移动载荷作用下浮冰的非线性动力学响应

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. 

NONLINEAR DYNAMIC RESPONSE OF A FLOATING ICE SHEET TO A MOVING LOAD1)

Vibrations of a floating ice cover on top of a two-dimensional ideal fluid of arbitrary depth are studied when the effects of nonlinearity, inertia, and damping are all considered. We reduce the fully nonlinear problem to a cubic-truncation system involving variables on the free surface by expanding the relevant pseudo-differential operators and retaining nonlinear terms up to the third order. To validate the accuracy of the reduced model, we focus on the free wavepacket solitary wave solutions. In the absence of damping, the normal form analysis is performed to derive the cubic nonlinear Schr?dinger equation, which predicts the existence of free wavepacket solitary waves in the primitive equations and the accuracy of the cubic-truncation model. The main advantage of the cubic-truncation approximation over the quadratic-truncation model is that the resultant NLS equation has correct coefficient of the nonlinear term, which allows a better approximation of dynamic responses of the ice cover near the phase speed minimum. Solitary waves are then numerically computed, and it is shown that the cubic-truncation approximation agrees well with the full Euler equations for bifurcation curves and wave profiles, indicating that the reduced model is more accurate than the quadratic truncation model. The nonlinear dynamic response of a floating ice sheet to a fully localized constant-moving load is investigated based on the cubic-truncation model. The time-dependent solutions are compared with the data from the field measurements, and good agreement is achieved between the numerical results and experimental records.