Eigenvalues and eigenvectors of the elastic systems with three identical parts.
Vlase, Sorin ; Purcarea, Ramona ; Scutaru, Maria Luminita 等
1. INTRODUCTION
In the case of mechanical systems showing certain symmetries, the
differential equations describing their evolution in time display a
series of specific properties--a consequence of the very existence of
these symmetries. The present work sets forth some of these properties,
for a system with three identical parts, which may be used in the
numerical resolution of these systems.
2. MOTION EQUATIONS
The equations describing the time-behavior, of an elastic
mechanical system with constant concentrated parameters are the
following (Tofan & Vlase, 1985), (Vlase, 2003), (Vlase, 2005),
(Mangeron et al, 1991) :
[M]{[??]}+[C]{[??]}+[K]{X}={F} (1)
where [M], [C], [k] are symmetrical matrices positively defined,
with constant elements, and {F} generally depends on t, {X} and {[??]}.
This type of equations results from many mechanical processes,
particularly by analysis via finite element method.
3. SYSTEMS WITH THREE IDENTICAL SUBSYSTEMS
The engineering practice has recorded many cases in which a
mechanical system (S) is composed by two or more identical sub-systems.
We refer these identical sub-systems with ([S.sub.I]). Be n the number
of independent coordinates of the system (S), [n.sub.1] the number of
independent coordinates of system ([S.sub.1]), [n.sub.2] = n -
[pn.sub.1] where p is the number of identical sub-systems. The evolution
of the system (S) containing two identical subsystems ([S.sub.1]) is
described by the equations:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)
where: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [0]
is zero matrix of [n.sub.1] x [n.sub.1] dimension. The matrices indexed
by 12 indicate the coupling of the identical sub-systems ([S.sub.1])
with the rest of the system (S).
The existences of symmetries facilitate problem resolution for such
system. In order to simplify the presentation of the properties of the
structure we shall either consider the system as non-damped ([C] = 0) or
assume that [C] meets the Caughey conditions.
4. PROPERTIES OF SYMMETRICAL BRANCHED SYSTEMS
Be {[PHI]} the solution of the eigenvector problem (Belmann, 1960):
[K]{[PHI]} = [p.sup.2.sub.i][M]{[THETA]} (3)
We have the following properties:
* P1. The eigenvalues of (S1) are also eigenvalues for (S)
* P2. If {[[PHI].sub.1]} is an eigenvector of ([S.sub.1]), then
[a[{[[PHI].sub.1]}.sup.T] [{A}.sup.T] [{B}.sup.T]].sup.T].
* P3. The other eigenvectors of (S) appear under the form of
[[{A}.sup.T] [{A}.sup.T] [{A}.sup.T] [{B}.sup.T]].sup.T].
The first two properties can be verified via direct calculation. In
order to demonstrate the third one, if we take the eigenvector under the
form [[{A}.sup.T] [{A'}.sup.T] [{B}.sup.T]].sup.T] we will
necessarily have:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)
where [p.sub.i] is not eigenvalue for the subsystem ([S.sub.1]).
Relation (4) can be written in an alternate form:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)
From the first three relations, because
det([[K.sub.1]]-[p.sup.2.sub.i][M.sub.i])[not equal to] 0 it results {A}
= {A'} = {A"}.
5. APPLICATION
We consider now a system with symmetries like in Fig.1. From this
system we can identify three identical parts.
The computed eigenvalues for this system are: 3717, 2765, 1512,
702, 18, 284, 2884, 2000, 2884, 2000, 116, 116 [rad/s].
[FIGURE 1 OMITTED]
In Fig. 2 are presented the eigenvectors for the system. It is
easily to see the proved properties.
[FIGURE 2 OMITTED]
5. REFERENCES
Belmann, R. (1960). Introduction to Matrix Analysis. Mc Graw-Hill
Book Company, Inc.
Mangeron, D., Goia, I., Vlase, S., (1991). Symmetrical Branched
Systems Vibrations. Memoriile sectiilor stiintifice ale Academiei (The
memoirss of the scientific sections of the Academy), seria IV (4th
series), Tomul XII, Nr.1, Bucuresti
Tofan, M., Vlase, S. (1985). Vibratiile sistemelor mecanice (The
vibrations of the mechanical systems), Universitatea TRANSILVANIA din Brasov (Transilvania University of Brasov), Brasov
Vlase, S. (2003). Mecanica III. Dinamica (Mechanics III. Dynamics).
Universitatea TRANSILVANIA din Brasov (Transilvania University of
Brasov), Brasov.
Vlase, S. (2005). Computational Mechanics, Ed. Infomarket, Brasov
