**5.3 Wavelet characteristics of double cylinders in tandem**

Results of the vertical bending moment corresponding to VIV with the wavelet analysis are discussed using the results in case of the model 1 in Table 1-b). Correspondence between the orbit and the wavelet pattern was similar to cases of the single cylinder. When the distance ratio increased, VIV behavior resembled the single cases. Therefore the results in case of the distance ratio *s*=2.0 are described here.

Figures 8 to 13 show results of the wavelet analysis, the orbit and a power spectrum of the vertical bending moment. Representations of "front" and "back" in the figures mean follows. The cylinder set on a side of starting direction of the forced oscillation corresponds "front" such as Fig. 3. "*f*" is a frequency Hz and "*f*0" is also a frequency Hz of the forced oscillation on the horizontal axis of the power spectra.

In Figs. 8, the behavior of vibration of both the cylinder is quite different. VIV is not induced strongly. VIV lock-in is confirmed in Figs. 9 to 12. However the behavior of the vibration is different each case from the orbit, even then a shape of the power spectrum of Figs. 9 to 12 are same very much. VIV occurs as bi-harmonic vibration to the frequency of the forced oscillation. The third harmonic vibration can be found on the power spectrum in Figs. 12-a). When the character of "8" is broken and becomes flat, a response component which is higher than the bi-harmonic frequency emerges.

Fig. 7. Patterns of wavelet analysis of Case 3

distance ratio *s*=2.0 are described here.

**5.3 Wavelet characteristics of double cylinders in tandem** 

oscillation on the horizontal axis of the power spectra.

higher than the bi-harmonic frequency emerges.

Results of the vertical bending moment corresponding to VIV with the wavelet analysis are discussed using the results in case of the model 1 in Table 1-b). Correspondence between the orbit and the wavelet pattern was similar to cases of the single cylinder. When the distance ratio increased, VIV behavior resembled the single cases. Therefore the results in case of the

Figures 8 to 13 show results of the wavelet analysis, the orbit and a power spectrum of the vertical bending moment. Representations of "front" and "back" in the figures mean follows. The cylinder set on a side of starting direction of the forced oscillation corresponds "front" such as Fig. 3. "*f*" is a frequency Hz and "*f*0" is also a frequency Hz of the forced

In Figs. 8, the behavior of vibration of both the cylinder is quite different. VIV is not induced strongly. VIV lock-in is confirmed in Figs. 9 to 12. However the behavior of the vibration is different each case from the orbit, even then a shape of the power spectrum of Figs. 9 to 12 are same very much. VIV occurs as bi-harmonic vibration to the frequency of the forced oscillation. The third harmonic vibration can be found on the power spectrum in Figs. 12-a). When the character of "8" is broken and becomes flat, a response component which is

a) on front cylinder

Fig. 8. Comparisons of orbit, power spectrum and wavelet pattern of vertical bending moment in case of *s*=2 and T=1.2 s

Application of Wavelet Analysis for the Understanding of Vortex-Induced Vibration 607

3.0 [102 ]

(kgf-cm)2sec

1.0

0

2.0

a) on front cylinder

*t* **second** 

0 10.0 20.0 30.0 40.0 50.0 60.0

3.0 [102 ]

(kgf-cm)2sec

1.0

0

2.0

1.0 2.0 3.0 4.0 5.0

bending moment

1.0 2.0 3.0 4.0 5.0

bending moment

*f/f0*

*f/f0*

b) on back cylinder

*t* **second** 

Fig. 10. Comparisons of orbit, power spectrum and wavelet pattern of vertical bending

0 10.0 20.0 30.0 40.0 50.0 60.0

moment in case of *s*=2 and T=2.2 s

4.0 3.0 2.0 1.0

*a* 

4.0 3.0 2.0 1.0

*a* 

a) on front cylinder

Fig. 9. Comparisons of orbit, power spectrum and wavelet pattern of vertical bending moment in case of *s*=2 and T=2.0 s

1.5 [102 ]

(kgf-cm)2sec

0.5

0

1.0

a) on front cylinder

*t* **second** 

0 10.0 20.0 30.0 40.0 50.0 60.0

0.2 0.4 0.6 0.8 1.0 [102 ]

(kgf-cm)2sec

0

1.0 2.0 3.0 4.0 5.0

1.0 2.0 3.0 4.0 5.0

bending moment

*f/f0*

bending moment

*f/f0*

b) on back cylinder

*t* **second** 

Fig. 9. Comparisons of orbit, power spectrum and wavelet pattern of vertical bending

0 10.0 20.0 30.0 40.0 50.0 60.0

moment in case of *s*=2 and T=2.0 s

4.0 3.0 2.0 1.0

4.0 3.0 2.0 1.0

*a* 

*a* 

a) on front cylinder

Fig. 10. Comparisons of orbit, power spectrum and wavelet pattern of vertical bending moment in case of *s*=2 and T=2.2 s

Application of Wavelet Analysis for the Understanding of Vortex-Induced Vibration 609

1.0 2.0 3.0 4.0 5.0 [101 ]

(kgf-cm)2sec

0

a) on front cylinder

*t* **second** 

0 10.0 20.0 30.0 40.0 50.0 60.0

8.0 [101 ]

(kgf-cm)2sec

2.0

0

4.0

6.0

1.0 2.0 3.0 4.0 5.0

bending moment

1.0 2.0 3.0 4.0 5.0

bending moment

*f/f0*

*f/f0*

b) on back cylinder

*t* **second** 

Fig. 12. Comparisons of orbit, power spectrum and wavelet pattern of vertical bending

0 10.0 20.0 30.0 40.0 50.0 60.0

moment in case of *s*=2 and T=2.8 s

4.0 3.0 2.0 1.0

4.0 3.0 2.0 1.0

*a* 

*a* 

a) on front cylinder

Fig. 11. Comparisons of orbit, power spectrum and wavelet pattern of vertical bending moment in case of *s*=2 and T=2.4 s

2.0 [102 ]

(kgf-cm)2sec

1.0

0

a) on front cylinder

*t* **second** 

0 10.0 20.0 30.0 40.0 50.0 60.0

2.0 [102 ]

(kgf-cm)2sec

1.0

0

1.0 2.0 3.0 4.0 5.0

bending moment

1.0 2.0 3.0 4.0 5.0

bending moment

*f/f0*

*f/f0*

b) on back cylinder

*t* **second** 

Fig. 11. Comparisons of orbit, power spectrum and wavelet pattern of vertical bending

0 10.0 20.0 30.0 40.0 50.0 60.0

moment in case of *s*=2 and T=2.4 s

4.0 3.0 2.0 1.0

4.0 3.0 2.0 1.0

*a* 

*a* 

a) on front cylinder

Fig. 12. Comparisons of orbit, power spectrum and wavelet pattern of vertical bending moment in case of *s*=2 and T=2.8 s

Application of Wavelet Analysis for the Understanding of Vortex-Induced Vibration 611

Above discussions can be explained from results of the orbits and the power spectra. However all of the wavelet patterns, comparing with results of each period of the forced oscillation, are clearly different. In particular, difference of the VIV behavior cannot be understood from only the power spectra drawing the bi-harmonic vibration such as Figs. 12 to 15. We can found the unique striped pattern on the wavelet contours. At first we notice that the striped pattern gets blurred when the orbit is more complex. When the stripe becomes clearer, in range from 1.5 to 2.5 in *a*, the orbit varies from the *Net* type to the 8 type through the *U* type. From the wavelet pattern, we can know vibration behavior of the cylinder including cross-flow and in-line vibration. In Figs. 16, it can seem that not only the third order but also the fourth order vibration component appear. At around 2.5 in *a*, the wavelet stripes of Fig. 16 get blurrier than that of Fig. 15. From these, it can be considered that a result with the wavelet analysis is higher resolution or more sensitive than that with the FFT analysis to frequency components. Therefore detail of vibration behaviors of cylindrical structures with VIV can be investigated by using the wavelet

In this paper, the wavelet transform was applied to the analysis of time histories of vibration of circular cylinders with the vortex induced vibration. From the results, the summary is as

 The orbit pattern of the cylinder roughly corresponds to the unique pattern of the wavelet contour. Therefore the vibration behavior can be known from time history data

of arbitral vibration with the wavelet analysis. However calibration is necessary. Results with the wavelet analysis are more sensitive than that with the FFT analysis to

 When VIV lock-in occurs, the pattern of the wavelet contour becomes to clear stripes. The Gabor's mother wavelet function is useful for analysis of VIV. In addition, the

Ikoma, T.; Masuda, K.; Maeda, H. & Hanazawa, S. (2007) *Behaviors of Drag and Inertia* 

Shi, C.; Manuel, L.; Tognarelli, M.A. & Botros, T. (2010) *On the Vortex-Induced Vibration* 

Proceedings of OMAE'10, CD-ROM OMAE2010-20991, ASME

*Coefficients of Circular Cylinders under Vortex-induced Vibrations with Forced Oscillation Tests in Still Water*, Proceedings of OMAE'07, CD-ROM OMAE2007-29473, ASME Khalak, A. & Williamson C.H.K. (1999) *Motions, Forces and Mode Transitions in Vortex-induced vibration at low mass-damping*, Journal of Fluids and Structures, Vol.13, pp.813-851 Masuda, K.; Ikoma, T.; Kondo, N. & Maeda, H. (2006) *Forced Oscillation Experiments for VIV* 

*of Circular Cylinders and Behaviors of VIV and Lock-in Phenomenon*, Proceedings of

*Response of a Model Riser and Lacatin of Sensors for Fatigue Damage Prediction*,

wavelet transform analysis is effective in order to investigate VIV detail.

OMAE'06, CD-ROM OMAE2006-92073, ASME

transform analysis.

frequency resolution.

**6. Conclusion** 

**7. References** 

follows:

a) on front cylinder

Fig. 13. Comparisons of orbit, power spectrum and wavelet pattern of vertical bending moment in case of *s*=2 and T=3.45 s

Above discussions can be explained from results of the orbits and the power spectra. However all of the wavelet patterns, comparing with results of each period of the forced oscillation, are clearly different. In particular, difference of the VIV behavior cannot be understood from only the power spectra drawing the bi-harmonic vibration such as Figs. 12 to 15. We can found the unique striped pattern on the wavelet contours. At first we notice that the striped pattern gets blurred when the orbit is more complex. When the stripe becomes clearer, in range from 1.5 to 2.5 in *a*, the orbit varies from the *Net* type to the 8 type through the *U* type. From the wavelet pattern, we can know vibration behavior of the cylinder including cross-flow and in-line vibration. In Figs. 16, it can seem that not only the third order but also the fourth order vibration component appear. At around 2.5 in *a*, the wavelet stripes of Fig. 16 get blurrier than that of Fig. 15. From these, it can be considered that a result with the wavelet analysis is higher resolution or more sensitive than that with the FFT analysis to frequency components. Therefore detail of vibration behaviors of cylindrical structures with VIV can be investigated by using the wavelet transform analysis.
