﻿ 波纹夹层板固有频率的一阶Zig-Zag理论计算方法
 舰船科学技术  2020, Vol. 42 Issue (8): 26-31    DOI: 10.3404/j.issn.1672-7649.2020.08.005 PDF

1. 中国舰船研究设计中心，湖北 武汉 430064;
2. 船舶振动噪声重点实验室，湖北 武汉 430064

Calculation method on corrugated core panel natrural frequency applying first order Zig-Zag theory
WANG Xiao-ming1,2, WEI Qiang1,2
1. China Ship Development and Design Center, Wuhan 430064, China;
2. National Key Laboratory on Ship Vibration and Noise, Wuhan 430064, China
Abstract: It is taken into account that corrugated core face sheets endure not only bending but also shearing, and core endure bending along corrugation generating line direction as well as shearing along double directions when the corrugated core panel tiny vibrates. Based on the core practical shape, geometry equations were demonstrated by applying first order Zig-zag theory to upper and lower face sheet and core. Tiny vibration diffrential equations were established according to Hamilton principle. Eigenvalue was determined and vibration frequency was calculated by solving the equations using double Fourier series according to boundary conditions. In the calculation example, the results of the first to eight order frequency of this proposal method are of good agreement with FEM or other literatures.
Key words: corrugated core panel     natrural frequency     Zig-Zag theory     variation principle     vibration analysis
0 引　言

 图 1 波纹夹层板结构示意图 Fig. 1 Corrugated sandwich structure

1 振动微分方程

 图 2 夹层板坐标系统 Fig. 2 Coordinate system of corrugated sandwich

1）夹层板的上下面板为普通薄板，考虑其抗剪作用，应用一阶剪切变形理论；

2）心层横向不可压缩，即假定 ${\varepsilon _z} = 0$ ${\sigma _z} = 0$

3）考虑心层的剪切作用，则心层中面法线在变形后保持为直线，但不再垂直于变形后的中面；

4）心层仅考虑其沿波纹方向的弯曲作用，忽略其垂直波纹方向的弯曲作用；

5）上下3层结构的面内位移沿板厚方向分段连续。

1.1 坐标系统与位移表达

 ${u_k}\left( {x,y,{z_k},t} \right) = {u_{ok}}\left( {x,y,t} \right) + {z_k}{\phi _{xk}}\left( {x,y,t} \right) {\text{，}}$ (1)
 ${v_k}\left( {x,y,{z_k},t} \right) = {v_{ok}}\left( {x,y,t} \right) + {z_k}{\phi _{yk}}\left( {x,y,t} \right) {\text{，}}$ (2)
 ${w_k}\left( {x,y,{z_k},t} \right) = w\left( {x,y,t} \right) {\text{。}}$ (3)

 ${u_c}\left( {x,y,{z_c},t} \right) = {u_{oc}}\left( {x,y,t} \right) + {z_c}{\phi _{xc}}\left( {x,y,t} \right) {\text{，}}$ (4)
 ${v_c}\left( {x,y,{z_c},t} \right) = {v_{oc}}\left( {x,y,t} \right) + {z_c}{\phi _{yc}}\left( {x,y,t} \right) {\text{，}}$ (5)
 ${w_c}\left( {x,y,{z_c},t} \right) = w\left( {x,y,t} \right) {\text{。}}$ (6)

 ${u_t}\left( {x,y,{z_t} = \frac{{{t_t}}}{2}} \right) = {u_c}\left( {x,y,{z_c} = - \frac{{{h_c}}}{2}} \right) {\text{，}}$ (7)
 ${v_t}\left( {x,y,{z_t} = \frac{{{t_t}}}{2}} \right) = {v_c}\left( {x,y,{z_c} = - \frac{{{h_c}}}{2}} \right) {\text{，}}$ (8)
 ${u_b}\left( {x,y,{z_b} = - \frac{{{t_b}}}{2}} \right) = {u_c}\left( {x,y,{z_c} = \frac{{{h_c}}}{2}} \right) {\text{，}}$ (9)
 ${v_b}\left( {x,y,{z_b} = - \frac{{{t_b}}}{2}} \right) = {v_c}\left( {x,y,{z_c} = \frac{{{h_c}}}{2}} \right) {\text{。}}$ (10)

 ${u_{ot}}\left( {x,y} \right) = {u_{oc}}\left( {x,y} \right) - \frac{{{h_c}}}{2}{\phi _{xc}}\left( {x,y} \right) - \frac{{{t_t}}}{2}{\phi _{xt}}\left( {x,y} \right) {\text{，}}$ (11)
 ${v_{ot}}\left( {x,y} \right) = {v_{oc}}\left( {x,y} \right) - \frac{{{h_c}}}{2}{\phi _{yc}}\left( {x,y} \right) - \frac{{{t_t}}}{2}{\phi _{yt}}\left( {x,y} \right) {\text{，}}$ (12)
 ${u_{ob}}\left( {x,y} \right) = {u_{oc}}\left( {x,y} \right) + \frac{{{h_c}}}{2}{\phi _{xc}}\left( {x,y} \right) + \frac{{{t_b}}}{2}{\phi _{xb}}\left( {x,y} \right) {\text{，}}$ (13)
 ${v_{ob}}\left( {x,y} \right) = {v_{oc}}\left( {x,y} \right) + \frac{{{h_c}}}{2}{\phi _{yc}}\left( {x,y} \right) + \frac{{{t_b}}}{2}{\phi _{yb}}\left( {x,y} \right) {\text{。}}$ (14)

 $\varepsilon _x^t = \frac{{\partial {u_t}}}{{\partial x}} = \frac{{\partial {u_{oc}}\left( {x,y} \right)}}{{\partial x}} - \frac{{{h_c}}}{2}\frac{{\partial {\phi _{xc}}\left( {x,y} \right)}}{{\partial x}} + \left( {{z_t} - \frac{{{t_t}}}{2}} \right)\frac{{\partial {\phi _{xt}}\left( {x,y} \right)}}{{\partial x}} {\text{，}}$ (15)
 $\begin{split}\varepsilon _y^t = \frac{{\partial {v_t}}}{{\partial y}} =& \frac{{\partial {v_{oc}}\left( {x,y} \right)}}{{\partial y}} - \frac{{{h_c}}}{2}\frac{{\partial {\phi _{yc}}\left( {x,y} \right)}}{{\partial y}} + \left( {{z_t} - \frac{{{t_t}}}{2}} \right)\\& \frac{{{\partial ^2}w(x,y)}}{{\partial {y^2}}}\frac{{\partial {\phi _{yt}}\left( {x,y} \right)}}{{\partial y}}{\text{，}}\end{split}$ (16)
 $\begin{split} \gamma _{xy}^t = & \frac{{\partial {u_t}}}{{\partial y}} + \frac{{\partial {v_t}}}{{\partial x}} = \frac{{\partial {u_{oc}}\left( {x,y} \right)}}{{\partial y}} - \frac{{{h_c}}}{2}\left( {\frac{{{\phi _{yc}}\left( {x,y} \right)}}{{\partial x}} + \frac{{\partial {\phi _{xc}}\left( {x,y} \right)}}{{\partial y}}} \right) + \\ & \left( {{z_t} - \frac{{{t_t}}}{2}} \right)\left( {\frac{{{\phi _{yt}}\left( {x,y} \right)}}{{\partial x}} + \frac{{\partial {\phi _{xt}}\left( {x,y} \right)}}{{\partial y}}} \right) {\text{，}}\\[-18pt] \end{split}$ (17)
 $\gamma _{xz}^t = \frac{{\partial {u_t}}}{{\partial {z_t}}} + \frac{{\partial w}}{{\partial x}} = {\phi _{xt}}\left( {x,y} \right) + \frac{{\partial w}}{{\partial x}}{\text{，}}$ (18)
 $\gamma _{yz}^t = \frac{{\partial {u_t}}}{{\partial {z_t}}} + \frac{{\partial w}}{{\partial x}} = {\phi _{yt}}\left( {x,y} \right) + \frac{{\partial w}}{{\partial y}}{\text{。}}$ (19)

 $\varepsilon _x^c = \frac{{\partial {u_c}\left( {x,y,{z_c}} \right)}}{{\partial x}} = \frac{{\partial {u_{oc}}\left( {x,y} \right)}}{{\partial x}} + {z_c}\frac{{\partial {\phi _{xc}}\left( {x,y} \right)}}{{\partial x}}{\text{，}}$ (20)
 $\begin{split}\gamma _s^c =& \left[ {\frac{{\partial {u_c}\left( {x,y,{z_c}} \right)}}{{\partial {z_c}}} + \frac{{\partial {w_c}\left( {x,y} \right)}}{{\partial x}}} \right]\sin \theta =\\&\left[ {{\phi _{xc}}\left( {x,y} \right) + \frac{{\partial w\left( {x,y} \right)}}{{\partial x}}} \right]\sin \theta{\text{，}} \end{split}$ (21)
 $\gamma _{yz}^c = \frac{{\partial {v_c}\left( {x,y,z} \right)}}{{\partial {z_c}}} + \frac{{\partial {w_c}\left( {x,y} \right)}}{{\partial y}} = {\phi _{yc}}\left( {x,y} \right) + \frac{{\partial w\left( {x,y} \right)}}{{\partial y}}{\text{。}}$ (22)

1.2 物理方程

 $\sigma _x^k = \frac{E}{{1 - {\mu ^2}}}\left( {\varepsilon _x^k + \mu \varepsilon _y^k} \right){\text{，}}$ (23)
 $\sigma _y^k = \frac{E}{{1 - {\mu ^2}}}\left( {\varepsilon _y^k + \mu \varepsilon _x^k} \right){\text{，}}$ (24)
 $\tau _{xy}^k = \frac{E}{{2\left( {1 + \mu } \right)}}\gamma _{xy}^k{\text{，}}$ (25)
 $\tau _{xz}^k = \frac{E}{{2\left( {1 + \mu } \right)}}\gamma _{xz}^k{\text{，}}$ (26)
 $\tau _{yz}^k = \frac{E}{{2\left( {1 + \mu } \right)}}\gamma _{yz}^k{\text{。}}$ (27)

 $\sigma _x^c = {E_c}\varepsilon _x^c{\text{，}}$ (28)
 $\tau _s^c = \frac{{{E_c}}}{{2\left( {1 + {\mu _c}} \right)}}\gamma _s^c{\text{，}}$ (29)
 $\tau _{yz}^c = G_{yz}^c\gamma _{yz}^c{\text{。}}$ (30)

 $G_{yz}^c = \frac{{{E_c}{t_c}{{\sin }^2}\theta \cos \theta }}{{{h_c}}}{\text{。}}$ (31)
1.3 能量原理

 $\delta \left( {T - U} \right) = 0{\text{。}}$ (32)

 $T = {T_t} + {T_b} + {T_c}{\text{，}}$ (33)
 $\small{T_k} = \frac{\rho }{2}\left\{ {{{\int_{ - {t_k}/2}^{{t_k}/2} {\left( {\frac{{\partial {u_k}}}{{\partial t}}} \right)} }^2}{\rm{d}}{z_k} + {{\int_{ - {t_k}/2}^{{t_k}/2} {\left( {\frac{{\partial {v_k}}}{{\partial t}}} \right)} }^2}{\rm{d}}{z_k} + {{\int_{ - {t_k}/2}^{{t_k}/2} {\left( {\frac{{\partial w}}{{\partial t}}} \right)} }^2}{\rm{d}}{z_k}} \right\}{\text{，}}$ (34)
 $\begin{split}{T_c} = &\frac{{{\rho _c}}}{2}\left\{ {{{\int_{{z_c} - {t_c}/(2\cos \theta )}^{{z_c} + {t_c}/(2\cos \theta )} {\left( {\frac{{\partial {u_c}}}{{\partial t}}} \right)} }^2}{\rm{d}}{z_c} + {{\int_{ - {t_c}/2}^{{t_c}/2} {\left( {\frac{{\partial {v_c}}}{{\partial t}}} \right)} }^2}{\rm{d}}{z_c}}+\right.\\& \left.{{\int_{ - {t_c}/2}^{{t_c}/2} {\left( {\frac{{\partial w}}{{\partial t}}} \right)} }^2}{\rm{d}}{z_c} \right\}{\text{，}}\\[-18pt]\end{split}$ (35)

 $U = {U_t} + {U_b} + {U_c}{\text{，}}$ (36)

 $\begin{split} {U_k} = & \frac{1}{2}\smallint _{ - {t_k}/2}^{{t_k}/2}{\text{d}}{z_k}\int {\int_A {} } \hfill \\ & \left( {\sigma _x^k\varepsilon _x^k + \sigma _y^k\varepsilon _y^k + \tau _{xy}^k\gamma _{xy}^k + \tau _{xz}^k\gamma _{xz}^k + + \tau _{yz}^k\gamma _{yz}^k} \right){\text{d}}x{\text{d}}y \end{split}$ (37)
 $\begin{split}{U_c} = &\frac{1}{2}\int\nolimits_{{z_c} - {t_c}/(2\cos \theta )}^{{z_c} + {t_c}/(2\cos \theta )} {{\rm{d}}{z_c}\iint\nolimits_A {\left( {\sigma _x^c\varepsilon _x^c + \tau _s^c\gamma _s^c} \right)dxdy}} + \\& \frac{1}{2}\int_{ - {h_c}/2}^{{h_c}/2} {{\rm{d}}{z_c}} \iint\limits_A {\tau _{yz}^c\gamma _{yz}^c{\rm{d}}x{\rm{d}}y}{\text{。}}\end{split}$ (38)

 $\begin{split}& - \frac{1}{2}\rho t_b^2\frac{{{\partial ^2}{\varphi _{xb}}}}{{\partial {t^2}}} + \frac{1}{2}\rho {h_c}\left( {{t_t} - {t_b}} \right)\frac{{{\partial ^2}{\varphi _{xc}}}}{{\partial {t^2}}} + \frac{1}{2}\rho t_t^2\frac{{{\partial ^2}{\varphi _{xt}}}}{{\partial {t^2}}} - \rho \left( {{t_t} + {t_b}} \right)\frac{{{\partial ^2}{u_{0c}}}}{{\partial {t^2}}} +\\& \frac{{Et_b^2}}{{4\left( {1 + \mu } \right)}}\frac{{{\partial ^2}{\varphi _{xb}}}}{{\partial {y^2}}} + \frac{{E{h_c}\left( {{t_b} - {t_t}} \right)}}{{4\left( {1 + \mu } \right)}}\frac{{{\partial ^2}{\varphi _{xc}}}}{{\partial {y^2}}} - \frac{{Et_t^2}}{{4\left( {1 + \mu } \right)}}\frac{{{\partial ^2}{\varphi _{xt}}}}{{\partial {y^2}}} +\\& \frac{{E\left( {{t_b} + {t_t}} \right)}}{{4\left( {1 + \mu } \right)}}\frac{{{\partial ^2}{u_{0c}}}}{{\partial {y^2}}} + \frac{{Et_b^2}}{{4\left( {1 - \mu } \right)}}\frac{{{\partial ^2}{\varphi _{yb}}}}{{\partial x\partial y}} + \frac{{E{h_c}\left( {{t_b} - {t_t}} \right)}}{{4\left( {1 - \mu } \right)}} \frac{{{\partial ^2}{\varphi _{yc}}}}{{\partial x\partial y}} -\\& \frac{{Et_t^2}}{{4\left( {1 - \mu } \right)}}\frac{{{\partial ^2}{\varphi _{yt}}}}{{\partial x\partial y}} + \frac{{E\left( {{t_b} + {t_t}} \right)}}{{2\left( {1 - \mu } \right)}}\frac{{{\partial ^2}{v_{0c}}}}{{\partial x\partial y}} + \frac{{Et_b^2}}{{2\left( {1 - {\mu ^2}} \right)}}\frac{{{\partial ^2}{\varphi _{xb}}}}{{\partial {x^2}}} + \\&\left[ {\frac{{E{h_c}\left( {{t_b} - {t_t}} \right)}}{{2\left( {1 - {\mu ^2}} \right)}} + \frac{{{E_c}{t_c}{z_c}}}{{\cos \theta }}} \right] \frac{{{\partial ^2}{\varphi _{xc}}}}{{\partial {x^2}}} - \frac{{Et_t^2}}{{2\left( {1 - {\mu ^2}} \right)}}\frac{{{\partial ^2}{\varphi _{xt}}}}{{\partial {x^2}}} + \\&\left[ {\frac{{E\left( {{t_b} + {t_t}} \right)}}{{1 - {\mu ^2}}} + \frac{{{E_c}{t_c}}}{{\cos \theta }}} \right]\frac{{{\partial ^2}{u_{0c}}}}{{\partial {x^2}}} = 0 {\text{，}}\\[-24pt] \end{split}$ (39)
 $\begin{split}& - \frac{1}{2}\rho t_b^2\frac{{{\partial ^2}{\varphi _{yb}}}}{{\partial {t^2}}} + \frac{1}{2}\rho {h_c}\left( {{t_t} - {t_b}} \right)\frac{{{\partial ^2}{\varphi _{yc}}}}{{\partial {t^2}}} + \frac{1}{2}\rho t_t^2\frac{{{\partial ^2}{\varphi _{yt}}}}{{\partial {t^2}}} -\\& \rho \left( {{t_t} + {t_b}} \right)\frac{{{\partial ^2}{v_{0c}}}}{{\partial {t^2}}} + \frac{{Et_b^2}}{{2\left( {1 - {\mu ^2}} \right)}}\frac{{{\partial ^2}{\varphi _{yb}}}}{{\partial {y^2}}} + \frac{{E{h_c}\left( {{t_b} - {t_t}} \right)}}{{2\left( {1 - {\mu ^2}} \right)}}\frac{{{\partial ^2}{\varphi _{yc}}}}{{\partial {y^2}}} -\\& \frac{{Et_t^2}}{{2\left( {1 - {\mu ^2}} \right)}}\frac{{{\partial ^2}{\varphi _{yt}}}}{{\partial {y^2}}}+ \frac{{E\left( {{t_b} + {t_t}} \right)}}{{1 - {\mu ^2}}}\frac{{{\partial ^2}{v_{0c}}}}{{\partial {y^2}}} + \frac{{Et_b^2}}{{4\left( {1 - \mu } \right)}}\frac{{{\partial ^2}{\varphi _{xb}}}}{{\partial x\partial y}} + \\&\frac{{E{h_c}\left( {{t_b} - {t_t}} \right)}}{{4\left( {1 - \mu } \right)}} \frac{{{\partial ^2}{\varphi _{xc}}}}{{\partial x\partial y}} - \frac{{Et_t^2}}{{4\left( {1 - \mu } \right)}}\frac{{{\partial ^2}{\varphi _{xt}}}}{{\partial x\partial y}} + \frac{{E\left( {{t_b} + {t_t}} \right)}}{{2\left( {1 - \mu } \right)}}\frac{{{\partial ^2}{u_{0c}}}}{{\partial x\partial y}} + \frac{{Et_b^2}}{{4\left( {1 + \mu } \right)}}\frac{{{\partial ^2}{\varphi _{yb}}}}{{\partial {x^2}}} + \\&\frac{{E{h_c}\left( {{t_b} - {t_t}} \right)}}{{4\left( {1 + \mu } \right)}} \frac{{{\partial ^2}{\varphi _{yc}}}}{{\partial {x^2}}} - \frac{{Et_t^2}}{{4\left( {1 + \mu } \right)}}\frac{{{\partial ^2}{\varphi _{yt}}}}{{\partial {x^2}}} + \frac{{E\left( {{t_b} + {t_t}} \right)}}{{2\left( {1 + \mu } \right)}}\frac{{{\partial ^2}{v_{0c}}}}{{\partial {x^2}}} = 0 {\text{，}}\\[-14pt] \end{split}$ (40)
 $\begin{split}& - \frac{{{E_c}t{}_c{{\sin }^2}\theta }}{{2\left( {1 + {\mu _c}} \right)\cos \theta }}{\varphi _{xc}} - \frac{1}{4}\rho {h_c}t_b^2\frac{{{\partial ^2}{\varphi _{xb}}}}{{\partial {t^2}}} - \frac{1}{4}\rho h_c^2\left( {{t_t} + {t_b}} \right)\frac{{{\partial ^2}{\varphi _{xc}}}}{{\partial {t^2}}} -\\& \frac{1}{4}\rho {h_c}t_t^2\frac{{{\partial ^2}{\varphi _{xt}}}}{{\partial {t^2}}} + \frac{1}{2}\rho {h_c}\left( {{t_t} - {t_b}} \right)\frac{{{\partial ^2}{u_{0c}}}}{{\partial {t^2}}} +\frac{{E{h_c}t_b^2}}{{8\left( {1 + \mu } \right)}}\frac{{{\partial ^2}{\varphi _{xb}}}}{{\partial {y^2}}} +\\& \frac{{Eh_c^2\left( {{t_b} + {t_t}} \right)}}{{8\left( {1 + \mu } \right)}}\frac{{{\partial ^2}{\varphi _{xc}}}}{{\partial {y^2}}} +\frac{{E{h_c}t_t^2}}{{8\left( {1 + \mu } \right)}}\frac{{{\partial ^2}{\varphi _{xt}}}}{{\partial {y^2}}} + \frac{{E{h_c}\left( {{t_b} - {t_t}} \right)}}{{4\left( {1 + \mu } \right)}}\frac{{{\partial ^2}{u_{0c}}}}{{\partial {y^2}}} -\\& \frac{{{E_c}{t_c}{{\sin }^2}\theta }}{{2\left( {1 + {\mu _c}} \right)\cos \theta }}\frac{{\partial w}}{{\partial x}} +\frac{{E{h_c}t_b^2}}{{8\left( {1 - \mu } \right)}}\frac{{{\partial ^2}{\varphi _{yb}}}}{{\partial x\partial y}} + \frac{{Eh_c^2\left( {{t_b} + {t_t}} \right)}}{{8\left( {1 - \mu } \right)}}\frac{{{\partial ^2}{\varphi _{yc}}}}{{\partial x\partial y}} +\\& \frac{{E{h_c}t_t^2}}{{8\left( {1 - \mu } \right)}}\frac{{{\partial ^2}{\varphi _{yt}}}}{{\partial x\partial y}} + \frac{{E{h_c}\left( {{t_b} - {t_t}} \right)}}{{4\left( {1 - \mu } \right)}}\frac{{{\partial ^2}{v_{0c}}}}{{\partial x\partial y}} + \frac{{E{h_c}t_b^2}}{{4\left( {1 - {\mu ^2}} \right)}}\frac{{{\partial ^2}{\varphi _{xb}}}}{{\partial {x^2}}} +\\& \left[ {\frac{{Eh_c^2\left( {{t_b} + {t_t}} \right)}}{{4\left( {1 - {\mu ^2}} \right)}} + \frac{{{E_c}{t_c}z_c^2}}{{\cos \theta }} + \frac{{{E_c}t_c^3}}{{12{{\cos }^3}\theta }}} \right] \times\\& \frac{{{\partial ^2}{\varphi _{xc}}}}{{\partial {x^2}}} + \frac{{E{h_c}t_t^2}}{{2\left( {1 - {\mu ^2}} \right)}}\frac{{{\partial ^2}{\varphi _{xt}}}}{{\partial {x^2}}} +\left[ {\frac{{E{h_c}\left( {{t_b} - {t_t}} \right)}}{{2\left( {1 - {\mu ^2}} \right)}} + \frac{{{E_c}{t_c}{z_c}}}{{\cos \theta }}} \right]\frac{{{\partial ^2}{u_{0c}}}}{{\partial {x^2}}} = 0{\text{，}} \end{split}$ (41)
 $\begin{split}& - 2G_{yz}^c{h_c}{\varphi _{yc}} - \frac{1}{4}\rho {h_c}t_b^2\frac{{{\partial ^2}{\varphi _{yb}}}}{{\partial {t^2}}} - \frac{1}{4}\rho h_c^2\left( {{t_t} + {t_b}} \right)\frac{{{\partial ^2}{\varphi _{yc}}}}{{\partial {t^2}}} -\\& \frac{1}{4}\rho {h_c}t_t^2\frac{{{\partial ^2}{\varphi _{yt}}}}{{\partial {t^2}}} +\frac{1}{2}\rho {h_c}\left( {{t_t} - {t_b}} \right)\frac{{{\partial ^2}{v_{0c}}}}{{\partial {t^2}}} - 2G_{yz}^c{h_c}\frac{{\partial w}}{{\partial y}} +\\& \frac{{E{h_c}t_b^2}}{{4\left( {1 - {\mu ^2}} \right)}}\frac{{{\partial ^2}{\varphi _{yb}}}}{{\partial {y^2}}} + \frac{{Eh_c^2\left( {{t_b} + {t_t}} \right)}}{{4\left( {1 - {\mu ^2}} \right)}}\frac{{{\partial ^2}{\varphi _{yc}}}}{{\partial {y^2}}} + \frac{{E{h_c}t_t^2}}{{4\left( {1 - {\mu ^2}} \right)}}\frac{{{\partial ^2}{\varphi _{yt}}}}{{\partial {y^2}}} +\\& \frac{{E{h_c}\left( {{t_b} - {t_t}} \right)}}{{2\left( {1 - {\mu ^2}} \right)}}\frac{{{\partial ^2}{v_{0c}}}}{{\partial {y^2}}} + \frac{{E{h_c}t_b^2}}{{8\left( {1 - \mu } \right)}}\frac{{{\partial ^2}{\varphi _{xb}}}}{{\partial x\partial y}} + \frac{{Eh_c^2\left( {{t_b} + {t_t}} \right)}}{{8\left( {1 - \mu } \right)}}\frac{{{\partial ^2}{\varphi _{yc}}}}{{\partial x\partial y}} +\\& \frac{{E{h_c}t_t^2}}{{8\left( {1 - \mu } \right)}}\frac{{{\partial ^2}{\varphi _{yt}}}}{{\partial x\partial y}} + \frac{{E{h_c}\left( {{t_b} - {t_t}} \right)}}{{4\left( {1 - \mu } \right)}}\frac{{{\partial ^2}{u_{0c}}}}{{\partial x\partial y}} + \frac{{E{h_c}t_b^2}}{{8\left( {1 + \mu } \right)}}\frac{{{\partial ^2}{\varphi _{yb}}}}{{\partial {x^2}}} + \\&\frac{{Eh_c^2\left( {{t_b} + {t_t}} \right)}}{{8\left( {1 + \mu } \right)}}\frac{{{\partial ^2}{\varphi _{yc}}}}{{\partial {x^2}}} + \frac{{E{h_c}t_t^2}}{{8\left( {1 + \mu } \right)}}\frac{{{\partial ^2}{\varphi _{yt}}}}{{\partial {x^2}}} + \frac{{E{h_c}\left( {{t_b} - {t_t}} \right)}}{{4\left( {1 + \mu } \right)}}\frac{{{\partial ^2}{v_{0c}}}}{{\partial {x^2}}} = 0{\text{，}} \end{split}$ (42)
 $\begin{split}& q\left( {x,y} \right) - \left[ {\rho \left( {{t_t} + {t_b}} \right) + \frac{{{\rho _c}{t_c}}}{{\cos \theta }}} \right]\frac{{{\partial ^2}w}}{{\partial {t^2}}} +\\& \frac{{E{t_b}}}{{2\left( {1 + \mu } \right)}}\left( {\frac{{\partial {\varphi _{xb}}}}{{\partial x}} + \frac{{\partial {\varphi _{yb}}}}{{\partial y}}} \right) + 2G_{yz}^c{h_c}\frac{{\partial {\varphi _{yc}}}}{{\partial y}} + \\&\frac{{E{t_t}}}{{2\left( {1 + \mu } \right)}}\left( {\frac{{\partial {\varphi _{xt}}}}{{\partial x}} + \frac{{\partial {\varphi _{yt}}}}{{\partial y}}} \right) + \left[ {2G_{yz}^c{h_c} + \frac{{E\left( {{t_b} - {t_t}} \right)}}{{2\left( {1 + \mu } \right)}}} \right]\frac{{{\partial ^2}w}}{{\partial {y^2}}} +\\& \frac{{{E_c}{t_c}{{\sin }^2}\theta }}{{2\left( {1 + {\mu _c}} \right)\cos \theta }} \times \frac{{\partial {\varphi _{xc}}}}{{\partial x}} +\\& \left[ {\frac{{E\left( {{t_b} + {t_t}} \right)}}{{2\left( {1 + \mu } \right)}}+ \frac{{{E_c}{t_c}{{\sin }^2}\theta }}{{2\left( {1 + {\mu _c}} \right)\cos \theta }}} \right]\frac{{{\partial ^2}w}}{{\partial {x^2}}} = 0 {\text{，}} \\[-18pt] \end{split}$ (43)
 $\begin{split}& - \frac{{E{t_t}}}{{2\left( {1 + \mu } \right)}}{\varphi _{xt}} - \frac{1}{4}\rho {h_c}t_t^2\frac{{{\partial ^2}{\varphi _{xc}}}}{{\partial {t^2}}} - \frac{1}{3}\rho t_t^3\frac{{{\partial ^2}{\varphi _{xt}}}}{{\partial {t^2}}} + \frac{1}{2}\rho t_t^2\frac{{{\partial ^2}{u_{0c}}}}{{\partial {t^2}}} + \\&\frac{{E{h_c}t_t^2}}{{8\left( {1 + \mu } \right)}}\frac{{{\partial ^2}{\varphi _{xc}}}}{{\partial {y^2}}} + \frac{{Et_t^3}}{{6\left( {1 + \mu } \right)}}\frac{{{\partial ^2}{\varphi _{xt}}}}{{\partial {y^2}}} - \frac{{Et_t^2}}{{4\left( {1 + \mu } \right)}}\frac{{{\partial ^2}{u_{0c}}}}{{\partial {y^2}}} -\\& \frac{{E{t_t}}}{{2\left( {1 + \mu } \right)}}\frac{{\partial w}}{{\partial x}} + \frac{{E{h_c}t_t^2}}{{8\left( {1 - \mu } \right)}}\frac{{{\partial ^2}{\varphi _{yc}}}}{{\partial x\partial y}} + \frac{{Et_t^3}}{{4\left( {1 - \mu } \right)}}\frac{{{\partial ^2}{\varphi _{yt}}}}{{\partial x\partial y}} - \\&\frac{{Et_t^2}}{{4\left( {1 - \mu } \right)}}\frac{{{\partial ^2}{v_{0c}}}}{{\partial x\partial y}} + \frac{{E{h_c}t_t^2}}{{4\left( {1 - {\mu ^2}} \right)}}\frac{{{\partial ^2}{\varphi _{xc}}}}{{\partial {x^2}}} + \frac{{Et_t^3}}{{3\left( {1 - {\mu ^2}} \right)}}\frac{{{\partial ^2}{\varphi _{xt}}}}{{\partial {x^2}}} -\\& \frac{{Et_t^2}}{{2\left( {1 - {\mu ^2}} \right)}}\frac{{{\partial ^2}{u_{0c}}}}{{\partial {x^2}}} = 0 {\text{，}}\\[-18pt] \end{split}$ (44)
 $\begin{split}& - \frac{{E{t_t}}}{{2\left( {1 + \mu } \right)}}{\varphi _{yt}} - \frac{1}{4}\rho {h_c}t_t^2\frac{{{\partial ^2}{\varphi _{yc}}}}{{\partial {t^2}}} - \frac{1}{3}\rho t_t^3\frac{{{\partial ^2}{\varphi _{yt}}}}{{\partial {t^2}}} + \frac{1}{2}\rho t_t^2\frac{{{\partial ^2}{v_{0c}}}}{{\partial {t^2}}} - \\&\frac{{E{t_t}}}{{2\left( {1 + \mu } \right)}}\frac{{\partial w}}{{\partial y}} + \frac{{E{h_c}t_t^2}}{{4\left( {1 - {\mu ^2}} \right)}}\frac{{{\partial ^2}{\varphi _{yc}}}}{{\partial {y^2}}} + \frac{{Et_t^3}}{{3\left( {1 - {\mu ^2}} \right)}}\frac{{{\partial ^2}{\varphi _{yt}}}}{{\partial {y^2}}} - \frac{{Et_t^2}}{{2\left( {1 - {\mu ^2}} \right)}}\frac{{{\partial ^2}{v_{0c}}}}{{\partial {y^2}}} + \\&\frac{{E{h_c}t_t^2}}{{8\left( {1 - \mu } \right)}}\frac{{{\partial ^2}{\varphi _{xc}}}}{{\partial x\partial y}} + \frac{{Et_t^3}}{{6\left( {1 - \mu } \right)}}\frac{{{\partial ^2}{\varphi _{xt}}}}{{\partial x\partial y}} - \frac{{Et_t^2}}{{4\left( {1 - \mu } \right)}}\frac{{{\partial ^2}{u_{0c}}}}{{\partial x\partial y}} + \frac{{E{h_c}t_t^2}}{{8\left( {1 + \mu } \right)}}\frac{{{\partial ^2}{\varphi _{yc}}}}{{\partial {x^2}}} +\\& \frac{{Et_t^3}}{{6\left( {1 + \mu } \right)}}\frac{{{\partial ^2}{\varphi _{yt}}}}{{\partial {x^2}}} - \frac{{Et_t^2}}{{4\left( {1 + \mu } \right)}}\frac{{{\partial ^2}{v_{0c}}}}{{\partial {x^2}}} = 0 {\text{，}}\\[-18pt] \end{split}$ (45)
 $\begin{split}& - \frac{{E{t_b}}}{{2\left( {1 + \mu } \right)}}{\varphi _{xb}} - \frac{1}{3}\rho t_b^3\frac{{{\partial ^2}{\varphi _{xb}}}}{{\partial {t^2}}} - \frac{1}{4}\rho {h_c}t_b^2\frac{{{\partial ^2}{\varphi _{xc}}}}{{\partial {t^2}}} - \frac{1}{2}\rho t_b^2\frac{{{\partial ^2}{u_{0c}}}}{{\partial {t^2}}} + \\&\frac{{Et_b^3}}{{6\left( {1 + \mu } \right)}}\frac{{{\partial ^2}{\varphi _{xb}}}}{{\partial {y^2}}} + \frac{{E{h_c}t_b^2}}{{8\left( {1 + \mu } \right)}}\frac{{{\partial ^2}{\varphi _{xc}}}}{{\partial {y^2}}} + \frac{{Et_b^2}}{{4\left( {1 + \mu } \right)}}\frac{{{\partial ^2}{u_{0c}}}}{{\partial {y^2}}} - \frac{{E{t_b}}}{{2\left( {1 + \mu } \right)}}\frac{{\partial w}}{{\partial x}} +\\& \frac{{Et_b^3}}{{6\left( {1 - \mu } \right)}}\frac{{{\partial ^2}{\varphi _{yb}}}}{{\partial x\partial y}} + \frac{{E{h_c}t_b^2}}{{8\left( {1 - \mu } \right)}}\frac{{{\partial ^2}{\varphi _{yc}}}}{{\partial x\partial y}} + \frac{{Et_b^2}}{{4\left( {1 - \mu } \right)}}\frac{{{\partial ^2}{v_{0c}}}}{{\partial x\partial y}} + \frac{{Et_b^3}}{{3\left( {1 - {\mu ^2}} \right)}}\frac{{{\partial ^2}{\varphi _{xb}}}}{{\partial {x^2}}} + \\&\frac{{E{h_c}t_b^2}}{{4\left( {1 - {\mu ^2}} \right)}}\frac{{{\partial ^2}{\varphi _{xc}}}}{{\partial {x^2}}} + \frac{{Et_b^2}}{{2\left( {1 - {\mu ^2}} \right)}}\frac{{{\partial ^2}{u_{0c}}}}{{\partial {x^2}}} = 0 {\text{，}}\\[-18pt] \end{split}$ (46)
 $\begin{split}& - \frac{{E{t_b}}}{{2\left( {1 + \mu } \right)}}{\varphi _{yb}} - \frac{1}{3}\rho t_b^3\frac{{{\partial ^2}{\varphi _{yb}}}}{{\partial {t^2}}} - \frac{1}{4}\rho {h_c}t_b^2\frac{{{\partial ^2}{\varphi _{yc}}}}{{\partial {t^2}}} - \frac{1}{2}\rho t_b^2\frac{{{\partial ^2}{v_{0c}}}}{{\partial {t^2}}} - \\&\frac{{E{t_b}}}{{2\left( {1 + \mu } \right)}}\frac{{\partial w}}{{\partial y}} + \frac{{Et_b^3}}{{3\left( {1 - {\mu ^2}} \right)}}\frac{{{\partial ^2}{\varphi _{yb}}}}{{\partial {y^2}}} + \frac{{E{h_c}t_b^2}}{{4\left( {1 - {\mu ^2}} \right)}}\frac{{{\partial ^2}{\varphi _{yc}}}}{{\partial {y^2}}} + \frac{{Et_b^2}}{{2\left( {1 - {\mu ^2}} \right)}}\frac{{{\partial ^2}{v_{0c}}}}{{\partial {y^2}}} +\\& \frac{{Et_b^3}}{{6\left( {1 - \mu } \right)}}\frac{{{\partial ^2}{\varphi _{xb}}}}{{\partial x\partial y}} + \frac{{E{h_c}t_b^2}}{{8\left( {1 - \mu } \right)}}\frac{{{\partial ^2}{\varphi _{xc}}}}{{\partial x\partial y}} + \frac{{Et_b^2}}{{4\left( {1 - \mu } \right)}}\frac{{{\partial ^2}{u_{0c}}}}{{\partial x\partial y}} + \frac{{Et_b^3}}{{6\left( {1 + \mu } \right)}}\frac{{{\partial ^2}{\varphi _{yb}}}}{{\partial {x^2}}} +\\& \frac{{E{h_c}t_b^2}}{{8\left( {1 + \mu } \right)}}\frac{{{\partial ^2}{\varphi _{yc}}}}{{\partial {x^2}}} + \frac{{Et_b^2}}{{4\left( {1 + \mu } \right)}}\frac{{{\partial ^2}{v_{0c}}}}{{\partial {x^2}}} = 0 {\text{。}}\\[-18pt] \end{split}$ (47)
2 边界条件与方程求解

x=0或a

 ${v_{0c}} = 0,\frac{{\partial {\phi _{xc}}}}{{\partial x}} = \frac{{\partial {\phi _{xk}}}}{{\partial x}} = 0,{\phi _{yc}} = {\phi _{yk}} = 0,w = 0,\frac{{{\partial ^2}w}}{{\partial {x^2}}} = 0{\text{，}}$ (48)

y=0或b

 ${u_{0c}} = 0,{\phi _{xc}} = {\phi _{xk}} = 0,\frac{{\partial {\phi _{yc}}}}{{\partial x}} = \frac{{\partial {\phi _{yk}}}}{{\partial x}} = 0,w = 0,\frac{{{\partial ^2}w}}{{\partial {y^2}}} = 0{\text{。}}$ (49)

 ${u_{0c}} = \sum\limits_{m = 1}^\infty {\sum\limits_{n = 1}^\infty {{u_{mn}}\cos \frac{{m\text{π} x}}{a}} } \sin \frac{{n\text{π} y}}{b}{e^{i\varpi t}}{\text{，}}$ (50)
 ${v_{0c}} = \sum\limits_{m = 1}^\infty {\sum\limits_{n = 1}^\infty {{v_{mn}}\sin \frac{{m\text{π} x}}{a}} } \cos \frac{{n\text{π} y}}{b}{e^{i\varpi t}}{\text{，}}$ (51)
 ${\phi _{xc}} = \sum\limits_{m = 1}^\infty {\sum\limits_{n = 1}^\infty {{\phi _{xmn}}\cos \frac{{m\text{π} x}}{a}} } \sin \frac{{n\text{π} y}}{b}{e^{i\varpi t}}{\text{，}}$ (52)
 ${\phi _{yc}} = \sum\limits_{m = 1}^\infty {\sum\limits_{n = 1}^\infty {{\phi _{ymn}}\sin \frac{{m\text{π} x}}{a}} } \cos \frac{{n\text{π} y}}{b}{e^{i\varpi t}}{\text{，}}$ (53)
 $w = \sum\limits_{m = 1}^\infty {\sum\limits_{n = 1}^\infty {{w_{mn}}\sin \frac{{m\text{π} x}}{a}} } \sin \frac{{n\text{π} y}}{b}{e^{i\varpi t}}{\text{，}}$ (54)
 ${\phi _{xk}} = \sum\limits_{m = 1}^\infty {\sum\limits_{n = 1}^\infty {{\phi _{xkmn}}\cos \frac{{m\text{π} x}}{a}} } \sin \frac{{n\text{π} y}}{b}{e^{i\varpi t}}{\text{，}}$ (55)
 ${\phi _{yk}} = \sum\limits_{m = 1}^\infty {\sum\limits_{n = 1}^\infty {{\phi _{ykmn}}\sin \frac{{m\text{π} x}}{a}} } \cos \frac{{n\text{π} y}}{b}{e^{i\varpi t}}{\text{。}}$ (56)

3 方法验证

1）方法1

2）方法2

3）方法3

4）方法4

 $C = G_{yz}^c{\left( {{h_c} + {t_t}} \right)^2}/{h_c} = {E_c}{t_c}{\sin ^2}\theta \cos \theta {\left( {{h_c} + {t_t}} \right)^2}/h_c^2{\text{。}}$ (57)

4 面板剪切作用的体现

 图 3 面板厚度与一阶频率关系曲线 Fig. 3 Face sheet thinkness and first order frequency relationship curves

 图 4 心层厚度与一阶频率关系曲线 Fig. 4 Core sheet thinkness and first order frequency relationship curves

5 结　语

 [1] KNOX.E.COWLING.M, M.J. Adhesively bonded steel corrugated core sandwich construction for marine application[J]. Marine Structure, 1998(11): 185-204. [2] Ye YU, Wen-bin Hou. Elastic constants for adhesively bonded corrugated core sandwich panels[J]. Composite Structures, 2017(176): 449-459. [3] KUJALA.P ,ROMANOFF. J . All steel sandwich panels – design challenges for practical applications on ships[C]. 9th Symposium on Practical Design of Ships and Other Floating Structures. [4] RAJAPAKSE Y D S, HUI D. Marine composite and sandwich structures[J]. Composites: Part B, 2008, 39: 1-4. [5] 中科院北京力学研究所. 夹层板的弯曲稳定和振动[M]. 北京: 科学出版社, 1977. [6] 吴晖, 俞焕然. 四边简支正交各向异性波纹型夹心矩形夹层板的固有频率[J]. 应用数学和力学, 2001, 22(9): 919-926. WU Hui, Yu Huan-ran. Natural frequency for rectangular orthotropic corrugated_core sandwich plates with all edges simply_supported[J]. Applied Mathematics and Mechanics, 2001, 22(9): 919-926. DOI:10.3321/j.issn:1000-0887.2001.09.005 [7] ZAMANIFAR H, SARRAMI-FOROUSHANI S, et al. Static and dynamic analysis of corrugated-core sandwich plates using finite strip method[J]. Engineering structures, 2019(183): 30-51. [8] ICARD U, SOLA F. Assessment of recent Zig-Zag theories for laminated and sandwich structure[J]. Composites Part B, 2019(97): 26-52. [9] FANG Taoxie, Yegao QU, et al. Nolinear areothermoelastic analysis of composite laminated panels using a general higher-order shear deformation zig-zag theory[J]. International Journal of Mechanical Sciences, 2019(150): 226-237. [10] AZHARI F, BOROOMAND B, et al.. Exponential basis function in the solution of laminated plates using a higer-order Zig-Zag theory[J]. Composite Structures, 2013(105): 398-407. [11] 白瑞祥, 张志锋, 陈浩然. 基于Zig-Zag变形假定的复合材料夹层板的自由振动[J]. 力学季刊, 2004, 25(4): 528-534. BAI Rui-xiang, ZHANG Zhi-feng, CHEN Hao-ran. Free vibration of composite sandwich plates based on Zig-Zag deformation assumption[J]. Chinese Quarterly of Mechanics, 2004, 25(4): 528-534. DOI:10.3969/j.issn.0254-0053.2004.04.015 [12] 何力. 船舶板架结构动力优化设计方法研究[D]. 武汉: 华中科技大学. 2011. HE Li. Dynamic Optimization of Ship Grillages[D]. Wuhan: Huazhong University of Science and Technology. 2011. [13] Hong-xia WANG, SAMUEL W.Chung.. Equivalent elastic constants of truss core sandwich plates[J]. Journal of Pressure Vessel Technology, 2011(133): 041203-1-041203-6.