Abstract
Eigenvalue problems of ordinary differential equation systems are discussed. The concept of the Green function matrix is introduced. By utilizing the Green function matrices the eigenvalue problems described by ordinary differential equation systems can be reduced to eigenvalue problems governed by homogeneous Fredholm integral equation systems. The solution algorithm presented in Chapter 8 is generalized for such eigenvalue problems. The applications are related to the vibration problems of Timoshenko beams.
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Notes
- 1.
See for instance
http://www.bu.edu/moss/mechanics-of-materials-bending-shear-stress.
References
S.P. Timoshenko, On the correction for shear of the differential equation for transverse vibrations of prismatic bars. Lond. Edinb. Dublin Philos. Mag. J. Sci. 41(245), 744–746 (1921). https://doi.org/10.1080/14786442108636264
F. Engesser, Die knickfestigkeit grader stëabe. Central Bauvervaltung 11, 483–486 (1981)
G.R. Cowper, The shear coefficient in Timoshenko’s beam theory. J. Appl. Mech. 33(2), 335–340. (1966). https://doi.org/10.1115/1.3625046
G.R. Cowper, On the accuracy of Timoshenko’s beam theory. ASCE, J. Eng. Mech. Div. 94(6), 1447–1453 (1968)
A. Marinetti, G. Oliveto, On the Evaluation of the Shear Correction Factors: a Boundary Element Approach (2009)
M. Levinson, D.W. Cooke, On the frequency spectra of Timohsenko beams. J. Sound Vib. 84(3), 319–324 (1982). https://doi.org/10.1016/0022-460X(82)90480-1
G. Szeidl, Effect of Change in Length on the Natural Frequencies and Stability of Circular Beams. in Hungarian. Ph.D. thesis. Department of Mechanics, University of Miskolc, Hungary (1975), 158 pp
J.G. Obádovics, On the Boundary and Initial Value Problems of Differential Equation Systems. Ph.D. thesis. Hungarian Academy of Sciences (1967), (in Hungarian)
L. Collatz, Eigenwertaufgaben mit Technischen Anwendungen. Russian Edition in 1968 (Akademische Verlagsgesellschaft Geest & Portig K.G., 1963)
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Szeidl, G., Kiss, L. (2020). Eigenvalue Problems of Ordinary Differential Equation Systems. In: Mechanical Vibrations. Foundations of Engineering Mechanics. Springer, Cham. https://doi.org/10.1007/978-3-030-45074-8_9
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DOI: https://doi.org/10.1007/978-3-030-45074-8_9
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