Wave based method is an efficient numerical technique that can be used to solve the structural dynamic problems, especially the vibration of plates. The approach expands the dynamic field variables to a weighted sum of wave functions, and the contribution factors are determined by minimizing the errors on the boundary conditions. While the governing equation is satisfied a priori, the boundary conditions are satisfied in the integral sense, using preferably a weighted residual formulation. However, the weighted residual formulation is not always suitable for all kinds of boundary conditions. In plate bending problems, the plate edges can be modelled as clamped, simply supported, free, symmetric or (visco) elastically restrained against translation or/and rotation, depending on how they are restrained in reality. But the currently available weighted residual formulations are suitable either only for the non-elastic restraints (clamped, simply supported, free and symmetric) or only for the general (visco) elastic restraints with both translational and rotational elasticities. In this paper, a general formulation is proposed for all the aforementioned types of edge restraints. It is equivalent to the conventional one when the plate edges are non-elastically restrained, but performs better with the introduction of corner residuals for the plate with general elastically restrained edges. Apart from the ideal non-elastic and the general full-elastic restraints, this modified formulation also works well for the partial-elastic ones with only either translational or rotational elasticity. Numerical cases are studied on a rectangular plate for all the restraint types, where the proposed formulation is validated. The restraint types can be identical or different among the plate edges, and how to apply the formulation to the non-uniformed situation is also demonstrated. Throughout the considered cases, this paper shows the accuracy and efficiency of the wave based method by comparing with the finite element method in computing the forced vibration of the thin plate. With the modified weighted residual formulation, the method further shows its great convenience in changing the restraint types or stiffness (and damping) parameters of plate edges.

基于波的方法是一种有效的数值技术，可用于解决结构动力问题，尤其是板的振动问题。该方法将动态场变量扩展为波函数的加权和，并且通过最小化边界条件上的误差来确定贡献因子。虽然先验地满足控制方程，但在积分意义上满足边界条件，最好使用加权残差公式。然而，加权残差公式并不总是适用于所有类型的边界条件。在板弯曲问题中，板边缘可以建模为夹紧、简单支撑、自由、对称或（粘滞）弹性约束以防止平移或/和旋转，具体取决于它们在现实中的约束方式。但是目前可用的加权残差公式仅适用于非弹性约束（夹紧、简支、自由和对称）或仅适用于具有平移和旋转弹性的一般（粘滞）弹性约束。在本文中，为所有上述类型的边缘约束提出了一个通用公式。当板边缘非弹性约束时，它等效于常规方法，但对于具有一般弹性约束边缘的板，引入角残差后性能更好。除了理想的非弹性约束和一般的全弹性约束外，这种修改后的公式也适用于仅具有平移或旋转弹性的偏弹性约束。对所有约束类型的矩形板上的数值案例进行了研究，并验证了所提出的公式。板边之间的约束类型可以相同或不同，并且还演示了如何将公式应用于非均匀情况。在所考虑的情况下，本文通过与有限元方法的比较，证明了基于波浪的方法在计算薄板受迫振动时的准确性和效率。通过改进的加权残差公式，该方法进一步显示了其在改变板边缘的约束类型或刚度（和阻尼）参数方面的极大便利。还演示了如何将公式应用于非制服情况。在所考虑的情况下，本文通过与有限元方法的比较，证明了基于波浪的方法在计算薄板受迫振动时的准确性和效率。通过改进的加权残差公式，该方法进一步显示了其在改变板边缘的约束类型或刚度（和阻尼）参数方面的极大便利。还演示了如何将公式应用于非制服情况。在所考虑的情况下，本文通过与有限元方法的比较，证明了基于波浪的方法在计算薄板受迫振动时的准确性和效率。通过改进的加权残差公式，该方法进一步显示了其在改变板边缘的约束类型或刚度（和阻尼）参数方面的极大便利。