一角点支撑另一对边固支正交各向异性矩形薄板弯曲的辛叠加解

数学物理学报 ›› 2021, Vol. 41 ›› Issue (3): 797-810.

一角点支撑另一对边固支正交各向异性矩形薄板弯曲的辛叠加解

寇天娇¹(),额布日力吐^1,*(),阿拉坦仓²()

¹ 内蒙古大学数学科学学院呼和浩特 010021
² 内蒙古师范大学呼和浩特 010022

收稿日期:2020-05-14 出版日期:2021-06-26 发布日期:2021-06-09
通讯作者: 额布日力吐 E-mail:1849208969@qq.com;ebu@imu.edu.cn;altanca@imu.edu.cn
作者简介:寇天娇, E-mail:1849208969@qq.com|阿拉坦仓, E-mail: altanca@imu.edu.cn
基金资助:
国家自然科学基金(11862019);国家自然科学基金(11761029);内蒙古自然科学基金(2020ZD01)

Symplectic Superposition Solution for the Bending of a Corner Point-Supported and the Other Opposite Edge Clamped Orthotropic Rectangular Thin Plate

Tianjiao Kou¹(), Eburilitu^1,*(), Alatancang²()

¹ School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021
² Inner Mongolia Normal University, Hohhot 010022

Received:2020-05-14 Online:2021-06-26 Published:2021-06-09
Contact: Eburilitu E-mail:1849208969@qq.com;ebu@imu.edu.cn;altanca@imu.edu.cn
Supported by:
the NSFC(11862019);the NSFC(11761029);the NSF of Inner Mongolia(2020ZD01)

摘要/Abstract

摘要：

该文研究均匀荷载下一角点支撑另一对边固支正交各向异性矩形薄板的弯曲问题.将该问题分解为两个对边滑支子问题和一个一边滑支对边简支子问题，再分别得到上述三个子问题所对应的Hamilton算子本征值及本征函数系，然后应用辛本征函数展开法分别求出这三个子问题的解，进而通过以上三个子问题解的叠加求解出原问题的辛叠加解.最后应用该文所得辛叠加解分别计算了各向同性和正交各向异性矩形薄板一些点处的挠度和弯矩值.

关键词: 正交各向异性矩形薄板, Hamilton算子, 本征函数, 角点支撑, 辛叠加解

Abstract:

The bending problem of a uniformly loaded orthotropic rectangular thin plate with a corner point-supported and the other opposite edge clamped is studied. First, we divide the bending problem into two sub-problems with two opposite edges slidingly clamped and another sub-problem with one edge slidingly clamped and its opposite edge simply supported. Then we obtain the eigenvalues and eigenfunctions of the Hamiltonian operator corresponding to the three sub-problems, respectively. Then the solutions of the above three sub-problems are solved by the method of symplectic eigenfunction expansion respectively. Finally, we obtain the symplectic superposition solution of the original bending problem by superposition of the solutions of the three sub-problems. In addition, we calculate the deflection and bending moment values at some points of the bending problems of isotropic rectangular plate and orthotropic rectangular plate by using the symplectic superposition solution obtained in this paper.

Key words: Orthotropic rectangular thin plate, Hamiltonian operator, Eigenfunction, Corner point-supported, Symplectic superposition solution

中图分类号:

O302

寇天娇,额布日力吐,阿拉坦仓. 一角点支撑另一对边固支正交各向异性矩形薄板弯曲的辛叠加解[J]. 数学物理学报, 2021, 41(3): 797-810.

Tianjiao Kou, Eburilitu, Alatancang. Symplectic Superposition Solution for the Bending of a Corner Point-Supported and the Other Opposite Edge Clamped Orthotropic Rectangular Thin Plate[J]. Acta mathematica scientia,Series A, 2021, 41(3): 797-810.

图/表 3

图 1

表 1

表 2

参考文献 18

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				$y = b/2$
$b/a$			$x = 0$	$x = a/4$	$x = a/2$	$x = 3a/4$	$x = a$
$1.0$	$Dw/(qa^{4})$	present	0.02943	0.02436	0.01842	0.01115	0.002789
		Ref.[10]	0.02949	0.02442	0.01847	0.01118	0.002783
	$M_y/(qa^{2})$	present	-0.01137	-0.008667	0.002233	0.01683	0.03146
		Ref.[10]	-0.01161	-0.008872	0.002031	0.01663	0.03130
$1.2$	$Dw/(qa^{4})$	present	0.05251	0.04247	0.03168	0.01956	0.006469
	$M_y/(qa^{2})$	present	0.01263	0.013099	0.022244	0.03556	0.05058
$1.4$	$Dw/(qa^{4})$	present	0.08433	0.06754	0.05032	0.03202	0.013093
	$M_y/(qa^{2})$	present	0.04384	0.041989	0.048879	0.06048	0.07534
$1.6$	$Dw/(qa^{4})$	present	0.12619	0.10093	0.07574	0.04992	0.023990
	$M_y/(qa^{2})$	present	0.08095	0.076966	0.081643	0.09153	0.10597
$1.8$	$Dw/(qa^{4})$	present	0.17962	0.14421	0.10954	0.07485	0.040737
	$M_y/(qa^{2})$	present	0.12328	0.117456	0.120189	0.12860	0.14255

				$y = b/2$
$b/a$			$x = 0$	$x = a/4$	$x = a/2$	$x = 3a/4$	$x = a$
$1.0$	$100E_{T}h^{3}Dw/(qa^{4})$	present	4.50209	4.33322	3.83644	2.59816	0.53881
	$M_y/(qa^{2})$	present	-0.269027	-0.247932	-0.206159	-0.101362	0.121649
$1.2$	$100E_{T}h^{3}Dw/(qa^{4})$	present	9.67375	8.79869	7.35414	4.77077	1.11120
	$M_y/(qa^{2})$	present	-0.382724	-0.341621	-0.264498	-0.106729	0.173482
$1.4$	$100E_{T}h^{3}Dw/(qa^{4})$	present	18.21916	15.81380	12.62913	7.97018	2.02218
	$M_y/(qa^{2})$	present	-0.499650	-0.437193	-0.320878	-0.104608	0.231455
$1.6$	$100E_{T}h^{3}Dw/(qa^{4})$	present	31.03660	26.01927	20.08335	12.44326	3.35009
	$M_y/(qa^{2})$	present	-0.612517	-0.528390	-0.372671	-0.096795	0.293216
$1.8$	$100E_{T}h^{3}Dw/(qa^{4})$	present	48.96215	40.04606	30.15277	18.44113	5.16290
	$M_y/(qa^{2})$	present	-0.716491	-0.609928	-0.416522	-0.084249	0.356611

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