SERIES ON STABILITY, VIBRATION AND CONTROL OF SYSTEMS Series A
Volume 11
Selected Topics in Vibrational Mechanics I di...
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SERIES ON STABILITY, VIBRATION AND CONTROL OF SYSTEMS Series A
Volume 11
Selected Topics in Vibrational Mechanics I dited by
lliya I. Blekhman
World Scientific
Selected Topics in Vibrational Mechanics
SERIES ON STABILITY, VIBRATION AND CONTROL OF SYSTEMS Founder and Editor: Ardeshir Guran Co-Editors: C. Christov, M. Cloud, F. Pichler & W. B. Zimmerman About the Series Rapid developments in system dynamics and control, areas related to many other topics in applied mathematics, call for comprehensive presentations of current topics. This series contains textbooks, monographs, treatises, conference proceedings and a collection of thematically organized research or pedagogical articles addressing dynamical systems and control. The material is ideal for a general scientific and engineering readership, and is also mathematically precise enough to be a useful reference for research specialists in mechanics and control, nonlinear dynamics, and in applied mathematics and physics. Selected
Volumes in Series B
Proceedings of the First International Congress on Dynamics and Control of Systems, Chateau Laurier, Ottawa, Canada, 5-7 August 1999 Editors: A. Guran, S. Biswas, L. Cacetta, C. Robach, K. Teo, and T. Vincent
Selected
Volumes in Series A
Vol. 2
Stability of Gyroscopic Systems Authors: A. Guran, A. Bajaj, Y. Ishida, G. D'Eleuterio, N. Perkins, and C. Pierre
Vol. 3
Vibration Analysis of Plates by the Superposition Method Author: Daniel J. Gorman
Vol. 4
Asymptotic Methods in Buckling Theory of Elastic Shells Authors: P. E. Tovstik and A. L Smirinov
Vol. 5
Generalized Point Models in Structural Mechanics Author: I. V. Andronov
Vol. 6
Mathematical Problems of Control Theory: An Introduction Author: G. A. Leonov
Vol. 7
Analytical and Numerical Methods for Wave Propagation in Fluid Media Author: K. Murawski
Vol. 8
Wave Processes in Solids with Microstructure Author: V. I. Erofeyev
Vol. 9
Amplification of Nonlinear Strain Waves in Solids Author: A. V. Porubov
Vol. 10 Spatial Control of Vibration: Theory and Experiments Authors: S. O. Reza Moheimani, D. Halim, and A. J. Fleming Vol. 11 Selected Topics in Vibrational Mechanics Editor: I. Blekhman
SERIES ON STABILITY, VIBRATION AND CONTROL OF SYSTEMS (x,x,t,cot)
(2.1)
where x is the n-dimensional vector of the generalized coordinates, m is a positive constant ("mass"), co is the positive parameter (in future the "large" parameter), F and in the right sides of the initial equation (2.1). What has been said can be formulated as the following statement: For the existence of the solutions of the initial differential equation (2.1) of type (2.2), or the corresponding solutions X and Y, of system (2.7), (2.8), satisfying conditions (2.15), (2.19) and such that y/ ~ enX (n being either an integer or zero), it is necessary that system (2.7), (2.8) should allow the notation in form (2.24) or (2.25).
2.5 The Main Equation of Vibrational Mechanics. Vibrational Forces, Observers O and V Let the main assumption of vibrational mechanics be satisfied, i. e. let there be a solution of equation (2.17) of type (2.2) or the corresponding solutions of system (2.7), (2.8) satisfying conditions (2.16). Let us assume further that the conditions of section 2.3 are satisfied. Then we will call equation (2.12), containing only the slow component of motion, the main equation of vibrational mechanics , or the equation of slow motions, and the expression V(X,X,f) for the additional force—the vibrational force. Equation (2.8) will then be called equation of fast motions. As has been mentioned in section 1.2, vibrational mechanics can be considered to be a mechanics in which the observer does not notice any fast forces and fast motions, and perceives only the slow forces and slow motions. Such an observer (the observer V) can be contrasted to the "ordinary" observer O who perceives both the slow and fast forces and motions. It is convenient to use these images, taking, as may be required, the positions of either the first or the second observer. In accordance with what has been said, the observer V, so as not to come in conflict with the laws of mechanics, must take into consideration not only the "ordinary" slow forces F, but also the additional slow forces i.e. the vibrational forces V. Note that according to equations (2.6), (2.9) and (2.11), the vibrational force is obtained by means of averaging the eigenfast force is relatively small, in solving Eq.(2.7) those values are being regarded as "frozen", i.e. as fixed parameters. Let Eq. (2.7), with X, X and t being frozen, admit either one or several almost-periodical (particularly 2K -periodical) with respect to z = ox solutions T = T*(X,X,r,r) satisfying condition (2.3) and asymptotically stable with respect to all fast generalized coordinates and velocities, while all the other fast variables changing and X, Xand t being frozen all over the region under consideration. This assumption is
20
2. Mathematical Apparatus. Direct Separation of Motions
usually checked up very easily and it is really valid under the conditions of section 2.3. Besides, Eq.(2.7) is such that the necessary condition (2.3) of the existence of almost-periodical (particularly 2n -periodical with respect to t) solutions is automatically fulfilled for it. To make sure of it, it is enough to average Eq.(2.7) with respect to T = ax ( X , X and t being frozen). What has been said holds the key to the way adopted by us of "splitting" the initial equation (2.1) into two equations (2.7) and (2.8). Substituting the definite solution *F = *F*(X,X,f,T) into expression (2.11), we will find an approximate expression for the vibrational force V(X,X,f) after which the main equation (2.12) can be composed. This equation will.of course, also be approximate and must be valid on the one hand with t > f0 where tQ is the time of achieving a steady state of the fast motions, and on the other hand—with t < TQ , where 7Q is the boundary of the interval of validity of the asymptotic approximation (it is naturally assumed that TQ » ? o )• The grounds of the described approximate method for the case when the solution of type (2.22) are being sought are given in the books [1,2] It is based on the scheme of averaging, suggested by V.M.Volosov [25]. For some other suppositions the grounds are given by O.Z.Malakhova [19] and for the case of systems with the constraints by V.I.Yudovich [27]; the results of their works being given below (sections 2.7 and 2.8). As for the grounds of the method for more a general case, see Part VI, written by A. Ja. Fidlin.
2.7 Important Special Case Let us consider the case when the initial system (2.1) can be presented in the following way (to simplify our reasoning and notation we will first consider x to be a scalar): mx = F(x,x,t) + ajQ>i(x,t,T), co»l
(2.29)
where the almost-periodic with respect to T function Oi has a zero average value with respect to this argument at the fixed x and t: (O,(x,r,T)) = 0
(2.30)
2.7. Important Special Case
21
Here we are searching for the solution of the type x = X(t)+ey/l(t,T),
(2.31)
e = l/ot)«l
In this case, considered by Malakhova [19], finding the function y/j and the expression for the vibrational force becomes much more simple and the ground of the approximate method stated in section 2.6 is obtained directly by means of using the first and second theorems of Bogolyubov. In this case the equation of fast motions (2.7) looks as mey/\ = F(X +ey\,X
+eyri,t)~(F(X
+ey/l,X
+8^,0)
+ -[O 1 (X+£Y' 1 ^T)-(0 1 (X+£l/A 1 ,f,T))] e or, considering (2.30) and with accuracy to the terms of a higher order with respect to e (2.32)
myrl=-TQl(X,t,T)
Looking ahead, we will note that the latter equation corresponds to the so-called purely inertial approximation (see section 2.9). Let y/j* = Y\{X,t,x) be a certain almost-periodic solution of equation (2.32), found at the frozen X and t. Then, according to the approximate method, stated in section 2.6, in the main equation of vibrational mechanics (2.12), i.e. in the equation mX = F(X,X,t)
+
V(X,X,t),
(2.33)
written with the same accuracy with respect to e as equation (2.32), the expression for the vibrational force (2.11) will be
V(X,X,t)
= (F x +
3VlT) aV(x,f,T)yaVr(x,*,T)
m 3T + £B(Y,X,t,T,£)},
drdt
dtdX
(2.35)
t' = £
where A and B are certain functions of the indicated arguments. Let functions Fand Oj be such that the right sides of equations (2.35) should satisfy the conditions of Bogolyubov's first theorem. Then applying to these equations the principle of averaging, and retaining the same designations for the averaged variables, we will obtain the following equation of the first approximation X' = EY,
Y' = £R(X,X,t),
t' = £.
(2.36)
where R(Y,X,t) = ±-((F + V))
7+
dy/i(X,t,T) -,X,t dx
+
80ig^W(X^)
It is obvious that the latter equations lead to equation (2.33) in which V(X,X,t) are determined according to (2.34), which is precisely what
2.7. Important Special Case
23
was to be proved. It should be noted that the relations of type (2.33), (2.34) were obtained by S. V. Chelomey [12] in a different way. Let us consider the question of stability of the positions of equilibrium Y = 0, X = X*, which are determined by equations (2.35), i.e. the so-called positions of quasi-equilibrium for the initial equation (2.29). Here we will assume that functions F and Ojdo not depend explicitly on t and consequently R = R(Y,X). The indicated positions of equilibrium, if they exist, are determined as solutions of the equation R(0,X) =—[F(0,X) + V(0,X)] = 0 m
(2.37)
Then, under more strict requirements to the right sides of equation (2.29) it is possible to apply Bogolyubov's second theorem [8]. If, according to this theorem, the roots of the equation
-X dR
a/?
dx y=o
dY
1
x=x*
•X
(2.38)
K=0 X=X*
have negative real parts, then the stationary solutions Y = 0, X = X* of the equations of the first approximation will be answered, at the sufficiently small s all over the interval -°O
where
atdx dx is the Euler operator, Q is the generalized force, and T is the kinetic energy of the system, then the equation of slow motions can be written in the following way (E(r x + T)) = (Qx+«p) where 7X+Y
an
(2-57)
d fix+T denote the result of substituting the values
x = X + Yand x = X + Yinto the corresponding expression. Then the equation of fast motions will be E(TX+¥)
- (E(r x + *))=QX + «F - (Q X+0
(4.15)
Hence the classical result follows directly: the solution «p0 -ax = 0 , corresponding to the lower position of the pendulum, is always stable; along with that, if the condition b2>2
(4.16)
is satisfied, or, if according to the designation (4.3), the conditions G2Cl2>2gl0
(4.17)
4.4. Floquet-Lyapunov's Theory
43
are satisfied, then the upper (overturned) position of the pendulum 2 =-ehQ 2 allows four stationary solutions a 3 _ 4 =±arccos(-2/i> 2 )
a j = 0 , a2=n,
(4.24)
Forming equations in variations for these solutions, it is easy to come to the conclusion that the solution <X\ = 0, corresponding to the lower position of the pendulum, will always be stable, while the upper ("overturned") position is stable provided condition (4.16) is satisfied, and the intermediate positions are unstable. In other words, we have come to the conclusions which agrees with the results of sections 4.3 and 4.4.
4.6 Method of Multiple Scales Using the method of multiple scales [22] we will consider the differential equation 2
—^- + £(£ + focosr0)sin(p = 0
(4.25)
which differs from the initial one (4.4) by the fact that the variable T = Qt is designated by TQ and to simplify the calculations the term efiQ(p, characterizing the viscous resistance, is omitted. In accordance with the procedure of the method, and confining ourselves to a two-scale decomposition, we will seek a periodic solution of equation (4.25) in the form of the series (p =