Modified (2+1)-dimensional displacement shallow water wave system and its approximate similarity solutions

Recently, a new （2＋1）-dimensional shallow water wave system, the （2＋1）-dlmenslonal displacement shallow water wave system （2DDSWWS）, was constructed by applying the variational principle of the analytic mechanics in the Lagrange coordinates. The disadvantage is that fluid viscidity is not considered in the 2DDSWWS, which is the same as the famous Kadomtsev-Petviashvili equation and Korteweg-de Vries equation. Applying dimensional analysis, we modify the 2DDSWWS and add the term related to the fluid viscidity to the 2DDSWWS. The approximate similarity solutions of the modified 2DDSWWS （M2DDSWWS） is studied and four similarity solutions are obtained. For the perfect fluids, the coefficient of kinematic viscosity is zero, then the M2DDSWWS will degenerate to the 2DDSWWS.

直角坐标下厚圆柱扁壳弯曲的一般解

基于直角坐标下考虑横向剪切变形情况下厚圆柱扁壳的几何方程、物理方程、平衡微分方程,建立了以3个中面位移和2个中面转角为独立变量的中厚圆柱扁壳弯曲的位移型基本微分方程。因该方程可退化为薄圆柱扁壳弯曲的基本微分方程,说明了其推导过程的正确性及一般性。此外,厚圆柱扁壳的位移型基本微分方程是一个10阶微分方程,对其使用双重三角级数进行了求解,进一步建立了厚圆柱扁壳小挠度弯曲的基本方程;然后应用MATLAB工具对简支矩形底厚圆柱扁壳弯曲的双重三角级数型基本方程进行求解;最后运用MATLAB工具分析厚圆柱扁壳,给出了3个中面位移和2个中面转角的解析解及挠度变化曲线。结果表明:双重三角级数法适用于任何形式的分布载荷,3个中面位移和2个中面转角的收敛性较好;级数的收敛性还取决于载荷的形式,其中均布载荷收敛性最好。