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## Über dieses Buch

This book considers methods of approximate analysis of mechanical, elec­ tromechanical, and other systems described by ordinary differential equa­ tions. Modern mathematical modeling of sophisticated mechanical systems consists of several stages: first, construction of a mechanical model, and then writing appropriate equations and their analytical or numerical ex­ amination. Usually, this procedure is repeated several times. Even if an initial model correctly reflects the main properties of a phenomenon, it de­ scribes, as a rule, many unnecessary details that make equations of motion too complicated. As experience and experimental data are accumulated, the researcher considers simpler models and simplifies the equations. Thus some terms are discarded, the order of the equations is lowered, and so on. This process requires time, experimentation, and the researcher's intu­ ition. A good example of such a semi-experimental way of simplifying is a gyroscopic precession equation. Formal mathematical proofs of its admis­ sibility appeared some several decades after its successful introduction in engineering calculations. Applied mathematics now has at its disposal many methods of approxi­ mate analysis of differential equations. Application of these methods could shorten and formalize the procedure of simplifying the equations and, thus, of constructing approximate motion models. Wide application of the methods into practice is hindered by the fol­ lowing. 1. Descriptions of various approximate methods are scattered over the mathematical literature. The researcher, as a rule, does not know what method is most suitable for a specific case. 2.

## Inhaltsverzeichnis

### Chapter I. Dimensional analysis and small parameters

Abstract
Scientific disciplines deal with quantities that are by nature qualitatively different and that are measured with the help of numbers.
Igor V. Novozhilov

### Chapter II. Regularly perturbed systems. Expansions of solutions

Abstract
In this section an approximate solution of a regularly perturbed system is constructed on a finite interval of normalized dimensionless time. In the examples considered in Sec. 2, dimensional time TT1 corresponds to this interval.
Igor V. Novozhilov

### Chapter III. Decomposition of motion in systems with fast phase

Abstract
Among the problems considered in the previous chapter special attention was paid to slowly changing motions parameters, for example, slowly increasing amplitudes in Sec. 4.2, and slow Magnus drifts in Sec. 4.3. The Magnus drifts were obtained by averaging the right-hand side of Eqs. (4.29) with respect to the time variable which is a part of high-frequency oscillating components. During the process of averaging, slow variables (ϕ1, (ϕ2 were assumed to be constant and equal to their initial values. This averaging procedure is correct only on bounded intervals of the dimensionless nutational time t = T/T1 ∽ 1 where the Poincaré theorem is valid and (ϕ1, (ϕ2 changes by values of the order ε according to (4.26). An attempt to use Magnus’s formula for large intervals of time t ∽ 1/ε contradicts the theorem. The foregoing procedure of averaging the right-hand sides of (4.29) loses its meaning because during the time interval t ∽ 1/ε the variables ϕ1, ϕ2 change by finite value.
Igor V. Novozhilov

### Chapter IV. Decomposition of motion in systems with boundary layer

Abstract
While analyzing the oscillator with high friction in Sec. 10.2 we met a new situation. Two kinds of dimensionless time were spoken about for the identical problem: slow time of slow quasistationary motion for x, and fast time for v inside the boundary layer.
Igor V. Novozhilov

### Chapter V. Decomposition of motion in systems with discontinuous characteristics

Abstract
The existence and uniqueness theorem does not define the solution in points where the right-hand sides of differential equations are discontinuous. Some methods of definition completing the solution in this case were considered in Filippov , Utkin , and Gerashenko and Gerashenko . In this section the method of definition from Novozhilov  is discussed, such that there exists a normalization which carries the problem into the Tikhonov form considered previously.
Igor V. Novozhilov

### Chapter VI. Correctness of limit models

Abstract
Passage to a limit with respect to a small parameter has already produced degenerate equations describing traditional mechanical models. In Sec. 12.3 a model of absolutely rigid body or, in other words, a model of holonomic constraint was obtained by passage to the limit. A model of kinematic constraint has been also obtained by passage to a limit in Sec. 15.2. In both these examples conditions under which the passage to a limit is possible have been discussed, and we have seen that there exist a lot of cases when these conditions do not hold.
Igor V. Novozhilov

### Backmatter

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