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线性化求解器/参考文献/Hansen-Improved modal dynamics of wind turbine/auto/Hansen-Improved modal dynamics of wind turbine.md
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@ -307,10 +307,21 @@ Figure 5 shows the FW and BW components of the flapwise and edgewise blade modes
At the operation speed, mode 9 must be characterized as the 1st FW edgewise mode with some content of flapwise whirling. This content is now suggested to explain the observed difference in aerodynamic damping of the BW and FW edgewise modes (modes 7 and 9). At the operation speed, mode 9 must be characterized as the 1st FW edgewise mode with some content of flapwise whirling. This content is now suggested to explain the observed difference in aerodynamic damping of the BW and FW edgewise modes (modes 7 and 9).
这种自旋模态对的固有频率分裂与陀螺效应相关的观测坐标系有关。图4中所示的固有频率 $\omega_{k}$ 是从地面固定参考系观测得到的。公式(17)表明，在共转叶片参考系中，对称风轮模态的频率仍然是 $\omega_{k}$，而BW（摆振）和FW（挥舞）模态的频率分别变为 $\omega_{k}+\Omega$ 和 $\omega_{k}-\Omega$。如果只有叶片模态数 $n$ 参与自旋模态，它们的频率在地面固定参考系中会围绕该叶片模态的固有频率 $\omega_{n}$ 分裂。在这种理想情况下，叶片上的观察者会同时测量到BW和FW模态的相同频率 $\omega_{n}=\omega_{k}^{\mathrm{BW}}+\Omega=\omega_{k}^{\mathrm{FW}}-\Omega$。
这种理想条件会受到涡轮结构的非对称性和涡轮模态中叶片模态的耦合的影响。当两个涡轮模态的固有频率接近时，这种模态相互作用可能发生。图4显示，第1个FW（挥舞）摆振模态（模态8）和第2个BW（摆振）挥舞模态（模态9）在 $\Omega=2{\cdot}5$ rad $\mathrm{s}^{-1}$ 左右变得接近。这两个模态相互作用，这可以通过它们模态形状中的摆振和挥舞自旋分量来显示。
图5显示了模态7–10中FW（挥舞）和BW（摆振）叶片模态的挥舞和摆振分量。这些自旋分量是使用公式(18)在多叶坐标系中从特征向量计算得出的。模态7和10的主导振幅表明，它们分别是BW（摆振）摆振模态和FW（挥舞）挥舞模态。然而，在整个转速范围内，模态8和9没有显著的模态振幅。似乎这些模态相互交换模态形状：在2.5 rad $\mathrm{s}^{-1}$ 以下的转速下，模态8可以定义为FW（挥舞）摆振模态，模态9为BW（摆振）挥舞模态，反之亦然。
在工作转速下，模态9必须表征为具有一些挥舞自旋内容的第1个FW（挥舞）摆振模态。现在建议这种内容来解释BW和FW摆振模态（模态7和9）观察到的气动阻尼差异。
# Why the Measured Difference in Aerodynamic Damping? # Why the Measured Difference in Aerodynamic Damping?
Thomsen et al.1 have estimated the total damping of the two edgewise whirling modes (see Figure 1). The results show that the FW edgewise mode (mode 9) is more damped than the BW edgewise mode (mode 7). The structural damping of these two modes is assumed to be the same, because their natural frequencies and mode shapes are almost identical. Thus the difference in total damping is assumed to be caused by a difference in aerodynamic damping. Thomsen et al.1 have estimated the total damping of the two edgewise whirling modes (see Figure 1). The results show that the FW edgewise mode (mode 9) is more damped than the BW edgewise mode (mode 7). The structural damping of these two modes is assumed to be the same, because their natural frequencies and mode shapes are almost identical. Thus the difference in total damping is assumed to be caused by a difference in aerodynamic damping.
Thomsen 等人1 已估算出了两个摆振模态的总阻尼（见图1）。结果表明，FW 摆振模态（模态9）比 BW 摆振模态（模态7）具有更大的阻尼。假设这两个模态的结构阻尼相同，因为它们的固有频率和模态形状几乎完全一致。因此，总阻尼的差异被认为是由气动阻尼的差异引起的。
![](b27d093788eedd61e8374571ef2a21361ad23a35d29ba894deefacc295c3a9c7.jpg) ![](b27d093788eedd61e8374571ef2a21361ad23a35d29ba894deefacc295c3a9c7.jpg)
Figure 5. Flapwise and edgewise whirling components of the modal blade amplitudes for modes 7–10 computed from the eigenvectors using equation (18) Figure 5. Flapwise and edgewise whirling components of the modal blade amplitudes for modes 7–10 computed from the eigenvectors using equation (18)
@ -318,9 +329,15 @@ The aim of the experiment was to measure the damping of edgewise blade vibration
Figure 6 shows a zoom in the Campbell diagram (Figure 4) on the natural frequencies of modes 7–9, together with lines of the $\pm1\mathrm{P}$ splitting about the edgewise blade frequency $(2{\cdot}94\ \mathrm{Hz})$ . These lines intersect with the natural frequencies of modes 7 and 9 at the operation speed, showing that the initial guesses on the experimental excitation frequencies are close. However, the modal analysis has also shown that mode 9 is not a pure edgewise whirling mode; it has some flapwise component. This component affects the direction in which the blades are vibrating relative to the rotor plane. Figure 6 shows a zoom in the Campbell diagram (Figure 4) on the natural frequencies of modes 7–9, together with lines of the $\pm1\mathrm{P}$ splitting about the edgewise blade frequency $(2{\cdot}94\ \mathrm{Hz})$ . These lines intersect with the natural frequencies of modes 7 and 9 at the operation speed, showing that the initial guesses on the experimental excitation frequencies are close. However, the modal analysis has also shown that mode 9 is not a pure edgewise whirling mode; it has some flapwise component. This component affects the direction in which the blades are vibrating relative to the rotor plane.
实验目的旨在通过在风轮舱盖上安装激励器激发相应的风轮模态，从而测量风轮运行期间摆振叶片阻尼。本次实验无法使用所呈现的风轮理论模态分析。因此，摆振模态的激发是基于将激励频率调谐以获得最大摆振叶片响应。然而，最初的两个猜测是基于理想频率分裂条件存在的前提，即激励频率应为摆振叶片模态的固有频率加上或减去运行速度（1P）。
图6放大了图4的坎贝尔图，显示了模态7-9的固有频率，以及关于摆振叶片频率（2·94 Hz）的±1P 分裂线。这些线与运行速度下的模态7和9的固有频率相交，表明实验激励频率的初始猜测较为接近。然而，模态分析还表明模态9不是一个纯摆振模态；它具有一些挥舞分量。这个分量影响了叶片相对于风轮平面的振动方向。
# Effective Direction of Blade Vibration # Effective Direction of Blade Vibration
Figure 7 shows how the blade cross-section at $90\%$ radius is moving in and out of the rotor plane during vibrations in the two edgewise whirling modes (modes 7 and 9). The motion of the cross-section includes the motion of the rotor support, i.e. the shaft and tower deformations add to the effective blade vibration. The traces show that the blades move more out of the rotor plane in the FW edgewise mode than in the Figure 7 shows how the blade cross-section at $90\%$ radius is moving in and out of the rotor plane during vibrations in the two edgewise whirling modes (modes 7 and 9). The motion of the cross-section includes the motion of the rotor support, i.e. the shaft and tower deformations add to the effective blade vibration. The traces show that the blades move more out of the rotor plane in the FW edgewise mode than in the BW edgewise mode. This difference in blade vibration for the two modes can explain why Thomsen et al. observed that the FW mode is more damped than the BW mode.
图 7 显示了在两个摆振模态（模态 7 和 9）振动期间，半径处 $90\%$ 的叶片横截面在风轮平面内进出运动的情况。该横截面的运动包括风轮支撑的运动，即轴和塔架的变形会增加叶片的有效振动。轨迹显示，在FW摆振模态下，叶片比在BW摆振模态下更多地离开风轮平面。这种两种模态叶片振动差异可以解释为什么 Thomsen 等人观察到挥舞模态比摆振模态更具阻尼。
![](fbc1b019314328084a7e66eb9b851b544a45506606be095b8746d98395d8f071.jpg) ![](fbc1b019314328084a7e66eb9b851b544a45506606be095b8746d98395d8f071.jpg)
Figure 6. A zoom in the Campbell diagram (Figure 4) on the natural frequencies of modes 7–9. The broken lines show the $\pm\Omega$ splitting about the edgewise blade frequency $\left(2{\cdot}94\,H z\right)$ Figure 6. A zoom in the Campbell diagram (Figure 4) on the natural frequencies of modes 7–9. The broken lines show the $\pm\Omega$ splitting about the edgewise blade frequency $\left(2{\cdot}94\,H z\right)$
@ -328,17 +345,18 @@ Figure 6. A zoom in the Campbell diagram (Figure 4) on the natural frequencies o
![](b8ac1f69e8e67bd89e848dd13bf3bc4bab5b265f8838cf15dc851116eb006242.jpg) ![](b8ac1f69e8e67bd89e848dd13bf3bc4bab5b265f8838cf15dc851116eb006242.jpg)
Figure 7. Motion of a blade cross-section at $90\%$ radius for the 1st BW edgewise mode (left) and the 1st FW edgewise mode (right). The motion includes the tower and shaft deformations Figure 7. Motion of a blade cross-section at $90\%$ radius for the 1st BW edgewise mode (left) and the 1st FW edgewise mode (right). The motion includes the tower and shaft deformations
BW edgewise mode. This difference in blade vibration for the two modes can explain why Thomsen et al.
observed that the FW mode is more damped than the BW mode.
A quasi-steady aerodynamic analysis in the Appendix shows that the aerodynamic damping of a blade cross-section is lowest for vibrations close to the rotor plane. It is presumed in the analysis that the blade cross-section is performing a small elliptical motion, with the major axes being parallel and perpendicular to the rotor plane. This type of motion is assumed to characterize the qualitative pattern of the traces in Figure 7, except for a slight tilt of the major axes which originates from the direction of blade vibration for the edgewise blade mode (see Figure 3). An effective direction of blade vibration is defined for the elliptical motion of a blade cross-section as A quasi-steady aerodynamic analysis in the Appendix shows that the aerodynamic damping of a blade cross-section is lowest for vibrations close to the rotor plane. It is presumed in the analysis that the blade cross-section is performing a small elliptical motion, with the major axes being parallel and perpendicular to the rotor plane. This type of motion is assumed to characterize the qualitative pattern of the traces in Figure 7, except for a slight tilt of the major axes which originates from the direction of blade vibration for the edgewise blade mode (see Figure 3). An effective direction of blade vibration is defined for the elliptical motion of a blade cross-section as
附录中的准静态气动分析表明，叶片截面的气动阻尼对于接近风轮平面的振动最为低。分析假设叶片截面执行一种小的椭圆运动，其长轴平行于且垂直于风轮平面。 这种运动被假定为表征图7中轨迹的定性模式，除了长轴存在轻微倾斜，该倾斜源于摆振叶片模态的振动方向（见图3）。 对于叶片截面的椭圆运动，定义了有效的叶片振动方向，如下所示：
$$ $$
\theta_{\mathrm{eff}}=\tan^{-1}\left({\frac{\mathrm{max.~amplitude~out~of~rotor~plane}}{\mathrm{max.~amplitude~in~rotor~plane}}}\right) \theta_{\mathrm{eff}}=\tan^{-1}\left({\frac{\mathrm{max.~amplitude~out~of~rotor~plane}}{\mathrm{max.~amplitude~in~rotor~plane}}}\right)
$$ $$
Because the elliptical motion is assumed to have the major axes parallel and perpendicular to the rotor plane, the maximum amplitudes are positive, yielding $0^{\circ}<\theta_{\mathrm{eff}}<90^{\circ}$ . Using this definition on the motion of the cross-section at $90\%$ radius in Figure 7, it is found that $\theta_{\mathrm{eff}}\approx9^{\circ}$ and $30^{\circ}$ for the 1st BW and FW edgewise modes respectively. Computations of $\theta_{\mathrm{eff}}$ along the blade show that the entire blade is vibrating more out of the rotor plane in the 1st FW edgewise mode than in the 1st BW edgewise mode. Hence, based on quasisteady aerodynamics, this behaviour can explain the measured difference in aerodynamic damping of these two modes. Because the elliptical motion is assumed to have the major axes parallel and perpendicular to the rotor plane, the maximum amplitudes are positive, yielding $0^{\circ}<\theta_{\mathrm{eff}}<90^{\circ}$ . Using this definition on the motion of the cross-section at $90\%$ radius in Figure 7, it is found that $\theta_{\mathrm{eff}}\approx9^{\circ}$ and $30^{\circ}$ for the 1st BW and FW edgewise modes respectively. Computations of $\theta_{\mathrm{eff}}$ along the blade show that the entire blade is vibrating more out of the rotor plane in the 1st FW edgewise mode than in the 1st BW edgewise mode. Hence, based on quasisteady aerodynamics, this behaviour can explain the measured difference in aerodynamic damping of these two modes.
由于假设椭圆运动的主要轴与风轮平面平行和垂直，最大振幅为正值，从而得到 $0^{\circ}<\theta_{\mathrm{eff}}<90^{\circ}$ 。根据图7中$90\%$半径的截面运动定义，发现1阶摆振（edgewise）和挥舞（flapwise）模态对应的$\theta_{\mathrm{eff}}$ 分别约为 $9^{\circ}$ 和 $30^{\circ}$。 叶片（blade）沿径向计算$\theta_{\mathrm{eff}}$ 表明，在1阶挥舞（flapwise）摆振（edgewise）模态下，整个叶片相对于风轮平面振动幅度更大，而1阶摆振（edgewise）模态则较小。因此，基于准稳态气动学，这种行为可以解释这两个模态气动阻尼差异的测量结果。
# Improved Modal Dynamics # Improved Modal Dynamics
The out-of-plane motion of the blades in the 1st FW edgewise mode is mainly due to a component of the flapwise blade mode through the previously described modal interaction with the 2nd BW flapwise mode (see Figure 5). The 1st BW edgewise mode is not interacting with a flapwise whirling mode; the blades are mainly vibrating in the edgewise blade mode (with the additional motion due to the tower and shaft deformations). Is it possible to add a flapwise component to the BW edgewise mode without removing the flapwise component in the FW edgewise mode, thereby increasing the overall aerodynamic damping of the turbine? The out-of-plane motion of the blades in the 1st FW edgewise mode is mainly due to a component of the flapwise blade mode through the previously described modal interaction with the 2nd BW flapwise mode (see Figure 5). The 1st BW edgewise mode is not interacting with a flapwise whirling mode; the blades are mainly vibrating in the edgewise blade mode (with the additional motion due to the tower and shaft deformations). Is it possible to add a flapwise component to the BW edgewise mode without removing the flapwise component in the FW edgewise mode, thereby increasing the overall aerodynamic damping of the turbine?