Theory of mechanical oscillations. Fundamentals of the theory of oscillations of mechanical systems

We have already considered the origin of classical mechanics, the strength of materials and the theory of elasticity. The most important component of mechanics is also the theory of vibrations. Vibrations are the main cause of the destruction of machines and structures. Already by the end of the 1950s. 80% of equipment accidents occurred due to increased vibrations. Fluctuations also have a harmful effect on people associated with the operation of machinery. They can also cause control systems to fail.

Despite all this, the theory of oscillations emerged as an independent science only at the turn of the 19th century. However, the calculations of machines and mechanisms up to the beginning XX century were held in a static setting. The development of mechanical engineering, the growth in the power and speed of steam engines while reducing their weight, the emergence of new types of engines - internal combustion engines and steam turbines, led to the need for strength calculations taking into account dynamic loads. As a rule, new problems in the theory of oscillations arose in technology under the influence of accidents or even catastrophes resulting from increased vibrations.

Oscillations is a movement or change of state, which has a certain degree of repetition.

The theory of oscillations can be divided into four periods.

Iperiod- the emergence of the theory of oscillations within the framework of theoretical mechanics (the end of the 16th century - the end of the 18th century). This period is characterized by the emergence and development of dynamics in the works of Galileo, Huygens, Newton, d "Alembert, Euler, D. Bernoulli and Lagrange.

Leonhard Euler became the founder of the theory of oscillations. In 1737, L. Euler, on behalf of the St. Petersburg Academy of Sciences, began research on the balance and movement of the ship, and in 1749 his book "Ship Science" was published in St. Petersburg. It was in this work of Euler that the foundations of the theory of static stability and the theory of oscillations were laid.

Jean Leron d "Alembert, in his numerous works, considered individual problems, such as small oscillations of a body around the center of mass and around the axis of rotation in connection with the problem of precession and nutation of the Earth, oscillations of a pendulum, a floating body, springs, etc. But the general theory Hesitation d "Alamber did not create.

The most important application of the methods of vibration theory was the experimental determination of the torsional rigidity of wire, carried out by Charles Coulomb. Empirically, Coulomb also established the property of isochronism of small oscillations in this problem as well. Investigating the damping of vibrations, this great experimenter came to the conclusion that its main cause is not air resistance, but losses from internal friction in the wire material.

A great contribution to the foundations of the theory of oscillations was made by L. Euler, who laid the foundations for the theory of static stability and the theory of small oscillations, d "Alembert, D. Bernoulli and Lagrange. In their works, the concepts of the period and frequency of oscillations, the form of oscillations were formed, the term small oscillations came into use , the principle of superposition of solutions was formulated, attempts were made to expand the solution into a trigonometric series.

The first tasks of the theory of oscillations were the problems of oscillations of a pendulum and a string. We have already spoken about the oscillations of the pendulum - the practical result of solving this problem was the invention of the clock by Huygens.

As for the problem of string vibrations, this is one of the most important problems in the history of the development of mathematics and mechanics. Let's consider it in more detail.

acoustic string it is an ideal smooth, thin and flexible thread of a finite length of hard material, stretched between two fixed points. In the modern interpretation, the problem of transverse vibrations of a string of length l reduces to finding a solution to the differential equation (1) in partial derivatives. Here x is the coordinate of the string point along the length, and y- its transverse displacement; H- string tension - its running mass. a is the speed of the wave. A similar equation also describes the longitudinal oscillations of the air column in the pipe.

In this case, the initial distribution of deviations of the string points from a straight line and their velocities must be specified, i.e. equation (1) must satisfy the initial conditions (2) and boundary conditions (3).

The first fundamental experimental studies of string vibrations were carried out by the Dutch mathematician and mechanic Isaac Beckmann (1614–1618) and M. Mersenne, who established a number of regularities and published his results in 1636 in the “Book of Consonances”:

Mersenne's regularities were theoretically confirmed in 1715 by Newton's student Brooke Taylor. He considers the string as a system of material points and makes the following assumptions: all points of the string simultaneously pass their equilibrium positions (coincide with the axis x) and the force acting on each point is proportional to its displacement y about the axis x. This means that it reduces the problem to a system with one degree of freedom - equation (4). Taylor correctly received the first natural frequency (fundamental tone) - (5).

D "Alembert in 1747 for this problem applied the method of reducing the problem of dynamics to the problem of statics (principle d" Alembert) and obtained a differential equation for vibrations of a homogeneous string in partial derivatives (1) - the first equation of mathematical physics. He was looking for the solution of this equation in the form of the sum of two arbitrary functions (6)

where and are periodic functions of period 2 l. When clarifying the question of the form of functions and d'Alembert takes into account the boundary conditions (1.2), assuming that at
the string coincides with the axis x. The meaning is
not specified in the task statement.

Euler considers a special case when
the string is deflected from the equilibrium position and released without initial velocity. It is essential that Euler does not impose any restrictions on the initial shape of the string, i.e. does not require that it can be given analytically, considering any curve that "can be drawn by hand". The final result obtained by the author: if
the shape of the string is described by the equation
, then the oscillations look like this (7). Euler revised his views on the concept of a function, in contrast to the previous idea of ​​it only as an analytic expression. Thus, the class of functions to be studied in analysis was expanded, and Euler came to the conclusion that "since any function will define a certain line, the converse is also true - curved lines can be reduced to functions."

The solutions obtained by d "Alembert and Euler represent the law of string vibrations in the form of two waves running towards each other. At the same time, they did not agree on the form of the function that defines the bending line.

D. Bernoulli, in studying the vibrations of a string, took a different path, breaking the string into material points, the number of which he considered to be infinite. He introduces the concept of a simple harmonic oscillation of a system, i.e. such its movement, in which all points of the system oscillate synchronously with the same frequency, but different amplitudes. Experiments performed with sounding bodies led D. Bernoulli to the idea that the most general movement of a string consists in the simultaneous execution of all movements available to it. This is the so-called superposition of solutions. Thus, in 1753, based on physical considerations, he obtained a general solution for string vibrations, presenting it as a sum of partial solutions, for each of which the string bends in the form of a characteristic curve (8).

In this series, the first form of oscillation is half a sinusoid, the second is a whole sinusoid, the third consists of three half-sinusoids, and so on. Their amplitudes are represented as functions of time and, in essence, are the generalized coordinates of the system under consideration. According to D. Bernoulli's solution, the string motion is an infinite series of harmonic vibrations with periods
. In this case, the number of nodes (fixed points) is one less than the natural frequency number. Restricting the series (8) to a finite number of terms, we obtain a finite number of equations for the continuum system.

However, D. Bernoulli's solution contains an inaccuracy - it does not take into account that the phase shift of each harmonic of oscillations is different.

D. Bernoulli, presenting the solution in the form of a trigonometric series, used the principle of superposition and expansion of the solution in terms of a complete system of functions. He rightly believed that with the help of various terms of formula (8) it is possible to explain the harmonic tones that the string emits simultaneously with its fundamental tone. He considered this as a general law, valid for any system of bodies that makes small vibrations. However, physical motivation cannot replace the mathematical proof, which was not presented then. Because of this, colleagues did not understand the solutions of D. Bernoulli, although as early as 1737 C. A. Clairaut used the expansion of functions in a series.

The presence of two different ways of solving the problem of string vibrations caused among the leading scientists of the 18th century. stormy controversy - "argument about the string." This dispute mainly concerned questions about the form of admissible solutions of the problem, about the analytical representation of a function, and whether it is possible to represent an arbitrary function in the form of a trigonometric series. One of the most important concepts of analysis, the concept of a function, was developed in the "string dispute".

D "Alamber and Euler did not agree that the solution proposed by D. Bernoulli could be general. In particular, Euler could not agree that this series could represent any "freely drawn curve", as he himself now defined the concept of function.

Joseph Louis Lagrange, entering into controversy, broke the string into small arcs of the same length with mass concentrated in the center, and investigated the solution of a system of ordinary differential equations with a finite number of degrees of freedom. Passing then to the limit, Lagrange obtained a result analogous to that of D. Bernoulli, without, however, postulating in advance that the general solution must be an infinite sum of particular solutions. At the same time, he refines the solution of D. Bernoulli, bringing it in the form (9), and also derives formulas for determining the coefficients of this series. Although the solution of the founder of analytical mechanics does not meet all the requirements of mathematical rigor, it was a noticeable step forward.

As regards the expansion of the solution into a trigonometric series, Lagrange believed that the series diverges under arbitrary initial conditions. After 40 years, in 1807, J. Fourier again found the expansion of the function into a trigonometric series for the third time and showed how this can be used to solve the problem, thereby confirming the correctness of D. Bernoulli's solution. A complete analytical proof of Fourier's theorem on the expansion of a single-valued periodic function in a trigonometric series was given in the integral calculus of Todgenter and in the "Treatise on Natural Philosophy" by Thomson (Lord Kelvin) and Tait.

Research into the free vibrations of a stretched string lasted two centuries, counting from the work of Beckmann. This problem served as a powerful stimulus for the development of mathematics. Considering the oscillations of continuum systems, Euler, d "Alembert and D. Bernoulli created a new discipline - mathematical physics. Mathematization of physics, i.e. presenting it through a new analysis, is Euler's greatest merit, thanks to which new paths in science were paved. The logical development of the results Euler and Fourier was the well-known definition of the function by Lobachevsky and Lejeune Dirichlet, based on the idea of ​​a one-to-one correspondence of two sets.Dirichlet also proved the possibility of expanding into a Fourier series of piecewise continuous and monotone functions.A one-dimensional wave equation was also obtained and the equality of its two solutions was established, which mathematically confirmed the relationship between vibrations and waves.The fact that a vibrating string generates sound prompted scientists to think about the identity of the process of sound propagation and the process of string vibration.The most important role of boundary and initial conditions in such problems was also revealed.For the development of mechanics, an important result was the the use of the d "Alembert principle for writing differential equations of motion, and for the theory of oscillations this task also played a very important role, namely, the principle of superposition and expansion of the solution in terms of natural modes of oscillations were applied, the basic concepts of the theory of oscillations were formulated - natural frequency and mode of oscillations.

The results obtained for free vibrations of a string served as the basis for creating the theory of vibrations of continuum systems. Further study of vibrations of inhomogeneous strings, membranes, and rods required finding special methods for solving the simplest hyperbolic equations of the second and fourth orders.

The problem of free vibrations of a stretched string interested scientists, of course, not for its practical application, the laws of these vibrations were known to one degree or another to craftsmen who made musical instruments. This is evidenced by the unsurpassed string instruments of such masters as Amati, Stradivari, Guarneri and others, whose masterpieces were created back in the 17th century. The interests of the greatest scientists who dealt with this problem, most likely, lay in the desire to bring a mathematical basis to the already existing laws of string vibration. In this question, the traditional path of any science manifested itself, starting with the creation of a theory that explains already known facts, in order to then find and investigate unknown phenomena.

IIperiod - analytical(late 18th century - late 19th century). The most important step in the development of mechanics was made by Lagrange, who created a new science - analytical mechanics. The beginning of the second period in the development of the theory of oscillations is associated with the work of Lagrange. In the book Analytical Mechanics, published in Paris in 1788, Lagrange summed up everything that had been done in mechanics in the 18th century and formulated a new approach to solving its problems. In the doctrine of equilibrium, he abandoned the geometric methods of statics and proposed the principle of possible displacements (the Lagrange principle). In dynamics, Lagrange, applying simultaneously the principle of d "Alembert and the principle of possible displacements, obtained a general variational equation of dynamics, which is also called the principle of d" Alembert - Lagrange. Finally, he introduced the concept of generalized coordinates into use and obtained the equations of motion in the most convenient form - the Lagrange equations of the second kind.

These equations became the basis for creating the theory of small oscillations described by linear differential equations with constant coefficients. Linearity is rarely inherent in a mechanical system, and in most cases is the result of its simplification. Considering small oscillations near the equilibrium position, which are carried out at low speeds, it is possible to discard the terms of the second and higher orders in the equations of motion with respect to generalized coordinates and velocities.

Applying the Lagrange equations of the second kind for conservative systems

we get the system s second-order linear differential equations with constant coefficients

, (11)

where I and C are, respectively, matrices of inertia and stiffness, the components of which will be inertial and elastic coefficients.

Particular solution (11) is sought in the form

and describes a monoharmonic oscillatory regime with a frequency k, which is the same for all generalized coordinates. Differentiating (12) twice with respect to t and substituting the result into equations (11), we obtain a system of linear homogeneous equations for finding the amplitudes in matrix form

. (13)

Since during system oscillations all amplitudes cannot be equal to zero, the determinant is equal to zero

. (14)

The frequency equation (14) was called the secular equation, since it was first considered by Lagrange and Laplace in the theory of secular perturbations of the elements of planetary orbits. It is the equation s-th degree relative to , the number of its roots is equal to the number of degrees of freedom of the system. These roots are usually arranged in ascending order, while they form the spectrum of natural frequencies. To every root corresponds to a particular solution of the form (12), the set s amplitudes represent the waveform, and the overall solution is the sum of these solutions.

Lagrange gave the statement of D. Bernoulli that the general oscillatory motion of a system of discrete points consists in the simultaneous execution of all its harmonic oscillations, the form of a mathematical theorem, using the theory of integration of differential equations with constant coefficients, created by Euler in the 40s of the 18th century. and the achievements of d "Alembert, who showed how systems of such equations are integrated. At the same time, it was necessary to prove that the roots of the secular equation are real, positive and not equal to each other.

Thus, in "Analytical Mechanics" Lagrange obtained the equation of frequencies in a general form. At the same time, he repeats the mistake made by d "Alembert in 1761, that the multiple roots of the secular equation correspond to an unstable solution, since supposedly in this case secular or secular terms appear in the solution, containing t not under the sign of sine or cosine. In this regard, both d'Alembert and Lagrange believed that the frequency equation cannot have multiple roots (d'Alembert-Lagrange's paradox). It was enough for Lagrange to consider at least a spherical pendulum or oscillations of a rod whose cross section is, for example, round or square, to make sure that multiple frequencies are possible in conservative mechanical systems. The mistake made in the first edition of Analytical Mechanics was repeated in the second edition (1812), which appeared during Lagrange's lifetime, and in the third (1853). The scientific authority of d "Alembert and Lagrange was so high that both Laplace and Poisson repeated this mistake, and corrected it only after almost 100 years independently of each other in 1858 by K. Weierstrass and in 1859 by Osip Ivanovich Somov , who made a great contribution to the development of the theory of oscillations of discrete systems.

Thus, to determine the frequencies and modes of free oscillations of a linear system without resistance, it is necessary to solve the secular equation (13). However, equations of degree higher than the fifth do not have an analytical solution.

The problem was not only the solution of the secular equation, but also, to a greater extent, its compilation, since the expanded determinant (13) has
terms, for example, for a system with 20 degrees of freedom, the number of terms is 2.4 10 18, and the time it takes to open such a determinant for the most powerful computer of the 1970s, performing 1 million operations per second, is approximately 1.5 million years , and for a modern computer "only" a few hundred years.

The problem of determining the frequencies and modes of free oscillations can also be considered as a problem of linear algebra and solved numerically. Rewriting equality (13) as

, (14)

note that the column matrix is an eigenvector of the matrix

, (15)

a its own meaning.

Solving the problem of eigenvalues ​​and vectors is one of the most attractive problems in numerical analysis. At the same time, it is impossible to propose a single algorithm for solving all problems encountered in practice. The choice of algorithm depends on the type of matrix, as well as on whether it is necessary to determine all eigenvalues ​​or only the smallest (largest) or close to a given number. In 1846, Carl Gustav Jacob Jacobi proposed an iterative rotation method to solve the complete eigenvalue problem. The method is based on such an infinite sequence of elementary rotations that, in the limit, transforms matrix (15) into a diagonal one. The diagonal elements of the resulting matrix will be the desired eigenvalues. In this case, to determine the eigenvalues, it is required
arithmetic operations, and for eigenvectors
operations. In this regard, the method in the XIX century. did not find application and was forgotten for more than a hundred years.

The next important step in the development of the theory of oscillations was the work of Rayleigh, especially his fundamental work The Theory of Sound. In this book, Rayleigh considers oscillatory phenomena in mechanics, acoustics and electrical systems from a unified point of view. Rayleigh owns a number of fundamental theorems of the linear theory of oscillations (theorems on stationarity and properties of natural frequencies). Rayleigh also formulated the principle of reciprocity. By analogy with kinetic and potential energy, he introduced a dissipative function, received the name of Rayleigh and represents half the rate of energy dissipation.

In The Theory of Sound, Rayleigh also offers an approximate method for determining the first natural frequency of a conservative system

, (16)

where
. In this case, some form of vibrations is taken to calculate the maximum values ​​of the potential and kinetic energies. If it coincides with the first mode of the system, we will get the exact value of the first natural frequency, otherwise this value is always overestimated. The method gives an accuracy that is quite acceptable for practice, if the static deformation of the system is taken as the first mode of vibration.

Thus, back in the 19th century, in the works of Somov and Rayleigh, a technique was formed for constructing differential equations that describe small oscillatory motions of discrete mechanical systems using the Lagrange equations of the second kind

where in the generalized force
all force factors must be included, with the exception of elastic and dissipative, covered by functions R and P.

The Lagrange equations (17) in matrix form, which describe the forced vibrations of a mechanical system, after substituting all the functions, look like this

. (18)

Here is the damping matrix, and
are column vectors of correspondingly generalized coordinates, velocities, and accelerations. The general solution of this equation consists of free and accompanying oscillations, which are always damped, and forced oscillations occurring at the frequency of the disturbing force. We confine ourselves to considering only a particular solution corresponding to forced oscillations. As an excitation, Rayleigh considered generalized forces that change according to a harmonic law. Many attributed this choice to the simplicity of the case under consideration, but Rayleigh gives a more convincing explanation - expansion in a Fourier series.

Thus, for a mechanical system with more than two degrees of freedom, the solution of a system of equations presents certain difficulties, which increase like an avalanche with an increase in the order of the system. Even with five to six degrees of freedom, the problem of forced oscillations cannot be manually solved in the classical way.

In the theory of oscillations of mechanical systems, small (linear) oscillations of discrete systems have played a special role. The spectral theory developed for linear systems does not even require the construction of differential equations, and to obtain a solution, one can immediately write systems of linear algebraic equations. Although methods for determining eigenvectors and eigenvalues ​​(Jacobi) and for solving a system of linear algebraic equations (Gauss) were developed in the middle of the 19th century, their practical application even for systems with a small number of degrees of freedom was out of the question. Therefore, before the advent of sufficiently powerful computers, many different methods were developed for solving the problem of free and forced oscillations of linear mechanical systems. Many outstanding scientists - mathematicians and mechanics dealt with these problems, they will be discussed below. The advent of powerful computing technology made it possible not only to solve linear problems of large dimensions in a fraction of a second, but also to automate the process of compiling systems of equations.

Thus, during the XVIII century. in the theory of small oscillations of systems with a finite number of degrees of freedom and oscillations of continual elastic systems, the basic physical schemes were developed and the principles essential for the mathematical analysis of problems were explained. However, to create the theory of mechanical oscillations as an independent science, there was not enough of a unified approach to solving problems of dynamics, and there were no technical demands for its faster development.

The growth of large-scale industry in the late 18th and early 19th centuries, caused by the widespread introduction of the steam engine, led to the separation of applied mechanics into a separate discipline. But until the end of the 19th century, strength calculations were carried out in a static formulation, since the machines were still low-powered and slow-moving.

By the end of the 19th century, with the increase in speeds and the decrease in the dimensions of machines, it became impossible to neglect vibrations. Numerous accidents that occurred from the onset of resonance or fatigue failure during vibrations forced engineers to pay attention to oscillatory processes. Of the problems that arose during this period, the following should be noted: the collapse of bridges from passing trains, torsional vibrations of shafting and vibrations of ship hulls, excited by the inertia forces of moving parts of unbalanced machines.

IIIperiod– formation and development of the applied theory of oscillations (1900–1960s). Developing mechanical engineering, improvement of locomotives and ships, the emergence of steam and gas turbines, high-speed internal combustion engines, automobiles, aircraft, etc. demanded a more accurate analysis of stresses in machine parts. This was dictated by the requirements of a more economical use of metal. The lightening of structures has given rise to vibration problems, which are increasingly becoming decisive in matters of machine strength. At the beginning of the 20th century, numerous accidents convincingly show what catastrophic consequences the neglect of vibrations or ignorance of them can lead to.

The advent of new technology, as a rule, poses new problems for the theory of oscillations. So in the 30s and 40s. new problems arose, such as stall flutter and shimmy in aviation, flexural and flexural-torsional vibrations of rotating shafts, etc., which required the development of new methods for calculating vibrations. At the end of the 1920s, first in physics, and then in mechanics, the study of nonlinear oscillations began. In connection with the development of automatic control systems and other technical demands, since the 1930s, the theory of motion stability has been widely developed and applied, the basis of which was the doctoral dissertation of A. M. Lyapunov “The general problem of motion stability”.

The absence of an analytical solution for the problems of the theory of oscillations, even in a linear formulation, on the one hand, and of computer technology, on the other, has led to the development of a large number of various numerical methods for solving them.

The need to carry out vibration calculations for various types of equipment led to the appearance in the 1930s of the first training courses in the theory of vibrations.

Transition to IVperiod(early 1960s - present) is associated with the era of scientific and technological revolution and is characterized by the emergence of new technology, primarily aviation and space, robotic systems. In addition, the development of power engineering, transport, etc. put forward the problems of dynamic strength and reliability in the first place. This is due to an increase in operating speeds and a decrease in material consumption with a simultaneous desire to increase the resource of machines. In the theory of oscillations, more and more problems are solved in a nonlinear setting. In the field of oscillations of continuum systems, under the influence of the demands of aviation and space technology, problems arise in the dynamics of plates and shells.

The greatest influence on the development of the theory of oscillations in this period is exerted by the emergence and rapid development of electronic computing technology, which led to the development of numerical methods for calculating oscillations.

oscillating motion Any movement or change of state is called, characterized by one or another degree of repetition in time of the values ​​of physical quantities that determine this movement or state. Fluctuations are characteristic of all natural phenomena: the radiation of stars pulses; the planets of the solar system rotate with a high degree of periodicity; winds excite vibrations and waves on the surface of the water; inside any living organism, various, rhythmically repeating processes continuously occur, for example, the human heart beats with amazing reliability.

In physics, vibrations are distinguished mechanical and electromagnetic. With the help of propagating mechanical fluctuations in the density and pressure of air, which we perceive as sound, as well as very fast fluctuations in electric and magnetic fields, which we perceive as light, we receive a large amount of direct information about the world around us. Examples of oscillatory motion in mechanics can be vibrations of pendulums, strings, bridges, etc.

The fluctuations are called periodical, if the values ​​of physical quantities that change in the process of oscillations are repeated at regular intervals. The simplest type of periodic oscillations are harmonic oscillations. Oscillations are called harmonic, in which the change in the oscillating quantity over time occurs according to the sine (or cosine) law:

where x is the displacement from the equilibrium position;

A - oscillation amplitude - maximum displacement from the equilibrium position;

- cyclic frequency;

- the initial phase of the oscillation;

- oscillation phase; it determines the offset at any point in time, i.e. determines the state of the oscillatory system.

In the case of strictly harmonic oscillations of the value A, and do not depend on time.

Cyclic frequency is related to the period T of oscillations and frequency ratio:

(2)

Period T oscillations is called the smallest period of time after which the values ​​of all physical quantities characterizing oscillations are repeated.

Frequency oscillations is called the number of complete oscillations per unit time, measured in hertz (1 Hz = 1
).

Cyclic frequency numerically equal to the number of oscillations made in 2 seconds.

An oscillation that occurs in a system that is not subject to the action of variable external forces, as a result of any initial deviation of this system from a state of stable equilibrium, is called free(or own).

If the system is conservative, then no energy dissipation occurs during oscillations. In this case, free vibrations are called undamped.

Speed point fluctuations are defined as the derivative of the time shift:

(3)

Acceleration oscillating point is equal to the derivative of the speed with respect to time:

(4)

Equation (4) shows that the acceleration during harmonic oscillations is variable, therefore, the oscillation is due to the action of a variable force.

Newton's second law allows you to write in general terms the relationship between the force F and acceleration with rectilinear harmonic oscillations of a material point with a mass
:

where
, (6)

k is the coefficient of elasticity.

Thus, the force that causes harmonic vibrations is proportional to the displacement and directed against the displacement. In this regard, we can give a dynamic definition of a harmonic oscillation: a harmonic oscillation is called an oscillation caused by a force that is directly proportional to the displacement x and directed against the displacement.

The restoring force can be, for example, an elastic force. Forces of a different nature than elastic forces, but also satisfying condition (5), are called quasi-elastic.

In the case of rectilinear oscillations along the x axis, the acceleration equals:

.

Substituting this expression for acceleration and the meaning of strength
into Newton's second law, we get the basic equation of rectilinear harmonic oscillations:


 or
(7)

The solution to this equation is equation (1).

The course program of the theory of vibrations for students 4 FACI course


The discipline is based on the results of such disciplines as classical general algebra, theory of ordinary differential equations, theoretical mechanics, theory of functions of a complex variable. A feature of the study of the discipline is the frequent use of the apparatus of mathematical analysis and other related mathematical disciplines, the use of practically important examples from the subject area of ​​theoretical mechanics, physics, electrical engineering, acoustics.


1. Qualitative analysis of motion in a conservative system with one degree of freedom

  • Phase plane method
  • Dependence of the oscillation period on the amplitude. Soft and hard systems

2. Duffing equation

  • Expression for the general solution of the Duffing equation in elliptic functions

3. Quasi-linear systems

  • Van der Pol variables
  • Averaging method

4. Relaxation vibrations

  • Van der Pol equation
  • Singularly perturbed systems of differential equations

5. Dynamics of Nonlinear Autonomous Systems of General Form with One Degree of Freedom

  • The concept of "roughness" of a dynamical system
  • Bifurcations of dynamical systems

6. Elements of Floquet's theory

  • Normal solutions and multipliers of linear systems of differential equations with periodic coefficients
  • Parametric resonance

7. Hill equation

  • Analysis of the behavior of solutions to a Hill-type equation as an illustration of the application of the Floquet theory to linear Hamiltonian systems with periodic coefficients
  • The Mathieu equation as a special case of the Hill-type equation. Ines-Strett diagram

8. Forced oscillations in a system with a nonlinear restoring force

  • Relationship between the amplitude of oscillations and the magnitude of the driving force applied to the system
  • Changing the mode of movement when changing the frequency of the driving force. The concept of "dynamic" hysteresis

9. Adiabatic invariants

  • Action-Angle Variables
  • Preservation of adiabatic invariants under a qualitative change in the nature of motion

10. Dynamics of multidimensional dynamical systems

  • The concept of ergodicity and mixing in dynamical systems
  • Poincaré mapping

11. Lorentz equations. strange attractor

  • Lorentz equations as a model of thermoconvection
  • Bifurcations of solutions of the Lorentz equations. Transition to chaos
  • Fractal structure of a strange attractor

12. One-dimensional mappings. Feigenbaum's versatility

  • Quadratic mapping - the simplest non-linear mapping
  • Periodic orbits of mappings. Bifurcations of Periodic Orbits

Literature (main)

1. Moiseev N.N. Asymptotic methods of nonlinear mechanics. – M.: Nauka, 1981.

2. Rabinovich M.I., Trubetskov D.I. Introduction to the theory of oscillations and waves. Ed. 2nd. Research Center "Regular and Chaotic Dynamics", 2000.

3. Bogolyubov N.N., Mitropolsky Yu.A. Asymptotic Methods in the Theory of Nonlinear Oscillations. – M.: Nauka, 1974.

4. Butenin N.V., Neimark Yu.I., Fufaev N.A. Introduction to the theory of nonlinear oscillations. – M.: Nauka, 1987.

5. Loskutov A.Yu., Mikhailov A.S. Introduction to synergetics. – M.: Nauka, 1990.

6. Karlov N.V., Kirichenko N.A. Oscillations, waves, structures .. - M .: Fizmatlit, 2003.

Literature (additional)

7. Zhuravlev V.F., Klimov D.M. Applied Methods in the Theory of Oscillations. Publishing house "Science", 1988.

8. Stoker J. Nonlinear vibrations in mechanical and electrical systems. - M .: Foreign literature, 1952.

9. V. M. Starzhinsky, Applied Methods of Nonlinear Oscillations. – M.: Nauka, 1977.

10. Hayashi T. Nonlinear oscillations in physical systems. – M.: Mir, 1968.

11. Andronov A.A., Witt A.A., Khaikin S.E. Theory of vibrations. – M.: Fizmatgiz, 1959.

The book acquaints the reader with the general properties of oscillatory processes occurring in radio engineering, optical and other systems, as well as with various qualitative and quantitative methods for studying them. Considerable attention is paid to the consideration of parametric, self-oscillating and other non-linear oscillatory systems.
The study of the oscillatory systems and processes described in the book is given by known methods of the theory of oscillations without a detailed presentation and substantiation of the methods themselves. The main attention is paid to the elucidation of the fundamental features of the studied oscillatory models of real systems using the most adequate methods of analysis.

Free oscillations in a circuit with a non-linear inductance.
Let us now consider another example of an electrical non-linear conservative system, namely, a circuit with an inductance that depends on the current flowing through it. This case has no illustrative and simple non-relativistic mechanical analogue, since the dependence of self-induction on the current is equivalent for mechanics to the case of the dependence of mass on velocity.

We encounter electrical systems of this type when cores of ferromagnetic material are used in inductors. In such cases, for each given core, it is possible to obtain a relationship between the magnetizing field and the flux of magnetic induction. The curve depicting this dependence is called the magnetization curve. If we neglect the phenomenon of hysteresis, then its approximate course can be represented by the graph shown in Fig. 1.13. Since the magnitude of the field H is proportional to the current flowing in the coil, the current can be plotted directly on the abscissa axis on the appropriate scale.

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