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Time-frequency-domain modal identification of ambient vibration structures using Wavelet Transform Numerical example

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Natural frequency & damping Frequency Time Cutting slide Frequency domain Time domain Damping Ratios Identification Natural Frequencies Identification

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Wavelet transform Continuous wavelet transform (CWT) is defined as convolution operator of signal X(t) and wavelet function : Wavelet function : Complex conjugate of wavelet function : Wavelet transform coefficient : Wavelet scale and translation parameters Info of time and frequency can be obtained. Relation of wavelet scale and Fourier frequency can be estimated s : Wavelet scale; f F : Fourier frequency f s : Sampling frequency; f : Central wavelet frequency

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Wavelet function The complex Morlet wavelet is commonly used in the CWT: : Fourier transform of complex Morlet wavelet : Fourier frequency and central wavelet frequency

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Damping & mode shapes Output displacements of the MDOF system can be decomposed in the structural normalized coordinates Wavelet transform coefficient of output response: Mode shape can be estimated via the wavelet coefficients of output displacements at point k and reference point: Decay envelope and logarithmic decrement can be extracted from this decay envelope and in tern of modulus: and

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Damped natural frequencies Wavelet transform (Floor1) Frequency domain Wavelet transform (Floor5) =80s Frequency domain 5.91Hz 9.12Hz 14.02Hz Difficulties in identifying high-order low-dominant frequencnies Difficulties in identifying high-order low-dominant frequencnies due to inflexible resolutions & used smoothing due to inflexible resolutions & used smoothing

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Refined by bandwidth filtering Filtered at frequency bandwidths Filtered at frequency bandwidths 1) 0-3.125Hz 1) 0-3.125Hz 2) 3.125-6.25Hz 2) 3.125-6.25Hz 3) 6.25-12.5Hz 3) 6.25-12.5Hz 4) 12.5-25Hz 4) 12.5-25Hz 5) 25-50Hz 5) 25-50Hz

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Refined wavelet transform Bandwidth 0-20Hz [Bandwidth 0-3.125Hz] [Bandwidth 3.125-6.25Hz] [Bandwidth 6.25-12.5Hz] Only 1 st mode dominated f1=1.72Hz f2=5.37Hz f3=8.99Hz Refined and localized by Refined and localized by multiresolution analysis multiresolution analysis Filtered at frequency Filtered at frequency bandwidths bandwidths (0-3.125Hz; 3.125-6.25Hz (0-3.125Hz; 3.125-6.25Hz 6.25-12.5Hz; 12.5-25Hz; 6.25-12.5Hz; 12.5-25Hz; 25-50Hz) 25-50Hz) Dominant for mode 1 Dominant for mode 3 Dominant for mode 2

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Refined wavelet transform [Bandwidth 0-3.125Hz] [Bandwidth 3.125-6.25Hz] [Bandwidth 6.25-12.5Hz] f1=1.72Hz f2=5.37Hz f3=8.99Hz Mode 1 Mode 2 Mode 3 [Slide 1] [Slide 2] [Slide 1] [Slide 2] [Slide 1] [Slide 2] 1.76Hz 5.49Hz 8.95Hz Amplitude envelope slop Damped Natural Frequencies (Hz) FEMFDDFDD-RDTWT mode 11.691.73 1.76 mode 25.225.355.345.47 mode 39.268.848.828.95 mode 413.613.6913.6713.72 mode 517.818.0518.0218.14

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