In this paper, new exact closed-form solutions for free longitudinal vibration of a one-parameter countable family of cantilever rods with one end tip mass are obtained. The analysis is based on the reduction of the equation governing the longitudinal vibration to the Sturm–Liouville canonical form and on the use of double Darboux transformations. The rods for which exact eigensolutions are provided are explicitly determined in terms of an initial rod with known closed-form eigensolutions. The method can be also extended to include longitudinally vibrating rods with tip mass at both ends.
Issue Section:
Research Papers
References
1.
Elishakoff
, I.
, 2005
, Eigenvalues of Inhomogeneous Structures: Unusual Closed-Form Solutions
, CRC Press
, Boca Raton, FL
.2.
Abrate
, S.
, 1995
, “Vibration of Non-Uniform Rods and Beams
,” J. Sound Vib.
, 185
(4
), pp. 703
–714
.3.
Kumar
, B. M.
, and Sujith
, R. I.
, 1997
, “Exact Solutions for the Longitudinal Vibration of Non-Uniform Rods
,” J. Sound Vib.
, 207
(5
), pp. 721
–729
.4.
Raj
, A.
, and Sujith
, R. I.
, 2005
, “Closed-Form Solutions for the Free Longitudinal Vibration of Inhomogeneous Rods
,” J. Sound Vib.
, 283
(3–5), pp. 1015
–1030
.5.
Yardimoglu
, B.
, and Aydin
, L.
, 2011
, “Exact Longitudinal Vibration Characteristics of Rods With Variable Cross-Sections
,” Shock Vib.
, 18
(4
), pp. 555
–562
.6.
Li
, Q. S.
, 2000
, “Exact Solutions for Free Longitudinal Vibration of Stepped Non-Uniform Rods
,” Appl. Acoust.
, 60
(1
), pp. 13
–28
.7.
Loya
, J. A.
, Aranda-Ruiz
, J.
, and Fernández-Sáez
, J.
, 2014
, “Torsion of Cracked Nanorods Using a Nonlocal Elasticity Model
,” J. Phys. D: Appl. Phys.
, 47
(11
), p. 115304
.8.
Elishakoff
, I.
, and Perez
, A.
, 2005
, “Design of a Polynomially Inhomogeneous Bar With a Tip Mass for Specified Mode Shape and Natural Frequency
,” J. Sound Vib.
, 287
(4–5), pp. 1004
–1012
.9.
Li
, Q. S.
, 2000
, “Exact Solutions for Free Longitudinal Vibrations of Non-Uniform Rods
,” J. Sound Vib.
, 234
(1
), pp. 1
–19
.10.
Darboux
, G.
, 1888
, “Sur la Répresentation Sphérique des Surfaces
,” Ann. Sci. Éc. Norm. Supér.
, 5
, pp. 79
–86
.11.
Gladwell
, G. M. L.
, and Morassi
, A.
, 1995
, “On Isospectral Rods, Horns and Strings
,” Inverse Probl.
, 11
(3
), pp. 533
–554
.12.
Pöschel
, J.
, and Trubowitz
, E.
, 1987
, Inverse Spectral Theory
, Academic Press
, London, UK
.Copyright © 2017 by ASME
You do not currently have access to this content.
