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摘要:
柔性机翼在气动载荷作用下产生较大变形，几何非线性因素不容忽视。利用机翼柔性特点，通常采用梁模型进行结构建模。从几何精确梁理论出发，结合Hamilton原理推导了几何非线性梁的动力学平衡方程。不同于经典的位移基有限元，采用梁广义应变作为插值变量，得到广义质量阵、广义阻尼阵、刚度阵及载荷列向量，建立非线性应变梁模型。结合Newmark数值算法和牛顿-拉夫森（Newdon-Raphson）迭代法建立了动力学方程求解算法。针对典型算例，开展静、动力学分析，并分别与有限元软件的仿真结果进行对比。结果表明：在相当计算精度下，所建模型收敛特性更好。为进一步验证所建模型在工程实际应用中的精度和有效性，开展针对典型大展弦比机翼的地面静力试验。试验表明：仿真结果与激光位移计和光纤传感设备测得的变形值具有较高的一致性，验证了所建模型具有较高精度。
Abstract:Under the action of aerodynamic forces, flexible wings undergo large deformations, for which geometric nonlinearity cannot be ignored. By taking advantage of their slenderness, flexible wings can be modeled as beams. This paper developed the dynamic equilibrium equations of nonlinear beams based on the geometrically exact beam theory and Hamilton principle. Different from the classical displacement-based finite element, this work took the generalized strains as interpolated variables and obtained the generalized mass matrix, damping matrix, stiffness matrix and force vectors, based on which a strain-based nonlinear beam model was proposed. The nonlinear dynamic equations were then solved by Newmark method combined with Newton-Raphson iterations with typical examples investigated from both static and dynamic perspectives. Next, the results were compared with those calculated by finite-element software, which proved that the strain-based beam model has better convergence with the comparable accuracy. The method was further verified by static ground tests of a large-aspect-ratio wing model, during which laser displacement sensor and fiber optic sensing technology were utilized to measure the structural deformation. The excellent agreement between numerical results calculated by the proposed method and the test results further validated the accuracy of the strain-based formulation.
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Key words:
- geometric nonlinearity /
- strain /
- beam /
- structural dynamics /
- finite-element method
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图 7 本文模型非线性静力学求解迭代收敛曲线
Figure 7. Iterative curve for solving nonlinear statics of proposed model
图 8 悬臂梁在定向载荷和随动载荷作用下的变形
Figure 8. Deformation of cantilevered beam subjected to dead force and follower force
图 9 端部定向正弦载荷作用下位移响应
Figure 9. Responses of tip displacement subject to sinusoidal dead force
图 10 端部随动正弦载荷作用下位移响应
Figure 10. Responses of tip displacement subject to sinusoidal follower force
表 1 悬臂梁模型物理属性
Table 1. Properties of cantilevered beam
参数 数值 长度L/m 1.00 弹性模量E/GPa 210 剪切模量G/GPa 84.2 横截面宽度b/mm 35 横截面高度h/mm 1.5 密度$\rho $/(kg·m−3) $7.75 \times {10^3}$ 扭转惯量${I_x}$/(kg·m) $5.37 \times {10^{ - 9}}$ 面外弯曲惯量${I_y}$/(kg·m) $9.84 \times {10^{ - 12}}$ 面内弯曲惯量${I_{\textit{z}}}$/(kg·m) $5.36 \times {10^{ - 9}}$ 
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表 2 不同端部载荷作用下端部垂向位移
Table 2. Tip vertical displacements with respect to different tip forces
端部载荷/N 端部垂向位移/m 相对误差/% Nastran仿真 本文模型 0.6 0.09584 0.09578 −0.063 1.2 0.18655 0.18643 −0.064 1.8 0.26852 0.26836 −0.060 2.4 0.34020 0.33999 −0.062 3.0 0.40165 0.40140 −0.062
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表 3 不同端部载荷作用下端部轴向位移
Table 3. Tip axial displacements with respect to different tip forces
端部载荷/N 端部轴向位移/m 相对误差/% Nastran仿真 本文模型 0.6 −0.005525 −0.005523 −0.040 1.2 −0.021120 −0.021110 −0.047 1.8 −0.044360 −0.044340 −0.045 2.4 −0.072380 −0.072350 −0.041 3.0 −0.102720 −0.102670 −0.049
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表 4 本文模型各迭代步端部垂向位移
Table 4. Tip vertical displacements of each iteration step in proposed model
迭代次数 端部垂向位移/m 1 0.447575 2 0.386251 3 0.405881 4 0.400023 5 0.401817 6 0.401272 7 0.401438 8 0.401388 9 0.401403 10 0.401398
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表 6 不同展向位置机翼垂向变形
Table 6. Vertical displacements at different positions
展向位置/m 试验值/m 垂向变形${{\textit{z}}_2}$/m ${{\textit{z}}_2} - {{\textit{z}}_1} $的
相对误差/%激光
位移计光纤
传感平均值${{\textit{z}}_1}$ 0.2 −0.01851 −0.02457 −0.02154 −0.02364 −0.21 0.4 −0.07833 −0.08559 −0.08196 −0.08243 −0.05 0.6 −0.15509 −0.16753 −0.16131 −0.16149 −0.02 0.8 −0.24355 −0.25978 −0.25167 −0.25058 0.11 1.0 −0.34 −0.35587 −0.34794 −0.34323 0.47
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