Abstract

This paper provides a generalization of the celebrated Merkin theorem. It provides new results on the destabilizing effect of circulatory forces on stable potential systems. Previous results are described and discussed, and the paper uncovers a deeper understanding of the fundamental reason for the destabilization. Instability results in terms of rank conditions that deal with the potential and circulatory matrices that describe the system are obtained, thereby generalizing this remarkable theorem. These new results are compared with those obtained earlier.

References

References
1.
Kirillov
,
O. N.
,
2013
,
Nonconservative Stability Problems of Modern Physics
,
De Gruyter
,
Berlin
.
2.
Bigoni
,
D.
, and
Kirillov
,
O. N.
,
2019
,
Dynamic Stability and Bifurcation in Nonconsevative Mechanics
,
CISM International Centre for Mechanical Sciences, 586
,
Springer
,
New York
.
3.
Udwadia
,
F. E.
,
2017
, “
Stability of Dynamical Systems With Circulatory Forces: Generalization of the Merkin Theorem
,”
AIAA J.
,
55
(
9
), pp.
2853
2858
. 10.2514/1.J056109
4.
Painleve
,
P.
,
1908
, “
Sur la Stabilite de L’equilibre
,”
C. R. Acad. Sci. Paris
,
138
, pp.
1555
1557
.
5.
Chetayev
,
N. G.
,
1962
,
Stability of Motion. Works in Analytical Mechanics
,
AN SSSR
,
Moscow
, (
in Russian
).
6.
Bulatovic
,
R.
,
1992
, “
On the Converse of the Lagrange–Dirichlet Theorem
,”
C. R. Acad. Sci. Paris
,
315
, pp.
1
6
.
7.
Merkin
,
D. R.
,
1997
,
Introduction to the Theory of Stability
,
Springer-Verlag
,
New York
.
8.
Merkin
,
D. R.
,
1956
,
Gyroscopic Systems
,
Gostekhizdat
,
Moscow
, (
in Russian
).
9.
Krechetnikov
,
R.
, and
Marsden
,
J. E.
,
2007
, “
Dissipation-Induced Instabilities in Finite Dimension
,”
Rev. Mod. Phys.
,
79
(
2
), pp.
519
553
. 10.1103/RevModPhys.79.519
10.
Bulatovic
,
R. M.
,
2011
, “
A Sufficient Condition for Instability of Equilibrium of Non-Conservative Undamped Systems
,”
Phys. Lett. A
,
375
(
44
), pp.
3826
3828
. 10.1016/j.physleta.2011.09.015
11.
Bulatovic
,
R. M.
,
1999
, “
On the Stability of Linear Circulatory Systems
,”
Z. Angew. Math. Phys.
,
50
(
4
), pp.
669
674
. 10.1007/s000330050172
12.
Udwadia
,
F. E.
, and
Bulatovic
,
R. M.
,
2020
, “
Stability of Potential Systems to General Positional Perturbations
,”
AIAA J.
,
58
(
9
), pp.
4106
4116
10.2514/1.J059241.
13.
Awrejcewicz
,
J.
,
Losyeva
,
N.
, and
Puzyrov
,
V.
,
2020
, “
Stability and Boundedness of the Solutions of Multi-Parameter Dynamical Systems with Circulatory Forces
,”
Symmetry
,
12
(
8
), p.
1210
.
14.
Bellman
,
R.
,
1970
,
Introduction to Matrix Analysis
,
McGraw-Hill
,
New York
.
15.
Shemesh
,
D.
,
1984
, “
Common Eigenvectors of Two Matrices
,”
Linear Algebra Appl.
,
62
, pp.
11
18
. 10.1016/0024-3795(84)90085-5
16.
Bulatovich
,
R. M.
,
1997
, “
Simultaneous Reduction of a Symmetric Matrix and a Skew-Symmetric One to Canonical Form
,”
Math. Montisnigri
,
8
, pp.
33
36
, (
in Russian
).
17.
Ikramov
,
K. D.
,
2000
, “
On a Quasidiagonalizability Criterion for Real Matrices
,”
Comput. Math. Math. Phys.
,
40
(
1
), pp.
4
17
.
You do not currently have access to this content.