124 Y. F. Xu et al. Acknowledgements The authors are grateful for the financial support from the National Science Foundation through Grant Nos. CMMI-1335024, CMMI-1763024, CMMI-1762917 and the College of Engineering and Information Technology at the University of Maryland, Baltimore County through a Strategic Plan Implementation Grant. The first author is also grateful for the faculty startup support from the Department of Mechanical and Materials Engineering at the University of Cincinnati. Appendix: Short-Time Fourier Transform The short-time Fourier transform of ˜z, denoted by ˜Vw(t,f), can be expressed by ˜Vw(t,ω) = ∞ −∞ ˜z(τ)g∗s (τ −t)e−jωτdτ (13.28) where gs is a window function with a scale s, the superscript ∗ denotes complex conjugation, and j = √ −1. The scale s determines the width of gs in the time domain, which should be smaller than that of a half-scan period. When ˜Vw at the i-th natural frequency of the structure becomes almost zero at an instant ti,0, the amplitude of ˜yi is considered to be zero. Note that in Eq. (13.28), ˜Vw(t,ω) is visualized by use of a spectrogram whose intensity denotes the power spectral density associated with ˜Vw(t,ω); gs is a Hamming function that can be expressed by gs (t) = 20.54−0.46cos 2πt s , 0 ≤t ≤s 0 , otherwise (13.29) References 1. Doebling, S.W., Farrar, C.R., Prime, M.B., et al.: A summary review of vibration-based damage identification methods. Shock Vib. Dig. 30(2), 91–105 (1998) 2. Fan, W., Qiao, P.: Vibration-based damage identification methods: a review and comparative study. Struct. Health Monit. 10(1), 83–111 (2011) 3. Ewins, D.J.: Modal Testing: Theory and Practice, vol. 15. Research Studies Press, Letchworth (1984) 4. Di Maio, D., Ewins, D.: Continuous scan, a method for performing modal testing using meaningful measurement parameters; Part I. Mech. Syst. Signal Process. 25(8), 3027–3042 (2011) 5. Chen, D.-M., Xu, Y.F., Zhu, W.D.: Damage identification of beams using a continuously scanning laser doppler vibrometer system. J. Vib. Acoust. 138(5), 051011 (2016) 6. Stanbridge, A.B., Ewins, D.J.: Modal testing using a scanning laser doppler vibrometer. Mech. Syst. Signal Process. 13(2), 255–270 (1999) 7. Stanbridge, A.B., Ewins, D., Khan, A.: Modal testing using impact excitation and a scanning LDV. Shock Vib. 7(2), 91–100 (2000) 8. Allen, M.S., Sracic, M.W.: A new method for processing impact excited continuous-scan laser doppler vibrometer measurements. Mech. Syst. Signal Process. 24(3), 721–735 (2010) 9. Sriram, P., Craig, J., Hanagud, S.: A scanning laser doppler vibrometer for modal testing. Int. J. Anal. Exp. Modal Anal. 5, 155–167 (1990) 10. Sriram, P., Hanagud, S., Craig, J.: Mode shape measurement using a scanning laser doppler vibrometer. Int. J. Anal. Exp. Modal Anal. 7(3), 169–178 (1992) 11. Stanbridge, A.B., Ewins, D.J.: Measurement of translational and angular vibration using a scanning laser doppler vibrometer. Shock Vib. 3(2), 141–152 (1996) 12. Yang, S., Allen, M.S.: Output-only modal analysis using continuous-scan laser doppler vibrometry and application to a 20 kw wind turbine. Mech. Syst. Signal Process. 31, 228–245 (2012) 13. Yang, S., Allen, M.S.: Lifting approach to simplify output-only continuous-scan laser vibrometry. Mech. Syst. Signal Process. 45(2), 267–282 (2014) 14. Wereley, N.M., Hall, S.R.: Linear time periodic systems: transfer function, poles, transmission zeroes and directional properties. In: American Control Conference, pp. 1179–1184. IEEE (1991) 15. Chen, D.-M., Xu, Y.F., Zhu, W.D.: Non-model-based multiple damage identification of beams by a continuously scanning laser doppler vibrometer system. Measurement 115, 185–196 (2018) 16. Chen, D.-M., Xu, Y.F., Zhu, W.D.: Experimental investigation of notch-type damage identification with a curvature-based method by using a continuously scanning laser doppler vibrometer system. J. Nondestruct. Eval. 36(2), 38 (2017) 17. Xu, Y.F., Chen, D.-M., Zhu, W.D.: Damage identification of beam structures using free response shapes obtained by use of a continuously scanning laser doppler vibrometer system. Mech. Syst. Signal Process. 92, 226–247 (2017) 18. Meirovitch, L.: Principles and Techniques of Vibrations, vol. 1. Prentice Hall, New Jersey (1997) 19. Caughey, T., O’Kelly, M.E.: Classical normal modes in damped linear dynamic systems. J. Appl. Mech. 32(3), 583–588 (1965) 20. Rao, S.S., Yap, F.F.: Mechanical Vibrations, vol. 4. Prentice Hall, Upper Saddle River (2011) 21. Hlawatsch, F., Auger, F.: Time-Frequency Analysis. Wiley-ISTE (2013) 22. Pandey, A., Biswas, M., Samman, M.: Damage detection from changes in curvature mode shapes. J. Sound Vib. 145(2), 321–332 (1991) 23. Xu, Y.F., Zhu, W.D., Liu, J., Shao, Y.: Identification of embedded horizontal cracks in beams using measured mode shapes. J. Sound Vib. 333(23), 6273–6294 (2014)
RkJQdWJsaXNoZXIy MTMzNzEzMQ==