## 1 Introduction  
A rectangular thin plate is considered to be the basic structural element in practical applications because of its relevance in various engineering fields such as civil and structural engineering, mechanical engineering, naval and aerospace, etc. The extensive application of such structures requires the investigation of the dynamic characteristics of thin plates to establish an accurate and reliable design. It is confirmed that free vibration of plates causes a reduction in structure stiffness which reduces its load carrying capacity, as a result may cause premature structure failure. Simultaneously, a clear understanding of natural frequencies and associated mode shapes is the basis for reliable design toward avoiding a plate's resonance problem. However, it is difficult to deal with the free vibration problem of plates under various boundary conditions, especially for the ones with complex non-classical boundaries. Numerous approaches have been developed to solve such types of intractable problems, all of them face the challenge of pursuing exact solutions that satisfy the governing partial differential equations (PDEs) under specific boundaries. Analytical methods are indispensable since they can provide theoretical, reliable and benchmark results, but are far from completed due to the complicated mathematical procedure involved. The literature review reveals that exact analytical solutions of plates under non-classical boundary restraints are not as easily available as those of Levy-type plates (two simply supported parallel edges). The boundary condition investigated in the present article is one of the most representative challenging cases, in which both the free corner and its opposite two adjacent rotationally restrained edges increase the solving difficulty of the vibration problem of plate. Accordingly, very few analytical benchmark results are available for the title problem. Consequently, seeking effective and simple analytical approaches becomes more and more crucial. The aforementioned situation motivates the development of numerical/approximate approaches. Since there are excellent capabilities of the advanced numerical approaches such as simplicity, versatility, high precision and good convergence, they have been continuously applied to solve intractable mechanical problems of plates. Here we introduce some typical studies to illustrate new developments in this field. [PERSON] [1] predicts the nonlinear dynamic behaviors of elastically restrained composite panels by using the classical Rayleigh-Ritz approach, the proposed solving procedure allowed one to transform the complicated nonlinear PDEs into handling several nonlinear ordinary differential equations. On basis of first-order shear deformation theory, [PERSON] [2] developed a novel Jacobi-Ritz approach to predict free vibration responses of laminated plates with different edge conditions, excellent convergent and precise approximate results were presented for further investigation. [PERSON] [3] provided new, precise buckling and vibration analysis for internally cracked square plates through combining the conventional Ritz method with the moving least square approach, in which novel enriched basis functions are selected as a solution for in-plane and out-of-plane displacements. It is promising to extend such an effective approach to investigate more intractable stability/dynamic problems of plates since it exhibits a powerful mathematical ability. [PERSON] [4] carried out the FEM analysis on the dynamic response of multilayer FG-GPLRC toroidal panels under non-classical boundaries. The present FEM results revealed that frequency parameters decrease with the decreasing of rotational spring factor and the GPLs volume fractions. [PERSON] [5] investigated vibration and buckling behavior of composite plates by applying the traditional finite strip approach, in which the effects of moisture and temperature are well illustrated.