## 1 Introduction  
Several researchers have extensively studied tapered beams to investigate the variations in the dynamic characteristics of beams experiencing various environments in many engineering applications, such as columns, girders, and rotor blades. The effects of elastic supports, concentrated mass, and axial loading have been included [1; 2; 3], and a combination of these effects has been discussed [4; 5; 6]. Such beam structures have received attention from many researchers [7; 8; 9] for analyzing the dynamic characteristics of many structures with intermediate or elastic supports at both ends where fully clamped boundary conditions cannot be employed. To obtain more accurate results, shape functions due to the cross-section varying along the length of the tapered beam were studied [10; 11; 12], and the solution of the power series to solve differential equations was employed [13]. The differential equation for the special case of a tapered Bernoulli-Euler beam is solved with Bessel's function [14; 15]. Employing classical beam theory, a cantilever uniform beam [16] and a cantilever tapered beam [17] attached to a concentrated mass at the free end were studied to analyze their effects on the natural frequencies. The characteristic equations for the unknown constants were determined by substituting the solution of the differential equation into the boundary and continuity conditions. The effects of constant axial and gravity loads on tapered beams have also been investigated [4; 18]. The effect of constant axial tensile and compressive loads on the eigenpairs of a uniform beam was analyzed in detail by considering different boundary conditions [19; 20], and when the compressive load reached the critical buckling loading, these effects were analyzed in detail for uniform beams with various end conditions. The buckling and post-buckling behaviors of a double-beam system supported on a Winkler-Pasternak elastic foundation under a compressive load were analyzed through numerous parametric studies [21]. To analyze the critical dynamic buckling loads, an axially compressed piezoelectric semiconductor rod simply supported at both ends was studied [22]. The transfer matrix method is the simplest and most powerful numerical tool for evaluating the dynamic characteristics of beam structures with elastic supports. It can be used simply by multiplying the change in state vectors (i.e., the effect of elastic support) during the transfer process of the state vectors [23; 24; 25]. After deriving a differential equation considering the elastic support, an improved transfer matrix was developed by introducing the concept of a multibody system [26]. The frequency determinant was determined using the general Rayleigh-Ritz method by substituting the bending displacements and slope of the bending curve for a beam with elastic supports into kinetic and potential energies [27; 28; 29]. The effects of the natural frequencies of elastically restrained cantilever beams with a concentrated mass at the free end were investigated [30; 31]. The frequency determinant is determined by substituting the bending displacement into the elastic boundary condition at the joint between the beam and elastic support [30; 31; 32; 33]. This technique is still a useful and powerful method for the dynamic analysis of beam structures with elastic supports [34; 35; 36]; however, the derivative of the frequency determinant owing to this procedure is complicated. The numerical expression was simplified because this process was omitted in this study. A study on the nonlinear control of a beam structure with elastic supports at both ends was conducted [37], and the dynamic characteristics of a beam structure with intermediate elastic supports were investigated [38]. In addition, many studies on elastic supports have been conducted to analyze the various dynamic characteristics of beam-like structures with elastic foundations [39; 40].