﻿ 基于弹性波理论的深水吊缆非线性运动研究
 舰船科学技术  2019, Vol. 41 Issue (1): 85-91 PDF

Research on deepwater hoisting rope nonlinear movement based on the theory of elastic wave
ZHAO Teng, ZHANG Rong, WANG Miao
Chongqing Jiaotong University, Chongqing 400074, China
Abstract: On the nonlinear deepwater hoisting cable movement characteristics, the hoisting rope of the nonlinear equations of motion were derived based on the elastic wave theory established the dynamic model of the lifting rope to solve the finite difference method. Comparing the test data to found that good agreement with the result, the accuracy and reliability of the method was verified. As the initial motivation to harmonic movement, the different water depths of crane load vertical motion response was calculated and the spectrum analysis was carried on, and the result show that hanging vertical motion response of the load is obvious nonlinear movement. Further more, the load calculation results of the vertical displacement when the length of cable is 4 500 and 3 000 was given, and it show that With the increase of incentive period, 3 000 m of crane load in vertical displacement amplitude is greater than 4 500 m, and the difference degree is bigger.
Key words: deepwater hoisting rope     elastic wave     finite difference     nonlinear
0 引　言

1 深水吊缆动力学建模

 图 1 吊缆动态几何构型示意图 Fig. 1 The dynamic geometry of hoisting cable

 $R(s,t) = {R^f}(s,t) - {R^i}(s)\text{。}$ (1)

$R(s,t)$ 分别沿法向 $\vec n$ 、切向 $\vec \tau$ 和副法向 $\vec b$ 分为3个分量 ${R_1}(s,t)$ ${R_2}(s,t)$ ${R_3}(s,t)$ ，可得

 $R\left( {s,t} \right) = {R_1}\left( {s,t} \right)\vec \tau + {R_2}\left( {s,t} \right)\vec n + {R_3}\left( {s,t} \right)\vec b\text{。}$ (2)

 ${e^f} = \frac{1}{2}\frac{{{{\left( {d{s^f}} \right)}^2} - {{\left( {d{s^{\circ}}} \right)}^2}}}{{{{\left( {d{s^{\circ}}} \right)}^2}}} = \frac{1}{2}\left( {\frac{{\partial {R^f}}}{{\partial {s^{\circ}}}} \cdot \frac{{\partial {R^f}}}{{\partial {s^{\circ}}}} - 1} \right)\text{。}$ (3)

 $\begin{split} \varPsi _s^f = \varPsi _s^i + \frac{1}{2}\int\limits_0^{{L^i}} {[2{P^i}\left( {{s^i},t} \right)\varepsilon \left( {{s^i},t} \right)} \times \\ \left( {1 + 2{e^i}} \right)\left. { + E{A^i}{\varepsilon ^2}{{\left( {1 + 2{e^i}} \right)}^2}} \right]{\rm{d}}{s^i} \end{split} \text{，}$ (4)

 $\varPsi _g^f = \varPsi _g^i + \int\nolimits_{{\varOmega ^i}} {\rho {A^i}g\left( { - {l_\tau }{R_1} - {l_n}{R_2}} \right)} {\rm d}{\varOmega ^i}\text{。}$ (5)

 $\varPsi _b^f = \varPsi _b^i + \int\nolimits_{{\varOmega ^i}}^{} {{\rho _w}{A^i}g\left( { - {l_t}{U_1} - {l_n}{U_2}} \right)} {\rm d}{\varOmega ^i}\text{。}$ (6)

 $\varPsi _k^f = \frac{1}{2}\int_0^{{L^i}} {\rho {A^i}\left( {{V^f} \cdot {V^f}} \right)} {\rm d}{s^i}\text{，}$ (7)

 $\begin{split} {V^f} = & \frac{{{\rm d}R\left( {{s^i},t} \right)}}{{{\rm d}t}} = {R_{1,t}}\left( {s,t} \right){e_1} + {R_{2,t}}\left( {s,t} \right){e_2}+ \\ & {R_{3,t}}\left( {s,t} \right){e_3} \text{。} \end{split}$ (8)

 $\varPsi _f^f = \int\nolimits_{{\varOmega ^i}}^{} {F \cdot U} {\rm d}{\varOmega ^i} = \int\nolimits_{{\varOmega ^i}}^{} {\left( {{F_1}{R_1} + {F_2}{R_2} + {F_3}{R_3}} \right)} {\rm d}{\varOmega ^i}\text{，}$ (9)

 $\delta \int\nolimits_{{t_1}}^{{t_2}} {\left( {\varPsi _k^f - \varPsi _s^f - \varPsi _g^f + \varPsi _b^f + \varPsi _f^f} \right)} {\rm d}t = 0\text{。}$ (10)

 $\begin{split} - \rho {A^i}{R_{1,tt}} = & {\left[ {\left( {{P^i} + E{A^i}\varepsilon } \right)\left( {1 + {R_{1,s}} - {\kappa ^i}{R_2}} \right)} \right]_{,s}} \\ & \left[ {\left( {{P^i} + E{A^i}\varepsilon } \right)\left( {1 + {R_{1,s}} - {\kappa ^i}{R_2}} \right)} \right] - \\ & {\kappa ^i}\left( {{P^i} + E{A^i}\varepsilon } \right)\left( {{R_{2,s}} - {\kappa ^i}{R_1}} \right) - \\ & \left( {\rho - {\rho _w}} \right){A^i}g{l_t} + {F_1} \text{，} \end{split}$ (11)

 $\begin{split} - \rho {A^i}{R_{2,tt}} = & {\left[ {\left( {{P^i} + E{A^i}\varepsilon } \right)\left( {{U_{2,s}} - {\kappa ^i}{R_1}} \right)} \right]_{,s}} - \\ & {\kappa ^i}\left( {{P^i} + E{A^i}\varepsilon } \right)\left( {1 + {R_{1,s}} - {\kappa ^i}{R_2}} \right) - \\ & \left( {\rho - {\rho _w}} \right){A^i}g{l_n} + {F_2} \text{，} \end{split}$ (12)

 $- \rho {A^i}{U_{3,tt}} = {\left[ {\left( {{P^i} + E{A^i}\varepsilon } \right){U_{3,s}}} \right]_{,s}} + {F_3}\text{。}$ (13)

 $\begin{split} - {R_{1,tt}} + & \frac{{{F_1}}}{{\rho A}} + \left[ { - 2g - \frac{E}{\rho }\lambda + \frac{E}{\rho }{\lambda ^3}{s^2}} + \right. \\ & \left. { g{\lambda ^2}{s^2}} \right]{R_{2,s}} + \left[ {\frac{{{P_0}}}{{\rho A}} + \frac{E}{\rho } + \frac{g}{2}\lambda {s^2}} + \right. \\ & \left. { 2\frac{E}{\rho }{R_{1,s}} + 2\frac{E}{\rho }{\lambda ^3}{s^2}{R_2} - 2\frac{E}{\rho }\lambda {R_2}} \right]{R_{1,ss}} + \\ & g\lambda s{R_{1,s}} + g\lambda s + \left( {2\frac{E}{\rho }{\lambda ^3}s + g{\lambda ^2}s} \right){R_2} +\\ & \left( {\frac{3}{2}g{\lambda ^3}{s^2} - g\lambda } \right){R_1} = 0 \text{，} \end{split}$ (14)
 $\begin{split} &- {R_{2,tt}} + \frac{{{F_2}}}{{\rho A}} + \left[ {\frac{{{P_0}}}{{\rho A}} + \frac{g}{2}\lambda {s^2} + \left( {{\lambda ^3}{s^2} - \lambda } \right)\frac{E}{\rho }{R_2}} + \right. \\ & \left. { \frac{E}{\rho }{R_{1,s}}} \right]{R_{2,ss}} - g{\lambda ^2}s{R_1} + g\lambda s{R_{2,s}} + \left( {g - \frac{g}{2}{\lambda ^2}{s^2}} \right) + \\ & \left( {2g - g{\lambda ^2}{s^2} - \frac{E}{\rho }{\lambda ^3}{s^2} + \frac{E}{\rho }\lambda } \right){R_{1,s}} + \\ & \left( { - \frac{E}{\rho }{\lambda ^2} - g\lambda + \frac{3}{2}g{\lambda ^3}{s^2}} \right){R_2} = 0 \text{。} \end{split}$ (15)
2 深水吊缆非线性运动响应数值求解 2.1 数值求解的有限差分法

 $\frac{{{\rm d}\bar u}}{{{\rm d}x}} = \frac{1}{{\left( {4!\Delta x} \right)}}\left[ {\begin{array}{*{20}{c}} { - 50}&{96}&{ - 72}&{32}&{ - 6} \\ { - 6}&{ - 20}&{36}&{ - 12}&2 \\ 2&{ - 16}&0&{16}&{ - 2} \\ { - 2}&{12}&{ - 36}&{20}&6 \\ 6&{ - 32}&{72}&{ - 96}&{50} \end{array}} \right]\bar u + {\rm O}\left( {\Delta {x^4}} \right)\text{，}$ (16)
 $\begin{array}{l} \displaystyle\frac{{{\rm d}{u^2}}}{{{\rm d}{x^2}}} = \displaystyle\frac{1}{{4!\Delta {x^2}}}\left[ {\begin{array}{*{20}{c}} { - \displaystyle\frac{{415}}{3}} & {192} & { - 72} & {\displaystyle\frac{{64}}{3}} & { - 3} & 0\\ {20} & { - 30} & { - 8} & {28} & { - 12} & 2\\ { - 2} & {32} & { - 60} & {32} & { - 2} & 0\\ {20} & { - 30} & { - 8} & {28} & { - 12} & 2\\ { - \displaystyle\frac{{415}}{3}} & {192} & { - 72} & {\displaystyle\frac{{64}}{3}} & { - 3} & 0 \end{array}} \right]u + \\ \left[ {\begin{array}{*{20}{c}} {100\displaystyle\frac{{{\rm d}u}}{{{\rm d}{x_1}}}}\\ 0\\ 0\\ 0\\ {100\displaystyle\frac{{{\rm d}u}}{{{\rm d}{x_n}}}} \end{array}} \right] + {\rm O}\left( {\Delta {x^4}} \right)\text{，} \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad(17) \end{array}$

 $\left\{ \begin{gathered} {\left( {{U_1}} \right)_j}^{i + 1} = {\left( {{U_1}} \right)_j}^i + \left( {{v_1}} \right)_j^i\Delta t + \frac{1}{2}\Delta {t^2}\left[ {{\beta _1}\left( {{a_1}} \right)_j^i + {\beta _2}\left( {{a_1}} \right)_j^{i + 1}} \right] \text{，} \\ {\left( {{U_2}} \right)_j}^{i + 1} = {\left( {{U_2}} \right)_j}^i + \left( {{v_2}} \right)_j^i\Delta t + \frac{1}{2}\Delta {t^2}\left[ {{\beta _1}\left( {{a_2}} \right)_j^i + {\beta _2}\left( {{a_2}} \right)_j^{i + 1}} \right] \text{，} \\ \left( {{v_1}} \right)_j^{i + 1} = \left( {{v_1}} \right)_j^i + \frac{1}{2}\Delta t\left[ {{\alpha _1}\left( {{a_1}} \right)_j^i + {\alpha _2}\left( {{a_1}} \right)_j^{i + 1}} \right] \text{，} \quad\quad\quad\;(18)\\ \left( {{v_2}} \right)_j^{i + 1} = \left( {{v_2}} \right)_j^i + \frac{1}{2}\Delta t\left[ {{\alpha _1}\left( {{a_2}} \right)_j^i + {\alpha _2}\left( {{a_2}} \right)_j^{i + 1}} \right] \text{。} \quad\quad\quad\quad\quad \end{gathered} \right.$

 $M{A^{i + 1}} + C\left| {{V^i}} \right|{V^i} + K{U^i} = {\left( {{F^{excit}}} \right)^i}\text{。}$ (19)

2.2 深水吊缆非线性运动响应分析

2.2.1 深水吊缆非线性运动响应计算及验证

 图 2 3 000 m时不同激励周期时吊缆张力值比较 Fig. 2 The comparison of cable tension with different inspiring period when the length is 3 000 m

 图 3 4 500 m时不同激励周期时吊缆张力值比较 Fig. 3 The comparison of cable tension with different inspiring period when the length is 4 500 m

2.2.2 深水吊缆非线性运动响应分析

 图 4 不同缆长时初始激励位移与吊载垂向位移时历 Fig. 4 The initial motivation displacement and load vertical displacement of cable under the different cable length

 图 5 不同缆长时吊载响应频谱分析 Fig. 5 The response spectrum analysis of hoisting load under the different cable length

 图 6 不同吊缆长度的吊载垂向位移随吊缆上端激励变化情况 Fig. 6 The variation of load vertical displacement with different inspiring period and cable length
3 结　语

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