# Experimental Analysis of the Space Ratio Influence on the Excitation Frequencies of One and Two Cylinders Free to Vibrate in Tandem Arrangement

^{*}

## Abstract

**:**

^{3}to 2.4 × 10

^{4}. Fourier transform, continuous wavelet transform, magnitude-square coherence, and wavelet coherence were applied to analyze the cylinder acceleration results for all L/D and wake velocity values studied. The results show that the amplitudes of vibration are below 1.5% of the diameter for all the cases, except for the lower L/D, where the amplitude increases. The first cylinder free to vibrate presents the highest amplitudes observed. Fourier and continuous wavelet analysis showed high energy associated with the two natural frequencies of the system and a third frequency, which may be associated with the flow excitation. In the second cylinder free to vibrate, energy spreads across the monitored spectrum, justifying the smaller amplitudes but the energy level increases with increasing L/D and may be associated with wake-induced vibration. The cases with both cylinders free to vibrate show that the relation between the assembly parameters of each cylinder is relevant to the vibration response and the excitation frequency range. The results showed that even with a clear excitation in a higher frequency, the main energy in the vibration signals is in the natural frequency range.

## 1. Introduction

## 2. Materials and Methods

^{3}and 2.4 × 10

^{4}. Details about the aerodynamic channel are presented in Figure 2a.

_{n1}= 7.8 Hz and f

_{n2}= 21.5 Hz. All the parameters were measured in the test section after the assembly of the system was free to vibrate in still air. The first frequency is related to movement of both blades to the same side and the second mode is related to the asymmetric movement of the blades. The information for each tested case is summarized in Table 1.

_{n1}, is based on the reference velocity obtained from the Pitot tube, the cylinder diameter, and the first natural frequency. The reduced velocity of the experiments presented values above 30 to explore the response over the resonance region. The uncertainties of the reduced velocities remain around 10.5%. The uncertainty associated with the frequency results is around ± 9%.

## 3. Results and Discussion

#### 3.1. Tandem Configurations—Non-Dimensional Amplitude and Main Frequencies

#### 3.2. Frequency Ranges and Flows Patterns in Tandem Configurations

#### 3.3. Tandem Configuration L/D = 1.26—Coherence between the Wake Velocity and the Acceleration

#### 3.4. Tandem Configuration L/D = 1.26—Two Cylinders Free to Vibrate

#### 3.4.1. Two Cylinders Free to Vibrate with the Same Natural Frequency Range

#### 3.4.2. Two Cylinders Free to Vibrate—The First Cylinder with Higher Natural Frequency Than the Second

#### 3.4.3. Two Cylinders Free to Vibrate—The First Cylinder with Lower Natural Frequency Than the Second

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

Symbol | Definition |

C1 | First Cylinder |

C2 | Second Cylinder |

D | Diameter [m] |

Db20 | Daubechies Wavelet |

f | Frequency [Hz] |

f_{n1} | First Natural Frequency [Hz] |

f_{n2} | Second Natural Frequency [Hz] |

g | Gravity [m/s^{2}] |

L | Longitudinal Pitch [m] |

L/D | Longitudinal Space Ratio [-] |

m | Mass per Length [kg/m] |

${m}^{*}=\frac{4m}{\rho \pi {D}^{2}}$ | Mass Ratio [-] |

Pxx | Power Spectrum |

Re = UD/v | Reynolds Number [-] |

St = fD/U | Strouhal Number [-] |

t | Time [s] |

U | Reference Velocity [m/s] |

Vr = U/(Df_{n1}) | Reduced Velocity [-] |

Y | Displacement Amplitude [m] |

Y/D | Non-Dimensional Displacement [-] |

Greek Letters | |

ζ | Damping ratio [-] |

v | Kinematic Viscosity [m^{2}/s] |

ρ | Density [kg/m^{3}] |

π | Pi Number |

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**Figure 2.**(

**a**) Aerodynamic channel (dimensions in millimeters), (

**b**) schematic detail of the assembly of the cylinder free to vibrate, and (

**c**) schematic top view of the test section with the configuration of the first cylinder and second cylinder and hot wire position.

**Figure 4.**Continuous wavelet and Fourier spectrum from acceleration signals on the cases: (

**a**) Case 2 with Vr = 72 for L/D =1.26, (

**b**) Case 3 with Vr = 72 for L/D =1.26, (

**c**) Case 4 with Vr = 72 for L/D =1.4, (

**d**) Case 5 with Vr = 72 for L/D =1.4, (

**e**) Case 6 with Vr = 72 for L/D =1.6, (

**f**) Case 7 with Vr = 72 for L/D =1.6, (

**g**) Case 8 with Vr = 72 for L/D = 3.52, (

**h**) Case 9 with Vr = 72 for L/D = 3.52.

**Figure 6.**Frequency peaks observed in the Fourier spectrum from acceleration signals for the tested cases with FV and SV in (

**a**) reduced velocity around Vr = 49, (

**b**) reduced velocity around Vr = 60, and (

**c**) reduced velocity around Vr = 70.

**Figure 7.**Continuous wavelet transforms of (

**a**) Case 10—Signal of the wake velocity behind the first cylinder free to vibrate, (

**b**) Case 11—Signal of the wake velocity behind the second cylinder free to vibrate, (

**c**) Case 10—Acceleration signal of the first cylinder free to vibrate and (

**d**) Case 11—Acceleration signal of the second cylinder free to vibrate.

**Figure 8.**(

**a**) First cylinder free to vibrate (Case 10)—Cross-coherence between wake velocity and cylinder acceleration for the reduced velocity equal to 63; (

**b**) Phase angle of Case 10. (

**c**) Second cylinder free to vibrate (Case 11)—Cross-coherence between wake velocity and cylinder acceleration for the reduced velocity equal to 63 and (

**d**) phase angle of Case 11.

**Figure 9.**Wavelet coherence between wake velocity and cylinder acceleration for the reduced velocity equal to 63. (

**a**) First cylinder free to vibrate and (

**b**) second cylinder free to vibrate.

**Figure 10.**Continuous wavelet and Fourier spectrum of acceleration signals of the Case 14 with two cylinders free to vibrate: (

**a**) reduced velocity = 35, (

**b**) reduced velocity = 46, and (

**c**) reduced velocity = 53.

**Figure 11.**(

**a**) Cross-coherence between acceleration signals from both cylinders free to vibrate for reduced velocity equal to 46, (

**b**) phase angle obtained in the cross-coherence, and (

**c**) wavelet coherence between acceleration from the first cylinder and the second cylinders free to vibrate for reduced velocity equal to 46.

**Figure 12.**Continuous wavelet and Fourier spectra of acceleration signals of the case with two cylinders free to vibrate: (

**a**) C1—first cylinder free to vibrate, Vr = 11; C2—second cylinder free to vibrate, Vr = 30; (

**b**) C1—first cylinder free to vibrate, Vr = 15; C2—second cylinder free to vibrate, Vr = 41; (

**c**) C1—first cylinder free to vibrate, Vr = 18; and C2—second cylinder free to vibrate, Vr = 52.

**Figure 13.**(

**a**) Cross-coherence of the acceleration signals from both cylinders free to vibrate for reduced velocity equal to Vr = 18 for the first cylinder and Vr = 52 for the second cylinder, (

**b**) phase angle obtained in the cross-coherence, and (

**c**) wavelet coherence of the acceleration from the first cylinder and acceleration from the second cylinders free to vibrate for equal reduced velocity.

**Figure 14.**Continuous wavelet and Fourier spectra from the acceleration signals on the case with two cylinders free to vibrate (

**a**) Vr C1 = 30, Vr C2 = 17; (

**b**) Vr C1 = 34, Vr C2 = 19; (

**c**) Vr C1 = 41, Vr C2 = 23.

**Figure 15.**(

**a**) Cross-coherence of the acceleration signals from both cylinders free to vibrate for reduced velocities equal to Vr C1 = 41 and Vr C2 = 23, (

**b**) phase angle obtained in the cross-coherence, and (

**c**) wavelet coherence of acceleration from the first cylinder and acceleration from the second cylinders free to vibrate for reduced velocities equal to Vr C1 = 41 and Vr C2 = 23.

Case | m* | ζ | L/D | f_{n1} | f_{n2} | Free to Vibrate | |
---|---|---|---|---|---|---|---|

Case 01 | Single Cylinder | 608 | 0.03 | - | 7.8 | 21.5 | SC |

Case 02 | FV-1.26 | 608 | 0.03 | 1.26 | 7.8 | 21.5 | FV |

Case 03 | SV-1.26 | 608 | 0.03 | 1.26 | 7.8 | 21.5 | SV |

Case 04 | FV-1.4 | 608 | 0.03 | 1.4 | 7.8 | 21.5 | FV |

Case 05 | SV-1.4 | 608 | 0.03 | 1.4 | 7.8 | 21.5 | SV |

Case 06 | FV-1.6 | 608 | 0.03 | 1.6 | 7.8 | 21.5 | FV |

Case 07 | SV-1.6 | 608 | 0.03 | 1.6 | 7.8 | 21.5 | SV |

Case 08 | FV-3.52 | 608 | 0.03 | 3.52 | 7.8 | 21.5 | FV |

Case 09 | SV-3.52 | 608 | 0.03 | 3.52 | 7.8 | 21.5 | SV |

Case 10 | FV-1.26-1 | 539 | 0.004 | 1.26 | 8.8 | 23.4 | FV |

Case 11 | SV-1.26-1 | 539 | 0.004 | 1.26 | 8.8 | 23.4 | SV |

Case 12-C1 | BV-C1-1.26-Config1 | 502 | 0.003 | 1.26 | 13.7 | 36.2 | BV-C1 |

Case 12-C2 | BV-C2-1.26-Config1 | 547 | 0.006 | 1.26 | 24.4 | 71 | BV-C2 |

Case 13-C1 | BV-C1-1.26-Config2 | 502 | 0.002 | 1.26 | 29.3 | 58.6 | BV-C1 |

Case 13-C2 | BV-C2-1.26-Config2 | 547 | 0.009 | 1.26 | 10.7 | 30.3 | BV-C2 |

Case 14-C1 | BV-C1-1.26-Config3 | 502 | 0.008 | 1.26 | 10.8 | 22.5 | BV-C1 |

Case 14-C2 | BV-C2-1.26-Config3 | 547 | 0.007 | 1.26 | 10.7 | 31.2 | BV-C2 |

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**MDPI and ACS Style**

Neumeister, R.F.; Petry, A.P.; Möller, S.V.
Experimental Analysis of the Space Ratio Influence on the Excitation Frequencies of One and Two Cylinders Free to Vibrate in Tandem Arrangement. *Vibration* **2022**, *5*, 770-791.
https://doi.org/10.3390/vibration5040045

**AMA Style**

Neumeister RF, Petry AP, Möller SV.
Experimental Analysis of the Space Ratio Influence on the Excitation Frequencies of One and Two Cylinders Free to Vibrate in Tandem Arrangement. *Vibration*. 2022; 5(4):770-791.
https://doi.org/10.3390/vibration5040045

**Chicago/Turabian Style**

Neumeister, Roberta Fátima, Adriane Prisco Petry, and Sergio Viçosa Möller.
2022. "Experimental Analysis of the Space Ratio Influence on the Excitation Frequencies of One and Two Cylinders Free to Vibrate in Tandem Arrangement" *Vibration* 5, no. 4: 770-791.
https://doi.org/10.3390/vibration5040045