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移动载荷作用下浮冰的非线性动力学响应

孟洋涵 王展

孟洋涵, 王展. 移动载荷作用下浮冰的非线性动力学响应. 力学学报, 2022, 54(4): 862-871 doi: 10.6052/0459-1879-22-040
引用本文: 孟洋涵, 王展. 移动载荷作用下浮冰的非线性动力学响应. 力学学报, 2022, 54(4): 862-871 doi: 10.6052/0459-1879-22-040
Meng Yanghan, Wang Zhan. Nonlinear dynamic response of a floating ice sheet to a moving load. Chinese Journal of Theoretical and Applied Mechanics, 2022, 54(4): 862-871 doi: 10.6052/0459-1879-22-040
Citation: Meng Yanghan, Wang Zhan. Nonlinear dynamic response of a floating ice sheet to a moving load. Chinese Journal of Theoretical and Applied Mechanics, 2022, 54(4): 862-871 doi: 10.6052/0459-1879-22-040

移动载荷作用下浮冰的非线性动力学响应

doi: 10.6052/0459-1879-22-040
基金项目: 中国科学院B类先导项目资助(XDB22040203)
详细信息
    作者简介:

    王展, 研究员, 主要研究方向: 水动力学. Email: zwang@imech.ac.cn

  • 中图分类号: O352

NONLINEAR DYNAMIC RESPONSE OF A FLOATING ICE SHEET TO A MOVING LOAD

  • 摘要: 本文考虑非线性、惯性和阻尼的影响, 研究了任意深度二维理想流体顶部浮冰的振动. 对相关的拟微分算子进行展开并将非线性项保留至三阶后, 完全非线性问题被简化为仅与自由面上的变量相关的三阶截断模型. 为了验证简化模型的准确性, 重点关注了自由孤立波解. 在不考虑阻尼的情况下, 采用多重尺度方法推导了三阶非线性薛定谔方程(NLS), 利用该方程预测了任意水深下原始欧拉方程中自由波包型孤立波解的存在性及三阶截断模型的准确性. 相比于Dinvay等所提出的二阶模型, 三阶截断模型的优势在于其对应的三阶NLS具有准确的非线性项系数, 能够在最小相速度附近更好地模拟冰层的动力学响应. 进一步地对自由孤立波解进行数值计算, 数值结果表明三阶截断模型在分岔曲线和孤立波波形上均与完全欧拉方程吻合良好, 准确性高于二阶截断模型. 基于三阶截断模型, 探究了匀速局域化载荷作用下的浮冰非线性动力学响应并将时间依赖解与实验测量数据进行比较, 数值计算结果与实验记录吻合良好.

     

  • 图  2  (a) 水深$ h = 250\;\mathrm{m} $, 惯性$ \alpha = 0.069 $时, 孤立波解分岔图. 实线: 完全欧拉方程的分岔曲线, 短划线: 三阶截断模型的分岔曲线, 点线: 二阶截断模型的分岔曲线. (b) 波速$ c = 1.287\;8 $时, 欧拉方程、三阶截断模型和二阶截断模型的孤立波解. 实线: 完全欧拉方程, 短划线: 三阶截断模型, 点线: 二阶截断模型

    Figure  2.  The bifurcation curves of wavepacket solitary waves for $ h = 250\;\text{m} $ and $ \alpha = 0.069 $. Solid line: full Euler equations, dashed line: third-truncation model, dotted line: quadratic-truncation model. (b) The typical profiles for $ c = 1.287\;8 $. Solid line: full Euler equations, dashed line: third-truncation model, dotted line: quadratic-truncation model

    图  1  (a) 水深$ h = 6.8\;\text{m} $, 惯性系数$ \alpha = 0.069 $时, 孤立波解分岔图. 实线: 完全欧拉方程的分岔曲线, 短划线: 三阶截断模型的分岔曲线, 点线: 二阶截断模型的分岔曲线. (b) 波速$c = 1.287\;8$时, 欧拉方程、三阶截断模型和二阶截断模型的孤立波解. 实线: 完全欧拉方程, 短划线: 三阶截断模型, 点线: 二阶截断模型

    Figure  1.  (a) Bifurcation curves of wavepacket solitary waves for $ h = 6.8\;\text{m} $ and $ \alpha = 0.069 $. Solid line: full Euler equations, dashed line: third-truncation model, dotted line: quadratic-truncation model. (b) The typical profiles for $c = 1.287\;8$. Solid line: full Euler equations, dashed line: third-truncation model, dotted line: quadratic-truncation model

    图  3  三阶截断模型的数值结果与文献[1]实验中所记录的冰层形变对比. 其中虚线代表Takizawa的实验结果, 红点代表实验中的运动载荷经过冰层形变测量仪的垂直位置. 实线表示三阶截断模型式(5)和式(7)的数值结果, 蓝点代表数值计算中载荷经过监测点的垂向位置

    Figure  3.  Comparisons of the numerical results of Eq. (3) the numerical results of cubic-truncation model and the experimental records of Ref. [1]. The dashed lines represent the experimental data, and the red dots indicates the position of the load as it passed the deflectometer. The solid lines shows the numerical results of Eqs. (5) and (7), and the blue dots indicates the z-position of the load as it passes the point where the time series are obtained

    图  4  三阶截断模型的数值结果与文献[2]实验中所测量的冰层应变对比, 图中应变值量级为$ 10^{-6} $. 虚线代表Squire的实验结果, 实线表示三阶截断模型式(5)和式(7)的数值结果

    Figure  4.  The comparison of microstrain between the numerical results of cubic-truncation model and numerical records of Ref. [2]. The microstrain is of $ 10^{-6} $. Dashed lines: experimental records, solid lines: numerical approximation of the cubic-truncation model Eqs. (5) and (7)

    表  1  浅水[1]及深水[2]实验中各物理参数的取值及单位

    Table  1.   Values and units of the physical parameters in the shallow-water[1] and deep-water[2] experiments

    Physical parameterSymbolLake SaromaMcMurdo sound
    Young's modulus$E/( { {\rm{N} } \cdot { {\rm{m} }^{ - 2} } } )$$5.1\times{10^8}$$4.2\times{10^9}$
    Poisson ratio$\nu$$0.3$$0.3$
    ice thickness$d/{\rm{m}}$$0.17$$1.6$
    water depth$h/{\rm{m}}$$6.8$$\infty$
    flexural rigidity$D/({ {\rm{N} } } \cdot { {\rm{m} } } )$$2.35\times{10^5}$$1.8\times{10^9}$
    water density$\rho_{w}/(\mathrm{kg \cdot m^{-3} })$$1024$$1026$
    ice density$\rho_{i}/(\mathrm{kg \cdot m^{-3} })$$917$$917$
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  • [1] Takizawa T. Response of a floating sea ice sheet to a steadily moving load. Journal of Geophysical Research, 1988, 93: 5100-5112 doi: 10.1029/JC093iC05p05100
    [2] Squire VA, Robinson WH, Langhorne PJ, et al. Vehicles and aircraft on floating ice. Nature, 1988, 33: 159-161
    [3] Kheisin DY. Moving load on an elastic plate which floats on the surface of an ideal fluid. Izv. AN SSSR, OTN, Mekh. i Mashinostroenie, 1963, 1: 178-180 (in Russian
    [4] Nevel DE. Moving loads on a floating ice sheet. US Army CRREL Report, 1970: 261
    [5] Kerr AD. The critical velocities of a load moving on a floating ice plate that is subjected to implane force. Cold Regions Science and Technology, 1983, 6: 267-274 doi: 10.1016/0165-232X(83)90047-2
    [6] Davys JW, Hosking RJ. Waves due to a steadily moving source on a floating ice plate. Journal of Fluid Mechanics, 1985, 158: 269-287 doi: 10.1017/S0022112085002646
    [7] Babaei H, van der Sanden JJ, Short NH, et al. Lake ice cover deflection induced by moving vehicles: comparing theoretical results with satellite observations//New Research and Developments in Road Safety Session of the 2016 Conference of the Transportation Association of Canada Toronto, ON, 2016
    [8] van der Sanden JJ, Short NH. Radar satellites measure ice cover displacements induced by moving vehicles. Cold Regions Science and Technology, 2017, 533: 56-62
    [9] Schulkes RMSM, Hosking RJ, Sneyd AD. Waves due to a steadily moving source on a floating ice plate. Part 2. Journal of Fluid Mechanics, 1987, 180: 297-318
    [10] Kheisin DY. Some non-stationary problems of dynamics of the ice cove//Studies in Ice Physics and Ice Engineering (Iakolev ed.), Israel Program for Scientific Translations, 1971
    [11] Schulkes RMSM, Sneyd AD. Time-dependent response of floating ice to a steadily moving load. Journal of Fluid Mechanics, 1988, 186: 25-46 doi: 10.1017/S0022112088000023
    [12] Nugroho WS, Wang K, Hosking RJ, et al. Time-dependent response of a floating flexible plate to an impulsively started steadily moving load. Journal of Fluid Mechanics, 1999, 381: 337-355 doi: 10.1017/S0022112098003875
    [13] Miles J, Sneyd AD. The response of a floating ice sheet to an accelerating line load. Journal of Fluid Mechanics, 2003, 497: 435-439 doi: 10.1017/S002211200300675X
    [14] Wilson JT. Coupling between moving loads and flexural waves in floating ice sheets. US Army SIPRE Report, 1955
    [15] Beltaos S. Field studies on the response of floating ice sheets to moving loads. Canadian Journal of Civil Engineering, 1981, 8: 1-8 doi: 10.1139/l81-001
    [16] Hosking RJ, Sneyd AD, Waugh DW. Viscoelastic response of a floating ice plate to a steadily moving load. Journal of Fluid Mechanics, 1988, 196: 409-430 doi: 10.1017/S0022112088002757
    [17] Wang K, Hosking RJ, Milinazzo F. Time-dependent response of a floating viscoelastic plate to an impulsively started moving load. Journal of Fluid Mechanics, 2004, 521: 295-317 doi: 10.1017/S002211200400179X
    [18] Părău EI, Dias F. Nonlinear effects in the response of a floating ice plate to a moving load. Journal of Fluid Mechanics, 2002, 460: 281-305 doi: 10.1017/S0022112002008236
    [19] Dinvay E, Kalisch H, Părău EI. Fully dispersive models for moving loads on ice sheets. Journal of Fluid Mechanics, 2019, 876: 122-149 doi: 10.1017/jfm.2019.530
    [20] Toland JF. Heavy hydroelastic travelling waves. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2007, 463: 2371-2397 doi: 10.1098/rspa.2007.1883
    [21] 浦俊, 卢东强. 三层流体中斜入射波作用下半无限板的水弹性响应. 力学学报, 2019, 51: 1614-1629 (Pu Jun, Lu Dongqiang. Hydroelastic response of a semi-infinite plate due to obliquely incident waves in a three-layer fluid. Chinese Journal of Theoretical and Applied Mechanics, 2019, 51: 1614-1629 (in Chinese) doi: 10.6052/0459-1879-19-081
    [22] Gao T, Vanden-Broeck JM, Wang Z. Numerical computations of two-dimensional flexural-gravity solitary waves on water of arbitrary depth. IMA Journal of Applied Mathematics, 2018, 83: 436-450 doi: 10.1137/S1064827597321532
    [23] Craig W, Sulem C. Numerical simulation of gravity waves. Journal of Computational Physics, 1993, 108: 73-83 doi: 10.1093/imamat/hxy007
    [24] Milewski PA, Tabak EG. A pseudospectral procedure for the solution of nonlinear wave equations with examples from free-surface flows. SIAM Journal on Scientific Computing, 1999, 21: 1102-1114 doi: 10.1006/jcph.1993.1164
    [25] Milewski PA, Wang Z. Three dimensional flexural-gravity waves. Studies in Applied Mathematics, 2013, 131: 135-148 doi: 10.1111/sapm.12005
    [26] Cho Y, Diorio JD, Akylas TR, et al. Resonantly forced gravity– capillary lumps on deep water. Part 2. Theoretical model. Journal of Fluid Mechanics, 2011, 672: 283-306
    [27] Squire VA. The breakp of shore fast sea ice. Cold Regions Science and Technology, 1993, 3: 211-218
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出版历程
  • 收稿日期:  2022-01-21
  • 录用日期:  2022-04-07
  • 网络出版日期:  2022-04-08
  • 刊出日期:  2022-04-18

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