Analysis of Lateral Fundamental Frequency of Monopile Offshore Wind Turbines Using Differential Transform Method
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摘要: 精准求解海上风机（OWTs）系统基频是海上风机结构研究的关键性难点之一。基于Euler–Bernoulli梁和Timoshenko梁理论，考虑风机塔筒变截面特性，根据是否考虑桩-土相互作用，分别建立底部弹簧模型与底部固接模型，采用微分变换法计算风机变截面梁横向振动方程，利用MATLAB软件求解风机系统基频。通过与实测结果进行比对，验证本文方法的有效性和精确性，并分析计算项数、塔筒高度、锥度比、弹簧刚度对风机基频的影响。研究结果表明，当计算项数为10及以上时，利用微分变换法计算基频满足精度要求；风机基频对塔筒高度变化的敏感性强，当塔筒高度增加时，基频下降趋势显著；变截面梁锥度比降低使塔筒底部直径变大，进而导致风机基频有所提升；考虑弹簧地基系统的风机基频随任一弹簧刚度的增加而提升，其中耦合弹簧刚度对风机基频的影响程度最大。
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关键词:
- Timoshenko梁 /
- Euler–Bernoulli梁 /
- 海上风机 /
- 系统基频 /
- 大直径单桩
Abstract: Based on the theories of Euler–Bernoulli and Timoshenko beams, and considering the characteristics of variable section towers and soil-structure interaction, both a comprehensive model with coupling springs and a fixed model were established. The differential transform method (DTM) was employed to calculate the transverse vibration equation of the variable section beam of offshore wind turbines (OWTs), and MATLAB software was used to solve for the fundamental frequency. The validity and accuracy of this method were verified by comparison with measured results. Additionally, the effects of tower height, taper ratio, and spring stiffness on the fundamental frequency were further analyzed. The research results indicate that the fundamental frequency is highly sensitive to changes in tower height; specifically, a decrease in tower height results in a significant decrease in fundamental frequency. As the taper ratio of the variable section beam decreases, the bottom diameter of the tower increases, leading to an improvement in the fundamental frequency. When considering the spring foundation system, the fundamental frequency increases with any increase in spring stiffness. Moreover, among the three-spring model components, the coupling spring has the greatest influence on the fundamental frequency. -
图 3 计算项数与塔筒高度对风机基频的影响
Figure 3. Influence of value of N and tower height on fundamental frequency
表 1 DTM基本变换法则
Table 1. Fundamental transformation theorems of DTM
原函数 转换函数 $ f(x) = g(x) \pm h(x) $ $ F(k) = G(k) \pm H(k) $ $ f(x) = c \cdot g(x) $ $ F(k) = c \cdot G(k) $ $ f(x) = g(x) \cdot h(x) $ $ F(k) = \displaystyle\sum\limits_{l = 0}^k {G(l) \cdot H(k - l)} $ $ f(x) = \dfrac{{{{\mathrm{d}}^n}g(x)}}{{{\mathrm{d}}{x^n}}} $ $ F(k) = (k + 1)(k + 2)\cdots(k + n)G(k + n) $ $ f(x) = {x^n} $ $ F(k) = \delta (k - n) = x ,\; \left\{ {x = 1,k = n;x = 0,k \ne n} \right\} $ 
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表 2 边界条件
Table 2. Boundary conditions table
边界条件 物理量 T-B理论 E-B理论 底部弹簧
模型弯矩 $ \dfrac{{E{I_{{\mathrm{b}}}}}}{L}{\varphi {'}}(0) - {K_{\text{R}}}\varphi (0) + {K_{{\text{LR}}}}LU(0) = 0 $ $ \dfrac{{E{I_{{\mathrm{b}}}}}}{{{L^{2}}}}{Y{'}}{'}(0) - \dfrac{{{K_{\text{R}}}}}{L}{Y{'}}(0) + {K_{{\text{LR}}}}Y(0) = 0 $ 剪力 $ \kappa {A_{{\mathrm{b}}}}G\left[ {{U{'}}(0) - \varphi (0)} \right] + {K_{{\text{LR}}}}\varphi (0) - {K_{\text{L}}}LU(0) = 0 $ $ \dfrac{{E{I_{{\mathrm{b}}}}}}{{{L^{3}}}}{Y{'''}}(0) + \dfrac{{{K_{{\text{LR}}}}}}{L}{Y{'}}(0) - {K_{\text{L}}}Y(0) = 0 $ 底部固接
模型位移 $ U\left( 0 \right) = 0 $ $ Y(0) = 0 $ 转角 $ \varphi \left( 0 \right) = 0 $ $ {Y{'}}(0) = 0 $ 风机顶部 弯矩 $ E{I_{{\mathrm{t}}}}{\varphi {'}}(1) = 0 $ $ E{I_{{\mathrm{t}}}}{Y{''}}(1) = 0 $ 剪力 $ \kappa {A_{t}}G\left[ {{U{'}}(1) - \varphi (1)} \right] + m{\omega ^2}U(1) = 0 $ $ E{I_{{\mathrm{t}}}}{Y{'''}}(1) + m{\omega ^2}Y(1) = 0 $ DTM底部
弹簧模型弯矩 $ \varPhi (1) - {\eta _{\text{R}}}\varPhi (0) + {\eta _{{\text{LR}}}}\nu (0) = 0 ,\;{\eta _{\text{R}}} = \dfrac{{{K_{\text{R}}}L}}{{E{I_{{\mathrm{b}}}}}} ,\; {\eta _{{\text{LR}}}} = \dfrac{{{K_{{\text{LR}}}}{L^2}}}{{E{I_{{\mathrm{b}}}}}}$ $ 2 Z(2) - {\eta _{\text{R}}}Z(1) + {\eta _{{\text{LR}}}}Z(0) = 0 ,\;{\eta _{\text{R}}} = \dfrac{{{K_{\text{R}}}L}}{{E{I_{{\mathrm{b}}}}}},\;{\eta _{{\text{LR}}}} = \dfrac{{{K_{{\text{LR}}}}{L^2}}}{{E{I_{{\mathrm{b}}}}}}$ 剪力 $ \nu (1) - \varPhi (0) + {\psi _{{\text{LR}}}}\varPhi (0) - {\psi _{\text{L}}}\nu (0) = 0,\;{\psi _{{\text{LR}}}} = \dfrac{{{K_{{\text{LR}}}}}}{{\kappa {A_{b}}G}} ,\; {\psi _{\text{L}}} = \dfrac{{{K_{\text{L}}}L}}{{\kappa {A_{b}}G}}$ $ 6 Z(3) + {\eta _{{\text{LR}}}}Z(1) - {\eta _{\text{L}}}Z(0) = 0 ,\;{\eta _{\text{L}}} = \dfrac{{{K_{\text{L}}}{L^3}}}{{E{I_{{\mathrm{b}}}}}} ,\;{\eta _{{\text{LR}}}} = \dfrac{{{K_{{\text{LR}}}}{L^2}}}{{E{I_{{\mathrm{b}}}}}} $ DTM底部
固接模型位移 $ \nu \left( 0 \right) = 0 $ $ Z(0) = 0 $ 转角 $ \varPhi \left( 0 \right) = 0 $ $ Z(1) = 0 $ DTM风机
顶部弯矩 $ \displaystyle\sum\limits_{k = 0}^N {(k + 1)\varphi (k + 1)} = 0 $ $ \displaystyle\sum\limits_{k = 0}^N {(k + 2)(k + 1)Z(k + 2)} = 0 $ 剪力 $ \displaystyle\sum\limits_{k = 0}^N {(k + 1)U(k + 1)} - \sum\limits_{k = 0}^N {\varphi (k)} + {\alpha _n}\sum\limits_{k = 0}^N {U(k)} = 0,\;{\alpha _n} = \dfrac{{m{\omega ^2}L}}{{\kappa {A_{{\mathrm{t}}}}G}} $ $ \displaystyle\sum\limits_{k = 0}^N {(k + 3)(k + 2)(k + 1)Z(k + 3)} + {\alpha _m}\sum\limits_{k = 0}^N {Z(k)} = 0 ,\;{\alpha _m} = \dfrac{{m{\omega ^2}{L^{3}}}}{{E{I_{{\mathrm{b}}}}}} $
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表 3 风机规格参数
Table 3. Specifications of Irene Vorrink23, 28 and LelyA2
规格参数 Irene Vorrink23 Irene Vorrink28 LelyA2 材料密度/（kg·m−3） 7 850 7 850 7 850 RNA质量/t 35.7 35.7 32.0 塔筒高/m 44.5 44.5 41.5 塔筒底部直径/m 3.5 3.5 3.2 塔筒顶部直径/m 1.7 1.7 1.9 塔筒壁厚/mm 13 13 13 材料杨氏模量/GPa 210 210 210 平台高/m 6.0 5.8 4.6
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表 4 风机系统基频结果验证
Table 4. Verification of OWTs fundamental frequency results
风机 基频/Hz 相对误差/% 文献实测频率 文献固接频率 T-B理论 E-B理论 T-B理论 E-B理论 弹簧 固接 弹簧 固接 弹簧 固接 弹簧 固接 LelyA2 0.634 0.713 0.642 0.691 0.644 0.688 1.26 3.09 1.57 3.51 Irene Vorrink23 0.563 0.590 0.582 0.616 0.584 0.618 3.37 4.41 3.73 4.75 Irene Vorrink28 0.560 0.586 0.580 0.614 0.581 0.615 3.57 4.79 3.75 4.95
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