ارتقای کارایی شناسایی پارامترهای فیزیکی سازه با استفاده از‌ کاهش تکینگی ماتریس‌های پاسخ در حل معکوس معادلات حرکت

نوع مقاله : پژوهشی

نویسندگان

1 دانشگاه آزاد اسلامی واحد قزوین

2 بین المللی امام خمینی(ره)

چکیده

در این مقاله شناسایی ماتریس‌های مشخصۀ سیستم خطی (شامل ماتریسهای جرم، میرایی و سختی) بااستفاده از حل معکوس معادلات حرکت، با تمرکز بر کاهش شرایط بدوضعی دستگاه معادلات حاصل، در حوزۀ فرکانس ارائه گردیده است. این روش با اندازه‌گیری پاسخهای شتاب در درجات آزادی سازه تحت یک تحریک اجباری، به شناسایی سیستم و تشخیص خرابیهای محتمل می‌پردازد. بدوضعی معادلات حرکت و وجود نویز در پاسخهای شتاب اندازه‌گیری‌شده به‌عنوان یک مشکل ناگزیر، موجب ناپایداری پاسخ مسئلۀ تشخیص سیستم، خطای فزاینده و عدم قابلیت اعتماد در ماتریس‌های مشخصۀ تعیین‌شده می‌شود. دراین مقاله الگوریتمی برای بهبود بخشیدن به مشکل بدوضعی حل معکوس مسئله در روند شناسایی سیستم ارائه گردیده است که یک روش خاص بالامثلثی‌سازی ماتریس است. الگوریتم پیشنهادی با شناسایی بردارهای موازی و شبه‌موازی موجود در ماتریس ضرایب دستگاه معادلات خطی به حذف بردارهای وابستۀ خطی می‌پردازد و با کاهش تکینگی ماتریس، موجب پایدارسازی و ارتقای پاسخ می‌گردد. در تخمین بهینۀ ماتریسهای مشخصۀ سیستم در روند شناسایی از روش حداقل مربعات و تابع جریمه استفاده شده است. به‌منظور بررسی کارایی الگوریتم پیشنهادی، ماتریس‌های مشخصۀ قاب هشت طبقۀ غیربرشی با میرایی نامتناسب با استفاده از الگوریتم کاهش تکینگی در شناسایی سیستم،تعیین شده‌اند. نتایج نشان می‌دهد استفاده از الگوریتم پیشنهادی در بهبود تخمین و پایداری پاسخ‌ها کاملاً مفید است.

کلیدواژه‌ها

عنوان مقاله [English]

Efficiency Improvement of the Structural System Identification by Reducing Singularity of the Response Matrixes in Inverse Solution of Equations of Motion

نویسندگان [English]

  • Majid Ghasemi 1
  • Babak Taghavi 1
  • Saeid Abbasbandy 2
1 Islamic Azad University Qazvin Branch
2 Imam Khomeini International University
چکیده [English]

This paper presents a method for identification of linear system physical parameters (structural mass, damping and stiffness matrices) using the inverse solution of equation of motion in the frequency domain, by focus on the reducing the illconditioning effect. The method utilizes the measured responses from the forced vibration test of structure in order to identify the system properties and detect the probable damages. Inputs and outputs data is gathered in an augmented matrix [A|b], that large number of this data is caused to ill condition problem. Moreover, as an inevitable problem, there is a noise in the measurement and makes discrepancy in result of identification. Ill conditioning causes instability in the result of identification, the instability and noisy result reduce validity of the results and accordingly will be valueless statistic methods in the system identification (SI). This paper is presented an algorithm to improve the ill conditioning problem that is a special upper triagularization matrix method. The proposed algorithm can identify parallel and pseudo parallel vectors in coefficient matrix of linear equations. By removing these linearly dependent vectors and thus reducing singularity of the matrix, stabilization is resulted which is a key objective in numerical linear algebra. In order to optimal estimation of identification results, least-squares and penalty function methods is used. The validity and efficiency of the reduce singularity of matrix method is tested on a eight non shear story frame structure by using direct model updating method. Aforementioned structures have a non-proportional damped matrix and subjected to sweep harmonic forces. The results show that the proposed algorithm improves the stability of the estimation and the answer is quite useful.

کلیدواژه‌ها [English]

  • System Identification
  • Inverse Problem in Equation of Motion
  • Ill-Condition Problem
  • Singularity of Matrix

مراجع

1. Morrasi,A., Vestroni,F.,"Dynamic Methods for Damage Detection in Structures, Springerwien", New York, (2008).
2. Isakov,Victor., "Inverse Problems For Partioal Differential Equations", Springer Science & Business Media, (1998).
3. Hanse,P.C., "Rank-Deficient and Discrete Ill-Posed Problems", Siam, Society for Industrial and Applied Mathematics, Philadelphia, (1998).
4. Rump, S.M., "Approximate inverses of almost singular matrices still contain useful information", Technical Report 90.1, Faculty for Information and Communication Sciences, Hamburg University of Technology, (1990).
5. Rump,S.M., "A class of arbitrarily ill-conditioned floating-point matrices", SIAM J. Matrix Anal. Appl., Vol. 12 (4), pp. 645–653, (1991).
6. Ogita,T., Rump,S.M., Oishi,S., "Accurate sum and dot product", SIAM J. Sci. Comput,. Vol. 26(6), pp. 1955–1988, (2005).
7. Ohta,T., Ogita,T., Rump,S.M., Oishi,S., "Numerical verification method for arbitrarily ill-conditioned linear systems", Trans. JSIAM, Vol. 15(3), pp. 269–287 (in Japanese), (2005).
8. Rump,S.M., Ogita,T., Oishi, S., "Accurate floating-point summation", 41 pages, 2006, submitted for publication, preprint is available from http://www.ti3.tu-harburg.de/publications/rump. Technical Report 05.12, Faculty for Information and Communication Sciences, Hamburg University of Technology, (2005).
9. Ohta,T., Ogita,T., Rump, S.M., Oishi, S., "Numerical verification method for arbitrarily ill-conditioned linear systems", Trans. JSIAM, Vol. 15 (3), pp. 269–287, (in Japanese),(2005).
10. Moszynski,K., "On solving linear algebraic equations with an ill-conditioned matrix", Appl. Math., Vol. 22 (4), pp. 499-513, (1995).
11. Strand,O., "Theory and methods related to the singular-function expansion and LandweberÕsiteration for integral equations of the first kind", SIAM. J. Numer. Anal., Vol. 11 (4), pp. 798-824, (1974).
12. Lu,M.,Liu,K., "Parallel algorithm for Householder transformation with applications to illconditioned problem", Int. J. Comput. Math., Vol. 64, pp. 89-101, (1997).
13. Golub,G.H.,and Kahan.,"Calculating the singular value and pseudo-inverse of a matrix”, SIAM J. Num. Anal. Vol. 2, pp. 205-224, (1965).
14. Businger,P.A., and Golub,G.H.,"Linear least squares solution by Householder transformation", (1965).
15. Golub,G.H., and Reinsch, C.,"Singular value decomposition and least squares solution", Numer. Math., Vol. 14 pp. 403-420, (1970).
16. Chan,T.F.,"An improved algorithm for computing the singular value decomposition" .ACM Trans., Vol. 8(1), pp. 72-83, (1982).
17. Demmel,J., and Kahan, W.,"Accurate singular value of bidiagonal matrices”, SIAM J. Sci. and Stat. Comput., Vol. 11(5), pp. 873-912, (1990).
18. Fernando,K.V., andParlet, B.N.,"Accurate singular and differential qd algorithms", Numerische Mathematik, Vol. 67(2), pp. 191-229, (1994).
19. Hansen., "A numerical method for solving Fredholm integral equations of the first kind using singular values",SIAM J. Numcr. Anal., Vol. 8(3), pp. 616-622, (1971).
20. Varah,J.M., "On the numerical solution of ill-conditioned linear systems with applications to ill-posed problems", SIAM J. Numcr. Anal., Vol. 10(2), pp. 257-267,(1973).
21. Varah,J.M., "A practical examination of some numerical methods for linear discrete ill-posed problems", SIAM Review, Vol. 21(1), pp. 100-111, (1979).
22. Hansen, P. C.,"Analysis of discrete ill-posed problems by means of the L-curve". SIAM Review, Vol. 34(4), pp. 561–80, (1992).
23. Hansen, P. C., O’Leary DP. "The use of the L-curve in the regularizationof discrete ill-posed problems". SIAM J Sci. Compute Vol. 14, pp. 1487–503, (1993).
24. Volokh,K. Y.,and Vilnay,O.,"Pin-pointing solution of ill-conditioned square systems of linear equations", Appl. Math. Lett. Vol. 13(7), pp. 119-124, (2000).
25. Tikhonov,A.N., "Solution of incorrectly formulated problems and the regularization method,Dokl. Akad. Nauk", SSSR 151(1963), 501-504, Soviet Math. Dokl. 4, pp. 1035-1038, (1963).
26. Leclere, Q., Pezerat, C., Laulagnet, B., Polac, L.,"Indirect measurement of main bearing loads in an operating dieselengine",Journal of Sound and Vibration, Vol. 286(1) pp. 341–361, (2005).
27. Nordberg, T.P., Gustafsson, I.,"Dynamic regularisation of input estimation problems by explicit block inversion", Computer Methods in Applied Mechanics and Engineering, Vol. 195(44), pp. 5877–5890, (2006).
28. Dicken, V., Menz, I., Maass, P., Niebsch, J., Ramlau, R.,"Inverse imbalance reconstruction for nonlinearly damped systems", Inverse Problems in Science and Engineering 2005, Vol. 13(5), pp. 507–543, (2005).
29. Yu, E., Taciroglu, E., and Wallace, J.W.,"Parameter Identification of Framed Structures using an Improved Finite Element Model-updating Method", Earthquake Engng Struct. Dyn. 36, pp. 619-639, (2007).
30. Titurus, B., and Friswell, M.I.,"Regularization in Model Updating",Int. J. Numer. Meth. Engng., Vol.75(4), pp. 440-478, (2008).
31. Ghafori–Ashtiany,M., and Ghasemi,M.,"System Identification Method by Using Inverse Solution of Equations of Motion in Frequency Domain",Journal of vibration and control, Vol. 19(11), pp. 1633-1645, (2012).
ارسال نظر در مورد این مقاله
CAPTCHA Image

فایل ها

هم رسانی

ارجاع به این مقاله

آمار