## 1 Introduction  
In Civil Engineering, many types of bridge decks and floor systems can be effectively modeled as orthotropic plates with opposite free edges. In practice, the other two edges of these structures are mostly not classical (i.e., neither simply supported nor fully clamped). In order to consider actual boundary conditions, these edges are intermediate between simply-supported and fully-clamped, which can be modelled as elastically restrained against rotation (i.e., rotationally restrained). However, vibration analysis of plate structures with non-classical boundary conditions involves complicated mathematical procedures. Therefore, in the design practice, the analyses of bridge decks and floors are often simplified based on a beam idealization. This simplification is not always suitable for short-span plate structures. Motivated by the extensive applications of structurally orthotropic plates in engineering structures, the present investigation deals with free transverse vibration of the rectangular orthotropic plates with two opposite edges rotationally restrained and leaving the remaining others free (i.e., R-F-R-F). A considerable amount of literature has been published on analytical solutions for vibration of orthotropic plates with classical boundary conditions such as simply supported, clamped,and free [1; 2; 3; 4]. In parallel, vibration of rectangular plates with elastically restrained edges has attracted considerable attention since the 1950s [5; 6; 7; 8; 9; 10; 11]. These studies focused primarily on the approximate estimation of natural frequencies by using Rayleigh method and Ritz method. Furthermore, most studies were devoted to isotropic plates with elastically restrained and simply supported edges. Limited research has been conducted to study orthotropic plates with elastically restrained and free edges. [PERSON] and his colleagues [12; 13] investigated orthotropic plates with three edges elastically restrained and the fourth edge free (i.e., R-R-R-F) and two adjacent edges elastically restrained and the other adjacent edges free (i.e., R-R-F-F) by Rayleigh method. Similarly, orthotropic plates with the edges elastically restrained against rotation and a free, straight corner cut-out was studied in [14]. The results reported in these three works are limited to the fundamental frequency. [PERSON] and [PERSON] [15] studied the dynamic response of orthotropic plate having clamped-simply supported and free-free boundary conditions (i.e., C-F-S-F). [PERSON] and [PERSON] [16] directly extended the semi-analytical finite difference method reported in [8] to free vibrations of orthotropic plates with elastically restrained and free edges. Some eigenvalues for R-F-R-F orthotropic plates have been reported. On top of that, to the best knowledge of the authors, no exact analytical solution is currently available for transverse vibration of R-F-R-F orthotropic plates. Although approximate solutions and numerical modelling may be sufficient for practical purposes, exact solutions are desirable to assess the effectiveness of these approximate solutions. Recently, an analytical method of finite integral transforms has been extensively applied to obtained the exact series solutions of bending of plates [17; 18; 19; 20]. By using this method, the authors [21; 22] obtained the exact solutions for bending of R-R-R-R and R-F-R-F orthotropic plates with integral kernel of \(\sin\alpha_{m}x\sin\beta_{n}y\) and \(\sin\alpha_{m}x\cos\beta_{n}y\), respectively. However, it has been found that the formulation of the finite integral transform method used for the bending problems of orthotropic plates cannot be directly applied to vibration problems [23], which involve solving a highly non-linear equation and would be quite difficult. In the present study, an alternative formulation of the finite integral transform method was developed for the free vibration of R-F-R-F rectangular orthotropic plates. Through a systematic solving procedure, the characteristic equation for eigenvalues can be derived.