 | | Mechanics of Solids
A Journal of Russian Academy of Sciences | | Founded
in January 1966
Issued 6 times a year
Print ISSN 0025-6544
Online ISSN 1934-7936 |
Archive of Issues
| Total articles in the database: | | 12804 |
| In Russian (Èçâ. ÐÀÍ. ÌÒÒ): | | 8044
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| In English (Mech. Solids): | | 4760 |
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| A. S. Bratus', "Multiple eigenvalues in nonconservative problems of stabilization of elastic systems," Mech. Solids. 36 (6), 120-127 (2001) |
| Year |
2001 |
Volume |
36 |
Number |
6 |
Pages |
120-127 |
| Title |
Multiple eigenvalues in nonconservative problems of stabilization of elastic systems |
| Author(s) |
A. S. Bratus' (Moscow) |
| Abstract |
The problem of the loss of stability of a nonconservative system
is considered. The behavior of the system depends on both the
instability parameter and distributions of stiffness properties of
the system. The case is analyzed where zero natural frequencies
appear in the system with a prescribed stiffness distribution for a
critical value of the instability parameter. It is shown that in
this case, the zero natural frequency is almost always a double
eigenvalue. The problem of stabilization of the system by choosing
appropriate stiffness distributions is formulated for a fixed
(critical) value of the instability parameter. The conditions under
which this problem has a solution are determined for various
implementations of double natural frequencies equal
to zero. As an example, the problems of stability for a simply
supported tube of variable cross section with a fluid flowing
inside the tube is considered for various definitions of
the dissipative forces. |
| References |
| 1. | V. V. Bolotin, Nonconservative Problems of Theory of
Elastic Stability [in Russian], Fizmatgiz, Moscow, 1961. |
| 2. | H. Leipholz, Stability of Elastic Systems, Sijthoff and
Noordhoff, Amsterdam, 1980. |
| 3. | M. A. Naimark, Linear Differential Operators [in Russian],
Nauka, Moscow, 1969. |
| 4. | V. M. Mikhailov, "On the Riesz bases in L2(0,1),"
Doklady AN SSSR, Vol. 144, No. 5, pp. 981-964, 1962. |
| 5. | A. S. Bratus, "On various cases of instability for elastic
nonconservative systems with damping," Intern. J. Solid and
Structures, Vol. 30, No. 24, pp. 3431-3441, 1993. |
| 6. | M. I. Vishik and L. A. Lyusternik, "Solution of some
perturbation problems in the case of matrices and self-adjoint
and nonself-adjoint differential equations," Uspekhi Mat. Nauk,
Vol. 15, No. 3, pp. 3-80, 1960. |
| 7. | T. Kato, Theory of Perturbations of Linear Operators [Russian
translation], Mir, Moscow, 1972. |
| 8. | N. V. Banichuk, A. S. Bratus', and A. D. Myshkis, "On the
effects of stabilization and destabilization in nonconservative
systems," PMM [Applied Mathematics and Mechanics], Vol. 53, No. 2,
pp. 206-214, 1989. |
| 9. | N. V. Banichuk and A. S. Bratus', "On the stability of
elastic nonconservative systems allowing divergence solutions,"
Izv. AN. MTT [Mechanics of Solids], No. 1, pp. 134-143, 1992. |
| 10. | B. N. Pshenichnyi, Necessary Conditions of Extremum [in
Russian], Nauka, Moscow, 1982. |
| 11. | A. G. Butkovskii, Methods of Control for Systems with
Distributed Parameters [in Russian], Nauka, Moscow, 1975. |
| 12. | Ya. G. Panovko and I. I. Gubanova, Stability and
Vibrations of Elastic Systems [in Russian], Nauka, Moscow, 1979. |
| 13. | A. S. Bratus' and A. P. Seiranyan, "Bimodal solutions in
problems of optimization of eigenvalues," PMM [Applied Mathematics and
Mechanics], Vol. 47, No. 4, pp. 546-554, 1983. |
| 14. | A. S. Bratus', "Condition of extremum for eigenvalues of
elliptic boundary-value problems," J. Optimiz. Theory and Appl.,
Vol. 68, No. 3, pp. 423-436, 1991. |
|
| Received |
17 April 1998 |
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