Nonlinear flutter modes and flutter suppression of an all-movable fin with freeplay
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摘要:
间隙非线性环节广泛存在于飞行器的结构当中，由其引发的极限环振荡（LCO）往往造成结构的疲劳破坏，但目前关于全动舵面非线性颤振机理和被动抑制方法的研究相对较少。针对典型含扭转间隙小展弦比全动舵面开展分析，提出了2种典型的非线性颤振模式，并探索相应的非线性颤振被动抑制方法。基于描述函数法和活塞理论建立舵面的动力学模型，考虑到间隙非线性环节导致刚度降低的效果，计算对比了2种不同扭转刚度下全动舵面的非线性颤振特性，进而提出了2种不同的非线性颤振模式，其中，模式Ⅱ不存在稳定的极限环振荡过程，可以有效抑制低于线性颤振边界的极限环振荡问题；研究了舵面根部弯曲扭转刚度和质量特性对非线性颤振模式和极限环初始动压的影响，提出相应的非线性颤振被动抑制策略。针对所提算例，数值计算结果表明：通过调整舵面根部弯曲扭转刚度或质量特性可以提高极限环初始动压，甚至改变非线性颤振模式，从而达到非线性颤振抑制的目的。
Abstract:Freeplay nonlinearity widely exists in the structure of aircraft, which can bring limit cycle oscillations (LCO), leading to the fatigue failure of the structure. There is relatively little research on the mechanism and passive suppression of the nonlinear flutter for all-movable fins. Therefore, this paper proposes two different nonlinear flutter modes and provides passive suppression methods. The dynamic model is developed using the describing function method and piston theory. Considering that the torsional stiffness is reduced due to the existence of the freeplay, nonlinear flutter characteristics under different torsional stiffnesses are compared and two different nonlinear flutter modes are defined, in which mode Ⅱ doesn’t contain stable LCO, effectively suppressing the LCO below the linear flutter boundary. Then the influence of bending and torsional stiffnesses and mass moments of inertia on nonlinear flutter modes and the dynamic pressure of LCO appearance is studied, and some nonlinear flutter passive suppression methods are proposed. The findings indicate that reasonable adjustment of the bending and torsional stiffnesses or mass moments of inertia can raise the dynamic pressure of LCO appearance and even change the nonlinear flutter mode, so as to suppress the nonlinear flutter.
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图 3 ${{{{\bar K}_{\text{T}}}} \mathord{\left/ {\vphantom {{{{\bar K}_{\text{T}}}} {{K_{\text{T}}}}}} \right. } {{K_{\text{T}}}}}$随$ {A \mathord{\left/ {\vphantom {A \delta }} \right. } \delta } $变化情况
Figure 3. ${{{{\bar K}_{\text{T}}}} \mathord{\left/ {\vphantom {{{{\bar K}_{\text{T}}}} {{K_{\text{T}}}}}} \right. } {{K_{\text{T}}}}}$ with variation of $ {A \mathord{\left/ {\vphantom {A \delta }} \right. } \delta } $
图 6 线性颤振动压随刚度比变化
Figure 6. Linear flutter dynamic pressure with variation of stiffness ratio
图 8 ${q_{\text{T}}}$时域响应结果（状态A）
Figure 8. Time domain response results of ${q_{\text{T}}}$ (State A)
图 10 ${q_{\text{T}}}$时域响应结果（状态B）
Figure 10. Time domain response results of ${q_{\text{T}}}$ (State B)
图 11 非线性颤振特性随刚度比变化（改变$ {K_{\text{T}}} $）
Figure 11. Nonlinear flutter characteristic with variation of stiffness ratio (Changing $ {K_{\text{T}}} $)
图 12 非线性颤振特性随刚度比变化（改变$ {K_{\text{B}}} $）
Figure 12. Nonlinear flutter characteristic with variation of stiffness ratio (Changing $ {K_{\text{B}}} $)
图 13 $ {\xi _C} $随配重位置变化（配重50 g）
Figure 13. $ {\xi _C} $ with variation of weight position (50 g)
图 14 $ {\xi _C} $随配重位置变化（配重150 g）
Figure 14. $ {\xi _C} $ with variation of weight position (150 g)
图 15 $ {\xi _C} $随配重位置变化（配重250 g）
Figure 15. $ {\xi _C} $ with variation of weight position (250 g)
图 16 ${Q_{{\text{L0}}}}$随配重位置变化（配重50 g）
Figure 16. ${Q_{{\text{L0}}}}$ with variation of weight position (50 g)
图 17 ${Q_{{\text{L0}}}}$随配重位置变化（配重150 g）
Figure 17. ${Q_{{\text{L0}}}}$ with variation of weight position (150 g)
图 18 ${Q_{{\text{L0}}}}$随配重位置变化（配重250 g）
Figure 18. ${Q_{{\text{L0}}}}$ with variation of weight position (250 g)
图 19 ${Q_{{\text{LF}}}}$随配重位置变化（配重50 g）
Figure 19. ${Q_{{\text{LF}}}}$ with variation of weight position (50 g)
图 20 ${Q_{{\text{LF}}}}$随配重位置变化（配重150 g）
Figure 20. ${Q_{{\text{LF}}}}$ with variation of weight position (150 g)
图 21 ${Q_{{\text{LF}}}}$随配重位置变化（配重250 g）
Figure 21. ${Q_{{\text{LF}}}}$ with variation of weight position (250 g)
图 23 模态频率、频率差和颤振动压随刚度比变化
Figure 23. Modal frequency, frequency difference and flutter dynamic pressure with variation of stiffness ratio
图 24 $ \eta $随配重位置变化（配重50 g）
Figure 24. $ \eta $ with variation of weight position (50 g)
图 25 临界模态节线位置与${Q_{{\text{L0}}}}$变化
Figure 25. ${Q_{{\text{L0}}}}$ with variation of weight position (250 g)
表 1 全动舵面结构参数
Table 1. Structural parameters of all-movable fin
参数 数值 质量$m/{\text{kg}}$ $ 3.794 $ 质心${ {\left[ { {x_{\text{c} } },{y_{\text{c} } } } \right]} \mathord{\left/ {\vphantom { {\left[ { {x_{\text{c} } },{y_{\text{c} } } } \right]} {\rm{m}}} } \right. } {\rm{m}}}$ $[0.012,{\text{ }}0.154]$ 转动惯量${ { {I_{xx} } } \mathord{\left/ {\vphantom { { {I_{xx} } } {\left( { {\text{kg} } \cdot { {\text{m} }^2} } \right)} } } \right. } {\left( { {\text{kg} } \cdot { {\text{m} }^2} } \right)} }$ $1.275 \times {10^{ - 1}}$ 转动惯量${ { {I_{xy} } } \mathord{\left/ {\vphantom { { {I_{xy} } } {\left( { {\text{kg} } \cdot { {\text{m} }^2} } \right)} } } \right. } {\left( { {\text{kg} } \cdot { {\text{m} }^2} } \right)} }$ $2.042 \times {10^{ - 2}}$ 转动惯量${ { {I_{yy} } } \mathord{\left/ {\vphantom { { {I_{yy} } } {\left( { {\text{kg} } \cdot { {\text{m} }^2} } \right)} } } \right. } {\left( { {\text{kg} } \cdot { {\text{m} }^2} } \right)} }$ $4.035 \times {10^{ - 2}}$ 弯曲弹簧刚度${ { {K_{\text{B} } } } \mathord{\left/ {\vphantom { { {K_{\text{B} } } } {\left( { {\text{N} }\cdot{\text{m} } \cdot {\text{ra} }{ {\text{d} }^{ - 1} } } \right)} } } \right. } {\left( { {\text{N} }\cdot{\text{m} }\cdot {\text{ra} }{ {\text{d} }^{ - 1} } } \right)} }$ $ 9.0 \times {10^3} $ 扭转弹簧刚度${ { {K_{\text{T} } } } \mathord{\left/ {\vphantom { { {K_{\text{T} } } } {\left( { {\text{N} }\cdot{\text{m} } \cdot {\text{ra} }{ {\text{d} }^{ - 1} } } \right)} } } \right. } {\left( { {\text{N} }\cdot{\text{m} }\cdot {\text{ra} }{ {\text{d} }^{ - 1} } } \right)} }$ $ 3.0 \times {10^3} $ 
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表 2 模态频率计算结果对比
Table 2. Comparison of modal frequency calculation results
Hz 模态阶数 简化模型 有限元模型 1 37.77 37.31 2 50.68 50.23 3 407.12 4 517.66 5 573.28
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表 3 线性颤振结果
Table 3. Linear flutter results
计算方法 ${\omega _{\rm{F}}/{\rm{Hz} } }$ ${V_{\text{F} } }/\left({\rm{m}\cdot {\rm{s} } }^{-1}\right)$ ${Q_{\text{F} } }/{\rm{kPa} }$ 本文方法 44.14 939.78 53.87 ZAERO(前2阶) 42.72 928.83 52.63 ZAERO(前5阶) 42.72 928.85 52.63
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