# Free Vibration Analysis of Curvilinearly Tapered Axially Functionally Graded Material Beams
## 3 Solution Methodology  
### Differential Transform Method  
Due to the variable coefficients, reflecting the dependence of material properties and cross-sectional geometry on the \(x\)-coordinate, the coupled nature of the differential equations in (9) presents mathematical challenges for finding a closed-form solution. To address this, we employ the differential transform method (DTM) for its versatility and simplicity. This method converts the differential equations into a set of recurrent algebraic equations, enabling the solution to be represented as an infinite power series. The fundamental aspects of the DTM can be found in [PERSON]'s original work [44].  
We reformulate system (9) into a more suitable form for applying the DTM as follows:  
\[\begin{array}{l}\frac{d^{2}\overline{\varphi}}{dx^{2}}+\overline{D}_{1}(x) \frac{d\overline{\varphi}}{dx}+\overline{D}_{2}(x)\frac{d\overline{\varphi}}{ dx}-\  
\[\overline{w}(x,\omega)=\Bigg{\{}1+\sum\limits_{p=0}^{N-2}B_{w}(p)(x-x_{0 })^{p+2}\Bigg{\}}W(0)+\Bigg{\{}\sum\limits_{p=0}^{N-2}C_{w}(p)(x-x_{0})^{p+2} \Bigg{\}}\Phi(0) \tag{14}\] \[+\Bigg{\{}(x-x_{0})+\sum\limits_{p=0}^{N-2}G_{w}(p)(x-x_{0})^{p+2 }\Bigg{\}}W(1)+\Bigg{\{}\sum\limits_{p=0}^{N-2}H_{w}(p)(x-x_{0})^{p+2}\Bigg{\}} \Phi(1),\] \[\overline{\varphi}(x,\omega)=\Bigg{\{}\sum\limits_{p=0}^{N-2}B_{ \varphi}(p)(x-x_{0})^{p+2}\Bigg{\}}W(0)+\Bigg{\{}1+\sum\limits_{p=0}^{N-2}C_{ \varphi}(p)(x-x_{0})^{p+2}\Bigg{\}}\[PERSON](0)+\Bigg{\{}\sum\limits_{p=0}^{N-2}G_{ \varphi}(p)(x-x_{0})^{p+2}\Bigg{\}}W(1)\] \[+\Bigg{\{}(x-x_{0})+\sum\limits_{p=0}^{N-2}H_{\varphi}(p)(x-x_{0 })^{p+2}\Bigg{\}}\Phi(1).\]  
The boundary conditions at the beam ends can include transverse displacement \(\overline{w}\), rotation angle \(\overline{\varphi}\), bending moment \(\overline{M}\), and shear force \(\overline{\mathcal{Q}}\), which are related by the following equations  
\[\overline{M}=D_{11}(x)\frac{d\overline{\varphi}}{dx},\,\overline{Q}=A_{55}(x) \bigg{(}\frac{d\overline{\overline{w}}}{dx}-\overline{\varphi}\bigg{)} \tag{15}\]  
Taking into account (14) and (15), we can formulate the eigenvalue problem in a form that allows for the incorporation of any constraints as follows:  
\[\begin{bmatrix}A_{11}(\omega)&A_{12}(\omega)&A_{13}(\omega)&A_{14}(\omega)\\ A_{21}(\omega)&A_{22}(\omega)&A_{23}(\omega)&A_{24}(\omega)\\ A_{31}(\omega)&A_{32}(\omega)&A_{33}(\omega)&A_{34}(\omega)\\ A_{41}(\omega)&A_{42}(\omega)&A_{43}(\omega)&A_{44}(\omega)\end{bmatrix} \begin{bmatrix}W(0)\\ \Phi(0)\\ W(1)\\ \end{bmatrix}=0. \tag{16}\]  
Here, the functions \(A_{ij}(\omega)\) are appropriate polynomials of the unknown angular frequency \(\omega\), which involve the recurrent coefficients and the power terms of \((x-x_{0})\).  
Finally, the eigenvalues are computed by solving the root-finding problem resulting from Equation [14]:  
\[\text{det}\big{|}A_{ij}(\omega)\big{|}=0. \tag{17}\]  
A computational program has been developed in the Matlab R2016b environment to apply the DTM for analyzing the free vibration of curvilinear tapered inhomogeneous Timoshenko beams. The eigenvalue problem is addressed using standard algorithms provided in the Matlab R2016b package [46].