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 哈尔滨工程大学学报  2018, Vol. 39 Issue (10): 1661-1667  DOI: 10.11990/jheu.201707053 0

引用本文

LI Hui, XUE Pengcheng, ZHOU Zhengxue, et al. Prediction of vibration response in fiber composite thin plate based on the multilevel correction method[J]. Journal of Harbin Engineering University, 2018, 39(10), 1661-1667. DOI: 10.11990/jheu.201707053.

文章历史

Prediction of vibration response in fiber composite thin plate based on the multilevel correction method
LI Hui, XUE Pengcheng, ZHOU Zhengxue, HAN Qingkai
School of Mechanical Engineering and Automation, Northeastern University, Shenyang 110819, China
Abstract: This research studied the problem of accurate prediction of vibration response in fiber composite thin plate using the multilevel correction technique. First, base excitation loading was considered in the formulation of the theoretical model of such a composite thin plate, clarifying the principle of prediction of vibration response by multilevel correction technique. Next, based on the measured natural frequencies, modal shapes, and frequency domain response data, dimensions including length, width, and thickness of the composite thin plate were modified at the first level of correction. The longitudinal and transverse elastic moduli, shear modulus and Poisson's ratio were then modified at the second level of correction, while loss factors in different fiber directions were modified at the third level of correction. A method to predict vibration response in composite thin plate was then proposed. Finally, a TC300 fiber/epoxy composite plate was taken as a study object to set up the vibration response testing system, verifying the predicted results with the experimental test data. It was found that the maximum predicted errors of resonance and non-resonance were no more than 15% compared with experimental results, which were within acceptable range, verifying the effectiveness of the proposed method.
Keywords: fiber-reinforced    composite thin plate    base excitation    multilevel    correction technique    response prediction    resonance response    non-resonant response

1 基础激励下纤维复合薄板振动响应求解

 Download: 图 1 基础激励下纤维复合薄板的理论模型 Fig. 1 Theoretical model of fiber composite plate under base excitation

 $y\left( t \right) = Y{{\rm{e}}^{{\rm{i}}\omega t}}$ (1)

 $E_1^ * = {{E'}_1}\left( {1 + {\rm{i}}{\eta _1}} \right),E_2^ * = {{E'}_2}\left( {1 + {\rm{i}}{\eta _2}} \right),$
 $G_{12}^ * = {{G'}_{12}}\left( {1 + {\rm{i}}{\eta _{12}}} \right)$

 $\left\{ \begin{array}{l} u\left( {x,y,z,t} \right) = {u_0}\left( {x,y,t} \right) - z\frac{{\partial {w_0}\left( {x,y,t} \right)}}{{\partial x}}\\ v\left( {x,y,z,t} \right) = {v_0}\left( {x,y,t} \right) - z\frac{{\partial {w_0}\left( {x,y,t} \right)}}{{\partial y}}\\ w\left( {x,y,z,t} \right) = {w_0}\left( {x,y,t} \right) \end{array} \right.$ (2)

 $\left\{ \begin{array}{l} {\varepsilon _x} = \frac{{\partial u}}{{\partial x}} = - z\frac{{{\partial ^2}{w_0}}}{{\partial {x^2}}}\\ {\varepsilon _y} = \frac{{\partial v}}{{\partial y}} = - z\frac{{{\partial ^2}{w_0}}}{{\partial {y^2}}}\\ {\gamma _{xy}} = \frac{{\partial u}}{{\partial y}} + \frac{{\partial v}}{{\partial x}} = - 2z\frac{{{\partial ^2}{w_0}}}{{\partial x\partial y}} \end{array} \right.$ (3)

 ${\kappa _x} = - \frac{{{\partial ^2}{w_0}}}{{\partial {x^2}}},{\kappa _y} = - \frac{{{\partial ^2}{w_0}}}{{\partial {y^2}}},{\kappa _{xy}} = - 2\frac{{{\partial ^2}{w_0}}}{{\partial x\partial y}}$ (4)

 ${\varepsilon _x} = z{\kappa _x},{\varepsilon _y} = z{\kappa _y},{\gamma _{xy}} = z{\kappa _{xy}}$

 $\left[ {\begin{array}{*{20}{c}} {{\sigma _1}}\\ {{\sigma _2}}\\ {{\sigma _6}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {Q_{11}^ * }&{Q_{12}^ * }&0\\ {Q_{21}^ * }&{Q_{22}^ * }&0\\ 0&0&{Q_{66}^ * } \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{\varepsilon _1}}\\ {{\varepsilon _2}}\\ {{\gamma _6}} \end{array}} \right]$ (5)

 $\left\{ \begin{array}{l} Q_{11}^ * = \frac{{E_1^ * }}{{1 - \mathit{\boldsymbol{\nu }}_{12}^ * \mathit{\boldsymbol{\nu }}_{21}^ * }},Q_{12}^ * = Q_{21}^ * = \frac{{\mathit{\boldsymbol{\nu }}_{12}^ * E_2^ * }}{{1 - \mathit{\boldsymbol{\nu }}_{12}^ * \mathit{\boldsymbol{\nu }}_{21}^ * }}\\ Q_{22}^ * = \frac{{E_2^ * }}{{1 - \mathit{\boldsymbol{\nu }}_{12}^ * \mathit{\boldsymbol{\nu }}_{21}^ * }},Q_{66}^ * = G_{12}^ * ,{\mathit{\boldsymbol{v}}_{21}} = {\mathit{\boldsymbol{v}}_{12}}\frac{{{{E'}_2}}}{{{{E'}_1}}} \end{array} \right.$ (6)

 ${\left[ {\begin{array}{*{20}{c}} {{\sigma _x}}\\ {{\sigma _y}}\\ {{\sigma _{xy}}} \end{array}} \right]^{\left( k \right)}} = {\left[ {\begin{array}{*{20}{c}} {\bar Q_{11}^ * }&{\bar Q_{12}^ * }&{\bar Q_{16}^ * }\\ {\bar Q_{12}^ * }&{\bar Q_{22}^ * }&{\bar Q_{26}^ * }\\ {\bar Q_{16}^ * }&{\bar Q_{26}^ * }&{\bar Q_{66}^ * } \end{array}} \right]^{\left( k \right)}}\left[ {\begin{array}{*{20}{c}} {{\sigma _x}}\\ {{\sigma _y}}\\ {{\sigma _{xy}}} \end{array}} \right]$ (7)

 $\begin{array}{*{20}{c}} {\left[ {\begin{array}{*{20}{c}} {{M_x}}\\ {{M_y}}\\ {{M_{xy}}} \end{array}} \right] = \sum\limits_{k = 1}^n {\int_{{z_k} - 1}^{{z_k}} {{{\left[ {\begin{array}{*{20}{c}} {{\sigma _x}}\\ {{\sigma _y}}\\ {{\sigma _{xy}}} \end{array}} \right]}^{\left( k \right)}}z{\rm{d}}z} } = }\\ {\left[ {\begin{array}{*{20}{c}} {D_{11}^ * }&{D_{12}^ * }&{D_{16}^ * }\\ {D_{12}^ * }&{D_{22}^ * }&{D_{26}^ * }\\ {D_{16}^ * }&{D_{26}^ * }&{D_{66}^ * } \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{\kappa _x}}\\ {{\kappa _y}}\\ {{\kappa _{xy}}} \end{array}} \right]} \end{array}$ (8)

 $q\left( t \right) = - \rho h\frac{{{{\rm{d}}^2}y\left( t \right)}}{{{\rm{d}}{t^2}}} = \rho hY{\omega ^2}{{\rm{e}}^{{\rm{i}}\omega t}}$ (9)

 $T = \frac{{\rho h}}{2}\iint_R {{{\left( {\frac{{\partial {w_0}}}{{\partial t}}} \right)}^2}{\text{d}}x{\text{d}}y}$ (10)

 $U = \frac{1}{2}\iint_R {\left[ {{\mathit{\boldsymbol{M}}_x}{\mathit{\boldsymbol{\kappa }}_x} + {\mathit{\boldsymbol{M}}_y}{\mathit{\boldsymbol{\kappa }}_y} + {\mathit{\boldsymbol{M}}_{xy}}{\mathit{\boldsymbol{\kappa }}_{xy}}} \right]{\text{d}}x{\text{d}}y}$ (11)

 ${W_q} = \iint_R {q\left( t \right){w_0}{\text{d}}x{\text{d}}y}$ (12)

 ${w_0}\left( {x,y,t} \right) = {{\text{e}}^{{\text{i}}\omega t}}W\left( {\xi ,\eta } \right)$ (13)

 $W\left( {\xi ,\eta } \right) = \sum\limits_{i = 1}^M {\sum\limits_{j = 1}^N {{a_{ij}}{p_i}\left( \xi \right){q_j}\left( \eta \right)} }$ (14)

 ${T_{\max }} = \frac{{\rho h{\omega ^2}}}{2}\iint_R {{W^2}{\text{d}}x{\text{d}}y}$ (15)
 $\begin{array}{*{20}{c}} {{U_{\max }} = \frac{1}{2}\iint_A {\left[ {D_{11}^ * {{\left( {\frac{{{\partial ^2}W}}{{\partial {x^2}}}} \right)}^2} + 2D_{12}^ * \frac{{{\partial ^2}W}}{{\partial {x^2}}}\frac{{{\partial ^2}W}}{{\partial {y^2}}} + } \right.}} \\ {D_{22}^ * {{\left( {\frac{{{\partial ^2}W}}{{\partial {y^2}}}} \right)}^2} + 4s\left( {D_{16}^ * \frac{{{\partial ^2}W}}{{\partial {x^2}}} + D_{26}^ * \frac{{{\partial ^2}W}}{{\partial {y^2}}}} \right)\frac{{{\partial ^2}W}}{{\partial x\partial y}} + } \\ {\left. {4D_{66}^ * {{\left( {\frac{{{\partial ^2}W}}{{\partial x\partial y}}} \right)}^2}} \right]{\text{d}}A} \end{array}$ (16)
 ${W_{q\max }} = \rho hY{\omega ^2}\iint_R {W{\text{d}}x{\text{d}}y}$ (17)

 $L = {T_{\max }} + {W_{q\max }} - {U_{\max }}$ (18)

 $\frac{{\partial L}}{{\partial {a_{mn}}}} = 0,m = 1,2, \cdots ,M,n = 1,2, \cdots ,N$ (19)

 $\left( {\mathit{\boldsymbol{K}} + i\mathit{\boldsymbol{C}} - {\omega ^2}\mathit{\boldsymbol{M}}} \right)\mathit{\boldsymbol{a}} = \mathit{\boldsymbol{F}}$ (20)

 $\left( {\mathit{\boldsymbol{K}} - {\omega ^2}\mathit{\boldsymbol{M}}} \right)\mathit{\boldsymbol{a}} = 0$ (21)

 $\lambda \left( {x,y,t} \right) = y\left( t \right) + {w_0}\left( {x,y,t} \right)$ (22)

2 基于多层次修正的基础激励下复合薄板振动响应预测原理

 Download: 图 2 基于多层次修正的复合薄板振动响应预测原理图 Fig. 2 Prediction principle diagram of vibration response of composite plate based on multilevel correction technique
3 纤维复合薄板振动响应预测流程

1) 建立复合薄板的理论模型。

2) 求解动能、应变能和外激励做功最大值。

3) 初步计算固有频率和振型。

4) 基于第一层次修正复合薄板尺寸参数。

5) 基于第二层次修正弹性模量和泊松比。

6) 基于第三层次修正复合材料损耗因子。

 Download: 图 3 采用多层次修正技术前、后获得的第一阶频域响应 Fig. 3 The 1st frequency domain responses obtained before and after using multilevel correction technique

7) 基于修正后模型预测关注的某阶振动响应。

4 实例研究

 Download: 图 4 理论预测和测试获得的复合薄板第一、三、五阶频域响应 Fig. 4 The 1st, 3rd and 5th frequency response of composite thin plate obtained by theoretical calculation and experiment test

5 结论

1) 建立了该类型复合薄板的理论模型，明确了基于多层次修正的基础激励下复合薄板振动响应预测原理。

2) 提出了复合薄板振动响应预测流程，主要包括七个关键步骤。

3) 以TC300碳纤维/树脂基复合薄板为对象，进行了实验验证。结果表明，预测获得的共振及非共振响应结果与测试结果的误差最大不超过15%，处于误差允许的范围内，进而证明了所提出的响应预测方法的正确性。

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