Abstract
The paper investigates changes in determinism of undamaged and cracked aluminium plates with respect to excitation frequencies. Harmonic excitation of frequencies corresponding to structural resonances has been used to vibrate the plates. Vibration responses have been analysed using recurrence plots and recurrence quantification analysis. The smallest sufficient embedding dimension has been estimated using the false nearest neighbour’s algorithm. Mutual information analysis has been applied to determine the relevant time delays. The results demonstrate that performed analysis indicates changes in dynamic behaviour of the plates with respect to various excitation frequencies and crack modes.
Similar content being viewed by others
References
Cawley, P., Adams, R.D.: The locations of defects in structures from measurements of natural frequencies. J. Strain Anal. 14, 49–57 (1979)
Adams, D.E., Nataraju, M.: A nonlinear dynamical systems framework for structural diagnosis and prognosis. Int. J. Eng. Sci. 40, 1919–1941 (2002)
Salawu, O.S.: Detection of structural damage through changes in frequency: a review. Eng. Struct. 19, 718–723 (1997)
Doebling, S.W., Farrar, C.R., Prime, M.B., Shevitz, D.W.: Damage identification and health monitoring of structural and mechanical systems from changes in their vibration characteristics: a literature review. Los Alamos National Laboratory report LA-13070-MS (1996)
Farrar, C.R., Doebling, S.W., Cornwell, P.J., Straser, E.G.: Variability of modal parameters measured on the Alamosa Canyon bridge. In: Proc. 15th Int. Modal Anal. Conf., Orlando, FL, pp. 257–263 (1997)
Staszewski, W.J.: Ultrasonic/guided waves for structural health monitoring. Key Eng. Mater. 293–294, 49–62 (2005)
Jhang, K.: Nonlinear ultrasonic techniques for nondestructive assessment of micro damage in material: a review. J. Precis. Eng. Manuf. 10, 123–135 (2009)
Qiu, Q., Xu, C., Wu, B.: Structural damage detection through chaotic interrogation and attractor analysis. Adv. Mater. Res. 163–167, 2515–2520 (2011)
Ghafari, S.H., Golnavaghi, F., Ismail, F.: Effects of localized faults on chaotic vibration of rolling element bearings. Nonlinear Dyn. 53, 287–301 (2008)
Nichols, J.M., Trickey, S.T., Todd, M.D., Virgin, L.N.: Structural health monitoring through chaotic interrogation. Meccanica 38, 239–250 (2003)
Yin, S.H., Epureanu, B.I.: Nonlinear feedback excitation for system interrogation by bifurcation morphing. AIAA J. 46, 2058–2065 (2008)
Yin, S.H., Epureanu, B.I.: Enhanced nonlinear dynamics and monitoring bifurcation morphing for the identification of parameter variations. J. Fluids Struct. 21, 543–559 (2005)
Staszewski, W.J.: Wavelets for Mechanical and Structural Damage Detection. Monograph 510/1469/2000. Studia, Materialy. Polish Academy of Science Press, Warsaw (2000)
Nichols, J.M., Trickey, S.T., Seaver, M.: Damage detection using multivariate recurrence quantification analysis. Mech. Syst. Signal Process. 20, 421–437 (2006)
Poincaré, H.: Sur la probleme des trios corps et les equations de la dynamique. Acta Math. 13, 1–27 (1890)
Marwan, N., Romano, M.C., Thiel, M., Kurths, J.: Recurrence plots for the analysis of complex systems. Phys. Rep. 438, 237–329 (2007)
Eckmann, J.P., Oliffson, K.S., Ruelle, D.: Recurrence plots of dynamical systems. Europhys. Lett. 5, 973–977 (1987)
Zbilut, J.P., Giuliani, A., Weber, C.L.: Recurrence quantification analysis and principal components in the detection of short complex signals. Phys. Lett. A 237, 131–135 (1998)
Zbilut, J.P., Weber, C.L.: Embedding and delays as derived from quantification of recurrence plot. Phys. Lett. A 171, 199–203 (1992)
Thiel, M., Romano, M.C., Kurths, J.: How much information is contained in a recurrence plot? Phys. Lett. A 330, 343–349 (2004)
Marwan, N., Kurths, J.: Line structures in recurrence plots. Phys. Lett. A 336, 349–357 (2005)
Ngamga, E.J., Nandi, A., Ramaswamy, R., Romano, M.C., Thiel, M., Kurths, J.: Recurrences of strange attractors. Pramana J. Phys. 70, 1039–1045 (2008)
Awrejcewicz, J., Krysko, V.A.: Chaos in Structural Mechanics. Springer, Berlin (2008)
Elwakil, A.S., Soliman, A.M.: Mathematical models of twin-T, Wien-bridge and family of minimum component electronic chaos generators with demonstrative recurrence plots. Chaos Solitons Fractals 10, 1399–1411 (1999)
Nichols, J.M., Trickey, S.T., Seaver, M.: Damage detection using multivariate recurrence analysis. Mech. Syst. Signal Process. 20, 421–437 (2006)
Fontaine, S., Dia, S., Renner, M.: Nonlinear friction dynamics on fibrous materials application to the characterization of surface quality. Part II: local characterization of phase space by recurrence plots. Nonlinear Dyn (online first 19.02.2011). doi:10.1007/s11071-011-9968-7
Kurths, J., Schwarz, U., Sonett, C.P., Parlitz, U.: Testing nonlinearity in radiocarbon data. Nonlinear Process. Geophys. 1, 72–75 (1994)
Zolotova, N.V., Ponyavin, D.I.: Phase asynchrony of the north-south sunspot activity. Astron. Astrophys. 449, L1–L4 (2006)
Manetti, C., Giuliani, A., Ceruso, M.A., Webber, C.L., Zbilut, J.P.: Recurrence analysis of hydration effects o nonlinear protein dynamics: multiplicative scaling and additive processes. Phys. Lett. A 281, 317–323 (2001)
Giuliani, A., Manetti, C.: Hidden pecularities in the potential energy time series of a tripeptide highlighted by a recurrence plot analysis: a molecular dynamics simulation. Phys. Rev. E 53, 6336–6340 (1996)
Marwan, N., Thiel, M., Nowaczyk, N.R.: Cross recurrence plot based synchronization of time series. Nonlinear Process. Geophys. 9, 325–331 (2002)
Zbilut, J.P., Koebbe, M., Loeb, H., Mayer-Kress, G.: Use of recurrence plots in the analysis of heart beat intervals. In: Proc. IEEE Conf. Comput. Cardiol. 1990, pp. 263–266. IEEE Computer Society Press, Chicago (1991)
Thomasson, N., Hoeppner, T.J., Webber, C.L., Zbilut, J.P.: Recurrence quantification in epileptic EEGs. Phys. Lett. A 279, 94–101 (2001)
Hołyst, J.A., Zebrowska, M., Urbanowicz, K.: Observations of deterministic chaos in financial time series by recurrence plots, can one control chaotic economy? Eur. Phys. J. B 20, 531–535 (2001)
Worden, K., Farrar, C.: Improving excitations for active sensing in structural health monitoring via evolutionary algorithms. J. Vib. Acoust. 129, 784–802 (2007)
Litak, G., Sawicki, J.T., Kasperek, R.: Cracked rotor detection by recurrence plots. Nondestruct. Test. Eval. 24, 347–381 (2009)
Masri, S.F., Caughey, T.K.: A nonparametric identification technique for nonlinear dynamic problems. J. Appl. Mech. 46, 433–447 (1979)
Crawley, E.F., O’Donnell, K.J.: Identification of nonlinear system parameters in joints using the force-state mapping technique. AIAA Pap. 86(1013), 659–667 (1986)
Crawley, E.F., Aubert, A.C.: Identification of nonlinear structural elements by force-state mapping. AIAA J. 24, 155–162 (1986)
Duym, S., Schoukens, J., Guillaume, P.: A local restoring surface method. Int. J. Anal. Exp. Modal Anal. 11, 116–132 (1996)
Worden, K., Tomlinson, G.R.: Application of restoring force method to nonlinear elements. In: Proc. 7th Int. Modal Anal. Conf., Las Vegas, NV, January 30–February 2, pp. 1347–1355 (1989)
Bouc, R.: Forced vibrations of mechanical systems with hysteresis. In: Proc. 4th Conf. Non-Linear Oscill., Prague, September 5–9, vol. 5, p. 315 (1967)
Wen, Y.K.: Method for random vibration of hysteretic systems. J. Eng. Mech. Div., Proc. Am. Soc. Civ. Eng. 102 (1976)
Dabrowski, A.: Estimation of the largest Lyapunov exponent from the perturbation vector and its derivative product. Nonlinear Dyn. (2011). doi:10.1007/s11071-011-9977-6
Yang, C., Qiong Wu, C.: A robust method on estimation of Lyapunov exponents from a noisy time series. Nonlinear Dyn. (2010). doi:10.1007/s11071-011-9860-x
Kantz, H., Schreiber, T.: Nonlinear Time Series Analysis. University Press, Cambridge (1997)
Cao, L.: Practical method for determining the minimum embedding dimension of a scalar time series. Physica D 110, 43–50 (1997)
Marwan, N.: Cross recurrence plot toolbox for Matlab, reference manual, version 5.15, release 28.6. http://tocsy.pik-potsdam.de (2010). Accessed 26 March 2010
Awrejcewicz, J., Krysko, V.A.: Nonclassical Thermoplastic Problems in Nonlinear Dynamics of Shells. Springer, Berlin (2003)
Klepka, A., Staszewski, W.J., Jenal, R.B., Szwedo, M., Iwaniec, J., Uhl, T.: Nonlinear acoustics for fatigue crack detection—experimental investigations of vibro-acoustic wave modulations. SHM. doi:10.1177/1475921711414236
Awrejcewicz, J., Krysko, V.A., Vakakis, A.F.: Nonlinear Dynamics of Continuous Elastic Systems. Springer, Berlin (2004)
Awrejcewicz, J., Andrianov, I.V., Manevitch, L.I.: Asymptotical Mechanics of Thin Walled Structures. A Handbook. Springer, Berlin (2004)
Klepka, A., Jenal, R.B., Szwedo, M., Staszewski, W.J., Uhl, T.: Experimental analysis of vibroacoustic modulations in nonlinear acoustics used for fatigue crack detection. In: Proc. 5th EWSHM 2010, Sorento, pp. 541–546 (2010)
Jenal, R.B., Staszewski, W.J., Klepka, A., Uhl, T.: Structural damage detection using laser vibrometer. In: Proc. 2nd Int. Symp. NDT Aerosp., Hamburg, pp. 1–8 (2010)
Delsanto, P.P.: Universality of Nonclassical Nonlinearity. Springer, New York (2010)
Haroon, M., Adams, D.E., Luk, Y.W.: A technique for estimating linear parameters of an automotive suspension system using nonlinear restoring force excitation in the absence of an input measurement. J. Vib. Acoust. 127, 483–492 (2005)
Facchini, A., Kantz, H., Tiezzi, E.: Recurrence plot analysis of nonstationary data: the understanding of curved patterns. Phys. Rev. E 72, 021915 (2005)
Kęcik, K., Warmiński, J.: Analysis of chaotic and regular motion of an autoparametric system by recurrence plots application. In: XXIV Symp. Vibrations in Physical Systems, Poznań–Będlewo, May 12–15, pp. 221–226. Poznan University of Technology, Poznan (2010)
Acknowledgements
The work presented in the paper was founded by the Foundation for Polish Science (under programme MISTRZ).
The authors are grateful to Professor Jerzy Warmiński from Lublin University of Technology, Poland, and Professor Keith Worden from Sheffield University, UK, for valuable comments and discussions.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Iwaniec, J., Uhl, T., Staszewski, W.J. et al. Detection of changes in cracked aluminium plate determinism by recurrence analysis. Nonlinear Dyn 70, 125–140 (2012). https://doi.org/10.1007/s11071-012-0436-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-012-0436-9