## 1 Introduction  
Sandwich structures with strength-to-weight ratios have been widely applied in fields, such as conveying structures, aeronautics, and astronautics. A three-layer sandwich beam is usually composed of a top face, core made of lightweight materials, and bottom face layers [1; 2; 3; 4; 5]. Investigating the dynamic characteristics of sandwich beams based on the behavior of the core is important. In particular, it is necessary to analyze the dynamic behavior of the face layer in relation to that of the core layer. Dynamic behavior analysis of sandwich beams has been performed by many investigators [6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26; 27; 28; 29; 30], and advanced techniques [7; 8; 9; 10; 11; 12] based on the classical sandwich beam theory have been developed for the vibration analysis of such beams. The static analysis of sandwich beams for distributed and sinusoidal loadings using the finite element model based on the higher-order zigzag shear deformation theory was investigated [13], and a sandwich beam element was studied to design sandwich beams with partially delaminated regions [14]. Studies on the bending, buckling, and free vibration analyses of laminated sandwich beams are discussed in a review paper by [PERSON] and [PERSON] [15]. The effect of height-to-length ratio on the natural frequencies of three-layered sandwich beams, with different boundary conditions, was examined using higher-order zigzag beam theory [16]. The bending-dominated vibration of the soft core in the three-layer sandwich beam was analyzed using the quadrature element method [17]. The dynamic stiffness method was used to analyze the dynamic characteristics of the out-of-plane bending vibrations of symmetric sandwich beams, and they investigated the independence of the bending and torsional modes in the free vibration of such beams [18]. The effect of the ratio of core to thickness to face thickness on the natural frequencies of soft-coresandwich beams was analyzed using tractable zig-zag beam theory [19]. A finite element model for the vibration analysis of sandwich beams with a soft core was developed by employing the Euler-Bernoulli theory for the face sheets and the Timoshenko beam theory for the core sheet [20]. The effect of the length-to-height ratio on the natural frequencies of sandwich beams with a soft core was studied using the zig-zag beam theory [21]. The dynamic behaviors of axially moving sandwich beams with a soft core were studied using the Galerkin method and complex mode method, which considered only the shear deformation for the core layer of such beams [22]. The effects of the core thickness and Poisson's ratio on the dynamic characteristics of sandwich beams were analyzed using Kriging-based finite element models based on different shear deformation theories [23]. The effect of thickness ratios and density ratios between the face and core layers on the natural frequencies of sandwich beams is investigated by obtaining the discrete solution of the differential equation using the Green function [24], the finite element method [25], and the Chebyshev series [27], as well as by analyzing the dynamic behaviors of sandwich beams in the core layer with the dynamic stiffness method, whereby the effect of bending displacements is neglected and only shear deformation is considered [26]. The problems presented in this study are similar to those considered in [24; 25; 26; 27]; however, the physical phenomena that were not discussed in previous studies have been analyzed using an analytical method that has been developed in the present study. The axial, bending, and shear deformations in all layers were considered in analyzing the natural frequencies and mode shapes of the sandwich beams [28]; yet, the method can be used in a limited manner for simply supported sandwich beams, and a fourth-order uncoupled governing equation was used.