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基于辛叠加方法的正交各向异性矩形悬臂薄板受迫振动解析解

王森林 李进宝 马红艳 李锐

王森林, 李进宝, 马红艳, 李锐. 基于辛叠加方法的正交各向异性矩形悬臂薄板受迫振动解析解[J]. 应用数学和力学, 2024, 45(9): 1117-1132. doi: 10.21656/1000-0887.440277
引用本文: 王森林, 李进宝, 马红艳, 李锐. 基于辛叠加方法的正交各向异性矩形悬臂薄板受迫振动解析解[J]. 应用数学和力学, 2024, 45(9): 1117-1132. doi: 10.21656/1000-0887.440277
WANG Senlin, LI Jinbao, MA Hongyan, LI Rui. Analytical Forced Vibration Solutions of Orthotropic Cantilever Rectangular Thin Plates With the Symplectic Superposition Method[J]. Applied Mathematics and Mechanics, 2024, 45(9): 1117-1132. doi: 10.21656/1000-0887.440277
Citation: WANG Senlin, LI Jinbao, MA Hongyan, LI Rui. Analytical Forced Vibration Solutions of Orthotropic Cantilever Rectangular Thin Plates With the Symplectic Superposition Method[J]. Applied Mathematics and Mechanics, 2024, 45(9): 1117-1132. doi: 10.21656/1000-0887.440277

基于辛叠加方法的正交各向异性矩形悬臂薄板受迫振动解析解

doi: 10.21656/1000-0887.440277
(我刊编委李锐来稿)
基金项目: 

国家自然科学基金 12372067

国家自然科学基金 1202220

国家自然科学基金 11972103

详细信息
    作者简介:

    王森林(1998—),男,硕士生(E-mail: 2719116256@mail.dlut.edu.cn)

    李进宝(1999—),男,硕士生(E-mail: jinbao_li@mail.dlut.edu.cn)

    马红艳(1968—),女,教授,博士,硕士生导师(E-mail: mahy@dlut.edu.cn)

    通讯作者:

    李锐(1985—),男,教授,博士,博士生导师(通讯作者. E-mail: ruili@dlut.edu.cn)

  • 中图分类号: O302

Analytical Forced Vibration Solutions of Orthotropic Cantilever Rectangular Thin Plates With the Symplectic Superposition Method

(Contributed by LI Rui, M.AMM Editorial Board)
  • 摘要: 基于辛叠加方法研究了正交各向异性矩形悬臂薄板在谐载载荷作用下的受迫振动问题. 首先从薄板受迫振动的基本方程出发,将问题导入到Hamilton体系,并将原问题拆分为若干子问题,然后在辛空间中利用分离变量和本征展开方法推导出子问题的解析解,最后通过叠加求解出悬臂薄板受迫振动的解析解. 辛叠加方法的主要优点是经过逐步严格推导获得解析解,不需要对解的形式做任何假设,突破了传统半逆解法的限制. 算例针对不同谐载载荷情况进行了数值计算,并将该文方法与有限元方法获得的结果进行比较,验证了该文方法的可靠性和精确性.
    1)  (我刊编委李锐来稿)
  • 图  1  正交各向异性矩形薄板示意图

    Figure  1.  Schematic diagram of an orthotropic rectangular thin plate

    图  2  矩形悬臂薄板受迫振动问题的辛叠加示意图

    Figure  2.  Symplectic superposition of the forced vibration of a cantilever rectangular thin plate

    图  3  不同谐载频率下正交各向异性悬臂板x=a/2处的无量纲挠度沿y轴变化曲线

    Figure  3.  Non-dimensional deflections along the y axis at x=a/2 of an orthotropic cantilever plate under different frequencies of harmonic loads

    图  4  正交各向异性悬臂板在不同弯曲刚度下的频率响应曲线

      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  4.  Frequency response curves of orthotropic cantilever plates with different flexural rigidities

    表  1  均布谐载下正交各向异性悬臂板的无量纲挠度和弯矩收敛性分析

    Table  1.   Convergence of non-dimensional deflections and bending moments of an orthotropic cantilever plate under a uniformly distributed harmonic load

    ω/ω11 N
    10 20 30 40 50 60 70 80 90 100
    $\bar{w}_{p}(0, 0)$ 0.3 0.282 1 0.282 0 0.282 0 0.282 0 0.282 0 0.282 0 0.282 0 0.282 0 0.282 0 0.282 0
    0.5 0.343 0 0.342 9 0.342 9 0.342 8 0.342 8 0.342 8 0.342 8 0.342 8 0.342 8 0.342 8
    0.8 0.718 4 0.718 0 0.717 9 0.717 9 0.717 9 0.717 9 0.717 9 0.717 9 0.717 9 0.717 9
    1.1 1.236 5 1.237 5 1.237 7 1.237 7 1.237 8 1.237 8 1.237 8 1.237 8 1.237 8 1.237 8
    $ \bar{w}_{p}(0, 0.4 b) $ 0.3 0.130 5 0.130 4 0.130 4 0.130 4 0.130 4 0.130 4 0.130 4 0.130 4 0.130 4 0.130 4
    0.5 0.157 8 0.157 7 0.157 7 0.157 7 0.157 7 0.157 7 0.157 7 0.157 7 0.157 7 0.157 7
    0.8 0.326 1 0.325 9 0.325 8 0.325 8 0.325 8 0.325 8 0.325 8 0.325 8 0.325 8 0.325 8
    1.1 0.550 0 0.550 4 0.550 5 0.550 5 0.550 5 0.550 5 0.550 5 0.550 5 0.550 5 0.550 5
    $ \bar{M}_{y}(0.5 a, b) $ 0.3 -0.603 0 -0.578 9 -0.593 2 -0.583 2 -0.590 7 -0.584 3 -0.584 3 -0.584 4 -0.584 4 -0.584 4
    0.5 -0.720 4 -0.691 5 -0.708 5 -0.696 5 -0.705 6 -0.697 9 -0.697 9 -0.698 0 -0.698 0 -0.698 0
    0.8 -1.443 6 -1.384 5 -1.418 7 -1.394 7 -1.412 7 -1.397 4 -1.397 6 -1.397 7 -1.397 8 -1.397 8
    1.1 -2.319 1 -2.225 1 -2.281 0 -2.242 8 -2.271 5 -2.247 4 -2.247 7 -2.247 9 -2.248 1 -2.248 1
    下载: 导出CSV

    表  2  中心点处作用集中谐载的正交各向异性悬臂板的无量纲挠度和弯矩收敛性分析

    Table  2.   Convergence of non-dimensional deflections and bending moments of an orthotropic cantilever plate under a concentrated harmonic load at the center

    ω/ω11 N
    10 30 50 70 90 110 120 130 140 150
    $\bar{w}_{p 0}(0, 0)$ 0.3 0.237 3 0.238 3 0.238 4 0.238 4 0.238 4 0.238 4 0.238 4 0.238 4 0.238 4 0.238 4
    0.5 0.290 5 0.291 8 0.291 9 0.291 9 0.291 9 0.291 9 0.291 9 0.291 9 0.291 9 0.291 9
    0.8 0.619 2 0.621 4 0.621 7 0.621 7 0.621 7 0.621 8 0.621 8 0.621 8 0.621 8 0.621 8
    1.1 1.092 4 1.098 0 1.098 5 1.098 6 1.098 7 1.098 7 1.098 7 1.098 7 1.098 7 1.098 7
    $\bar{w}_{p 0}(0, 0.4 b)$ 0.3 0.114 8 0.115 1 0.115 2 0.115 2 0.115 2 0.115 2 0.115 2 0.115 2 0.115 2 0.115 2
    0.5 0.138 7 0.139 1 0.139 2 0.139 2 0.139 2 0.139 2 0.139 2 0.139 2 0.139 2 0.139 2
    0.8 0.286 1 0.287 0 0.287 1 0.287 1 0.287 1 0.287 1 0.287 1 0.287 1 0.287 1 0.287 1
    1.1 0.480 8 0.483 3 0.483 6 0.483 6 0.483 7 0.483 7 0.483 7 0.483 7 0.483 7 0.483 7
    $\bar{M}_{y 0}(0.5 a, b)$ 0.3 -0.630 6 -0.621 5 -0.619 1 -0.613 3 -0.613 4 -0.613 4 -0.613 5 -0.613 5 -0.613 5 -0.613 5
    0.5 -0.733 8 -0.723 2 -0.720 5 -0.713 6 -0.713 7 -0.713 8 -0.713 8 -0.713 8 -0.713 8 -0.713 8
    0.8 -1.367 9 -1.348 5 -1.343 4 -1.329 9 -1.330 1 -1.330 3 -1.330 4 -1.330 3 -1.330 4 -1.330 4
    1.1 -1.924 8 -1.902 7 -1.895 5 -1.875 1 -1.875 5 -1.875 7 -1.876 0 -1.875 9 -1.876 0 -1.876 0
    下载: 导出CSV

    表  3  各向同性悬臂板的无量纲挠度和弯矩

    Table  3.   Non-dimensional deflections and bending moments of an isotropic cantilever plate

    ω/ω11 method $\bar{w}_{p}\left(\bar{w}_{p 0}\right), x=0$ $\bar{w}_{p}\left(\bar{w}_{p 0}\right), y=0$ $\bar{M}_{y}\left(\bar{M}_{y 0}\right), y=b$
    y=0 y=0.1b y=0.3b x=0.1a x=0.3a x=0.5a x=0.1a x=0.3a x=0.5a
    uniformly distributed harmonic load 0.3 present 0.137 9 0.119 3 0.082 4 0.047 8 0.138 3 0.138 8 0.139 0 -0.548 4 -0.562 5
    FEM 0.137 9 0.119 3 0.082 4 0.047 8 0.138 3 0.138 8 0.139 0 -0.538 0 -0.554 5
    ref. [20] 0.142 2 0.123 0 0.084 8 0.049 1 0.142 5 0.142 9 0.143 1 -0.581 1 -0.569 7
    0.5 present 0.167 7 0.144 9 0.099 9 0.057 7 0.168 1 0.168 8 0.169 0 -0.653 4 -0.671 1
    FEM 0.167 7 0.144 9 0.099 9 0.057 7 0.168 1 0.168 8 0.169 1 -0.641 3 -0.662 0
    ref. [20] 0.174 2 0.150 5 0.102 5 0.059 7 0.174 5 0.175 0 0.175 2 -0.696 8 -0.684 7
    0.8 present 0.350 9 0.302 7 0.207 3 0.118 6 0.351 9 0.353 5 0.354 0 -1.298 9 -1.339 5
    FEM 0.350 9 0.302 7 0.207 3 0.118 6 0.351 9 0.353 5 0.354 0 -1.276 1 -1.322 4
    ref. [20] 0.381 6 0.329 0 0.225 0 0.128 5 0.382 5 0.383 8 0.384 2 -1.448 1 -1.430 7
    1.1 present 0.605 5 0.520 7 0.353 3 0.199 2 0.607 4 0.610 4 0.611 4 -2.067 1 -2.145 7
    FEM 0.605 8 0.521 0 0.353 5 0.199 3 0.607 7 0.610 6 0.611 7 -2.035 0 -2.123 1
    concentrated harmonic load at the center 0.3 present 0.114 1 0.099 4 0.070 0 0.041 4 0.115 7 0.118 2 0.119 3 -0.492 3 -0.581 7
    FEM 0.114 0 0.099 4 0.070 0 0.041 4 0.115 6 0.118 2 0.119 3 -0.483 5 -0.572 9
    ref. [20] 0.116 3 0.101 3 0.071 3 0.042 1 0.117 8 0.120 4 0.121 5 -0.451 1 -0.568 7
    0.5 present 0.140 1 0.121 8 0.085 3 0.050 1 0.141 8 0.144 5 0.145 6 -0.584 4 -0.677 1
    FEM 0.140 0 0.121 8 0.085 3 0.050 1 0.141 7 0.144 5 0.145 6 -0.574 1 -0.667 3
    ref. [20] 0.143 8 0.125 0 0.087 4 0.051 3 0.145 4 0.148 1 0.149 2 -0.536 8 -0.664 7
    0.8 present 0.300 4 0.259 8 0.179 3 0.103 4 0.302 5 0.306 0 0.307 5 -1.149 9 -1.262 9
    FEM 0.300 3 0.259 8 0.179 3 0.103 4 0.302 5 0.306 0 0.307 4 -1.130 2 -1.246 2
    ref. [20] 0.322 1 0.278 4 0.191 9 0.110 5 0.324 1 0.327 5 0.328 9 -1.090 0 -1.283 4
    1.1 present 0.536 6 0.460 7 0.311 3 0.174 7 0.537 0 0.537 4 0.537 5 -1.794 1 -1.784 9
    FEM 0.536 9 0.461 0 0.311 4 0.174 8 0.537 3 0.537 6 0.537 6 -1.766 0 -1.767 2
    下载: 导出CSV

    表  4  均布谐载下正交各向异性悬臂板的无量纲挠度和弯矩(b=a)

    Table  4.   Non-dimensional deflections and bending moments of orthotropic cantilever plates under uniformly distributed harmonic loads, with b=a

    Dy/Dx ω/ω11 method $\bar{w}_p, x=0$ $\bar{w}_p, y=0$ $\bar{M}_y, y=b$
    y=0 y=0.4b y=0.8b x=0.1a x=0.3a x=0.5a x=0.1a x=0.3a x=0.5a
    0.5 0.3 present 0.282 0 0.130 4 0.016 3 0.282 8 0.283 9 0.284 4 -0.541 8 -0.585 8 -0.584 4
    FEM 0.282 0 0.130 4 0.016 3 0.282 8 0.284 0 0.284 4 -0.524 1 -0.577 5 -0.580 9
    0.5 present 0.342 8 0.157 7 0.019 5 0.343 8 0.345 3 0.345 8 -0.644 5 -0.699 2 -0.698 1
    FEM 0.342 8 0.157 7 0.019 5 0.343 8 0.345 3 0.345 8 -0.623 6 -0.689 6 -0.694 0
    0.8 present 0.717 9 0.325 8 0.039 0 0.720 1 0.723 5 0.724 7 -1.276 4 -1.398 2 -1.397 8
    FEM 0.717 5 0.325 6 0.039 0 0.719 7 0.723 1 0.724 3 -1.235 7 -1.379 5 -1.390 3
    1.1 present 1.237 8 0.550 5 0.062 7 1.242 0 1.248 5 1.251 0 -2.015 1 -2.243 2 -2.248 1
    FEM 1.239 0 0.551 0 0.062 7 1.243 2 1.249 7 1.252 2 -1.957 3 -2.220 2 -2.242 7
    1.0 0.3 present 0.141 1 0.064 6 0.007 7 0.141 6 0.142 4 0.142 7 -0.527 7 -0.586 8 -0.591 8
    FEM 0.141 1 0.064 6 0.007 7 0.141 6 0.142 4 0.142 7 -0.523 8 -0.580 7 -0.583 4
    0.5 present 0.171 5 0.078 1 0.009 1 0.172 2 0.173 1 0.173 5 -0.627 3 -0.700 6 -0.706 9
    FEM 0.171 5 0.078 1 0.009 1 0.172 1 0.173 1 0.173 5 -0.622 8 -0.693 5 -0.697 2
    0.8 present 0.359 1 0.161 1 0.018 2 0.360 5 0.362 8 0.363 6 -1.240 6 -1.401 7 -1.416 5
    FEM 0.359 2 0.161 1 0.018 2 0.360 6 0.362 8 0.363 7 -1.233 1 -1.389 1 -1.398 8
    1.1 present 0.619 9 0.272 2 0.029 0 0.622 7 0.627 1 0.628 8 -1.956 6 -2.254 3 -2.283 9
    FEM 0.620 0 0.272 2 0.029 0 0.622 8 0.627 2 0.628 9 -1.948 2 -2.237 9 -2.259 7
    下载: 导出CSV

    表  5  均布谐载下正交各向异性悬臂板的无量纲挠度和弯矩(b=2a)

    Table  5.   Non-dimensional deflections and bending moments of orthotropic cantilever plates under uniformly distributed harmonic loads, with b=2a

    Dy/Dx ω/ω11 method $\bar{w}_p, x=0$ $\bar{w}_p, y=0$ $\bar{M}_y, y=b$
    y=0 y=0.4b y=0.8b x=0.1a x=0.3a x=0.5a x=0.1a x=0.3a x=0.5a
    0.5 0.3 present 4.615 4 2.159 0 0.282 4 4.616 6 4.618 5 4.619 1 -1.974 6 -2.402 5 -2.481 2
    FEM 4.606 3 2.154 8 0.281 9 4.607 5 4.609 3 4.610 0 -2.008 7 -2.385 6 -2.433 1
    0.5 present 5.610 5 2.611 4 0.337 9 5.612 0 5.614 4 5.615 3 -2.347 3 -2.866 4 -2.962 1
    FEM 5.616 9 2.614 2 0.338 3 5.618 4 5.620 7 5.621 5 -2.395 1 -2.854 6 -2.913 1
    0.8 present 11.753 5 5.403 1 0.680 3 11.757 1 11.762 6 11.764 6 -4.645 4 -5.727 6 -5.928 0
    FEM 11.754 3 5.403 3 0.680 3 11.757 9 11.763 5 11.765 5 -4.739 4 -5.700 9 -5.827 5
    1.1 present 20.257 0 9.139 5 1.102 1 20.264 0 20.274 9 20.278 9 -7.316 8 -9.167 9 -9.513 3
    FEM 20.526 7 9.262 1 1.117 2 20.534 0 20.545 3 20.549 5 -7.585 9 -9.262 4 -9.492 9
    1.0 0.3 present 2.320 6 1.080 4 0.134 7 2.321 1 2.321 8 2.322 0 -2.052 6 -2.441 8 -2.469 3
    FEM 2.320 9 1.080 5 0.134 7 2.321 5 2.322 2 2.322 5 -1.986 0 -2.415 4 -2.469 5
    0.5 present 2.820 3 1.306 3 0.161 0 2.821 0 2.821 9 2.822 2 -2.438 1 -2.913 1 -2.948 3
    FEM 2.818 8 1.305 6 0.160 9 2.819 5 2.820 5 2.820 8 -2.358 3 -2.880 3 -2.947 0
    0.8 present 5.905 2 2.700 3 0.323 1 5.906 8 5.909 0 5.909 8 -4.814 4 -5.819 3 -5.902 5
    FEM 5.915 0 2.704 7 0.323 6 5.916 6 5.919 0 5.919 9 -4.669 6 -5.768 4 -5.914 1
    1.1 present 10.204 8 4.577 2 0.522 5 10.208 1 10.212 7 10.214 3 -7.581 3 -9.343 8 -9.511 6
    FEM 10.147 8 4.551 3 0.519 4 10.151 2 10.156 1 10.158 1 -7.308 0 -9.201 4 -9.465 7
    下载: 导出CSV

    表  6  中心点处作用集中谐载的正交各向异性悬臂板的无量纲挠度和弯矩(b=a)

    Table  6.   Non-dimensional deflections and bending moments of orthotropic cantilever plates under concentrated harmonic loads at the center, with b=a

    Dy/Dx ω/ω11 method $\bar{w}_{p 0}, x=0$ $\bar{w}_{p 0}, y=0$ $\bar{M}_{y 0}, y=b$
    y=0 y=0.4b y=0.8b x=0.1a x=0.3a x=0.5a x=0.1a x=0.3a x=0.5a
    0.5 0.3 present 0.238 4 0.115 2 0.014 7 0.240 1 0.242 7 0.243 7 -0.505 7 -0.594 5 -0.613 5
    FEM 0.239 8 0.116 3 0.015 0 0.241 2 0.243 3 0.243 7 -0.497 0 -0.589 5 -0.608 3
    0.5 present 0.291 9 0.139 2 0.017 5 0.293 8 0.296 7 0.297 7 -0.596 3 -0.694 7 -0.713 8
    FEM 0.293 3 0.140 4 0.017 8 0.294 9 0.297 3 0.297 7 -0.585 0 -0.688 6 -0.708 3
    0.8 present 0.621 8 0.287 1 0.034 7 0.624 7 0.629 2 0.630 9 -1.152 9 -1.310 4 -1.330 4
    FEM 0.623 3 0.288 3 0.035 0 0.625 9 0.629 9 0.630 9 -1.125 0 -1.297 1 -1.322 2
    1.1 present 1.098 7 0.483 7 0.054 7 1.101 4 1.105 6 1.107 2 -1.740 2 -1.891 4 -1.876 0
    FEM 1.097 1 0.482 3 0.054 4 1.100 1 1.105 0 1.107 3 -1.679 9 -1.866 6 -1.871 2
    1.0 0.3 present 0.119 6 0.055 1 0.006 2 0.121 0 0.123 2 0.124 0 -0.466 2 -0.608 5 -0.650 0
    FEM 0.120 8 0.056 0 0.006 4 0.122 0 0.123 8 0.124 1 -0.472 5 -0.606 9 -0.640 4
    0.5 present 0.146 5 0.067 0 0.007 5 0.148 1 0.150 5 0.151 3 -0.554 7 -0.709 6 -0.752 4
    FEM 0.147 8 0.068 0 0.007 7 0.149 1 0.151 1 0.151 4 -0.560 9 -0.707 4 -0.741 6
    0.8 present 0.312 7 0.140 6 0.015 6 0.315 0 0.318 5 0.319 7 -1.098 9 -1.331 8 -1.382 0
    FEM 0.314 3 0.141 7 0.015 8 0.316 3 0.319 2 0.319 8 -1.103 3 -1.325 1 -1.363 9
    1.1 present 0.554 6 0.243 2 0.026 3 0.556 0 0.558 4 0.559 4 -1.733 2 -1.905 5 -1.894 0
    FEM 0.552 2 0.241 6 0.026 0 0.554 1 0.557 3 0.559 4 -1.708 7 -1.884 1 -1.874 8
    下载: 导出CSV

    表  7  中心点处作用集中谐载的正交各向异性悬臂板的无量纲挠度和弯矩(b=2a)

    Table  7.   Non-dimensional deflections and bending moments of orthotropic cantilever plates under concentrated harmonic loads at the center, with b=2a

    Dy/Dx ω/ω11 method $\bar{w}_{p 0}, x=0$ $\bar{w}_{p 0}, y=0$ $\bar{M}_{y 0}, y=b$
    y=0 y=0.4b y=0.8b x=0.1a x=0.3a x=0.5a x=0.1a x=0.3a x=0.5a
    0.5 0.3 present 1.937 8 0.980 7 0.139 4 1.938 2 1.938 6 1.938 8 -0.983 3 -1.185 2 -1.221 7
    FEM 1.934 2 0.979 0 0.139 2 1.934 5 1.935 0 1.935 1 -1.000 2 -1.177 4 -1.198 6
    0.5 present 2.370 1 1.177 5 0.163 6 2.370 5 2.371 2 2.371 5 -1.146 1 -1.387 7 -1.431 5
    FEM 2.373 6 1.179 1 0.163 8 2.374 1 2.374 7 2.375 0 -1.169 0 -1.382 2 -1.408 2
    0.8 present 5.039 4 2.391 4 0.312 6 5.040 8 5.042 8 5.043 5 -2.147 3 -2.633 3 -2.722 3
    FEM 5.042 4 2.392 7 0.312 7 5.043 8 5.045 8 5.046 6 -2.190 6 -2.622 1 -2.677 7
    1.1 present 8.874 9 3.928 8 0.461 8 8.878 2 8.883 3 8.885 2 -3.051 2 -3.838 5 -3.985 4
    FEM 8.824 1 3.905 7 0.458 9 8.827 4 8.832 4 8.834 2 -3.097 0 -3.799 0 -3.896 9
    1.0 0.3 present 0.978 4 0.490 4 0.066 4 0.978 5 0.978 5 0.978 5 -1.020 8 -1.205 4 -1.217 4
    FEM 0.978 6 0.490 5 0.066 4 0.978 6 0.978 7 0.978 7 -0.988 0 -1.192 4 -1.217 5
    0.5 present 1.196 1 0.589 0 0.077 9 1.196 3 1.196 4 1.196 4 -1.189 6 -1.411 7 -1.427 1
    FEM 1.195 5 0.588 7 0.077 8 1.195 6 1.195 7 1.195 8 -1.151 0 -1.395 9 -1.426 4
    0.8 present 2.540 8 1.197 0 0.148 6 2.541 3 2.542 0 2.542 2 -2.227 0 -2.680 6 -2.717 1
    FEM 2.545 7 1.199 2 0.148 9 2.546 3 2.547 0 2.547 3 -2.161 1 -2.657 6 -2.722 6
    1.1 present 4.483 7 1.975 6 0.219 9 4.485 2 4.487 5 4.488 3 -3.171 7 -3.927 2 -4.001 6
    FEM 4.459 3 1.964 5 0.218 6 4.461 0 4.463 4 4.464 4 -3.059 2 -3.867 5 -3.981 4
    下载: 导出CSV

    表  8  (a/4, 0)处作用集中谐载的正交各向异性悬臂板的无量纲挠度和弯矩(b=a)

    Table  8.   Non-dimensional deflections and bending moments of orthotropic cantilever plates under concentrated harmonic loads at (a/4, 0), with b=a

    Dy/Dx ω/ω11 method $\bar{w}_{p 0}, x=0$ $\bar{w}_{p 0}, y=0$ $\bar{M}_{y 0}, y=b$
    y=0 y=0.4b y=0.8b x=0.1a x=0.3a x=0.5a x=0.1a x=0.3a x=0.5a
    0.5 0.3 present 0.897 5 0.390 6 0.046 0 0.878 7 0.831 3 0.758 7 -1.399 6 -1.379 7 -1.235 3
    FEM 0.907 3 0.394 0 0.046 4 0.887 1 0.835 6 0.762 2 -1.384 1 -1.370 2 -1.232 8
    0.5 present 1.058 0 0.463 1 0.054 5 1.038 9 0.990 6 0.915 9 -1.671 8 -1.677 9 -1.525 7
    FEM 1.068 3 0.466 7 0.055 0 1.047 8 0.995 3 0.920 3 -1.652 2 -1.663 6 -1.522 5
    0.8 present 2.032 0 0.901 2 0.105 8 2.013 8 1.965 8 1.884 4 -3.344 3 -3.484 5 -3.318 1
    FEM 2.044 6 0.905 9 0.106 4 2.024 9 1.972 4 1.894 0 -3.260 5 -3.452 4 -3.310 8
    1.1 present 2.962 7 1.334 0 0.152 7 3.001 2 3.072 0 3.163 9 -5.033 3 -5.829 0 -6.038 9
    FEM 2.958 2 1.332 7 0.152 5 2.998 4 3.085 1 3.182 2 -4.878 6 -5.767 2 -6.024 8
    1.0 0.3 present 0.543 1 0.236 5 0.026 5 0.519 4 0.460 6 0.382 2 -1.624 4 -1.525 2 -1.255 8
    FEM 0.555 5 0.240 8 0.027 0 0.529 8 0.467 3 0.383 4 -1.641 7 -1.524 5 -1.245 4
    0.5 present 0.632 3 0.276 9 0.031 2 0.606 9 0.543 9 0.461 2 -1.926 3 -1.839 7 -1.550 3
    FEM 0.645 5 0.281 6 0.031 7 0.618 0 0.551 4 0.462 7 -1.945 1 -1.838 0 -1.537 1
    0.8 present 1.144 1 0.506 7 0.057 0 1.114 4 1.043 5 0.947 7 -3.687 3 -3.715 0 -3.364 2
    FEM 1.164 5 0.514 5 0.057 9 1.132 2 1.054 5 0.951 8 -3.666 5 -3.699 4 -3.338 2
    1.1 present 1.276 3 0.557 2 0.057 1 1.332 5 1.453 3 1.590 3 -4.208 2 -5.501 4 -6.133 2
    FEM 1.264 1 0.552 6 0.056 4 1.323 9 1.453 9 1.600 2 -4.137 6 -5.445 4 -6.072 5
    下载: 导出CSV

    表  9  (a/4, 0)处作用集中谐载的正交各向异性悬臂板的无量纲挠度和弯矩(b=2a)

    Table  9.   Non-dimensional deflections and bending moments of orthotropic cantilever plates under concentrated harmonic loads at (a/4, 0), with b=2a

    Dy/Dx ω/ω11 method $\bar{w}_{p 0}, x=0$ $\bar{w}_{p 0}, y=0$ $\bar{M}_{y 0}, y=b$
    y=0 y=0.4b y=0.8b x=0.1a x=0.3a x=0.5a x=0.1a x=0.3a x=0.5a
    0.5 0.3 present 6.504 9 2.821 9 0.340 9 6.445 4 6.316 0 6.164 4 -2.284 7 -2.648 6 -2.596 4
    FEM 6.513 0 2.822 7 0.340 8 6.453 1 6.322 1 6.167 5 -2.330 2 -2.634 0 -2.554 8
    0.5 present 7.784 6 3.403 4 0.412 3 7.724 8 7.594 9 7.565 5 -2.764 4 -3.242 9 -3.210 4
    FEM 7.813 0 3.413 4 0.413 3 7.752 9 7.621 3 7.465 6 -2.827 4 -3.234 8 -3.168 1
    0.8 present 15.669 6 6.986 2 0.851 7 15.610 9 15.481 9 15.393 4 -5.716 1 -6.909 8 -7.006 0
    FEM 15.702 4 6.998 3 0.853 1 15.643 3 15.512 5 15.355 2 -5.842 0 -6.887 8 -6.903 6
    1.1 present 25.350 4 11.650 1 1.432 2 25.425 1 25.579 9 25.609 7 -9.617 3 -12.187 7 -12.795 1
    FEM 25.238 0 11.601 5 1.426 2 25.313 2 25.469 9 25.640 0 -9.777 2 -12.088 8 -12.533 9
    1.0 0.3 present 3.572 0 1.558 5 0.183 2 3.486 9 3.306 3 3.090 6 -2.634 7 -2.827 9 -2.597 5
    FEM 3.586 2 1.561 9 0.183 5 3.500 2 3.316 4 3.109 3 -2.550 5 -2.810 1 -2.603 6
    0.5 present 4.224 1 1.854 6 0.218 1 4.137 2 3.952 7 3.730 5 -3.145 4 -3.439 0 -3.209 9
    FEM 4.237 2 1.857 6 0.218 3 4.149 3 3.961 6 3.750 3 -3.044 6 -3.414 2 -3.215 1
    0.8 present 8.211 3 3.660 0 0.429 0 8.120 2 7.926 2 7.700 7 -6.236 8 -7.185 1 -6.995 2
    FEM 8.245 7 3.672 5 0.430 4 8.153 5 7.956 4 7.734 4 -6.057 4 -7.143 7 -7.022 5
    1.1 present 12.410 5 5.649 1 0.650 7 12.517 1 12.738 7 12.967 4 -9.583 4 -12.238 8 -12.794 6
    FEM 12.353 7 5.626 1 0.648 1 12.461 7 12.687 2 12.932 3 -9.266 6 -12.068 7 -12.757 3
    下载: 导出CSV

    表  10  正交各向异性悬臂板的挠度和弯矩分布云图

    Table  10.   Contour plots of deflections and bending moments of the orthotropic cantilever plate

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  • 收稿日期:  2023-09-20
  • 修回日期:  2023-10-26
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