Eigenvalue formulas for the uniform Timoshenko beam: the free-free problem
Authors:
Bruce Geist and Joyce R. McLaughlin
Journal:
Electron. Res. Announc. Amer. Math. Soc. 4 (1998), 12-17
MSC (1991):
Primary 34Lxx; Secondary 73Dxx
DOI:
https://doi.org/10.1090/S1079-6762-98-00041-9
Published electronically:
March 20, 1998
MathSciNet review:
1611829
Full-text PDF Free Access
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Additional Information
Abstract: This announcement presents asymptotic formulas for the eigenvalues of a free-free uniform Timoshenko beam.
- Bruce Geist, The asymptotic expansion of the eigenvalues of the Timoshenko beam, Ph.D. Dissertation, Rensselaer Polytechnic Institute, Troy, NY, 1994.
- Bruce Geist and J. R. McLaughlin, Double eigenvalues for the uniform Timoshenko beam, Applied Mathematics Letters, 10 (1997), 129–134.
- T. C. Huang, The effect of rotatory inertia and of shear deformation on the frequency and normal mode equations of uniform beams with simple end conditions, Trans. ASME Ser. E. J. Appl. Mech. 28 (1961), 579–584. MR 134526
- Edwin T. Kruszewski, Effect of transverse shear and rotary inertia on the natural frequency of a uniform beam, National Advisory Committee for Aeronautics, Technical Note no. 1909, July 1949.
- Stephen Timoshenko, On the correction for shear of the differential equation for transverse vibrations of prismatic bars, Philisophical Magazine 41 (1921), 744–746.
- ---, On the transverse vibrations of bars of uniform cross-section, Philisophical Magazine 43 (1922), 125–131.
- R. W. Trail-Nash and A. R. Collar, The effects of shear flexibility and rotatory inertia on the bending vibrations of beams, Quart. J. Mech. Appl. Math. 6 (1953), 186–222.
- Bruce Geist, The asymptotic expansion of the eigenvalues of the Timoshenko beam, Ph.D. Dissertation, Rensselaer Polytechnic Institute, Troy, NY, 1994.
- Bruce Geist and J. R. McLaughlin, Double eigenvalues for the uniform Timoshenko beam, Applied Mathematics Letters, 10 (1997), 129–134.
- T. C. Huang, The effect of rotatory inertia and of shear deformation on the frequency and normal mode equations of uniform beams with simple end conditions, Journal of Applied Mechanics 28 (1961), 579–584.
- Edwin T. Kruszewski, Effect of transverse shear and rotary inertia on the natural frequency of a uniform beam, National Advisory Committee for Aeronautics, Technical Note no. 1909, July 1949.
- Stephen Timoshenko, On the correction for shear of the differential equation for transverse vibrations of prismatic bars, Philisophical Magazine 41 (1921), 744–746.
- ---, On the transverse vibrations of bars of uniform cross-section, Philisophical Magazine 43 (1922), 125–131.
- R. W. Trail-Nash and A. R. Collar, The effects of shear flexibility and rotatory inertia on the bending vibrations of beams, Quart. J. Mech. Appl. Math. 6 (1953), 186–222.
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Additional Information
Bruce Geist
Affiliation:
Unisys Corporation, 41100 Plymouth Road, Plymouth, MI 48170
Email:
Bruce.Geist@unisys.com
Joyce R. McLaughlin
Affiliation:
Department of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, NY 12180
Email:
mclauj@rpi.edu
Keywords:
Timoshenko beam,
asymptotic distribution of eigenvalues,
boundary value problems
Received by editor(s):
January 5, 1998
Published electronically:
March 20, 1998
Additional Notes:
The work of both authors was completed at Rensselaer Polytechnic Institute, and was partially supported by funding from the Office of Naval Research, grant number N00014-91J-1166. The work of the first author was also partially supported by the Department of Education fellowship grant number 6-28069. The work of the second author was also partially supported by the National Science Foundation, grant number DMS-9410700.
Communicated by:
Michael Taylor
Article copyright:
© Copyright 1998
American Mathematical Society
