PSI - Issue 14

J. Prawin et al. / Procedia Structural Integrity 14 (2019) 234–241

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response spectrum map is not considered to identify the number of kernel components. It may be noted that the response spectrum maps cannot be plotted as the amplitude is constrained. However, due to bilinear nature, it is adequate enough to choose the number of kernel components as 3 (linear component, opening state and closing state component). The memory depth is chosen as 3 based on convergence studies.The various response components are computed using the identified kernel coefficients obtained from the model. The separated response components x 1 (t) to x 4 (t) and their corresponding power spectra are presented in Figure3. It can be observed from Figure 3 (b) that the first component power spectrum shows a peak at excitation frequency (i.e., first harmonic). The second order component (Figure 3 (d)) shows a single peak at second harmonic while the fourth order component (Figure 3(f)) shows peaks at the second and fourth harmonic even under measurement noise. It can also be observed that the second and fourth component contributes to the crack opening and crack closure phenomena respectively. Since the bilinear phenomenon is approximated with a polynomial form of nonlinearity, the identified components cannot exhibit the structured harmonic nature of the components. This can be easily explained by the fact that the Volterra kernels identified are usually functions of system parameters alone which in turn indirectly depends on the amplitude dependent coefficients. 4. Conclusion Numerical simulation studies have been carried out to test the presented adaptive filter algorithm by solving a bilinear oscillator (a beam with a breathing crack). The investigations presented in this paper clearly indicate that the proposed adaptive filter has the ability to separate the opening and closing states of the cracked structure. Further, the application of an adaptive algorithm for Volterra filters is quite simple due to the linearity of the input-output relation according to the kernels/filter coefficients. The nonlinearity is reflected only by multiple products between the delayed versions of the input signal. The proposed algorithm also has the ability to predict the system’s response to any arbitrary input using Volterra kernel coefficients alone without any knowledge about the system parameters. Acknowledgements This paper is being published with the permission of The Director, CSIR-Structural Engineering Research Centre, Taramani, Chennai - 600113. Billings, S. A., Boaghe, O. M., 1999. The response spectrum map, Volterra series representations and the Duffing equation. The Department of Automatic Control & Systems Engineering, The University of Sheffield. Billings, S. A., Boaghe, O. M., 2001. The response spectrum map, a frequency domain equivalent to the bifurcation diagram. Int. Journal of Bifur. Chaos App. Science and Eng, 11, 1961–1975. Budra, G., Botoca, C., 2005. Nonlinearities identification using the LMS Volterra filter. Proceedings of the WSEAS international conference on Dynamical systems and control, 148-153. Chatterjee Animesh, Vyas, N. S., 2001. Stiffness nonlinearity classification through structured response component analysis using Volterra Mech. Syst. Signal Process, 15(2), 323-336. Hickey, D., Worden, K., 2009. Higher-order spectra for identification of nonlinear modal coupling. Mechanical Systems and Signal Processing 23, 1037-1061. Jeffreys, H., Jeffreys, B. S., 1988. Methods of Mathematical Physics. 3rd ed. Cambridge, England: Cambridge University Press. Paulo, S. R. D., 2008. Adaptive Filtering Algorithms and Practical Implementation. Third Edition, Kluwer Academic Publishers. Peng, Z. K., Lang, Z.Q., Billings, S.A., Lu, Y., 2007. Analysis of bilinear oscillators under harmonic loading using nonlinear output frequency response functions. Int. J. Mech. Sciences 49 (11), 1213 – 1225. Prawin, J., A. Rama Mohan Rao, A., 2016. Development of Polynomial Model for Cantilever Beam with Breathing Crack. Procedia Engineering, 144, 1419-1425. Prawin, J., Rama Mohan Rao, A., 2017. Nonlinear identification of MDOF systems using Volterra series approximation. Mechanical Systems and Signal Processing, 84, 58-77. Prawin, J., Rama Mohan Rao, A., 2018. Nonlinear Structural Damage Detection Based on Adaptive Volterra Filter Model. International Journal of Structural Stability and Dynamics, 18(2), 1871003(12 pages). Tomilson, G. R., Manson, G., 1996. A simple criterion for establishing an upper limit to the harmonic excitation level of the Duffing oscillator using the Volterra series. J Sound Vib, 190, 751–762. References