## 2 Theory
 \tag{31}\]  
The constant \(\mathbf{A}\) can be obtained as  
\[\mathbf{A}=\mathbf{Q}^{-1}\mathbf{Z}_{0}. \tag{32}\]  
The state vector \(\mathbf{Z}_{1}\) at \(\overline{x}=1\) for \(n=1\) and \(n=2\) can be expressed in matrix form as follows:  
\[\mathbf{Z}_{1}=\mathbf{H}\mathbf{A} \tag{33}\]  
where \(\mathbf{Z}_{1}=[W,\ \Phi,\ M,V]_{\overline{x}=L}^{T}\), and the components of the \(\mathbf{H}\) matrix for \(n=1\) can be defined as  
\[ \begin{array}{c}H_{1j}=\sum\limits_{j=1}^{4}f(c,j-1),\ H_{2j}=\xi\sum \limits_{j=1}^{4}f^{\prime}(c,j-1),\\ H_{3j}=N_{1}\sum\limits_{j=1}^{4}\Big{\{}(1-c)^{3}f^{\prime\prime}(c,j-1)+ \overline{D}_{7}f^{\prime}(c,j-1)\Big{\}},\\ H_{4j}=-N_{2}\sum\limits_{j=1}^{4}\Big{\{}(1-c)^{3}f^{\prime\prime}(c,j-1)-3(1 -c)^{2}f^{\prime\prime}(c,j-1)-\Big{\{}\overline{D}_{8}-\overline{D}_{1}(1-c )^{3}\Big{\}}f^{\prime}(c,j-1)+\big{(}\overline{D}_{9}-\overline{D}_{10}\big{)} f(c,j-1)\Big{\}},\\ \end{array}  \tag{34}\]  
and for \(n=2\),  
\[ \begin{array}{c}H_{3j}=N_{1}\sum\limits_{j=1}^{4}\Big{\{}(1-c)^{4}f^{\prime \prime}(c,j-1)+\overline{D}_{8}f^{\prime}(c,j-1)\Big{\}},\\ H_{4j}=-N_{2}\sum\limits_{j=1}^{4}\Big{\{}(1-\zeta)^{4}f^{\prime\prime}(c,j-1)-4(1 -c)^{3}f^{\prime\prime}(c,j-1)-\Big{\{}\overline{D}_{9}-\overline{D}_{1}(1-c )^{4}\Big{\}}f^{\prime}(c,j-1)+\big{(}\overline{D}_{10}-\overline{D}_{11}\big{)} f(c,j-1)\Big{\}}.\\ \end{array}  \tag{35}\]  
From Equations (32) and (33), the relationship between \(\mathbf{Z}_{0}\) and \(\mathbf{Z}_{1}\) is obtained as follows:  
\[\mathbf{Z}_{1}=\mathbf{T}\mathbf{Z}_{0} \tag{36}\]  
where \(\mathbf{T}=\mathbf{H}\mathbf{Q}^{-1}\), and \(\mathbf{T}\) is the transfer matrix of a beam element. By applying the free-free end condition in Equation (36), the natural frequencies and mode shapes of the elastically restrained tapered Rayleigh beams can be calculated. \(K_{\Phi 1}\)\(K_{\Phi 2}\), \(K_{v1}\), and \(K_{w2}\) in the free-free end conditions are zero. The clamped end condition can be expressed by adjusting the spring constants related to the elastic supports. The spring values for this end condition are investigated in detail in the results and discussion section. The dynamic characteristics of tapered beams utilizing a single element can be evaluated by considering the effects of the axial loading, concentrated mass, and elastic supports at both ends. In addition, to evaluate the effects between ends, they can be assembled generally as a transfer matrix. The frequency determinant for the free-free end condition can be obtained as follows:  
At \(\overline{x}=0\) and \(\overline{x}=1\),  
\[M,\ V=0 \tag{37}\]  
By substituting Equation (37) into Equation (36),  
\[ \begin{bmatrix}T_{31}&T_{32}\\ T_{41}&T_{44}\end{bmatrix} \begin{Bmatrix}W\\ \Phi\end{bmatrix}_{x=0}=0, \tag{38}\]  
and the frequency determinant for the free-free end condition is expressed as  
\[ \begin{vmatrix}T_{31}&T_{32}\\ T_{41}&T_{42}\end{vmatrix} =0. \tag{39}\]  
Assuming that the bending displacement (\(W\)) is 1, the slope of the deformation curve can be defined as  
\[\Phi=-\frac{T_{31}}{T_{32}}\ or-\frac{T_{41}}{T_{42}} \tag{40}\]  
Therefore, the state vector \(\mathbf{Z}_{0}\) is given by  
\[\mathbf{Z}_{0}=[1,-T_{31}/T_{32},0,0]. \tag{41}\]  
By substituting Equation (41) into Equation (32), constant \(\mathbf{A}\) can be computed. Therefore, the modal shapes can be calculated with Equation (26).