## 3 Conventional Analysis  
This study presents a combination of case I and II boundary conditions, termed clamped free (CF) boundary conditions, commonly used in prismatic cantilever beam applications and evaluated as examples to approximate the transcendental roots using the proposed methodology. A beam is a slender horizontal structural member that appears in various forms and is used in various artifacts, such as supporting members in high-rise buildings, railways, long-span bridges, flexible satellites, gun barrels, robot arms, airplane wings, balconies and many other areas. Therefore, this section contains mathematical modelling for CF boundary conditions using only a conventional approach based on the work of [PERSON] [3]. A general 3D view of the prismatic cantilever beam, with beam geometric characteristics such as Young's Modulus \(E\), moment of inertia \(I\), mass density \(\rho\) with uniform cross-sectional area \(A\), width \(W\), length \(L\) and thickness \(t\), is shown in Figure 2.  
The equation of motion for transverse vibration of the prismatic beam with homogenous material parameters and a constant cross-sectional area is derived using the Euler-Bernoulli beam theory [3]:  
\[\rho A(x)\frac{\partial^{2}\psi(t,x)}{\partial t^{2}}+\frac{\partial^{2}}{ \partial x^{2}}\left[EI(x)\frac{\partial^{2}\psi(t,x)}{\partial x^{2}} \right]=0 \tag{1}\]  
where \(t\) is time and \(\psi\) is the transverse displacement. Figure 1: Different boundary conditions for a beam without mass system. Equation (1) can be represented in the following simplified form:  
\[\frac{\partial^{2}\psi}{\partial t^{2}}(t,x)+\gamma^{2}\frac{\partial^{4}\psi}{ \partial x^{4}}(t,x)=0 \tag{2}\]  
where  
\[\gamma=\sqrt{\frac{IE}{A\rho}} \tag{3}\]  
Moreover, Equation (2) is solved by the separation of variables method, defined as:  
\[\psi(t,x)=T(t)W(x) \tag{4}\]  
Substituting Equation (4) in Equation (2) after some mathematical reconfiguration yields  
\[\frac{\gamma^{2}}{W(x)}\frac{d^{4}W(x)}{dx^{4}}=-\frac{1}{T(t)}\frac{d^{2}T(t) }{dt^{2}}=a=\omega^{2} \tag{5}\]  
where \(a=\omega^{2}\) can be demonstrated as a positive constant. Equation (5) can be represented as two equations as:  
\[\frac{d^{4}W(x)}{dx^{4}}-\beta^{4}W(x)=0 \tag{6}\]  
\[\frac{d^{2}T(t)}{dt^{2}}-\omega^{2}T(t)=0 \tag{7}\]  
where  
\[\beta^{4}=\frac{\omega^{2}}{\gamma^{2}}=\frac{\rho A\omega^{2}}{EI} \tag{8}\]  
The solution to Equation (7) is given by  
\[\rho A(x)\frac{\partial^{2}\psi(t,x)}{\partial t^{2}}+\frac{\partial^{2}}{ \partial x^{2}}\left[EI(x)\frac{\partial^{2}\psi(t,x)}{\partial x^{2}} \right]=0 \tag{1}\]0  
Initial conditions can be used to determine the constants \(A\) and \(B\). Similarly, the general solution of Equation (6) can be expressed as:  
\[ \begin{array}{ll}W(x)=&Z_{1}(\cosh\beta l+\cos\beta l)+Z_{2}(\cos\beta l- \cosh\beta l)\\ &+Z_{3}(\sinh\beta l+\sin\beta l)+Z_{4}(\sin\beta l-\sinh\beta l)\end{array}  \tag{10}\]  
The function \(W(x)\) is known as the characteristic function or normal mode of the beam, where \(Z_{1}\), \(Z_{2}\), \(Z_{3}\) and \(Z_{4}\) are the constants and can be determined based on different  
Figure 2: Geometry of prismatic cantilever beam. boundary conditions. The beam with the CF case is fixed at one end at \(x=0\) and free at the other at \(x=l\); hence, transverse deflection and its slope must be zero at \(x=0\), and the bending moment and shear force must also be zero.