# On the Added Modal Coefficients of a Rotating Submerged Cylinder Induced by a Whirling Motion—Part 1: Experimental Investigation

^{*}

## Abstract

**:**

## 1. Introduction

^{4}to 5.1 × 10

^{5}for rotating speed ratios from 0 to 2. They concluded that the evolution of the drag coefficient and the Magnus effect depended on the flow state. For instance, in the subcritical flow state, the drag coefficient decreased at rotating speed ratios below one whilst it increased at rotating speed ratios above one whenever the speed ratio was increased. On the other hand, the Magnus effect continuously increased.

## 2. Experimental Setup

#### 2.1. Test Rig

#### 2.2. Instrumentation

^{−4}mm and the total uncertainty was ±0.09 mm when measuring the maximum displacement of 30 mm. Similarly, the pressure sensors had a linearity uncertainty of 0.1%, a repeatability uncertainty of 0.11%, and an accuracy of 0.1%. Assuming again a confidence interval of 95%, the random uncertainty of the pressure sensors was ±0.03 mbar and the total uncertainty was ±0.77 mbar when measuring the maximum pressure of 200 mbar.

## 3. Mathematical Model

#### 3.1. Non-Rotating Cylinder in Air

#### 3.2. Cylinder Rotating in Air

#### 3.3. Non-Rotating Cylinder in Water

#### 3.4. Cylinder Rotating in Water

## 4. Experimental Methodology

#### 4.1. Measurement of Natural Frequencies, Damping Ratios, and Whirling Directions during Steady Conditions

#### 4.2. Fluid-Added Modal Coefficient Calculation during Steady Conditions

#### 4.3. Fluid-Added Modal Coefficient Assessment during Transient Conditions

_{max}, with a resolution of 0.016 Hz, as shown in Figure 8b, using a Hanning window with an FFT block size of 50,000 points and a sliding time segment with a 5% time increment between two consecutive FFT blocks.

## 5. Results

#### 5.1. Natural Frequencies and Damping Ratios of the Cylinder in Air and Water during Steady Conditions

#### 5.2. Natural Frequencies, Damping Ratios, and Whirling Directions of the Cylinder in Water during Steady Conditions

#### 5.3. Fluid-Added Coefficients during Steady Conditions

#### 5.4. Lock-in between the FW Natural Frequency and the Rotating Frequency during Transient Conditions

_{max}values of P0 during ramps performed at different accelerations, with an enlargement of the frequency range from 1 to 2.4 Hz in order to visualize the lock-in amplitude in detail. The lock-in amplitude of the 1 s ramp is 180 times lower than that of the 100 s ramp, suggesting that the lock-in requires a certain amount of time to develop. The lock-in peaks during ramps with higher accelerations are wider than those observed during ramps with lower accelerations, indicating that an increase in the ramp acceleration may also imply an increase in the damping ratio. The reduction in lock-in amplitudes observed with higher accelerations could therefore occur due to both of the following conditions: (i) the lock-in is not fully developed and (ii) the damping ratio of the system is higher.

_{max}values of P0, normalized by the lock-in amplitude, during ramps performed at different accelerations, with an enlargement of the frequency range of 1 to 2.4 Hz in order to visualize the exact lock-in frequency in detail. During ramps of 1 s, plotted in blue, the lock-in frequencies are 1.77 Hz for downward ramps and 1.87 Hz for upward ramps. However, during ramps of 100 s, plotted in black, the lock-in frequencies are 2.12 Hz for downward ramps and 2.18 Hz for upward ramps. This represents lock-in frequency increases of 19.8% for downward ramps and 16.6% for upward ramps. The decrease in the ramp acceleration results in a continuous increase in the lock-in frequency. In both upward and downward ramps, the acceleration may alter the modal coefficients of the system and consequently ${f}_{FW}{}_{f}$. Higher ramp accelerations imply a lower ${f}_{FW}{}_{f}$ and consequently a lower lock-in frequency due to an increase in the ${M}_{f}$ and/or a decrease in ${A}_{f}$ and/or ${K}_{f}$.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Lateral view of the positions of the sensors (

**a**). Section A-A view of the positions of the sensors (

**b**). D0 and P0 sensors (

**c**).

**Figure 4.**Spectrum of the D0 displacement showing the ${f}_{FW}{}_{f}$ and ${f}_{BW}{}_{f}$ when the cylinder rotates at 1.25 Hz in water.

**Figure 5.**D0 filtered signal (blue) and curve-fitting signal (orange) of the BW when the cylinder rotates at 2.5 Hz in water.

**Figure 6.**Orbit plot corresponding to the FW when the cylinder rotates at 1.25 Hz in water. The red dot indicates the beginning of the orbit.

**Figure 8.**Waterfall obtained from the STFT analysis of a ramp-up test with a duration of 100 s (

**a**). Maximum spectrum of the STFT (

**b**).

**Figure 9.**Natural frequencies in air (black lines) and in water (red lines) as a function of $\Omega $.

**Figure 10.**D0 filtered signal (blue) and curve-fitting signal (orange) of the response of ${q}_{1}$ to a push release in air (

**a**) and water (

**b**).

**Figure 11.**Natural frequencies of FW and BW in water as well as $\Omega $ as a function of $\Omega $.

**Figure 12.**D0 filtered signal (blue) and curve-fitting signal (orange) of the response of the BW (

**a**) and FW (

**b**) to push release when the cylinder rotates at 0.42 Hz.

**Figure 13.**${\xi}_{BW}{}_{f}$ (in black) and ${\xi}_{FW}{}_{f}$ (in red) as a function of $\Omega $.

**Figure 14.**Orbits corresponding to the FW (

**a**) and BW (

**b**) at $\Omega $ of 1.25 and to the FW (

**c**) and BW (

**d**) at $\Omega $ of 5.4 Hz. The red dot indicates the beginning of the orbit.

**Figure 15.**Evolution of ${A}_{f}$ (in black) and $\overline{{A}_{f}}$ (in red) (

**a**) and ${K}_{f}$ (

**b**) as a function of $\Omega $.

**Figure 16.**Colormap of the P0 signal during an upward ramp (

**a**) and a downward ramp (

**b**) of the rotating frequency of the cylinder, accelerating from 0 to 6.25 Hz with a duration of 375 s.

**Figure 17.**Spectra of upward ramps (in black) and downward ramps (in red) performed with durations of 1, 2.5, 10, 50 and 100 s (

**a**) and with durations of 1, 2.5, 5 and 10 s (

**b**) with an enlargement of the frequency range from 1 to 2.4 Hz.

**Figure 18.**Spectra of upward ramps (

**a**) and downward ramps (

**b**), normalized by the maximum lock-in amplitude, performed with durations of 1, 2.5, 10, 50, and 100 s with an enlargement of the frequency range from 1 to 2.4 Hz.

**Table 1.**Mesh sensitivity study of the numerical structural modal analysis with $\u2206{f}_{s}$ (%) indicating a reduction in the natural frequency relative to that calculated using the previous mesh.

Node Number (-) | 88,073 | 135,514 | 359,954 | 567,761 | 1,254,242 | 2,771,720 | 3,427,578 |

${f}_{s}$ (Hz) | 3.461 | 3.385 | 3.303 | 3.300 | 3.299 | 3.298 | 3.298 |

$\u2206{f}_{s}$ (%) | −2.20 | −2.42 | −0.09 | −0.03 | −0.03 | 0 |

**Table 2.**Values of ${M}_{f}$, $\overline{{M}_{f}}$, ${C}_{f}$, $\overline{{C}_{f}}$, ${A}_{f}$, $\overline{{A}_{f}}$, ${K}_{f}$, and ${B}_{f}$ when the cylinder is not rotating and it is oscillating at 1.2 Hz in water.

${\mathit{M}}_{\mathit{f}}$ (kg) | $\overline{{\mathit{M}}_{\mathit{f}}}$ (kg) | ${\mathit{C}}_{\mathit{f}}$ (Ns/m) | $\overline{{\mathit{C}}_{\mathit{f}}}$ (Ns/m) | ${\mathit{A}}_{\mathit{f}}$ (Ns/m) | $\overline{{\mathit{A}}_{\mathit{f}}}$ (Ns/m) | ${\mathit{K}}_{\mathit{f}}$ (N/m) | ${\mathit{B}}_{\mathit{f}}$ (N/m) |
---|---|---|---|---|---|---|---|

13.1 | 38 | 4.23 | 11.9 | 0 | 0 | 0 | 0 |

$\mathit{\Omega}$ (Hz) | ${\mathit{C}}_{\mathit{f}}$ (Ns/m) | ${\mathit{B}}_{\mathit{f}}$ (N/m) |
---|---|---|

0 | 4.23 | 0 |

0.4 | 5.45 | 5.76 |

1.25 | 9.26 | 24 |

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**MDPI and ACS Style**

Roig, R.; Sánchez-Botello, X.; Jou, E.; Escaler, X.
On the Added Modal Coefficients of a Rotating Submerged Cylinder Induced by a Whirling Motion—Part 1: Experimental Investigation. *J. Mar. Sci. Eng.* **2023**, *11*, 1758.
https://doi.org/10.3390/jmse11091758

**AMA Style**

Roig R, Sánchez-Botello X, Jou E, Escaler X.
On the Added Modal Coefficients of a Rotating Submerged Cylinder Induced by a Whirling Motion—Part 1: Experimental Investigation. *Journal of Marine Science and Engineering*. 2023; 11(9):1758.
https://doi.org/10.3390/jmse11091758

**Chicago/Turabian Style**

Roig, Rafel, Xavier Sánchez-Botello, Esteve Jou, and Xavier Escaler.
2023. "On the Added Modal Coefficients of a Rotating Submerged Cylinder Induced by a Whirling Motion—Part 1: Experimental Investigation" *Journal of Marine Science and Engineering* 11, no. 9: 1758.
https://doi.org/10.3390/jmse11091758