Математика/1.Дифференциальные и интегральные уравнения
Kaftaikina E.N., Dokukova
N.А., Zenkovich V.V., Konon P.N.
Belarusian State University, Belarus
GENERAL CHARACTERISTIC EQUATION OF DYNAMIC MULTIELEMENT SYSTEM
Algebraic
conditions V.V. Voronov [1] based
on the analysis of the general characteristic
equation of a mathematical system are used to study the
stability of multi-element oscillating mechanical
systems. They allow to establish criteria for the
selection of rational physical and geometrical parameters of
the original mechanical system. Methods of calculating the
dynamics of multi-element mechanical system was
introduced in [2,
3]. They theoretically
and mathematically proved the
main provisions of improper calculation
of dynamical fluctuations systems with
an arbitrary number of degrees of freedom.
Improper vibration multielement mechanical
systems described by N-dependent linear inhomogeneous differential
equations of second order
(1)
Let 
denote the set of all combinations of
the set 
by k elements. For example: 
. Assume
that the set of components in each element of the set 
in ascending order, i.e. for every 
- component
elements 
in ascending order,
i.e. 
.
For each 
introduce Cartesian product 
as follows [4,
5]: 
(2)
Using results obtained in [2] for any 
introduce the
matrix 
and 
, obtained
from B and C by the following rule:
,
The system of equations (1) becomes:
, (3)
Introduce the
following amount: 
, that is the sum of the determinants 
of the
form: when k=0: 
, when k=n: 
, when 0<k<n: 
is the sum of the determinants of matrices, each of which is equal to the matrix 
with the exception of k rows, corresponding
rows of the matrix 
. Or, on the other, 
is the sum of the determinants of matrices, each of which is 
, except for n - k rows of the matrix 
rows. For
example:
.
Next, consider the following determinant:
(4)
where 
.
Example 1. SupposeN=2, then formula (4) becomes:
Example 2. Suppose N≥3, then formula (4) will be the
(
through 
and 
denote the matrix 
and 
in which the crossed 1st row
and m-th column).
Here the determinant is
decomposed in the first row. Next, the brackets and changing the order of
summation, we obtain:
(5)
Based on this general formula (5) it
can be argued that the determinant of the
system as a sum for N≥2
(6)
A general formula for the
inhomogeneous decoupled system of differential
equations (6),
similar to the dynamics of
the vibrational motions of mechanisms with two degrees of freedom and
three degrees of freedom.
(7)
The right sides 
for each i, determined
from the system of equations written in
terms of operators in matrix form. Characteristic equation is
written for the linear differential equation (7) with constant
coefficients of order 2N
(8)
The calculation method
of the coefficients of the characteristic equation (8) connected to the
system of second order differential equations
(1) is developed. It is used to calculate the physical parameters of
the mechanical system described by a system of N-linear second
order differential equations.
References
1. Voronov ,V.V. Sustainability
Indicators of robust control systems / / Izv. Academy
of Sciences. Theory and control systems. 1995. Number
6. S. 49-54
2. Dokukova,
N.A. General patterns of improper vibrations of dynamical
systems with arbitrary number of degrees of freedom / N. A. Dokukova,
E.N. Kaftaikina, V.V. Zenkovich // News on
the scientific progression -2011: The material for the 7-and
international scientific and practical conference, Sofia, 17-25 August 2011.: 9 tons / Editorial Board.: M. Petkov. -
Sofia: GRAD-BG Ltd.,
2011.-. T. 9. -
S. 56 -64.3.N. A. 3.Dokukova
and P. N. Konon General laws governing in mechanical vibratory systems//
Journal of Engineering Physics and Thermophysics, 2006, Volume 79, Number 4,
Pages 824-831, Publisher Springer New York, ISSN: 1062-0125.
4.Bugrov Y.S., Nikolskiy
S.M. Higher Mathematics. Differential equations. Multiple
integrals. Series. Functions of a complex variable. - Moscow:
Nauka. Ch. Ed. nat. mat. Lit., 1989.- 464 p.
5.Vekua, N.P. Some
questions in the theory of differential equations and
applications in mechanics. - Moscow:
Nauka. Ch. Ed. nat. mat. Lit., 1991.- 256 p.