# Nonlinear Multi-Frequency Dynamics of Wind Turbine Components with a Single-Mesh Helical Gear Train

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## Abstract

**:**

## 1. Introduction

#### 1.1. Background

#### 1.2. Literature Survey

#### 1.3. Contribution of This Study

#### 1.4. Organization of the Paper

## 2. Methods

#### 2.1. Mesh Stiffness

#### 2.2. External Excitation

#### 2.3. Generator Torque Equation

#### 2.4. Backlash Equation

#### 2.5. Rotational Equations of Motion

#### 2.6. Equations for Axial Vibrations

## 3. Results and Discussion

## 4. Conclusions

## 5. Future Work

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

${T}_{t}$, ${T}_{m}$ | the turbine and generator torques |

${I}_{i}$ | the moment of inertia of the wheels |

${m}_{i}$ | the masses of the wheels |

${\theta}_{i}$ | the angular positions of the wheels |

${K}_{l}$, ${K}_{h}$, ${K}_{2}$, ${K}_{3}$ | the low speed shaft torsional stiffness, high speed shaft torsional stiffness |

and bearing stiffness for Gear 1 and Gear 2, respectively | |

${C}_{l}$, ${C}_{h}$, ${C}_{2}$, ${C}_{3}$ | the low speed shaft damping, high speed shaft damping |

and bearing damping for Gear 1 and Gear 2, respectively | |

$\beta $ | the helical or pressure angle |

${C}_{m}$ and ${K}_{m}$ | the mesh damping and stiffness, respectively |

RK4 | Runge-Kutta of order 4 |

## References

- Perdana, A. Dynamic Models of Wind Turbines. A Contribution towards the Establishment of Standardized Models of Wind Turbines for Power System Stability Studies; Avancez: Goteborg, Sweden, 2008; pp. 29–32. ISBN 978-91-7385-226-5. [Google Scholar]
- Wasynczuk, O.; Man, D.; Sullivan, J. Dynamic behavior of a class of wind turbine generators during random wind fluctuations. IEEE Trans. Power Syst. Appar. Syst.
**1981**, PAS-100, 2837–2845. [Google Scholar] [CrossRef] - Hinrichsen, E.; Nolan, P. Dynamics and stability of wind turbine genera- tors. IEEE Trans. Power Appar. Syst.
**1982**, 101, 2640–2648. [Google Scholar] [CrossRef] - Zhao, M.; Ji, J.C. Nonlinear torsional vibrations of a wind turbine gearbox. Appl. Math. Modell.
**2015**, 39, 4928–4950. [Google Scholar] [CrossRef] - Zhao, M.; Ji, J. Dynamic Analysis of Wind Turbine Gearbox Components. Energies
**2016**, 9, 110. [Google Scholar] [CrossRef] - Yang, F.; Shi, Z.; Meng, J. Nonlinear dynamics and load sharing of double-mesh helical gear train. J. Eng. Sci. Technol. Rev.
**2013**, 6, 29–34. [Google Scholar] - Helsen, J.; Vanhollebeke, F.; Marrant, B.; Vandepitte, D.; Desmet, W. Multibody modeling of varying complexity for modal behavior analysis of wind turbine gearboxes. Renew. Energy
**2011**, 36, 3098–3113. [Google Scholar] [CrossRef] - Heege, A.; Betran, J.; Radovcic, Y. Fatigue load computation of wind turbine gearboxes by coupled finite element, multi-body system and aerodynamic analysis. Wind Energy
**2007**, 10, 395–413. [Google Scholar] [CrossRef] - Zhao, X.; Maißer, P.; Wu, J. A new multibody modeling methodology for wind turbine structures using a cardanic joint beam element. Renew. Energy
**2007**, 32, 532–546. [Google Scholar] [CrossRef] - Kim, T.; Hansen, A.M.; Branner, K. Development of an anisotropic beam finite element for composite wind turbine blades in multibody system. Renew. Energy
**2013**, 59, 172–183. [Google Scholar] [CrossRef] [Green Version] - Pappalardo, C.M. A natural absolute coordinate formulation for the kinematic and dynamic analysis of rigid multibody systems. Nonlinear Dyn.
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**2017**, 87, 1647–1665. [Google Scholar] [CrossRef] - Girsang, I.P.; Dhupia, J.S.; Eduard, M.; Mohit, S. Gearbox and Drivetrain Models to Study Dynamic Effects of Modern Wind Turbines. National Renewable Energy Laboratory (NREL). Available online: www.nrel.gov/publications (accessed on 28 February 2018).
- Parker, R.G.; Lin, J. Mesh phasing relationships in planetary and epicyclic gears. J. Mech. Des.
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**Figure 5.**A plot of wind turbine rotational speed (${\dot{\theta}}_{1}$) vs. time (t) to study the effect of higher damping and torsional stiffness. (

**a**,

**b**) are the time series and the corresponding FFT frequency spectrum plots for low damping and torsional stiffness at ${K}_{l}=1.0$, ${K}_{h}=0.01$, ${K}_{2}={K}_{3}=0.005$, ${C}_{l}=0.05$, ${C}_{h}=0.009$, ${C}_{2}={C}_{3}=0.01$, ${C}_{m}=0.001$, ${D}_{t}=0.01$, ${D}_{g}=0.009$ and $\beta =\pi /6$. In (

**c**,

**d**), the damping and torsional terms have been increased by a factor of 10.

**Figure 6.**A plot of generator rotational speed (${\dot{\theta}}_{4}$) vs. time (t) to study the effect of higher damping and torsional stiffness. (

**a**,

**b**) are the time series and the corresponding FFT frequency spectrum plots for low damping and torsional stiffness at ${K}_{l}=1.0$, ${K}_{h}=0.01$, ${K}_{2}={K}_{3}=0.005$, ${C}_{l}=0.05$, ${C}_{h}=0.009$, ${C}_{2}={C}_{3}=0.01$, ${C}_{m}=0.001$, ${D}_{t}=0.01$, ${D}_{g}=0.009$ and $\beta =\pi /6$. In (

**c**,

**d**), the damping and torsional terms have been increased by a factor of 10. The remaining parameters are fixed and presented in Table 1.

**Figure 7.**The effect of a bigger helical angle on wind turbine rotational speed (${\dot{\theta}}_{1}$). (

**a**,

**b**) are the time series and the corresponding FFT frequency spectrum plots for the case $\beta =\pi /6$. In (

**c**,

**d**), the helical angle has been set to $\beta =\pi /12$. The other parameters are fixed and listed in Table 2.

**Figure 8.**The effect of a bigger helical angle on generator rotational speed (${\dot{\theta}}_{4}$). (

**a**,

**b**) are the time series and the corresponding FFT frequency spectrum plots for the case $\beta =\pi /6$. In (

**c**,

**d**), the helical angle has been set to $\beta =\pi /12$. The other parameters are fixed and listed in Table 2.

**Figure 9.**The effect of a smaller mesh frequency on wind turbine and generator rotational speed. (

**a**,

**b**) are the turbine and generator FFT frequency spectrum respectively for ${\omega}_{m}=0.05$. In (

**c**,

**d**), the mesh frequency has been set to ${\omega}_{m}=0.5$. The other parameters used for these plots are presented in Table 2.

**Figure 10.**The effect of external excitation frequency on wind turbine rotational speed. (

**a**) is for the case ${\omega}_{i}=0.02$; and (

**b**) is for the case ${\omega}_{i}=0.2$. The other parameters used for these plots are presented in Table 2.

Name | |
---|---|

Parameters | Values |

Inertia (Kgm^{2}) | ${I}_{1}={I}_{2}={I}_{3}={I}_{4}=0.09$ |

mass | ${m}_{1}={m}_{2}={m}_{3}={m}_{4}=2.0$ |

Radius | ${r}_{1}={r}_{2}=1.0$, ${r}_{3}={r}_{4}=0.2$ |

Pressure angle | $\beta =\pi /6$ |

Torsional stiffness | ${K}_{l}=1.0$, ${K}_{h}=0.01$, |

Damping | ${K}_{2}={K}_{3}=0.005$ |

${C}_{l}=0.05$, ${C}_{h}=0.009$, | |

${C}_{2}={C}_{3}=0.01$, | |

${C}_{m}=0.001$, ${D}_{t}=0.01$, ${D}_{r}=0.009$ | |

Torque | $T\left(0\right)=50.0$, ${T}_{m}\left(0\right)=5.0$ |

Excitation frequencies | ${\omega}_{i}=0.2$, ${\omega}_{m}=0.5$ |

Name | |
---|---|

Parameters | Values |

Inertia (Kgm^{2}) | ${I}_{1}={I}_{2}={I}_{3}={I}_{4}=0.09$ |

mass | ${m}_{1}={m}_{2}={m}_{3}={m}_{4}=2.0$ |

Radius | ${r}_{1}={r}_{2}=1.0$, ${r}_{3}={r}_{4}=0.2$ |

Torsional stiffness | ${K}_{l}=10.0$, ${K}_{h}=0.1$, |

Damping | ${K}_{2}={K}_{3}=0.05$ |

${C}_{l}=0.5$, ${C}_{h}=0.09$, | |

${C}_{2}={C}_{3}=0.1$, | |

${C}_{m}=0.01$, ${D}_{t}=0.1$, ${D}_{r}=0.09$ | |

Torque | $T\left(0\right)=50.0$, ${T}_{m}\left(0\right)=5.0$ |

Excitation frequencies | ${\omega}_{i}=0.2$, ${\omega}_{m}=0.5$ |

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**MDPI and ACS Style**

Ayuketang Arreyndip, N.; Moise Dikande, A.; Joseph, E.
Nonlinear Multi-Frequency Dynamics of Wind Turbine Components with a Single-Mesh Helical Gear Train. *Math. Comput. Appl.* **2018**, *23*, 12.
https://doi.org/10.3390/mca23010012

**AMA Style**

Ayuketang Arreyndip N, Moise Dikande A, Joseph E.
Nonlinear Multi-Frequency Dynamics of Wind Turbine Components with a Single-Mesh Helical Gear Train. *Mathematical and Computational Applications*. 2018; 23(1):12.
https://doi.org/10.3390/mca23010012

**Chicago/Turabian Style**

Ayuketang Arreyndip, Nkongho, Alain Moise Dikande, and Ebobenow Joseph.
2018. "Nonlinear Multi-Frequency Dynamics of Wind Turbine Components with a Single-Mesh Helical Gear Train" *Mathematical and Computational Applications* 23, no. 1: 12.
https://doi.org/10.3390/mca23010012