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TECHNICAL PAPERS

# On the Fundamental Transverse Vibration Frequency of a Free-Free Thin Beam With Identical End Masses

[+] Author and Article Information
A. Erturk1

Center for Intelligent Material Systemsand Structures,Department of Engineering Science and Mechanics, Virginia Tech, Blacksburg, Virgina 24061erturk@vt.edu

D. J. Inman

Center for Intelligent Material Systems and Structures, Department of Mechanical Engineering, Virginia Tech, Blacksburg, Virgina 24061dinman@vt.edu

It is the first nonzero root of the transcendental equation $1−cosλcoshλ=0$ (see, for instance, Blevins (5)).

See the exact ranges given by Eq. 4.

1

Corresponding author.

J. Vib. Acoust 129(5), 656-662 (May 23, 2007) (7 pages) doi:10.1115/1.2776341 History: Received January 20, 2007; Revised May 23, 2007

## Abstract

Current research in vibration-based energy harvesting and in microelectromechanical system technology has focused renewed attention on the vibration of beams with end masses. This paper shows that the commonly accepted and frequently quoted fundamental natural frequency formula for a beam with identical end masses is incorrect. It is also shown that the higher mode frequency expressions suggested in the referred work (Haener, J., 1958, “Formulas for the Frequencies Including Higher Frequencies of Uniform Cantilever and Free-Free Beams With Additional Masses at the Ends  ,” ASME J. Appl. Mech.25, pp. 412) are also incorrect. The correct characteristic (frequency) equation is derived and nondimensional comparisons are made between the results of the previously published formula and the corrected formulation using Euler–Bernoulli beam assumptions. The previous formula is shown to be accurate only for the extreme case of very large end mass to beam mass ratios. Curve fitting is used to report alternative first order and second order polynomial ratio expressions for the first natural frequency, as well as for the frequencies of some higher modes.

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## Figures

Figure 1

Dimensionless frequency parameter of the first vibration mode calculated from the relation suggested by Haener (4) versus end mass to beam mass ratio

Figure 2

Dimensionless frequency parameter of the first mode calculated from the eigensolution and from the relation suggested by Haener (4) versus end mass to beam mass ratio

Figure 3

Relative percentage error due to using the relation suggested by Haener (4) (a) in the dimensionless frequency parameter and (b) in the fundamental natural frequency

Figure 4

Dimensionless frequency parameters of the higher modes calculated from the eigensolution and from the relations suggested by Haener (4) versus end mass to beam mass ratio

Figure 5

Dimensionless frequency parameters versus end mass to beam mass ratio for the first five modes

Figure 6

Relative percentage errors (a) in the dimensionless frequency parameters and (b) in the natural frequencies due to using the first order polynomial ratios given by Eq. 30

Figure 7

Relative percentage errors (a) in the dimensionless frequency parameters and (b) in the natural frequencies due to using the second order polynomial ratios given by Eq. 31

Figure 8

Dimensionless frequency parameters versus end mass to beam mass ratio; (a) the first order polynomial ratio fit with the exact solution and (b) the second order polynomial ratio fit with the exact solution (solid line, curve fit; circle, 엯 exact)

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