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

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

大连理工大学工程力学系工业装备结构分析优化与CAE软件全国重点实验室，辽宁大连 116024

(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
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- 文章访问数: 147
- HTML全文浏览量: 47
- PDF下载量: 64
- 被引次数: 0
出版历程
- 收稿日期: 2023-09-20
- 修回日期: 2023-10-26
- 刊出日期: 2024-09-01

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

State Key Laboratory of Structural Analysis, Optimization and CAE Software for Industrial Equipment, Department of Engineering Mechanics, Dalian University of Technology, Dalian, Liaoning 116024, P.R.China

(Contributed by LI Rui, M.AMM Editorial Board)

摘要

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.
- 辛叠加方法 /
- 正交各向异性薄板 /
- 解析解 /
- 受迫振动
Abstract: The forced vibrations of orthotropic cantilever rectangular thin plates under harmonic loadings were investigated with the symplectic superposition method. The basic equations for the forced vibration of thin plates were introduced into the Hamiltonian system. The original problem was divided into some fundamental subproblems, and the analytical solutions of the subproblems were derived with the method of separation of variables and through eigenvector expansion in the symplectic space. The solution of the original problem was finally obtained by superposition. The main advantage of the symplectic superposition method is that the analytical solution can be obtained by step-by-step rigorous derivation, without any assumptions on the form of the solution, which breaks through the limitations of traditional semi-inverse methods. The numerical results calculated corresponding to different harmonic loads were compared with those obtained via the finite element method to verify the reliability and accuracy of the proposed method.
- symplectic superposition method /
- orthotropic thin plate /
- analytical solution /
- forced vibration
edited-by

edited-by
注释:

1) (我刊编委李锐来稿)

HTML全文

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

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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

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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

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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

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

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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
	ω/ω₁₁	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

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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
	ω/ω₁₁	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

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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	$\bar{w}_{p}\left(\bar{w}_{p 0}\right), x=0$			$\bar{w}_{p}\left(\bar{w}_{p 0}\right), y=0$			$\bar{M}_{y}\left(\bar{M}_{y 0}\right), y=b$
	ω/ω₁₁	method	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
	1.1	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
	1.1	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

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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

D_y/D_x	ω/ω₁₁	method	$\bar{w}_p, x=0$			$\bar{w}_p, y=0$			$\bar{M}_y, y=b$
D_y/D_x	ω/ω₁₁	method	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
	0.3	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
	0.5	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
	0.8	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
	1.1	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
	0.3	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
	0.5	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
	0.8	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
	1.1	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

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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

D_y/D_x	ω/ω₁₁	method	$\bar{w}_p, x=0$			$\bar{w}_p, y=0$			$\bar{M}_y, y=b$
D_y/D_x	ω/ω₁₁	method	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
	0.3	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
	0.5	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
	0.8	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
	1.1	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
	0.3	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
	0.5	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
	0.8	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
	1.1	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

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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

D_y/D_x	ω/ω₁₁	method	$\bar{w}_{p 0}, x=0$			$\bar{w}_{p 0}, y=0$			$\bar{M}_{y 0}, y=b$
D_y/D_x	ω/ω₁₁	method	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
	0.3	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
	0.5	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
	0.8	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
	1.1	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
	0.3	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
	0.5	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
	0.8	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
	1.1	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

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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

D_y/D_x	ω/ω₁₁	method	$\bar{w}_{p 0}, x=0$			$\bar{w}_{p 0}, y=0$			$\bar{M}_{y 0}, y=b$
D_y/D_x	ω/ω₁₁	method	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
	0.3	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
	0.5	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
	0.8	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
	1.1	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
	0.3	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
	0.5	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
	0.8	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
	1.1	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

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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

D_y/D_x	ω/ω₁₁	method	$\bar{w}_{p 0}, x=0$			$\bar{w}_{p 0}, y=0$			$\bar{M}_{y 0}, y=b$
D_y/D_x	ω/ω₁₁	method	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
	0.3	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
	0.5	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
	0.8	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
	1.1	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
	0.3	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
	0.5	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
	0.8	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
	1.1	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

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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

D_y/D_x	ω/ω₁₁	method	$\bar{w}_{p 0}, x=0$			$\bar{w}_{p 0}, y=0$			$\bar{M}_{y 0}, y=b$
D_y/D_x	ω/ω₁₁	method	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
	0.3	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
	0.5	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
	0.8	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
	1.1	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
	0.3	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
	0.5	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
	0.8	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
	1.1	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

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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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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

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)

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

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留言板

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

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)

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出版历程

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

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出版历程

目录

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)