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Variational and Projection Methods for Solving Vibration Theory Equations

  • Vladimir FridmanEmail author
Chapter
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Part of the Foundations of Engineering Mechanics book series (FOUNDATIONS)

Abstract

Exact solutions of equations of the theory of vibrations can only be constructed for a limited class of problems under homogeneous properties of an elastic body. However, if the elastic, inertial and dissipative properties are variable in coordinate, then there is a need to use approximate methods to solve equations of the theory of vibrations.

Keywords

Mixed Variational Principle Forced Harmonic Oscillator Ritz Formula Free Oscillations Displacement Equations 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Fridman, V. M. (1956). On an approximate method for determining the natural frequencies of vibrations. Academy of Sciences of the U.S.S.R., Vibrations in Turbomachinery, pp. 69–76 (in Russian).Google Scholar
  2. 2.
    Kukishev, V. L., & Fridman, V. M. (1976). Variational difference method in the theory of elastic vibrations, based on the Reissner variational principle. Academy of Sciences of the U.S.S.R., Mechanics of Solids, 5, 112–119 (in Russian).Google Scholar
  3. 3.
    Lurie, A. I. (2005). Theory of elasticity (1050 pp). Berlin: Springer.Google Scholar
  4. 4.
    Mikhlin, S. G. (1964). Variational methods in mathematical physics (p. 510). Oxford: Pergamon Press.zbMATHGoogle Scholar
  5. 5.
    Petrov, G. I. (1940). Application of the Galerkin method to the problem of viscous fluid flow stability. Academy of Sciences of the USSR, Applied Mathematics and Mechanics, 4(3), 3–11 (in Russian).Google Scholar
  6. 6.
    Reissner, E. O. (1961). On some variational theorems of the theory of elasticity. Academy of Sciences of the U.S.S.R., Problems of Continuum Mechanics, 328–337 (in Russian).Google Scholar
  7. 7.
    Smirnov, V. I. (1964). Course of higher mathematics (Vol. IV, 336 pp). Oxford: Pergamon Press.Google Scholar
  8. 8.
    Ritz, W. (1909). Über eine neue Methode zur Lösung gewisser Variationsprobleme der mathematischen Physik. Journal für die reine und angewandte Mathematik (Grelle), 135(1), 1–61.zbMATHGoogle Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  1. 1.Los AngelesUSA

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