## 1 Introduction  
Research on transverse vibration of the beam can date back to the 18 th century. [PERSON] and [PERSON] presented the Bernoulli-Euler beam theory based on famous classical assumptions [1]. It was believed that the bending effect is the most significant factor for a transversely vibrating beam. Later, this beam model was gradually improved by [PERSON] [2], who took the effect of rotary inertia caused by the flexural deformation into account. The [PERSON] beam theory partially revises the distortion of natural frequencies calculated by the Bernoulli-Euler beam theory. Despite this modification, the natural frequencies remain distorted and are rather severe when a beam is not slim enough. Afterwards, the shear model took the shear deformation into account based on the Bernoulli-Euler model. It is recognized that the shear stress is evenly distributed along the beam height without considering the rotary inertia caused by the bending effect. In contrast, the pure shear model [3] only includes the shear deformation but excludes an important factor, the bending effect. The difference between them should be noted. Considering shear deformation based on the Bernoulli-Euler beam, the precision estimation of natural frequencies increases conspicuously. The Timoshenko beam theory [4, 5] was proposed by [PERSON], who considered the effect of shear deformation and the rotary inertia caused by the flexural deformation. Accuracy of the natural frequencies of the Timoshenko beam model is significantly improved, and therefore, it is applied widely. Many researches have been devoted since the Timoshenko beam theories were presented. [PERSON] and [PERSON] [6] finished a complete study where the derivation process for the four beam models with four various boundary conditions was described in detail in 1953. The orthogonality conditions of the Timoshenko beam were presented by [PERSON] [7], who also solved the initial and boundary-value problem of the Timoshenko beam. [PERSON] [8] derived the orthogonality conditions of the Timoshenko beam with the modal expansion method and provided an analytical solution of the forced Timoshenko beam in 1955. [PERSON] and [PERSON] [9] investigated the dynamic response of the semi-infinite beams under four given boundary conditions using the Laplace inverse method. [PERSON] [10] derived partial differential equations for flexural vibrations of six common types of simple, finite beams. [PERSON] and [PERSON] [11] put forward a solution to the dynamic response of Timoshenko beams with the given initial conditions and external forces using the eigenfunction expansion method. Since the classical theory of the Timoshenko beam was applied to scientific research and practical engineering, more controversies followed. The major point of contention is the second spectrum of the Timoshenko beam. [PERSON] and [PERSON] pointed out the possibility of a second spectrum of the Timoshenko beam in 1953, and many efforts have been devoted to this study. [PERSON] [12] and [PERSON] analyzed the vibration characteristics of the simply supported Timoshenko beam, and two sets of independent spectrum curves of the [PERSON] beam were proposed. [PERSON] [13] also claimed the existence of a second spectrum using the dynamic discrete element method. However, there are still many scholars who hold opposite opinions. [PERSON] and [PERSON] [14] used relatively elementary analysis, showing that there is no second spectrum for a [PERSON] beam with any boundary conditions. [PERSON] and [PERSON] [3,15] believed the nonexistence of the second spectrum except for a hinged-hinged beam by using the finite element method. [PERSON] [16, 17, 18] compared the second spectrum with a solution derived from the plane stress elastic theory and found that there is no relationship between them as well as a physical meaning of the second spectrum. In 2006, he finished a further appraisal on the second spectrum of the [PERSON] beam and thought it should be neglected. Other than the uniform beam, [PERSON] et al.