作旋转运动功能梯度材料矩形Mindlin板的刚柔耦合动力学建模$^{1)}$

图1 作大范围运动FGM矩形板

$R$为中心刚体半径,$\mathbf r_0$为连体坐标系原点在惯性坐标系中的矢径, $\mathbf \rho_0$为点$P_0$在连体坐标系中的位置。现假定$P_{0}$点为矩形板变形前板内任意一点,变形后为$P$,变形矢量为$\mathbf u$,在连体坐标系下的分量为($u_{1} $,$u_{2} $,$u_{3} )$。其中纵向位移$u_{1}$,$u_{2} $可表示为

(2)$\begin{eqnarray} \left. \begin{array}{l} u_{1} =w_{1} -\dfrac{1}{2}\displaystyle\int_0^x {\left( {\dfrac{\partial u_{3} }{\partial \xi }} \right)^{2}{\rm d}\xi +z\varphi_{x} } \\[3mm] u_{2} =w_{2} -\dfrac{1}{2}\displaystyle\int_0^y {\left( {\dfrac{\partial u_{3} }{\partial \eta }} \right)^{2}{\rm d}\eta +z\varphi_{y} } \\ \end{array} \right\} \end{eqnarray}$

式中,$w_{1} $和$w_{2}$分别为板中面沿$x$和$y$方向的实际伸长量,而$-\dfrac{1}{2}\displaystyle\int_0^x {\left({\dfrac{\partial u_{3} }{\partial \xi }} \right)^{2}{\rm d}\xi } $和$-\dfrac{1}{2}\displaystyle\int_0^y {\left( {\dfrac{\partial u_{3} }{\partial \eta }}\right)^{2}{\rm d}\eta } $是由板的横向变形引起的纵向缩短量,即二次耦合变形量,而传统的零次模型是建立在结构动力学小变形假设上,即舍去该项。$\varphi_{x} $和$\varphi_{y}$分别为横截面相对于$y$轴和$x$轴的转角。在浮动坐标系下

(3)$\begin{eqnarray} \left. {\begin{array}{l} \mathbf V_{O} =v_{1} \mathbf e_{1} +v_{2} \mathbf e_{2} +v_{3} \mathbf e_{3} \\ \mathbf \omega_{A} =\omega_{1} \mathbf e_{1} +\omega_{2} \mathbf e_{2} +\omega_{3} \mathbf e_{3} \\ \mathbf \rho_{0} =x\mathbf e_{1} +y\mathbf e_{2} +z\mathbf e_{3} \\ \mathbf u=u_{1} \mathbf e_{1} +u_{2} \mathbf e_{2} +u_{3} \mathbf e_{3} \\ \mathbf V_{PA} =\dot{{u}}_{1} \mathbf e_{1} +\dot{{u}}_{2} \mathbf e_{2} +\dot{{u}}_{3} \mathbf e_{3} \\ \end{array}} \right\} \end{eqnarray}$

式中,$\mathbf V_{O} $和$\mathbf \omega_{A}$分别为连体坐标系相对于惯性坐标$O$-$XYZ$的速度和角速度矢量,$\mathbf V_{PA}$为$P$点相对连体坐标系的速度矢量。则$P$点相对于惯性坐标系$O$-$XYZ$的速度矢量为

(4)$\begin{aligned} \boldsymbol{V}_{P}=& \boldsymbol{V}_{O}+\omega_{A}\left(\rho_{0}+\boldsymbol{u}\right)+\boldsymbol{V}_{P A}=\\ &\left[v_{1}+\dot{u}_{1}+\omega_{2}\left(z+u_{3}\right)-\omega_{3}\left(y+u_{2}\right)\right] e_{1}+\\ &\left[v_{2}+\dot{u}_{2}+\omega_{3}\left(x+u_{1}\right)-\omega_{1}\left(z+u_{3}\right)\right] e_{2}+\\ &\left[v_{3}+\dot{u}_{3}+\omega_{1}\left(y+u_{2}\right)-\omega_{2}\left(x+u_{1}\right)\right] e_{3} \end{aligned}$

根据一阶剪切变形理论,矩形板内任意一点的应变$\varepsilon$,其分量为

(5)$\begin{eqnarray} \left.\begin{array}{l} \varepsilon_{x} =\dfrac{\partial w_{1} }{\partial x}+z\dfrac{\partial \varphi_{x} }{\partial x} \\[2mm] \varepsilon_{y} =\dfrac{\partial w_{2} }{\partial y}+z\dfrac{\partial \varphi_{y} }{\partial y} \\[2mm] \gamma_{xy} \approx \dfrac{\partial w_{1} }{\partial y}+\dfrac{\partial w_{2} }{\partial x}+z\left(\dfrac{\partial \varphi_{x} }{\partial y}+\dfrac{\partial \varphi_{y} }{\partial x}\right) \\[2mm] \gamma_{xz} =\dfrac{\partial u_{3} }{\partial x}+\varphi_{x} \\[2mm] \gamma_{yz} =\dfrac{\partial u_{3} }{\partial y}+\varphi_{y} \\[2mm] \end{array}\right\} \end{eqnarray}$

由本构关系得应力为

(6)$\begin{eqnarray} {\mathbf \sigma }=\dfrac{E(z)}{1-\mu^{2}}\cdot\\&&\hspace{-7mm}\quad\left[ {{\begin{array}{*{20}c} 1 & \mu & 0 & 0 & 0 \\ \mu & 1 & 0 & 0 & 0 \\ 0 & 0 & {(1-\mu )/2} & 0 & 0 \\ 0 & 0 & 0 & {\kappa (1-\mu )/2} & 0 \\ 0 & 0 & 0 & 0 & {\kappa (1-\mu )/2} \\ \end{array} }} \right]{\mathbf \varepsilon } \end{eqnarray}$

其中$\kappa $为剪切修正系数,取值为$\kappa ={5}/{6}$。

1.3 系统的动能和势能

FGM矩形板的动能为

(7)$\begin{eqnarray} T=\dfrac{1}{2}\iiint_V \rho (z)\mathbf V_{P}^{\rm T}\mathbf V_P {\rm d}V \end{eqnarray}$

将式(4)代入得

(8)$\begin{eqnarray} T=\dfrac{1}{2}\iiint_V {\rho (z)} [(v_{1}^{2}+v_{2} ^{2}+v_{3}^{2})+\\(\dot{{u}}_{1}^{2}+\dot{{u}}_{2}^{2}+\dot{{u}}_{3} ^{2})+ (\omega_{1}^{2}+\omega_{2}^{2})(z+u_{3} )^{2}+\\(\omega_{2} ^{2}+\omega_{3}^{2})(x+u_{1} )^{2} - 2\omega_{2} \omega_{3} (y+u_{2} )\cdot \\(u_{3} +z)+(\omega_{1} ^{2}+\omega_{3}^{2})(y+u_{2} )^{2}+\\ 2(v_{1} \dot{{u}}_{1} +v_{2} \dot{{u}}_{2} +v_{3} \dot{{u}}_{3} )+2(v_{2} \omega_{3} -v_{3} \omega_{2} )\cdot \\(x+u_{1} ) + 2(v_{3} \omega_{1} -v_{1} \omega_{3} )(y+u_{2} )+\\2(v_{1} \omega_{2} -v_{2} \omega_{1} )(u_{3} +z)+ 2(\omega _{3} \dot{{u}}_{2} -\omega_{2} \dot{{u}}_{3} )\cdot \\(x+u_{1} )-2(\omega_{1} \dot{{u}}_{3} -\omega_{3} \dot{{u}}_{1} )(y+u_{2} ) -\\ 2(\omega_{2} \dot{{u}}_{1} -\omega_{1} \dot{{u}}_{2} )(z+u_{3} )-2\omega _{1} \omega_{3} (x+u_{1} )\cdot \\(u_{3} +z)-2\omega_{1} \omega_{2} (x+u_{1} )(y+u_{2} )]{\rm d}V \end{eqnarray}$

FGM矩形板的势能为

(9)$\begin{eqnarray} U=\dfrac{1}{2}\iiint_V {{\mathbf \varepsilon }^{\rm T}{\mathbf \sigma }{\rm d}V} \end{eqnarray}$

将式(5)和式(6)代入式(9)得

(10)$\begin{eqnarray} U=\dfrac{1}{2}\iiint_V \dfrac{E(z)}{1-\mu^{2}}\bigg[\varepsilon _{x}^{2}+2\mu \varepsilon_{x} \varepsilon_{y} +\varepsilon_{y} ^{2}+ \dfrac{1-\mu }{2}\gamma_{xy}^{2} +\dfrac{\kappa (1-\mu )}{2}\gamma _{xz}^{2}+\dfrac{\kappa (1-\mu )}{2}\gamma_{yz}^{2}\bigg]{\rm d}V= U_{1} +U_{2} \end{eqnarray}$

式中$U_{1}$为板面内变形能,$U_{2}$为横向弯曲变形能,即

(11)$\begin{eqnarray} U_{1} =\dfrac{1}{2}\iiint_V \dfrac{E(z)}{1-\mu^{2}}\bigg[\bigg(\dfrac{\partial w_{1} }{\partial x}\bigg)^{2}+\\[0.3mm]\bigg(\dfrac{\partial w_{2} }{\partial y}\bigg)^{2}+ 2\mu \bigg(\dfrac{\partial w_{1} }{\partial x}\bigg)\bigg(\dfrac{\partial w_{2} }{\partial y}\bigg)+\\[0.3mm] \dfrac{1-\mu }{2}\bigg(\dfrac{\partial w_{1} }{\partial y}+\dfrac{\partial w_{2} }{\partial x}\bigg)^{2}\bigg]{\rm d}V+\\[0.3mm]\iiint_V {\dfrac{zE(z)}{1-\mu^{2}}} \bigg[\dfrac{\partial w_{1} }{\partial x}\dfrac{\partial \varphi_{x} }{\partial x}+\dfrac{\partial w_{2} }{\partial y}\dfrac{\partial \varphi_{y} }{\partial y}+\\[0.3mm] \mu \bigg(\dfrac{\partial w_{1} }{\partial x}\dfrac{\partial \varphi_{y} }{\partial y}+\dfrac{\partial w_{2} }{\partial y}\dfrac{\partial \varphi_{x} }{\partial x}\bigg)+\\[0.3mm]\dfrac{1-\mu }{2}\bigg(\dfrac{\partial w_{1} }{\partial y}+\dfrac{\partial w_{2} }{\partial x}\bigg)\bigg(\dfrac{\partial \varphi_{x} }{\partial y}\dfrac{\partial \varphi_{y} }{\partial x}\bigg)\bigg]{\rm d}V \end{eqnarray}$

(12)$\begin{eqnarray} U_{2} =\dfrac{1}{2}\iiint_V {\dfrac{z^{2}E(z)}{1-\mu^{2}}} \bigg[\bigg(\dfrac{\partial^{2}u_{3} }{\partial x^{2}}\bigg)^{2}+\bigg(\dfrac{\partial^{2}u_{3} }{\partial y^{2}}\bigg)^{2}+\\[0.3mm]2\mu \bigg(\dfrac{\partial^{2}u_{3} }{\partial x^{2}}\bigg)\bigg(\dfrac{\partial^{2}u_{3} }{\partial y^{2}}\bigg)+ \dfrac{1-\mu }{2}\bigg(\dfrac{\partial^{2}u_{3} }{\partial x\partial y}\bigg)^{2}\bigg]{\rm d}V+\\[0.3mm]\dfrac{1}{2}\iiint_V {\dfrac{\kappa E(z)}{2(1+\mu )}} \bigg[\bigg(\dfrac{\partial u_{3} }{\partial x}+\varphi_{x} \bigg)^{2}+\\[0.3mm]\bigg(\dfrac{\partial u_{3} }{\partial y}+\varphi_{y} \bigg)^{2}\bigg]{\rm d}V \end{eqnarray}$

1.4 FGM矩形板变形场的离散

常用的矩形板变形场离散方法有假设模态法和有限元法,而本文采用无网格法中的径向基点插值法(radial point interpolation method, RPIM)。对于问题域$\varOmega$中的连续场函数${u}({\mathbf x})$,可以用在该问题域中相关点的基函数表示如下

(13)$\begin{eqnarray} {u}({\mathbf x})=\sum\limits_{i=1}^n {R_{i} } ({\mathbf x})a_{i} +\sum\limits_{j=1}^m {p_{j} ({\mathbf x})b_{j} } =\\ {\mathbf R}^{\rm T}({\mathbf x}){\mathbf a}+{\mathbf p}^{\rm T}({\mathbf x}){\mathbf b} \end{eqnarray}$

式中${\mathbf x}$为计算点坐标,$R_{i} ({\mathbf x})$为径向基函数,$n$为计算点支持域中的节点数,$p_{j} ({\mathbf x})$为空间坐标${\mathbf x}^{\rm T}=[x,y]$中的单项式,$m$为多项式基函数的个数,$a_{i}$和$b_{j} $为待定常数,假设$m=6$,则${\mathbf p}^{\rm T}=\left\{{1,x,y,x^{2},xy,y^{2}} \right\}$。

在径向基函数$R_{i} ({\mathbf x})$中,仅有表示计算点${\mathbf x}$和问题域中节点${\mathbf x}_{i} $之间距离的变量$r_{i}$,本文节点距离变量表达式为：$r_{i} =\sqrt{(x-x_{i} )^{2}+(y-y_{i} )^{2}}$,常用的径向基函数见文献[24]。本文选取的径向基函数为2次函数,其中形状参数$\alpha _{\rm c} $和$q$分别为4和1.03。2次函数表达式为

(14)$\begin{eqnarray} R_{i} (x,y)=[r_{i}^{2}+(\alpha_{\rm c} d_{\rm c})^{2}]^{q} \end{eqnarray}$

式中$d_{\rm c} $为计算点${\mathbf x}$同其局部支持域中的节点的间距的均值。

矩形板问题域中任意一点的变形可以表示为

(15)$\begin{eqnarray} \left. {{\begin{array}{*{20}c} {w_{1} (x,y,t)={\mathbf \varPhi }_{1} (x,y){\mathbf q}_{1} (t)} \\ {w_{2} (x,y,t)={\mathbf \varPhi }_{2} (x,y){\mathbf q}_{2} (t)} \\ {u_{3} (x,y,t)={\mathbf \varPhi }_{3} (x,y){\mathbf q}_{3} (t)} \\ {\varphi_{x} (x,y,t)=\mathbf \phi_{x} (x,y){\mathbf q}_{3} (t)} \\ {\varphi_{y} (x,y,t)=\mathbf \phi_{y} (x,y){\mathbf q}_{3} (t)} \\ \end{array} }} \right\} \end{eqnarray}$

其中所对应的形函数${\mathbf \varPhi }_{1} $,${\mathbf \varPhi }_{2} $,${\mathbf \varPhi }_{3}$,$\mathbf \phi_{x} $,$\mathbf \phi_{y} $的定义可见文献[21]。${\mathbf q}_{1} (t)$和${\mathbf q}_{2} (t)$为对应节点的纵向变形列阵,${\mathbf q}_{3}(t)$为相应节点横向变形和截面转角组成的列阵,对应$i$节点的表达式为：${\mathbf q}_{3i} (t)=\big[\begin{array}{*{20}c} {u_{3i} } & {\varphi_{xi} } & {\varphi_{yi} } \end{array}\big]^{\rm T}$

令${\mathbf \varPhi }_{4} =z\mathbf \phi_{x} $,${\mathbf \varPhi }_{5} =z\mathbf \phi_{y}$,将式(15)代入式({2})中,得变量$u_{1} $,$u_{2} $及其速度为

(16)$\begin{eqnarray} &&\left. \begin{array}{*{20}c} {u_{1} ={\mathbf \varPhi }_{1} {\mathbf q}_{1} -\dfrac{1}{2}{\mathbf q}_{3}^{\rm T} {\mathbf H}_{1} (x,y)\mathbf q_{3} +{\mathbf \varPhi }_{4} {\mathbf q}_{3} } \\[2mm] {u_{2} ={\mathbf \varPhi }_{2} {\mathbf q}_{2} -\dfrac{1}{2}{\mathbf q}_{3}^{\rm T} {\mathbf H}_{2} (x,y){\mathbf q}_{3} +{\mathbf \varPhi }_{5} {\mathbf q}_{3} } \\ \end{array} \right\}\end{eqnarray}$

(17)$\begin{eqnarray} \left. \begin{array}{*{20}c} {\dot{{u}}_{1} ={\mathbf \varPhi }_{1} {\dot{{\mathbf q}}}_{1} -{\mathbf q}_{3}^{\rm T} {\mathbf H}_{1} (x,y)\dot{{{\mathbf q}}}_{3} +{\mathbf \varPhi }_{4} {\dot{{\mathbf q}}}_{3} } \\ {\dot{{u}}_{2} ={\mathbf \varPhi }_{2} {\dot{{\mathbf q}}}_{2} -{\mathbf q}_{3}^{\rm T} {\mathbf H}_{2} (x,y)\dot{{{\mathbf q}}}_{3} +{\mathbf \varPhi }_{5} {\dot{{\mathbf q}}}_{3} } \\ \end{array} \right\} \end{eqnarray}$

式中,${\mathbf H}_{1} (x,y)$和${\mathbf H}_{2} (x,y)$为耦合函数,表达式为

(18)$\begin{eqnarray} \left. {{\begin{array}{*{20}c} {{\mathbf H}_{1} (x,y{\mathbf )}=\int_0^x {{\mathbf \varPhi }_{3,x}^{\rm T} {\mathbf \varPhi }_{3,x} {\rm d}x} } \\[3mm] {{\mathbf H}_{2} (x,y{\mathbf )}=\int_0^y {{\mathbf \varPhi }_{3,y}^{\rm T} {\mathbf \varPhi }_{3,y} {\rm d}y} } \\ \end{array} }} \right\} \end{eqnarray}$

式中下标","表示对坐标求偏导数。

1.5 FGM矩形板刚柔耦合动力学方程

取广义坐标${\mathbf q}=\big[{{\mathbf q}_{1} } \ \ {{\mathbf q}_{2} } \ \ {{\mathbf q}_{3}} \big]^{\rm T}$,将动能和势能代入第二类拉格朗日方程

(19)$\begin{eqnarray} \dfrac{\rm d}{{\rm d}t}\left(\dfrac{\partial T}{\partial {\dot{{\mathbf q}}}}\right)-\dfrac{\partial T}{\partial {\mathbf q}}+\dfrac{\partial U}{\partial {\mathbf q}}={\mathbf 0} \end{eqnarray}$

得到FGM板的刚柔耦合动力学方程为

(20)$\begin{eqnarray} \left[ {{\begin{array}{*{20}c} {\mathbf M_{11} } & {\mathbf0} & {\mathbf M_{13} } \\ {\mathbf0} & {\mathbf M_{22} } & {\mathbf M_{23} } \\ {\mathbf M_{31} } & {\mathbf M_{32} } & {\mathbf M_{33} } \\ \end{array} }} \right]\left[ {{\begin{array}{*{20}c} {\ddot{{\mathbf q}}_{1} } \\ {\ddot{{\mathbf q}}_{2} } \\ {\ddot{{\mathbf q}}_{3} } \\ \end{array} }} \right]+\\ \left[ {{\begin{array}{*{20}c} {\mathbf0} & {\mathbf G_{12} } & {\mathbf G_{13} } \\ {\mathbf G_{21} } & {\mathbf0} & {\mathbf G_{23} } \\ {\mathbf G_{31} } & {\mathbf G_{32} } & {\mathbf G_{33} } \\ \end{array} }} \right]\left[ {{\begin{array}{*{20}c} {\dot{{\mathbf q}}_{1} } \\ {\dot{{\mathbf q}}_{2} } \\ {\dot{{\mathbf q}}_{3} } \\ \end{array} }} \right]+\\ \left[ {{\begin{array}{*{20}c} {\mathbf K_{11} } & {\mathbf K_{12} } & {\mathbf K_{13} } \\ {\mathbf K_{21} } & {\mathbf K_{22} } & {\mathbf K_{23} } \\ {\mathbf K_{31} } & {\mathbf K_{32} } & {\mathbf K_{33} } \\ \end{array} }} \right]\left[ {{\begin{array}{*{20}c} {\mathbf q_{1} } \\ {\mathbf q_{2} } \\ {\mathbf q_{3} } \\ \end{array} }} \right]=\left[ {{\begin{array}{*{20}c} {\mathbf Q_{1} } \\ {\mathbf Q_{2} } \\ {\mathbf Q_{3} } \\ \end{array} }} \right] \end{eqnarray}$

式中各分块矩阵为

(21)$\mathbf M_{11} =\mathbf W_{11},\ \ \mathbf M_{22} =\mathbf W_{22}$

(22)$\mathbf M_{33} =\mathbf W_{33} +\underline{\mathbf W_{44} +\mathbf W_{55} }$

(23)$ \underline{\mathbf M_{31} =\mathbf M_{13}^{\rm T} =\mathbf W_{41} },\ \ \underline{\mathbf M_{32} =\mathbf M_{23}^{\rm T} =\mathbf W_{52}}$

(24)$\mathbf G_{12} =-\mathbf G_{21}^{\rm T} =-2\omega_{3} \mathbf W_{12}$

(25)$\mathbf G_{23} =-\mathbf G_{32}^{\rm T} =-2\omega_{1} \mathbf W_{23}$

(26)$\mathbf G_{13} =-\mathbf G_{31}^{\rm T} =2(\omega_{2} \mathbf W_{13} -\underline{\omega_{3} \mathbf W_{15} })$

(27)$\mathbf G_{33} =\underline{2\omega_{2} (\mathbf W_{43} -\mathbf W_{34} )+2\omega_{3} (\mathbf W_{54} -\mathbf W_{45} )+}\\\underline{2\omega_{1} (\mathbf W_{35} -\mathbf W_{53} )}$

(28)$\mathbf K_{11} =\mathbf K_{f11} -(\omega_{2}^{2} +\omega_{3}^{2} )\mathbf W_{11}$

(29)$\mathbf K_{12} =\mathbf K_{f12} +(\omega_{1} \omega_{2} -\dot{{\omega }}_{3} )\mathbf W_{12}$

(30)$\mathbf K_{13} =(\omega_{1} \omega_{3} +\dot{{\omega }}_{2} )\mathbf W_{13}-\\\underline{(\omega_{2}^{2} +\omega_{3}^{2} )\mathbf W_{14} +}\underline{(\omega _{1} \omega_{2} -\dot{{\omega }}_{3} )\mathbf W_{15} }$

(31)$\mathbf K_{21} =\mathbf K_{f12}^{\rm T} +(\omega_{1} \omega_{2} +\dot{{\omega }}_{3} )\mathbf W_{21}$

(32)$\mathbf K_{22} =\mathbf K_{f22} -(\omega_{1}^{2} +\omega_{3}^{2} )\mathbf W_{22}$

(33)$\mathbf K_{23} =(\omega_{2} \omega_{3} -\dot{{\omega }}_{1} )\mathbf W_{23} +\underline{(\omega_{1} \omega_{2} +\dot{{\omega }}_{3} )\mathbf W_{24} }\underline{-(\omega_{1}^{2} +\omega_{3}^{2} )\mathbf W_{25} }$

(34)$\mathbf K_{31} =(\omega_{1} \omega_{3} -\dot{{\omega }}_{2} )\mathbf W_{31} -\underline{(\omega_{2}^{2} +\omega_{3}^{2} )\mathbf W_{41} +}\underline{(\omega _{1} \omega_{2} +\dot{{\omega }}_{3} )\mathbf W_{51} }$

(35)$\mathbf K_{32} =(\omega_{2} \omega_{3} +\dot{{\omega }}_{1} )\mathbf W_{32} +\underline{(\omega_{1} \omega_{2} -\dot{{\omega }}_{3} )\mathbf W_{42} }\underline{-(\omega_{1}^{2} +\omega_{3}^{2} )\mathbf W_{52} }$

(36)$\begin{eqnarray}\mathbf K_{33} =\mathbf K_{f33} -(\omega_{1}^{2} +\omega_{2}^{2} )\mathbf W_{33} -\underline{(\omega_{2}^{2} +\omega_{3}^{2} )\mathbf W_{44} -(\omega_{1}^{2} +\omega_{3}^{2} )\mathbf W_{55}} +\underline{\dot{{\omega }}_{2} (\mathbf W_{43} -\mathbf W_{34} -\mathbf D_{31} )}+\\ \underline{\dot{{\omega }}_{1} (\mathbf W_{35} -\mathbf W_{53} +\mathbf D_{32} )+2\omega _{1} \omega_{3} \mathbf W_{43} +2\omega_{1} \omega_{2} \mathbf W_{45} }+ \underline{2\omega_{2} \omega_{3} \mathbf W_{53} +\dot{{\omega }}_{3} (\mathbf W_{54} -\mathbf W_{45} )}+\\ \underline{\underline {(\omega_{2}^{2} +\omega_{3}^{2} )\mathbf D_{11} }}+ \underline{\underline {(\omega_{1}^{2} +\omega_{3}^{2} )\mathbf D_{22} -(\omega_{1} \omega_{2} +\dot{{\omega }}_{3} )\mathbf D_{12} -(\omega_{1} \omega _{2} -\dot{{\omega }}_{3} )\mathbf D_{21} }} -\\ \underline{\underline{a_{01} \mathbf C_{1} -a_{02} \mathbf C_{2} }}\underline{\underline {-\omega_{1} \omega_{3} \mathbf D_{31} -\omega_{2} \omega_{3} \mathbf D_{32} }}\end{eqnarray}$

(37)$\mathbf Q_{1} =(\omega_{2}^{2} +\omega_{3}^{2} )\mathbf S_{11}^{\rm T} -(\omega_{1} \omega _{2} -\dot{{\omega }}_{3} )\mathbf S_{21}^{\rm T} -(\omega_{1} \omega_{3} +\dot{{\omega }}_{2} )\mathbf S_{31}^{\rm T} -a_{01} \mathbf Y_{1}^{\rm T}$

(38)$\mathbf Q_{2} =(\omega_{1}^{2} +\omega_{3}^{2} )\mathbf S_{22}^{\rm T} -\omega_{1} \omega _{2} \mathbf S_{12}^{\rm T} -\dot{{\omega }}_{3} \mathbf S_{13}^{\rm T} +\dot{{\omega }}_{1} \mathbf S_{32}^{\rm T} -a_{02} \mathbf Y_{2}^{\rm T}$

(39)$\begin{eqnarray}\mathbf Q_{3} =(\omega_{1}^{2} +\omega_{2}^{2} )\mathbf S_{33}^{\rm T} +(\omega_{1}^{2} +\omega_{3}^{2} )\mathbf S_{25}^{\rm T} +(\omega_{2}^{2} +\omega_{3}^{2} )\mathbf S_{14}^{\rm T} - \omega_{2} \omega_{3} (\mathbf S_{35}^{\rm T} +\mathbf S_{23}^{\rm T} )-\omega_{1} \omega_{3} (\mathbf S_{34}^{\rm T} +\mathbf S_{13}^{\rm T} )-\\\omega_{1} \omega_{2} (\mathbf S_{15}^{\rm T} +\mathbf S_{24}^{\rm T} ) + \dot{{\omega }}_{1}\mathbf S_{35}^{\rm T} -\mathbf S_{23}^{\rm T}+\dot{{\omega }}_{2} (\mathbf S_{13}^{\rm T} -\mathbf S_{34}^{\rm T})+\dot{{\omega }}_{3}(\mathbf S_{24}^{\rm T} -\mathbf S_{15}^{\rm T})- a_{03} \mathbf Y_{3}^{\rm T} - a_{02} \mathbf Y_{4}^{\rm T} -a_{01} \mathbf Y_{5}^{\rm T}\end{eqnarray}$

式中,$a_{01} $,$a_{02} $,$a_{03}$为基点加速度在连体坐标系下的分量,表达式为

(40)$a_{01} =\dot{{v}}_{1} +(\omega_{2} v_{3} -\omega_{3} v_{2} )$

(41)$a_{02} =\dot{{v}}_{2} +(\omega_{3} v_{1} -\omega_{1} v_{3} )$

(42)$a_{03} =\dot{{v}}_{3} +(\omega_{1} v_{2} -\omega_{2} v_{1} )$

各常数阵为

(43)$\mathbf W_{ij} =\iiint_V {\rho (z)} {\mathbf \varPhi }_{i}^{\rm T} {\mathbf \varPhi}_{j} {\rm d}V \\[.6mm](i=1,2,3,4,5;\ \ j=1,2,3,4,5)$

(44)$\mathbf C_{i} =\iiint_V {\rho (z)} \cdot \mathbf H_{i} {\rm d}V\ (i=1,2)$

(45)$\mathbf D_{1i} =\iiint_V {\rho (z)} \cdot x\cdot \mathbf H_{i} {\rm d}V\ (i=1,2)$

(46)$\mathbf D_{2i} =\iiint_V {\rho (z)} \cdot y\cdot \mathbf H_{i} {\rm d}V\ (i=1,2)$

(47)$\mathbf D_{3i} =\iiint_V {\rho (z)} \cdot z\cdot \mathbf H_{i} {\rm d}V\ (i=1,2)$

(48)$\mathbf S_{1i} =\iiint_V {\rho (z)} \cdot x\cdot {\mathbf \varPhi }_{i} {\rm d}V\ (i=1,2,3,4,5)$

(49)$\mathbf S_{2i} =\iiint_V {\rho (z)} \cdot y\cdot {\mathbf \varPhi }_{i} {\rm d}V\ (i=1,2,3,4,5)$

(50)$\mathbf S_{3i} =\iiint_V {\rho (z)} \cdot z\cdot {\mathbf \varPhi }_{i} {\rm d}V\ (i=1,2,3,4,5)$

(51)$\mathbf Y_{i} =\iiint_V {\rho (z)} \cdot {\mathbf \varPhi }_{i} {\rm d}V\ (i=1,2,3,4,5)$

(52)$\mathbf K_{f11} =\iiint_V {\dfrac{E(z)}{1-\mu^{2}}} \bigg({\mathbf \varPhi }_{1,x}^{\rm T} {\mathbf \varPhi }_{1,x} +\\[.6mm] \dfrac{1-\mu }{2}{\mathbf \varPhi }_{1,y}^{\rm T} {\mathbf \varPhi }_{1,y} \bigg){\rm d}V$

(53)$\mathbf K_{f12} =\mathbf K_{f21}^{\rm T} =\iiint_V {\dfrac{E(z)}{1-\mu^{2}}}\bigg(\mu {\mathbf \varPhi }_{1,x}^{\rm T} {\mathbf \varPhi }_{2,y} +\\[.6mm]\dfrac{1-\mu }{2}{\mathbf \varPhi }_{1,y}^{\rm T} {\mathbf \varPhi }_{2,x} \bigg){\rm d}V$

(54)$\mathbf K_{f22} =\iiint_V {\dfrac{E(z)}{1-\mu^{2}}} \bigg(\mu {\mathbf \varPhi }_{2,y}^{\rm T} {\mathbf \varPhi }_{2,y} +\\[.6mm]\dfrac{1-\mu }{2}{\mathbf \varPhi }_{2,x}^{\rm T} {\mathbf \varPhi }_{2,x} \bigg){\rm d}V$

(55)$\begin{eqnarray}\mathbf K_{f33} =\iiint_V {\dfrac{E(z)}{1-\mu^{2}}} \bigg[{\mathbf \varPhi }_{4,x}^{\rm T} {\mathbf \varPhi }_{4,x} +{\mathbf \varPhi }_{5,y}^{\rm T} {\mathbf \varPhi }_{5,y} +\\[.6mm] \dfrac{1-\mu }{2}({\mathbf \varPhi }_{4,y}^{\rm T} {\mathbf \varPhi }_{4,y} +{\mathbf \varPhi }_{5,x}^{\rm T} {\mathbf \varPhi }_{5,x} +{\mathbf \varPhi }_{4,y}^{\rm T} {\mathbf \varPhi }_{5,x} +\\[.6mm] {\mathbf \varPhi }_{5,x}^{\rm T} {\mathbf \varPhi }_{4,y} )+ \mu ({\mathbf \varPhi }_{4,x}^{\rm T} {\mathbf \varPhi }_{5,y} + {\mathbf \varPhi }_{5,y}^{\rm T} {\mathbf \varPhi }_{4,x} )\bigg]{\rm d}V+\\[.6mm]\iiint_V {\dfrac{\kappa E(z)}{2(1+\mu )}}\cdot (\underline{{\mathbf \varPhi }_{3,x}^{\rm T} {\mathbf \varPhi }_{3,x} +{\mathbf \varPhi }_{3,y}^{\rm T} {\mathbf \varPhi }_{3,y} +}\\[.6mm]\underline{\mathbf \phi_{x}^{\rm T} \mathbf \phi_{x} +\mathbf \phi_{y}^{\rm T} \mathbf \phi_{y} +{\mathbf \varPhi }_{3,x}^{\rm T} \mathbf \phi_{x} +{\mathbf \varPhi }_{3,y}^{\rm T} \mathbf \phi_{y} +}\\[.6mm]\underline{\mathbf \phi_{x}^{\rm T} {\mathbf \varPhi }_{3,x} +\mathbf \phi_{y}^{\rm T} {\mathbf \varPhi }_{3,y} }){\rm d}V\end{eqnarray}$

各分块式中双下划线项为考虑二次耦合变形量带来的附加动力刚度项。传统的零次近似模型建模时忽略了这些项,即这些项都为零。单下划线为考虑横向剪切变形所推导出的剪切变形项,若不考虑该项,则动力学方程退化为经典薄板理论。

2 数值仿真

2.1 功能梯度板"动力刚化"研究

本节仿真一作大范围已知运动的悬臂FGM矩形板,hub-FGM板系统的材料参数：长$L=1.828 8$ m,宽$H=1.219 2$ m,厚度$h=0.02$ m,$\rho _{\rm c} =3000$ kg/m$^{3}$,$\rho_{\rm m} =2707$ kg/m$^{3}$,$E_{\rm c}=151$ GPa,$E_{\rm m} =70$ GPa,泊松比$\mu =0.3$,中心刚体半径$R=0$。矩形板节点划分沿$x$和$y$轴分别以16和8个节点均匀分布,积分网格沿$x$和$y$轴分别均匀划分8格和6格,假设模态法(assumed mode method, AMM)和三角形单元有限元法(triangle finite element method,TR-FEM)分别取$5\times7$阶模态和$2\times 16\times 8$个单元。给定角速度规律为：$\omega_{1} =\omega_{3} =\dot{{\omega }}_{1} =\dot{{\omega }}_{3} =0$,$\omega_{2} =\omega $,$\dot{{\omega }}_{2} =\dot{{\omega }}$。其中

(56)$\begin{eqnarray} \omega =\left\{ {\begin{array}{ll} \dfrac{\varOmega }{T}t-\dfrac{\varOmega }{2\pi}\sin \left(\dfrac{2\pi }{T}t\right), & 0\leqslant t\leqslant T \\ \varOmega, & t>T \\ \end{array}} \right. \end{eqnarray}$

式中,时间$T=30$ s。

若只考虑FGM板沿厚度方向上的横向振动,系统的动力学方程可简化为

(57)$\begin{eqnarray} \mathbf M_{33} \ddot{{\mathbf q}}_{3} +\mathbf G_{33} \dot{{\mathbf q}}_{3} +\mathbf K_{33} \mathbf q_{3} =\mathbf Q_{3} \end{eqnarray}$

式中

(58)$\mathbf M_{33} =\mathbf W_{33} +\mathbf W_{44} +\mathbf W_{55}$

(59)$\mathbf K_{33} =\mathbf K_{f33} -\omega_{2}^{2}(\mathbf W_{33} +\mathbf W_{44} )-\\\dot{{\omega }}_{2} (\mathbf W_{34} -\mathbf W_{43} )+\underline{\underline{\omega_{2}^{2}\mathbf D_{11} -\dot{{\omega }}_{2} \mathbf D_{31} }}\qquad$

(60)$\mathbf G_{33} =2\omega_{2} (\mathbf W_{43} -\mathbf W_{34} )$

(61)$\mathbf Q_{3} =\dot{{\omega }}_{2} \mathbf S_{13}^{\rm T} -\dot{{\omega }}_{2} \mathbf S_{34}^{\rm T} +\omega_{2}^{2} (\mathbf S_{33}^{\rm T} +\mathbf S_{14}^{\rm T} )$

图2和图3是角速度分别取$\varOmega=3$ Hz和10 Hz,板功能梯度系数$N$分别取1和2的零次近似模型和一次近似模型计算得到的板外侧角点的横向变形图。从图2中可以看出,本文模型的计算结果与文献[9]基本吻合,但是在最大变形处,计算结果存在微小差异。这是因为文献[9]是基于经典薄板理论,而本文是基于考虑剪切变形的Mindlin板理论,因而系统更显柔性,所以最大变形处偏大。比较图3(a)和图3(b)可以看出,$\varOmega=3$ Hz低速运动时,不考虑附加刚度项的零次近似模型计算结果收敛,当运动为高速运动$\varOmega=10$ Hz时,零次近似模型的计算结果则发散,而一次近似模型的结果依然收敛。说明未考虑附加刚度项的零次近似模型只适用于低转速工况,而一次近似模型既适用于低速,又适用于高转速工况。

图2

图2 $N=1, \varOmega=3$ Hz时矩形板外侧角点横向变形

图3

图3 $N=2$时矩形板外侧角点横向变形

图4和图5是$N=2$,角速度$\varOmega =3$ Hz和10 Hz时一次近似模型得到板外侧角点横向变形速度。从两图中可以看出三种离散方法的仿真结果基本一致。进一步说明了本文所建模型的正确性。

图4

图4 $N=2$,$\varOmega =3$ Hz时板外侧角点横向变形速度

图5

图5 $N=2$,$\varOmega =10$ Hz时板外侧角点横向变形速度

图6为$\varOmega =10$ Hz,$h=0.002$ m,$E_{\rm c} =15.1$ GPa,$N=1$,其余参数不变的FGM矩形板外侧角点横向变形图。从图中可以看出,TR-FEM和RPIM的结果在前10 s基本一致,但随着时间的增大,TR-FEM结果发散。而RPIM的结果仍然收敛,说明了RPIM有更好的计算稳定性,在同等计算条件下比TR-FEM更有计算优势。

图6

图6 矩形板外侧角点横向变形图 ($\varOmega =10$ Hz,$h=0.002$ m,$E_{\rm c} =15.1$ GPa,$N=1$)

图7分别给出了在$\varOmega =3$ Hz和$\varOmega=10$ Hz时板外侧角点的横向变形随功能梯度系数的变化情况。从图中可以看出,当$N=0$时,功能梯度板退化为均质板,板的最大变形最小,随着$N$的增大,板外侧角点变形也增大,说明功能梯度系数能影响板的柔性,功能梯度系数越大,板的柔性越大。

图7

图7 不同功能梯度系数下矩形板外侧角点的横向变形

2.2 功能梯度板的固有频率研究

假设中心刚体半径为$R$,中心刚体以角速度$\varOmega$绕$y$轴匀速转动,则在连体坐标系下,原点$O$的速度矢量为：$(v_{1},v_{2},v_{3})=(0,0,-R\varOmega )$,$(\omega_{1},\omega_{2},\omega_{3})=(0,\varOmega,0)$。

若只考虑矩形板的横向振动,则基于Mindlin板理论推导的作定轴转动矩形板的自由振动方程为

(62)$\begin{eqnarray} \mathbf M_{33} \ddot{{\mathbf q}}_{3} +[\varOmega^{2}(R\mathbf C_{1} +\mathbf D_{11})-\\\varOmega^{2}(\mathbf W_{33} +\mathbf W_{44} )+\mathbf K_{f33} ]\mathbf q_{3} ={\mathbf0} \end{eqnarray}$

考虑到矩形板几何尺寸对固有频率的影响,定义如下无量纲变量：$\tau ={t}/{T}$,$\chi ={x}/{L}$,$\zeta ={y}/{H}$,$Z={q_{3} }/{L}$,$\delta ={L}/{H}$,$\eta ={h}/{L}$,$\sigma ={R}/{L}$,$\lambda ={E_{\rm c} }/{E_{\rm m} }$,$\vartheta ={\rho_{\rm c} }/{\rho_{\rm m} }$,$\varpi =\omega T$,$\gamma =\varOmega T$。

将上述无量纲变量代入式({62})得

(63)$\begin{eqnarray} \overline{\mathbf W}_{33} \ddot{{\mathbf Z}}+(\overline{\mathbf W}_{44} +\overline{\mathbf W}_{55} )\ddot{{\mathbf Z}}+[\gamma^{2}(\sigma \overline{\mathbf C}_{1} +\overline{\mathbf D}_{11} )-\\\gamma ^{2}(\overline{\mathbf W}_{33} +\overline{\mathbf W}_{44} )+\beta_{1} \mathbf K_{f_{1} 33} +\\ \beta _{2} \mathbf K_{f_{2} 33} +\beta_{3} \mathbf K_{f_{3} 33} ]\mathbf Z={\mathbf0} \end{eqnarray}$

式中,$T=\sqrt {(1-\mu^{2})\rho_{\rm m}L^{2}/E_{\rm m}}$。$\lambda $为无量纲弹性模量,$\vartheta $为无量纲密度,$\delta$为矩形板的纵横比,$\eta $为矩形板的厚长比,$\sigma$为中心刚体半径与矩形板纵向边长之比,$\varpi$为无量纲固有频率,$\gamma$为无量纲转动角速度,$N$为功能梯度指数。

$\begin{eqnarray} \beta_{1} =\dfrac{N^{3}+3N^{2}(\lambda +1)+N(3\lambda +8)+6\lambda}{N^{3}+3N^{2}(\vartheta +1)+N(3\vartheta +8)+6\vartheta }\nonumber\\ \beta_{2} =\dfrac{(1-\mu )\kappa (\lambda +N)}{2(\vartheta +N)}\nonumber\\ \beta_{3} =\dfrac{6(1-\mu )\kappa (\lambda +N)(N+3)(N+2)}{\eta^{2}(N^{3}+3N^{2}(\vartheta +1)+N(3\vartheta +8)+6\vartheta )}\nonumber\\\end{eqnarray}$

(64)$\overline{\mathbf W}_{33} =\int_0^1 {\int_0^1 {\overline {{\mathbf \varPhi }}_{3}^{\rm T} } } \overline {{\mathbf \varPhi }}_{3} \mbox{d}\chi \mbox{d}\zeta$

(65)$\overline{\mathbf W}_{44} =\int_0^1 {\int_0^1 {\overline{\mathbf \phi}_{\chi}^{\rm T} } } \overline{\mathbf \phi}_{\zeta } \mbox{d}\chi \mbox{d}\zeta\\$

(66)$\overline{\mathbf W}_{55} =\int_0^1 {\int_0^1 {\overline{\mathbf \phi}_{\chi}^{\rm T} } } \overline{\mathbf \phi}_{\zeta } \mbox{d}\chi \mbox{d}\zeta$

(67)$\overline{\mathbf C}_{1} =\int_0^1 {\int_0^1 {\overline{\mathbf H}_{1} } }(\chi,\zeta )\mbox{d}\chi \mbox{d}\zeta$

(68)$\overline{\mathbf D}_{11} =\int_0^1 {\int_0^1 {\chi \overline{\mathbf H}_{1} } } (\chi,\zeta )\mbox{d}\chi \mbox{d}\zeta$

(69)$\begin{eqnarray}\mathbf K_{f_1,33} =\int_0^1 {\int_0^1 {[\overline{\mathbf \phi} _{\chi,\chi }^{\rm T} } } \mathbf \phi_{\chi,\chi } +\delta^{2}\overline{\mathbf \phi} _{\zeta,\zeta }^{\rm T} \mathbf \phi_{\zeta,\zeta } +\\[2mm] \dfrac{1-\mu }{2}(\delta ^{2}\overline{\mathbf \phi}_{\chi,\zeta }^{\rm T} \mathbf \phi_{\chi,\zeta } +\overline{\mathbf \phi}_{\zeta,\chi }^{\rm T} \mathbf \phi_{\zeta,\chi } +\delta \overline{\mathbf \phi}_{\chi,\zeta }^{\rm T} \mathbf \phi_{\zeta,\chi } +\\[2mm]\delta \overline{\mathbf \phi}_{\zeta,\chi }^{\rm T} \mathbf \phi_{\chi,\zeta } ) + \mu \delta (\overline{\mathbf \phi}_{\chi,\chi }^{\rm T} \mathbf \phi_{\zeta,\zeta } +\\[2mm]\overline{\mathbf \phi}_{\zeta,\zeta }^{\rm T} \mathbf \phi_{\chi,\chi } )]\mbox{d}\chi \mbox{d}\zeta\end{eqnarray}$

(70)$\mathbf K_{f_2,33} =\int_0^1 {\int_0^1 {[\overline {{\mathbf \varPhi}}_{3,\chi }^{\rm T} } } \overline {{\mathbf \varPhi }}_{3,\chi } +\overline {{\mathbf \varPhi }}_{3,\zeta }^{\rm T} \overline {{\mathbf \varPhi }}_{3,\zeta }^{\rm T} ]\mbox{d}\chi \mbox{d}\zeta$

(71)$\mathbf K_{f_3,33} =\int_0^1 \int_0^1 [\overline{\mathbf \phi} _{\chi }^{\rm T} \mathbf \phi_{\chi } +\overline{\mathbf \phi}_{\zeta }^{\rm T} \mathbf \phi _{\zeta } ]\mbox{d}\chi \mbox{d}\zeta$

为进一步验证本文所建模型的正确性,将功能梯度板退化为匀质薄板,同文献[25]中的计算结果进行比较。表1为$\delta=1$,$\eta =0.01$,$\sigma =0$,中心刚体悬臂板无量纲角速度$\gamma=1$的前五阶无量纲固有频率。由表1可知,本文结果同文献[25]基本吻合,且由于考虑剪切变形,系统更偏柔性,固有频率比文献中的结果都偏小。表2为$\delta=1$,$\eta =0.01$,$\sigma =0$,$N=2$,不同无量纲角速度$\gamma $下中心刚体FGM悬臂板前三阶无量纲固有频率。由该表知,随着无量纲角速度的增大,三种方法的无量纲固有频率皆增大,且基于一阶剪切变形理论的RPIM结果总是比基于经典薄板理论的假设模态法的数值小,说明忽略剪切变形的经典薄板理论总是高估了系统的固有频率。表3为$\delta=1$,$\eta =0.01$,$\sigma =0$,$\gamma=10$,$N$取不同值时,中心刚体FGM悬臂板前五阶无量纲固有频率。结果表明,随着$N$的增大,系统无量纲固有频率减小,进一步说明功能梯度系数越大,系统的柔性越大。

表1 作定轴转动中心刚体悬臂板的前五阶无量纲固有频率($\delta=1$,$\eta=1$,$\sigma=0$,$\gamma=1$)

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表2 作定轴转动中心刚体FGM悬臂板的前三阶无量纲固有频率($\delta=1$,$\eta=0.01$,$\sigma=0$,$N=2$)

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表3 作定轴转动中心刚体FGM悬臂板的前五阶无量纲固有频率($\delta =1$,$\eta =0.01$,$\sigma =0$,$\gamma =10)$

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图8所示为$\delta =5$,$\eta =0.01$,$\sigma =1$时,均质悬臂板前五阶无量纲固有频率随无量纲角速度的变化图。从图中可以看出,各阶无量纲固有频率皆随无量纲角速度的增大而增大,而且在第二阶和第三阶无量纲固有频率有频率转向现象。从放大图8(b)可以看出,第二阶和第三阶固有频率没有发生交叉。

图8

图8 均质悬臂板前五阶无量纲固有频率($\delta =5$,$\eta =0.01$,$\sigma =1)$

图9为$\delta =5$,$\eta =0.01$,$\sigma=1$,$N=2$时,FGM悬臂板前八阶无量纲固有频率随无量纲角速度变化图。从图9(a)中能观察到随着无量纲角速度的增大,多阶无量纲固有频率发生了显著的频率转向现象,而从频率转向区域的局部放大图9(b)~9(d)中可知,发生频率转向的相邻两阶频率并没有交叉。

图9

图9 FGM悬臂板前八阶无量纲固有频率的变化情况($\delta =5$,$\eta =0.01$,$\sigma =1$,$N=2$)

图10为$\delta =1$,$\eta =0.01$,$\sigma=1$,$N=2$时,FGM悬臂板第六阶到第八阶无量纲固有频率随无量纲角速度变化图和局部放大图,从图10(a)看到第七阶固有频率同第六阶和第八阶固有频率发生了多次转向现象,说明板的纵横比对系统固有频率的频率转向影响较大。

图10

图10 FGM悬臂板第六阶到第八阶无量纲固有频率的变化情况($\delta =1$,$\eta =0.01$,$\sigma =1$,$N=2$)

3 结论

(1)采用RPIM描述柔性板的变形场,基于一阶剪切变形理论,建立了旋转FGM矩形板的刚柔耦合一次近似模型。该模型既适用于低转速情况,又适用于高转速情况,既适用于薄板问题,又适用于中厚板问题。在同等计算条件下,RPIM相比TR-FEM更具有计算优越性。

(2)随着功能梯度指数的增大,旋转FGM矩形板的横向弯曲变形变大,固有频率减小,说明功能梯度指数的增大使系统的柔性增大,可通过改变功能梯度指数的取值范围以达到控制变形及频率的目的。

(3)基于经典薄板理论的无量纲固有频率总是大于基于一阶剪切变形理论的无量纲固有频率,说明忽略剪切变形的Kirchhoff假设使结构更偏刚性,总是高估了系统的固有频率。

(4)系统的无量纲固有频率会随着转速和板纵横比的变化而发生频率转向现象。不仅有单次转向,还有多次转向现象,且发生频率转向的相邻两阶频率并不相交。

参考文献

原文顺序

文献年度倒序

文中引用次数倒序

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振动工程学报, 2003(2): 43-47

Liu

Jinyang

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Jiazhen

Dynamic modeling theory and finite element method for a rectangular plate undergoing large overall motion

Journal of Vibration Engineering, 2003(2): 43-47 (in Chinese)

[4]

Yoo

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Modal analysis of rotating composite cantilever plates

Journal of Sound & Vibration, 2002, 258(2): 233-246

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Nonlinear vibrations of functionally graded graphene reinforced composite cylindrical panels

Applied Mathematical Modelling, 2022, 101: 1-18

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Large amplitude thermally induced vibrations of temperature dependent annular FGM plates

Composites Part B: Engineering, 2019, 163: 371-383

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Alireza

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Nonlinear transient thermoelastic response of FGM plate under sudden cryogenic cooling

Ocean Engineering, 2021, 226: 108875

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Free vibration analysis of rotating functionally graded rectangular plates

Composite Structures, 2016, 136: 493-504

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黎亮, 章定国, 洪嘉振.

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动力学与控制学报, 2013, 11(4): 329-335

[本文引用: 3]

Liang

, Zhang

Dingguo

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Jiazhen

Dynamic characteristics of FGM rectangular thin plate for large range motion

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曲线加筋Kirchhoff-Mindlin板自由振动分析

力学学报, 2017, 49(4): 929-939

DOI [本文引用: 1]

相比传统加筋板,曲线加筋板能够更充分地发挥材料力学性能.在加筋板力学分析中,厚板通常采用Reissner-Mindlin理论,然而当板厚较薄时易出现剪切自锁,离散的Kirchhoff-Mindlin理论采用假设剪切应变场可避免该问题.针对曲线加筋Kirchhoff-Mindlin板自由振动分析,采用离散的Kirchhoff-Mindlin三角形单元和Timoshenko曲梁单元分别模拟板和加强筋,根据板的位移插值函数及筋板交界面的位移协调条件,建立基于板单元位移自由度的有限元方程.为了验证方法的有效性和准确性,采用直线加筋薄板、曲线加筋薄板和厚板3种模型进行算例研究,通过收敛性和精度分析来选择合理的有限元网格密度.直线加筋薄板前20阶固有频率均与文献结果吻合良好；曲线加筋板算例中,本文方法满足收敛条件的板单元数目为2469,Nastran模型板单元数目为6243；本文所得曲线加筋板固有频率与Nastran计算结果最大误差为3.4%.研究结果表明,本文方法无需筋板单元共节点,可使用较少的有限元网格数量,并能够保证计算精度；在离散Kirchhoff-Mindlin三角形板单元基础上构造Timoshenko梁单元可同时适用于曲线加筋薄板与厚板自由振动分析.

Liu

Jingze

, Jiang

Dong

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Xiaolin

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Free vibration analysis of curvilinearly stiffened Kirchhoff-Mindlin plates

Chinese Journal of Theoretical and Applied Mechanics, 2017, 49(4): 929-939 (in Chinese)

DOI [本文引用: 1]

Compared with traditional stiffened plates, curvilinearly stiffened plates can deliver the mechanical properties of materials more adequately. In mechanical analysis of stiffened thick plates, Reissner-Mindlin theory is usually adopted. However, difficulties are encountered in connection with shear locking when the plate thickness approaches zero. In order to avoid the above problem, the discrete Kirchhoff-Mindlin theory was investigated by employing the assumption of shear strain field. An efficient finite element approach for free vibration analysis of curvlinearly stiffened KirchhoffMindlin plates is presented in this paper. The discrete Kirchhoff-Mindlin triangular (DKMT) element and the Timoshenko curved beam element are employed for modeling the plate and the stiffeners, respectively. The finite element equation is established through the displacement interpolation function of plate and the displacement compatibility conditions at the plate-stiffener interfaces. In order to verify the efficiency and accuracy of the present method, linearly stiffened thin plate and curvilinearly stiffened thin and thick plates are used as numerical examples. The reasonable finite element mesh density is selected by convergence and accuracy analysis. The first 20 natural frequencies of the linearly stiffened plate are in good agreement with the literature. In the examples of the curvilinearly stiffened plate, the number of plate elements satisfying the convergence condition is 2469, while the number in Nastran model is 6243. The maximum error of the natural frequency between the present method and Nastran is 3.4%. Results show that present approach can guarantee the accuracy of calculation with less number of elements. The present method can be applied to the free vibration analysis of both stiffened thin and thick plates.

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Genyong

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Dynamic modeling for a composite plate undergoing large over motion

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杨兴, 刘仁伟, 侯鹏

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基于一阶剪切板理论的FGM板刚柔耦合动力学建模与仿真

动力学与控制学报, 2020, 18(4): 33-43

Yang

Xing

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Dynamic modeling and simulation of functionally graded materials plates based on first order shear plate theory

Journal of Dynamics and Control, 2020, 18(4): 33-43 (in Chinese)

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力学学报, 2015, 47(2): 279-288

DOI [本文引用: 2]

将无网格径向基点插值法用于旋转柔性梁的动力学分析. 利用无网格方法对柔性梁的变形场进行离散,考虑梁的纵向拉伸变形和横向弯曲变形,并计入横向弯曲变形引起的纵向缩短,即非线性耦合项,运用第二类拉格朗日方程推导得到系统刚柔耦合动力学方程. 将无网格径向基点插值法的仿真结果有限元法和假设模态法进行比较分析,说明假设模态法的局限性,并表明其作为一种柔性体离散方法在刚柔耦合多体系统动力学的研究中具有可推广性,并讨论了径向基形状参数的影响. 同时运用3 种求解系统动力学方程的方法:纽马克方法、4阶龙格库塔法、亚当姆斯预报校正法,并比较各方法的计算效率, 结果表明纽马克方法最快.

Chaofan

, Zhang

Dingguo

, Hong

Jiazhen

A meshfree method based on radial point interpolation method for the dynamic analysis of rotating flexible beams

Chinese Journal of Theoretical and Applied Mechanics, 2015, 47(2): 279-288 (in Chinese)

DOI [本文引用: 2]

A meshfree method based on radial point interpolation method (RPIM) is proposed for dynamic analysis of a rotating flexible beam. The RPIM is used to describe the deformation of the flexible beam. The longitudinal and transverse deformations of the beam are both considered, and the coupling term of the deformation which is caused by the transverse deformation is included in the longitudinal deformation of the beam. The rigid-flexible coupling dynamic equations of the system are derived via employing Lagrange's equations of the second kind. Simulation results of the RPIM are compared with those obtained by using finite element method (FEM) and assumed modes method and show the limitations of assumed modes method. It is demonstrated that the meshfree method as a discrete method of the flexible body can be extended in the field of multibody system dynamics. Meanwhile, the influence of the radial basis shape parameters is discussed. What's more, three integration methods (Newmark method, fourth-order Runge—Kutta method and Adams prediction correction method) are used to solve the dynamic equations, and it shows that the Newmark method is the fastest with the computational effciency.

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Liu

Guirong

, Gu

Yuantong

. Meshless method theory and program design. Wang Jianming, Zhou Xuejun, transl. Jinan: Shandong University Press, 2007 (in Chinese)