# A Hybrid Multistep Procedure for the Vibroacoustic Simulation of Noise Emission from Wind Turbines

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Hybrid Analytical-Computational Procedure

#### 2.1. Angle-Dependent Mesh Stiffness (Step 1)

#### 2.2. Consideration of Time-Varying Operating Conditions (Step 2)

#### 2.3. Transient Gear Dynamics (Step 3)

#### 2.4. Structural Response and Transfer Behavior Using Dynamic Finite Element Analysis (Step 4)

#### 2.5. Response at Operating Conditions (Step 5)

## 3. Numerical Results and Experimental Validation

#### 3.1. Application of the Integrated Computational Procedure

#### 3.2. Computational Costs

^{7}DoFs in both stages. Mesh sizes were determined by examining the convergence of the rotational deflection, with a particular focus on the regions of potential contact with significantly smaller element sizes. The average duration of the calculation is about 10 minutes per angular position, using preconditioned conjugate gradient solvers and parallel computations on 16 cores. As described above, both gear stages are analyzed in 75 equidistant angular positions, with a total computational time of approx. 25 $\mathrm{h}$ for the 150 separate FE simulations. The computational time to obtain the TVMS for varying operational speeds in step 2 amounts to about $0.5$ $\mathrm{s}$. The transient simulation of the gear dynamic model in step 3 takes approx. 10 $\mathrm{s}$. The FE model of the gearbox structure in step 4 consists of 463,034 quadratic hexahedral, tetrahedral, and wedge elements with 796,539 nodes and 2.39 × 10

^{6}DoFs in total. The mesh sizes were determined here by evaluating the quantitative assurance of the operational deflection results for each frequency response with a reference model with very fine mesh size (2,227,510 nodes). This new approach will be described in more detail in a future publication. The time-harmonic analysis is performed for TE excitation at IS and HS stage with 2000 frequency increments each. The solution of the reduced FE model requires $6.70$ $\mathrm{GB}$ RAM and $40.50$ $\mathrm{min}$ of computing time, while using distributed sparse matrix direct solvers and parallel computations on 16 cores. The creation of the reduced model takes additional $26.25$ $\mathrm{min}$. In the fifth step, the calculation and output of the operational response spectra for 84 considered degrees of freedom takes about $12.09$ $\mathrm{s}$. From the above, it can be observed that the execution of the entire proposed sequential procedure has a total duration of approx. $26.80$ $\mathrm{h}$ for the presented benchmark analysis.

#### 3.3. Comparison with Measurements

#### 3.4. Influence of Varying Operating Conditions

#### 3.5. Influence of the Internal Components

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Schematic overview of the integrated computational procedure. In the first step, static FE simulations of gears in contact provide the angle-dependent mesh stiffness ${k}_{\mathrm{m}}\left(\phi \right)$. The angle-dependent mesh stiffness is then transformed to the TVMS ${k}_{\mathrm{m}}\left(t\right)$, while varying operating speeds are considered by both kinematic and kinetic effects. The TVMS ${k}_{\mathrm{m}}\left(t\right)$ is applied in an analytical two-body model of the transient gear dynamics to obtain the dynamic transmission error $\delta $. The frequency spectra of the operational gearbox response are calculated by an analytical combination of the excitation spectra $\delta \left(\omega \right)$ and transfer functions, which have been computed independently using a detailed FE model of the gearbox.

**Figure 2.**The angle-dependent variation in the mesh stiffness is a significant cause of gear vibration and depends on the specific gear properties, as the comparison for a spur gear, see subplot (

**a**), and a helical gear, see subplot (

**b**), over one mesh cycle shows. For better comparability, the values are related to the respective average value over one mesh cycle ${\overline{k}}_{\mathrm{m}}$. Spur gears generally have a low contact ratio, caused by alternating single and double tooth contact, leading to an abrupt change in the mesh stiffness. Helical gears, in contrast, generally have a high contact ratio due to multiple simultaneous tooth contacts at all times, and thereby an angle-dependent mesh stiffness variation with relative fluctuations of a smaller magnitude.

**Figure 3.**Schematic overview of the finite element setup to determine the mesh stiffness of two mating helical gears in a specific angular position within a mesh cycle.

**Figure 4.**For a sequence of n different angular positions ${\phi}_{i}$, static FE simulations of a gear stage are performed, in order to obtain a set of n corresponding mesh stiffness values ${k}_{\mathrm{m}}\left({\phi}_{i}\right)$.

**Figure 5.**The operating speed ${\Omega}_{\mathrm{d}}\left(t\right)$ during a considered time range is defined at m discrete time steps ${t}_{j}$.

**Figure 6.**Procedure to determine the time-varying mesh stiffness ${k}_{\mathrm{m}}\left(t\right)$ using ${k}_{\mathrm{m}}\left(\phi \right)$ from step 1 and time-depending operating speeds ${\Omega}_{\mathrm{d}}\in \Omega $, where $\Omega $ contains the operational speed at m time instances of a pre-defined speed profile. Therefore, the angle dependent mesh stiffness ${k}_{\mathrm{m},j}\left(\phi \right)$ is expanded cyclically to m cycles with $j=1,2,\dots ,m$. Subsequently, the angle-dependent description of the mesh stiffness can be converted into a time-dependent relationship ${k}_{\mathrm{m}}\left(t\right)$ with the kinematic relation $\frac{\partial \phi}{\partial t}={\Omega}_{\mathrm{d}}\left({t}_{j}\right)$ and the assumption of a constant rotational speed within each mesh cycle.

**Figure 7.**Schematic illustration of the conversion from angle-dependent mesh stiffness ${k}_{\mathrm{m}}\left(\phi \right)$ to time-dependent mesh stiffness ${k}_{\mathrm{m}}\left(t\right)$ for constant (lower left) and varying operating speed (lower right).

**Figure 8.**Gear dynamic model consisting of two rigid discs, representing the moments of inertia, connected at the corresponding base circle radii ${r}_{\mathrm{b}1}$ and ${r}_{\mathrm{b}2}$ by a spring element representing the mesh stiffness ${k}_{\mathrm{m}}\left(t\right)$, and a viscous damper representing the mesh damping with the damping coefficient ${c}_{\mathrm{m}}\left(t\right)$.

**Figure 9.**Numerical results for the dynamic transmission error $\delta $ for the gear system discussed by Zhou et al. [28]. Subplot (

**a**) shows the obtained transient results. Due to the constant operating speed, the TE frequency spectrum is characterized by narrow banded peaks at the gear meshing frequency orders, see subplot (

**b**).

**Figure 11.**FE model with internal components of the three-staged gearbox with a low-speed (LS) planetary stage and a intermediate speed (IS) as well as a high-speed helical stage.

**Figure 12.**Bearing approximation by ${\mathbf{K}}_{\mathrm{b}}$ to couple the relative motion of the interfaces on shaft (red) and housing (red), lumped with RBE (green).

**Figure 13.**Approximation of internal components and gear contact, excited by harmonically varying transmission error ${\delta}_{\mathrm{harm}}\left(\omega \right)$.

**Figure 14.**Approach for the dynamic response of any DoF r under operating conditions to l different gear-induced vibration sources. The dynamic transmission error spectra ${\delta}_{s}\left(\omega \right)$ of the the s-th gear stage is calculated from an analytical model of the transient gear dynamics. A finite element model of the gearbox is used for harmonic response analyses, to obtain transfer functions ${H}_{sr}(i\omega )$ from TE excitation of the s-th gear stage to the r-th DoF of the gearbox, as shown in Equation (25). The response under operating conditions of the r-th DoF ${Y}_{r}\left(\omega \right)$ is calculated as a linear combination of excitation spectra and transfer functions. In case of multiple excitation sources, the operational response spectrum is obtained by superposition of the specific source spectra.

**Figure 15.**Results for the varying mesh stiffness of the IS (

**left**) and HS gear stage (

**right**), obtained from series of static simulations using detailed FE-models of the specific gear stages.

**Figure 16.**Angular frequency spectra of the transmission error $\delta $ for the IS (

**left**) and HS stage (

**right**), with vertical lines indicating the average mesh frequency of the HS stage ${\overline{\omega}}_{\mathrm{m},\mathrm{HS}}$ and its multiples.

**Figure 17.**Response spectra are measured using accelerometers on a wind turbine gearbox during operation.

**Figure 18.**Operational shape to TE excitation at HS stage with $\omega =6300\mathrm{rad}{\mathrm{s}}^{-1}$ and selected sensor positions A-H (order in increasing distance to vibration sources).

**Figure 19.**Operating speed of the HS shaft during measurement runs with different operating conditions.

**Figure 20.**Influence of sensor position: Response spectra from measurement (grey) and simulation (black) for SPs A to H, corresponding FRAC values and dashed vertical lines for the average mesh frequency of the HS stage ${\overline{\omega}}_{\mathrm{m},\mathrm{HS}}$ and its multiples. The spectral results are compared over a frequency range of more than 10,000 $\mathrm{rad}$ ${\mathrm{s}}^{-1}$.

**Figure 21.**FRAC for the response spectra regarding the first run, considering all SPs and axes, evaluated in dependence on the DoF and frequency.

**Figure 22.**The angle-dependent mesh stiffness ${k}_{\mathrm{m}}\left(\phi \right)$, see subplot (

**a**), and the corresponding excitation spectra $\delta \left(\omega \right)$, see subplot (

**b**), of the HS stage are calculated for different loads T. The variation of the load level does not significantly influence the frequency spectrum $\delta \left(\omega \right)$ of the transmission error. (

**a**) Angle-dependent mesh stiffness ${k}_{\mathrm{m}}\left(\phi \right)$ for different torques T. (

**b**) Excitation spectra resulting from the different ${k}_{\mathrm{m}}\left(\phi \right)$ of subplot (

**a**). Vertical dashed lines indicate the average meshing frequency ${\overline{\omega}}_{\mathrm{m},\mathrm{HS}}$ and its multiples.

**Figure 23.**Influence of the operating speed: The dominant frequency peak at $2{\omega}_{\mathrm{m},\mathrm{HS}}$ for different speed profiles and comparison of measurement (grey) and hybrid approach (black) results for the response spectra at the reference position R in a frequency range of $1000\pi $ $\mathrm{rad}$ ${\mathrm{s}}^{-1}$. Of particular interest is the frequency range $2{\overline{\omega}}_{\mathrm{m},\mathrm{HS}}\pm 2{\sigma}_{\Omega}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}c$ around the peak, where $c=4\pi \frac{N}{60\mathrm{s}/\mathrm{min}}$ accounts for the transformation from operational speed to the second order meshing frequency of the HS stage. Despite a negative offset for run 1 and 2, the model results are in good agreement with the measurement results regarding the position, width and relative shape of the dominant frequency peak.

**Figure 24.**Calculation of the response of the housing to a unit bearing force with (

**top**) and without (

**bottom**) consideration of internal components.

**Figure 25.**Influence of the internal components: Dynamic transfer functions $H\left(\omega \right)$ at the sensors positions A–F for an excitation by a bearing force ${F}_{\mathrm{B}}$, calculated with (grey) and without (black) internal components. The comparison shows significant influence of the internal components. Dashed lines indicate ${\overline{\omega}}_{\mathrm{m},\mathrm{HS}}$ and its multiples.

**Table 1.**Computational time required for each step of the integrated procedure for an application to a $2.5$ $\mathrm{M}$$\mathrm{W}$ wind turbine gearbox.

Step | Analysis Task | Method | Time |
---|---|---|---|

1 | Angle-varying mesh stiffness | Static FEM | 10 $\mathrm{min}$ |

2 | Time-varying mesh stiffness | Analytical | $0.5$ $\mathrm{s}$ |

3 | Transmission Error (TE) | Runge-Kutta scheme | 10 $\mathrm{s}$ |

4 | Frequency Response | Time-harmonic FEM | $61.1$ $\mathrm{min}$ |

5 | Linear Combination and Superposition | Analytical | $12.1$ $\mathrm{s}$ |

**Table 2.**Operating speeds ${\Omega}_{\mathrm{d}}$ of the HS stage during measurement runs 1–3, regarding mean value ${\overline{\Omega}}_{\mathrm{d}}$, standard derivation ${\sigma}_{\Omega}$, and range ${R}_{\Omega}$.

Run | ${\overline{\Omega}}_{\mathbf{d}}$ | ${\mathit{\sigma}}_{\Omega}$ | ${\mathit{R}}_{\Omega}$ |
---|---|---|---|

[${\mathbf{min}}^{-1}$] | [${\mathbf{min}}^{-1}$] | [${\mathbf{min}}^{-1}$] | |

1 | 290.1 | 9.6 | 42.3 |

2 | 315.7 | 14.6 | 48.2 |

3 | 280 | 0.6 | 2.2 |

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## Share and Cite

**MDPI and ACS Style**

Zarnekow, M.; Grätsch, T.; Ihlenburg, F.
A Hybrid Multistep Procedure for the Vibroacoustic Simulation of Noise Emission from Wind Turbines. *Acoustics* **2023**, *5*, 1-27.
https://doi.org/10.3390/acoustics5010001

**AMA Style**

Zarnekow M, Grätsch T, Ihlenburg F.
A Hybrid Multistep Procedure for the Vibroacoustic Simulation of Noise Emission from Wind Turbines. *Acoustics*. 2023; 5(1):1-27.
https://doi.org/10.3390/acoustics5010001

**Chicago/Turabian Style**

Zarnekow, Marc, Thomas Grätsch, and Frank Ihlenburg.
2023. "A Hybrid Multistep Procedure for the Vibroacoustic Simulation of Noise Emission from Wind Turbines" *Acoustics* 5, no. 1: 1-27.
https://doi.org/10.3390/acoustics5010001