**2.3 Experimental conditions**

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

L

d

a) in case of one cylinder

D

b) in case of two cylinders

D

Doppler

current-meter

Three-

dimensionalcurrent meter

Load cell

Plate spring

Flat spring

Model of circular

*d*

Fig. 2. Side views of experimental setup system

Flat spring *l*

cylinder

Potentiometer

Load cell

cylinder

Model of circular

Potentio

meter

Detail of the cylinder models is described in the paper (Ikoma et al., 2007). Length of the flat spring is expressed as "*l*" in Table 1. Natural periods *Tn* of cross-flow vibration of a suspended cylinder were obtained with the plucked decay test in still water.

Water depth is set to 1.0 m. The amplitude of forced oscillation is 7.2 cm, the Keulegan-Carpenter (*KC*) number accordingly corresponds to 5.7 and 9.0 in the experiments.

In case of double cylinders, the cylinders are straightly suspended, and then the distance *ld* between the center to center of both the cylinders is varied such as Table 2. The distance ratio *s* is defined as follows,

$$\mathbf{s} = \frac{l\_d}{D} \,. \tag{1}$$

The front cylinder and the back cylinder are defined as Fig. 3.

Photo 2. Experimental models filled with sand

Application of Wavelet Analysis for the Understanding of Vortex-Induced Vibration 599

The K-C number and the period of *Ts* are defined as same as them (Ikoma et al., 2007) as

*D <sup>U</sup> <sup>T</sup> <sup>K</sup> <sup>O</sup>*

*D*

where *UO* is the maximum velocity of forced oscillation, *T* stands for the period of the forced oscillation. The forced oscillation is simple harmonic motion in this study, hence eqn. (2) can

> *<sup>a</sup> KC* <sup>2</sup>

The range of periods of the forced oscillation is from about 0.4 seconds to 4.6 seconds, the Reynolds (*Re*) numbers accordingly correspond to about 5.0e+3 to 6.0e+5 if the maximum

The natural frequency of transverse vibration varies due to the length of a flat spring. The experimental conditions of each case are shown in Table 1. '*St*' in Table 1 is the Strouhal

> *t U*

*O s*

in which *fs* is the frequency of vortex shedding. The Strouhal number has been well known as about 0.2 in range of *Re*>1.0e+3. In this paper, the Strouhal number is approximately fixed

*<sup>C</sup>* , (2)

Current meter

, (3)

*<sup>f</sup> <sup>D</sup> <sup>S</sup>* , (4)

Fig. 3. Distance ratio between two cylinders

*D* 

follows,

to,

be rewritten as follows,

**2.4 Definitions of nominal period and nominal frequency** 

Front Back

*ld*

in which *a* is amplitude of the forced oscillation in inline direction.

velocity *UO* of the forced oscillations are applied to the calculation.

number and is defined in this study as follows,


a) for single cylinder


b) for double cylinders

Table 1. Principal particulars of cylinder model setting


Table 2. Variation of distance ratio of straight cylinders

Fig. 3. Distance ratio between two cylinders

filled with water filled with sand

measured natural period in water *Tn*

period of forced oscillation in case of *St* =0.2,

period of forced oscillation in case of *St* =0.2, *Ts*

period of forced

oscillation in case

1 8cm 30cm 10cm 0.86s 0.95s 0.93s 1.05s 2 8cm 80cm 13cm 3.28s 3.70s 2.72s 3.10s 3 5cm 30cm 10cm 0.54s 1.15s 0.57s 1.05s 4 5cm 80cm 13cm 2.15s 3.90s 2.08s 3.75s 5 5cm 80cm 10cm 1.86s 3.35s 1.85s 3.35s 6 5cm 60cm 10cm 1.21s 2.20s 1.28s 2.30s 7 8cm 60cm 10cm 1.91s 2.15s 1.79s 2.00s 8 8cm 60cm 4cm 1.18s 1.35s 1.20s 1.35s 9 8cm 80cm 4cm 1.91s 2.15s 1.83s 2.10s 10 5cm 80cm 4cm 1.20s 2.20s 1.23s 2.25s

a) for single cylinder

measured natural period in water *T <sup>n</sup>*

1 5cm 60cm 10cm 1.28 s 2.00 s

model of *St* =0.2, *T <sup>s</sup>*

*l*

2 5cm 80cm 10cm 1.85 s 3.35 s

3 8cm 60cm 10cm 1.79 s 2.00 s

4 8cm 80cm 10cm 2.52 s 2.85 s

b) for double cylinders

*D* 5 cm 8 cm *ld* cm 10 13 15 18 20 16 20 S 2.0 2.5 3.0 3.5 4.0 2.0 2.5

measured natural period in water *Tn*

case *Ts*

*l*

diameter *D*

draft *d*

diameter *D*

draft *d*

Table 1. Principal particulars of cylinder model setting

Table 2. Variation of distance ratio of straight cylinders

#### **2.4 Definitions of nominal period and nominal frequency**

The K-C number and the period of *Ts* are defined as same as them (Ikoma et al., 2007) as follows,

$$K\_c = \frac{\mathcal{U}\_o T}{D} \, \, \, \, \tag{2}$$

where *UO* is the maximum velocity of forced oscillation, *T* stands for the period of the forced oscillation. The forced oscillation is simple harmonic motion in this study, hence eqn. (2) can be rewritten as follows,

$$K\_c = 2\pi \frac{a}{D} \,\prime \tag{3}$$

in which *a* is amplitude of the forced oscillation in inline direction.

The range of periods of the forced oscillation is from about 0.4 seconds to 4.6 seconds, the Reynolds (*Re*) numbers accordingly correspond to about 5.0e+3 to 6.0e+5 if the maximum velocity *UO* of the forced oscillations are applied to the calculation.

The natural frequency of transverse vibration varies due to the length of a flat spring. The experimental conditions of each case are shown in Table 1. '*St*' in Table 1 is the Strouhal number and is defined in this study as follows,

$$S\_s = \frac{f\_s D}{\mathcal{U}\_o},\tag{4}$$

in which *fs* is the frequency of vortex shedding. The Strouhal number has been well known as about 0.2 in range of *Re*>1.0e+3. In this paper, the Strouhal number is approximately fixed to,

$$S\_{\iota} \approx 0.2\,\,. \tag{5}$$

Application of Wavelet Analysis for the Understanding of Vortex-Induced Vibration 601

A sampling frequency of the experimental recording has been 500 Hz, which corresponds to 2.0e-3 seconds in the sampling time. '0.4 seconds' of *b* in the time sifting for the wavelet analysis corresponds to 200 sampling data skipping. In addition, the shortest natural period of cross-flow vibration in the experimental models in Tables 1a) and 1b) is 0.86 seconds. If the bi-harmonic vibration in VIV in this case occurs, the vibration period is 0.43 seconds. The resolution would be thereby enough 0.4 seconds. Using *b*=0.4, variation pattern of the wavelet would be able to be reproduced. The step of *a* is now 0.2. The parameter *a* corresponds to a resolution of the frequency component. The cross-flow vibration appears relatively simply from the FFT analysis so that the resolution of 0.2 may be reasonable.

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.

VIV

VIV

c) Pattren 3 (two peaks type) d) Pattern 4 (bi-harmonic type)

From the experiment using the single cylinder, orbit patterns can be classified to six patterns such as Fig. 5. The net type is specified to *N*1 and *N*2. It can be considered that response of

1 2 *<sup>o</sup> f* / *f*

very small power

1 2 *<sup>o</sup> f* / *f*

b) Pattern 2 (three peaks type)

largel power

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

VIV

VIV

**5. Results and discussion** 

**5.1 Orbit patterns** 

1 2 *<sup>o</sup> f* / *f*

a) Pattern 1 (general type)

1 2 *<sup>o</sup> f* / *f*

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

the type *U* and the type 8 corresponds to VIV lock-in.

'*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*. '*Ts*' is calculated with following equations,

$$\frac{f\_s D}{\mathcal{U}\_o} = 0.2 \,\text{\AA} \tag{6}$$

$$f\_s = 0.2 \frac{\mathcal{U}\_o}{D} \,\prime \tag{7}$$

$$T\_s = \frac{1}{f\_s}.\tag{8}$$

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

#### **3. Wavelet analysis**

The wavelet analysis is the time-frequency analysis for time histories such like the Hilbert transform. The wavelet transform is defined as follows,

$$\mathcal{W}\_{\tau}(b, a) = \frac{1}{\sqrt{|a|}} \int\_{-\epsilon}^{\epsilon} f(t) \cdot \nu \left(\frac{t - b}{a}\right) dt \,, \tag{9}$$

where *f*(*t*) is a time history, *a* stands for a dilation parameter and *b* stands for a location parameter. "(*t*)" is the mother wavelet function. The Gabor's mother wavelet is applied such as follows in this study,

$$\psi(t) = \frac{1}{\sqrt{2\pi\sigma^2}} \exp\left[-\frac{t^2}{2\sigma^2}\right] \cdot e^{i\alpha t} \,, \tag{10}$$

where is a damping parameter of the mother wavelet function and 0 is a principle angular frequency. Half of the natural angular frequency of each model is applied in this study.

When the damping parameter becomes smaller, the mother function be attenuated soon. 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 scaling parameter, is individual to the resolution parameter . The parameters are consequently individual each other so that the tuning of the parameters in order to draw the wavelet contour is not difficult.

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 is 1.0.

A sampling frequency of the experimental recording has been 500 Hz, which corresponds to 2.0e-3 seconds in the sampling time. '0.4 seconds' of *b* in the time sifting for the wavelet analysis corresponds to 200 sampling data skipping. In addition, the shortest natural period of cross-flow vibration in the experimental models in Tables 1a) and 1b) is 0.86 seconds. If the bi-harmonic vibration in VIV in this case occurs, the vibration period is 0.43 seconds. The resolution would be thereby enough 0.4 seconds. Using *b*=0.4, variation pattern of the wavelet would be able to be reproduced. The step of *a* is now 0.2. The parameter *a* corresponds to a resolution of the frequency component. The cross-flow vibration appears relatively simply from the FFT analysis so that the resolution of 0.2 may be reasonable.
