# Dynamic Characteristics of a Segmented Supercritical Driveline with Flexible Couplings and Dry Friction Dampers

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## Abstract

**:**

## 1. Introduction

## 2. The Mathematical Formulation for Segmented Helicopter Driveline with Flexible Coupling

_{r}Y

_{r}Z

_{r}is the rotating coordinate frame with X

_{r}coinciding with X, while Y

_{r}and Z

_{r}rotate about the X

_{r}axis at the same angular velocity as the shaft. $u(x,t),v(x,t),w(x,t)$ are dynamic deflection in the X, Y, and Z axial directions in the inertia-fixed coordinate system caused by the eccentricity of the shaft and external load produces. ${u}_{s}(x,t),{v}_{s}(x,t),{w}_{s}(x,t)$ are dynamic deflection in the rotating coordinate system. The torque and inertia induce twist are ${\phi}_{1}(x,t),{\phi}_{2}(x,t)$ along X. The shafts are connected with flexible diaphragm couplings.

#### 2.1. Slender Flexible Shaft with Viscous Internal Damping

_{L}, and the speed variation caused by a misalignment between the shafts is small, i.e., ${\phi}_{1}(x,t)={\phi}_{2}(x,t)={\phi}_{r}(t)=\Omega t+\varphi (x,t)$. J and N can be moved to both ends of two intermediate shafts. The sleeve or disc in the shaft moves with the shaft segment, and its length is negligible relative to the shaft. According to the disk kinetic energy presented in ref. [5], the vibration kinetic energy expression for a sleeve or disc, or a short segment in the shaft, is

#### 2.2. Flexible Diaphragm Couplings Subject to Misalignment

_{cp}is the distance from the diaphragm center to the bolt. ${\phi}_{Lcp}$ is the rotation angle of bolt hole A as it spins around central point O′. We have

## 3. Vibration Suppression of the Dry Friction Damper and Equations of Motion

#### 3.1. Multiple Stages

#### 3.2. Dual Rub-Impact Model

#### 3.2.1. The First Rub-Impact with Variable DOFs

#### 3.2.2. Radical Impact Stiffness of the First Rub Impact

- Linearization of the local surface stiffness

- 2.
- Impact stiffness of the damping ring

#### 3.2.3. The Second Rub-Impact with Nonlinear Restricted Stiffness

#### 3.3. Equations of Motion

## 4. Simulations and Results

#### 4.1. Dynamics of Multiple Vibration Suppression

_{h}) and ring are investigated in detail. To eliminate the interference of misalignment, the static angular and parallel misalignment is set to 0. The natural frequencies of the two shafts are close to each other, and the first critical speed is delayed due to the restriction of the damping ring. The analysis in this part is at a speed that is slightly later than the natural frequency.

#### 4.2. Misalignment Effect of Flexible Diaphragm Coupling

#### 4.2.1. Angular and Parallel Misalignment

#### 4.2.2. Static Misalignment

#### 4.3. Coeffect of Misalignment and Vibration Suppression

## 5. Experimental Verification

^{4}Hz sampling frequency, and the results of the data analysis are shown in Figure 24.

## 6. Conclusions

- (1)
- The vibration response in every vibration suppression stage is analyzed. The vibration suppression of the two dampers is not synchronized for the same parameter settings. Single rub impact occurs in the 1st and 2nd stage, dual rub-impact with interaction occurs in the 3rd stage. The amplitudes of shaft 1, shaft 2, damper 1, and damper 2 have step increases and step decreases. Full annual rub between the sleeve and damping ring occurs in the 1st and 3rd stages, while partial rub occurs in the 2nd stage. The amplitude bifurcation spanning the critical speed indicates that the transformation conditions are consistent with the theoretical analysis. The analytical model developed in this work can reflect the practical system. Even if the damper is more greatly affected by rub impact than the shaft, the degree of chaos is mild due to the low friction coefficient on its surface.
- (2)
- Parallel misalignment and angular misalignment result in 2nd and 3rd harmonic frequencies, respectively. In addition, they are not intercoupled with each other. Resonances also appear at the $1/3,1/2$ first critical speed due to misalignment. The vibration energy in the case of only static misalignment is smaller than that coexisting of static and dynamic misalignment, but the characteristics are similar.
- (3)
- In the case of the coexistence of rub impact and misalignment, both of them can stimulate each other and increase their components relative to the eccentricity. However, misalignment still accounts for most of the frequency spectrum of the shaft. In addition, more severe instability and more serious rub impact can be demonstrated from more high-frequency components appearing in the damper.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

Nomenclature | |||

$\rho ,E$ | density, elastic modulus | $\delta $ | clearance |

I | cross-sectional transverse moment of inertia | $T,D,V$ | kinetic, dissipation, and strain energy |

$K$ | stiffness | J | lumped inertia |

$L$ | length | $\gamma $ | unit pulse function |

$u,v,w$ | vibration displacement in X Y Z | $\mathit{F}$ | force, vector |

$F$ | force, scalar | $\mathit{Q}$ | amplitude vector of deflection |

N | torque | $\upsilon $ | poisson’s ratio |

$c$ | damping coefficient | $\vartheta $ | angular misalignment |

$\Omega ,\varphi $ | rotating speed and rotation angle of the shaft | $\alpha $ | the angle between the tangent of contact point A in the fillet on the plate and the vertical line |

$\phi $ | the rotation angle of rotating coordinate | $\epsilon $ | the ratio of static friction to dynamic friction coefficient on the surface of the ring |

$O$ | center | $\Phi $ | modal function |

R | radius | $\theta $ | the angle around Z-axis |

$e$ | eccentricity | $\Theta $ | Heaviside function, |

$m$ | quality | ${\omega}_{w}$ | whirling angular velocity of the shaft |

${\mu}_{dh}$ | the friction coefficient between the damping hole and sleeve | $d,{r}_{m}$ | the distance from pipe to contact point A and arc radius of the fillet on the plate. |

${\mu}_{1},{\mu}_{2}$ | static friction coefficients between the ring and plate, the ring and base | ${\mu}_{3},{\mu}_{4},{\mu}_{5}$ | friction coefficients between bushing and plate, bushing and base, pipe and plate, respectively. |

Subscript | |||

1 or 2 | shaft 1 or shaft 2 | I, II, III | the 1st, 2nd,3rd stage |

st | static misalignment | cp | coupling |

X Z Y | in X, Z, Y direction | r | modal number of the shaft |

vi | viscous internal damping | n | bearing block |

h | sleeve | rt | in the rotating coordinate frame |

s | shaft | d | damper |

N0 | pre-tightening | sp | spring |

$\varphi $ | torsion around X-axis | cn | contact |

N | normal direction | T | tangential direction |

a | angular misalignment | p | parallel misalignment |

## Appendix A

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**Figure 2.**Profile of segmented supercritical driveline with flexible diaphragm couplings and dry friction dampers.

**Figure 5.**Three damping stages of the damper (the diameters of the shaft and sleeve are exaggeratedly reduced). Detail: The chamfer on the bushing is in contact with the fillet on the plate.

**Figure 6.**Rub impact between the sleeve and damper ring, (

**left**) the first stage, (

**right**) the third stage. Detail: The chamfer on the sleeve impacts the fillet on the plate.

**Figure 7.**The deformation of the dry friction damper when the collision point is at the top of the damping ring obtained by the FEA method.

**Figure 9.**Displacement orbits and time-domain waveforms at the Z-axis of (

**a**) shaft 1 and (

**b**) shaft 2, ${F}_{pr}=200N,1.08{\omega}_{n1}$.

**Figure 10.**Displacement orbits, frequency spectrum, Poincaré map and time-domain waveform at the Z-axis: (

**a**) shaft 1, (

**b**) shaft 2, (

**c**) damper 1, and (

**d**) damper 2, ${F}_{pr}=110N,1.08{\omega}_{n1}$.

**Figure 11.**Displacement orbits and frequency spectra of (

**a**) shaft 1 and shaft 2, (

**b**) damper 1 and damper 2, ${F}_{pr}=20N,1.08{\omega}_{n1}$.

**Figure 12.**The amplitudes of (

**a**) shaft 1 and damper 1 and (

**b**) shaft 2 and damper 2, ${F}_{pr}=20N$. (s−shaft, d−damper, s0—no contact stage, s1—the 1st stage, s2—the 2nd stage, s3—3rd stage, r1—the first rub−impact, r2—the second rub−impact).

**Figure 13.**Bifurcation diagrams of (

**a**) shaft 1 and (

**b**) damper 1 in the Z-axis direction, ${F}_{pr}=110N$.

**Figure 14.**Bifurcation diagrams of (

**a**) shaft 2 and (

**b**) damper 2 in the Z-axis direction, ${F}_{pr}=110N$.

**Figure 15.**Vibration with parallel misalignment ${v}_{i}=0.1\mathrm{mm},{w}_{i}=0.2\mathrm{mm}$ of shaft 1 at $5{\omega}_{n1}/6$.

**Figure 16.**Vibration with angular misalignment ${\vartheta}_{Zi}=0.005\mathrm{rad},{\vartheta}_{Yi}=0.01\mathrm{rad}$ of shaft 1 at $5{\omega}_{n1}/6$.

**Figure 17.**Vibration with angular and parallel misalignment ${v}_{i}=0.1\mathrm{mm},{w}_{i}=0.2\mathrm{mm}$ and ${\vartheta}_{Zi}=0.005\mathrm{rad},{\vartheta}_{Yi}=0.01\mathrm{rad}$ of shaft 1 at $5{\omega}_{n1}/6$.

**Figure 18.**Resonance of shaft 1 at (

**a**) ${\omega}_{n1}/3$ and (

**b**) ${\omega}_{n1}/2$ due to parallel and angular misalignment with ${v}_{i}=0.2\mathrm{mm},{w}_{i}=0.4\mathrm{mm}$ and ${\vartheta}_{Zi}=0.007\mathrm{rad},{\vartheta}_{Yi}=0.014\mathrm{rad}$.

**Figure 19.**Vibration with static angular and parallel misalignment for shaft 1 at $5{\omega}_{n1}/6$.

**Figure 20.**Displacement orbits, Poincaré map and time-domain waveform, power spectrum or frequency spectrum at the Z-axis for (

**a**) shaft 1 (

**b**) shaft 2 (

**c**) damper 1 and (

**d**) damper 2, where ${F}_{pr}=200N$, ${v}_{i}=0.1\mathrm{mm},{w}_{i}=0.2\mathrm{mm}$, ${\vartheta}_{Zi}=0.007\mathrm{rad},{\vartheta}_{Yi}=0.014\mathrm{rad}$ at $1.08{\omega}_{n1}$.

**Figure 21.**Waterfall plot of the vibration suppression of (

**a**) shaft 1 and (

**b**) damper 1 for ${F}_{N0}=200N$.

**Figure 22.**Waterfall plot of the misalignment of shaft 1 for ${v}_{i}=0.1\mathrm{mm},{w}_{i}=0.2\mathrm{mm}$ ${\vartheta}_{Zi}=0.005\mathrm{rad},{\vartheta}_{Yi}=0.01\mathrm{rad}$.

**Figure 23.**Waterfall plot of the coeffect of misalignment and vibration suppression of (

**a**) shaft 1 and (

**b**) damper 1 for ${v}_{i}=0.1\mathrm{mm},{w}_{i}=0.2\mathrm{mm}$ ${\vartheta}_{Zi}=0.005\mathrm{rad},{\vartheta}_{Yi}=0.01\mathrm{rad}$ ${F}_{N0}=200N$.

**Figure 25.**Waterfall plot of (

**a**) misalignment, (

**b**) vibration suppression, and (

**c**) coeffect of misalignment and vibration suppression from the Z-axis direction of shaft 2.

$\mathit{\theta}$ | 0 | 90 | 180 | 270 |
---|---|---|---|---|

${K}_{de\theta}$ | 1.9 | 1 | 1.95 | 4.6 |

Parameter | Value/Unit | Parameter | Value/Unit |
---|---|---|---|

${L}_{1},{L}_{2}$ | 3653, 3514 mm | ${L}_{s1},{L}_{s2}$ | 1965, 1757 mm |

${L}_{c1},{L}_{c2}$ | 3642, 11 mm | ${L}_{n1},{L}_{n2}$ | 76, 76 mm |

${E}_{1},{E}_{2}$ | 72 GPa | ${A}_{1},{A}_{2}$ | 747 mm^{2} |

${c}_{s}$ | 0.0001 N s/m | ${I}_{1},{I}_{2}$ | 1.633 × 10^{6} mm^{4} |

${\rho}_{sh}$ | 7850 kg/M^{3} | ${\rho}_{1},{\rho}_{2}$ | 2700 kg/m^{3} |

${e}_{r1},{e}_{r2}$ | 0.05%, 0.04% | $a$ | 7.47 × 10^{−4} m^{2} |

${\delta}_{1,1},{\delta}_{1,2}$ | 2 mm | ${\delta}_{1,2},{\delta}_{2,2}$ | 1.44 mm |

${C}_{d1},{C}_{d2}$ | 160 N s/m | ${R}_{s}$ | 0.06705 m |

${K}_{n1},{K}_{n2}$ | 150 kN/m | ${m}_{d1},{m}_{d2}$ | 0.4925 kg |

${K}_{sp}$ | 23.17 kN/m | ${m}_{h1},{m}_{h2}$ | 0.3033 kg |

${m}_{eh1},{m}_{eh1}$ | 0 | ${e}_{h1},{e}_{h2}$ | 0 |

${c}_{f}$ | 0.8 | ${I}_{sz1}$ | 2.055 × 10^{−3} m^{4} m^{2} |

${E}_{h}$ | 1.7 × 10^{11} Pa | ${\upsilon}_{h}^{}$ | 0.3 |

${R}_{h}$ | 70 mm | ${E}_{d}$ | 9 × 10^{8} Pa |

${\upsilon}_{d}$ | 0.4 | ${R}_{d}$ | 72 mm |

${k}_{p}$ | 1 × 10^{4} kN/m | ${k}_{a}$ | 1.1 × 10^{4} kN m/rad |

${\mu}_{1},{\mu}_{2},{\mu}_{3},{\mu}_{4},{\mu}_{5},{\mu}_{hd}$ | 0.19, 0.19, 0.25, 0.2, 0.2, 0.15 |

Components | Material Composition | Value |
---|---|---|

Shaft 1 and shaft 2 | steel | Φ10 × 700 mm, Φ10 × 1000 mm, elastic modulus 211 GPa, Poisson’s ratio 0.31, and internal damping 0.001 N s/m |

Disc and sleeve | steel | Φ78 × 34 mm, Φ16 × 20 mm, |

Hexagon diaphragm coupling | steel | 8 × 10^{4} N/m 6 × 10^{4} N m/rad |

damping ring | graphite, POB and PTFE | R 16.2 × T 3.6 × W 5.2 mm |

Bearing | Left and right: stiffness 40 kN/m, damping 25 N s/m | |

Unbalance in the disc | Left and right: 2.5 g, eccentricity distance 35 mm | |

motor and control | constant acceleration from 10 to 2000 rpm in 20 s |

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**MDPI and ACS Style**

Huang, Z.; Tan, J.; Liu, C.; Lu, X.
Dynamic Characteristics of a Segmented Supercritical Driveline with Flexible Couplings and Dry Friction Dampers. *Symmetry* **2021**, *13*, 281.
https://doi.org/10.3390/sym13020281

**AMA Style**

Huang Z, Tan J, Liu C, Lu X.
Dynamic Characteristics of a Segmented Supercritical Driveline with Flexible Couplings and Dry Friction Dampers. *Symmetry*. 2021; 13(2):281.
https://doi.org/10.3390/sym13020281

**Chicago/Turabian Style**

Huang, Zhonghe, Jianping Tan, Chuliang Liu, and Xiong Lu.
2021. "Dynamic Characteristics of a Segmented Supercritical Driveline with Flexible Couplings and Dry Friction Dampers" *Symmetry* 13, no. 2: 281.
https://doi.org/10.3390/sym13020281