## 2 Theory  
Consider an axially loaded tapered beam with concentrated masses and elastic supports on both sides, as illustrated in Figure 1. \(XYZ\) is the coordinate system, and \(L\) is the total length of the beam. The height \(h_{0}\) and width \(b_{0}\) of the cross-section are assumed to be reduced with a taper ratio \(c\), and \(h(x)\) and \(b(x)\) are the cross-section dimensions at arbitrary locations. \(m_{p1}\) and \(m_{p2}\) are the concentrated masses, and \(F_{0}\) is the axial force with a constant value. The elastic supports at both sides of a tapered beam are modeled as rotational springs \(K_{\Phi 1}\) and \(K_{\Phi 2}\), and translational springs \(K_{W1}\) and \(K_{W2}\), respectively. The beam model considered the effect of rotational inertia; however, the effect of shear deformation was ignored. When considering only one element, the strain (\(\mathcal{U}\)) and kinetic (\(\mathcal{T}\)) energy T can be expressed as follows:  
\[\mathcal{U}=\frac{1}{2}\int_{0}^{L}EI(x)\big{(}w^{\prime\prime}\big{)}^{2}dx+ \frac{1}{2}\int_{0}^{L}F_{0}(w\prime)^{2}dx+\frac{1}{2}K_{W1}(w_{x=0})^{2}+ \frac{1}{2}K_{W2}(w_{x=L})^{2}+\frac{1}{2}K_{\Phi 1}(w\prime_{x=0})^{2}+ \frac{1}{2}K_{\Phi 2}(w\prime_{x=L})^{2}, \tag{1}\]  
and  
\[\mathcal{T}=\frac{1}{2}\int_{0}^{L}\rho A(x)\big{(}\ddot{w}\big{)}^{2}dx+ \frac{1}{2}\int_{0}^{L}\rho I(x)\big{(}\dot{\theta}\big{)}^{2}dx+\frac{1}{2}m _{p1}\big{(}\dot{w}_{x=0}\big{)}^{2}+\frac{1}{2}m_{p2}\big{(}\dot{w}_{x=L} \big{)}^{2}, \tag{2}\]  
where \(EI(x)\), \(\rho A(x)\), and \(\rho I(x)\) are the bending stiffness, mass per unit length, and rotary inertia varying along the length of the beam by the taper ratio, respectively. The primes and dots indicate the differentiation with respect to distance \(x\) and time \(t\), respectively. \(K_{W1}\) and \(K_{W2}\), \(K_{\Phi 1}\) and \(K_{\Phi 2}\), and \(m_{p1}\) and \(m_{p2}\) are the translational spring, rotational spring, and concentrated masses at \(x=0\) and \(x=L\), respectively; the terms containing them in the energies are utilized to deal with the boundary conditions at the joint. \(\theta\) and \(w\) denote the bending rotation and bending displacement, respectively. The cross-sectional shapes varying along the length of the beam can be defined by  
\[I(x)=I_{0}(1-\zeta)^{n+2},\ A(x)=A_{0}(1-\zeta)^{n}, \tag{3}\]  
where \(\zeta=c\overline{x}\), and \(\overline{x}(=x/L)\) is a nondimensional length coordinate. \(A_{0}\) and \(I_{0}\) denote the area and moment of inertia, respectively, at \(c=0\). There are \(n=1\) and \(n=2\) for the single-tapered and double-tapered beams, respectively. The differential equation of motion from [PERSON]'s principle, considering the kinetic and strain energies, is given by:  
\[(EI(x)w^{\prime\prime})^{n}-F_{0}w^{\prime\prime}-\Big{(}\rho I(x)\ddot{w}^{ \prime}\Big{)}^{\prime}+\rho A(x)\ddot{w}=0. \tag{4}\]  
Figure 1: Geometry of elastically restrained tapered beam with concentrated mass and axial loading. The bending moment and shear force at the end of the beam must be balanced by the sum of the resisting forces, such as the linear springs and mass. Therefore, the following relationships can be determined using previous techniques [45;47]:  
Bending moment,  
\[EI(x)w^{\prime\prime}=\alpha K_{\Phi i}w^{\prime} \tag{5}\]  
and shear force,  
\[-(EI(x)w^{\prime\prime})^{\prime}+F_{0}w^{\prime}+\rho I(x)\ddot{w}^{\prime}= \alpha\big{(}K_{Wi}w+m_{pi}\ddot{w}\big{)} \tag{6}\]  
where the terms on the right side of Equations (5) and (6) represent the bending moment and shear force of the beam at the joint, respectively, and the left-hand terms represent the resistance forces generated by the elastic supports and mass. \(\alpha=-1\) for the left end and \(\alpha=+1\) for the right end of a tapered beam. In addition, \(K_{\Phi i}\), \(K_{Wi}\), and \(m_{pi}\) are \(K_{\Phi 1}\), \(K_{W1}\), and \(m_{pi}\) for the left end, and \(K_{\Phi 2}\), \(K_{W2}\), and \(m_{pi2}\) for the right end of a tapered beam, respectively.