Identification of structural parameters and boundary conditions using a minimum number of measurement points

Abstract

This article proposes a novel methodology that uses mathematical and numerical models of a structure to build a data set and determine crucial nodes that possess the highest sensitivity. Regression surfaces between the structural parameters and structural output features, represented by the natural frequencies of the structure and local transmissibility, are built using the numerical data set. A description of a possible experimental application is provided, where sensors are mounted at crucial nodes, and the natural frequencies and local transmissibility at each natural frequency are determined from the power spectral density and the power spectral density ratios of the sensor responses, respectively. An inverse iterative process is then applied to identify the structural parameters by matching the experimental features with the available parameters in the myriad numerical data set. Three examples are presented to demonstrate the feasibility and efficacy of the proposed methodology. The results reveal that the method was able to accurately identify the boundary coefficients and physical parameters of the Euler-Bernoulli beam as well as a highway bridge model with elastic foundations using only two measurement points. It is expected that the proposed method will have practical applications in the identification and analysis of restored structural systems with unknown parameters and boundary coefficients.

Keywords

structural model validation / eigenvalue problem / response surface / inverse problems

References

[1] Zong Z, Lin X, Niu J. Finite element model validation of bridge based on structural health monitoring—Part I: Response surface-based finite element model updating. Journal of Traffic and Transportation Engineering, 2015, 2(4): 258–278

[2] Castro-Triguero R, Murugan S, Gallego R, Friswell M I. Robustness of optimal sensor placement under parametric uncertainty. Mechanical Systems and Signal Processing, 2013, 41(1–2): 268–287

[3] Sehgal S, Kumar H. Structural dynamic model updating techniques: A state-of-the-art review. Archives of Computational Methods in Engineering, 2016, 23(3): 515–533

[4] Fang S E, Zhang Q H, Ren W X. An interval model updating strategy using interval response surface models. Mechanical Systems and Signal Processing, 2015, 60–61: 909–927

[5] Ren W X, Chen H B. Finite element model updating in structural dynamics by using the response surface method. Engineering Structures, 2010, 32(8): 2455–2465

[6] Zapico J L, González M P, Friswell M I, Taylor C A, Crewe A J. Finite element model updating of a small scale bridge. Journal of Sound and Vibration, 2003, 268(5): 993–1012

[7] Park Y S, Kim S, Kim N, Lee J J. Finite element model updating considering boundary conditions using neural networks. Engineering Structures, 2017, 150: 511–519

[8] Park Y S, Kim S, Kim N, Lee J J. Evaluation of bridge support condition using bridge responses. Structural Health Monitoring, 2019, 18(3): 767–777

[9] Cui Y, Lu W, Teng J. Updating of structural multi-scale monitoring model based on multi-objective optimisation. Advances in Structural Engineering, 2019, 22(5): 1073–1088

[10] Gordis J H. Artificial boundary conditions for model updating and damage detection. Mechanical Systems and Signal Processing, 1999, 13(3): 437–448

[11] Zhou L, Wang L, Chen L, Ou J. Structural finite element model updating by using response surfaces and radial basis functions. Advances in Structural Engineering, 2016, 19(9): 1446–1462

[12] Mottershead J E, Link M, Friswell M I. The sensitivity method in finite element model updating: A tutorial. Mechanical Systems and Signal Processing, 2011, 25(7): 2275–2296

[13] Jin S S, Cho S, Jung H J, Lee J J, Yun C B. A new multi-objective approach to finite element model updating. Journal of Sound and Vibration, 2014, 333(11): 2323–2338

[14] Jang J, Smyth A W. Model updating of a full-scale FE model with nonlinear constraint equations and sensitivity-based cluster analysis for updating parameters. Mechanical Systems and Signal Processing, 2017, 83: 337–355

[15] Nanthakumar S S, Lahmer T, Zhuang X, Zi G, Rabczuk T. Detection of material interfaces using a regularized level set method in piezoelectric structures. Inverse Problems in Science and Engineering, 2016, 24(1): 153–176

[16] Mao X, Dai H. A quadratic inverse eigenvalue problem in damped structural model updating. Applied Mathematical Modelling, 2016, 40(13–14): 6412–6423

[17] Tsai S H, Ouyang H, Chang J Y. Inverse structural modifications of a geared rotor-bearing system for frequency assignment using measured receptances. Mechanical Systems & Signal Processing, 2018, 110(Sep): 59–72

[18] Anitescu C, Atroshchenko E, Alajlan N, Rabczuk T. Artificial neural network methods for the solution of second order boundary value problems. Computers, Materials, & Continua, 2019, 59(1): 345–359

[19] Nehete D V, Modak S V, Gupta K. Experimental studies in finite element model updating of vibro-acoustic cavities using coupled modal data and FRFs. Applied Acoustics, 2019, 150: 113–123

[20] Catbas F N, Ciloglu S K, Hasancebi O, Grimmelsman K, Aktan A E. Limitations in structural identification of large constructed structures. Journal of Structural Engineering, 2007, 133(8): 1051–1066

[21] Zhang J, Maes K, De Roeck G, Reynders E, Papadimitriou C, Lombaert G. Optimal sensor placement for multi-setup modal analysis of structures. Journal of Sound and Vibration, 2017, 401: 214–232

[22] Mallardo V, Aliabadi M. Optimal sensor placement for structural, damage and impact identification: A review. Structural Durability and Health Monitoring, 2013, 9(4): 287–323

[23] Guo H Y, Zhang L, Zhang L L, Zhou J X. Optimal placement of sensors for structural health monitoring using improved genetic algorithms. Smart Materials and Structures, 2004, 13(3): 528–534

[24] Sun H, Büyüköztürk B. Optimal sensor placement in structural health monitoring using discrete optimization. Smart Materials and Structures, 2015, 24(12): 125034

[25] Jaishi B, Ren W X. Structural finite element model updating using ambient vibration test results. Journal of Structural Engineering, 2005, 131(4): 617–628

[26] Avitabile P. Modal Testing (A Practitioner’s Guide). Hoboken, NJ: John Wiley & Sons, 2018

[27] Marler R T, Arora J S. The weighted sum method for multi-objective optimization: New insights. Structural and Multidisciplinary Optimization, 2010, 41(6): 853–862

[28] Shahidi S G, Pakzad S N. Generalized response surface model updating using time domain data. Journal of Structural Engineering, 2014, 140(8): A4014001

[29] Rennen G, Husslage B, Van Dam E R, Den Hertog D. Nested maximin Latin hypercube designs. Structural and Multidisciplinary Optimization, 2010, 41(3): 371–395

[30] Brincker R, Ventura C. Introduction to Operational Modal Analysis. Hoboken, NJ: John Wiley & Sons, 2015

[31] Weijtjens W, Lataire J, Devriendt C, Guillaume P. Dealing with periodical loads and harmonics in operational modal analysis using time-varying transmissibility functions. Mechanical Systems and Signal Processing, 2014, 49(1–2): 154–164

[32] Devriendt C, Guillaume P. The use of transmissibility measurements in output-only modal analysis. Mechanical Systems and Signal Processing, 2007, 21(7): 2689–2696

[33] Devriendt C, Guillaume P. Identification of modal parameters from transmissibility measurements. Journal of Sound and Vibration, 2008, 314(1–2): 343–356

[34] Yan W J, Ren W X. An enhanced power spectral density transmissibility (EPSDT) approach for operational modal analysis: Theoretical and experimental. Engineering Structures, 2015, 102: 108–119

[35] Yan W J, Zhao M Y, Sun Q, Ren W X. Transmissibility-based system identification for structural health Monitoring: Fundamentals, approaches, and applications. Mechanical Systems and Signal Processing, 2019, 117: 453–482

[36] Brandt A. Noise and Vibration Analysis: Signal Analysis and Experimental Procedures. Hoboken, NJ: John Wiley & Sons, 2011

[37] Marwala T. Finite element model updating using response surface method. In: Proceedings of the 45th Collection of Technical Papers-AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference. California: Palm Springs, 2004, 5165–5173

[38] Box G E P, Draper N R. Empirical Model Building and Response Surface. Hoboken, NJ: John Wiley & Sons, 1987

[39] Myers R H, Montgomery D C, Anderson-Cook C M. Response Surface Methodology: Process and Product Optimization Using Designed Experiments. Hoboken, NJ: John Wiley & Sons, 2016

[40] Khuri A I, Mukhopadhyay S. Response surface methodology. Wiley Interdisciplinary Reviews: Computational Statistics, 2010, 2(2): 128–149

[41] Raseman W J, Jacobson J, Kasprzyk J R. Parasol: An open source, interactive parallel coordinates library for multi-objective decision making. Environmental Modelling & Software, 2019, 116: 153–163

[42] Huang T H, Huang M L, Jin J S. Parallel rough set: Dimensionality reduction and feature discovery of multi-dimensional data in visualization. In: Lu B L, Zhang L, Kwok J, eds. Neural Information Processing. ICONIP 2011. Lecture Notes in Computer Science, 7063. Berlin, Heidelberg: Springer, 2011.

[43] Inselberg A. The plane with parallel coordinates. Visual Computer, 1985, 1: 69–91

[44] Papazafeiropoulos G, Muñiz-Calvente M, Martínez-Pañeda E. Abaqus2Matlab: A suitable tool for finite element post-processing. Advances in Engineering Software, 2017, 105: 9–16

[45] Bishop R E D, Johnson D B. The Mechanics of Vibration. Cambridge: Cambridge University Press, 1960.

[46] Digilov R M, Abramovich H. Flexural vibration test of a beam elastically restrained at one end: A new approach for Young’s modulus determination. Advances in Materials Science and Engineering, 2013, 2013: 329530

[47] Leissa A W, Qatu M S. Vibration of Continuous Systems. New York: McGraw Hill Professional, 2011

[48] Fahy F, Walker J. Advanced Applications in Acoustics, Noise and Vibration. London: Taylor & Francis, 2005

[49] Gonçalves P J P, Brennan M J, Peplow A, Tang B. Calculation of the natural frequencies and mode shapes of a Euler-Bernoulli beam which has any combination of linear boundary conditions. Journal of Vibration and Control, 2019, 25(18): 2473–2479

[50] Li Z, Tang D, Li W. Analysis of vibration frequency characteristic for elastic support beam. Advanced Materials Research, 2013, 671–674: 1324–1328

[51] Karnovsky I A, Lebed O I. Free Vibrations of Beams and Frames: Eigenvalues and Eigenfunctions. New York: McGraw Hill Professional, 2004

[52] Dawson C B, Cha P D. A sensitivity-based approach to solving the inverse eigenvalue problem for linear structures carrying lumped attachments. International Journal for Numerical Methods in Engineering, 2019, 120(5): 537–566

[53] Devriendt C, De Sitter G, Vanlanduit S, Guillaume P. Operational modal analysis in the presence of harmonic excitations by the use of transmissibility measurements. Mechanical Systems and Signal Processing, 2009, 23(3): 621–635

[54] Araújo I G, Laier J E. Operational modal analysis using SVD of power spectral density transmissibility matrices. Mechanical Systems and Signal Processing, 2014, 46(1): 129–145

[55] Montgomery D C. Design and Analysis of Experiments. Hoboken, NJ: John Wiley & Sons, 2017

[56] Sansen J, Richer G, Jourde T, Lalanne F, Auber D, Bourqui R. Visual exploration of large multidimensional data using parallel coordinates on big data infrastructure. Informatics (MDPI), 2017, 4(3): 21

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