加筋板结构在航空航天、船舶等工程领域中应用广泛，对其固有振动特性展开研究具有重要的理论价值和应用价值。针对矩形加筋板固有模态的求解问题，本文基于Rayleigh商变和Rayleigh-Ritz法提出了一种半解析分析方法。在该方法中，封闭形式的薄板模态函数作为加筋板模态函数的基函数，然后根据Rayleigh商得到加筋板频率方程和模态函数。此外，对于四边简支单筋加筋板的固有振动问题，由Rayleigh商得到其控制微分方程，采用分离变量方法求得其精确解。通过把所得结果与有限元结果和文献结果进行比较，验证了本文方法的有效性与精确性。所提出方法可以用于加筋板的理论分析和参数化设计。

Stiffened plate structures are widely used in aerospace, shipbuilding and other engineering fields, so the research on its vibration characteristics has significant academic and practical value. Based on the Rayleigh quotient and Rayleigh-Ritz method, this paper presents a semi-analytical method to solve the natural modes of rectangular stiffened plates. In this method, the closed-form mode functions of thin plates are used as the basis functions of the mode functions of stiffened plates, and the natural frequency equation and the mode functions of rectangular stiffened plates are derived according to the Rayleigh quotient. In addition, the governing differential equation of the simply supported stiffened rectangular plate with single stiffener is derived according to the Rayleigh quotient, and the exact solution is obtained using the separation-of-variable method. The effectiveness and accuracy of the proposed method are verified by comparing the present results with those of the finite element method and literature. The proposed method in this work can be used for theoretical analysis and parametric design of stiffened plates.