NATURAL VIBRATION ANALYSIS OF TWO-DIMENSIONAL FLEXIBLE WING BASED ON NON-UNIFORM BEAM MODEL
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摘要: 变体飞行器通过光滑连续的结构变形改变气动特性, 从而提高飞行器的飞行性能, 具有很大的应用前景. 由于这类新概念飞行器主要通过改变自身结构形状以获得最佳工作性能的需求, 因此具有变形大、质量轻等特点, 较容易发生结构振动响应. 本文研究了一种以柔性变后缘作为变体形式的二维柔性机翼等效建模方法, 基于非均匀梁模型假设, 建立了该柔性翼的动力学模型. 通过利用Frobenius方法得到解析解及固有频率, 并用有限元方法进行对比验证, 发现前4阶固有频率的误差均在1%以内, 每阶固有频率对应的振型一致. 通过3D打印工程塑料ABS和硅胶蒙皮材料制备了柔性机翼结构件, 并通过动态测量法和拉伸试验分别测定了打印材料和硅胶蒙皮材料的杨氏模量, 搭建振动响应实验平台对制备的柔性机翼试验件进行振动试验. 对比发现模型振动试验获得的基频与理论模型结果一致, 并与有限元方法误差在3%以内. 本文通过理论分析和实验验证, 建立了二维柔性机翼等效建模方法, 研究结果将为柔性变后缘结构动力学特性分析及其控制应用方面提供理论支持.Abstract: In order to improve the flight performance of the aircraft, morphing technologies are used to change aerodynamic characteristics through smooth and continuous structural deformation. Since this new concept requires changing the structural shape to obtain the best performance, its inherent dynamic characteristics will be affected and even change its aeroelastic performance. In this paper, an equivalent modelling method of the two-dimensional flexible wing with camber morphing is developed. The dynamic model of the flexible wing is established based on the hypothesis of a non-uniform beam model. The analytical solution and natural frequencies are obtained by the method of Frobenius and verified by comparison with the finite element method solution. The errors of the first four natural frequencies are within 1% and the corresponding modes are consistent. The flexible wing is prepared by 3D printing engineering plastic (ABS) and silicone rubber skin. The Young's modulus of the 3D printing material and silicone rubber are respectively measured by dynamic measurement method and tensile test. The vibration response test platform is built to carry out vibration test of the flexible wing. It is found that the fundamental frequency obtained by vibration test is consistent with the theoretical model results, and the error is less than 3% compared with the finite element method. The equivalent modelling method of a two-dimensional flexible wing is established through theoretical analysis and experimental verification. The research results will provide theoretical support for applying the flexible trailing edge structures.
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Keywords:
- FishBAC /
- camber variation /
- nonuniform beam /
- natural vibration /
- power series solution
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引 言
变体飞行器(morphing aircraft)由于外形可变性, 可以实时地适应多种任务需求, 使其在各种飞行环境中始终保持最佳效率和性能, 已经成为未来先进飞行器重要特征和发展方向之一[1-2]. 在变体飞行器发展早期, 主要通过机械机构设计改变机翼形状、面积等参数改善飞行器的性能[3], 但是随之而来的附加重量增加、复杂的机械系统以及严苛的驱动需求却限制了其发展[4]. 直到20世纪80年代, 随着智能材料、先进驱动器、仿生学等学科的发展, 智能材料与结构、柔顺机构、分布式驱动等新的理念逐渐改善了变体飞行器结构设计方法[5-6]. 相关研究表明翼型沿弦向平滑变形可获得更好的气动性能, 在起降阶段通过调整机翼弯度使飞机实现增升减阻, 在巡航阶段自适应改变机翼弯度使其始终保持最优气动效率, 因此变弯度自适应机翼逐渐成为研究热点[7-9]. 此外, 为了综合考虑减轻结构质量、降低功耗, 柔顺机构也因此被应用于自适应机翼[6].
柔性机翼结构的设计主要通过驱动力克服结构内力来实现变形. Wang等[10-12]提出了基于波纹板结构的变体翼尖, 通过使用复合波纹板提供的刚度不对称性来实现变形, 如图1所示. 张盛等[13]提出了一种多变形控制点偏心梁驱动结构, 通过驱动与柔性后缘连接的偏心梁实现变形, 如图2所示. Woods等[14]提出了一种鱼骨主动变弯度机翼FishBAC, 如图3所示, 由朝向前缘的D型翼梁和朝向后缘的鱼骨柔性段所组成. 鱼骨柔性段则由细长、可弯曲的横梁及其上均匀分布的纵梁和预拉伸后的弹性基体复合材料EMC蒙皮构成.
但是由于传统的柔性结构具有变形大、质量轻等特点, 该特性将会导致柔性机翼较低的固有频率, 容易发生结构振动响应. 因此, 柔性机翼的固有振动特性在气动弹性分析[15-17]、动态特性分析[18]等领域具有重要意义. 目前针对机翼结构固有振动分析主要采用有限元法[19-20], 虽然该方法可以给出高精度的结果, 但是其存在计算量大、效率低的问题, 同时在飞机设计初期若对机翼参数进行大量修改将会带来一系列不便[21]. 柔性机翼相比于传统机翼结构, 为了实现通过改变自身结构形状以获得最佳工作性能, 结构分布形式存在着非均匀的截面特性和质量分布. 二维机翼其动力学特性仅存在弯曲模态, 而真实机翼具有一定展弦比且存在多种复杂模态. 但是在针对变后缘柔性翼的研究中, 变后缘柔性部分一般处理为固支的悬臂板, 对于该模型其展弦比对于弯曲频率影响较小, 因此对于二维结构的分析可以近似推广到三维结构的弯曲固有频率. 本文拟通过针对该类变后缘柔性机翼进行简化, 将基于非均匀梁假设, 建立一类柔性翼的动力学模型, 进而研究一种求解二维柔性机翼固有振动特性的高效方法.
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1. 计算原理
1.1 变后缘柔性机翼结构模型
在典型变后缘柔性机翼结构建模时一般简化为梁模型[10,22-24], 然后通过有限元或实验方法进行验证. Woods等[22,24]在分析FishBAC结构特性[22]和静气动弹性特性[24]时, 利用平行轴定理将结构简化为非均匀悬臂梁进行分析. 为了更加准确地计算柔性机翼的气动弹性特性, Zhang等[25]基于FishBAC建立了一种刚柔耦合的结构模型, 将柔性段等效为轴向流作用下的柔性二维板, 计算结果与实验数据比较取得十分理想的结果. 本文将基于平行轴定理将该类变后缘柔性翼简化为非均匀梁并应用于分析结构的动力学特性, 以提供准确高效的结构固有频率与模态分析.
1.2 非均匀梁的幂级数解法
非均匀梁的固有振动方程为变系数常微分方程, 基本无法获得精确解, 常采用Frobenius方法[26]求解近似解析解. 该方法基于弹性力学的连续性假设, 也就是结构内的一些物理量可以由连续函数来表示[27].
Naguleswaran [28]提出了一种求解受线性轴力作用的等截面欧拉梁广义幂级数解法. 徐腾飞等[29]将该方法推广至截面特性、轴力、质量分布任意连续变化的欧拉梁. 但是没有考虑梁剪切变形和截面转动惯性的影响, 欧拉梁在求解短粗梁振动问题和高阶固有频率时将得不到满意的结果[30], 因此许多学者将剪切梁理论引入以获得更精确的结果.
目前基于非均匀剪切梁的研究主要集中在功能梯度材料梁FGM, 其材料属性和截面面积在域内非均匀分布. 杜运兴等[31]使用一种幂级数法对基于剪切梁理论的变截面FGM梁固有频率、振型进行了求解. Şimşek[32]使用不同高阶梁理论对FGM的基频进行了分析, 结果表明使用各种高阶梁理论所得结果差异很小. 本文将上述方法推广至变后缘柔性机翼的固有振动分析中, 以FishBAC为例将其柔性段等效为非均匀悬臂梁分析其动力学特性.
2. 理论分析
2.1 柔性机翼的等效力学模型
鱼骨柔性翼段模型参数如表1所示. 在FishBAC柔性段中, 只有横梁、纵梁以及蒙皮对结构刚度和质量具有贡献, 且总刚度和总质量为三部分刚度和质量线性叠加. 横梁视为等矩形截面梁且中性轴与翼型中弧线重合. 虽然蒙皮的弹性模量较小, 但是蒙皮由于贴附在纵梁上与结构中性轴有较大距离, 根据平行轴定理[33]可知蒙皮对结构刚度的贡献不可忽略. 纵梁由于在机翼厚度方向上尺寸较大且弦向尺寸较小, 相较于前两者刚度大得多而质量小得多, 在结构分析中可以将其处理为刚体且质量忽略不计. 因此可以得到截面抗弯刚度EI(x)和质量线密度ρS(x)
表 1 鱼骨柔性翼段模型参数Table 1. Model parameters of the FishBAC morphing conceptParameters Value baseline airfoil NACA0012 chord c/mm 305 span b/mm 150 start of morph xS/mm 107 end of morph xE/mm 260 number of stringer pairs n 14 stringer thickness tst/mm 0.8 skin thickness tsk/mm 1.5 spine thickness tbs/mm 2 stringer modulus Est/GPa 2.14 spine modulus Ebs/GPa 2.14 spine Poisson's ratio νbs 0.4 spine density ρbs/(kg·m−3) 1010 skin modulus Esk/MPa 4.56 skin Poisson's ratio νsk 0.4 skin density ρsk/(kg·m−3) 1010 $$ \left. \begin{gathered} EI\left( x \right) = E{I_{{\text{bs}}}} + E{I_{{\text{sk}}}}\left( x \right) \\ \rho S\left( x \right) = \rho {S_{{\text{bs}}}} + \rho {S_{{\text{sk}}}}\left( x \right) \\ \end{gathered} \right\} $$ (1) 式中下标bs和sk分别代表横梁和蒙皮. 如图4所示, 进一步简化为截面积不变, 即宽度b和厚度h均与横梁相同, 材料的弹性模量E(x)和密度ρ(x)在弦向非均匀分布的悬臂梁, 可以得到
$$ \left. \begin{gathered} E\left( x \right) = {E_{{\text{bs}}}} + {E_{{\text{sk}}}}\frac{{{I_{{\text{sk}}}}\left( x \right)}}{{{I_{{\text{bs}}}}}} \\ \rho \left( x \right) = {\rho _{{\text{bs}}}} + {\rho _{{\text{sk}}}}\frac{{{S_{{\text{sk}}}}\left( x \right)}}{{{S_{{\text{bs}}}}}} \\ \end{gathered} \right\} $$ (2) $$ G = \frac{E}{{2\left( {1 + \nu } \right)}} $$ (3) 对于线弹性材料, 材料的剪切模量G由式(3)可得[33]. 由于蒙皮的剪切模量远小于横梁, 因此可以忽略, 即等效非均匀梁剪切模量等于横梁的剪切模量.
2.2 基于欧拉梁理论的幂级数解法
欧拉梁固有振动方程[34]为
$$ \frac{{{{\text{d}}^2}}}{{{\text{d}}{x^2}}}\left( {EI(x)\frac{{{{\text{d}}^2}y}}{{{\text{d}}{x^2}}}} \right) - \rho S(x){\omega ^2}y = 0 $$ (4) 式中ω为梁的振动频率. 由1.2节可知, 非均匀梁的弹性模量和质量分布均可采用多项式表示, 可得
$$ \left. \begin{gathered} EI(x) = \sum\limits_{i = 0}^{{N_1}} {{\alpha _i}} {x^i} \\ \rho S(x) = \sum\limits_{i = 0}^{{N_2}} {{\beta _i}} {x^i} \\ \end{gathered} \right\} $$ (5) 式中αi和βi由非均匀梁的性质可得, 为了简化计算, 本文使用三次多项式拟合非均匀梁的弹性模量和质量分布, 即多项式次数N1 = N2 = 3.
根据Frobenius方法, 假设方程(4)的精确解具有如下形式
$$ y(x,c) = {x^c}\sum\limits_{n = 0}^\infty {{a_{n + 1}}(c){x^n}} $$ (6) 不失一般性, 令
$$ {a_1}\left( c \right) = 1 $$ (7) 将式(5)和式(6)代入式(4), 并利于无穷级数的性质
$\displaystyle\sum\limits_{n = k}^\infty {{a_n}} = \displaystyle\sum\limits_{n = k - 1}^\infty {{a_{n + 1}}}$ 将所有的项的系数调整至c + n – 3, 利用多项式相等可得$$ \begin{split} & {a_{n + 2}} = - \sum\limits_{i = 0}^2 { {\frac{{(c + n - 1 - i)(c + n - i){\alpha _{1 + i}}}}{{(c + n + 1)(c + n){\alpha _0}}}{a_{n + 1 - i}}} } + \\ &\qquad {\omega ^2}\frac{{\displaystyle\sum\limits_{i = 0}^3 { {{\beta _i}{a_{n - 2 - i}}} } }}{{\displaystyle\prod\limits_{i = 0}^3 {(c + n + 1 - i)} {\alpha _0}}}\quad \left( {n \geqslant - 1} \right) \end{split} $$ (8) 式中当k≤0时, ak = 0. 令n = −1, 为了使式(7)成立, 必须满足
$$ c(c - 1)(c - 2)(c - 3) = 0 $$ 所以c = 0, 1, 2, 3, 根据Frobenius理论方程的解为
$$ y(x,c) = {C_0}\overline y (x,0) + {C_1}\overline y (x,1) + {C_2}\overline y (x,2) + {C_3}\overline y (x,3) $$ (9) 式中y(x,c)为解的基础解系, c取不同值对应着不同边界条件, 记为
$$ \overline y (x,c) = {x^c}\sum\limits_{n = 0}^\infty {{a_{n + 1}}(c){x^n}} $$ (10) 在计算过程中, 为了加快级数收敛, 对坐标进行规范化处理, 即X = x/l ≤1, Y = y/l.
首先利用其中一端边界条件确定所需的基本解系, 通过迭代动态调整级数解阶数n确保级数收敛, 并利用收敛的级数解作为近似解代替精确解. 在确保收敛后, 将方程中频率ω从0开始设置合适的步长进行增加, 代入式(8)中计算得到对应频率的基础解系. 然后, 判断基础解系是否满足另一边界条件, 若满足另一边界条件即为方程(4)的某一阶固有频率. 若需要得到更精确的结果可以根据计算得到初步结果缩小频率范围, 设置更小的步长可以得到更加精确的结果, 进一步可得固有频率对应的模态.
2.3 基于剪切梁理论的幂级数解法
由于需要考虑剪切变形和截面转动惯性, 因此需要在欧拉梁广义位移挠度y的基础上增加一个广义位移截面转角θ, 如图4虚线方框内所示. 根据剪切梁理论[34], 梁的位移函数可以表示为
$$ \left. \begin{gathered} u\left( {x,z,t} \right) = - \theta z \\ y\left( {x,z,t} \right) = y \\ \end{gathered} \right\} $$ (11) 式中, u和y分别是梁沿x轴和z轴方向的位移分量. 根据几何方程[27]可以得到相应的正应变ε和剪应变γ为
$$ \left. \begin{gathered} \varepsilon = - z\frac{{\partial \theta }}{{\partial x}} \\ \gamma = \frac{{\partial y}}{{\partial x}} - \theta \\ \end{gathered} \right\} $$ (12) 由于材料满足线弹性假设, 所以梁的应变能U和动能K为
$$ \left. \begin{split} & U = \frac{1}{2}\int_0^L {\int_A^{} {\left( {E\left( x \right)\varepsilon _{}^2 + \kappa G\left( x \right)\gamma _{}^2} \right)} } {\text{ d}}A{\text{d}}x = \\ &\qquad {\frac{1}{2}\int_0^L {\left[ {I{{\left( {\frac{{\partial \theta }}{{\partial x}}} \right)}^2} + D{{\left( {\frac{{\partial y}}{{\partial x}} - \theta } \right)}^2}} \right]} {\text{d}}x} \\ & K = \frac{1}{2}\int_0^L {\int_A^{} {\rho \left( x \right)\left[ {{{\left( {\frac{{\partial u}}{{\partial t}}} \right)}^2} + {{\left( {\frac{{\partial y}}{{\partial t}}} \right)}^2}} \right]} } {\text{ d}}A{\text{d}}x = \\ &\qquad {\frac{1}{2}\int_0^L {\left[ {{J_1}{{\left( {\frac{{\partial \theta }}{{\partial t}}} \right)}^2} + {J_2}{{\left( {\frac{{\partial y}}{{\partial t}}} \right)}^2}} \right]} {\text{d}}x}\end{split} \right\} $$ (13) 式中κ为剪切系数[27], 其他项分别为
$$ \left. \begin{split} & I = \int_A {E\left( x \right){\text{d}}A} = bhE\left( x \right) \\ & D = \int_A {\kappa G\left( x \right){\text{d}}A} = \kappa bhG\left( x \right) \\ & {J_1} = \int_A {{z^2}\rho \left( x \right){\text{d}}A} = \frac{{b{h^3}}}{{12}}\rho \left( x \right) \\ & {J_2} = \int_A {\rho \left( x \right){\text{d}}A} = bh\rho \left( x \right)\end{split} \right\} $$ (14) 假定梁的两端边界受到2个作用力, 分别为剪力Q和弯矩M, 因此外力做功W可以表述为
$$ W = {\left( {M\theta + Qy} \right)_{x = 1}} - {\left( {M\theta + Qy} \right)_{x = 0}} $$ (15) $$ \int_{{t_1}}^{{t_2}} {\left( {\delta K - \delta U + \delta W} \right){\text{d}}t = 0} $$ (16) $$ \left. \begin{gathered} \theta (x,t) = \varTheta (x){{\rm{e}}^{{\rm{j}}\omega t}} \\ y(x,t) = Y(x){{\rm{e}}^{{\rm{j}}\omega t}}\end{gathered} \right\} $$ (17) 利用哈密顿原理, 将式(13) ~ 式(15)代入式(16)并结合式(17)进行分离变量, 式中j是虚数. 得到相应的固有振动方程组和边界条件为
$$ \left. \begin{aligned} & {J_2}(x)Y(x){\omega ^2} + \frac{{\text{d}}}{{{\text{d}}x}}\left[ {D(x)\left( {\frac{{{\text{d}}Y(x)}}{{{\text{d}}x}} - \varTheta (x)} \right)} \right] = 0 \\ & {J_1}(x)\varTheta (x){\omega ^2} + \frac{{\text{d}}}{{{\text{d}}x}}\left( {I\left( x \right)\frac{{{\text{d}}\varTheta (x)}}{{{\text{d}}x}}} \right)+ \\ &\qquad D(x)\left( {\frac{{{\text{d}}Y(x)}}{{{\text{d}}x}} - \varTheta (x)} \right) = 0 \end{aligned} \right\} $$ (18) $$ \left. \begin{gathered} Q = D\left( {\frac{{\partial y}}{{\partial x}} - \theta } \right) \\ M = I\frac{{\partial \theta }}{{\partial x}} \\ \end{gathered} \right\} $$ (19) 将(18)无量纲化可得
$$ \left. \begin{aligned} & {s^2}\frac{{\text{d}}}{{{\text{d}}X}}\left[ {D(X)\left( {\frac{{{\text{d}}Y(X)}}{{{\text{d}}X}} - {{\varTheta }}(X)} \right)} \right] + {\beta ^2}{J_2}(X)W(X) = 0 \\ & \frac{{\text{d}}}{{{\text{d}}X}}\left( {I(X)\frac{{{{{\rm{d}}\varTheta }}(X)}}{{{\text{d}}X}}} \right) + {s^2}D(X)\left( {\frac{{{\text{d}}Y(X)}}{{{\text{d}}X}} - {{\varTheta }}(X)} \right) + \\ &\qquad {\beta ^2}{\mu ^2}{J_1}(X){{\varTheta }}(X) = 0 \\ \end{aligned} \right\} $$ (20) 无纲量化参数为
$$ \left. \begin{gathered} X = \frac{x}{l},\;\;W(X) = \frac{{W(x)}}{l},\;\;{\mu ^2} = {\left( {\frac{{{h_0}}}{l}} \right)^2} \\ {s^2} = \frac{{\kappa {G_0}{A_0}{l^2}}}{{{E_0}{I_0}}},\;\;\beta = \sqrt {\frac{{{\rho _0}{\omega ^2}{l^4}}}{{{E_0}h_0^2}}} \\ {I_0} = {b_0}h_0^3,\;\;{A_0} = {b_0}{h_0} \\ \end{gathered} \right\} $$ (21) 式中, 下标为0的项均为X = 0时非均匀梁对应的材料或截面参数. 将非均匀梁的弹性模量、剪切模量和质量分布均采用多项式形式表达, 即
$$ \left. \begin{gathered} E\left( X \right) = \sum\limits_{i = 0}^{{N_1}} {{\alpha _i}{X^i}} \\ \rho \left( X \right) = \sum\limits_{i = 0}^{{N_2}} {{\beta _i}{X^i}} \\ G\left( X \right) = \sum\limits_{i = 0}^{{N_3}} {{\eta _i}{X^i}} \\ \end{gathered} \right\} $$ (22) 采用Frobenius方法求解该微分方程组, 其精确解表示成幂级数形式
$$ \left.\begin{gathered} W(X) = \sum\limits_{i = 0}^\infty {{a_i}{X^i}} \\ \varTheta (X) = \sum\limits_{i = 0}^\infty {{b_i}{X^i}} \\ \end{gathered}\right\} $$ (23) 将式(14)、式(22)和式(23)代入式(20), 同2.2节类似即可得到级数系数关系式. 考虑本文模型简化分析, 取N1 = 3, N2 = N3 = 1, 级数系数关系式(i > 2)为
$$ \left. \begin{split} & {a_i} = \frac{{{b_{i - 1}}}}{{\left( {i - 1} \right)}} - \frac{{{\beta ^2}{a_{i - 2}}}}{{{s^2}\left( {i - 1} \right)\left( {i - 2} \right)}} \\ & {b_i} = - \frac{{\displaystyle\sum\limits_{j = 1}^3 {\left( {i - 1 - j} \right){\alpha _j}{b_{i - j}}} }}{{\left( {i - 1} \right){\alpha _0}}} - \\ &\qquad 12\frac{{{s^2}\left( {i - 2} \right){a_{i - 1}} + \left( {\dfrac{{{\beta ^2}{\mu ^2}}}{{12}} - {s^2}} \right){b_{i - 2}}}}{{\left( {i - 1} \right)\left( {i - 2} \right)}}\end{split} \right\} $$ (24) 式中当k≤0时, ak = bk = 0. 进一步, 令a1, a2, b1, b2分别为1, 其余项为0, 即可计算出给定频率下的基础解系, 记为y1, y2, θ1, θ2. 不同的基础解对应着不同的边界条件, 此方法后续计算过程与2.2节中欧拉梁计算方法类似，在此不再赘述.
3. 结果与讨论
3.1 算法正确性及可行性验证
为了验证算法的正确性, 首先计算等截面悬臂梁, 两种不同梁理论的级数解均选取前1000阶确保级数稳定收敛, 并与文献[34]中的精确解和NASTRAN结果(使用400个Beam单元)进行比较, 相关参数详见表2.
表 2 验证算例参数Table 2. Parameters of verification exampleParameters Value length l/mm 152.5 density ρ/(kg·m−3) 2700 width b/mm 1 thickness t/mm 2 Poisson's ratio ν 0.4 modulus of elasticity E/GPa 2.14 表3和图5中为计算所得频率与模态, 图5中W为无量纲位移. 计算结果误差均不超过1%, 验证了算法的正确性与可行性. 同时发现, 由于剪切梁理论考虑了结构的剪切变形和截面转动惯性，使得其计算结果小于基于欧拉梁理论的计算结果, 且由于NASTRAN中的BEAM单元不考虑转动惯量的影响使其结果介于两种梁理论之间, 这与文献[30]中的论述一致.
表 3 等截面悬臂梁的频率(Hz)Table 3. Frequency of cantilever beam with equal section (Hz)Methods Vibration order 1 2 3 4 Euler beam solution 12.3679 77.5085 217.025 425.283 Euler exact solution 12.3678 77.3732 217.022 425.280 shear beam series solution 12.3656 77.4313 216.513 423.434 FEM (NASTRAN) 12.3665 77.4470 216.619 423.817 3.2 对柔性机翼的验证与分析
根据表1的数据, 采用商用软件Patran 2020对鱼骨柔性翼柔性段原模型和等效非均匀梁模型进行建模. 柔性段原模型中横梁与纵梁采用Shell单元建模, 如图6所示; 非均匀梁模型采用15个Beam单元建模. 边界条件均为柔性段起始位置固支.
两种不同梁理论的级数解均选取前1000阶确保级数稳定收敛, 不同模型计算所得频率和模态如表4和图7、图8所示, 图6所示结果为在判断频率是否为固有频率时是否符合另一边界条件判断值绝对值的对数, 当判断值为0时出现峰值, 对应频率为某一阶固有频率, 图8中Shell单元的有限元计算结果为横梁中轴面的模态.
表 4 FishBAC柔性段的频率(Hz)Table 4. Frequency of the FishBAC (Hz)Methods Vibration order 1 2 3 4 FEM (shell) 22.7621 118.163 292.228 — FEM (beam) 24.4647 136.962 368.522 709.733 Euler beam series solution 24.5482 138.507 373.935 725.166 shear beam series solution 24.5339 138.151 371.708 717.439 通过上述计算可以得到简化的非均匀梁模型在计算FishBAC柔性段基频和较低阶模态时具有较高精度. 该模型计算在较高阶频率具有误差的原因为非均匀梁模型的等效相较于原模型刚度有所提高, 使得频率计算结果偏大.
图9分析了级数解阶数n和刚度拟合阶数N对频率计算的影响. 由于计算原理略有不同, 基于剪切梁理论的级数解法比基于欧拉梁理论的级数解法收敛更快, 在n = 30阶前4阶模态即可完成全部收敛. 对于越高阶的频率随着n的增加收敛越滞后, 则是由于拟合模态函数的级数越多拟合高阶模态的能力越强, 可进一步在方程中求出对应阶固有频率. 刚度拟合阶数越高一定程度会提高级数解求解的精确性, 但是过高的刚度拟合阶数一定程度会影响收敛效率和求解的稳定性.
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图 9 级数解与刚度拟合的阶数对频率的影响Figure 9. Effect of order of series solution and stiffness fitting polynomial on frequency![]()
9 级数解与刚度拟合的阶数对频率的影响(续)9. Effect of order of series solution and stiffness fitting polynomial on frequency (continued)3.3 柔性翼结构振动实验
在前文作者使用了有限元方法对FishBAC柔性段的等效力学模型进行了数值验证. 本节将进一步通过比较鱼骨柔性段的基频解析解和实验值对等效力学模型及算法进行验证. 实验样件鱼骨部分采用3D打印制作, 材料为ABS塑料; 蒙皮材料为白色硅胶, 两种材料的杨氏模量均由实验测得.
考虑到3D打印机精度受限会使试验件存在材料不均匀, 将会导致材料打印后力学特性与基材存在一定差异. 针对该问题, 利用动态法测量杨氏模量, 即通过测量3D打印的悬臂梁试件的固有频率并根据试件几何参数测得材料的杨氏模量, 固有频率的测量通过给予悬臂梁初始扰动并利用激光传感器采集结构响应, 对结构响应进行快速傅里叶分析(FFT)即可获得固有频率. 硅胶材料采用单向拉伸实验方法测得力学性能, 按照国标GB/T 528-2009加工3个实验样件并在实验样件等距涂若干标记点, 采用基于图片识别的方法测量加载过程中产生的形变, 通过应变-应力曲线测得弹性模量. 两种实验方案如图10、图11所示, 实验结果如表5、图12所示.
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图 10 ABS塑料动态法测杨氏模量实验实物图Figure 10. Dynamic measurement test of Young's modulus of ABS plastic表 5 材料特性实验结果Table 5. Experimental results of material propertiesParameters Value modulus of elasticity (ABS plastics) E1/GPa 1.823 modulus of elasticity (silicone rubber) E2/MPa 2.919 ![]()
图 12 硅胶单向拉伸实验应变-应力曲线Figure 12. Strain-stress curve of silicone rubber in uniaxial tensile test采用动态法测量FishBAC的固有频率与基于不同梁理论求解及有限元模型的结果进行比较, FishBAC试验件与试验方案如图13所示, 测量结果如表6和图14所示.
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图 13 动态法测FishBAC固有频率实验实物图Figure 13. Dynamic measurement test of the natural frequency of FishBAC flexible section表 6 FishBAC柔性段的固有频率(Hz)Table 6. The natural frequency of FishBAC (Hz)Methods Frequency dynamic measurement test 19.4521 FEM (shell element) 18.5898 Euler beam series solution 19.9821 shear beam series solution 19.9654 ![]()
图 14 动态法测量FishBAC固有频率的实验结果Figure 14. Dynamic measurement test results of the natural frequency of FishBAC通过计算发现级数解对于结构固有频率的计算误差在3%以内, 因此本文使用的等效力学模型和级数解算法可以用于柔性机翼结构的固有频率分析.
4. 结论
本文通过将二维柔性机翼简化为等效非均匀梁模型, 并基于欧拉梁和剪切梁理论结合Frobenius方法, 提出了一种快速准确求解二维柔性机翼固有振动特性的建模方法, 并结合有限元和实验进行验证, 主要结论如下:
(1) 建立了FishBAC柔性段等效力学模型, 可以快速准确地计算变后缘柔性段的固有频率, 相较于修改较为复杂的有限元方法, 可以在设计初期提供快速分析其动力学特性的方法;
(2) 通过分析发现了基于剪切梁理论分析所得到的固有频率计算精度更高, 且收敛速度大于欧拉梁理论;
(3) 基于此方法求得的模态整体具有较高的精度, 可以应用于后续分析, 例如基于刚柔耦合的机翼颤振特性分析;
(4) 该方法可以推广到其他类型的变体飞行器非均匀结构动力学特性分析中.
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图 9 级数解与刚度拟合的阶数对频率的影响
Figure 9. Effect of order of series solution and stiffness fitting polynomial on frequency
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9 级数解与刚度拟合的阶数对频率的影响(续)
9. Effect of order of series solution and stiffness fitting polynomial on frequency (continued)
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图 10 ABS塑料动态法测杨氏模量实验实物图
Figure 10. Dynamic measurement test of Young's modulus of ABS plastic
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图 12 硅胶单向拉伸实验应变-应力曲线
Figure 12. Strain-stress curve of silicone rubber in uniaxial tensile test
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图 13 动态法测FishBAC固有频率实验实物图
Figure 13. Dynamic measurement test of the natural frequency of FishBAC flexible section
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图 14 动态法测量FishBAC固有频率的实验结果
Figure 14. Dynamic measurement test results of the natural frequency of FishBAC
表 1 鱼骨柔性翼段模型参数
Table 1 Model parameters of the FishBAC morphing concept
Parameters Value baseline airfoil NACA0012 chord c/mm 305 span b/mm 150 start of morph xS/mm 107 end of morph xE/mm 260 number of stringer pairs n 14 stringer thickness tst/mm 0.8 skin thickness tsk/mm 1.5 spine thickness tbs/mm 2 stringer modulus Est/GPa 2.14 spine modulus Ebs/GPa 2.14 spine Poisson's ratio νbs 0.4 spine density ρbs/(kg·m−3) 1010 skin modulus Esk/MPa 4.56 skin Poisson's ratio νsk 0.4 skin density ρsk/(kg·m−3) 1010 
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表 2 验证算例参数
Table 2 Parameters of verification example
Parameters Value length l/mm 152.5 density ρ/(kg·m−3) 2700 width b/mm 1 thickness t/mm 2 Poisson's ratio ν 0.4 modulus of elasticity E/GPa 2.14
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表 3 等截面悬臂梁的频率(Hz)
Table 3 Frequency of cantilever beam with equal section (Hz)
Methods Vibration order 1 2 3 4 Euler beam solution 12.3679 77.5085 217.025 425.283 Euler exact solution 12.3678 77.3732 217.022 425.280 shear beam series solution 12.3656 77.4313 216.513 423.434 FEM (NASTRAN) 12.3665 77.4470 216.619 423.817
下载: 导出CSV
表 4 FishBAC柔性段的频率(Hz)
Table 4 Frequency of the FishBAC (Hz)
Methods Vibration order 1 2 3 4 FEM (shell) 22.7621 118.163 292.228 — FEM (beam) 24.4647 136.962 368.522 709.733 Euler beam series solution 24.5482 138.507 373.935 725.166 shear beam series solution 24.5339 138.151 371.708 717.439
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表 5 材料特性实验结果
Table 5 Experimental results of material properties
Parameters Value modulus of elasticity (ABS plastics) E1/GPa 1.823 modulus of elasticity (silicone rubber) E2/MPa 2.919
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表 6 FishBAC柔性段的固有频率(Hz)
Table 6 The natural frequency of FishBAC (Hz)
Methods Frequency dynamic measurement test 19.4521 FEM (shell element) 18.5898 Euler beam series solution 19.9821 shear beam series solution 19.9654
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