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## Über dieses Buch

Many books on stability theory of motion have been published in various lan­ guages, including English. Most of these are comprehensive monographs, with each one devoted to a separate complicated issue of the theory. Generally, the examples included in such books are very interesting from the point of view of mathematics, without necessarily having much practical value. Usually, they are written using complicated mathematical language, so that except in rare cases, their content becomes incomprehensible to engineers, researchers, students, and sometimes even to professors at technical universities. The present book deals only with those issues of stability of motion that most often are encountered in the solution of scientific and technical problems. This allows the author to explain the theory in a simple but rigorous manner without going into minute details that would be of interest only to specialists. Also, using appropriate examples, he demonstrates the process of investigating the stability of motion from the formulation of a problem and obtaining the differential equations of perturbed motion to complete analysis and recommendations. About one fourth of the examples are from various areas of science and technology. Moreover, some of the examples and the problems have an independent value in that they could be applicable to the design of various mechanisms and devices. The present translation is based on the third Russian edition of 1987.

## Inhaltsverzeichnis

### Introduction

Abstract
Problems of stability appear for the first time in mechanics during the investigation of an equilibrium state of a system. A simple reflection may show that some equilibrium states of a system are stable with respect to small perturbations, whereas other balanced states, although available in principal, cannot be realized in practice. Thus, for instance, when a pendulum is in its lowest position any small perturbations will result only in its oscillation about this position. However, if after some effort we can set the pendulum at its highest position, then any push will cause its downfall. Certainly, the question of stability in this case is resolved in an elementary manner, but in general, the conditions under which the equilibrium state of a system will be stable are not always as clear. The criterion for stability of rigid bodies in equilibrium under gravitational forces was formulated by E. Torricelli in 1644. In 1788, G. Lagrange proved a theorem that defines sufficient conditions for stability of equilibrium of any conservative system (see Section 3.1).
David R. Merkin

### 1. Formulation of the Problem

Abstract
We denote the real variables characterizing the state of a mechanical, electromechanical, or any other system by y1,⋯, y n . These variables may be coordinates, velocities, currents, voltages, temperatures, etc., or functions of these parameters. It is assumed that the number of variables y1,⋯,y n is finite and the system’s motion, i.e., the process of the changing of y1,⋯, y n in time, is described by ordinary differential equations resolved with respect to the time derivatives.1
David R. Merkin

### 2. The Direct Liapunov Method. Autonomous Systems

Abstract
One of the most effective methods for studing stability is the direct Liapunov method (most often called the Second Liapunov Method). In this chapter the direct method is presented for autonomous systems (nonautonomous systems are considered in Chapter 7).
David R. Merkin

### 3. Stability of Equilibrium States and Stationary Motions of Conservative Systems

Abstract
We consider a mechanical system with holonomic and scleronomic constraints. The state of the system can be described by s generalized independent coordinates q1,⋯, q s . It is well known that at a state of equilibrium all generalized forces Q k of such a system are equal to zero:
$${Q_1} = 0, \ldots, {Q_s} = 0$$
(3.1)
.
David R. Merkin

### 4. Stability in First Approximation

Abstract
In many cases, especially in practical problems, stability of a motion is investigated using equations of first approximation. This is justified not only because of the relative simplicity of the method, but also by the fact that quite often we are able to determine accurately only the first order linear terms that define processes occurring in real systems. However, as was shown in Example 1.1, in investigating stability of a motion, the conclusions arrived at on the basis of equations of first approximation are sometimes absolutely incorrect. Therefore, it is essential to formulate and determine those conditions under which the equations of first approximation will correctly answer the question about the stability of a motion. In general, this problem can be formulated in the following way: Equations of a perturbed motion are given as
$$\begin{gathered} {{\dot{x}}_1} = {a_{{11}}}{x_1} + \cdots + {a_{{1n}}}{x_n} + {X_1}, \hfill \\ \cdots = \cdots \hfill \\ \dot{x} = {a_{{n1}}}{x_1} + \cdots + {a_{{nn}}}{x_n}, \hfill \\ \end{gathered}$$
(4.1)
where the nonlinear terms X1, ..., X n are terms of order higher than one in x1, ..., x n (in this chapter we will write just X k instead of X k * ).
David R. Merkin

### 5. Stability of Linear Autonomous Systems

Abstract
In this chapter we continue our consideration of various methods for analyzing the stability of motion of linear autonomous systems. The standard form of the differential equations of a perturbed motion are (see equations (1.14))
$$\begin{array}{*{20}{c}} {\dot{x} = {a_{{11}}}{x_1} + \ldots + {a_{{1n}}}{x_n},} \\ { \ldots = \ldots \,} \\ {\dot{x} = {a_{{n1}}}{x_1} + \ldots + {a_{{nn}}}{x_n},} \\ \end{array}$$
(5.1)
where the coefficients a kj are constant real numbers.
David R. Merkin

### 6. The Effect of Force Type on Stability of Motion

Abstract
Liapunov’s methods of investigating stability of motion are powerful methods due to their generality and universality. However, they cannot address the possible effects of various physical factors on stability of motion. Meanwhile, in many cases such an analysis, in a rather general sense, may be very useful. In this chapter we present the effect of various types of forces on stability of motion.
David R. Merkin

### 7. The Stability of Nonautonomous Systems

Abstract
Before we start to define Liapunov functions for nonautonomous systems, we briefly discuss some problems associated with the direct method.
David R. Merkin

### 8. Application of the Direct Method of Liapunov to the Investigation of Automatic Control Systems

Abstract
In the majority of cases automatic control systems involve complicated devices, consisting of objects to be controlled (plants or processes) and controllers. The task of a controller is to support continuously either the stationary operating conditions or those conditions of the plant that change according to a given law. All deviations from the desired conditions that may arise in the control system must be reduced to zero with time. In other words, the control system must be asymptotically stable.
David R. Merkin

### 9. The Frequency Method of Stability Analysis

Abstract
The frequency method of stability analysis of linear and nonlinear systems is highly convenient for engineering calculations. This is so because a frequency characteristic is invariant in nonsingular linear coordinate transformations, and it may be obtained in a simple manner either from system equations or experimentally. Moreover, this method is applicable to a wider class of systems.
David R. Merkin

### Backmatter

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