0 引　言

夹层板是由强度大的上下面板和密度较小的心层组成，通常上下面板可以是钢板，碳纤维或者玻璃钢等，心层可以是木材，铝蜂窝或者泡沫塑料。心层的设计还有一种思路来降低等效密度，可以把心层设计成空心的波纹状结构，通过粘结^[1-2]或者激光焊接^[3-4]与上下面板相连，这就是波纹夹层板（见图1）。这样组成的新结构具有比强度高、比刚度高的优点，经常被应用到航空航天工程，船舶海洋工程，建筑桥梁工程和车辆工程等。

对于夹层板固有频率的研究，最经典的著作是文献[5]，其中以Hoff理论研究了夹层板的振动频率，分别研究了不同边界条件下，固有频率的计算方法。吴晖把夹层板看作正交异性体，借鉴文献[5]中的结论，认为垂直波纹方向的剪切刚度无限大，推导出固有频率的解析表达式^[6]；Hossein Zamanifar等运用精细切片法研究了夹层板的自由振动和强迫振动，通过等效参数，将三维的夹层板通过等效参数转化成二维正交异性板，求解了无量纲的自由振动频率，与前人的研究成果吻合一致^[7]。Zig-Zag理论在复合材料或层合板领域应用很广泛^[8-10]。白瑞祥等以3层均质复合材料（或等效成3层均质复合材料）夹层板为研究对象，引入一阶Zig-Zag理论，建立了夹层板自由振动的有限元模型，采用子空间迭代法求解自由振动固有频率^[11]。对于上下面板之间是非连续介质的心层夹层板，目前采用一阶Zig-Zag理论来建立运动方程的文献很少。本文提出一种考虑波纹夹层板心层不连续形状的情况下，上下3层都应用一阶Zig-Zag理论建立夹层板的运动方程的方法。

1 振动微分方程

如图2所示，波纹夹层板由上下面板和中间心层组成。在波纹夹层板中建立坐标系统。

图中：t_t为上面板厚度，t_b为下面板厚度，t_c为中间心层的厚度；心层的静高度为h_c，心层的周期间距为l_c。在线弹性理论分析，做以下基本假设：

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

根据假设1~假设4和图2中夹层板的坐标系统，上下面板的位移函数可以表示为：

${u_k}\left( {x,y,{z_k},t} \right) = {u_{ok}}\left( {x,y,t} \right) + {z_k}{\phi _{xk}}\left( {x,y,t} \right) {\text{，}}$

(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)

式中：k取t，表示上面板；k取b，表示下面板。

心层的位移函数可以表示为：

${u_c}\left( {x,y,{z_c},t} \right) = {u_{oc}}\left( {x,y,t} \right) + {z_c}{\phi _{xc}}\left( {x,y,t} \right) {\text{，}}$

(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)

其中： ${u_k}$ ， ${v_k}$ 和 ${w_k}$ （k=t或b，以下表述与此同）分别表示上下面板的面内x，y，z 三个方向的位移函数； ${u_{0k}}$ ， ${v_{0k}}$ 分别表示上下面板中面内x，y方向的位移函数， ${\phi _{xk}}$ 和 ${\phi _{yk}}$ 分别表示上下面板中面法线在xz平面内的转角和上下面板中面法线在yz平面内的转角； ${z_k}$ 表示上下面板垂向坐标，原点分别对应上下面板的中面，向下为正方向。 ${u_c}$ ， ${v_c}$ 为心层在x，y方向的位移函数； ${\phi _{xc}}$ 和 ${\phi _{yc}}$ 分别表示心层中面法线在xz平面内的转角和在yz平面内的转角， ${z_c}$ 表示心层垂向坐标，原点位于心层的中面上，向下为正方向。 $w$ 为夹层板在z向的位移函数。坐标系统如图2所示。需要说明的是， ${\phi _{xk}}$ ， ${\phi _{yk}}$ ， ${\phi _{xc}}$ 和 ${\phi _{yc}}$ 都是以x轴或y轴转向远离z轴的方向为正。

根据假设5，可以列出位移连续条件如下：

${u_t}\left( {x,y,{z_t} = \frac{{{t_t}}}{2}} \right) = {u_c}\left( {x,y,{z_c} = - \frac{{{h_c}}}{2}} \right) {\text{，}}$

(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)

将式（1）~式（5）代入式（7）~式（10），求出：

${u_{ot}}\left( {x,y} \right) = {u_{oc}}\left( {x,y} \right) - \frac{{{h_c}}}{2}{\phi _{xc}}\left( {x,y} \right) - \frac{{{t_t}}}{2}{\phi _{xt}}\left( {x,y} \right) {\text{，}}$

(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)

将式（11）~式（14）代入式（1）和式（2），联合式（3）~式（6），则所有的位移函数都用心层的中面位移函数表达。

应用弹性理论的应变位移关系，对于上面板：

$\varepsilon _x^t = \frac{{\partial {u_t}}}{{\partial x}} = \frac{{\partial {u_{oc}}\left( {x,y} \right)}}{{\partial x}} - \frac{{{h_c}}}{2}\frac{{\partial {\phi _{xc}}\left( {x,y} \right)}}{{\partial x}} + \left( {{z_t} - \frac{{{t_t}}}{2}} \right)\frac{{\partial {\phi _{xt}}\left( {x,y} \right)}}{{\partial x}} {\text{，}}$

(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)

对于下面板也可以类似推导。

对于心层，正应变和剪应变分别为：

$\varepsilon _x^c = \frac{{\partial {u_c}\left( {x,y,{z_c}} \right)}}{{\partial x}} = \frac{{\partial {u_{oc}}\left( {x,y} \right)}}{{\partial x}} + {z_c}\frac{{\partial {\phi _{xc}}\left( {x,y} \right)}}{{\partial x}}{\text{，}}$

(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)

其中， $\varepsilon _x^c$ 和 $\gamma _s^c$ 为心层沿x方向的正应变和沿心层表面的剪应变， $\gamma _{yz}^c$ 为心层在yz平面内的剪应变。

1.2 物理方程

上下面板和心层的每一个构件材料，认为它是各向同性的，则由弹性理论可以求出应力。

$\sigma _x^k = \frac{E}{{1 - {\mu ^2}}}\left( {\varepsilon _x^k + \mu \varepsilon _y^k} \right){\text{，}}$

(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)

其中， $E,\mu $ 分别对应上下面板的弹性模量和泊松比。心层的应力同样求得如下：

$\sigma _x^c = {E_c}\varepsilon _x^c{\text{，}}$

(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)

根据文献[12]和文献[13]的结论，心层yz平面的剪切弹性模量 $G_{yz}^c$ 可表达为：

$G_{yz}^c = \frac{{{E_c}{t_c}{{\sin }^2}\theta \cos \theta }}{{{h_c}}}{\text{。}}$

(31)

1.3 能量原理

哈密尔顿（Hamilton）原理提出稳定系统的动能和势能之差（动势）必然是最小值。对于振动分析的波纹夹层板而言，系统存在动能T和势能U，但是没有外力做功，势能仅包括系统的应变能。根据变分法，则应该使得

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

(32)

系统的动能包括3部分，分别是上下面板动能和心层动能。

$T = {T_t} + {T_b} + {T_c}{\text{，}}$

(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)

系统的势能主要是指构件的应变能，应变能也包括3部分，分别是上下面板应变能和心层应变能。

$U = {U_t} + {U_b} + {U_c}{\text{，}}$

(36)

其中：

$\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)

将式（33）~（38）代入式（32），应用变分法中的拉格朗日–欧拉方程，经过复杂的推导、运算，得出以下振动微分方程：

$\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

对于简支的夹层板，其边界条件为：

在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)

在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)

根据边界条件式（48）和式（49），可以取双傅里叶级数为式（39） ~ 式（47）的解，则

${u_{0c}} = \sum\limits_{m = 1}^\infty {\sum\limits_{n = 1}^\infty {{u_{mn}}\cos \frac{{m\text{π} x}}{a}} } \sin \frac{{n\text{π} y}}{b}{e^{i\varpi t}}{\text{，}}$

(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)

将式（50）~式（56）代入式（39）~式（47），然后比较两边的系数，则可以列出关于 ${u_{mn}}$ ， ${v_{mn}}$ ， ${\phi _{xmn}}$ ， ${\phi _{ymn}}$ ，w， ${\phi _{xkmn}}$ 和 ${\phi _{ykmn}}$ 的方程组，要使得这些系数有非零解，必然使得系数行列式为零。这个系数行列式就是关于固有角频率 $\varpi $ 的十八次多项式，解出这个十八次方程，取最小的正根就是所要求的固有角频率，则固有频率 $f = \varpi /2\text{π} $ 。

3 方法验证

夹层板四边简支，长边a=2 000 mm，短边b=1 500 mm，上下面板厚度t_t=t_b=3 mm，心层厚度t_c=2 mm，心层静高度h_c=40 mm，心层周期间距l_c=50 mm，上下面板和夹心都是同样材料，弹性模量E=E_c=2.1×10⁵ MPa，泊松比 $\mu = {\mu _c} = 0.3$ 。计算其横向微振动的固有频率。

为了形成对比，验证方法的准确性，分别用4种方法分别计算这个算例。

1）方法1

按照文献[5]所列公式计算固有频率（即Hoff理论），不过其中的剪切刚度不能采用连续介质心层的公式，而是采用等效剪切刚度，即采用式（56）；

2）方法2

按照文献[6]中所列公式求解（即简化的Reisser理论），其中的剪切刚度也是采用方法1中的等效剪切刚度；

3）方法3

采用Ansys软件，用shell181单元模拟面板和心层，三维有限元计算固有频率；

4）方法4

本文方法。

表1列出了这4种方法计算的前9阶固有频率。从表1看出关于前8阶固有频率，上述4种方法的计算结果都非常接近，误差小于10%。因此本文的计算方法是有效可信的。

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

(57)

4 面板剪切作用的体现

当面板厚度较小的时候，忽略面板的剪切作用造成的误差是很小的。但是当面板厚度增加，剪切作用就表现比较明显。以第4节中算例的参数为基础，固定其他变量，分别以面板厚度和心层厚度为自变量，一阶频率为因变量，图3为不同方法计算的夹层板一阶频率随面板厚度变化曲线（上下面板相等，且同步变化），图4为不同方法计算的夹层板一阶频率随心层厚度变化曲线。

从图3和图4看出，面板厚度较小，不同方法计算的结果差别很小，面板厚度大于5 mm后，差别则很明显。同时，因为各类方法都考虑了心层的剪切作用，不管心层厚度如何变化，不同计算方法计算的固有频率都很接近。

5 结　语

通过应用一阶Zig-zag理论，同时考虑心层的实际形状，建立波纹夹层板的振动微分方程。用算例验证了本方法计算的前8阶固有频率的计算误差都在10%以内。

波纹夹层板面板板厚较小，忽略面板剪切作用的影响，导致的计算误差可以忽略；当面板厚度大于心层厚度1/8时，则误差是明显的，并且厚度越大，误差也越大。

因为Hoff理论中考虑了心层的剪切作用，因此波纹心层厚度的增加，不管是用Hoff理论计算还是简化的Hoff理论计算，都与本文方法计算结果极为接近。