Knowledge of the flexural vibration frequencies of thin rectangular cantilever plates forms the basis for numerous applications in sensing and instrumentation. Despite the seemingly simple nature of the problem, an accurate formula for the fundamental resonant frequency that is valid for all aspect ratios and Poisson’s ratios is notably lacking in the literature. In this article, we present such a result using a variational and singular perturbation formulation. This yields a simple analytical formula that exhibits a maximum error of 2%.
Issue Section:
Research Papers
1.
Reissner
, E.
, and Stein
, M.
, 1951, “Torsion and Transverse Bending of Cantilever Plates
,” N.A.C.A. Technical Note 2369.2.
Fritz
, J.
, Baller
, M. K.
, Lang
, H. P.
, Rothuizen
, H.
, Vettiger
, P.
, Meyer
, E.
, Guentherodt
, H. J.
, Gerber
, C.
, and Gimzewski
, J. K.
, 2000, “Translating Biomolecular Recognition Into Nanomechanics
,” Science
0036-8075, 288
(5464
), pp. 316
–318
.3.
Ilic
, B.
, Craighead
, H. G.
, Krylov
, S.
, Senaratne
, W.
, Ober
, C.
, and Neuzil
, P.
, 2004, “Attogram Detection Using Nanoelectromechanical Oscillators
,” J. Appl. Phys.
0021-8979, 95
(7
), pp. 3694
–3703
.4.
Giessibl
, F. J.
, 1995, “Atomic-Resolution of the Silicon (111)-(7×7) Surface by Atomic-Force Microscopy
,” Science
0036-8075, 267
(5194
), pp. 68
–71
.5.
Levy
, M.
, 1899, “Sur l’Équilibre Élastique d’Une Plaque Rectangulaire
,” C. R. Acad. Sci. Hebd Seances Acad. Sci. D
0567-655X, 129
, p. 535
.6.
Jaramillo
, T. J.
, 1950, “Deflections and Moments Due to a Concentrated Load on a Cantilever Plate of Infinite Length
,” ASME J. Appl. Mech.
0021-8936, 17
(1
), pp. 67
–72
.7.
Warburton
, G. B.
, 1954, “The Vibration of Rectangular Plates
,” Proc. Inst. Mech. Eng.
0020-3483, 168
, pp. 371
–384
.8.
Leissa
, A. W.
, 1973, “Free Vibration of Rectangular-Plates
,” J. Sound Vib.
0022-460X, 31
(3
), pp. 257
–293
.9.
Dickinson
, S. M.
, 1978, “Buckling and Frequency of Flexural Vibration of Rectangular Isotropic and Orthotropic Plates Using Rayleigh’s Method
,” J. Sound Vib.
0022-460X, 61
(1
), pp. 1
–8
.10.
Kim
, C. S.
, and Dickinson
, S. M.
, 1985, “Improved Approximate Expressions for the Natural Frequencies of Isotropic and Orthotropic Rectangular-Plates
,” J. Sound Vib.
0022-460X, 103
(1
), pp. 142
–149
.11.
Jänich
, R.
, 1962, “Die naherungsweise Berechnung der Eigenfrequenzen von rechteckigen Platten bei verschiedenen Randbedingungen
,” Die Bautechnik
, (3
), pp. 93
–99
.12.
Feynman
, R. P.
, Leighton
, R. B.
, and Sands
, M.
, 1963, The Feynman Lectures on Physics
, Addison-Wesley
, Reading, MA
.13.
Smirnov
, V. I.
, 1964, A Course of Higher Mathematics
, Vol. 4
, Pergamon
, Oxford
.14.
Timoshenko
, S. P.
, and Woinowsky-Krieger
, S.
, 1959, Theory of Plates and Shells
, McGraw-Hill
, New York
.15.
Hinch
, E. J.
, 1991, Perturbation Methods
, Cambridge University Press
, Cambridge
.16.
Sader
, J. E.
, 1998, “Frequency Response of Cantilever Beams Immersed in Viscous Fluids With Applications to the Atomic Force Microscope
,” J. Appl. Phys.
0021-8979, 84
(1
), pp. 64
–76
.17.
Van Eysden
, C. A.
, and Sader
, J. E.
, 2007, “Frequency Response of Cantilever Beams Immersed in Viscous Fluids With Applications to the Atomic Force Microscope: Arbitrary Mode Order
,” J. Appl. Phys.
0021-8979, 101
, p. 044908
.18.
Mansfield
, E. H.
, 1964, The Bending and Stretching of Plates
, Pergamon
, Oxford
.Copyright © 2008
by American Society of Mechanical Engineers
You do not currently have access to this content.
