13 KiB
The distributed properties of the tower bring about generalized inertia forces and generalized active forces associated with tower elasticity, tower damping, tower aerodynamics, and tower weight. Note that I eliminated the tower mass tuners, since it is redundant to have both mass and stiffness tuners when trying to tune tower frequencies (to tune the frequencies for individual modes, all that is needed is to tune the mass or the stiffness for the individual modes, but not both). Note also that I eliminated the tower stiffness tuner’s effects on the gravitational destiffening loads. It is also beneficial to eliminate the tower mass tuners because the tower mass density is needed to compute the tower base loads and thus these tuners affect the tower base loads directly—this makes the form of the tower base load equations considerably more complex and considerably less intuitive. Since the tower elastic stiffness does not directly influence the tower base loads in a fundamental way, the retention of the tower stiffness tuners is much more favorable than the retention of the tower mass tuners (recall that only one set of tuners needs to be retained in order to permit the user to match natural frequencies). The elimination of the tower stiffness tuner’s effects on the gravitational destiffening was done for the same reason (i.e., the gravity loads directly affect the tower base loads, and thus, tower stiffness tuners make the form of the tower base load equations considerable more complex and considerably less intuitive). The fact that the gravitational destiffening of the tower is small compared to the overall stiffness of the tower is another reason this elimination of stiffness tuning effects should not be of significant concern.
塔架的分布式特性带来了与塔架弹性、塔架阻尼、塔架气动特性和塔架重量相关的广义惯性力和广义主动力。值得注意的是，我消除了塔架质量调谐器，因为在尝试调谐塔架频率时，同时拥有质量和刚度调谐器是多余的（为了调谐单个模态的频率，所需的只是调谐单个模态的质量或刚度，而不是两者都调谐）。另请注意，我消除了塔架刚度调谐器对重力去刚化载荷的影响。消除塔架质量调谐器也是有益的，因为计算塔基载荷需要塔架质量密度，因此这些调谐器直接影响塔基载荷——这使得塔基载荷方程的形式变得相当复杂且直观性大大降低。由于塔架弹性刚度从根本上不直接影响塔基载荷，因此保留塔架刚度调谐器比保留塔架质量调谐器更有利（回想一下，为了允许用户匹配固有频率，只需保留一套调谐器）。消除塔架刚度调谐器对重力去刚化的影响也是出于同样的原因（即，重力载荷直接影响塔基载荷，因此，塔架刚度调谐器使得塔基载荷方程的形式变得相当复杂且直观性大大降低）。塔架的重力去刚化相对于塔架的整体刚度而言很小，这是这种消除刚度调谐影响的做法不应引起重大关注的另一个原因。
F_r^* \Big|_T = - \int_0^{TwFlexL} \mu^T (h) \cdot {}^E v_r^T (h) \cdot {}^E a^T (h)dh - YawBrMass {}^E v_r^O \cdot {}^E a^O \quad (r = 1, 2, \ldots, 22) \\
\text{where} \quad \mu^T (h) = AdjTwMa \cdot TMassDen(h)\tag{1}
or,
F_r^* \Big|_T = - \int_0^{TwrFlexL} \mu^T (h) \cdot {}^E \mathbf{v}_r^T (h) \cdot \left\{ \left( \sum_{i=1}^{10} {}^E \mathbf{v}_i^T (h) \ddot{q}_i \right) + \left[ \sum_{i=4}^{10} \frac{d}{dt} \left( {}^E \mathbf{v}_i^T (h) \right) \dot{q}_i \right] \right\} dh - YawBrMass {}^E \mathbf{v}_r^O \cdot \left\{ \left( \sum_{i=1}^{10} {}^E \mathbf{v}_i^O \ddot{q}_i \right) + \left[ \sum_{i=4}^{10} \frac{d}{dt} \left( {}^E \mathbf{v}_i^O \right) \dot{q}_i \right] \right\} \quad (r=1,2,\ldots,10)\tag{2}
Thus,
[C(\mathbf{q},t)]_T (Row, Col) = \int_0^{TwrFlexL} \mu^T (h) {}^E \mathbf{v}_{Row}^T (h) \cdot {}^E \mathbf{v}_{Col}^T (h)dh - YawBrMass {}^E \mathbf{v}_{Row}^O \cdot {}^E \mathbf{v}_{Col}^O \quad (Row, Col = 1, 2, \ldots, 10)\tag{3}
\left\{ - f(\dot{q},q,t) \right\} \Big|_T (Row) = - \int_0^{TwrFlexL} \mu^T (h) {}^E \mathbf{v}_{Row}^T (h) \cdot \left( \sum_{i=4}^{10} \frac{d}{dt} \left[ {}^E \mathbf{v}_i^T (h) \right] \dot{q}_i \right) dh - YawBrMass {}^E \mathbf{v}_{Row}^O \cdot \left[ \sum_{i=4}^{10} \frac{d}{dt} \left( {}^E \mathbf{v}_i^O \right) \dot{q}_i \right] \quad (Row = 1,2,\ldots,10)\tag{4}
F_{r \Big| ElasticT} = - \frac{\partial V'^T}{\partial q_r} \quad (r = 1,2,\ldots,22)\tag{5}
so,
F_{r \Big| ElasticT} = \left\{
\begin{array}{ll}
-k_{11}^{'TFA} q_{TFA1} - k_{12}^{'TFA} q_{TFA2} & \text{for } r=TFA1 \\
-k_{11}^{'TSS} q_{TSS1} - k_{12}^{'TSS} q_{TSS2} & \text{for } r=TSS1 \\
-k_{21}^{'TFA} q_{TFA1} - k_{22}^{'TFA} q_{TFA2} & \text{for } r=TFA2 \\
-k_{21}^{'TSS} q_{TSS1} - k_{22}^{'TSS} q_{TSS2} & \text{for } r=TSS2 \\
0 & \text{otherwise}
\end{array}
\right.\tag{6}
where k_{\ i j}^{\,\prime T F A} and k_{{i j}}^{\,\prime{T S S}} are the generalized stiffnesses of the tower in the fore-aft and side-to-side directions respectively when gravitational destiffening effects are not included as follows:
其中，k_{\ i j}^{\,\prime T F A} 和 k_{{i j}}^{\,\prime{T S S}} 分别是塔筒在前后和侧向方向上的广义刚度，此时不包括重力去刚化效应，如下所示：
k_{ij}^{'TFA} = \sqrt{FAStTunr(i) FAStTunr(j)} \int_0^{TwrFlexL} EI^{TFA}(h) \frac{d^2 \phi_i^{TFA}(h)}{dh^2} \frac{d^2 \phi_j^{TFA}(h)}{dh^2} dh \quad (i, j = 1,2) \text{ (which is symmetric)}\tag{7}
where
EI^{TFA}(h) = AdjFASt \cdot TwFAStif(h)\tag{8}
k_{ij}^{'TSS} = \sqrt{SSStTunr(i) SSStTunr(j)} \int_0^{TwrFlexL} EI^{TSS}(h) \frac{d^2 \phi_i^{TSS}(h)}{dh^2} \frac{d^2 \phi_j^{TSS}(h)}{dh^2} dh \quad (i, j = 1,2) \text{ (which is symmetric)}\tag{9}
where
EI^{TSS}(h) = AdjSSSt \cdot TwSSStif(h)\tag{10}
The coefficient in front of the integral in these generalized stiffnesses represents the individual modal stiffness tuning, which allows the user to vary the stiffness of the tower between the individual modes to permit better matching of tower frequencies. To be precise, the tuner coefficient only really makes sense when working with a generalized stiffness of a single mode (i.e., k_{\;\;I I}^{\;\prime T F A},\;k_{\;\;22}^{\;\prime T F A},\;k_{\;\;I I}^{\;\prime T S S} , or k\,_{\,22}^{\prime T S S} ), in which case the coefficient for mode i is simply F A S t T u n r(i) or SSStTunr (i) . However, since the cross-correlation elements of the generalized stiffness matrix will, in general, not vanish, the coefficient in the form above permits the tuning to apply to these terms in a consistent fashion.
这些广义刚度中的积分系数代表了各个模态刚度调谐，允许用户在各个模态之间改变塔的刚度，从而更好地匹配塔的固有频率。 值得注意的是，调谐系数的意义主要体现在针对单一模态的广义刚度时（例如，k_{\;\;I I}^{\;\prime T F A},\;k_{\;\;22}^{\;\prime T F A},\;k_{\;\;I I}^{\;\prime T S S} , 或 k\,_{\,22}^{\prime T S S} ），此时第 i 阶模态的系数仅仅是 F A S t T u n r(i) 或 SSStTunr (i) 。 然而，由于广义刚度矩阵的互相关元素通常不会消失，因此上述形式的系数允许以一致的方式将调谐应用于这些项。
Similarly, when using the Rayleigh damping technique where the damping is assumed proportional to the stiffness, then 同样地，当使用瑞利阻尼技术（其中阻尼被假定与刚度成比例）时，则
F_r |_{\text{Damp}T} = \begin{cases}
- \frac{\zeta_1^{TFA} k_{11}^{'TFA}}{\pi f_1^{'TFA}} \dot{q}_{TFA1} - \frac{\zeta_2^{TFA} k_{12}^{'TFA}}{\pi f_2^{'TFA}} \dot{q}_{TFA2} & \text{for } r=TFA1 \\
- \frac{\zeta_1^{TSS} k_{11}^{'TSS}}{\pi f_1^{'TSS}} \dot{q}_{TSS1} - \frac{\zeta_2^{TSS} k_{12}^{'TSS}}{\pi f_2^{'TSS}} \dot{q}_{TSS2} & \text{for } r=TSS1 \\
- \frac{\zeta_1^{TFA} k_{21}^{'TFA}}{\pi f_1^{'TFA}} \dot{q}_{TFA1} - \frac{\zeta_2^{TFA} k_{22}^{'TFA}}{\pi f_2^{'TFA}} \dot{q}_{TFA2} & \text{for } r=TFA2 \\
- \frac{\zeta_1^{TSS} k_{21}^{'TSS}}{\pi f_1^{'TSS}} \dot{q}_{TSS1} - \frac{\zeta_2^{TSS} k_{22}^{'TSS}}{\pi f_2^{'TSS}} \dot{q}_{TSS2} & \text{for } r=TSS2 \\
0 & \text{otherwise}
\end{cases}\tag{11}
where \zeta_{i}^{T F A} and \zeta_{i}^{T S S} represent the structural damping ratio of the tower for the i^{\mathrm{th}} mode in the fore-aft and side-to-side directions, T w r F A D m p(i)/l O O and T w r S S D m p(i)/l O O respectively. Also, f_{\ i}^{\,\prime T F A} and f_{\ i}^{\prime T S S} represent the natural frequency of the tower for the i^{\mathrm{th}} mode in the fore-aft and side-to-side directions respectively without tower-top mass or gravitational destiffening effects. That is,
其中，\zeta_{i}^{T F A} 和 \zeta_{i}^{T S S} 分别表示塔针对第 i 阶模态在前后向和侧向上的结构阻尼比，分别记为 T w r F A D m p(i)/100 和 $T w r S S D m p(i)/100$。 此外，f_{\ i}^{\,\prime T F A} 和 f_{\ i}^{\prime T S S} 分别表示塔针对第 i 阶模态在前后向和侧向上的固有频率，不考虑塔顶质量和重力软化效应。 即，
f_{\;\;i}^{\;\prime^{T F A}}=\frac{1}{2\pi}\sqrt{\frac{k_{\;i i}^{\prime^{T F A}}}{m_{\;i i}^{\prime^{T F A}}}}\;\;\;\;\;\mathrm{and}\;\;\;\;\;\;\;f_{\;i}^{\;\prime^{T S S}}=\frac{1}{2\pi}\sqrt{\frac{k_{\;i i}^{\prime T S S}}{m_{\;i i}^{\prime T S S}}}\tag{12}
where m_{\;i i}^{\;\prime T F A} and m_{\ i i}^{\prime T S S} represent the generalized mass of the tower for the i^{\mathrm{th}} mode in the fore-aft and side-to-side directions respectively without tower-top mass effects as follows:
其中，m_{\;i i}^{\;\prime T F A} 和 m_{\ i i}^{\prime T S S} 分别表示在塔顶质量效应忽略的情况下，第 i 阶模态在前后方向和左右方向上的广义质量，具体如下：
m_{\ i j}^{\prime^{T F A}}=\int_{0}^{T w r F l e x L}\mu^{T}\left(h\right)\phi_{i}^{T F A}\left(h\right)\phi_{j}^{T F A}\left(h\right)d h\quad\left(i,j=1,2\right)\tag{13}
and
m_{\it i j}^{\prime T S S}=\int_{0}^{T w r F l e x L}\mu^{T}\left(h\right)\phi_{i}^{T S S}\left(h\right)\phi_{j}^{T S S}\left(h\right)d h\quad\left(i,j=1,2\right)\tag{14}
Thus,
[C(q,t)]|_{\text{Elastic}T} = 0\tag{15}
\{-f(\dot{q},q,t)\}|_{\text{Elastic}T} = \begin{Bmatrix}
\vdots \\
\vdots \\
\vdots \\
-k_{11}^{'TFA} q_{TFA1} - k_{12}^{'TFA} q_{TFA2} \\
-k_{11}^{'TSS} q_{TSS1} - k_{12}^{'TSS} q_{TSS2} \\
-k_{21}^{'TFA} q_{TFA1} - k_{22}^{'TFA} q_{TFA2} \\
-k_{21}^{'TSS} q_{TSS1} - k_{22}^{'TSS} q_{TSS2} \\
\vdots \\
\vdots \\
\vdots \\
\end{Bmatrix}\tag{16}
[C(q,t)]|_{\text{Damp}T} = 0\tag{17}
\{-f(\dot{q},q,t)\}|_{\text{Damp}T} = \begin{Bmatrix}
\vdots \\
\vdots \\
\vdots \\
- \frac{\zeta_1^{TFA} k_{11}^{'TFA}}{\pi f_1^{'TFA}} \dot{q}_{TFA1} - \frac{\zeta_2^{TFA} k_{12}^{'TFA}}{\pi f_2^{'TFA}} \dot{q}_{TFA2} \\
- \frac{\zeta_1^{TSS} k_{11}^{'TSS}}{\pi f_1^{'TSS}} \dot{q}_{TSS1} - \frac{\zeta_2^{TSS} k_{12}^{'TSS}}{\pi f_2^{'TSS}} \dot{q}_{TSS2} \\
- \frac{\zeta_1^{TFA} k_{21}^{'TFA}}{\pi f_1^{'TFA}} \dot{q}_{TFA1} - \frac{\zeta_2^{TFA} k_{22}^{'TFA}}{\pi f_2^{'TFA}} \dot{q}_{TFA2} \\
- \frac{\zeta_1^{TSS} k_{21}^{'TSS}}{\pi f_1^{'TSS}} \dot{q}_{TSS1} - \frac{\zeta_2^{TSS} k_{22}^{'TSS}}{\pi f_2^{'TSS}} \dot{q}_{TSS2} \\
\vdots \\
\vdots \\
\vdots
\end{Bmatrix}\tag{18}
F_r |_{\text{Grav}T} = \int_{0}^{\text{TwrFlexL}} \! {}^E \mathbf{v}_r^T(h) \cdot [-\mu^T(h) \mathbf{g}\mathbf{z}_2] \, dh + {}^E \mathbf{v}_r^O \cdot (-\text{YawBrMass} \cdot \mathbf{g}\mathbf{z}_2) \quad (r=3,4,...,10)\tag{19}
Thus,
[C(q,t)]|_{\text{Grav}T} = 0\tag{20}
\{-f(\dot{q},q,t)\}|_{\text{Grav}T} (\text{Row}) = - \int_{0}^{\text{TwrFlexL}} \! \mu^T(h) g {}^E \mathbf{v}_{\text{Row}}^T(h) \cdot \mathbf{z}_2 \, dh - \text{YawBrMass} \cdot g {}^E \mathbf{v}_{\text{Row}}^O \cdot \mathbf{z}_2 \quad (Row=3,4,...,10)\tag{21}
F_r |_{\text{Aero}T} = \int_{0}^{\text{TwrFlexL}} \! \left[ {}^E \mathbf{v}_r^T(h) \cdot \mathbf{F}_{\text{Aero}T}^T(h) + {}^E \boldsymbol{\omega}_r^F(h) \cdot \mathbf{M}_{\text{Aero}T}^F(h) \right] dh \quad (r=1,2,...,10)\tag{22}
where F_{A e r o T}^{T}\left(h\right) and M_{A e r o T}^{F}\left(h\right) are aerodynamic forces and moments applied to point \mathrm{T} on the tower respectively expressed per unit height. Thus,
[C(q,t)]|_{\text{Aero}T} = 0\tag{23}
\{-f(\dot{q},q,t)\}|_{\text{Aero}T} (\text{Row}) = \int_{0}^{\text{TwrFlexL}} \! \left[ {}^E \mathbf{v}_{\text{Row}}^T(h) \cdot \mathbf{F}_{\text{Aero}T}^T(h) + {}^E \boldsymbol{\omega}_{\text{Row}}^F(h) \cdot \mathbf{M}_{\text{Aero}T}^F(h) \right] dh \quad (Row=1,2,...,10)\tag{24}