## 1 Introduction
### The Analytical Solution  
We obtain the solution to Equation (5) by employing the Fourier transform, which is defined as follows:  
Figure 1: The diagrammatic representation of an infinite E–B beam on a foundation subjected to a moving load. \[ \begin{array}{rcl}\hat{w}(\xi)&=&\int_{-\infty}^{+\infty}w(y)e^{-i\xi y}dy,\\ w(y)&=&\frac{1}{2\pi}\int_{-\infty}^{+\infty}\hat{w}(\xi)e^{i\xi y}d\xi.\end{array}  \tag{6}\]  
By applying the Fourier transform to Equation (5), we can derive the following result:  
\[\hat{w}=\int_{-\infty}^{+\infty}w(y)e^{-i\xi y}dy=\frac{\beta_{4}}{\xi^{4}- \beta_{1}\xi^{2}+i\beta_{2}\xi+\beta_{3}}. \tag{7}\]  
To apply the residue theorem for calculating the inverse Fourier transform of the above equation, the root configuration of the denominator in Equation (7) must be determined. Let \(f(\xi)\) denote the denominator in Equation (7), and \(f_{d}(\xi)\) denote its derivative, as follows:  
\[ \begin{array}{l}f(\xi)=\xi^{4}-\beta_{1}\xi^{2}+i\,\beta_{2}\xi+\beta_{3}, \\ f_{d}(\xi)=4\xi^{3}-2\beta_{1}\xi+i\beta_{2}.\end{array}  \tag{8}\]  
To simplify the analysis, substituting \(\xi=i\kappa\) into \(f(\xi)\) gives  
\[g_{1}(\kappa)=f(i\kappa)=\kappa^{4}+\beta_{1}\kappa^{2}-\beta_{2}\kappa+\beta _{3}. \tag{9}\]  
All coefficients of \(g_{1}(\kappa)\) are real. If \(g_{1}(\kappa)\) has a pair of complex roots \(M_{1}\pm iM_{2}\), then \(\mp M_{2}+iM_{1}\) must be two complex roots of \(f(\xi)\). To further simplify the analysis for Equation (9), substituting \(\kappa=\sqrt{\beta_{1}}\mu\) into Equation (9) transforms it into  
\[g_{2}(\mu)=g_{1}(\sqrt{\beta_{1}}\mu)=\beta_{1}^{2}(\mu^{4}+\mu^{2}+\gamma_{1} \mu+\gamma_{2})\equiv\beta_{1}^{2}g_{3}(\mu), \tag{10}\]  
where \(\gamma_{1}=-\beta_{2}\beta_{1}^{-\frac{3}{2}}\), \(\gamma_{2}=\beta_{3}\beta_{1}^{-2}\). Differentiating \(g_{3}(\mu)\) with respect to \(\mu\) yields  
\[ \begin{array}{l}g_{41}\equiv\frac{dg_{3}(\mu)}{d\mu}=4(\mu^{3}+\frac{1}{2} \mu+\frac{\gamma_{1}}{4}),\\ g_{42}\equiv\frac{d^{2}g_{3}(\mu)}{d\mu^{2}}=12\mu^{2}+2.\end{array}  \tag{11}\]  
From [PERSON]'s formula, the discriminant of the polynomial \(g_{41}\) can be determined as  
\[\Delta=\Big{(}\frac{\gamma_{1}}{8}\Big{)}^{2}+\Big{(}\frac{1}{6}\Big{)}^{3}.\]  
Evidently, \(\Delta>0\) for any \(\gamma_{1}\), so the polynomial \(g_{41}\) has one real root and a pair of conjugate complex roots. Suppose \(\mu=\mu_{0}\) is the unique real root of \(g_{41}\), as follows:  
\[g_{41}(\mu_{0})=\mu_{0}^{3}+\frac{1}{2}\mu_{0}+\frac{\gamma_{1}}{4}=0. \tag{12}\]  
\(\mu_{0}\) can be explicitly written as  
\[\mu_{0}=-\frac{1}{6}\left(e_{1}(\gamma_{1})+e_{2}(\gamma_{1})\right). \tag{13}\]  
where \(e_{1}(\gamma_{1})=\sqrt[3]{27\,\gamma_{1}+3\,\sqrt{81\,\gamma_{1}^{2}+24}}\), \(e_{2}(\gamma_{1})=\sqrt[3]{27\,\gamma_{1}-3\,\sqrt{81\,\gamma_{1}^{2}+24}}\). It is straightforward to verify that \(\mu_{0}<0\) from the equation above. Given that \(g_{42}>0\) for any \(\mu\) (as per the second equation in Equation (11)), when \(\mu<\mu_{0}\), \(g_{41}<0\), whereas when \(\mu>\mu_{0}\), \(g_{41}>0\). Consequently, the function \(g_{3}(\mu)\) is strictly increasing (or decreasing) for \(\mu>\mu_{0}\) (or \(\mu<\mu_{0}\)). Moreover, \(g_{3}(\mu)\) attains its minimum value at \(\mu=\mu_{0}\). \[g_{\mu_{0}}\equiv g_{3}(\mu_{0})=\mu_{0}^{4}+\mu_{0}^{2}+\gamma_{1}\mu_{0}+ \gamma_{2}. \tag{14}\]Substituting Equation (13) into Equation (14), one has  
\[g_{\mu_{0}}=(-\frac{1}{2}\mu_{0}^{2}-\frac{1}{4}\gamma_{1}\mu_{0})+\mu_{0}^{2}+ \gamma_{1}\mu_{0}+\gamma_{2}=\frac{1}{2}\mu_{0}^{2}+\frac{3}{4}\gamma_{1}\mu_{0 }+\gamma_{2}. \tag{15}\]  
The root configurations of the polynomial \(g_{3}(\mu)\) can be expressed as follows:  
\[g_{3}(\mu)=\left\{ \begin{array}{ll}[\mu-(d_{1}+id_{2})][\mu-(d_{1}-id_{2})]( \mu-\mu_{1})(\mu-\mu_{2}),&g_{\mu_{0}}<0,\\ [\mu-(d_{3}+id_{4})][\mu-(d_{3}-id_{4})](\mu-\mu_{0})^{2},&g_{\mu_{0}}=0,\\ [\mu-(d_{5}+id_{6})][\mu-(d_{5}-id_{6})][\mu-(d_{7}+id_{8})][\mu-(d_{7}-id_{8}) ],&g_{\mu_{0}}>0,\end{array} \right.