A wavelet-based approach for vibration analysis of framed structures

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-@b, Wilson-@q 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.

[1]  Weiwei Zhao,et al.  Haar wavelet operational matrix of fractional order integration and its applications in solving the fractional order differential equations , 2010, Appl. Math. Comput..

[2]  Ülo Lepik,et al.  Haar wavelet method for solving stiff differential equations , 2009 .

[3]  M. Dokainish,et al.  A survey of direct time-integration methods in computational structural dynamics—I. Explicit methods , 1989 .

[4]  S. H. Mahdavi,et al.  Optimum time history analysis of SDOF structures using free scale of Haar wavelet , 2013 .

[5]  Yuanlu Li,et al.  Solving a nonlinear fractional differential equation using Chebyshev wavelets , 2010 .

[6]  Anil K. Chopra,et al.  Dynamics of Structures: Theory and Applications to Earthquake Engineering , 1995 .

[7]  Esmail Babolian,et al.  Numerical computation method in solving integral equations by using Chebyshev wavelet operational matrix of integration , 2007, Appl. Math. Comput..

[8]  Jiunn-Lin Wu,et al.  A wavelet operational method for solving fractional partial differential equations numerically , 2009, Appl. Math. Comput..

[9]  SOLUTION OF DYNAMIC RESPONSE OF FRAMED STRUCTURE USING PIECEWISE BIRKHOFF POLYNOMIAL , 2002 .

[10]  Jintai Chung,et al.  A new family of explicit time integration methods for linear and non‐linear structural dynamics , 1994 .

[11]  K. Bathe.  Finite Element Procedures , 1995 .

[12]  L. Fox,et al.  Chebyshev polynomials in numerical analysis , 1970 .

[13]  Ülo Lepik,et al.  Numerical solution of differential equations using Haar wavelets , 2005, Math. Comput. Simul..

[14]  Gérard Rio,et al.  Comparative study of numerical explicit time integration algorithms , 2002, Adv. Eng. Softw..

[15]  Edward L. Wilson,et al.  Nonlinear dynamic analysis of complex structures , 1972 .

[16]  Robert M. Ebeling,et al.  Accuracy of Response of Single-Degree-of-Freedom Systems to Ground Motion , 1997 .

[17]  Carlo Cattani.  Haar wavelets based technique in evolution problems , 2004, Proceedings of the Estonian Academy of Sciences. Physics. Mathematics.

[18]  E. Salajegheh,et al.  TIME HISTORY DYNAMIC ANALYSIS OF STRUCTURES USING FILTER BANKS AND WAVELET TRANSFORMS , 2005 .

[19]  Shuenn-Yih Chang,et al.  Explicit Pseudodynamic Algorithm with Unconditional Stability , 2002 .

[20]  Ulo Lepik,et al.  Haar wavelet method for solving higher order differential equations , 2008 .

[21]  Esmail Babolian,et al.  Numerical solution of differential equations by using Chebyshev wavelet operational matrix of integration , 2007, Appl. Math. Comput..

[22]  T. J. Rivlin.  The Chebyshev polynomials , 1974 .

[23]  Shuenn-Yih Chang,et al.  A new family of explicit methods for linear structural dynamics , 2010 .