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

1
Time-frequency-domain modal identification of
ambient vibration structures using Wavelet
Transform
Numerical example
2
Natural frequency damping
Damping Ratios Identification
Natural Frequencies Identification
Frequency domain
Time domain
Cutting slide
Frequency
Time
3
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 fF Fourier frequency
fs Sampling frequency fy Central wavelet
frequency
4
Wavelet function

• The complex Morlet wavelet is commonly used in
  the CWT

Fourier transform of complex Morlet wavelet
Fourier frequency and central wavelet frequency
5
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
6
Damped natural frequencies
Wavelet transform (Floor1)
Frequency domain
t80s
14.02Hz
9.12Hz
5.91Hz
Wavelet transform (Floor5)
Frequency domain
t80s

• Difficulties in identifying high-order
  low-dominant frequencnies
• due to inflexible resolutions used
  smoothing

7
Refined by bandwidth filtering

• Filtered at frequency bandwidths
• 1) 0-3.125Hz
• 2) 3.125-6.25Hz
• 3) 6.25-12.5Hz
• 4) 12.5-25Hz
• 5) 25-50Hz

8
Refined wavelet transform
Bandwidth 0-20Hz
Bandwidth 0-3.125Hz
Dominant for mode 1
f11.72Hz
Only 1st mode dominated
Bandwidth 3.125-6.25Hz
f25.37Hz
Dominant for mode 2

• Refined and localized by
• multiresolution analysis
• Filtered at frequency
• bandwidths
• (0-3.125Hz 3.125-6.25Hz
• 6.25-12.5Hz 12.5-25Hz
• 25-50Hz)

Bandwidth 6.25-12.5Hz
Dominant for mode 3
f38.99Hz
9
Refined wavelet transform
Bandwidth 0-3.125Hz
Mode 1
Amplitude envelope slop
1.76Hz
Slide 1
Slide 2
f11.72Hz
Damped Natural Frequencies (Hz) Damped Natural Frequencies (Hz) Damped Natural Frequencies (Hz) Damped Natural Frequencies (Hz)
  FEM FDD FDD-RDT WT
mode 1 1.69 1.73 1.73 1.76
mode 2 5.22 5.35 5.34 5.47
mode 3 9.26 8.84 8.82 8.95
mode 4 13.6 13.69 13.67 13.72
mode 5 17.8 18.05 18.02 18.14
Mode 2
Bandwidth 3.125-6.25Hz
5.49Hz
Slide 1
Slide 2
f25.37Hz
Bandwidth 6.25-12.5Hz
Mode 3
Slide 1
Slide 2
8.95Hz
f38.99Hz