Mathematical Modeling of Uncertain Natural Frequencies of a Thin Beam via Polynomial Chaos Expansion

Year 2018, Volume: 20 Issue: 60, 852 - 862, 15.09.2018

Murat Kara Abdullah Seçgin

Abstract

Uncertainty is generally defined as uncontrollable
variability in the dynamic responses. This study introduces mathematical
modeling of uncertain natural frequencies of a thin beam having uncertain
elasticity modulus and specific volume. The uncertain variables are assumed to
have shifted normal distribution. The uncertain variables and the uncertain
natural frequencies corresponding to the uncertain variables are modeled via
polynomial chaos expansion (PCE). Hermitian polynomials are selected as
polynomial type. Discrete singular convolution method (DSC) is utilized to
solve differential equation solver and it is seen that DSC presents a unique
advantage in determining PCE coefficients of natural frequencies. First thirty
natural frequencies of the beam are considered and different PCE coefficients
are obtained for each natural frequency. Monte Carlo
simulation is performed for the validation. Results show that PCE application
via DSC is a powerful alternative to Monte Carlo
simulation. 

Keywords

Polynomial Chaos Expansion, Uncertain thin beam, Discrete Singular Convolution, Natural frequency

İnce Bir Çubuğun Belirsiz Doğal Frekanslarının Çokterimli Kaos Açılımı ile Matematiksel Olarak Modellenmesi

Abstract

Belirsizlik genellikle dinamik cevaplardaki kontrol
edilemeyen değişkenlikler olarak tanımlanır. Bu çalışma, belirsiz elastisite
modülü ve özgül hacme sahip ince bir çubuğun belirsiz doğal frekanslarının
matematiksel olarak modellenmesini içerir. Belirsiz değişkenlerin ötelenmiş
Normal dağılıma sahip olduğu kabul edilmiştir. Belirsiz değişkenler ve bu
değişkenlere karşılık gelen belirsiz doğal frekanslar çok terimli kaos (ÇKA)
ile modellenmiştir. Çokterimli tipi olarak Hermite çokterimlisi seçilmiştir. Ayrık
tekil konvolüsyonu (ATK) diferansiyel denklem çözücü olarak kullanılmış ve
ATK’nın doğal frekansların ÇKA katsayılarını belirlemekte oldukça avantajlı
olduğu görülmüştür. Çubuğun ilk otuz doğal frekansı göz önüne alınarak her bir
doğal frekans için farklı ÇKA katsayıları elde edilmiştir. Doğrulama çalışması
için Monte Carlo simülasyonu gerçekleştirilmiştir. Sonuçlar, ATK ile ÇKA
uygulamasının Monte Carlo simülasyonuna oldukça güçlü bir alternatif olduğunu
göstermektedir. 

Keywords

Çokterimli Kaos açılımı, Belirsiz ince çubuk, Ayrık Tekil Konvolüsyonu, Doğal frekans

References

• [1] Elishakoff, I., Haftka, R.T., Fang, J. 1994. Structural design under bounded uncertainty—Optimization with anti-optimization, Computers & Structures, Cilt. 53, s. 1401–5. doi:10.1016/0045-7949(94)90405-7.
• [2] Moore, R.E. 1979. Methods and applications of interval analysis. 2nd, Siam.
• [3] Hanss, M. 2013. Fuzzy Arithmetic for Uncertainty Analysis. Springer, Berlin, Heidelberg, s. 235–40.
• [4] Kumar, V., Schuhmacher, M. 2005. Fuzzy uncertainty analysis in system modelling, Computer Aided Chemical Engineering, Cilt. 20, s. 391–6. doi:10.1016/S1570-7946(05)80187-7.
• [5] Evans, M., Swartz, T. 2000. Approximating Integrals via Monte Carlo and Deterministic Methods. OUP, Oxford.
• [6] Rubinstein, R.Y., Kroese, D.P. 2016. Simulation and the Monte Carlo Method. John Wiley & Sons.
• [7] Lyon, R., DeJong, R. 1995. Statistical Energy Analysis of Dynamics Systems: Theory and Applications. 2nd, MIT Press.
• [8] Ghanem, R.G., Spanos, P.D. 2003. Stochastic Finite Elements: A Spectral Approach. Courier Corporation.
• [9] Sepahvand, K., Marburg, S., Hardtke, H-J. 2007. Numerical solution of one-dimensional wave equation with stochastic parameters using generalized polynomial chaos expansion, Journal of Computational Acoustics, Cilt. 15, s.579–93. doi:10.1142/S0218396X07003524.
• [10] Sepahvand, K., Marburg, S., Hardtke, H-J. 2010. Uncertainty quantification in stochastic systems using polynomial chaos expansion, International Journal of Applied Mechanics, Cilt. 2, s. 305–53. doi:10.1142/S1758825110000524.
• [11] Wei, G.W. 1999. Discrete singular convolution for the solution of the Fokker–Planck equation, The Journal of Chemical Physics, Cilt. 110, s.8930–42. doi:10.1063/1.478812.
• [12] Wei, G.W. 2000. Solving quantum eigenvalue problems by discrete singular convolution, Journal of Physics B: Atomic, Molecular and Optical Physics, Cilt.33, s.343–52. doi:10.1088/0953-4075/33/3/304.
• [13] Wei, G.W. 2001. A new algorithm for solving some mechanical problems, Computer Methods in Applied Mechanics and Engineering, Cilt.190, s. 2017–30. doi:10.1016/S0045-7825(00)00219-X.
• [14] Wei, G.W. 2001. Discrete singular convolution for beam analysis, Engineering Structures, Cilt. 23, s. 1045–53. doi:10.1016/S0141-0296(01)00016-5.
• [15] Wei, G.W. 2001. Vibration analysis by discrete singular convolution, Journal of Sound and Vibration, Cilt. 244, s. 535–53. doi:10.1006/jsvi.2000.3507.
• [16] Seçgin, A., Sarıgül, A.S. 2009. A novel scheme for the discrete prediction of high-frequency vibration response: Discrete singular convolution–mode superposition approach, Journal of Sound and Vibration, Cilt. 320, s. 1004–22. doi:10.1016/j.jsv.2008.08.031.
• [17] Seçgin, A., Sarıgül, A.S. 2008. Free vibration analysis of symmetrically laminated thin composite plates by using discrete singular convolution (DSC) approach: Algorithm and verification, Journal of Sound and Vibration, Cilt. 315, s. 197–211. doi:10.1016/j.jsv.2008.01.061.
• [18] Rao, S.S. 2011. Mechanical vibrations. Prentice Hall.
• [19] Wei, G.W, Zhao, Y.B., Xiang, Y. 2002, Discrete singular convolution and its application to the analysis of plates with internal supports, Part 1: Theory and algorithm, International Journal for Numerical Methods in Engineering, Cilt. 55, s. 913–46. doi:10.1002/nme.526.
• [20] Wang, X, Xu, S. 2010. Free vibration analysis of beams and rectangular plates with free edges by the discrete singular convolution, Journal of Sound and Vibration, Cilt. 329, s. 1780–92. doi:10.1016/j.jsv.2009.12.006.
• [21] Zhao, S., Wei, G.W., Xiang, Y. 2005. DSC analysis of free-edged beams by an iteratively matched boundary method, Journal of Sound and Vibration, Cilt. 284, s. 487–93. doi:10.1016/j.jsv.2004.08.037.