Free vibration characteristics of corrugated sandwich plates under different boundary conditions
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摘要: 波纹夹芯板作为一种特殊的复合材料结构，边界条件对其振动特性有重要影响。根据不同剪切方式下的剪切变形理论和基尔霍夫经典板理论(CLPT)，利用Hamilton原理建立波纹夹芯板的动力学方程。其中，波纹芯层等效成各向异性均质体。根据四边简支、四边固支、对边简支和固支、一边固支三边简支的边界条件，推导出位移形式的偏微分动力学方程。求解得到波纹夹芯板在不同边界条件下自由振动的固有频率，与有限元仿真结果进行对比，验证了理论结果的正确性。在此基础上，基于指数剪切变形理论(ESDT)，分析了不同边界条件下波纹夹芯板的基频随材料参数和结构几何参数的变化规律。结果表明，材料和几何参数对不同边界条件下波纹夹芯板的振动特性有重要影响。相关研究结果将对波纹夹芯板在工程应用中的减振设计及优化分析提供一定的理论依据。Abstract: As a special composite structure, the vibration characteristics of corrugated sandwich panel are greatly influenced by the boundary conditions. According to the shear deformation theory of different shear modes and Kirchhoff's classical plate theory(CLPT), the dynamic equation of corrugated sandwich plates was established by Hamilton principle. Among them, the corrugated core layer was equivalent to an anisotropic homogeneous body. According to the boundary conditions of four sides simply supported, four sides clamped, opposite sides simply supported and clamped, one side fixed and three edges clamped, the partial differential dynamic equation relative to the displacements was derived. By solving the equation, the natural frequencies of the corrugated sandwich plates under different boundary conditions were obtained. Compared with the finite element simulation results, the correctness of the theoretical results was verified. On this basis, based on the exponential shear deformation theory(ESDT), the variation of fundamental frequency of the corrugated sandwich plate with material parameters and structural geometric parameters under different boundary conditions was analyzed. The results show that the material and structural geometric parameters have an important influence on the vibration characteristics of the corrugated sandwich plates under different boundary conditions. Relevant research results will provide a theoretical basis for the vibration reduction design and optimization analysis of corrugated sandwich plates in engineering application.
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图 1 波纹夹芯板的模型(a)和波纹单胞示意图(b)
Figure 1. Model diagrams of corrugated sandwich panel (a) and corrugated cell (b)
a—Length of corrugated sandwich panel; b—Width of corrugated sandwich panel; lc—Length of the hypotenuse; hc—Height of core layer; tc—Wall thickness; θ—Corrugation angle; p—Length of bottom side
图 3 波纹夹芯板前五阶振型图
Figure 3. The first five mode shapes of the corrugated sandwich plates
图 4 不同边界条件下波纹夹芯板一阶振型图
Figure 4. The first mode shapes of corrugated sandwich plates under different boundary conditions
图 5 波纹与面板夹角对波纹夹芯板基频的影响
Figure 5. Influence of the angle between corrugation and panel on the fundamental frequency of corrugated sandwich panel
图 6 波纹芯层高度占比对波纹夹芯板基频的影响
Figure 6. Influence of corrugated core height ratio on the fundamental frequency of corrugated sandwich panel
图 7 波纹壁厚度对波纹夹芯板基频的影响
Figure 7. Influence of corrugated wall thickness on the fundamental frequency of corrugated sandwich panel
图 8 波纹夹芯板厚度
$h$ 对波纹夹芯板基频的影响Figure 8. Influence of corrugated plate thickness on the fundamental frequency of corrugated sandwich panel
图 9 比例因子
$k$ 对波纹夹芯板基频的影响Figure 9. Influence of scale factor on the fundamental frequency of corrugated sandwich panel
表 1 不同板理论对应的剪切形状函数
Table 1. Shear shape functions corresponding to different plate theories
Shear theory Function f (${\textit{z}} $) CLPT $f({\textit{z}}) = 0$ FSDT $f({\textit{z} }) = {\textit{z}}$ SSDT $f({\textit{z}}) = \dfrac{h}{{\text{π}}}\sin \left(\dfrac{{{\text{π}}{\textit{z}}} }{h}\right)$ TSDT $f({\textit{z}}) = {\textit{z}}\left[1 - \dfrac{4}{3}{\left(\dfrac{{\textit{z}}}{h}\right)^2}\right]$ ESDT $f({\textit{z}}) = {\textit{z}}{ {\rm{e} }^{ - 2{ {\left( {\dfrac{{\textit{z}}}{h} } \right)}^2} } }$ Notes: CLPT—Classical plate theory; FSDT—First-order shear plate theory; SSDT—Sinusoidal shear deformation theory; TSDT—Third-order shear deformation theory; ESDT—Exponential shear deformation theory. 
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表 2 不同边界条件下的函数Xm(x) 和 Yn(y)
Table 2. Functions Xm(x) and Yn(y) for different boundary conditions
Boundary conditions Function Xm(x) Function Yn(y) SSSS $\sin \alpha x$ $\sin \beta x$ CCCC $1 - \cos 2\alpha x$ $1 - \cos 2\beta x$ CCSS $1 - \cos 2\alpha x$ $\sin \beta x$ CSSS $\sin \alpha x(\cos \alpha x - 1)$ $\sin \beta x$ Notes: SSSS—Four sides simply supported; CCCC—Four sides clamped; CCSS—Opposite sides simply supported and clamped; CSSS—One side fixed and three edges clamped; ${ {\alpha = m{\text{π}}} / a}$; ${ {\beta = n{\text{π}} } / b}$; m, n—Half-wave numbers in two orthogonal coordinate directions respectively.
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表 3 四边简支波纹夹芯板在不同理论下的固有频率理论解与有限元仿真结果
Table 3. Theoretical solutions and finite element simulation results of natural frequency of simply supported corrugated sandwich plates using different theories
Mode ABAQUS CLPT FSDT SSDT TSDT ESDT Result Error/% Result Error/% Result Error/% Result Error/% Result Error/% (1,1) 31.18 31.72 1.75 31.41 0.77 31.00 −0.57 31.01 −0.54 30.99 −0.58 (2,1) 65.39 69.52 6.32 68.09 4.13 65.82 0.66 65.88 0.75 65.79 0.61 (1,2) 86.48 88.51 2.36 86.23 −0.29 83.74 −3.16 83.78 −3.12 83.74 −3.17 (2,2) 116.13 125.70 8.24 121.21 4.38 115.60 −0.45 115.72 −0.35 115.57 −0.48 (3,1) 118.46 132.06 11.48 127.11 7.30 119.24 0.66 119.45 0.84 119.12 0.56
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表 4 不同边界条件和板理论下波纹夹芯板的基频理论解与有限元仿真结果
Table 4. Theoretical solutions and finite element simulation results of fundamental frequency of corrugated sandwich plates with different boundary conditions and plate theories
Boundary conditons ABAQUS CLPT FSDT SSDT TSDT ESDT Result Error/% Result Error/% Result Error/% Result Error/% Result Error/% SSSS 31.18 31.72 1.75 31.41 0.77 31.00 −0.57 31.01 −0.54 30.99 −0.58 CCCC 56.42 60.46 7.17 58.90 4.40 56.99 1.01 57.03 1.08 56.98 0.99 CCSS 40.47 43.19 6.71 42.44 4.87 41.23 1.88 41.26 1.96 41.22 1.84 CSSS 39.50 42.14 6.70 41.56 5.22 40.66 2.94 40.68 2.99 40.65 2.91
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表 5 不同材料组合下波纹夹芯板的基频
Table 5. Fundamental frequencies of corrugated sandwich panels with different material combinations
Boundary conditon Al-Al-Al Ti-Ti-Ti Al-Ti-Al Ti-Al-Ti SSSS 30.99 38.72 29.07 40.53 CCCC 56.98 71.16 54.31 71.99 CCSS 41.22 51.46 39.96 52.38 CSSS 40.65 50.76 39.03 52.17
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