**4. Orbit patterns and power spectrum patterns**

600 Advances in Wavelet Theory and Their Applications in Engineering, Physics and Technology

'*Ts*' in Table 1 corresponds to the period of the forced oscillation which corresponds to about 5.0 in the nominal reduced velocity. The lock-in phenomenon of VIV is therefore expected in each experimental case when the model is in forced inline oscillations with the period of *Ts*.

0.2

*D*

*s*

Therefore, the frequency of vortex shedding *fs* is not an actual frequency, but is a nominal

The wavelet analysis is the time-frequency analysis for time histories such like the Hilbert

where *f*(*t*) is a time history, *a* stands for a dilation parameter and *b* stands for a location

 *<sup>i</sup> <sup>t</sup> e <sup>t</sup> <sup>t</sup>* <sup>0</sup>

<sup>2</sup> 2 2

angular frequency. Half of the natural angular frequency of each model is applied in this

And then, resolution of frequency is high although resolution of time gets worse. It is a merit to select the Gabor's wavelet because the dilation parameter *a*, which corresponds to a

consequently individual each other so that the tuning of the parameters in order to draw the

In this study, *b* is set at 0.4 seconds, *a* is carried out from 0.0 to 3.0 with resolution of 0.2 and

*<sup>t</sup> <sup>b</sup> <sup>f</sup> <sup>t</sup>*

(*t*)" is the mother wavelet function. The Gabor's mother wavelet is applied

2 2

 

*dt*

 

exp , (10)

becomes smaller, the mother function be attenuated soon.

0 is a principle

. The parameters are

<sup>1</sup> , , (9)

*a*

1

is a damping parameter of the mother wavelet function and

*a*

*O s U f D*

> *s f*

*<sup>U</sup> <sup>f</sup> <sup>O</sup>*

'*Ts*' is calculated with following equations,

transform. The wavelet transform is defined as follows,

*W b a <sup>T</sup>*

scaling parameter, is individual to the resolution parameter

frequency in this study.

**3. Wavelet analysis** 

such as follows in this study,

When the damping parameter

wavelet contour is not difficult.

parameter. "

where

study.

is 1.0.

*St* 0.2 . (5)

*<sup>s</sup>* 0.2 , (7)

*<sup>T</sup>* <sup>1</sup> . (8)

, (6)

In case of a single cylinder experiment (Ikoma & Masuda et al., 2006, 2007), the four patterns of the power spectrum of VIV have been found such like Fig. 4. In addition, there was an adequate correlation between the orbit pattern and the spectrum pattern in the paper (Ikoma et al., 2007). However both the power spectrum patterns of the orbit patterns of the type *U* and the type 8 correspond to the pattern 4 which is bi-harmonic type. Therefore detail of VIV behavior cannot be understood from a result with the FFT analysis of VIV.

Fig. 4. Classifications of power spectrum patterns of VIV [1]
