Discrete Singular Convolution Method for Dynamic Stability Analysis of Beams under Periodic Axial Forces
Publication: Journal of Engineering Mechanics
Volume 141, Issue 10
Abstract
In the present study, the discrete singular convolution (DSC) method is used to analyze the dynamic stability of beams subjected to different boundary conditions under periodic axial forces. A unified DSC algorithm is established to solve the governing equations of beam motion under different boundary conditions. The regularized Shannon’s delta kernel is selected as singular convolution to illustrate the present algorithm. The matched interface and boundary method is applied to the treatment of elastic restraint boundary conditions. The effects of a constant term in the periodic axial force and damping on dynamic instability regions are also investigated. The obtained numerical results are compared with those of Bolotin’s method. Numerical results indicate that the DSC method is a reliable method for dynamic stability analysis of beams.
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References
Bolotin, V. V. (1965). The dynamic stability of elastic systems, Holden Day, New York.
Briseghella, L., Majorana, C. E., and Pellegrino, C. (1998). “Dynamic stability of elastic structures: A finite element approach.” Comput. Struct., 69(1), 11–25.
Civalek, Ö. (2007a). “A parametric study of the free vibration analysis of rotating laminated cylindrical shells using the method of discrete singular convolution.” Thin Wall. Struct., 45(7-8), 692–698.
Civalek, Ö. (2007b). “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, Ö. (2008). “Vibration analysis of conical panels using the method of discrete singular convolution.” Commun. Numer. Methods Eng., 24(3), 169–181.
Civalek, Ö. (2009). “Fundamental frequency of isotropic and orthotropic rectangular plates with linearly varying thickness by discrete singular convolution method.” Appl. Math. Model., 33(10), 3825–3835.
Civalek, Ö., Korkmaz, A., and Demir, C. (2010). “Discrete singular convolution approach for buckling analysis of rectangular Kirchhoff plates subjected to compressive loads on two-opposite edges.” Adv. Eng. Software, 41(4), 557–560.
Evan-Iwanowski, R. M. (1965). “On the parametric response of structures.” Appl. Mech. Rev., 18(9), 699–702.
Huang, J. S., and Hung, L. H. (1984). “Dynamic stability for a simply supported beam under periodic axial excitation.” Int. J. Non Linear Mech., 19(4), 287–301.
Iwatsubo, T., Saigo, M., and Sugiyama, Y. (1973). “Parametric instability of clamped– clamped and clamped-simply supported columns under periodic axial load.” J. Sound Vib., 30(1), 65–IN2.
Iwatsubo, T., Sugiyama, Y., and Ishihara, K. (1972). “Stability and non-stationary vibration of columns under periodic loads.” J. Sound Vib., 23(2), 245–257.
Kar, R. C., and Sujata, T. (1990). “Parametric instability of an elastically restrained cantilever beam.” Comput. Struct., 34(3), 469–475.
Kim, J. H., and Choo, Y. S. (1998). “Dynamic stability of a free-free Timoshenko beam subjected to a pulsating follower force.” J. Sound Vib., 216(4), 623–636.
Lai, S. K., and Xiang, Y. (2009). “DSC analysis for buckling and vibration of rectangular plates with elastically restrained edges and linearly varying in-plane loading.” Int. J. Struct. Stab. Dyn., 9(3), 511–531.
Lim, C. W., Li, Z. R., Xiang, Y., Wei, G. W., and Wang, C. M. (2005). “On the missing modes when using the exact frequency relationship between Kirchhoff and Mindlin plates.” Adv. Vib. Eng., 4(3), 221–248.
Phan-Dao, H. H., Nguyen-Xuan, H., Thai-Hoang, C., Nguyen-Thoi, T., and Rabczuk, T. (2013). “An edge-based smoothed finite element method for analysis of laminated composite plates.” Int. J. Comput. Methods, 10(1), 1340005.
Song, Z. W., Li, W., and Liu, G. R. (2012). “Stability and non-stationary vibration analysis of beams subjected to periodic axial forces using discrete singular convolution.” Struct. Eng. Mech., 44(4), 487–499.
Svensson, I. (1996). “Dynamic buckling of a beam with transverse constraints.” Nonlinear Dyn., 11(4), 315–328.
Svensson, I. (2001). “Dynamic instability regions in a damped system.” J. Sound Vib., 244(5), 779–793.
Tang, J. S., and Liu, Z. Y. (2004). “Quasi-wavelet solution of diffusion problems.” Commun. Numer. Methods Eng., 20(12), 877–888.
Timoshenko, S. (1974). Vibration problems in engineering, Wolfenden, New York.
Wan, D. C., and Wei, G. W. (2000). “The study of quasi-wavelets based numerical method applied to Burgers’ equations.” Appl. Math. Mech., 21(10), 1099–1110.
Wan, D. C., Zhou, Y. C., and Wei, G. W. (2002). “Numerical solution of incompressible flows by discrete singular convolution.” Int. J. Numer. Methods Fluids, 38(8), 789–810.
Wang, S. H., and Ni, P. P. (2014). “Application of block theory modeling on spatial block topological identification to rock slope stability analysis.” Int. J. Comput. Methods, 11(1), 1350044.
Wang, X. W., Xu, C. L., and Xu, S. M. (2010). “The discrete singular convolution for analyses of elastic wave propagations in one-dimensional structures.” Appl. Math. Model., 34(11), 3493–3508.
Wang, X. W., and Xu, S. M. (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. (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. (2002b). “A novel approach for the analysis ofhigh-frequency vibrations.” J. Sound Vib., 257(2), 207–246.
Wei, G. W., Zhao, Y. B., and Xiang, Y. (2002a). “Discrete singular convolution and its application to the analysis of plates with internal supports. Part i: Theory and algorithm.” Int. J. Numer. Methods Eng., 55(8), 913–946.
Yu, S. N., Xiang, Y., and Wei, G. W. (2009). “Matched interface and boundary (MIB) method for the vibration analysis of plates.” Commun. Numer. Methods Eng., 25(9), 923–950.
Zhao, S., and Wei, G. W. (2004). “High order FDTD methods via derivative matching for Maxwell’s equations with material interfaces.” J. Comput. Phys., 200(1), 60–103.
Zhao, S., and Wei, G. W. (2009). “Matched interface and boundary (MIB) for the implementation of boundary conditions in high-order central finite differences.” Int. J. Numer. Methods Eng., 77(12), 1690–1730.
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 nonuniform boundary conditions.” J. Sound Vib., 255(2), 203–228.
Zhao, Y. B., Wei, G. W., and Xiang, Y. (2002). “Discrete singular convolution for the prediction of high frequency vibration of plates.” Int. J. Solids Struct., 39(1), 65–88.
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Published In
Journal of Engineering Mechanics
Volume 141 • Issue 10 • October 2015
Copyright
© 2015 American Society of Civil Engineers.
History
Received: Nov 7, 2014
Accepted: Jan 15, 2015
Published online: May 4, 2015
Published in print: Oct 1, 2015
Discussion open until: Oct 4, 2015
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