Nonlinear acoustic vibrations. Theory of Nonlinear Oscillations Nonlinear Oscillations

1. The hypothesis of an infinitesimal perturbation used above in linear analysis does not allow us to consider the development of real perturbations. In the linear theory, as can be seen, the perturbation amplitude is either not defined at all (on the stability boundary) or grows indefinitely (in the instability zone), which is obtained as a consequence of its initial positions. In fact, at a certain amplitude of perturbations, nonlinear effects become significant, which prevent an infinite increase in the amplitude and lead to a limiting cycle of oscillations.

Nonlinearity begins to manifest itself only for perturbations with a certain (critical) amplitude: at a lower amplitude, according to the nonlinear theory, the oscillations die out, at a larger one, the so-called nonlinear instability takes place (instability in the large, impulse instability). The non-linearities of the oscillatory process in a solid propellant rocket motor are determined by the non-linearity of the combustion process and wave motion in the chamber, which manifests itself in an increase in the curvature of pressure waves, dispersion of perturbations, and in the occurrence of shock waves.

Despite the fact that linear theories provide a fairly complete understanding of the problem of solid propellant rocket instability, they cannot solve the question of the most dangerous for the engine and for the entire aircraft oscillations of large amplitude, which is extremely important for practice. Therefore, more and more attention is paid to the study of such nonlinear oscillations. At present, it is possible to indicate a narrow circle of already solved nonlinear problems.

2. Initial Equations . Let us consider in the following formulation the problem of nonlinear acoustic oscillations for a one-dimensional flow. The system of nonlinear differential equations for such a case can be represented as follows:

gas mass conservation equation

particle mass conservation equation

; (5.85)

momentum conservation equation

; (5.86)

energy conservation equation

where index " l » means mass flow per unit length; v- per unit volume; other indices and values are the same.

3. Key Assumptions . To solve these equations, we make the following assumptions:

There is no afterburning, i.e. E = 0; Q = 0;

Energy exchange is represented by heat exchange between particles and gas in the CS;

The cross section of the charge channel is unchanged, i.e. F= const;

At z= 0 velocity of gas and particles of the region zero;

For a two-phase flow in the nozzle, a constant lag of the heavy fraction is assumed;

The operating mode of the nozzle is quasi-stationary;

The characteristics of transient combustion are determined by the sensitivity function in the form

. (5.88)

therefore, the combustion characteristic assumes linearity;

The relationship between the burning rate and pressure is taken into account, in some cases - with the flow rate;

Particles are considered only one size, and using a linear and non-linear drag coefficient.

4. Numerical solution results . Numerical methods for solving nonlinear stability problems include the method of characteristics, the "discretization" method, etc. In the latter case, the solution of the problem is approximated under the assumption that the nonlinearity is satisfied at a finite number of discrete points. The system of equations presented (5.84) ... (5.87) can be solved, for example, by the method of characteristics. Such a solution, obtained by F. Kulik, gives the dependence of the amplitude of perturbations on time. Examples of the results of numerical calculations by F. Kulik are shown in Fig.7. The initial conditions were set in the form of a standing wave of the fundamental frequency of the chamber. The initial perturbation was an equal part of the first and second modes, but after three cycles the pressure almost did not contain the second harmonic. The influence of the connection with transient combustion in this case obviously plays a decisive role; sensitivity function at accepted A And IN shows this to a strong extent for the fundamental frequency and to a weak extent for the second mode. It can also be noted that the pressure amplitude does not begin to increase immediately; moreover, even some of its attenuation after one cycle is observed. This can be explained by the fact that the combustion rate only after several cycles reaches a value corresponding to the resulting pressure perturbations.

Nonlinear effects can manifest themselves in many different ways. A classic example is a non-linear spring, in which the restoring force is non-linearly dependent on extension. In the case of symmetric nonlinearity (the same response under compression and tension), the equation of motion takes the form

If there is no damping and there are periodic solutions in which, for , the natural frequency increases with amplitude.

Rice. 1.7. The classical resonance curve of a non-linear oscillator with a rigid spring in the case when the oscillations are periodic and have the same period as the driving force (a and are defined in equation (1.2.4)).

This model is often called the Duffing equation after the mathematician who studied it.

If a periodic force acts on the system, then in the classical theory it is believed that the response will also be periodic. The resonance of a non-linear spring at a response frequency coinciding with the frequency of the force is shown in fig. 1.7. As shown in this figure, with a constant driving force amplitude, there is a range of driving frequencies in which three different response amplitudes are possible. It can be shown that the dashed line in Fig. 1.7 is unstable, and as the frequency increases and decreases, hysteresis occurs. This phenomenon is called overshoot, and it has been observed in experiments with many mechanical and electrical systems.

There are other periodic solutions, such as subharmonic and superharmonic oscillations. If the driving force has the form , then subharmonic oscillations can have the form plus higher harmonics ( - integer). As we will see below, subharmonics play an important role in prechaotic oscillations.

The theory of nonlinear resonance is based on the assumption that a periodic action causes a periodic response. However, it is precisely this postulate that is challenged by the new theory of chaotic oscillations.

Self-excited oscillations are another important class of nonlinear phenomena. These are oscillatory movements that occur in systems without periodic external influences or periodic forces. On fig. 1.8 shows a few examples.

Rice. 1.8. Examples of self-excited oscillations: a - dry friction between the mass and the moving rheum; b - aeroelastic forces acting on a thin wing; c - negative resistance in the circuit with the active element.

In the first example, the vibrations are caused by the friction created by the relative motion of the mass and the moving belt. The second example illustrates a whole class of aeroelastic oscillations, in which stationary oscillations are caused by a stationary fluid flow behind a solid body on an elastic suspension. In the classic electrical example shown in Fig. 1.9 and investigated by Van der Pol, the circuit includes an electron tube.

In all these examples, the system contains a stationary energy source and a source of dissipation, or a non-linear damping mechanism. In the case of the Van der Pol oscillator, the energy source is a constant voltage.

Rice. 1.9. Diagram of a vacuum tube circuit oscillating in a limit cycle of the same type that van der Pol investigated.

In the mathematical model of this circuit, the energy source enters in the form of a negative resistance:

Energy can enter the system at small amplitudes, but as the amplitude increases, its growth is limited by nonlinear damping.

In the case of a Froude pendulum (see, for example, ), energy is supplied by stationary rotation of the axis. For small oscillations, nonlinear friction plays the role of negative damping; meanwhile, for strong oscillations, the amplitude of oscillations is limited by the nonlinear term

The oscillatory motions of such systems are often called limit cycles. On fig. 1.10 shows the trajectories of the Van der Pol oscillator on the phase plane. Small fluctuations spiral, approaching a closed asymptotic trajectory, and large-amplitude motions contract spirally to the same limit cycle (see Figs. 1.10 and 1.11, where ).

When studying such problems, two questions often arise. What is the amplitude and frequency of oscillations on the limit cycle? At what values of the parameters are there stable limit cycles?

Rice. 1.10. Solution with a limit cycle for the Van der Pol oscillator depicted on the phase plane.

Rice. 1.11. Relaxation oscillations of the Van der Pol oscillator.

In the case of the van der Pol equation, it is convenient to normalize the space variable to and the time to , so that the equation takes the form

Where . For small, the limit cycle is a circle of radius 2 on the phase plane, i.e.

where are the harmonics of the third and higher orders. At large, the motion takes the form of relaxation oscillations shown in Figs. 1.11, with a dimensionless period of about 1.61 at

The problem with a periodic force in the Van der Pol system is more complicated:

Since this system is non-linear, the principle of superposition of free and forced oscillations is inapplicable. Instead, the resulting periodic motion is captured at the driving frequency when the latter is close to the limit cycle frequency. With a weak external action, there are three periodic solutions, but only one of them is stable (Fig. 1.12). For large values of the force amplitude, there is only one solution. In any case, with increasing detuning - at fixed, the captured periodic solution turns out to be unstable and other types of motion become possible.

Rice. 1.12. Amplitude curves for the forced motion of the Van der Pol oscillator (1.2.9).

With large differences between the driving and natural frequencies in the van der Pol system, a new phenomenon appears - combination oscillations, sometimes called almost periodic or quasi-periodic solutions. Combination oscillations have the form

When the frequencies and are incommensurable, i.e. - an irrational number, the solution is called quasi-periodic. For the Van der Pol equation , where is the frequency of the limit cycle of free oscillations (see, for example, ).

fluctuations in physical systems described by nonlinear systems of ordinary differential equations

Where contains terms not lower than the 2nd degree in terms of vector components - the vector function of time is a small parameter (or and ). Possible generalizations are related to the consideration of discontinuous systems, actions with discontinuous characteristics (for example, such as hysteresis), delay and random actions, integro-differential and differential-operator equations, oscillatory systems with distributed parameters described by partial differential equations, as well as with using methods of optimal control of nonlinear oscillatory systems. The main general tasks of N. to .: finding equilibrium positions, stationary modes, in particular periodic. movements, self-oscillations and the study of their stability, the problems of synchronization and stabilization of N. to.

All physical. systems are, strictly speaking, non-linear. One of the most characteristic features of N. k. is the violation in them of the principle of superposition of oscillations: the result of each of the influences in the presence of another turns out to be different than in the absence of another influence.

Quasilinear systems - systems (1) at . The main research method is small parameter method. First of all, this is the Poincaré-Lindstedt method for determining the periodicity. solutions of quasilinear systems that are analytic in the parameter at sufficiently small values, either in the form of power series (see Chap. IX) or in the form of power series and - additions to the initial values of the vector components (see Chapter III). For further development of this method, see, for example, in -.

Another of the small parameter methods is the method averaging. At the same time, new methods also penetrated into the study of quasilinear systems: asymptotic. methods (see , ), the method of K-functions (see ), based on the fundamental results of A. M. Lyapunov - N. G. Chetaeva, and others.

Essentially nonlinear systems, in which there is no pre-assigned small parameter . For Lyapunov systems

moreover, among the eigenvalues of the matrix there are no multiples of the root - analytical vector function X, the expansion of which begins with terms of at least 2nd order, and there is an analytic first integral of a special form, A. M. Lyapunov (see § 42) proposed a method for finding periodic. solutions in the form of a series in powers of an arbitrary constant c (which can be taken as the initial value of one of the two critical variables or ).

For systems close to Lyapunov systems,

where of the same form as in (2) - analytical. vector-function and small parameter , continuous and -periodic in t, also proposed a method for determining the periodic. solutions (see Chapter VIII). Lyapunov-type systems (2), in which the matrix has l zero eigenvalues with simple elementary divisors, two purely imaginary eigenvalues, and no eigenvalues that are multiples of - the same as in (2), can be reduced to Lyapunov systems (see IV.2). N. to. in the Lyapunov systems and in the so-called. Lyapunov systems with damping, and also solved the general problem of energy transfer in them (see Chapters I, III, IV).

Let an essentially nonlinear autonomous system be reduced to the Jordan form of its linear part

where the vector, by assumption, has at least one non-zero component; , are equal to zero or one, respectively, in the absence or presence of non-simple elementary divisors of the matrix of the linear part, - coefficients; the set of vector values with integer components is:

Then there is a normalizing transformation:

leading (3) to the normal form of differential equations

and such that if . Thus, the normal form (5) contains only resonance terms, i.e., the coefficients can differ from zero only for those for which the resonance equation

which plays an essential role in the theory of oscillations. The convergence and divergence of the normalizing transformation (4) has been studied (see Part I, Chapters II, III); the calculation of the coefficients (by means of their symmetrization) is given (see § 5.3). In a number of problems on the N. c. of essentially nonlinear autonomous systems, the method of normal forms has proved to be effective (see, Ch. VI-VIII).

Of the other methods for studying essentially nonlinear systems, the method of point mappings (see , ), strobosconic. method and functional-analytical. methods .

Qualitative methods of N. to. Initial here are the studies of the form of integral curves of nonlinear ordinary differential equations, carried out by A. Poincare (N. Poincare, see). Applications for problems of N. to., described by autonomous systems of the 2nd order, see,. Questions of the existence of periodic solutions and their stability in the large for multidimensional systems; Almost periodic N. c. are considered. Applications of the theory of ordinary differential equations with a small parameter for certain derivatives to problems of relaxation N. c.

Important aspects of N. to. and lit. see articles Perturbation theory, Oscillation theory.

Lit.: Poincaré A., Fav. works, trans. from French, vol. 1, M., 1971; Andronov A. A., Witt A. A., Khaikin S. E., Theory of Oscillations, 2nd ed., M., 1959; Bulgakov B. V., Fluctuations, M., 1954; Malkin I. G., Some problems in the theory of nonlinear oscillations, M., 1956: Bogolyubov N. N., Izbr. works, vol. 1, K., 1969; [b] N. N. Bogolyubov, Yu. A. Mitropolsky, Asymptotic Methods in the Theory of Nonlinear Oscillations, 4th ed., M-, 1974; Kamenkov G.V., Izbr. works, vol. 1-2, M., 1971-72; Lyapunov A. M., Sobr. soch., vol. 2, M.-L., 195B, p. 7-263; Starzhinsky V. M., Applied Methods of Nonlinear Oscillations, M., 1977; Bruno A.D., "Proceedings of the Moscow Mathematical Society", 1971, v. 25, p. 119-262; 1972, v. 26, p. 199-239; Neimark Yu. I., Method of point mappings in the theory of nonlinear oscillations, M., 1972; Minorsky N., Introduction to non-linear mechanics, Ann Arbor, 1947; Krasnoselsky M. A., Burd V. Sh., Kolesov Yu. S., Nonlinear almost periodic oscillations, M., 1970; Poincare A., On curves defined by differential equations, trans. from French, M.-L., 1947; Butenin N. V., Neimark Yu. I., Fufaev N. A., Introduction to the theory of nonlinear oscillations, M., 1976; Plise V. A., Non-local problems of the theory of oscillations, M.-L., 1964; Mishchenko E. F., Rozov N. Kh., Differential Equations with a Small Parameter and Relaxation Oscillations, Moscow, 1975.

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Great Soviet Encyclopedia

“VOCATIONS” IN DEFINITIONS

author Golovin Boris Nikolaevich

"VOCATIONS" OF DEFINITIONS At the lesson, the students were given an exercise: to enter a definition in the phrase five workers. The students quickly offered their examples: five young workers, five old workers, five skilled workers... There was no difficulty.

§ 1 Economic fluctuations

From the book Fundamentals of Economics author Borisov Evgeny Filippovich

§ 1 Economic fluctuations When looking for the truth, we come across a paradox (an unexpected phenomenon that does not correspond to conventional ideas). What the undulating movement of the economy looks like

Kitaygorodsky Alexander Isaakovich

V. Vibrations Balance In some cases, balance is very difficult to maintain - try walking a tightrope. At the same time, no one awards applause to the person sitting in the rocking chair. But he also maintains his balance. What is the difference in these

fluctuations

From the book Course of Russian History (Lectures XXXIII-LXI) author Klyuchevsky Vasily Osipovich

Fluctuations In answering this question, we will go over all the most prominent phenomena of our inner life. They are very complex, they go in different, often intersecting and sometimes oncoming currents. But you can see their common

Far from any fluctuations, the restoring force is proportional to the deviation (i.e., it changes according to the law (- kx)). Consider, for example, the spring shown in Figure 2.74. It consists of several plates. With small deformations, only long plates bend. Under heavy loads, shorter (and more rigid) plates are also subject to bending. The restoring force can now be described as follows:

battery mode switches to aperiodic, when the oscillations disappear and the body just slowly approaches the equilibrium position (Fig. 2.72, b, c).

Enter instead of the line where dots are placed (t, x), line where dots will be placed ( x,v), and obtain phase portraits of damped oscillations for different friction. You can also use one of the ready-made programs Phaspdem* or Phport* from those available in the PAKPRO package. Diagrams of the type shown in Figure 2.73 should be obtained.

For it to be returning, i.e. F And X always had different signs, it should be expanded in a series in odd powers X. Because the potential energy U related to strength by the formula F = - dU/dx, it means that

i.e., oscillations occur in a potential well with walls steeper than those of a parabola (Fig. 2.75, a). The friction of the plates against each other provides the damping needed to dampen the oscillations.

Oscillations are also possible in an asymmetric well, when

(Fig. 2.75, b). The restoring force will be equal to

When solving problems for nonlinear oscillations, the use of a computer is inevitable, since there are no analytical solutions. On a computer, the solution is not at all difficult. It is only necessary in the line where the speed increase is performed (v = v + F At/m), write the full expression for F, for example -kh-gh 2 - px 3 .

Example. The program for drawing a graph of non-linear oscillations is given in the PAKPRO package under the name Nlkol. Put her to work. You should get a series of curves for different initial deviations. When x 0 is greater than a certain value, the oscillating particle leaves the potential well, having overcome the potential barrier.

Try also the programs Ncol* And Nlosc.*, available in the PAKPRO package, as well as programs that can be used to obtain phase portraits of nonlinear oscillations: Phaspnl*, Phportnl*.

Note that, strictly speaking, almost any oscillations are non-linear. Only at small amplitudes can they be considered linear (neglect the terms c x 2 , x 3 , etc. in formulas like (2.117)).

Let the oscillator, in addition to the restoring force, which provides natural oscillations with a frequency C00, also be affected by an external force, which changes periodically with a frequency co, equal or not equal to (Oo. This force will swing the body with a frequency co. The resulting oscillations are called forced.

The equation of motion in this case will be:

First, there is a process of establishing oscillations. From the first push, the body begins to oscillate with its own frequency from 0. Then, gradually, the natural oscillations fade, and the driving force begins to control the process. Forced oscillations are established no longer with a frequency (00, but with a frequency of the driving force ω. The transient process is very complicated, there is no analytical solution. When solving the problem by a numerical method, the program will be no more complicated than, say, the program for damped oscillations. line, where, in accordance with the equation of motion, the speed is increased, add the driving force in the form FobiH = Focos(cot).

Example. The PACG1RO package contains an example of a program for obtaining a graph of forced oscillations on a computer screen. See also programs Ustvcol.pas And UstvcoW.pas. The resulting x(?) graph and phase diagram v(x) shown in Figure 2.76. With a successful selection of parameters, it is clearly seen how forced oscillations are gradually established. It is also interesting to observe the establishment of forced oscillations in the phase diagram (program Phpforce.pas).

When oscillations with frequency ω have already been established, we can find the solution of equation (2.118) in the form

Here Jo is the amplitude of steady oscillations. If we substitute (2.119) into (2.118), having previously found the time derivatives X" And X" and given that To= coo 2 m, then it turns out that (2.119) will be a solution of equation (2.118) provided that

Friction was not taken into account, coefficient A assumed to be zero. It can be seen that the amplitude of the oscillations increases sharply as ω approaches C0 (Fig. 2.77). This phenomenon is called resonance.

If there really were no friction, the amplitude at co = (Oo) would be infinitely large. In reality, this does not happen. The same figure 2.77 shows how the resonant curve changes with increasing friction. But still, if co and coo coincide, the amplitude can become tens and hundreds of times more than with F COo. In engineering, this phenomenon is dangerous, since the driving vibrations of the engine can get into resonance with the natural frequency of any parts of the machine, and it can collapse.

NONLINEAR OSCILLATIONS

fluctuations in physical systems described by nonlinear systems of ordinary differential equations

Essentially nonlinear systems, in which there is no pre-assigned small parameter . For Lyapunov systems

moreover, among the eigenvalues of the matrix there are no multiples of the root - analytical vector function X, expansion of which begins with terms not lower than 2nd order, and there is an analytical special form, A. M. Lyapunov (see § 42) proposed a method for finding periodic. solutions in the form of a series in powers of an arbitrary constant c (which can be taken as the initial value of one of the two critical variables or ).

For systems close to Lyapunov systems,

where of the same form as in (2) - analytical. vector-function and small parameter , continuous and -periodic in t, also proposed a method for determining the periodic. solutions (see Chapter VIII). Systems of Lyapunov type (2), in which it has l zero eigenvalues with simple elementary divisors, two purely imaginary eigenvalues, and no eigenvalues that are multiples of - the same as in (2), can be reduced to Lyapunov systems (see IV.2). N. to. in the Lyapunov systems and in the so-called. Lyapunov systems with damping, and also solved the general problem of energy transfer in them (see Chapters I, III, IV).

Let the essentially nonlinear part be reduced to the Jordan form of its linear part

Then there is a normalizing transformation:

leading (3) to the normal form of differential equations

and such that if . Thus, (5) contains only , i.e., the coefficients can differ from zero only for those for which the resonance equation

Of the other methods for studying essentially nonlinear systems, the method of point mappings (see , ), strobosconic. method and functional-analytical. methods .

Important aspects of N. to. and lit. see articles Perturbations, Oscillation theory.

V. M. Starzhinsky.

Mathematical encyclopedia. - M.: Soviet Encyclopedia. I. M. Vinogradov. 1977-1985.

See what "NON-LINEAR OSCILLATIONS" is in other dictionaries:

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