A wavelet-based approach for vibration analysis of framed structures

Mahdavi, S.H. and Abdul Razak, H. (2013) A wavelet-based approach for vibration analysis of framed structures. Applied Mathematics and Computation, 220. pp. 414-428. ISSN 00963003, DOI https://doi.org/10.1016/j.amc.2013.06.026.

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Abstract

In this paper, an explicit time integration scheme is proposed for structural vibration analysis by using wavelet functions. Initially, the differential equation of vibration governing SDOF (single-degree of freedom) systems has been solved by wavelet operators, and later the proposed approach has been generalized for MDOF (multi-degrees of freedom) systems. For this purpose, two different types of wavelet functions have been exemplified including, complex Chebyshev wavelet functions and simple Haar wavelet functions. In the proposed approach, a straightforward formulation has been derived from the numerical approximation of response through the wavelet definition. Emphasizing on frequency-domain approximation, a simple step-by-step algorithm has been implemented and improved to calculate the response of MDOF systems. Moreover, stability and accuracy of results have been evaluated. The effectiveness of the proposed approach is demonstrated using three examples compared with some of the existing numerical integration schemes such as family of Newmark-β, Wilson-θ and central difference method. In all the procedures, computation time involved has also been considered. Finally, it is concluded that the vibration analysis of structures is improved by lesser computation time and high accuracy of proposed approach, particularly, in large-scaled systems. © 2013 Elsevier Inc. All rights reserved.

Item Type:	Article
Funders:	UNSPECIFIED
Additional Information:	Export Date: 16 December 2013 Source: Scopus CODEN: AMHCB Language of Original Document: English Correspondence Address: Abdul Razak, H.; StrucHMRS Group, Department of Civil Engineering, University of Malaya, 50603 Kuala Lumpur, Malaysia; email: hashim@um.edu.my References: Dokainish, M.A., Subbaraj, K., Survey of direct time-integration methods in computational structural dynamics. I. Explicit methods (1989) Computers and Structures, 32 (6), pp. 1371-1386. , DOI 10.1016/0045-7949(89)90314-3; Bathe, K.J., (1996) Finite Element Procedures, , Prentice-Hall Englewood Cliffs, New Jersey; Chang, S.Y., Explicit pseudo-dynamic algorithm with unconditional stability (2002) J. Eng. Mech. ASCE, 128 (9), pp. 935-947; Chang, S.Y., A new family of explicit methods for linear structural dynamics (2010) Comput. Struct., 88, pp. 755-772; Chung, J., Lee, J.M., A new family of explicit time integration methods for linear and non-linear structural dynamics (1994) Int. J. Numer. Methods Eng., 37, pp. 3961-3976; Rio, G., Soive, A., Grolleau, V., Comparative study of numerical explicit time integration algorithms (2005) Advances in Engineering Software, 36 (4), pp. 252-265. , DOI 10.1016/j.advengsoft.2004.10.011, PII S0965997804002029; Wilson, E.L., Farhoomand, I., Bathe, K.J., Nonlinear dynamics analysis of complex structures (1973) Earthquake Eng. Struct. Dyn., 1, pp. 241-252; Chopra, A.K., (1995) Dynamic of Structures: Theory and Applications to Earthquake Engineering, , Prentice-Hall Englewood Cliffs, New Jersey; Hughes, T.J.R., (1987) The Finite Element Method, Linear Static and Dynamic Finite Element Analysis, , Prentice-Hall Englewood Cliffs, New Jersey; Liu, J.L., Solution of dynamic response of framed structure using piecewise Birkhoff polynomial (2002) Journal of Sound and Vibration, 251 (5), pp. 847-857. , DOI 10.1006/jsvi.2001.4014; Lepik, U., Haar wavelet method for solving higher order differential equations (2008) Int. J. Math. Comput., 1, pp. 84-94; Yuanlu, L., Zhao, W., Haar wavelet operational matrix of fractional order integration and its applications in solving the fractional order differential equations (2010) Appl. Math. Comput., 216 (8), pp. 2276-2285; Wu, J.L., A wavelet operational method for solving fractional partial differential equations numerically (2009) Appl. Math. Comput., 214 (1), pp. 31-40; Lepik, U., Haar wavelet method for solving stiff differential equations (2009) Int. J. Math. Comput., 14 (1), pp. 467-481; Yuanlu, L., Solving a nonlinear fractional differential equations using Chebyshev wavelet (2010) Commun. Nonlinear Sci. Numer. Simul., 15 (9), pp. 2284-2292; Babolian, B., Fatahzadeh, F., Numerical solution of differential equations by using Chebyshev wavelet operational matrix of integration (2010) Appl. Math. Comput., 188, pp. 417-426; Babolian, B., Fatahzadeh, F., Numerical computation method in solving integral equations by using Chebyshev wavelet operational matrix of integration (2007) Appl. Math. Comput., 188 (1), pp. 1016-1022; Lepik, U., Numerical solution of differential equations using Haar wavelets (2005) Mathematics and Computers in Simulation, 68 (2), pp. 127-143. , DOI 10.1016/j.matcom.2004.10.005, PII S0378475404002757; Mahdavi, S.H., Shojaee, S., Optimum time history analysis of SDOF structures using free scale of Haar wavelet (2013) Struct. Eng. Mech. Int. J., 45 (1), pp. 95-110; Salajeghe, E., Heidari, A., Time history dynamic analysis of structures using filter bank and wavelet transform (2006) Comput. Struct., 83 (1), pp. 53-68; Cattani, C., Haar wavelets based technique in evolution problems (2004) Proc. Estonian Acad. Sci. Phys. Math., 53 (1), pp. 45-63; Fox, L., Parker, I.B., (1968) Chebyshev Polynomials in Numerical Analysis, , Oxford University Press London; Mason, J.C., Handscomb, D.C., (2003) Chebyshev Polynomials, , Chapman and Hall/CRC; Ebeling, R.M., Green, R.A., French, S.E., Accuracy of response of single-degree-of-freedom systems to ground motion (1997) Technical Report ITL-97-7, , US Army Crops of Engineers, December
Uncontrolled Keywords:	Numerical approximation, Optimum time history analysis, Vibration analysis, Wavelet function, Central difference method, Explicit time integration scheme, Multi-degrees of freedoms, Numerical approximations, Numerical integration scheme, Step-by-Step algorithm, Time history analysis, Approximation algorithms, Differential equations, Structural analysis, Structural dynamics
Subjects:	T Technology > TA Engineering (General). Civil engineering (General)
Divisions:	Faculty of Engineering
Depositing User:	Mr Jenal S
Date Deposited:	24 Jan 2014 14:48
Last Modified:	29 Oct 2014 06:29
URI:	http://eprints.um.edu.my/id/eprint/9065

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