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E. А. Энгель, И. В. Ковалев
ИСПОЛЬЗОВАНИЕ ИНТЕЛЛЕКТУАЛЬНЫХ МЕТОДОВ ДЛЯ ОБРАБОТКИ ИНФОРМАЦИИ
НА ПРИМЕРЕ РЕШЕНИЯ ЗАДАЧ WCCI 2010
Рассмотрена актуальная проблема выбора стратегии решения слабоформализованных задач, предполагающих обработку как количественных, так и качественных данных, высокую размерность и пропуски в данных.
Представлен детальный анализ моделей прогноза для обработки данных. Эксперименты подтверждают эффективность интеллектуальных алгоритмов, разработанных авторами.
Ключевые слова: обработка данных, слабоформализованные задачи, интеллектуальные алгоритмы.
© Engel E. А., Kovalev I. V., 2011
УДК 517.95
M. B. Frost
DETERMINING THE SOURCE OF TRANSVERSE OSCILLATIONS OF AN ELASTIC ROD
Solvability of an inverse problem for the equations of transverse oscillations of an elastic rod (determining the source of oscillations on the basis of the rod deflection at the final time) is proved.
Keywords: elastic rod, inverse problem, source of oscillations, solvability.
Many issues of engineering, geophysics, and medicine involve problems whose sought quantities are elements of initial-boundary problems for differential equations. These unknown elements are determined on the basis of additional information. Such problems are called inverse problems for differential equations. The advanced theory
of inverse problems and numerous publications can be found in [1].
Let us consider an inverse problem of determining the source of transverse oscillations of an elastic rod. The initial-boundary problem of transverse oscillations of a simply supported rod with a constant cross section and a
length l under the action of a distributed load q (x, t) is posed [2] as follows:
ut, + a 2uxxxx = q (xt), (1)
u (x,0) = m (x), ut (x,0) = n (x), (2)
U (0, t) = u (l, t) = uxx (0, t) = uxx (l, t) = 0 , (3)
Here, u = u (x, t), 0 < x < l, 0 < t < T is the
deflection, a is a constant, m (x) and n (x), 0 < x < l
are given functions satisfying the conditions
m (0 ) = n (0 ) = m (l ) = n (l ) = 0. (4)
Let us consider an inverse problem of determining the source of oscillations. We assume that
q (x, t) = ro (t)X(x) , (5)
where ro (t), 0 < t < T , is a known function of time, and X (x), 0 < x < l is an unknown function determined with
the use of an additional condition (deflection at the final time t = T ):
u (x, T ) = p (x), 0 < x < l, p (0) = p (l ) = 0, T > 0. (6)
We integrate Eq. (1) with respect to t from 0 to t, and then from 0 to T . Taking into account equalities (2-5), we obtain
p (x)-m (x)-Tn (x) + a2 jj j U;œa (x T)d x о V о
t ( ‘
= X(x )jl j ro (x)d t
dt =
dt.
Let us assume that
h = j (T -T)ro(T)dt Ф 0 .
о
Then, equality (7) yields
X(x) = h-1 (p (x) - m (x)- Tn (x) +
T '
+ a2 J(T -T)Uxxxx ( T)dT .
0 )
Substituting Eq. (9) into Eq. (1), we obtain
T
■ a"Uxxxx - h~la'ro (t) J(T - T)Uxxxx (x, T) dT =
0
= hTl (p (x) - m (x) - Tn (x)) ro (t).
u„ +1
Let us study solvability of the initial-boundary problem (10), (2-4) under the following assumptions:
- the function is ro (t) > 0 (or ro (t) < 0 ) for 0 < t < T ,
ro ( 0) — 0, ro (T ) ^ 0 ; the function ro (t ) is continuously differentiable on [0, T] ;
- the following expansions are valid:
/ \ ^ . ( k nx I, I m
m(x) = Zmk sin| — I, Kl <-
k=1
Kx )=£
n, sin
l
k nx
~T
k7
n
(11)
p (x )=j=;p, sin [ ^ ], m <
(e, m , n and p* are positive constants).
We seek for the solution of problem (10), (2-4) by the Fourier method of separation of variables in the form
u (^ 0=Ê Uk (t)sin V —П— j.
(12)
Substituting Eq. (12) into Eq. (10) and taking into account Eq. (2), (11), we obtain an infinite system of ordinary differential equations with initial conditions for the functions Uk (t), k = 1,2,...:
d 2Uk dt2
+ ^u— (t) = ekro(t). k = U—
(7)
Uk (0) =
dUk
m.
dt
ak2 n2 = nk , ^ k = ——
(13)
(14)
(8)
ek = h-1 (pk -mk -nkT) + h~lV-k jUk (T)(T~T)dT. (15)
0
Solving Eq. (13) with the initial condition (14) for k = 1,2,..., we obtain
(9)
u— (t ) = — j ro (t) sin (^k (t - t)) d t -
H-k о
n
+ mk cos(Hkt)+-Lsin(Hkt).
Hk
(16)
Let us use t = T , Uk (t ) = pt in Eq. (16) and assume
that
= j ro (t) sin (k (T - t)^ d t Ф 0, k = 1,2,.... (17)
(10)
The following equality is valid:
Thus, the problem of determining the function u ( x, t )
is reduced to the initial-boundary problem for the loaded equation of oscillations (10) under conditions (2-4). In this case, condition (6) is satisfied automatically. The function X (x), 0 < x < l is then determined from equality (9).
.
ro.
Pk - mk c°s(-kT)—k-sin(-kT) —k
(18)
As |ak with k , then, by virtue of the as-
sumptions made for the function ro(t), we obtain [3] for large values of k
I | 12ro(T)
an k
(19)
k=1
t=0
k
Therefore, the series determining the function u (x, t)
by equality (12) with allowance for Eqs. (16, 18, 19), is absolutely converging, and it can be differentiated four times with respect to x and two times with respect to t . It satisfies Eq. (10) and conditions (1-4).
This solution is unique in the class of functions that admit twofold differentiation with respect to t and fourfold differentiation with respect to x . The function X (x) for 0 < x < l is uniquely determined by equality (9) in the form of an absolutely converging series - / \ • k nx
X(x) = L ek sin— .
k=1 l
Equality (6) is also satisfied. The constructed functions u (x, t) and X (x) uniquely determine the solution of the
inverse problem (1-6).
Note that condition (17) is always satisfied for the linear function ro (t) = at for 0 < t < T , where a is a constant, because
sin (u.kT)
1-------> 0 for all |ak, k = 1,2,...
VkT
The algorithm of the numerical solution of problem (1-6) can be constructed as follows. Let n = 0,1,2,... be the number of the iteration. We seek for the function un (x, t) numerically, using the grid technique or the
finite-element method [4] to solve the equation of rod oscillations with a known right side determined via the function un-1 (x, t)
К + a2ulxx = г ®(* )j <1 (x T)(T-T)d T +
h 0
+hTx (p (x)- m (x)- Tn (x))со (t),
(20)
and satisfying conditions (2-4). An arbitrary function satisfying conditions (2-4) can be used as the initial approximation. The calculations are performed until
l i T Y
Jl K(^t)-uxx-x(xt))(t-T)dt dx<8,
0 v 0 y
where S is an arbitrary, rather small number.
This condition means a small value of the integral
l 2 j(Xn (x)-Xn-1 (x)) dx,
0
where Xn (x) is calculated via un (x, t) by Eq. (9).
In conclusion, we can mention that the Fourier method can be used in a similar manner to prove unique solvability of the inverse problem of determining the source of oscillations of an elastic rectangular constantthickness plate with sides a and b , which is simply supported on its edges. In this case, u = u (x, y, t), 0 < x < a , 0 < y < b , 0 < t < T ; u^ in Eq. (1) should
be replaced by a biharmonic operator A2u for the variables x and y [5]; the transverse load should be
taken in the form q (x, y, t ) = rop )X(x, y), where ro(t) is
a known function of time and X(x, y) is an unknown
function of coordinates, which is determined from the known value of the deflection at the final time t = T .
An inverse problem for the equation of transverse oscillations of an elastic rod with a constant cross section (determining the source of oscillations on the basis of the deflection at the final time) is considered in the paper. The problem is reduced to solving an initial-boundary problem for the loaded equation of oscillations containing a functional of the solution. Unique solvability of the initial-boundary problem for the loaded equation and then of the inverse problem is proved by the Fourier method. An algorithm of the numerical solution of the inverse problem is proposed.
References
1. Prilepko A. J., Orlovsky D. G., Vasin J. A. Methods for solving inverse problems in mathematical physics. Monographs and textbooks in pure and applied mathematics. N.-Y. : M. Dekker, 2000.
2. Krylov A. N. On some differential equations of mathematical physics, which have engineering applications. M.-L. : Gos. Izdat. Tekhn. - Teor. Lit., 1950.
3. Kopson E. T. Asymptotic Expansions. Cambridge : Cambridge Univ. Press, 1965.
4. Basov K. A. Ansys in Examples and Problems. М. : Computer Press, 2002.
5. Panovko Ya. G. Fundamentals of the Applied Theory of Oscillations and Impacts. L. : Politekhnika, 1990.
М. Б. Фрост
ОПРЕДЕЛЕНИЕ ИСТОЧНИКА ПОПЕРЕЧНЫХ КОЛЕБАНИЙ УПРУГОГО СТЕРЖНЯ
Рассмотрена задача определения источника поперечных колебаний упругого стержня по дополнительной информации - значению прогиба в финальный момент времени.
Ключевые слова: упругий стержень, обратная задача, источник поперечных колебаний, разрешимость.
© Frost M. B., 2011
