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

Published electronically:
March 20, 1998

MathSciNet review:
1611829

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Abstract | References | Similar Articles | Additional Information

Abstract: This announcement presents asymptotic formulas for the eigenvalues of a free-free uniform Timoshenko beam.

**1.**Bruce Geist,*The asymptotic expansion of the eigenvalues of the Timoshenko beam*, Ph.D. Dissertation, Rensselaer Polytechnic Institute, Troy, NY, 1994.**2.**Bruce Geist and J. R. McLaughlin,*Double eigenvalues for the uniform Timoshenko beam*, Applied Mathematics Letters,**10**(1997), 129-134. CMP**97:15****3.**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**0134526****4.**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.**5.**Stephen Timoshenko,*On the correction for shear of the differential equation for transverse vibrations of prismatic bars*, Philisophical Magazine**41**(1921), 744-746.**6.**-,*On the transverse vibrations of bars of uniform cross-section*, Philisophical Magazine**43**(1922), 125-131.**7.**R. W. Traill-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. MR**0066205**

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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

DOI:
http://dx.doi.org/10.1090/S1079-6762-98-00041-9

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