A Wave Finite Element Method for Free Vibration Analysis of Lattice Core Sandwich Cylindrical Shells
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摘要: 针对点阵夹芯圆柱壳的自由振动分析发展了波有限元法. 首先基于自由波的传播规律，建立了点阵夹芯圆柱壳胞元的二维波有限元控制方程，相比于全尺寸有限元模型，显著缩减了控制方程的自由度规模；其次，基于Neumann级数推导了约束动刚度矩阵求逆的显式表达式，不仅可以提高计算效率，而且使得固有频率从控制方程中分离出来，从而将点阵夹芯圆柱壳的固有频率求解转化为单个胞元自由度规模的二次特征值问题；最后，根据结构振动模态与自由波的关系，给出了圆柱壳轴向和周向的波传播参数的表达式，进而求得点阵夹芯圆柱壳的固有频率和模态. 数值算例考虑了多种边界条件下的点阵夹芯圆柱壳自由振动问题，验证了该方法的正确性和高效性.Abstract: A wave finite element method was developed for the free vibration analysis of lattice core sandwich cylindrical shells. Firstly, based on the propagation law of free waves, governing equations for a core element of the lattice core sandwich cylindrical shell was established. Compared with the full-scale finite element model, degrees of freedom of the governing equations for a core element are significantly reduced. Secondly, an explicit expression for the inverse of the constrained dynamic stiffness matrix was derived based on the Neumann series, which not only improves computation efficiency but also separates the natural frequency from the governing equations, thereby transforming the natural frequency solution of the lattice core sandwich cylindrical shell into a quadratic eigenvalue problem of a core element. Finally, according to the relationship between the structural vibration mode and the free wave, the expressions of the axial and circumferential wave propagation parameters of the cylindrical shell were given, and the natural frequencies and modes of the lattice core sandwich cylindrical shell were obtained. Numerical examples of the free vibration analysis on a lattice core sandwich cylindrical shell under different boundary conditions verify the validity and efficiency of the proposed method.
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Key words:
- lattice core /
- cylindrical shell /
- free vibration /
- wave finite element /
- wave propagation
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图 1 点阵夹芯圆柱壳胞元的有限元模型
注 为了解释图中的颜色，读者可以参考本文的电子网页版本，后同.
Figure 1. The finite element model for a core element in a lattice core sandwich cylindrical shell
图 2 点阵夹芯圆柱壳及其胞元示意图
Figure 2. The lattice core sandwich cylindrical shell and a pyramidal lattice truss core element
图 3 本文方法和有限元方法计算的模态对比(m=4, n=3)
Figure 3. Natural mode shapes (m=4, n=3) obtained with the 2 methods
表 1 两端简支边界条件下(SS)点阵夹芯圆柱壳的自由振动频率对比
Table 1. Comparison of natural frequencies of the lattice core sandwich shell with simply supported boundary conditions at both ends
m n=2 n=3 n=4 n=5 present F/Hz FEM F/Hz error δ/% present F/Hz FEM F/Hz error δ/% present F/Hz FEM F/Hz error δ/% present F/Hz FEM F/Hz error δ/% 1 63.85 63.84 0.015 72.41 72.36 0.069 123.59 123.55 0.032 192.13 192.09 0.021 2 188.95 188.97 -0.011 127.99 127.91 0.063 145.66 145.51 0.103 204.58 204.41 0.083 3 321.37 321.43 -0.019 217.41 217.32 0.041 194.79 194.54 0.129 232.03 231.71 0.138 4 431.87 431.95 -0.019 311.86 311.77 0.029 261.98 261.67 0.118 275.34 274.87 0.171 5 516.00 516.07 -0.014 398.52 398.42 0.025 335.31 334.96 0.104 330.19 329.62 0.173 
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表 2 两端固支边界条件下(CC)点阵夹芯圆柱壳的自由振动频率对比
Table 2. Comparison of natural frequencies of the lattice core sandwich shell with clamped boundary conditions at both ends
m n=2 n=3 n=4 n=5 present F/Hz FEM F/Hz error δ/% present F/Hz FEM F/Hz error δ/% present F/Hz FEM F/Hz error δ/% present F/Hz FEM F/Hz error δ/% 1 122.00 101.70 19.961 93.63 87.42 7.104 131.34 127.91 2.682 196.78 193.72 1.580 2 256.90 212.26 21.031 170.71 151.09 12.986 167.13 158.03 5.758 216.22 210.19 2.869 3 380.00 331.64 14.582 265.10 235.26 12.684 226.97 210.42 7.865 251.90 241.61 4.259 4 477.01 436.91 9.178 356.53 323.48 10.217 298.44 276.14 8.076 301.67 286.81 5.181 5 549.70 520.28 5.655 437.62 406.78 7.581 371.90 347.02 7.170 360.24 341.75 5.410
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表 3 两端自由边界条件下(FF)点阵夹芯圆柱壳的自由振动频率对比
Table 3. Comparison of natural frequencies of the lattice core sandwich shell with free boundary conditions at both ends
m n=2 n=3 n=4 n=5 present F/Hz FEM F/Hz error δ/% present F/Hz FEM F/Hz error δ/% present F/Hz FEM F/Hz error δ/% present F/Hz FEM F/Hz error δ/% -1/2(λx=0) 22.80 22.71 0.396 63.59 63.33 0.411 119.66 119.10 0.470 189.09 188.16 0.494 1 28.18 24.04 17.221 64.68 65.04 -0.554 120.42 120.84 -0.348 189.79 189.79 0.000 2 122.00 129.83 -6.031 93.63 97.27 -3.742 131.34 134.23 -2.153 196.78 199.29 -1.259 3 256.90 269.57 -4.700 170.71 177.66 -3.912 167.13 172.35 -3.029 216.22 220.94 -2.136 4 380.00 396.43 -4.144 265.10 275.78 -3.873 226.97 234.54 -3.228 251.90 258.51 -2.557 5 477.01 492.12 -3.070 356.53 369.24 -3.442 298.44 308.07 -3.126 301.67 309.92 -2.662
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表 4 一端固支一端简支边界条件下(CS)点阵夹芯圆柱壳的自由振动频率对比
Table 4. Comparison of natural frequencies of the lattice core sandwich shell with one end clamped and the other end simplify supported
m n=2 n=3 n=4 n=5 present F/Hz FEM F/Hz error δ/% present F/Hz FEM F/Hz error δ/% present F/Hz FEM F/Hz error δ/% present F/Hz FEM F/Hz error δ/% 1 91.16 82.72 10.203 81.16 78.88 2.890 126.75 125.35 1.117 194.12 192.80 0.685 2 223.16 201.60 10.694 148.62 139.91 6.225 155.54 151.58 2.612 209.89 207.13 1.332 3 351.49 326.67 7.598 241.27 226.64 6.455 210.42 202.59 3.865 241.48 236.58 2.071 4 406.80 434.46 -6.367 288.68 317.77 -9.154 244.21 269.05 -9.232 263.21 280.86 -6.284 5 497.23 518.16 -4.039 377.88 402.64 -6.149 316.87 341.08 -7.098 315.70 335.73 -5.966
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表 5 一端固支一端自由边界条件下(CF)点阵夹芯圆柱壳的自由振动频率对比
Table 5. Comparison of natural frequencies of the lattice core sandwich shell with one end clamped and the other end free
m n=2 n=3 n=4 n=5 present F/Hz FEM F/Hz error δ/% present F/Hz FEM F/Hz error δ/% present F/Hz FEM F/Hz error δ/% present F/Hz FEM F/Hz error δ/% 1 28.18 31.89 -11.634 64.68 64.82 -0.216 120.42 119.95 0.392 189.79 189.01 0.413 2 122.00 110.66 10.248 93.63 91.28 2.574 131.34 130.96 0.290 196.78 196.49 0.148 3 256.90 239.50 7.265 170.71 163.47 4.429 167.13 164.90 1.352 216.22 215.57 0.302 4 380.00 363.26 4.608 265.10 254.43 4.194 226.97 221.95 2.262 251.90 249.97 0.772 5 477.01 465.45 2.484 356.53 345.96 3.055 298.44 291.65 2.328 301.67 298.22 1.157
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