The approximation function of bridge deck vibration derived from the measured eigenmodes

Open access


This article deals with a method of how to acquire approximate displacement vibration functions. Input values are discrete, experimentally obtained mode shapes. A new improved approximation method based on the modal vibrations of the deck is derived using the least-squares method. An alternative approach to be employed in this paper is to approximate the displacement vibration function by a sum of sine functions whose periodicity is determined by spectral analysis adapted for non-uniformly sampled data and where the parameters of scale and phase are estimated as usual by the least-squares method. Moreover, this periodic component is supplemented by a cubic regression spline (fitted on its residuals) that captures individual displacements between piers. The statistical evaluation of the stiffness parameter is performed using more vertical modes obtained from experimental results. The previous method (Sokol and Flesch, 2005), which was derived for near the pier areas, has been enhanced to the whole length of the bridge. The experimental data describing the mode shapes are not appropriate for direct use. Especially the higher derivatives calculated from these data are very sensitive to data precision.

Baili, J., Lahouar, S., Hergli, M., Amimi, A. and Besbes, K. (2009). GPR signal de-noising by discrete wavelet transform, Nondestructive Testing and Evaluation International 42: 696-703.

Clough, R. and Penzien, J. (1993). Dynamics of Structures, McGraw-Hill, Singapore.

De Roeck, G., Peeters, B. and Maeck, J. (2000). Dynamic monitoring of civil engineering structures, Computational Methods for Shells and Spatial Structures, IASS-IACM, Athens, Greece, pp. 1-24.

Farrar, C., Doebling, S. and Nix, D. (2001). Vibration-based structural damage identification, Philosophical Transactions of the Royal Society A359: 131-149.

Flesch, R., Stebernjak, B. and Freytag, B. (1999). System identification of bridge Warth/Austria, Structural Dynamics-EURODYN 99, Balkema, Rotterdam, pp. 813-818.

Hastie, T., Tibshirani, R. and Friedman, J. (2009). The Elements of Statistical Learning, Springer, New York, NY.

James, G., Witten, D., Hastie, T. and Tibshirani, R. (2013). An Introduction to Statistical Learning, 6th Edn., Springer, New York, NY.

Janiszowski, K.B. (2014). Approximation of a linear dynamic process model using the frequency approach and a non-quadratic measure of the model error, International Journal of Applied Mathematics and Computer Science 24(1): 99-109, DOI: 10.2478/amcs-2014-0008.

Joannin, C., Chouvion, B., Thouverez, F.and Mbaye, M.O.J. (2015). Nonlinear modal analysis of mistuned periodic structures subjected to dry friction, ASME Journal of Engineering for Gas Turbines and Power 138(7): 072504-072504-12.

Leite, F., Montagne, R., Corso, G., Vasconcelos, G. and Lucena, L. (2008). Optimal wavelet filter for suppression of coherent noise with an application to seismic data, Physica A 387: 1439-1445.

Li, P., Liu, C., Tian, Q., Hu, H. and Song, Y. (2016). Dynamics of a deployable mesh reflector of satellite antenna: Form-finding and modal analysis, ASME Journal of Computational and Nonlinear Dynamics 11(4): 041017-041017-12.

Liu, X. (2013). A new method for calculating derivatives of eigenvalues and eigenvectors or discrete structural systems, Journal of Sound and Vibration 332(7): 1859-1867.

Maeck, J. and De Roeck, G. (1999a). Detection of damage in civil engineering structures by direct stiffness derivation, 4th European Conference of Structural Dynamics, EURODYN 99, Prague, Czech Republic, pp. 485-490.

Maeck, J. and De Roeck, G. (1999b). Dynamic bending and torsion stiffness derivation from modal curvatures and torsion rates, Journal of Sound and Vibration 225(1): 153-170.

Martinček, G. (1981). Soil investigation according to mechanical impedance method, Die Strasse 21(11): 374-379, (in German).

Martinček, G. (1994). Dynamics of Pavement Structures, E & FN Spon., London. Microsoft (2013). Microsoft Excel 2013, Redmond, WA.

Murthy, D. and Haftka, R.T. (1988). Derivatives of eigenvalues and eigenvectors of general complex matrix, International Journal for Numerical Methods in Engineering 26: 293-311.

Roshan, P., Kumar, A., Tewatia, D. and Pal, S. (2015). Review paper on structural health monitoring: Its benefit and scope in India, Journal of Civil Engineering and Environmental Technology 2(2): 109-112.

Scargle, J.D. (1982). Studies in astronomical time series analysis. II: Statistical aspects of spectral analysis of unevenly spaced data, The Astrophysical Journal 263: 835-853.

Silva, T., Loja, L., Maia, N. and Barbosa, J. (2015). A hybrid procedure to identify the optimal stiffness coefficients of elastically restrained beams, International Journal of Applied Mathematics and Computer Science 25(2): 245-257, DOI: 10.1515/amcs-2015-0019.

Sokol, M., Aroch, R., Venglar, M., Fabry, M. and Zivner, T. (2015). Experience with structural damage identification of an experimental bridge model, Applied Mechanics and Materials 769: 192-199.

Sokol, M. and Flesch, R. (2005). Assessment of soil stiffness properties by dynamic tests on bridges, ASCE Journal of Bridge Engineering 10(1): 77-86.

Stoica, P., Li, J. and He, H. (2009). Spectral analysis of nonuniformly sampled data: A new approach versus the periodogram, IEEE Transactions on Signal Processing 57(3): 843-858.

Sutter, T., Gamarda, C.J., Walsh, J.L. and Adelman, H.M. (1988). Comparison of several methods for calculating vibration mode shape derivatives, AIAA Journal 26(12): 1506-1511.

Vio, R., Diaz-Trigo, M. and Andreani, P. (2013). Irregular time series in astronomy and the use of the Lomb-Scargle periodogram, Astronomy and Computing 1: 5-16.

Wenzel, H. (2009). Health Monitoring of Bridges, Wiley & Sons, Chichester.

Wolfram Research, Inc. (2015). Mathematica 10.2, Champaign, IL.

Yang, S. and Sultan, C. (2016). Free vibration and modal analysis of a tensegrity-membrane system, 12th International Conference on Multibody Systems, Nonlinear Dynamics, and Control, Charlotte, NC, USA, Paper no. DETC2016-59292, pp. V006T09A023.

Yi, T., Li, H. and Zhao, X. (2012). Noise smoothing for structural vibration test signals using an improved wavelet thresholding technique, Sensors 12: 11205-11220.

Yu, M., Liu, Z. and Wang, D. (1997). Comparison of several approximate modal methods for computing mode shape derivatives, Computers & Structures 62(2): 381-393.

ZhangPing, L. and JinWu, X. (2007). Novel modal method for efficient calculation of complex eigenvector derivatives, AIAA Journal 45(6): 1406-1414.

International Journal of Applied Mathematics and Computer Science

Journal of the University of Zielona Góra

Journal Information

IMPACT FACTOR 2017: 1.694
5-year IMPACT FACTOR: 1.712

CiteScore 2017: 2.20

SCImago Journal Rank (SJR) 2017: 0.729
Source Normalized Impact per Paper (SNIP) 2017: 1.604

Mathematical Citation Quotient (MCQ) 2017: 0.13


All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 183 145 5
PDF Downloads 76 71 6