Numerical Analysis of Free Longitudinal Vibration of Nonuniform Rods: Discrete Singular Convolution Approach
This article has been corrected.
VIEW CORRECTIONPublication: Journal of Engineering Mechanics
Volume 140, Issue 8
Abstract
In this paper, the problem of longitudinal free vibration of nonuniform rods is investigated for the first time by using the discrete singular convolution (DSC). This method is a relatively novel and powerful numerical approach especially in analyzing vibration problems. The governing equation of this problem is nonlinear and has analytical solutions only in a few cases, so the analysis uses DSC as a numerical method. The regularized Shannon delta kernel is used in the present algorithm. The nondimensional natural frequencies of rods with various-area cross sections have been studied in this paper. Results show good agreement with the previous analytical solutions. This study indicates the efficiency of the DSC in solving nonuniform rod problems.
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References
Abrate, S. (1993). “Wave propagation during high velocity impacts on composite materials.” Final Rep. for Summer Faculty Research Program, Air Force Office of Scientific Research, Arlington, VA, 15.1–15.19.
Abrate, S. (1995). “Vibration of non-uniform rods and beams.” J. Sound Vib., 185(4), 703–716.
Al Kaisy, A. M. A., Esmaeel, R. A., and Nassar, M. M. (2007). “Application of the differential quadrature method to the longitudinal vibration of non-uniform rods.” Eng. Mech., 14(5), 303–310.
Arndt, M., Machado, R. D., and Scremin, A. (2010). “An adaptive generalized finite element method applied to free vibration analysis of straight bars and trusses.” J. Sound Vib., 329(6), 659–672.
Bapat, C. N. (1995). “Vibration of rods with uniformly tapered sections.” J. Sound Vib., 185(1), 185–189.
Civalek, O. (2006a). “An efficient method for free vibration analysis of rotating truncated conical shells.” Int. J. Press. Vessels Piping, 83(1), 1–12.
Civalek, O. (2006b). “Free vibration analysis of composite conical shells using the discrete singular convolution algorithm.” Steel Compos. Struct., 6(4), 353–366.
Civalek, O. (2007a). “Numerical analysis of free vibrations of laminated composite conical and cylindrical shells: Discrete singular convolution (DSC) approach.” J. Comput. Appl. Math., 205(1), 251–271.
Civalek, O. (2007b). “Three-dimensional vibration, buckling and bending analyses of thick rectangular plates based on discrete singular convolution method.” Int. J. Mech. Sci., 49(6), 752–765.
Civalek, O. (2008). “Vibration analysis of conical panels using the method of discrete singular convolution.” Commun. Numer. Methods Eng., 24(3), 169–181.
Eisenberger, M. (1991). “Exact longitudinal vibration frequencies of a variable cross-section rod.” Appl. Acoust., 34(2), 123–130.
Hou, Y. G., Wei, G. W., and Xiang, Y. (2005). “DSC-Ritz method for the free vibration analysis of Mindlin plates.” Int. J. Numer. Methods Eng., 62(2), 262–288.
Inaudi, J. A., and Matusevich, A. E. (2010). “Domain-partition power series in vibration analysis of variable-cross-section rods.” J. Sound Vib., 329(21), 4534–4549.
Kumar, B. M., and Sujith, R. I. (1997). “Exact solutions for the longitudinal vibration of non-uniform rods.” J. Sound Vib., 207(5), 721–729.
Li, Q. S. (2000). “Exact solutions for free longitudinal vibration of non-uniform rods.” J. Sound Vib., 234(1), 1–19.
Li, Q. S., Cao, H., and Li, G. (1994). “Analysis of free vibrations of tall buildings.” J. Eng. Mech., 1861–1876.
Li, Q. S., Cao, H., and Li, G. (1996). “Static and dynamic analysis of straight bars with variable cross-section.” Comput. Struct., 59(6), 1185–1191.
Lim, C. W., Li, Z. R., and Wei, G. W. (2005a). “DSC-Ritz method for high-mode frequency analysis of thick shallow shells.” Int. J. Numer. Methods Eng., 62(2), 205–232.
Lim, C. W., Li, Z. R., Xiang, Y., Wei, G. W., and Wang, C. M. (2005b). “On the missing modes when using the exact frequency relationship between Kirchhoff and Mindlin plates.” Adv. Vib. Eng., 4(3), 221–248.
Matsuda, H., Sakiyama, T., Morita, C., and Kawakami, M. (1995). “Longitudinal impulsive response analysis of variable cross section bars.” J. Sound Vib., 181(3), 541–551.
Meek, J. W., and Wolf, J. P. (1992a). “Cone models for homogeneous soil. I.” J. Geotech. Engrg., 667–685.
Meek, J. W., and Wolf, J. P. (1992b). “Cone models for rigid rock. II.” J. Geotech. Engrg., 686–703.
Meek, J. W., and Wolf, J. P. (1993). “Why cone models can represent the elastic half-space.” Earthquake Eng. Struct. Dynam., 22(9), 759–771.
Ng, C. H. W., Zhao, Y. B., and Wei, G. W. (2004). “Comparison of discrete singular convolution and generalized differential quadrature for the vibration analysis of rectangular plates.” Comput. Methods Appl. Mech. Eng., 193(23-26), 2483–2506.
Raj, A., and Sujith, R. I. (2005). “Closed-form solutions for the free longitudinal vibration of inhomogeneous rods.” J. Sound Vib., 283(3–5), 1015–1030.
Subrahmanyam, K. B., and Leissa, A. W. (1985). “An improved finite difference analysis of uncoupled vibrations of cantilevered beams.” J. Sound Vib., 98(1), 1–11.
Wang, G. Y. (1978). Vibration of building and structures, Science and Technology Press, Beijing.
Wang, X., and Xu, S. (2010). “Free vibration analysis of beams and rectangular plates with free edges by the discrete singular convolution.” J. Sound Vib., 329(10), 1780–1792.
Wei, G. W. (1999). “Discrete singular convolution for the solution of the Fokker–Planck equations.” J. Chem. Phys., 110(18), 8930–8942.
Wei, G. W. (2000). “Solving quantum eigenvalue problems by discrete singular convolution.” J. Phys. At. Mol. Opt. Phys., 33(3), 343–352.
Wei, G. W. (2001a). “A new algorithm for solving some mechanical problems.” Comput. Methods Appl. Mech. Eng., 190(15–17), 2017–2030.
Wei, G. W. (2001b). “Discrete singular convolution for beam analysis.” Eng. Struct., 23(9), 1045–1053.
Wei, G. W. (2001c). “Vibration analysis by discrete singular convolution.” J. Sound Vib., 244(3), 535–553.
Wei, G. W., Zhao, Y. B., and Xiang, Y. (2001). “The determination of natural frequencies of rectangular plates with mixed boundary conditions by discrete singular convolution.” Int. J. Mech. Sci., 43(8), 1731–1746.
Wei, G. W., Zhao, Y. B., and Xiang, Y. (2002a). “A novel approach for the analysis of high-frequency vibrations.” J. Sound Vib., 257(2), 207–246.
Wei, G. W., Zhao, Y. B., and Xiang, Y. (2002b). “Discrete singular convolution and its application to the analysis of plates with internal supports. Part 1: Theory and algorithm.” Int. J. Numer. Methods Eng., 55(8), 913–946.
Xiang, Y., Zhao, Y. B., and Wei, G. W. (2002). “Discrete singular convolution and its application to the analysis of plates with internal supports. Part 2: Applications.” Int. J. Numer. Methods Eng., 55(9), 947–971.
Zeng, H., and Bert, C. W. (2001). “Vibration analysis of a tapered bar by differential transformation.” J. Sound Vib., 242(4), 737–739.
Zhao, S., Wei, G. W., and Xiang, Y. (2005). “DSC analysis of free-edged beams by an iteratively matched boundary method.” J. Sound Vib., 284(1–2), 487–493.
Zhao, Y. B., and Wei, G. W. (2002). “DSC analysis of rectangular plates with non-uniform boundary conditions.” J. Sound Vib., 255(2), 203–228.
Zhao, Y. B., Wei, G. W., and Xiang, Y. (2002a). “Discrete singular convolution for the prediction of high frequency vibration of plates.” Int. J. Solids Struct., 39(1), 65–88.
Zhao, Y. B., Wei, G. W., and Xiang, Y. (2002b). “Plate vibration under irregular internal supports.” Int. J. Solids Struct., 39(5), 1361–1383.
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Published In
Journal of Engineering Mechanics
Volume 140 • Issue 8 • August 2014
Copyright
© 2014 American Society of Civil Engineers.
History
Received: Aug 23, 2013
Accepted: Jan 17, 2014
Published online: Mar 7, 2014
Published in print: Aug 1, 2014
Discussion open until: Aug 7, 2014
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