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 中国舰船研究  2017, Vol. 12 Issue (4): 102-109  DOI: 10.3969/j.issn.1673-3185.2017.04.016 0

### 引用本文 [复制中英文]

[复制中文]
WANG M H, LI K, QIU Y K, et al. Free vibration characteristics analysis of rectangular plate with rectangular opening based on Fourier series method[J]. Chinese Journal of Ship Research, 2017, 12(4): 102-109. DOI: 10.3969/j.issn.1673-3185.2017.04.016.
[复制英文]

### 文章历史

1 华中科技大学船舶与海洋工程学院, 湖北 武汉 430074;
2 中国舰船研究设计中心, 湖北 武汉 430064;
3 船舶与海洋水动力湖北省重点实验室, 湖北 武汉 430074;
4 高新船舶与深海开发装备协同创新中心, 上海 200240

Free vibration characteristics analysis of rectangular plate with rectangular opening based on Fourier series method
WANG Minhao1 , LI Kai2 , QIU Yongkang1 , LI Tianyun1,3,4 , ZHU Xiang1,3
1 School of Naval Architecture and Ocean Engineering, Huazhong University of Science and Technology, Wuhan 430074, China;
2 China Ship Development and Design Center, Wuhan 430064, China;
3 Hubei Key Laboratory of Naval Architecture and Ocean Engineering Hydrodynamics, Wuhan 430074, China;
4 Collaborative Innovation Center for Advanced Ship and Deep-Sea Exploration, Shanghai 200240, China
Abstract: Plate structures with openings are common in many engineering structures. The study of the vibration characteristics of such structures is directly related to the vibration reduction, noise reduction and stability analysis of an overall structure. This paper conducts research into the free vibration characteristics of a thin elastic plate with a rectangular opening parallel to the plate in an arbitrary position. We use the improved Fourier series to represent the displacement tolerance function of the rectangular plate with an opening. We can divide the plate into an eight zone plate to simplify the calculation. We then use linear springs, which are uniformly distributed along the boundary, to simulate the classical boundary conditions and the boundary conditions of the boundaries between the regions. According to the energy functional and variational method, we can obtain the overall energy functional. We can also obtain the generalized eigenvalue matrix equation by studying the extremum of the unknown improved Fourier series expansion coefficients. We can then obtain the natural frequencies and corresponding vibration modes of the rectangular plate with an opening by solving the equation. We then compare the calculated results with the finite element method to verify the accuracy and effectiveness of the method proposed in this paper. Finally, we research the influence of the boundary condition, opening size and opening position on the vibration characteristics of a plate with an opening. This provides a theoretical reference for practical engineering application.
Key words: arbitrary opening position    rectangular plate with an opening    improved Fourier series method    energy variational method

0 引言

Li[12]提出了任意支撑梁振动分析的改进傅里叶级数方法，并随后被拓展应用到矩形板[13]和圆柱壳[14]等结构的振动分析之中；文献[13]表明任意边界条件下板的假定振型函数可以不变地用一种改进傅里叶级数形式表示。本文将引入改进傅里叶级数方法建立任意边界条件下开口矩形板的振动分析模型，并应用区域划分思想将开口在任意位置的开口板划分为8块区域板，采用沿边界均匀分布的位移约束弹簧和转角约束弹簧模拟板的边界条件，然后运用能量泛函变分方法对结构振动问题进行求解，并与有限元仿真运算结果进行对比分析以说明文中方法的准确性，最后讨论边界条件、开口尺寸和开口位置对开口板振动特性的影响，以便为工程应用提供理论指导。

1 理论分析 1.1 开口矩形板模型描述

 图 1 开口矩形板示意图 Figure 1 Rectangular plate with a rectangular opening

 图 2 开口板物理模型描述图 Figure 2 Physical model of rectangular plate with an opening

 ${w^{\left[ j \right]}}\left( {x,y,t} \right) = \left( {\sum\limits_{m = 1}^M {\sum\limits_{n = 1}^N {A_{mn}^{\left[ j \right]}\phi _m^{\left[ j \right]}\left( x \right)\psi _n^{\left[ j \right]}\left( y \right)} } } \right){{\rm{e}}^{{\rm{i}}\omega t}}$ (1)

 $\left\{ \begin{array}{l} {\phi _m}\left( {{x^ * }} \right) = \sin \left( {m{\rm{\pi }}{x^ * }} \right)\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;0 < m < 5\\ {\phi _m}\left( {{x^ * }} \right) = \cos \left[ {\left( {m - 5} \right){\rm{\pi }}{x^ * }} \right]\;\;\;\;\;\;\;m \ge 5 \end{array} \right.$ (2)
 $\left\{ \begin{array}{l} {\psi _n}\left( {{y^ * }} \right) = \sin \left( {n{\rm{\pi }}{y^ * }} \right)\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;0 < n < 5\\ {\psi _n}\left( {{y^ * }} \right) = \cos \left[ {\left( {n - 5} \right){\rm{\pi }}{y^ * }} \right]\;\;\;\;\;\;\;n \ge 5 \end{array} \right.$ (3)

1.2 含开孔板振动固有特征的能量分析

 $\begin{array}{*{20}{c}} {L = \left( {V_{\rm{p}}^{\left[ 1 \right]} + V_{\rm{p}}^{\left[ 2 \right]} + \cdots + V_{\rm{p}}^{\left[ 8 \right]}} \right) + \left( {V_{\rm{s}}^{\left[ 1 \right]} + V_{\rm{s}}^{\left[ 2 \right]} + \cdots + V_{\rm{s}}^{\left[ 8 \right]}} \right) - }\\ {\left( {{T^{\left[ 1 \right]}} + {T^{\left[ 2 \right]}} + \cdots + {T^{\left[ 8 \right]}}} \right)} \end{array}$ (4)

 $\begin{array}{*{20}{c}} {V_{\text{p}}^{\left[ j \right]} = } \\ {\frac{{{D_{\text{p}}}}}{2}\iint {\left[ \begin{gathered} {\left( {\frac{{{\partial ^2}{w^{\left[ j \right]}}}}{{\partial {x^2}}}} \right)^2} + {\left( {\frac{{{\partial ^2}{w^{\left[ j \right]}}}}{{\partial {y^2}}}} \right)^2} + 2\mu \left( {\frac{{{\partial ^2}{w^{\left[ j \right]}}}}{{\partial {x^2}}}} \right)\left( {\frac{{{\partial ^2}{w^{\left[ j \right]}}}}{{\partial {y^2}}}} \right) \hfill \\ + 2\left( {1 - \mu } \right){\left( {\frac{{{\partial ^2}{w^{\left[ j \right]}}}}{{\partial x\partial y}}} \right)^2} \hfill \\ \end{gathered} \right]{\text{d}}x{\text{d}}y}} \end{array}$ (5)

 ${T^{\left[ j \right]}} = \frac{{\rho h}}{2}\iint {{{\left( {\frac{{\partial {w^{\left[ j \right]}}}}{{\partial t}}} \right)}^2}{\text{d}}x{\text{d}}y}$ (6)

 $\begin{array}{*{20}{c}} {V_{\rm{s}}^{\left[ 1 \right]} = \frac{1}{2}\int_{{a_1}}^{{a_2}} {\left\{ \begin{array}{l} {\left[ {{k_1}{{\left( {{w^{\left[ 1 \right]}}} \right)}^2} + {K_1}{{\left( {\frac{{\partial {w^{\left[ 1 \right]}}}}{{\partial y}}} \right)}^2}} \right]_{y = 0}}\\ + {\left[ {{k_{21}}{{\left( {{w^{\left[ 1 \right]}}} \right)}^2} + {K_{21}}{{\left( {\frac{{\partial {w^{\left[ 1 \right]}}}}{{\partial y}}} \right)}^2}} \right]_{y = {b_1}}} \end{array} \right\}{\rm{d}}x} + }\\ {\frac{1}{2}\int_0^{{b_1}} {\left\{ \begin{array}{l} {\left[ {{k_{14}}{{\left( {{w^{\left[ 1 \right]}} - {w^{\left[ 2 \right]}}} \right)}^2} + {K_{14}}{{\left( {\frac{{\partial {w^{\left[ 1 \right]}}}}{{\partial x}} - \frac{{\partial {w^{\left[ 2 \right]}}}}{{\partial x}}} \right)}^2}} \right]_{x = {a_1}}}\\ + {\left[ {{k_{13}}{{\left( {{w^{\left[ 1 \right]}} - {w^{\left[ 8 \right]}}} \right)}^2} + {K_{13}}{{\left( {\frac{{\partial {w^{\left[ 1 \right]}}}}{{\partial x}} - \frac{{\partial {w^{\left[ 8 \right]}}}}{{\partial x}}} \right)}^2}} \right]_{x = {a_2}}} \end{array} \right\}{\rm{d}}y} } \end{array}$ (7)

 $\left\{ \begin{array}{l} \frac{{\partial L}}{{\partial A_{mn}^{\left[ 1 \right]}}} = \frac{{\partial V_{\rm{p}}^{\left[ 1 \right]}}}{{\partial A_{mn}^{\left[ 1 \right]}}} + \frac{{\partial V_{\rm{s}}^{\left[ 1 \right]}}}{{\partial A_{mn}^{\left[ 1 \right]}}} + \frac{{\partial V_{\rm{s}}^{\left[ 2 \right]}}}{{\partial A_{mn}^{\left[ 1 \right]}}} + \frac{{\partial V_{\rm{s}}^{\left[ 8 \right]}}}{{\partial A_{mn}^{\left[ 1 \right]}}} - \frac{{\partial {T^{\left[ 1 \right]}}}}{{\partial A_{mn}^{\left[ 1 \right]}}} = 0\\ \frac{{\partial L}}{{\partial A_{mn}^{\left[ j \right]}}} = \frac{{\partial V_{\rm{p}}^{\left[ j \right]}}}{{\partial A_{mn}^{\left[ j \right]}}} + \frac{{\partial V_{\rm{s}}^{\left[ {j - 1} \right]}}}{{\partial A_{mn}^{\left[ j \right]}}} + \frac{{\partial V_{\rm{s}}^{\left[ j \right]}}}{{\partial A_{mn}^{\left[ j \right]}}} + \frac{{\partial V_{\rm{s}}^{\left[ {j + 8} \right]}}}{{\partial A_{mn}^{\left[ j \right]}}} - \frac{{\partial {T^{\left[ j \right]}}}}{{\partial A_{mn}^{\left[ j \right]}}} = 0\\ \frac{{\partial L}}{{\partial A_{mn}^{\left[ 8 \right]}}} = \frac{{\partial V_{\rm{p}}^{\left[ 8 \right]}}}{{\partial A_{mn}^{\left[ 8 \right]}}} + \frac{{\partial V_{\rm{s}}^{\left[ 8 \right]}}}{{\partial A_{mn}^{\left[ 8 \right]}}} + \frac{{\partial V_{\rm{s}}^{\left[ 7 \right]}}}{{\partial A_{mn}^{\left[ 8 \right]}}} + \frac{{\partial V_{\rm{s}}^{\left[ 8 \right]}}}{{\partial A_{mn}^{\left[ 8 \right]}}} - \frac{{\partial {T^{\left[ 8 \right]}}}}{{\partial A_{mn}^{\left[ 8 \right]}}} = 0 \end{array} \right.$ (8)

 $\left( {\mathit{\boldsymbol{K}} - {\omega ^2}\mathit{\boldsymbol{M}}} \right)\mathit{\boldsymbol{A}} = 0$ (9)

 $\mathit{\boldsymbol{A}} = {\left\{ {A_{mn}^{\left[ 1 \right]},A_{mn}^{\left[ 2 \right]}, \cdots ,A_{mn}^{\left[ 8 \right]}} \right\}^{\rm{T}}}$ (10)
 $\mathit{\boldsymbol{A}}_{mn}^{\left[ j \right]} = {\left[ {A_{mn}^{\left[ j \right]},A_{mn}^{\left[ j \right]}, \cdots ,A_{mn}^{\left[ j \right]},A_{m,n + 1}^{\left[ j \right]}, \cdots ,A_{MN}^{\left[ j \right]}} \right]^{\rm{T}}}$ (11)

2 数值计算与分析

2.1 收敛性及有效性验证

 图 3 开口板前4阶振型对比图 Figure 3 Comparison of the first four order modes of rectangular plate with a rectangular opening

2.2 不同边界条件、开口尺寸和开口位置对开口板振动性能的影响分析

 图 4 不同边界下开口板前6阶固有频率对比图 Figure 4 Natural frequencies of rectangular plate with square opening in different boundaries

 图 5 SS-SS边界下开口板前6阶固有频率对比图 Figure 5 Natural frequencies of rectangular plate with different rectangular openings in SS-SS boundary
 图 6 F-F边界下开口板前6阶固有频率对比图 Figure 6 Natural frequencies of rectangular plate with different rectangular openings in F-F boundary

 图 7 开口位置分布描述图 Figure 7 The opening position of rectangular plate
 图 8 C-F边界下开口板前3阶固有频率对比图 Figure 8 Natural frequencies of rectangular plate with different opening positions in C-F boundary

3 结论

1）随着傅里叶级数截断项以及模拟弹簧刚度值的增加，计算结果收敛性良好，数值稳定性很好。

2）文中方法对于处理任意边界条件下开口在任意位置的开口矩形板的自由振动问题具有较高的精度。

3）随着边界约束刚度的增加，开口板的同阶固有频率不断增大；在边界约束刚度较大时，振动特性以刚度影响为主，开口板的同阶固有频率随开口尺寸的增加而增大，在边界约束刚度较小时，振动特性以质量影响为主，开口板的同阶固有频率随开口尺寸的增加而减小；在C-F边界下，开口板的首阶固有频率随着开口位置向板中心的靠近而增大。

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