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Research Papers

Exact Eigensolutions for a Family of Nonuniform Rods With End Point Masses

[+] Author and Article Information
Lourdes Rubio

Department of Mechanical Engineering,
University Carlos III of Madrid,
Madrid 28911, Spain
e-mail: lrubio@ing.uc3m.es

José Fernández-Sáez

Professor
Department of Continuum Mechanics and
Structural Analysis,
University Carlos III of Madrid,
Madrid 28911, Spain
e-mail: ppfer@ing.uc3m.es

Antonino Morassi

Professor
Polytechnic Department of
Engineering and Architecture,
University of Udine,
Udine 33100, Italy
e-mail: antonino.morassi@uniud.it

1Corresponding author.

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received December 22, 2016; final manuscript received March 15, 2017; published online July 13, 2017. Assoc. Editor: Michael Leamy.

J. Vib. Acoust 139(5), 051018 (Jul 13, 2017) (7 pages) Paper No: VIB-16-1608; doi: 10.1115/1.4036467 History: Received December 22, 2016; Revised March 15, 2017

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.

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References

Elishakoff, I. , 2005, Eigenvalues of Inhomogeneous Structures: Unusual Closed-Form Solutions, CRC Press, Boca Raton, FL.
Abrate, S. , 1995, “ Vibration of Non-Uniform Rods and Beams,” J. Sound Vib., 185(4), pp. 703–714. [CrossRef]
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. [CrossRef]
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. [CrossRef]
Yardimoglu, B. , and Aydin, L. , 2011, “ Exact Longitudinal Vibration Characteristics of Rods With Variable Cross-Sections,” Shock Vib., 18(4), pp. 555–562. [CrossRef]
Li, Q. S. , 2000, “ Exact Solutions for Free Longitudinal Vibration of Stepped Non-Uniform Rods,” Appl. Acoust., 60(1), pp. 13–28. [CrossRef]
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. [CrossRef]
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. [CrossRef]
Li, Q. S. , 2000, “ Exact Solutions for Free Longitudinal Vibrations of Non-Uniform Rods,” J. Sound Vib., 234(1), pp. 1–19. [CrossRef]
Darboux, G. , 1888, “ Sur la Répresentation Sphérique des Surfaces,” Ann. Sci. Éc. Norm. Supér., 5, pp. 79–86. [CrossRef]
Gladwell, G. M. L. , and Morassi, A. , 1995, “ On Isospectral Rods, Horns and Strings,” Inverse Probl., 11(3), pp. 533–554. [CrossRef]
Pöschel, J. , and Trubowitz, E. , 1987, Inverse Spectral Theory, Academic Press, London, UK.

Figures

Grahic Jump Location
Fig. 1

Examples of isospectral rods under cantilever conditions with Â=1, M̂1=0.25, a(x)(=A(x)) and M1 as in Eqs. (12) and (13), respectively, and c=−0.9,−0.3,0,0.5,1.0: (a) m = 1, (b) m = 2, and (c) m = 3

Grahic Jump Location
Fig. 2

Examples of isospectral rods under cantilever conditions with Â=1, M̂1=0.50, a(x)(=A(x)) and M1 as in Eqs. (12) and (13), respectively, and c=−0.9,−0.3,0,0.5,1.0: (a) m = 1, (b) m = 2, and (c) m = 3

Grahic Jump Location
Fig. 3

Examples of isospectral rods under cantilever conditions with Â=1, M̂1=1.00, a(x)(=A(x)) and M1 as in Eqs. (12) and (13), respectively, and c=−0.9,−0.3,0,0.5,1.0: (a) m = 1, (b) m = 2, and (c) m = 3

Grahic Jump Location
Fig. 4

Behavior of the ratio M1/M̂1 as in Eq. (13) with respect to c for the isospectral rods under cantilever conditions with Â=1 and m = 1, 2, 3: (a) M̂1=0.25, (b) M̂1=0.50, and (c) M̂1=1.00

Grahic Jump Location
Fig. 5

First three eigenfunctions kn of the isospectral rods under cantilever conditions with Â=1, M̂1=0.25, m = 1 and c=−0.9,−0.3,1.0, where kn has the expression (14): (a) n = 1, (b) n = 2, and (c) n = 3

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