﻿ 1:1内共振对随机振动系统可靠性的影响
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 力学学报  2015, Vol. 47 Issue (5): 807-813  DOI: 10.6052/0459-1879-15-058 0

### 引用本文 [复制中英文]

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Wang Haoyu, Wu Yongjun. THE INFLUENCE OF ONE-TO-ONE INTERNAL RESONANCE ON RELIABILITY OF RANDOM VIBRATION SYSTEM[J]. Chinese Journal of Ship Research, 2015, 47(5): 807-813. DOI: 10.6052/0459-1879-15-058.
[复制英文]

### 文章历史

2015-02-11 收稿
2015-07-08 录用
2015–07–10 网络版发表
1:1内共振对随机振动系统可靠性的影响

1 平均伊藤随机微分方程

 $\left.\!\! \ddot {X}_1 + (\beta _{10} + \beta _{11} X_1^2 + \beta _{12} X_2^2 )\dot {X}_1 +\\ \qquad \omega _{01}^2 X_1 + \alpha _1 X_1^3 + \eta _{12} \dot {X}_2 = W_1 (t) \\ \ddot {X}_2 + (\beta _{20} + \beta _{21} X_1^2 + \beta _{22} X_2^2 )\dot {X}_2 +\\ \qquad \omega _{02}^2 X_2 + \alpha _2 X_2^3 + \eta _{21} \dot {X}_1 = W_2 (t) \right \}$ (1)

 $\left. X_i (t) = A_i \cos \varPhi _i (t) \\ \dot {X}_i (t) = - A_i \nu _i (A_i ,\varPhi _i )\sin \varPhi _i (t) \right\}$ (2)

 $\left. \omega _i (A_i ) = (\omega _{0i}^2 + {3\alpha _i A_i^2 }/ 4)^{1/2}(1 - {\lambda _i^2 }/{16}) \\ \lambda _i = {\alpha _i A_i^2 } \Big / {\left[{4\left( {\omega _{0i}^2 + 3{\alpha _i A_i^2 }/4} \right)} \right]} \right\}$ (3)

1.1 内共振情况

 $\dfrac{\omega _1 (A_1 )}{\omega _2 (A_2 )} = 1\pm \varepsilon$ (4)

 $\left. \begin{array}{l} {\rm{d}}{A_1} = \bar m_1^{(1)}({A_1},{A_2},\Gamma ){\rm{d}}t + \bar \sigma _{11}^{(1)}{\rm{d}}{B_1}(t) + \bar \sigma _{12}^{(1)}{\rm{d}}{B_2}(t)\\ {\rm{d}}{A_2} = \bar m_2^{(1)}({A_1},{A_2},\Gamma ){\rm{d}}t + \bar \sigma _{21}^{(1)}{\rm{d}}{B_1}(t) + \bar \sigma _{22}^{(1)}{\rm{d}}{B_2}(t)\\ {\rm{d}}\Gamma = \bar m_1^{(2)}({A_1},{A_2},\Gamma ){\rm{d}}t + \bar \sigma _{11}^{(2)}{\rm{d}}{B_1}(t) + \bar \sigma _{12}^{(2)}{\rm{d}}{B_2}(t) \end{array} \right\}$ (5)

1.2 无内共振情况

 $\left.\!\! {\rm{d}} A_1 = \tilde {m}_1^{(1)} \left( {A_1 ,A_2 } \right) {\rm{d}} t + \tilde {\sigma }_{11}^{(1)} {\rm{d}} B_1 (t) + \tilde {\sigma }_{12}^{(1)} {\rm{d}} B_2 (t) \\ {\rm{d}} A_2 = \tilde {m}_2^{(1)} \left( {A_1 ,A_2 } \right) {\rm{d}} t + \tilde {\sigma }_{21}^{(1)} {\rm{d}} B_1 (t) + \tilde {\sigma }_{22}^{(1)} {\rm{d}} B_2 (t) \!\!\right\}$ (6)

2 后向柯尔莫哥洛夫方程

2.1 内共振情况

 $R(t | A_{10} ,A_{20} ,\Gamma _0 ) = \\ \qquad P\,\Big\{A_1 (s) \in [0,A_{1{\rm c}} ) \cap A_2 (s) \in [0,A_{2{\rm c}} ),s \in (0,t] \\ \qquad | A_{i0} \in [0,A_{i{\rm c}} ),\Gamma _0 \in [0,2\pi]\,,\ i = 1,2\Big\} \hskip 17mm$ (7)

 $\dfrac{\partial R}{\partial t} = \bar {m}_{10}^{(1)} \dfrac{\partial R}{\partial A_{10} } + \bar {m}_{20}^{(1)} \dfrac{\partial R}{\partial A_{20} } + \bar {m}_{10}^{(2)} \dfrac{\partial R}{\partial \Gamma _{10} } +\\ \qquad \dfrac{1}{2}\bar {b}_{110}^{(11)} \dfrac{\partial ^2R}{\partial A_{10}^2 } + \dfrac{1}{2}\bar {b}_{220}^{(11)} \dfrac{\partial ^2R}{\partial A_{20}^2 } + \dfrac{1}{2}\bar {b}_{110}^{(22)} \dfrac{\partial ^2R}{\partial \Gamma _{10}^2 }$ (8)

 $\left.\!\! R(0 | A_{10} ,A_{20} ,\Gamma _0 ) = 1 \\ A_{i0} \in [0,A_{i{\rm c}} )\,,\ \ \Gamma _0 \in [0,2\pi] \\ \dfrac{\partial R}{\partial A_{i0} } \Big |_{A_{i0} = 0} = 0 \,,\ \ \dfrac{\partial R}{\partial \Gamma _0 } \Big |_{A_{i0} = 0} = 0 \,,\ \ \dfrac{\partial ^2R}{\partial \Gamma _0^2 } \Big |_{A_{i0} = 0} = 0 \\ R(t | A_{10} ,A_{20} ,\Gamma _0 ) = 0 \,,\ \ A_{i0} = A_{i{\rm c}} \ \ (i = 1,2) \\ R(t | A_{10} ,A_{20} ,0) = R(t | A_{10} ,A_{20} ,2\pi ) \!\!\right\}$ (9)

2.2 无内共振情况

 $R(t | A_{10} ,A_{20} ) = \\ \qquad P \,\Big \{A_1 (s) \in [0,A_{1{\rm c}} ) \cap A_2 (s) \in [0,A_{2{\rm c}} ),s \in (0,t] \\ \qquad | A_{i0} \in [0,A_{i{\rm c}} ),i = 1,2 \Big\} \hskip 35mm$ (10)

 $\dfrac{\partial R}{\partial t} = \tilde {m}_{10}^{(1)} \dfrac{\partial R}{\partial A_{10} } + \tilde {m}_{20}^{(1)} \dfrac{\partial R}{\partial A_{20} } + \\ \qquad \dfrac{1}{2}\tilde {b}_{110}^{(11)} \dfrac{\partial ^2R}{\partial A_{10}^2 } + \dfrac{1}{2}\tilde {b}_{220}^{(11)} \dfrac{\partial ^2R}{\partial A_{20}^2 }$ (11)

 $\left. \begin{array}{l} R(0|{A_{10}},{A_{20}}) = 1{\mkern 1mu} ,\;\;{A_{i0}} \in [0,{A_{i{\rm{c}}}})\\ {\left. {\frac{{\partial R}}{{\partial {A_{i0}}}}} \right|_{{A_{i0}} = 0}} = 0\\ R(t|{A_{10}},{A_{20}}) = 0{\mkern 1mu} ,\;\;{A_{i0}} = {A_{i{\rm{c}}}}\;\;\;(i = 1,2) \end{array} \right\}$ (12)

2.3 内共振情况与无内共振情况结果对比

 图 1 内共振情况与无内共振情况的条件可靠性函数 Fig. 1 Conditional reliability function in the case of internal resonance and non-internal resonance ($X_{10} = \dot {X}_{10} = X_{20} = \dot {X}_{20} = 0,\Gamma _0 = 0)$

3 庞德辽金方程 3.1 内共振情况

 $\begin{array}{l} - 1 = \bar m_{10}^{(1)}\frac{{\partial \mu }}{{\partial {A_{10}}}} + \bar m_{20}^{(1)}\frac{{\partial \mu }}{{\partial {A_{20}}}} + \bar m_{10}^{(2)}\frac{{\partial \mu }}{{\partial {\Gamma _0}}} + \\ \;\;\;\;\;\;\;\frac{1}{2}\bar b_{110}^{(11)}\frac{{{\partial ^2}\mu }}{{\partial A_{10}^2}} + \frac{1}{2}\bar b_{220}^{(11)}\frac{{{\partial ^2}\mu }}{{\partial A_{20}^2}} + \frac{1}{2}\bar b_{110}^{(22)}\frac{{{\partial ^2}\mu }}{{\partial \Gamma _0^2}} \end{array}$ (13)

 $\left. \begin{array}{l} \frac{{\partial \mu }}{{\partial {A_{i0}}}}{|_{{A_{i0}} = 0}} = 0{\mkern 1mu} ,\;\;\frac{{\partial \mu }}{{\partial {\Gamma _0}}}{|_{{A_{i0}} = 0}} = 0{\mkern 1mu} ,\;\;\frac{{{\partial ^2}\mu }}{{\partial \Gamma _0^2}}{|_{{A_{i0}} = 0}} = 0\\ \mu ({A_{10}},{A_{20}},{\Gamma _0}) = 0{\mkern 1mu} ,\;\;{A_{i0}} = {A_{i{\rm{c}}}}\;\;\;(i = 1,2)\\ \mu ({A_{10}},{A_{20}},0) = \mu ({A_{10}},{A_{20}},2\pi ) \end{array} \right\}$ (14)

3.2 无内共振情况

 $\begin{array}{l} - 1 = \tilde m_{10}^{(1)}\frac{{\partial \mu }}{{\partial {A_{10}}}} + \tilde m_{20}^{(1)}\frac{{\partial \mu }}{{\partial {A_{20}}}} + \\ \;\;\;\;\;\;\;\frac{1}{2}\tilde b_{110}^{(11)}\frac{{{\partial ^2}\mu }}{{\partial A_{10}^2}} + \frac{1}{2}\tilde b_{220}^{(11)}\frac{{{\partial ^2}\mu }}{{\partial A_{20}^2}} \end{array}$ (15)

 $\left. \begin{array}{l} {\left. {\frac{{\partial \mu }}{{\partial {A_{i0}}}}} \right|_{{A_{i0}} = 0}} = 0\\ \mu ({A_{10}},{A_{20}}) = 0{\mkern 1mu} ,\;\;{A_{i0}} = {A_{i{\rm{c}}}}\;\;\;(i = 1,2) \end{array} \right\}$ (16)

3.3 内共振情况与无内共振情况结果对比

 图 2 内共振情况与无内共振情况平均首次穿越时间 Fig. 2 Mean first-passage time in the case of internal resonance and non-internal resonance

 图 3 内共振情况平均首次穿越时间, $\omega_{02}=2.01$ Fig. 3 Mean first-passage time in the case of internal resonance, $\omega_{02}=2.01$
 图 4 无内共振情况平均首次穿越时间, $\omega_{02}=1$ Fig. 4 Mean first-passage time in the case of non-internal resonance, $\omega_{02}=1$
4 结 论

 $\begin{array}{l} \bar m_i^{(1)} = \bar m_{i1}^{(1)} + \bar m_{i2}^{(1)}\\ \bar m_{i1}^{(1)} = \left\{ { - 4{A_i}\left[{4{\beta _{i0}} + A_i^2{\beta _{ii}} + } \right.2\left. {{\beta _{ij}}A_j^2} \right]} \right. + \frac{{12A_i^3{D_i}\alpha _i^2}}{{{{\left( {A_i^2{\alpha _i} + \omega _{0i}^2} \right)}^3}}} - \frac{{26{D_i}{A_i}{\alpha _i}}}{{{{\left( {A_i^2{\alpha _i} + \omega _{0i}^2} \right)}^2}}} + \\ \qquad [16{D_i} + A_i^6{\alpha _i}{\beta _{ii}} + 3A_i^4{\alpha _i}\left( {2{\beta _{i0}} + A_j^2{\beta _{ij}}} \right)]/\left. {\left[{{A_i}\left( {A_i^2{\alpha _i} + \omega _{0i}^2} \right)} \right]} \right\}/32\\ \bar m_{i2}^{(1)} = {A_i}A_j^2{\beta _{ij}}\left( {{\alpha _i}A_i^2 + 2\omega _{0i}^2} \right)\cos \left( {2\Gamma } \right)/\left[{16\left( {{\alpha _i}A_i^2 + \omega _{0i}^2} \right)} \right] - \\ \qquad {A_j}\left[{2{b_{10}}\left( {{A_1}} \right) - {b_{12}}\left( {{A_1}} \right)} \right]\left[{2{b_{20}}\left( {{A_2}} \right) - {b_{22}}\left( {{A_2}} \right)} \right]{\eta _{ij}}\cos \Gamma /\left[{8\left( {{\alpha _i}A_i^2 + \omega _{0i}^2} \right)} \right] - \\ \qquad {A_j}\left[{{b_{12}}\left( {{A_1}} \right) - {b_{14}}\left( {{A_1}} \right)} \right]\left[{{b_{22}}\left( {{A_2}} \right) - {b_{24}}\left( {{A_2}} \right)} \right]{\eta _{ij}}\cos 3\Gamma /\left[{8\left( {{\alpha _i}A_i^2 + \omega _{0i}^2} \right)} \right] - \\ \qquad {A_j}\left[{{b_{14}}\left( {{A_1}} \right) - {b_{16}}\left( {{A_1}} \right)} \right]\left[{{b_{24}}\left( {{A_2}} \right) - {b_{26}}\left( {{A_2}} \right)} \right]{\eta _{ij}}\cos 5\Gamma /\left[{8\left( {{\alpha _i}A_i^2 + \omega _{0i}^2} \right)} \right] - \\ \qquad {A_j}{b_{16}}\left( {{A_1}} \right){b_{26}}\left( {{A_2}} \right){\eta _{ij}}\cos 7\Gamma /\left[{8\left( {{\alpha _i}A_i^2 + \omega _{0i}^2} \right)} \right]{\mkern 1mu} ,\\ \qquad \left( {i,j = 1,2{\mkern 1mu} ;\;\;i \ne j{\mkern 1mu} ;\;{\rm{no}}\;{\rm{summation}}\;{\rm{about}}\;i} \right)\\ \bar m_1^{(2)} = {b_{10}}({A_1}) - {b_{20}}({A_2}) - A_2^2{\beta _{12}}\left( {3{\alpha _1}A_1^2 + 4\omega _{01}^2} \right)\left. {\sin \left( {2\Gamma } \right)} \right]/\left[{32\left( {{\alpha _1}A_1^2 + \omega _{01}^2} \right)} \right] - \\ \qquad A_1^2{\beta _{21}}\left( {3{\alpha _2}A_2^2 + 4\omega _{02}^2} \right)\sin \left( {2\Gamma } \right)/\left[{32\left( {{\alpha _2}A_2^2 + \omega _{02}^2} \right)} \right] + \\ \qquad {A_2}\left[{2{b_{10}}\left( {{A_1}} \right) + {b_{12}}\left( {{A_1}} \right)} \right]\left[{2{b_{20}}\left( {{A_2}} \right) - {b_{22}}\left( {{A_2}} \right)} \right]{\eta _{12}}\sin \Gamma /\left[{8{A_1}\left( {{\alpha _1}A_1^2 + \omega _{01}^2} \right)} \right] + \\ \qquad {A_2}\left[{{b_{12}}\left( {{A_1}} \right) + {b_{14}}\left( {{A_1}} \right)} \right]\left[{{b_{22}}\left( {{A_2}} \right) - {b_{24}}\left( {{A_2}} \right)} \right]{\eta _{12}}\sin 3\Gamma /\left[{8{A_1}\left( {{\alpha _1}A_1^2 + \omega _{01}^2} \right)} \right] + \\ \qquad {A_2}\left[{{b_{14}}\left( {{A_1}} \right) + {b_{16}}\left( {{A_1}} \right)} \right]\left[{{b_{24}}\left( {{A_2}} \right) - {b_{26}}\left( {{A_2}} \right)} \right]{\eta _{12}}\sin 5\Gamma /\left[{8{A_1}\left( {{\alpha _1}A_1^2 + \omega _{01}^2} \right)} \right] + \\ \qquad {A_2}{b_{16}}\left( {{A_1}} \right){b_{26}}\left( {{A_2}} \right){\eta _{12}}\sin 7\Gamma /\left[{8{A_1}\left( {{\alpha _1}A_1^2{\rm{ + }}\omega _{01}^2} \right)} \right] + \\ \qquad {A_1}\left[{2{b_{10}}\left( {{A_1}} \right) - {b_{12}}\left( {{A_1}} \right)} \right]\left[{2{b_{20}}\left( {{A_2}} \right) + {b_{22}}\left( {{A_2}} \right)} \right]{\eta _{21}}\sin \Gamma /\left[{8{A_2}\left( {{\alpha _2}A_2^2 + \omega _{02}^2} \right)} \right] + \\ \qquad {A_1}\left[{{b_{12}}\left( {{A_1}} \right) - {b_{14}}\left( {{A_1}} \right)} \right]\left[{{b_{22}}\left( {{A_2}} \right) + {b_{24}}\left( {{A_2}} \right)} \right]{\eta _{21}}\sin 3\Gamma /\left[{8{A_2}\left( {{\alpha _2}A_2^2 + \omega _{02}^2} \right)} \right] + \\ \qquad {A_1}\left[{{b_{14}}\left( {{A_1}} \right) - {b_{16}}\left( {{A_1}} \right)} \right]\left[{{b_{24}}\left( {{A_2}} \right) + {b_{26}}\left( {{A_2}} \right)} \right]{\eta _{21}}\sin 5\Gamma /\left[{8{A_2}\left( {{\alpha _2}A_2^2 + \omega _{02}^2} \right)} \right] + \\ \qquad {A_1}{b_{16}}\left( {{A_1}} \right){b_{26}}\left( {{A_2}} \right){\eta _{21}}\sin 7\Gamma /\left[{8{A_2}\left( {{\alpha _2}A_2^2 + \omega _{02}^2} \right)} \right]\\ {b_{i0}}\left( {{A_i}} \right) = {\left( {\omega _{0i}^2 + 3{\alpha _i}A_i^2/4} \right)^{1/2}}\left( {1 - \lambda _i^2/16} \right)\\ {b_{i2}}\left( {{A_i}} \right) = {\left( {\omega _{0i}^2 + 3{\alpha _i}A_i^2/4} \right)^{1/2}}\left( {{\lambda _i}/2 + \lambda _i^3/64} \right)\\ {b_{i4}}\left( {{A_i}} \right) = {\left( {\omega _{0i}^2 + 3{\alpha _i}A_i^2/4} \right)^{1/2}}\left( { - \lambda _i^2/16} \right)\\ {b_{i6}}\left( {{A_i}} \right) = {\left( {\omega _{0i}^2 + 3{\alpha _i}A_i^2/4} \right)^{1/2}}\left( {\lambda _i^3/64} \right)\\ {\lambda _i} = {\alpha _i}A_i^2/\left[{4\left( {\omega _{0i}^2 + 3{\alpha _i}A_i^2/4} \right)} \right]\\ \bar b_{ii}^{(11)} = \bar \sigma _{i1}^{(1)}\bar \sigma _{i1}^{(1)} + \bar \sigma _{i2}^{(1)}\bar \sigma _{i2}^{(1)} = {D_i}\omega _{0i}^2/{\left( {A_i^2{\alpha _i} + \omega _{0i}^2} \right)^2} + 5A_i^2{D_i}{\alpha _i}/\left[{8{{\left( {A_i^2{\alpha _i} + \omega _{0i}^2} \right)}^2}} \right]{\mkern 1mu} ,\\ \qquad (i = 1,2,\;{\rm{no}}\;{\rm{summation}}\;{\rm{about}}\;i)\\ \bar b_{11}^{(22)} = \bar \sigma _{11}^{(2)}\bar \sigma _{11}^{(2)} + \bar \sigma _{12}^{(2)}\bar \sigma _{12}^{(2)} = {D_1}\left( {7A_1^2{\alpha _1} + 8\omega _{01}^2} \right)/\left[{8{{\left( {A_1^3{\alpha _1} + \omega _{01}^2{A_1}} \right)}^2}} \right] \end{array}$

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THE INFLUENCE OF ONE-TO-ONE INTERNAL RESONANCE ON RELIABILITY OF RANDOM VIBRATION SYSTEM
Wang Haoyu, Wu Yongjun
Department of Engineering Mechanics, Shanghai Jiao Tong University, Shanghai 200240, China
Fund: The project was supported by the National Natural Science Foundation of China (11272201, 11372271, 11132007).
Abstract: Based on first-passage model, the reliability problem of two degrees-of-freedom random vibration system under Gaussian white noise excitations is studied analytically. In the case of 1:1 internal resonance, the equations of motion of the original system are reduced to a set of Itô stochastic differential equations after averaging. The backward Kolmogorov equation and the Pontryagin equation, which determine the conditional reliability function and the mean first-passage time of the random vibration systems, are constructed under appropriate boundary and (or) initial conditions, respectively. To study the influence of the internal resonance on the reliability, the averaged Itô stochastic differential equations, the backward Kolmogorov equation and the Pontryagin equation in the case of non-internal resonance are also derived. Numerical solutions of high-dimensional backward Kolmogorov equation and Pontryagin equation are obtained. The results of resonant case and non-resonant case are compared. It is shown that 1:1 internal resonance can greatly reduce the reliability. All the analytical results are validated by Monte Carlo digital simulation.
Key words: 1:1 internal resonance    averaging method    reliability function    mean first-passage time    digital simulation