## 2 Free Vibration of a Ship Mast  
The dynamic behavior of a ship mast approaches a uniform cross-section and cantilever beam, neglecting rotary inertia and shear effects, and can be described by a second-order differential equation system. In order to achieve this, a fourth-order partial differential equation is initially utilized, incorporating both \(x\) (distance) and \(t\) (time) as independent variables. However, this equation is subsequently reduced to a second-order ordinary differential equation with respect to \(t\). The fourth-order differential equation governing the deflection \(w\)(\(x\),\(t\)) of the beam is provided as follows [26]:  
\[EI\frac{\partial^{4}}{\partial x^{4}}w(x,t)+\rho A\frac{\partial^{2}}{\partial t ^{2}}w(x,t)=q(x,t) \tag{1}\]  
while the vibration of a beam in the case of a non-uniform cross-section is expressed by the equation [26]  
\[\frac{\partial^{2}}{\partial x^{2}}\bigg{(}EI(x)\frac{\partial^{2}w(x,t)}{ \partial x^{2}}\bigg{)}+\rho A(x)\frac{\partial^{2}w(x,t)}{\partial t^{2}}=q (x,t) \tag{2}\]  
where \(E\) represents the Young's modulus of the beam material, \(I\) is the second moment of area of the cross-section, \(\rho\) is the density, \(A\) the cross-sectional area, and \(q\)(\(x\),\(t\)) signifies the force per unit length acting in the \(y\) direction. Considering the free vibration of the beam, where \(q\)(\(x\),\(t\)) is equal to zero, the given partial differential equation can be solved using a separation-of-variables method. \[w(x,t)=Y(x)W(t) \tag{3}\]  
Consequently, the application of this method leads to the formulation of two separate ordinary differential equations. These equations serve as important components in the analysis and subsequent solution of the given problem. \[\frac{d^{2}W(t)}{dt^{2}}+\omega_{n}^{2}W(t)=0 \tag{4}\]  
\[\frac{d^{4}w(x)}{dx^{4}}-\beta^{4}w(x)=0 \tag{5}\]  
where \(\omega_{n}^{2}\) denotes the square of the \(n\)-th natural frequency and \(\beta^{4}=\frac{\omega_{n}^{2}m}{EI}\), assuming in first approximation the non-consideration of damping effects. The natural deflection shapes or modes of the beam are obtained by solving Equation (4) with the relevant boundary conditions and the following general solution:  
\[w(x)=C_{1}\sin\beta x+C_{2}\cos\beta x+C_{3}\sinh\beta x+C_{4}\cosh\beta x \tag{6}\]  
The constants \(C_{1}\), \(C_{2}\), \(C_{3}\), \(C_{4}\), and \(\beta\) in the general solution are determined by imposing the appropriate boundary conditions to the cantilever beam,  
\[w(0)=0:\text{no displacement at the fixed end} \tag{7}\]  
\[\frac{d}{dx}w(x)|_{x=0}=0:\text{no slope at the fixed end} \tag{8}\]\[M(L)=EI\frac{d^{2}}{dx^{2}}w(x)\big{|}_{x=L}=0:\text{no bending moment at the free end} \tag{9}\]  
\[V(L)=EI\frac{d^{3}}{dx^{2}}w(x)\big{|}_{x=L}=0:\text{no shear force at the free end} \tag{10}\]  
with \(L\) the length of the beam, which leads to the expression  
\[\frac{\partial^{2}}{\partial x^{2}}\bigg{(}EI(x)\frac{\partial^{2}w(x,t)}{ \partial x^{2}}\bigg{)}+\rho A(x)\frac{\partial^{2}w(x,t)}{\partial t^{2}}=q (x,t) \tag{2}\]0  
In order to determine the allowable values of \(\beta\), the transcendental equation must be solved numerically. This equation has an infinite number of numerical solutions, each representing a specific mode of vibration.