Natural oscillations of mechanical systems and the spectral theory of self-adjoint operators

In this paper we investigate certain aspects of the multiparameter spectral theory of systems of singular ordinary differential operators. Such systems arise in various contexts. For instance, separation of variables for a partial differential equation on an unbounded domain leads to a multiparameter system of ordinary differential equations, some of which are defined on unbounded intervals. The spectral theory of systems of regular differential operators has been studied in many recent papers, e.g. [1, 3, 6, 9, 19, 21], but the singular case has not received so much attention. Some references for the singular case are [7, 8, 10, 13, 14, 18, 20], in addition general multiparameter spectral theory for self adjoint operators is discussed in [3, 9, 19].

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Feynman’s Operational Calculi: Spectral Theory for Noncommuting Self-adjoint Operators

Mathematical Physics Analysis and Geometry ◽

10.1007/s11040-007-9021-8 ◽

2007 ◽

Vol 10 (1) ◽

pp. 65-80 ◽

Cited By ~ 2

Author(s):

Brian Jefferies ◽

Gerald W. Johnson ◽

Lance Nielsen

Keyword(s):

Spectral Theory ◽

Adjoint Operators ◽

Feynman’S Operational Calculi

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Methods for Determinning Critical Values of Nonconservative Loads in Problems of Stability of Mechanical Systems

Proceedings of Higher Educational Institutions Маchine Building ◽

10.18698/0536-1044-2019-10-3-13 ◽

2019 ◽

pp. 3-13 ◽

Cited By ~ 1

Author(s):

V.P. Radin ◽

V.P. Chirkov ◽

A.V. Shchugorev ◽

V.N. Shchugorev

Keyword(s):

Complex Function ◽

Mechanical Systems ◽

Critical Values ◽

The Finite Element Method ◽

Natural Oscillations ◽

Flowing Liquid ◽

Stability Problems ◽

Distributed Parameters ◽

The Stability ◽

Systems With Distributed Parameters

Methods for determining critical values of nonconservative loads in stability problems of mechanical systems with distributed parameters are considered in this work. Based on a dynamic approach to stability problems, the method of direct integration of the linearized equation of perturbed motion is proposed, and the problem of determining critical loads is reduced to the problem of minimizing a complex function of several variables. As a second method, the method of decomposition of the solution of the equation of perturbed motion in the forms of natural oscillations is presented. The fundamentals of the application of the finite element method to the problems of stability under the action of non-conservative loads are also described. The methods are illustrated on classical problems: the stability of the cantilever rod under the action of potential and tracking forces and the stability of the pipeline section with flowing liquid. The accuracy and convergence of the latter two methods are analyzed depending on the number of members in the series and the number of finite elements.

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