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

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