﻿ 超空泡运动体圆柱薄壳的非线性动力屈曲分析
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 力学学报  2016, Vol. 48 Issue (1): 181-191  DOI: 10.6052/0459-1879-15-222 0

### 引用本文 [复制中英文]

[复制中文]
Wang Jiefang, An Hai, An Weiguang. THE NONLINEAR DYNAMIC BUCKLING ANALYSIS OF A THIN-WALLED CYLINDRICAL SHELL OF SUPERCAVITATING VEHICLES[J]. Chinese Journal of Ship Research, 2016, 48(1): 181-191. DOI: 10.6052/0459-1879-15-222.
[复制英文]

### 文章历史

2015-06-16收稿
2015-09-10录用
2015-10-12网络版发表

1. 哈尔滨工程大学 航天工程系, 哈尔滨 150001;
2. 南昌工程学院 土木与建筑工程学院, 南昌 330099

1 圆柱薄壳的非线性动力屈曲微分方程

 \left. \begin{aligned} & {\varepsilon _1} = \displaystyle\frac{1}{R}\frac{{\partial u}}{{\partial \alpha }} \,{\rm{ + }} \,\frac{{\rm{1}}}{{{\rm{2}}{R^2}}}{\left( {\frac{{\partial w}}{{\partial \alpha }}} \right)^2}\\ & {\varepsilon _2} = \displaystyle\frac{1}{R}\left( {\frac{{\partial v}}{{\partial \beta }} + w} \right) \,{\rm{ + }} \,\frac{{\rm{1}}}{{{\rm{2}}{R^2}}}{\left( {\frac{{\partial w}}{{\partial \beta }}} \right)^2}\\ & {\gamma _{12}} = \displaystyle\frac{1}{R}\left( {\frac{{\partial v}}{{\partial \alpha }} + \frac{{\partial u}}{{\partial \beta }}} \right) \,{\rm{ + }} \,\frac{{\rm{1}}}{{{R^2}}}\frac{{\partial w}}{{\partial \alpha }} \cdot \frac{{\partial w}}{{\partial \beta }}\\ & {\kappa _1} = - \,\displaystyle\frac{1}{{{R^2}}}\frac{{{\partial ^2}w}}{{\partial {\alpha ^2}}}\\ & {\kappa _2} = - \,\displaystyle\frac{1}{{{R^2}}}\left( {\frac{{{\partial ^2}w}}{{\partial {\beta ^2}}} + w} \right)\\ & {\kappa _{12}} = \displaystyle - \,\frac{1}{{2{R^2}}}\left( {\frac{{\partial u}}{{\partial \beta }} - \frac{{\partial v}}{{\partial \alpha }} + 2\frac{{{\partial ^2}w}}{{\partial \alpha \partial \beta }}} \right) \end{aligned} \right\} (1)

 \left. {\begin{aligned} & {{N_1} = \displaystyle\frac{{Eh}}{{1 - {\nu ^2}}}\left( {{\varepsilon _1} + \nu {\varepsilon _2} + \frac{{{h^2}}}{{12R}}{\kappa _1}} \right)}\\ & {{N_2} = \displaystyle\frac{{Eh}}{{1 - {\nu ^2}}}\left( {{\varepsilon _2} + \nu {\varepsilon _1} - \frac{{{h^2}}}{{12R}}{\kappa _2}} \right)}\\ & {{N_{12}} = \displaystyle\frac{{Eh}}{{2\left( {1 + \nu } \right)}}\left( {{\gamma _{12}} + \frac{{{h^2}}}{{12R}}{\kappa _{12}}} \right)}\\ & {{N_{21}} = \displaystyle\frac{{Eh}}{{2\left( {1 + \nu } \right)}}\left( {{\gamma _{12}} - \frac{{{h^2}}}{{12R}}{\kappa _{12}}} \right)}\\ & {{M_1} = - \,\displaystyle\frac{{E{h^3}}}{{12\left( {1 - {\nu ^2}} \right)}}\left[{{\kappa _1} + \nu {\kappa _2} + \frac{1}{R}\left( {{\varepsilon _1} + \nu {\varepsilon _2}} \right)} \right]}\\ & {{M_2} = - \,\displaystyle\frac{{E{h^3}}}{{12\left( {1 - {\nu ^2}} \right)}}\left( {{\kappa _2} + \nu {\kappa _1}} \right)}\\ & {{M_{12}} = \displaystyle\frac{{E{h^3}}}{{24\left( {1 + \nu } \right)}}\left( {2{\kappa _{12}} + \frac{1}{R}{\gamma _{12}}} \right)}\\ & {{M_{21}} = \displaystyle\frac{{E{h^3}}}{{12\left( {1 + \nu } \right)}}{\kappa _{12}}} \end{aligned}} \right\} (2)

 \left. \begin{aligned} && \displaystyle\frac{{\partial {N_1}}}{{\partial \alpha }} + \frac{{\partial {N_{21}}}}{{\partial \beta }} \,{\rm{ + }} \,RX = 0\\ && \displaystyle\frac{{\partial {N_{12}}}}{{\partial \alpha }} + \frac{{\partial {N_2}}}{{\partial \beta }} + {Q_2}\,{\rm{ + }}\,RY = 0\\ && \displaystyle\frac{{\partial {Q_1}}}{{\partial \alpha }} + \frac{{\partial {Q_2}}}{{\partial \beta }} - {N_2} \,{\rm{ + }} \,RZ = 0\\ && {Q_1} = \displaystyle\frac{1}{R}\left( {\frac{{\partial {M_{21}}}}{{\partial \beta }} - \frac{{\partial {M_1}}}{{\partial \alpha }}} \right)\\ && {Q_2} = \displaystyle\frac{1}{R}\left( {\frac{{\partial {M_{12}}}}{{\partial \alpha }} - \frac{{\partial {M_2}}}{{\partial \beta }}} \right) \end{aligned} \right\} (3)

 \left. {\begin{aligned} & {{L_{11}}\left( u \right) + {L_{12}}\left( v \right) + {L_{13}}\left( w \right) + {\varDelta _1} + \displaystyle\frac{{{R^2}\left( {1 - {\nu ^2}} \right)}}{{Eh}}X = 0}\\ & {{L_{21}}\left( u \right) + {L_{22}}\left( v \right) + {L_{23}}\left( w \right) + {\varDelta _2} + \displaystyle\frac{{{R^2}\left( {1 - {\nu ^2}} \right)}}{{Eh}}Y = 0}\\ & {{L_{31}}\left( u \right) + {L_{32}}\left( v \right) + {L_{33}}\left( w \right) + {\varDelta _3} - \displaystyle\frac{{{R^2}\left( {1 - {\nu ^2}} \right)}}{{Eh}}Z = 0} \end{aligned}} \right\} (4)

 \left. \begin{aligned} & \displaystyle{L_{11}} = \frac{{{\partial ^2}}}{{\partial {\alpha ^2}}} + \frac{{1 - \nu }}{2}\frac{{{\partial ^2}}}{{\partial {\beta ^2}}} \\ & {L_{12}} = \displaystyle \frac{{1 + \nu }}{2}\frac{{{\partial ^2}}}{{\partial \alpha \partial \beta }}\\ & {L_{13}} = \displaystyle \nu \frac{\partial }{{\partial \alpha }} - {c^2}\frac{{{\partial ^3}}}{{\partial {\alpha ^3}}} + \frac{{1 - \nu }}{2}{c^2}\frac{{{\partial ^3}}}{{\partial \alpha \partial {\beta ^2}}} \end{aligned} \right\} (5)
 \left. \begin{aligned} & \displaystyle{L_{21}} = \frac{{1 + \nu }}{2}\frac{{{\partial ^2}}}{{\partial \alpha \partial \beta }} \\ & {L_{22}} = \displaystyle \frac{{1 - \nu }}{2}\frac{{{\partial ^2}}}{{\partial {\alpha ^2}}} + \frac{{{\partial ^2}v}}{{\partial {\beta ^2}}}\\ & {L_{23}} =\displaystyle \frac{\partial }{{\partial \beta }} - \frac{{3 - \nu }}{2}{c^2}\frac{{{\partial ^3}}}{{\partial {\alpha ^2}\partial \beta }} \end{aligned} \right\} (6)
 \left. \begin{aligned} & {L_{31}} = \displaystyle \nu \frac{\partial }{{\partial \alpha }} - {c^2}\left( {\frac{{{\partial ^3}}}{{\partial {\alpha ^3}}} - \frac{{1 - \nu }}{2}\frac{{{\partial ^3}}}{{\partial \alpha \partial {\beta ^2}}}} \right)\\ & {L_{32}} =\displaystyle \frac{\partial }{{\partial \beta }} - {c^2}\frac{{3 - \nu }}{2}\frac{{{\partial ^3}}}{{\partial {\alpha ^2}\partial \beta }}\\ & {L_{33}} = \displaystyle {c^2}\left( {{\nabla ^2}{\nabla ^2} + 2\frac{{{\partial ^2}}}{{\partial {\beta ^2}}} + 1} \right) + 1 \end{aligned} \right\} (7)

 $${\nabla ^2}{\nabla ^2} = \frac{{{\partial ^4}}}{{\partial {\alpha ^4}}} + 2\frac{{{\partial ^4}}}{{\partial {\alpha ^2}\partial {\beta ^2}}} + \frac{{{\partial ^4}}}{{\partial {\beta ^4}}}$$ (8)
 $${c^2} = \frac{{{h^2}}}{{12{R^2}}}$$ (9)

 $$w\left( {\alpha ,\beta ,t} \right) = f(t)\sin (n\alpha) \cos (k\beta)$$ (10)

 \begin{align} {\varDelta _1} = & \,\left( {\nu {k^2} - {n^2}} \right)\displaystyle\frac{{n{f^2}}}{{2R}}\sin (2n\alpha) \; {\cos ^2}(k\beta) \,- \notag \quad\quad\quad \\ & \displaystyle\frac{{\left( {1 + \nu } \right){k^2}}}{2}\frac{{n{f^2}}}{{2R}}\sin (2n\alpha) \cos (2k\beta) \end{align} (11)
 \begin{align} {\varDelta _2} = & \,\left( {{k^2} - \nu {n^2}} \right)\frac{{k{f^2}}}{{2R}}\sin (2k\beta) \; {\sin ^2}(n\alpha) \,- \quad\quad\quad\,\,\,\notag \\ & \frac{{\left( {1 + \nu } \right){n^2}}}{2}\frac{{k{f^2}}}{{2R}}\sin (2k\beta) \cos (2n\alpha) \end{align} (12)
 \begin{align} {\varDelta _3} = & \,\frac{{{c^2}{n^2}{f^2}}}{R}\cos (2n\alpha) \Big[{{n^2}{{\cos }^2}(k\beta) } - {k^2}{\sin ^2}(k\beta) \,+ \notag \\ & { \frac{{\left( {1 - \nu } \right){k^2}}}{2}} \Big] + \frac{{{f^2}{k^2}}}{{{\rm{2}}R}}{\sin ^2}(n\alpha) \; {\sin ^2}(k\beta) \,+ \notag \\ & \frac{{\nu {n^2}{f^2}}}{{{\rm{2}}R}}{\cos ^2}(n\alpha) \; {\cos ^2}(k\beta) \end{align} (13)

 \left. \begin{aligned} u = & \,{A_1}\sin (2n\alpha) \; {\cos ^2}(k\beta) + {B_1}\sin (2n\alpha) \cos (2k\beta) + {u_0}\\ v = & \,{A_2}\sin (2k\beta) \; {\sin ^2}(n\alpha) + {B_2}\sin (2k\beta) \cos (2n\alpha) + {v_0}\\ w = & \,{w_0} = f(t)\sin (n\alpha) \cos (k\beta) \end{aligned} \right\} (14)

 \left. \begin{aligned} {L_{11}}\left( u \right) + {L_{12}}\left( v \right) + {L_{13}}\left( w \right) = 0\\ {L_{21}}\left( u \right) + {L_{22}}\left( v \right) + {L_{23}}\left( w \right) = 0 \end{aligned} \right\} (15)

 \left. \begin{aligned} {A_1} = & \,\displaystyle\frac{{{f^2}\left( {\nu {k^2} - {n^2}} \right)}}{{8nR}},\,\,\,{B_1} = - \frac{{\nu {k^2}{f^2}}}{{16nR}}\\ {A_2} = & \,\displaystyle\frac{{{f^2}\left( {{k^2} - \nu {n^2}} \right)}}{{8kR}},\,\,\,{B_2} = - \frac{{\nu {n^2}{f^2}}}{{16kR}} \end{aligned} \right\} (16)

 \left. \begin{aligned} {N_1} = & \,\displaystyle\frac{{Eh\nu {k^2}{f^2}}}{{8\left( {1 - {\nu ^2}} \right){R^2}}} \! \left[{1 + \frac{{{n^2}}}{{\nu {k^2}}} + \frac{{\left( {1 - {\nu ^2}} \right){n^2}}}{{\nu {k^2}}}\cos (2k\beta) } \! \right] + N_1^0\\ {N_2} = & \,\displaystyle\frac{{Eh\nu {n^2}{f^2}}}{{8\left( {1 - {\nu ^2}} \right){R^2}}} \! \left[{1 + \frac{{{k^2}}}{{\nu {n^2}}} - \frac{{\left( {1 - {\nu ^2}} \right){k^2}}}{{\nu {n^2}}}\cos (2n\alpha) } \! \right] + N_2^0 \end{aligned} \right\} (17)

 \left. \begin{aligned} N_1^0 = \displaystyle\frac{{Eh}}{{\left( {1 - {\nu ^2}} \right)R}}\left[{\frac{{\partial {u_0}}}{{\partial \alpha }} + \nu \frac{{\partial {v_0}}}{{\partial \beta }} + \left( {\nu + {c^2}{n^2}} \right){w_0}} \right]\\ N_2^0 = \displaystyle\frac{{Eh}}{{\left( {1 - {\nu ^2}} \right)R}}\left[{\nu \frac{{\partial {u_0}}}{{\partial \alpha }} + \frac{{\partial {v_0}}}{{\partial \beta }} + \left( {1 - {k^2}{c^2}} \right){w_0}} \right] \end{aligned} \right\} (18)

 \left. {\begin{aligned} & {{u_0}\left( {\alpha ,\beta ,t} \right) = U\cos (n\beta) \cos (k\beta) }\\ & {{v_0}\left( {\alpha ,\beta ,t} \right) = V\sin (n\alpha) \sin (k\beta) }\\ & {{w_0}\left( {\alpha ,\beta ,t} \right) = f\sin (n\alpha) \cos (k\beta) } \end{aligned}} \right\} (19)

 \left. {\begin{aligned} U = & \,\displaystyle\frac{{n \left[{\nu {n^2} - {k^2} + {c^2}\left( {{n^4} - {k^4}} \right)} \right]f}}{{{{\left( {{n^2} + {k^2}} \right)}^2}}}\\ V = & \,\displaystyle - \frac{{k \left[{\left( {\nu + 2} \right){n^2} + {k^2} + 2{c^2}{n^2}\left( {{n^2} + {k^2}} \right)} \right]f}}{{{{\left( {{n^2} + {k^2}} \right)}^2}}} \end{aligned}} \right\} (20)

 $$\int_0^{2{\pi} } {{N_1}} \left( {\alpha ,\,\beta } \right){\rm{d}}\beta = - \frac{{{P_0} + {P_t}\cos (\theta t)}}{R}$$ (21)

 {\left. \begin{aligned} {N_1} = & - \frac{{{P_0} + {P_t}\cos (\theta t)}}{{2{\pi} R}} + \frac{{Eh{f^2}}}{{8{R^2}}}{n^2}\cos (2k\beta) \,+ \\ & \frac{{Eh{n^2}{k^2}}}{{\left( {1 - {\nu ^2}} \right)R}} \left[{\frac{{\left( {1 - \nu } \right)\left( {1 + \nu + 2{c^2}{k^2} + 2{c^2}{n^2}} \right)}}{{{{\left( {{n^2} + {k^2}} \right)}^2}}}} \right] \cdot \\ & f\sin (n\alpha) \cos (k\beta) \\ {N_2} = & \,\frac{{Eh\left( {\nu {n^2} + {k^2}} \right){f^2}}}{{8\left( {1 - {\nu ^2}} \right){R^2}}} - \frac{{Eh{f^2}}}{{8{R^2}}}{k^2}\cos (2n\alpha) \,+ \\ & \frac{{Eh}}{{\left( {1 - {\nu ^2}} \right)R}}\left[{\frac{{\left( {1 - {\nu ^2}} \right){n^4} + \left( {\nu - 4} \right){c^2}{n^2}{k^4}}}{{{{\left( {{n^2} + {k^2}} \right)}^2}}} \,- } \right.\\ & \left. { \displaystyle\frac{{{c^2}\nu {n^6} + 3{c^2}{n^4}{k^2} + {c^2}{k^6}}}{{{{\left( {{n^2} + {k^2}} \right)}^2}}}} \right]f\sin (n\alpha) \cos (k\beta) \end{aligned} \right\} } (22)

 $$Z = \frac{1}{{{R^2}}}\left[{{N_1}\frac{{{\partial ^2}w}}{{\partial {\alpha ^2}}} + {N_2}\left( {\frac{{{\partial ^2}w}}{{\partial {\beta ^2}}} + w} \right)} \right] - m\frac{{{\partial ^2}w}}{{\partial {t^2}}}$$ (23)

 \begin{align} & \displaystyle\bigg[{f'' + {{ {{{\omega '}_{n,k}}} }^2}f - {n^2}\frac{ {{P_0} + {P_t}\cos (\theta t)} }{{2\pi m{R^3}}}f} \bigg]\sin (n\alpha) \cos (k\beta) \,+ \notag \\ & \quad\quad \displaystyle\frac{{EhB}}{{m{R^2}\left( {1 - {\nu ^2}} \right)}}{f^2} + \left[{\frac{{\left( {1 - {\nu ^2}} \right){n^4}}}{{8{R^2}}}\cos (2k\beta) \,+ } \right.\notag \\ & \quad\quad \displaystyle\left. {\frac{{\left( {1 - {\nu ^2}} \right){k^2}\left( {1 - {k^2}} \right)}}{{8{R^2}}}\cos (2n\alpha) - \frac{{\left( {1 - {k^2}} \right)\left( {\nu {n^2} + {k^2}} \right)}}{{8{R^2}}}} \right] \cdot \notag \\ & \quad\quad \displaystyle{f^3}\sin (n\alpha) \cos (k\beta) = 0 \end{align} (24)

 \begin{align} B = & \displaystyle\frac{{{\nu ^2}{k^2}}}{{8R}}\cos (2n\alpha) + \frac{{{k^2}}}{{4R}}{\sin ^2}(n\alpha) + \frac{{\nu {n^2}}}{{8R}} \,{\rm{ + }} \,\frac{{{n^2}}}{R} \cdot \notag \\ & \displaystyle\frac{{\left( {1 - \nu } \right){n^2}{k^2}\left( {1 + \nu + 2{c^2}{k^2} + 2{c^2}{n^2}} \right)}}{{{{\left( {{n^2} + {k^2}} \right)}^2}}} {\sin ^2}(n\alpha) \; {\cos ^2}(k\beta) \,- \notag \\ & \displaystyle\frac{{\left( {1 - {\nu ^2}} \right){n^4} - {c^2}\nu {n^6} + \left( {\nu - 4} \right){c^2}{n^2}{k^4} - 3{c^2}{n^4}{k^2} - {c^2}{k^6}}}{{{{\left( {{n^2} + {k^2}} \right)}^2}}} \cdot \notag \\ & \displaystyle\frac{{\left( {1 - {k^2}} \right)}}{R} {\sin ^2}(n\alpha) \; {\cos ^2}(k\beta) \end{align} (25)

${\omega '_{n,\,k}}$ 是振型为 ($n,\,k$) 时圆柱薄壳的自由振动固有频率

 $${\omega '_{n,k}}^2 = \frac{{Eh{c^2}g'\left( {n,k} \right)}}{{m{R^2}\left( {1 - {\nu ^2}} \right)}}$$ (26)

 \begin{align} g'\left( {n,k} \right) = & \,\displaystyle\frac{{{{\left( {{n^2} + {k^2} - 1} \right)}^2}{{\left( {{n^2} + {k^2}} \right)}^2}}}{{{{\left( {{n^2} + {k^2}} \right)}^2}}} \,{\rm{ + }} \notag \\ & \displaystyle\frac{\Big[{2\left( {1 - \nu } \right)\left( {{n^6} - {n^2}{k^4}} \right) + \displaystyle\frac{{1 - {\nu ^2}}}{{{c^2}}}{n^4}}\Big]}{{{{\left( {{n^2} + {k^2}} \right)}^2}}} \end{align} (27)

$g'\left( {n,\,k} \right)$ 与文献[30] 中的表达式完全一致,经验证,文献[30] 中的线性问题得到的自由振动固有频率 ${\omega _{n,k}}$ 与非线性问题中 ${\omega '_{n,k}}$ 的表达式一致,这说明非线性微分方程式 (24) 中的线性部分与文献[30] 中的线性微分方程式是一致的,也进一步验证了圆柱薄壳非线性横向振动微分方程式 (24) 的正确性.

 \begin{align} \displaystyle f'' + & { {{{\omega '}_{n,k}}} ^2}f - {n^2}\frac{{ {{P_0} + {P_t}\cos (\theta t)} }}{{2\pi m{R^3}}}f \,{\rm{ + }} \notag \\ & \displaystyle \frac{{\left[{{n^4} + {k^2}\left( {{k^2} - 1} \right) - 2\left( {1 - {k^2}} \right)\left( {\nu {n^2} + {k^2}} \right)} \right]Eh}}{{16m{R^4}}}{f^3} = 0 \end{align} (28)

 $${P_{*n,k}} = \frac{{2\pi m{R^3}{{\omega '}_{n,k}}^2}}{{{n^2}}} = \frac{{2\pi Dg'\left( {n,\,k} \right)}}{{{n^2}R}}$$ (29)

 $${\gamma _{n,k}} = \frac{{\left[{{n^4} + {k^2}\left( {{k^2} - 1} \right) - 2\left( {1 - {k^2}} \right)\left( {\nu {n^2} + {k^2}} \right)} \right]Eh}}{{16m{R^4}}}$$ (30)

 $$f'' + { {{{\omega '}_{n,k}}} ^2}\left[{1 - \frac{{{P_0} + {P_t}\cos (\theta t)}}{{{P_{*n,k}}}}} \right]f + {\gamma _{n,k}}\,{f^3} = 0$$ (31)

 $$f'' + \varOmega _{n,k}^2\left[{1 - 2{\mu _{n,k}}\cos (\theta t)} \right]f + {\gamma _{n,k}}\,{f^3} = 0$$ (32)

 $${\varOmega _{n,k}} = {\omega '_{n,k}}\sqrt {1 - \frac{{{P_0}}}{{{P_{*n,k}}}}} ,\quad {\mu _{n,k}} = \frac{{{P_t}}}{{2\left( {{P_{*n,k}} - {P_0}} \right)}}$$ (33)
2 圆柱薄壳的非线性动力屈曲分析

2.1 第一阶不稳定区域内的定态振动振幅

 $$f\left( t \right) = {a_1}\sin \frac{{\theta t}}{2} + {b_1}\cos \frac{{\theta t}}{2}$$ (34)

 \left. \begin{aligned} \left[{\varOmega _{n,k}^2\left( {1 + {\mu _{n,k}}} \right) - \displaystyle\frac{{{\theta ^2}}}{4}} \right]{a_1} + {\varPhi _1} = 0\\ \left[{\varOmega _{n,k}^2\left( {1 + {\mu _{n,k}}} \right) - \displaystyle\frac{{{\theta ^2}}}{4}} \right]{b_1} + {\varPsi _1} = 0 \end{aligned} \right\} (35)

 \left. {\begin{aligned} {{\varPhi _k} = \displaystyle\frac{\theta }{{2{\pi} }}\int_0^{\textstyle \frac{{4{\pi} }}{\theta }} {{\gamma _{n,k}}{f^3}\sin \frac{{k\theta t}}{2}{\rm{d}}t} }\\ {{\varPsi _k} = \displaystyle\frac{\theta }{{2{\pi} }}\int_0^{\textstyle \frac{{4{\pi} }}{\theta }}{{\gamma _{n,k}}{f^3}\cos \frac{{k\theta t}}{2}{\rm{d}}t} } \end{aligned} \left( {k = 1,2,3,\cdots } \right)} \right\} (36)

 \left. \begin{aligned} \left( 1 + {\mu _{n,k}} - \displaystyle\frac{{{\theta ^2}}}{{4\varOmega _{n,k}^2}} \right) {a_1} + \displaystyle\frac{{3{\gamma _{n,k}}A_1^2}}{{4\varOmega _{n,k}^2}}{a_1} = 0 \\ \left( 1 - {\mu _{n,k}} - \displaystyle\frac{{{\theta ^2}}}{{4\varOmega _{n,k}^2}} \right) {b_1} + \displaystyle\frac{{3{\gamma _{n,k}}A_1^2}}{{4\varOmega _{n,k}^2}}{b_1} = 0 \end{aligned} \right\} (37)

 $${\begin{vmatrix} {1 + \mu \! - \! \displaystyle{{\left(\frac{\theta }{{2{\varOmega _{n,k}}}} \right)}^2} \!\! + \frac{{3\gamma A_1^2}}{{4\varOmega _{n,k}^2}}} \!\!\! & \!\! 0\\ 0 \!\! & \!\! {1 \! - \mu - \displaystyle{{\left( {\frac{\theta }{{2{\varOmega _{n,k}}}}} \right)}^2} \!\! + \frac{{3\gamma A_1^2}}{{4\varOmega _{n,k}^2}}} \end{vmatrix}} = \! 0 \!$$ (38)

 \left. {\begin{aligned} & {{A_{1{\rm{1}}}} = \displaystyle\frac{{2{\varOmega _{n,k}}}}{{\sqrt {3\gamma } }}\sqrt {{{\left( {\frac{\theta }{{2{\varOmega _{n,k}}}}} \right)}^2} - 1 \,{\rm{ + }} \,\mu } }\\ & {{A_{1{\rm{2}}}} = \displaystyle\frac{{2{\varOmega _{n,k}}}}{{\sqrt {3\gamma } }}\sqrt {{{\left( {\frac{\theta }{{2{\varOmega _{n,k}}}}} \right)}^2} - 1 - \mu } } \end{aligned}} \right\} (39)

2.2 第二阶不稳定区域内的定态振动振幅

 $$f\left( t \right) = {b_0} + {a_2}\sin (\theta t) + {b_2}\cos (\theta t)$$ (40)

 \left. {\begin{aligned} & \ {\varOmega _{n,k}^2\left( {{b_0} - {\mu _{n,k}}{b_2}} \right) + {\varPsi _0} = 0}\\ & {\left( {\varOmega _{n,k}^2 - {\theta ^2}} \right){a_2} + {\varPhi _2} = 0}\\ & {\left( {\varOmega _{n,k}^2 - {\theta ^2}} \right){b_2} - 2{\mu _{n,k}}\varOmega _{n,k}^2{b_0} + {\varPsi _2} = 0} \end{aligned}} \right\} (41)

 $${\varPsi _0} = \frac{\theta }{{4{\pi} }}\int_0^{\textstyle \frac{{4{\pi} }}{\theta }} {{\gamma _{n,k}}\,{f^3}{\rm{d}}t}$$ (42)

 \left. {\begin{aligned} & {\left( {{b_0} - {\mu _{n,k}}{b_2}} \right) + \frac{{3{\gamma _{n,k}}{b_0}}}{{\varOmega _{n,k}^2}}\left( {\frac{2}{3}{b_0}^2 + {A_2}^2} \right) = 0}\\ & {\left( {1 - \frac{{{\theta ^2}}}{{\varOmega _{n,k}^2}}} \right){a_2} + \frac{{3{\gamma _{n,k}}{a_2}}}{{4\varOmega _{n,k}^2}}\left( {\frac{1}{4}{b_0}^2 + {A_2}^2} \right) = 0}\\ & {\left( {1 - \frac{{{\theta ^2}}}{{4\varOmega _{n,k}^2}}} \right){b_2} - 2{\mu _{n,k}}{b_0} + \frac{{3{\gamma _{n,k}}{b_2}}}{{4\varOmega _{n,k}^2}}\left( {\frac{1}{4}{b_0}^2 + {A_2}^2} \right) = 0} \end{aligned}} \right\} (43)

 \begin{align} & \left( {1 - \displaystyle\frac{{{\theta ^2}}}{{\varOmega _{n,k}^2}} + \frac{{3{\gamma _{n,k}}}}{{4\varOmega _{n,k}^2}}{A_2}^2} \right) \cdot \notag \\ & \quad\quad {\begin{vmatrix} {1 + \displaystyle\frac{{3{\gamma _{n,k}}}}{{\varOmega _{n,k}^2}}{A_2}^2} & { - {\mu _{n,k}}} \\ { - 2{\mu _{n,k}}} & {1 - \displaystyle\frac{{{\theta ^2}}}{{\varOmega _{n,k}^2}} + \frac{{3{\gamma _{n,k}}}}{{4\varOmega _{n,k}^2}}{A_2}^2} \end{vmatrix}} = {\bf 0} \end{align} (44)

 \left. {\begin{aligned} & {{A_{2{\rm{1}}}} = \frac{{2{\varOmega _{n,k}}}}{{\sqrt {3{\gamma _{n,k}}} }}\sqrt {{{\left( {\frac{\theta }{{{\varOmega _{n,k}}}}} \right)}^2} - 1} }\\ & {{A_{22}} = \frac{{{\varOmega _{n,k}}}}{{\sqrt {6{\gamma _{n,k}}} }}\sqrt {\frac{{4{\theta ^2}}}{{\varOmega _{n,k}^2}} - 5 \,{\rm{ + }}\sqrt {{{\left( {\frac{{4{\theta ^2}}}{{\varOmega _{n,k}^2}} - 3} \right)}^2} + 32\mu _{n,k}^2} } }\\ & {{A_{23}} = \frac{{{\varOmega _{n,k}}}}{{\sqrt {6{\gamma _{n,k}}} }}\sqrt {\frac{{4{\theta ^2}}}{{\varOmega _{n,k}^2}} - 5 - \sqrt {{{\left( {\frac{{4{\theta ^2}}}{{\varOmega _{n,k}^2}} - 3} \right)}^2} + 32\mu _{n,k}^2} } } \end{aligned}} \right\} (45)

3 超空泡运动体圆柱薄壳舱段的数值算例

3.1 超空泡运动体受力分析

 $${F_D} = \frac{1}{2}{\rho _w}{A_c}{C_x}{V^2}$$ (46)

 $${C_x} = {C_{x0}}\left( {1 + \sigma } \right)$$ (47)
 $${C_{x0}} = 0.5 + 1.81( \varphi /360 - 0.25) - 2{( \varphi /360 - 0.25)^2}$$ (48)
 $$\sigma = \frac{{2\left( {{p_\infty } - {p_c}} \right)}}{{{\rho _w}{V^2}}}$$ (49)

 $${p_\infty } = {\rho _w}gH + p$$ (50)

 $$p\left( t \right) = {P_0} + {P_t} = {p_0}\left[{1 + \delta \cos (\theta t)} \right]$$ (51)
 $${P_0} = {F_D} = \frac{\pi{\rho _w}{C_x}d_n^2{V^2}}{8} \qquad \qquad$$ (52)

3.2 圆柱薄壳舱段非线性动力屈曲计算结果及分析

3.2.1 给定速度和载荷比例系数时,不同振型下的非线性参数共振曲线

 图 1 第一阶参数共振曲线图(a) Fig.1 The first order parametric resonance curves (a)
 图 2 第一阶参数共振曲线图(b) Fig.2 The first order parametric resonance curves (b)
 图 3 第一阶参数共振曲线图(c) Fig.3 The first order parametric resonance curves (c)

(1) 当圆柱薄壳所受的外载荷的频率处于激发区以外且在小于激发区一侧 (图 1 中的 $BC$ 段) 时,圆柱薄壳没有横向振动.

(2) 考虑几何非线性的圆柱薄壳舱段,其各阶参数共振曲线都向大于激发区频率的一侧倾斜. 因此,

① 外载荷频率从小于激发下界的一侧开始逐渐增大 (图 1 中的路线为 $BCGF$) 时,圆柱薄壳的横向振动振幅沿着 $BCHD$ 增大.

② 外载荷频率从大于激发上界的一侧开始逐渐减小 (路线为 $FGCB$) 时：

(3) 对比图 1图 2图 3 的纵坐标可知,当 $k$ = 2 或 $k$ = 3 时,不稳定区域内的定态振动振幅与壳体的厚度 ($h$ = 3 mm) 是同一个数量级的,但当 $k$ = 1 时,不稳定区域内的定态动振幅比 $k$ = 2 或 $k$ = 3 时高出两个数量级.

(1) 在参数振动的线性理论中,不存在参数共振曲线的“倾斜”问题,激发区域只有图 1 中的 $CG$ 段,但在考虑几何非线性的参数振动非线性理论中,由于参数共振曲线向大于激发区频率一侧“倾斜”,导致激发区不仅有 $CG$ 段,还包括 $GF$ 段. 因此,考虑几何非线性后,激发区的扩大会导致外载荷安全频率范围的缩小和结构动力稳定可靠性的降低.

(2) 参数振动的线性理论认为激发区域内振动的振幅是无限大的,这不符合工程实际. 考虑几何非线性,激发区域内的振动振幅实际上为有限振幅. 对于超空泡运动体而言,结构的变形量直接影响其沾湿面积,而沾湿面积决定了空泡的稳定性,这是一个典型的流固耦合问题. 因此,考虑非线性因素,准确地获得激发区域内的参数振动振幅,是进行流固耦合分析的必要前提.

3.2.2 给定的振型 ($i$ = 2,$k$ = 2) 时,速度和载荷比例系数对非线性参数共振曲线的影响

 图 4 不同航行速度下的第一阶参数共振曲线图 Fig.4 The first order parametric resonance curves with different speed
 图 5 不同载荷比例系数下的第一阶参数共振曲线图 Fig.5 The first order parametric resonance curves with different proportional coefficient of load

(1) 激发区的范围随着超空泡运动体的航行速度 $V$ 和载荷比例系数 $\delta$ 增大而增大; 激发区内的定态振动的振幅随着超空泡运动体的航行速度 $V$ 和载荷比例系数 $\delta$ 增大而增大.

(2) 结论“(1)”与线性理论的结论相同 [30],事实上,从式 (38) 和文献[31] 中的线性临界频率方程式的对比中可知,若令式 (38) 中的 ${A_1}$ = 0,那么,式 (38) 和线性临界频率方程式是一致的,也就是说,图 1 中的 $CG$ 段的激发区域在线性理论和非线性理论中是相同的. 不同之处在于,非线性理论中要将 $GF$ 段也计入激发区域范围内,使激发区比线性理论中给出的激发区扩大.

 图 6 第一阶和第二阶非线性参数共振曲线对比图 Fig.6 The comparison chart of the nonlinear resonance between the firstand the second-order.

4 结论

(1) 考虑几何非线性因素后,参数共振曲线向大于激发区频率的一侧倾斜,使得激发区的范围变大,从而导致外载荷安全频率范围的缩小和结构动力稳定可靠性的降低.

(2) 考虑非线性因素,能准确地获得激发区域内的参数振动振幅,是进行超空泡运动体流固耦合分析的基础.

(3) 激发区的范围随着超空泡运动体的航行速度 $V$ 和载荷比例系数 $\delta$ 增大而增大; 激发区内的定态振动的振幅随着超空泡运动体的航行速度 V 和载荷比例系数 δ 增大而增大.

(4) 当 k = 2 或 k = 3 时,不稳定区域内的定态振动振幅与壳体的厚度 (h = 3 mm) 是同一个数量级的,但当 $k$ = 1 时,不稳定区域内的定态振动振幅比 $k$ = 2 或 $k$ = 3 时高出两个数量级.