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IssuesArchive of Issues2001-6pp.120-127

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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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