Let's consider special cases of this equation:

1. Oscillation phase difference α= 0. At the same time those. or This is the equation of a straight line, and the resulting oscillation occurs along this straight line with amplitude (Fig. 1.1.10).a.

its acceleration is equal to the second derivative of the displacement with respect to time then the force acting on the oscillating point, according to Newton’s second law, is equal to

That is, the force is proportional to the displacement X and is directed against the displacement to the equilibrium position. This force is called restoring force. In the case of a load on a spring, the restoring force is the elastic force; in the case of a mathematical pendulum, it is a component of the gravity force.

The restoring force in nature obeys Hooke's law F= -kx, Where

– restoring force coefficient. Then the potential energy of the oscillating point is:

(the integration constant is chosen equal to zero, so that when X).

ANHARMONIC OSCILLATOR

Let's consider a simple physical system– a material point capable of oscillating on a horizontal surface without friction under the influence of the Hooke force (see Fig. 2).

If the displacement of the load is small (much less than the length of the undeformed spring), and the spring stiffness is equal to k, then the only force acting on the load is the Hooke force. Then the equation

movement of the load (Newton's Second Law) has the form

Moving the terms to the left side of the equality and dividing by the mass of the material point (we neglect the mass of the spring in comparison with m), we obtain the equation of motion

(*) ,

period of oscillation.

Then, taking the function

and having differentiated it with respect to time, we are convinced, firstly, that the speed of movement of the load is equal to

and secondly, after repeated differentiation,

that is, X(t) is really a solution to the equation of a load on a spring.

Such a system, in general, any system, mechanical, electrical or other, that has an equation of motion (*), is called a harmonic oscillator. A function of type X(t) is called the law of motion of a harmonic oscillator, the quantity
are called amplitude,cyclical or natural frequency,initial phase. The natural frequency is determined by the parameters of the oscillator, the amplitude and initial phase are specified by the initial conditions.

The law of motion X(t) represents free oscillations. Such oscillations are performed by undamped pendulums (mathematical or physical), current and voltage in an ideal oscillatory circuit, and some other systems.

Harmonic vibrations can add up both in one and in different directions. The result of addition is also a harmonic oscillation, for example,

This is the principle of superposition (superposition) of vibrations.

Mathematicians have developed a theory of series of this kind, which are called Fourier series. There are also a number of generalizations such as Fourier integrals (frequencies can vary continuously) and even Laplace integrals that work with complex frequencies.

§15. Damped oscillator. Forced vibrations.

Real mechanical systems always have at least a little friction. The simplest case is liquid or viscous friction. This is friction, the magnitude of which is proportional to the speed of movement of the system (and is directed, naturally, against the direction of movement). If the motion occurs along the X axis, then the equation of motion can be written (for example, for a weight on a spring) in the form

Where – coefficient of viscous friction.

This equation of motion can be transformed to the form

Here
– attenuation coefficient, – is still the natural frequency of the oscillator (which can no longer be called harmonic; it is a damped oscillator with viscous friction).

Mathematicians can solve such differential equations. It was shown that the solution is the function

The last formula uses the following notation: – initial amplitude, frequency of weakly damped oscillations
,
. In addition, other parameters characterizing attenuation are often used: logarithmic attenuation decrement
, system relaxation time
, system quality factor
, where the numerator is the energy stored by the system, and the denominator is the energy loss over the period T.

In case of strong attenuation
the solution has an aperiodic form.

There are often cases when, in addition to frictional forces, an external force acts on the oscillator. Then the equation of motion is reduced to the form

the expression on the right is often called the reduced force, the expression itself
called coercive force. For an arbitrary driving force, it is not possible to find a solution to the equation. Usually a harmonic driving force of the type is considered
. Then the solution represents a damped part of type (**), which tends to zero for large times, and steady (forced) oscillations

Amplitude of forced oscillations

and the phase of forced oscillations

Note that as the natural frequency approaches the frequency of the driving force, the amplitude of the forced oscillations increases. This phenomenon is known as resonance. If the damping is large, then the resonant increase is not large. This resonance is called “dull”. At low attenuations, the amplitude of the “sharp” resonance can increase quite significantly. If the system is ideal and there is no friction in it, then the amplitude of forced oscillations increases unlimitedly.

Note also that at the driving force frequency

The maximum value of the amplitude of the driving force is achieved, equal to