# Get PDF Some problems in the theory of nonlinear oscillations

The zero subscript on avt and on the partial derivatives denotes evaluation at q10 t , In the expres-. In Subsection 1. The kinetic energy K is positive and therefore equations 1, 1. Kinetic and potential energies. The first spring is fixed at point 0, and each subsequent spring is. We express the potential energy II of the linear elastic forces of the springs and of the forces of gravity as f. It is readily demonstrated, by analogy with Subsection 1.

The theorem of Subsection 1. One possible solution of equations 1. We assume this solution to be the unperturbed motion 1. The instability of unperturbed motion is determined by the instability of the trivial solution of the first group 1, 1. We consider a two-link oscillatory chain where the first link is absolutely rigid and the second is elastic with force constant c. The notation is clear from Fig. The static elongation of the spring is 1. Formulas 2. Wide instability regions those where the angle between the tangents is distinct from zero may be contiguous only to those points of the half-axis f.

The results of the calculations are given below. The regions of the first 1 and second 2 principal resonances Fig. A 2 from the inequalities. Pendulum subject to elastic free suspension. Let a simple pendulum of mass m 2 and rod length l 2 be suspended from a hinge of mass m 1 Fig. A spring of unstressed length l 1 and with force 0.

These vertical oscillations of masses m 1 and m 2 are taken as unperturbed motion. The first group of variational equations 1, 1. The instability of unperturbed motion 4. Formulas V, 2. The calculation of the slopes x'f of the tangents to these regions will not be discussed here. Pendulum subject to elastic guided suspension. Let the motion of mass m1 be constrained by vertical guides Fig. This oscillatory chain is not FIG. The kinetic and potential energies are now given by 1. The instability of unperturbed motion 5. The literature devoted to Mathieu's equation is quite extensive; for our purposes it is sufficient to apply equation 3.

Equation 3. This region i. This chapter treats several problems in mechanics and physics. Computational aspects of the Poincare method [a] are presented in the first two sections; as for the method of averaging, only the simplest applications in the sen:-e of Vander Pol [] will be given. More complicated problems some of them are pointed out in Section 4 are likely to require application, and possibly modification, of algorithms evolved in the fundamental investigations of Bogolyubov [22, ], Mitropolskii [ d], and Samoilenko [24].

It is possible to outline a general approach to the problem of energy transfer Sections The first step of the solution is based on the mathematical theory of parametric resonance, which is used to fmd the initial usually trivial periodic mode and to determine its instability regions in the space of the system's parameters. This approach has already been employed in Chapter II, Subsection 1. The second step of the solution consists in determining periodic modes that appear at critical values of the parameters and are different from the initial mode.

This step is based on the transformations given in Chapter I, Subsection 1. Other methods of small parameter can also be applied to the transformed system, for example, the method of averaging, which enables us to realize the third step of the solution, namely, an analysis of the transient process, which is often referred to as energy transfer. The three steps of the solution are illustrated for mechanical systems Sections 1, 3 and for a physical system Section 2 with two degrees of freedom.

Loss of Stability of Vertical Vibrations of a Spring-Loaded Pendulum For certain values of the parameters, vertical vibrations of a point mass suspended from a spring become unstable as the result of any. This section presents a mathematical treatment of this process. Step 1. We consider the motion of mass m suspended from a weightless spring of unstressed length l and with force constant c see Fig. Hook's law is assumed valid. The equations of motion then become. Since the system is conservative and the constraints do not depend explicitly on time, there exists the energy integral.

General formulas for transforming systems of second-order equations were given in Chap-. The trivial solution of equation 1. In positions of maximum elongation or contraction the potential energy II of the mass is equal, according to 1. The stability of 1. Step 2. We wish to determine the periodic solutions of equation 1. To achieve this, let us use the method of small parameter for nonautonomous systems with one degree of freedom in the form suggested by Poincare [a], vol.

We seek this solution in the form of the series see I, 2. Equation 2. Solution 2. Therefore, solution 2. Therefore, equation 1. Any positive rational number is either expressed by 2. Taking into account equations 2. This is the equation determining the first correction in of the 2pn-periodic solution of equation 2.

The nonhomogeneous part of equation 2. Let us fmd out when one of these frequencies coincides with the cyclic frequency of the generating solution a. In case c , equality is impossible. In cases a and b , equation 2. Formula 2. The equations for generating amplitudes become identities for. Finally, we shall analyze case a in which equation 1. Formulas I, 1, 2. It should also be noted that the second formula of 2.

Step 3. Next we study the loss of stability of vertical vibrations. We consider, therefore, the equation. Unfortunately, the substitution of this integral into the first equation of 3. Of course, the truncated Van der Pol equations give only the first approximation of the solution to 3.

Consequently, we limit our analysis to an approximate integration of system 3. The second 2 V2 and 0 for I a I equation of 3. The f1rst equation of 3. Note that the transient process covers a time interval of the order of 0 f.! The two signs in 3. On Coupling of Radial and Vertical Oscillations of Particles in Cyclic Accelerators In this section we define pure radial side-to-side oscillations and analyze their stability.

Oscillation equations are then transformed and vertical radial up-and-down oscillations are found by means of the Poincare method of small parameter. There exists a unique value of the principal physical parameter for which the reduced amplitude of up-and-down oscillations is unique, whereas pure side-to-side oscillations of any amplitude, no matter how small, are unstable at this value of the parameter. Then we describe the transient process and find the time of transition from pure side-to-side to up-and-down oscillations and point out an analogy with vibrations of a spring-loaded pendulum in transient processes.

The equations of betatron oscillations of particles in cyclic accelerators with weak focusing are given by [], 4. Integral I, 1, 2. Before transforming system 1. As follows from integral 1. To analyze the stability of these oscillations, we set in 1. Hence, the instability of side-to-side oscillations 1. The principal physical parameter is n see 1. Therefore, in the first approximation the only wide instability region 1. By using this analogy, we can show that equation 2. The set of values of n given by 2. Formula 1, 2. G are unstable for any amplitude 1-l no matter how small see 1.

Analogously to 1, 2. The value of! The truncated Van der Pol equations take the form of 1, 3. The expression for the Van der Pol amplitude is found in a manner similar to that of Subsection 1. The duration 7 of the process of transition from pure side-to-side oscillations 1. The initial periodic mode is a vertical oscillation II, 2, 5. Determination of nontrivial periodic modes Step 2.

By employing formulas II, 2, 5. Expanding the right-hand sides of the system of equations II, 2, 5. Comparing this with I, 1, 2. In the case under discussion, system 1, 1, 2. By calculations similar to those of Subsections 1. Together with formulas 1, 1, 2. The truncated Van der Pol equations obtained by averaging over the independent variable explicitly included in 2. An attempt to solve equations 2. Of course, the truncated Van der Pol equations 2. Therefore we are satisfied with approximate integration of system 2.

The second equation of 2. Va for I a I 2 2 we recall that 2 V2i is the amplitude of the generating solution 1. The first equation of 2. Note that the transition time is of the order of 0! Transformation of equations of motion. We expand the righthand sides of system II, 2, 4. The primes denote the derivatives with respect to -r, and the expansions of the functions Y, X, and C!

The energy integral I, 1, 2. Periodic solutions. The general solution of the generating system 1. An analysis of the first equation in I, 2, 1. Lyapunov Systems with Damping An unperturbed nonlinear autonomous Lyapunov system of order 2k 2 is perturbed by an analytic damping of sufficiently small norm. A solution of the latter system is assumed known for a sufficiently small square root as compared with unity of the initial reduced energy of the system. The first and all subsequent corrections of the corresponding solution of the perturbed system i.

According to Poincare's results [a], vol. II the integration of a system of variational equations can be reduced to quadratures if the general integral of the unperturbed system is known. Subsections 1. We consider a class of Lyapunov systems with damping where each system is represented by a second-order equation. We assume that the unperturbed system 1. In the linear case condition 1. The Lyapunov substitution. Taking 1. The unperturbed system 1. Cnmplete system of variational equations in the Poincare parameter and its solution.

Let us represent system 1. Let us assume that we know a solution x 0 'l't of the unperturbed system 2. On the basis of the Poincare theorem [a , vol. II , we seek a solution of system 2. Subtracting identity 2. In general, these operators are noncommutative, so that, say, the terms with x 1 x 2 and x 2 x 1 are considered distinct. If e is a linear element of 2. Successive integration of equations 2. If, however, the general integral of the unperturbed system 2. The solution of the first system of 2. In order to fi. Vibrations in mechanical systems with one degree of freedom and different types of nonlinearity.

We consider a mechanical system with one degree of freedom that is subject to restoring and resistance forces. Both of these functions are assumed to be analytic functions of the coordinates and velocity, respectively. II , which in general is ineffective. As for the restoring force, its nonlinear part determines the upper bound on the parameter f.

The general solution suggested in the fi. With a suitable time scale 'T, the equation of motion of the system in question can be written as. The unperturbed equation 3. We assume henceforth that for all e and pin the rectangle 0 ;; e 80 and 0 p r 0 and for any 'l't the following inequality holds. Together with the first two formulas of 1. Under condition 3.

The integral of equation 3. On the basis of the Poincare theorem [a], vol. II , we seek a solution of 3. A sequence of the differential equations 2. We are now interested in the first correction p1 'l't; It ; an appropriate differential equation for this correction is a variational equation in the. Poincare parameter the first equation of 2.

Let us derive it directly. Recasting 3. Dividing 3. Therefore it is preferable to resort to the Poincare method [a], vol. II presented at the end of Subsection 1. In our case equation 3. Then a solution of 3. The Duffing equation with linear damping. As an example, we consider the equation. The energy integral 3. The substitution. The general solution of equation 4. Spring-loaded pendulum with linear damping. We consider a plane spring-loaded pendulum of mass m suspended from a weightless spring of unstressed length l and with force constant c Fig.

The unperturbed system 5. In the example above, the vectors a and x 0 are. In formula 2. Omitting obvious computations in 2. Finally, the solution of system 5. On Lyapunov-Type Systems Definition 1. A nearly Lyapunov system is a real system of the type 0. Definition 2. A Lyapunov-type system is a real system of the type dx. Lyapunov-type systems are shown in this section to be reducible to Lyapunov systems. We present a Lyapunov-type system in detailed form, using notation distinct from that of 0. We assume that in addition to the Lyapunov scalar integral y, y. Expansions of 'ljl, cp 1 ,.

Transformation of Lyapunov-type systems. Lemma I. The proof of the lemma is based on the Lyapunov transformation [a], Sec. Namely, we perform a substitution of variables in 1. Substitution of the new variables yields new nonlinear terms are denoted by and. It follows from 2. System 2. We assume that system 1. We resolve the vector integral 1. Substitution of 2.

Lemma II. Given a vector c in integral 1. Let us differentiate integral 2. The determinant of the homogeneous system 2. Owing to continuity, these equalities must also hold for the vector c sufficiently small in norm. The lemma is therefore proved. The periodic solutions of a Lyapunov system 2. Other aspects of system 2. The problem of reducing an autonomous system of ordinary differential equations to its simplest, normal form by means of a change of variables was formulated by Poincare [a] and later developed by Lyapunov [a] and others.

The results of the most general character were obtained by Brjuno [j]; the relevant references can be found in his publication. Sections 1 and 2 of this chapter describe the problem on the basis of Brjuno's results, but adjust the presentation to the problems of nonlinear oscillations governed by the systems of equations in question, thus avoiding the most general approach and the most general situations.

Some of the proofs in Section 2 are omitted. By virtue of the Weierstrass theorem see, for example [80], Sec. The problem consists in reducing system 1. In particular, Brjuno [a] introduced the following notation for the transformed system. Normal forms are defined in the following theorem, which we formulate in terms of this subsection. The Fundamental Brjuno theorem [j], Ch.

There exists a reversible transformation 1. Here Yvhv y , xvfv x , and Yvgv y are power series containing no zero- and first-power terms. Transformation 2. This is true for the first and second terms in the right-hand side of 2. Equalities 2. Here CvQ stands for the right-hand side of 2. As a result, gvQ and hvQ are determined from Q in the manner described, thus proving the theorem. As stipulated by b , only 2. If for q1 for a finite number of values of the initial parameters of the system, then it is logical that hvQ be based, if possible, on continuity.

The Poincare theorem [a]. The existence of a unique formal transformation of 1. We write A, Q , using 1. By condition 3. This completes the proof; the reader will fmd an analysis of the conwrgence of transformation 1. In the real-variable case discussed above, the spectrum J.. Therefore, condition 2 is satisfied not only by H but by a straight line H symmetric with respect to the real axis, and thus by the imaginary axis of the complex plane 'A as well. Consequently, it is sufficient to eheck condition 2 for the imaginary axis.

Additional Information 2. Some properties of normalizing transformations. At the end of the proof of the fundamental Brjuno theorem Subsection 1. Q of the normalizing transformation 1, 2. At the same time, although the structure of the normal form is fixed by the established numbering of the variables, its coefficients gvQ depend on the choice of coefficients for the normalizing transformation. It is logical to assume that subsequent 1ransformations of the variables, if carried out for resonant terms only, will transform one normal form into another.

This clarifies the meaning of the Brjuno theorem [j], Ch. We also assume that series 1. As follows from the third Brjuno theorem ljl, Ch. Classification of normal forms; integrable normal forms. We consider the case when in 1, 1. We now number the variables so that.

For instance, from 2. The Brjuno theorem [j], Ch. Under the previous assumptions, a normal form can be written as. The first sum in 2. In the normal form 1, 2. This last equation resonant equation is equivalent, owing to 2. In order to emphasize the number of the equation of the normal form to which Q corresponds, we give Q an appropriate subscript. The SP"Ond equation of 2. The terms given in 2. Let us analyze several special cases of the above theorem. This normal form was derived by Dulac [ Equations have a finite number of solutions and no solutions in the hypothesis of the Poincare theorem of Subsection 1.

Equations 2. The integration of the normal form 2. Concept of power transformations. We wish to discuss the possibility of lowering the order of the normal form 1, 2. The term yQ is transformed by 3. System 3. But if it is analytic in wme neighbourhood of zero, and the initial ystem is a normal form, then 3. The possibility of lowering the order of the form is established by the Brjuno theorem [j], Ch. There exists a birational transformation 3. The first d equations of this system form a system of order d. Methods for efficiently constructing A can be found in [23Hcl.

The Brjuno theorem on convergence and divergence of normalizing transformations. In the normalizing transformation 1. We assume Wk. For condition A, we recall that inequalities 2. As in Subsection 2. Condition A in the case considered above when inequalities 2.

The Brjuno theorem on convergence and divergence of normalizing transformations [jl, Ch. II, III. To illustrate this, we return to the Poincare theorem Subsection 1. Obviously, in this case condition A is trivial automatically satisfied and Wk. Moreover, in the hypothesis of the Poincare theorem, a normalizing transformation is also single-valued, so that according to statement 1 of the above theorem this transformation is convergent in fome neighbourhood of its zero values. Fundamental identities.

We assume that an oscillatory system is described by a real autonomous system of nth-order differential equations. Here and henceforth a repeated subscript indicates summation and takes on the values 1, According to the fundamental Brjuno theorem see Subsection 1. Summation in 1. Substitution of 1. By dropping the terms with powers above three, we obtain the formal identities the derivatives with respect to 't' are primed.

By changing the summation indices and symmetrizing the coefficients in the sums, we derive the fundamental identities. Computational alternative. The following alternative is true: 1 Suppose that the values taken on by v, l, m, and p and by the real parameters of the initial oscillatory system on which Av, A1, 'Am, and 'Ap depend are such that the expressions in parentheses in the fourth and fifth sums in the left-hand side of 1.

Comparing the terms containing YzYm in the left- and right-hand sides of the fundamental identities 1. Indeed, 1. The absence of the term containing YzYmYp in the second sum of 1. Equating the coefficients of the quadratic terms in 1. We wish to emphasize that expressions 2.

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This has been already mentioned see condition b at the end of the proof of the fundamental Brjuno theorem Subsections 1. Comparison of the terms containing YzYm and YzYmYp in the left- and right-hand sides of the fundamental identities 1. It should Le emphasized that formulas 2. A general procedure for determining the coefficients of a normalizing transformation and a normal form is contained in the proof of the fundamental Brjuno theorem Subsection 1.

The method we have outlined above seems to us better suited for applications to oscillation problems; hence, the term "practical" in the section title. Fundamental identities in general form and their transformation. Unification of notation in the initial diagonal system 1. The subscripts repeated twice denote summation from 1 to n, summation being independent of other subscripts. We also use the concise notation. Substitution of 3. Nonsymmetric coefficients generated in the course of the calculations must be symmetrized since the coefficients in question are subject to conditions 3.

We replace the summation indices in each addend of the sum over fL in the second term in the left-hand side of 3. It becomes obvious that all the addends of the sum over fL are identical, so it can be expressed as one addend times x. Thus we have completed the transformations. Note that the numbers jp 1 ,. As for the subscripts of i or j, all of them are distinct; this is why combinations are relevant in this analysis.

We now take up the transformation of the second term in the right-hand side of 3. We replace the summation indices i! LI , then all combinations PJJ. L 1 natural numbers from 1, L 2 , and so on, until we come to the combinations Pk-JJ. X -JJ. J denotes summation over all combinations PJJ. L 2 at a time, and so on. The following alternative is valid: 1 Suppose that v, it, Indeed, 4. Let us clarify the notation used in 4. Formulas 4. Remark on the transition from symmetrized coefficients to ordinary ones.

The number N of distinct permutations of these subscripts is k! Formulas for coefficients of fourth-power variables. Here S denotes the sum over all combinations from numbers 1, 2, 3, 4 of subscripts of j in the first cofactor. Finally, 4. Case of composite elementary divisors of the matrix of the linear part. Let the linear part of an arbitrary autonomous analytic system of ordinary differential equations be transformed to the Jordan form All coefficients of power series as well as A. An are assumed complex, and the series themselves are assumed convergent in some neighbourhood of the origin.

The coefficients of terms that include variables to a power greater than one are assumed symmetrized see Subsection 3. We represent the normalizing transformation in the symmetrized form 3. Jt transforms system 7. Similarly to the initial steps traced in Subsection 3. We select in these identities the terms with variables to the kth power; in view of 7. As a result, identities 7.

We have already emphasized that all coefficients of interest are symmetrized. From the set of equal coefficients corresponding to distinct permutations of subscripts. Equating the coefficients of Yit Yik in identities 7. System 7. Note that this assumption means cpj Indeed, owing to representation 7. For this term, A, Q. The last equation of 7. This means, first, that a! Second, the last equation of 3. In treating the last-but-one equation of 7. Solutions of the subsequent equations of 7.

We begin by specifying a class of problems for which the Poincare theorem V, 1. This covers damped oscillatory systems asymptotically stable in linear approximation with analytic nonlinearities of a general type. The results are illustrated in Section 2 for mechanical systems with one and two degrees of freedom. Damped Oscillatory Systems 1. Reduction to diagonal form. We reduce the linear part of system 1. Calculation of coefficients of normalizing transformation. We limit ourselves to computing the quadratic terms of the normalizing transformation, substituting 2.

For coefficients of the quadratic terms of expansion 2. We shall also need the inverse transformation of 2. General solution of the initial system. Let us find an approximation of the general. Formulas 3. Formulas 1, 2. The solution of the Cauchy problem for equation 1. Oscillations of a spring-suspended mass with linear damping.

Equations IV, 1, 5. The location of the eigenvalues of the matrix representing the linear part of a third-order system in the closed left half-plane of the complex variable '. Typical patterns are illustrated in Fig. A case similar to a 1 was discussed in Chapter VI, Section 1. Case a 2 and its. Cases b and c are discussed in Sections 1 and 2; they are of predominant importance. Case d is treated in Section 3. Section 1 concludes with an example from electromechanics. The results of this chapter can also be applied to electromagnetic oscillations of two coupled oscillators when the free oscillations of one of them are described by a nonlinear firstorder equation.

Reduction to normal form. In general, the coefficients ajh, bjhk, By the fundamental Brjuno theorem V, 1. The normal form 1. Calculation of coefficients of normalizing transformation and normal form. It was demonstrated in Subsection 1. This can also be verified from formulas V, 3, 2. Consequently, formula V, 3, 2. Now we wish to calculate the coefficients of third-power terms.

From formulas V, 3, 2. It follows from the proof of the fundamental Brjuno theorem V, 1. This is discussed in more detail in V, 2. Before we discuss the convergence of 1. This means that in series V, 2, 1. We now take up the conditions of convergence of the normalizing transformation 1.

If condition A' V, 2. This follows from the form of the first and second equations of 1. Hence, conditions w and A hold if 2. What if not all of gi, are pure imaginary? In this general case the normalizing transformation is, by Brjuno's hypothesis 1 [i], smooth infinitely differentiable. When conditions 2. Substitution of this integral into the third equation of 1. Application of power transformation. The number of linearly independent solutions of equation 1. Consequently, by the Brjuno theorem V, 2. The last two equations of 1. The matrix A of the power transformation for these equations is.

The integration in 3. The same conclusion concerning system 1. Our purpose was to illustrate Subsecti'On 2. Finally, we shall analyze the case when, as in Subsection 1. Then 3. If, however, 3. Within the framework of our approximation with condition 3. Free oscillations of an electric servodrive.

A schematic of an electric servodrive is shown in Fig. The controlled element shaft is rotated by a servomotor coupled to a reducing gearbox. To be specific, the servomotor is a separately excited de electric motor. The current in the motor armature is controlled by an amplifier with input voltage V, which is a function of the displacement. If we neglect the signal delay, the play in each pair of engaged gear wheels, and friction, the equations of motion of the system can be written as.

We introduce the dimensionless variables I. The matrix S defines the transformation of system 4. Condition 3. Using formulas V, 3, 2. In order to express the initial values Yo 0 and y 1 0 in terms of x 0 0 and x1 0 , we must invert formulas 4. It is now possible, by using formulas 4. We shall not do that here, however, but shall single out the principal part of the solution. The principal part of a solution is defined, within the approximation made, as the solution of a normal form up to the accuracy of third-power terms that has been transformed by a normalizing transformation up to the accuracy of second-power terms.

The latter is given by formulas 4. Case of Neutral Linear Approximation 2. Normal form. Let us return to Subsection 1. When all solutions of the resonant equation 1. This is not only a complication of notation. Second, all the solutions of the resonant equation 1. This means that the number d of linearly independent solutions of the resonant equation is two. By the Brjuno theorem of Chapter V, Subsection 2. Let us follow the alternative of Chapter V, Subsection 3. By virtue of the notation of V, 3, 2. For the remaining quadratic coefficients of the normalizing transformation we obtain from formula V, 3, 2.

## Oscillations, theory of

V, 3, 2. From formula V, 3, 2. Taking into account 2. The coefficients here are given by formulas 2. Remark on convergence. Following the Brjuno theorem V, 2. Condition w is satisfied see V, 2, 4. With these conditions satisfied, the normalizing transformation in question is convergent in some neighbourhood of zero if all its arbitrarily chosen coefficients are set to zero.

Conclusions on stability. Multiplying the second equation by y1 and the third by y1 and adding, we obtain the truncated system of equations. Note that system 4. In order to reveal the instability of the trivial solution of system 4. We consider those solutions of system 4. The second equation of 4. As follows from the first equation of 4. Dividing the first equation of 4. The first equation of 4. Recapitulating, the trivial solution of system 4.

The critical case of one vanishing and two pure imaginary roots is discussed for steady motions by Kamenkov [83], vol. It should be emphasized that system 4. It cannot be stated, however, that a normalizing transformation and hence, a normal form can be chosen analytic in some neighbourhood of zero. This consideration precludes the generalizing of conclusions on the stability of the trivial solution of system 4. This corresponds to hypothesis 2 [i], which has not yet been proved.

This hypothesis must be applied to the solution p 0. If, however, conditions 4. Integration of normal form in quadratic approximation. Returning to the truncated system 4. The solution of system 4. Each solution of system 4. In this case integration of system 4. We consider now the general case when all three coefficients in system 4.

The first integral of system 4. An analysis of cases 1 - 7 shows that only in case 4 are all the solutions of system 4. This coincides with the conclusions of Subsection 2. The normalizing transformation 2. It is also possible to determine conditional stability regions in the space of initial values in cases when the trivial solution of system 4. We consider the equation. The eigennlues of the matrix A are 0, i, -i.

Since there are no quadratic terms in system 6. Multiplying the second equation of 6. Integration, however, is not necessary in this particular case, since it is immediately apparent that the trivial solution of 6. Normal form and normalizing transformation. Let us return again to system 1, 1. The resonant equation 1, 1.

The corresponding quadratic coefficients of the normal form are, by formulas V, 3, 2. The coefftcients here are given by formulas 1. Integration of normal form. The second and third equations of 1. The trivial solution of system 1. Let us consider the sufficient conditions for convergence of a normalizing transformation and normal form stated in the Brjuno theorem V, 2.

Conditions ffi and A are therefore satisfied if 3. Conseqnently, the normalizing transformation discussed in this snbsection is convergent in some neighbourhood of zero, provided that all its arbitrary coefficients are set to zero and conditions 3. Free oscillations in a tracking system with a TV sensor. Let us consider a system comprising a gyroscope with two degrees of freedom, to whose inner frame an optical TV device is rigidly f1xed [59]. The image of an object tracked in the optical system is projected onto a CRT screen covered by a rectangular raster of lightsensitive point elements.

Together with formulas 1, 1, 2. The truncated Van der Pol equations obtained by averaging over the independent variable explicitly included in 2. An attempt to solve equations 2. Of course, the truncated Van der Pol equations 2. Therefore we are satisfied with approximate integration of system 2. The second equation of 2. Va for I a I 2 2 we recall that 2 V2i is the amplitude of the generating solution 1. The first equation of 2.

Note that the transition time is of the order of 0! Transformation of equations of motion. We expand the righthand sides of system II, 2, 4. The primes denote the derivatives with respect to -r, and the expansions of the functions Y, X, and C! The energy integral I, 1, 2. Periodic solutions. The general solution of the generating system 1. An analysis of the first equation in I, 2, 1.

Lyapunov Systems with Damping An unperturbed nonlinear autonomous Lyapunov system of order 2k 2 is perturbed by an analytic damping of sufficiently small norm. A solution of the latter system is assumed known for a sufficiently small square root as compared with unity of the initial reduced energy of the system.

The first and all subsequent corrections of the corresponding solution of the perturbed system i. According to Poincare's results [a], vol. II the integration of a system of variational equations can be reduced to quadratures if the general integral of the unperturbed system is known. Subsections 1.

We consider a class of Lyapunov systems with damping where each system is represented by a second-order equation. We assume that the unperturbed system 1. In the linear case condition 1. The Lyapunov substitution. Taking 1. The unperturbed system 1. Cnmplete system of variational equations in the Poincare parameter and its solution. Let us represent system 1.

Let us assume that we know a solution x 0 'l't of the unperturbed system 2. On the basis of the Poincare theorem [a , vol. II , we seek a solution of system 2. Subtracting identity 2. In general, these operators are noncommutative, so that, say, the terms with x 1 x 2 and x 2 x 1 are considered distinct. If e is a linear element of 2. Successive integration of equations 2. If, however, the general integral of the unperturbed system 2. The solution of the first system of 2. In order to fi. Vibrations in mechanical systems with one degree of freedom and different types of nonlinearity. We consider a mechanical system with one degree of freedom that is subject to restoring and resistance forces.

Both of these functions are assumed to be analytic functions of the coordinates and velocity, respectively. II , which in general is ineffective. As for the restoring force, its nonlinear part determines the upper bound on the parameter f. The general solution suggested in the fi. With a suitable time scale 'T, the equation of motion of the system in question can be written as. The unperturbed equation 3. We assume henceforth that for all e and pin the rectangle 0 ;; e 80 and 0 p r 0 and for any 'l't the following inequality holds. Together with the first two formulas of 1.

Under condition 3. The integral of equation 3.

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On the basis of the Poincare theorem [a], vol. II , we seek a solution of 3. A sequence of the differential equations 2. We are now interested in the first correction p1 'l't; It ; an appropriate differential equation for this correction is a variational equation in the. Poincare parameter the first equation of 2. Let us derive it directly. Recasting 3. Dividing 3. Therefore it is preferable to resort to the Poincare method [a], vol. II presented at the end of Subsection 1.

In our case equation 3.

## Starzhinskii-Applied-Methods-in-the-Theory-of-Nonlinear-Oscillations-Mir by Balja - Issuu

Then a solution of 3. The Duffing equation with linear damping. As an example, we consider the equation. The energy integral 3. The substitution. The general solution of equation 4. Spring-loaded pendulum with linear damping. We consider a plane spring-loaded pendulum of mass m suspended from a weightless spring of unstressed length l and with force constant c Fig.

The unperturbed system 5. In the example above, the vectors a and x 0 are. In formula 2. Omitting obvious computations in 2. Finally, the solution of system 5. On Lyapunov-Type Systems Definition 1. A nearly Lyapunov system is a real system of the type 0. Definition 2.

A Lyapunov-type system is a real system of the type dx. Lyapunov-type systems are shown in this section to be reducible to Lyapunov systems. We present a Lyapunov-type system in detailed form, using notation distinct from that of 0. We assume that in addition to the Lyapunov scalar integral y, y. Expansions of 'ljl, cp 1 ,. Transformation of Lyapunov-type systems. Lemma I. The proof of the lemma is based on the Lyapunov transformation [a], Sec. Namely, we perform a substitution of variables in 1.

Substitution of the new variables yields new nonlinear terms are denoted by and. It follows from 2. System 2. We assume that system 1. We resolve the vector integral 1. Substitution of 2. Lemma II. Given a vector c in integral 1.

Let us differentiate integral 2. The determinant of the homogeneous system 2. Owing to continuity, these equalities must also hold for the vector c sufficiently small in norm. The lemma is therefore proved. The periodic solutions of a Lyapunov system 2. Other aspects of system 2. The problem of reducing an autonomous system of ordinary differential equations to its simplest, normal form by means of a change of variables was formulated by Poincare [a] and later developed by Lyapunov [a] and others.

The results of the most general character were obtained by Brjuno [j]; the relevant references can be found in his publication. Sections 1 and 2 of this chapter describe the problem on the basis of Brjuno's results, but adjust the presentation to the problems of nonlinear oscillations governed by the systems of equations in question, thus avoiding the most general approach and the most general situations. Some of the proofs in Section 2 are omitted. By virtue of the Weierstrass theorem see, for example [80], Sec.

The problem consists in reducing system 1. In particular, Brjuno [a] introduced the following notation for the transformed system. Normal forms are defined in the following theorem, which we formulate in terms of this subsection. The Fundamental Brjuno theorem [j], Ch. There exists a reversible transformation 1. Here Yvhv y , xvfv x , and Yvgv y are power series containing no zero- and first-power terms. Transformation 2. This is true for the first and second terms in the right-hand side of 2. Equalities 2. Here CvQ stands for the right-hand side of 2.

As a result, gvQ and hvQ are determined from Q in the manner described, thus proving the theorem. As stipulated by b , only 2. If for q1 for a finite number of values of the initial parameters of the system, then it is logical that hvQ be based, if possible, on continuity. The Poincare theorem [a]. The existence of a unique formal transformation of 1.

We write A, Q , using 1. By condition 3. This completes the proof; the reader will fmd an analysis of the conwrgence of transformation 1. In the real-variable case discussed above, the spectrum J.. Therefore, condition 2 is satisfied not only by H but by a straight line H symmetric with respect to the real axis, and thus by the imaginary axis of the complex plane 'A as well. Consequently, it is sufficient to eheck condition 2 for the imaginary axis. Additional Information 2.

Some properties of normalizing transformations. At the end of the proof of the fundamental Brjuno theorem Subsection 1. Q of the normalizing transformation 1, 2. At the same time, although the structure of the normal form is fixed by the established numbering of the variables, its coefficients gvQ depend on the choice of coefficients for the normalizing transformation. It is logical to assume that subsequent 1ransformations of the variables, if carried out for resonant terms only, will transform one normal form into another.

This clarifies the meaning of the Brjuno theorem [j], Ch. We also assume that series 1. As follows from the third Brjuno theorem ljl, Ch. Classification of normal forms; integrable normal forms. We consider the case when in 1, 1. We now number the variables so that. For instance, from 2. The Brjuno theorem [j], Ch. Under the previous assumptions, a normal form can be written as. The first sum in 2. In the normal form 1, 2. This last equation resonant equation is equivalent, owing to 2.

In order to emphasize the number of the equation of the normal form to which Q corresponds, we give Q an appropriate subscript. The SP"Ond equation of 2. The terms given in 2. Let us analyze several special cases of the above theorem. This normal form was derived by Dulac [ Equations have a finite number of solutions and no solutions in the hypothesis of the Poincare theorem of Subsection 1. Equations 2. The integration of the normal form 2. Concept of power transformations.

We wish to discuss the possibility of lowering the order of the normal form 1, 2. The term yQ is transformed by 3. System 3. But if it is analytic in wme neighbourhood of zero, and the initial ystem is a normal form, then 3. The possibility of lowering the order of the form is established by the Brjuno theorem [j], Ch. There exists a birational transformation 3. The first d equations of this system form a system of order d. Methods for efficiently constructing A can be found in [23Hcl. The Brjuno theorem on convergence and divergence of normalizing transformations. In the normalizing transformation 1.

We assume Wk. For condition A, we recall that inequalities 2. As in Subsection 2. Condition A in the case considered above when inequalities 2. The Brjuno theorem on convergence and divergence of normalizing transformations [jl, Ch. II, III. To illustrate this, we return to the Poincare theorem Subsection 1.

Obviously, in this case condition A is trivial automatically satisfied and Wk. Moreover, in the hypothesis of the Poincare theorem, a normalizing transformation is also single-valued, so that according to statement 1 of the above theorem this transformation is convergent in fome neighbourhood of its zero values. Fundamental identities. We assume that an oscillatory system is described by a real autonomous system of nth-order differential equations.

Here and henceforth a repeated subscript indicates summation and takes on the values 1, According to the fundamental Brjuno theorem see Subsection 1. Summation in 1. Substitution of 1. By dropping the terms with powers above three, we obtain the formal identities the derivatives with respect to 't' are primed.

By changing the summation indices and symmetrizing the coefficients in the sums, we derive the fundamental identities. Computational alternative. The following alternative is true: 1 Suppose that the values taken on by v, l, m, and p and by the real parameters of the initial oscillatory system on which Av, A1, 'Am, and 'Ap depend are such that the expressions in parentheses in the fourth and fifth sums in the left-hand side of 1.

Comparing the terms containing YzYm in the left- and right-hand sides of the fundamental identities 1. Indeed, 1. The absence of the term containing YzYmYp in the second sum of 1. Equating the coefficients of the quadratic terms in 1. We wish to emphasize that expressions 2. This has been already mentioned see condition b at the end of the proof of the fundamental Brjuno theorem Subsections 1. Comparison of the terms containing YzYm and YzYmYp in the left- and right-hand sides of the fundamental identities 1. It should Le emphasized that formulas 2. A general procedure for determining the coefficients of a normalizing transformation and a normal form is contained in the proof of the fundamental Brjuno theorem Subsection 1.

The method we have outlined above seems to us better suited for applications to oscillation problems; hence, the term "practical" in the section title. Fundamental identities in general form and their transformation.

Unification of notation in the initial diagonal system 1. The subscripts repeated twice denote summation from 1 to n, summation being independent of other subscripts. We also use the concise notation. Substitution of 3. Nonsymmetric coefficients generated in the course of the calculations must be symmetrized since the coefficients in question are subject to conditions 3. We replace the summation indices in each addend of the sum over fL in the second term in the left-hand side of 3. It becomes obvious that all the addends of the sum over fL are identical, so it can be expressed as one addend times x.

Thus we have completed the transformations. Note that the numbers jp 1 ,. As for the subscripts of i or j, all of them are distinct; this is why combinations are relevant in this analysis. We now take up the transformation of the second term in the right-hand side of 3. We replace the summation indices i! LI , then all combinations PJJ.

L 1 natural numbers from 1, L 2 , and so on, until we come to the combinations Pk-JJ. X -JJ. J denotes summation over all combinations PJJ. L 2 at a time, and so on. The following alternative is valid: 1 Suppose that v, it, Indeed, 4. Let us clarify the notation used in 4. Formulas 4. Remark on the transition from symmetrized coefficients to ordinary ones. The number N of distinct permutations of these subscripts is k! Formulas for coefficients of fourth-power variables.

Here S denotes the sum over all combinations from numbers 1, 2, 3, 4 of subscripts of j in the first cofactor. Finally, 4. Case of composite elementary divisors of the matrix of the linear part. Let the linear part of an arbitrary autonomous analytic system of ordinary differential equations be transformed to the Jordan form All coefficients of power series as well as A.

An are assumed complex, and the series themselves are assumed convergent in some neighbourhood of the origin. The coefficients of terms that include variables to a power greater than one are assumed symmetrized see Subsection 3. We represent the normalizing transformation in the symmetrized form 3. Jt transforms system 7. Similarly to the initial steps traced in Subsection 3. We select in these identities the terms with variables to the kth power; in view of 7. As a result, identities 7. We have already emphasized that all coefficients of interest are symmetrized.

From the set of equal coefficients corresponding to distinct permutations of subscripts. Equating the coefficients of Yit Yik in identities 7. System 7. Note that this assumption means cpj Indeed, owing to representation 7. For this term, A, Q. The last equation of 7. This means, first, that a! Second, the last equation of 3. In treating the last-but-one equation of 7.

Solutions of the subsequent equations of 7. We begin by specifying a class of problems for which the Poincare theorem V, 1. This covers damped oscillatory systems asymptotically stable in linear approximation with analytic nonlinearities of a general type. The results are illustrated in Section 2 for mechanical systems with one and two degrees of freedom. Damped Oscillatory Systems 1. Reduction to diagonal form. We reduce the linear part of system 1. Calculation of coefficients of normalizing transformation. We limit ourselves to computing the quadratic terms of the normalizing transformation, substituting 2.

For coefficients of the quadratic terms of expansion 2. We shall also need the inverse transformation of 2. General solution of the initial system. Let us find an approximation of the general. Formulas 3. Formulas 1, 2. The solution of the Cauchy problem for equation 1. Oscillations of a spring-suspended mass with linear damping. Equations IV, 1, 5. The location of the eigenvalues of the matrix representing the linear part of a third-order system in the closed left half-plane of the complex variable '.

Typical patterns are illustrated in Fig. A case similar to a 1 was discussed in Chapter VI, Section 1. Case a 2 and its. Cases b and c are discussed in Sections 1 and 2; they are of predominant importance. Case d is treated in Section 3. Section 1 concludes with an example from electromechanics. The results of this chapter can also be applied to electromagnetic oscillations of two coupled oscillators when the free oscillations of one of them are described by a nonlinear firstorder equation.

Reduction to normal form. In general, the coefficients ajh, bjhk, By the fundamental Brjuno theorem V, 1. The normal form 1. Calculation of coefficients of normalizing transformation and normal form. It was demonstrated in Subsection 1. This can also be verified from formulas V, 3, 2. Consequently, formula V, 3, 2. Now we wish to calculate the coefficients of third-power terms. From formulas V, 3, 2. It follows from the proof of the fundamental Brjuno theorem V, 1. This is discussed in more detail in V, 2. Before we discuss the convergence of 1. This means that in series V, 2, 1.

We now take up the conditions of convergence of the normalizing transformation 1. If condition A' V, 2. This follows from the form of the first and second equations of 1. Hence, conditions w and A hold if 2. What if not all of gi, are pure imaginary? In this general case the normalizing transformation is, by Brjuno's hypothesis 1 [i], smooth infinitely differentiable. When conditions 2. Substitution of this integral into the third equation of 1. Application of power transformation. The number of linearly independent solutions of equation 1.

Consequently, by the Brjuno theorem V, 2. The last two equations of 1. The matrix A of the power transformation for these equations is. The integration in 3. The same conclusion concerning system 1. Our purpose was to illustrate Subsecti'On 2. Finally, we shall analyze the case when, as in Subsection 1. Then 3. If, however, 3. Within the framework of our approximation with condition 3.

Free oscillations of an electric servodrive. A schematic of an electric servodrive is shown in Fig. The controlled element shaft is rotated by a servomotor coupled to a reducing gearbox. To be specific, the servomotor is a separately excited de electric motor. The current in the motor armature is controlled by an amplifier with input voltage V, which is a function of the displacement.

If we neglect the signal delay, the play in each pair of engaged gear wheels, and friction, the equations of motion of the system can be written as. We introduce the dimensionless variables I. The matrix S defines the transformation of system 4. Condition 3. Using formulas V, 3, 2. In order to express the initial values Yo 0 and y 1 0 in terms of x 0 0 and x1 0 , we must invert formulas 4.

It is now possible, by using formulas 4. We shall not do that here, however, but shall single out the principal part of the solution. The principal part of a solution is defined, within the approximation made, as the solution of a normal form up to the accuracy of third-power terms that has been transformed by a normalizing transformation up to the accuracy of second-power terms. The latter is given by formulas 4. Case of Neutral Linear Approximation 2. Normal form. Let us return to Subsection 1. When all solutions of the resonant equation 1.

This is not only a complication of notation. Second, all the solutions of the resonant equation 1. This means that the number d of linearly independent solutions of the resonant equation is two. By the Brjuno theorem of Chapter V, Subsection 2. Let us follow the alternative of Chapter V, Subsection 3. By virtue of the notation of V, 3, 2.

For the remaining quadratic coefficients of the normalizing transformation we obtain from formula V, 3, 2. V, 3, 2. From formula V, 3, 2. Taking into account 2. The coefficients here are given by formulas 2. Remark on convergence. Following the Brjuno theorem V, 2. Condition w is satisfied see V, 2, 4. With these conditions satisfied, the normalizing transformation in question is convergent in some neighbourhood of zero if all its arbitrarily chosen coefficients are set to zero.

Conclusions on stability. Multiplying the second equation by y1 and the third by y1 and adding, we obtain the truncated system of equations. Note that system 4. In order to reveal the instability of the trivial solution of system 4. We consider those solutions of system 4. The second equation of 4. As follows from the first equation of 4. Dividing the first equation of 4.

The first equation of 4. Recapitulating, the trivial solution of system 4. The critical case of one vanishing and two pure imaginary roots is discussed for steady motions by Kamenkov [83], vol. It should be emphasized that system 4. It cannot be stated, however, that a normalizing transformation and hence, a normal form can be chosen analytic in some neighbourhood of zero. This consideration precludes the generalizing of conclusions on the stability of the trivial solution of system 4.

This corresponds to hypothesis 2 [i], which has not yet been proved. This hypothesis must be applied to the solution p 0. If, however, conditions 4. Integration of normal form in quadratic approximation. Returning to the truncated system 4. The solution of system 4. Each solution of system 4. In this case integration of system 4. We consider now the general case when all three coefficients in system 4. The first integral of system 4. An analysis of cases 1 - 7 shows that only in case 4 are all the solutions of system 4. This coincides with the conclusions of Subsection 2.

The normalizing transformation 2. It is also possible to determine conditional stability regions in the space of initial values in cases when the trivial solution of system 4. We consider the equation. The eigennlues of the matrix A are 0, i, -i. Since there are no quadratic terms in system 6. Multiplying the second equation of 6. Integration, however, is not necessary in this particular case, since it is immediately apparent that the trivial solution of 6. Normal form and normalizing transformation. Let us return again to system 1, 1. The resonant equation 1, 1.

The corresponding quadratic coefficients of the normal form are, by formulas V, 3, 2. The coefftcients here are given by formulas 1. Integration of normal form. The second and third equations of 1. The trivial solution of system 1. Let us consider the sufficient conditions for convergence of a normalizing transformation and normal form stated in the Brjuno theorem V, 2. Conditions ffi and A are therefore satisfied if 3.

Conseqnently, the normalizing transformation discussed in this snbsection is convergent in some neighbourhood of zero, provided that all its arbitrary coefficients are set to zero and conditions 3. Free oscillations in a tracking system with a TV sensor. Let us consider a system comprising a gyroscope with two degrees of freedom, to whose inner frame an optical TV device is rigidly f1xed [59].

The image of an object tracked in the optical system is projected onto a CRT screen covered by a rectangular raster of lightsensitive point elements. An illuminated spot appears on the raster when the object is within the field of view of the optical system. The object is tracked by means of a special TV sensor [59] with a rec-. The TV system superimposes the centre of the window on the centroid of the image on the raster. The tracking system must achieve this with minimum error.

Let us consider the case when the object's image is larger than the tracking window. This situation arises when the distance from the. Moreover, if the spot covers the window completely, the output of the TV sensor is zero. Figure 14 gives the structural diagram of the tracking system in question for the situation described above. The matrix S, which reduces A to diagonal form, has the eigenvectors of A as its elements. This matrix and its inverse are.

Since there are no quadratic terms in 4. The third equation of 4. In order to integrate the truncated normal form, we find from the first equation of 4. Within cubic terms, the normalizing transformation 1, 1. The coefficients here are given by 1. Obviously, the inversion of formulas 4.

This completes the solution of the Cauchy problem in general form for equation 4. The solution is given as the sequence of formulas 4. According to the definition given in Subsection 1. By virtue of the definition of the principal part of a solution and by formulas 4. Fourth-Order Systems The first subsection treats systems of arbitrary order. The second and third subsections, based on the results of Chapter V, analyze resonances and normal forms of real analytic autonomous in general, nonconservative fourth-order systems with two pairs of distinct pure imaginary eigenvalues of the matrix of the linear part.

Stability of the trivial solution is analyzed in Subsection 1. Remark on coefficients of systems of diagonal form. We consider the oscillations in a system with k degrees of freedom that is described by the vector equation v. We assume that the eigenvalues of the matrix P. Let S be a nonsingular matrix it can be chosen orthogonal that reduces P to diagonal form, that is, p. Let us rewrite 1.

Let us distinguish between the following cases: a If the right-hand side of 1. G ; the sixth sum in 1. We consider a real autonomous fourth-order system under the assumption that the linear part of the system, which has two pairs of distinct pure imaginary conjugate eigenvalues, is reduced to diagonal form. By the fundamental Brjuno theorem see V, 1. Hence, the normal form includes only the resonant terms whose exponents satisfy the resonant equation 2. We wish to determine whether 2. If equations 2. Calculation of coefficients of normalizing transformation and normal forms.

Next we follow the alternative of Chapter V, Subsection 3. We choose Formula 3. The coefficient of the quadratic terms appearing in 2. Now we consider cubic terms. Formulas V, 3, 2. Finally, we consider case c of Subsection 1. In addition to 3. We choose A-1 1",. For any "A E 0, 1 , the calculations start with formulas 3. The Molchanov criterion of oscillation stability. Kamenkov [83 and Malkin [c investigated stability in the critical case of two pairs of pure imaginary roots. In this chapter it is more convenient to use the Molchanov criterion [b , as it reflects the specifics of normal forms and the accompanying resonances.

By definition, instability is established if we find one unstable trajectory. The Molchanov criterion [b]. Unstable systems 4. Summing equations 4. We recall that v1 and v2 are nonnegative variables. Therefore the conditions for the negative definiteness of the matrix A are sufficient conditions for asymptotic stability of the trivial solution of system 4.

These last conditions result in the inequalities 4. Let us demonstrate that the second condition of 4. We consider the function. By the Lyapunov theorem on asymptotic stability under condition 4. Let us prove the first half. We seek solutions of system 4. The degree of "amplification" in such systems can be roughly evaluated by I a I I b 1. Clearly, these systems may prove practically unstable in cases of great "amplification". The Bihikov-Piiss criterion. Let us rewrite the second and fourth equations of system 2.

System 5. Applied to system 5. The Ishlinskii Problem The presentation in Subsections 2. Reduction of equations of motion to the Lyapunov form. Under assumptions 17 , 41 , and 47 , and an additional assumption concerning the uniform motion of the suspension point on a sphere, these equations are da.

As in expressions 45 , 19 , and 20 , we set in 1. The eigenvalues of the matrix A of the linear part of system 1. The matrix S1 of the linear transformation that reduces the linear part of system 1. We can readily obtain the matrix S 2 of the linear transformation that reduces the diagonal system to the Lyapunov skew-symmetric form. Now we can make a linear change of variables in system 1o3 the inverse transformation is given in parentheses.

Under assumption 1.