## 2 Problem Statement  
Let us consider the transverse vibrations of a beam with constant bending stiffness \(EI\) and mass \(\mu\) along span \(L\), as shown in Figure 1. It is assumed that the effects of rotary inertia of the cross-section and shear deformation on bending are negligible in the vibration analysis. Furthermore, taking into account the assumption that the energy dissipation linearly depends on the mass and stiffness of the structure, the partial differential equation for the considered problem of the transverse vibrations of the beam can be defined as follows [43; 44]:  
\[\mu\frac{\partial^{2}\
u(x,t)}{\partial t^{2}}+2\beta\mu\frac{\partial\
u(x,t)}{ \partial t}+Ela\frac{\partial}{\partial t}\left(\frac{\partial^{4}\
u(x,t)}{ \partial x^{4}}\right)+El\frac{\partial^{4}\
u(x,t)}{\partial x^{4}}=p(x,t). \tag{1}\]  
In Equation (1), \(\
u(x,t)\) represents the deflection at point \(x\) and time \(t\). The load on the bridge, whether stationary or moving at a constant speed (\(c=\text{const. }\)), is described by the force \(p(x,t)\), which, without considering its inertial effect, is defined as a deterministic disturbance force. The second and third terms in Equation (1) represent damping forces, where the damping constant \(\beta\) expresses the proportionality of the damping force with respect to the mass \(\mu\) and constant \(\alpha\) expresses that with the stiffness \(El\) of the system. Using modal analysis and considering the orthogonality of eigenfunctions, Equation (1) can be reduced to a system of independent equations of the following form:  
\[\ddot{\eta}_{r}(t)+2\zeta_{r}\omega_{r}\dot{\eta}_{r}(t)+\omega_{r}^{2}\eta_{r }(t)=\frac{1}{M_{r}}F_{r}(t), \tag{2}\]  
where \(\eta_{r}(t)\) denotes a principal coordinate of the \(r\)-th mode, \(\omega_{r}\) is the natural circular frequency of the \(r\)-th mode of the system, and \(\zeta_{r}\) represents the relative damping of the \(r\)-th mode. In Equation (2) \(M_{r}\) stands for the generalized mass of the \(r\)-th mode of the system, which is defined as follows:  
\[M_{r}=\mu\int_{0}^{L}V_{r}^{2}(x)dx, \tag{3}\]  
and \(F_{r}\) is the generalized force of the \(r\)-th mode:  
\[F_{r}(t)=\int_{0}^{L}p(x,t)V_{r}(x)dx. \tag{4}\]  
Applying the Laplace transform to Equation (2), the most general expression for the dynamic deflection of a beam, as a solution of Equation (1), based on which various specific cases can be analyzed, results in the following [43]:  
\[ \begin{array}{ll}\upsilon(x,t)=\sum\limits_{r=1}^{\infty}\frac{V_{r}(x)}{M_ {r}}&\int_{0}^{t}\Bigl{[}\int_{0}^{L}p(u,\tau)V_{r}(u)du\Bigr{]}\frac{sin \omega_{dr}(t-\tau)}{\omega_{dr}}e^{-\zeta_{r}\omega_{r}(t-\tau)}d\tau\\ &+\sum\limits_{r=1}^{\infty}\frac{V_{r}(x)}{\int_{0}^{L}V_{r}^{2}(x)dx} \Bigl{[}cos\omega_{dr}t\int_{0}^{L}v_{0r}(u)V_{r}(u)du\\ &+sin\omega_{dr}t\int_{0}^{L}\frac{\zeta_{r}\omega_{r}v_{0r}(u)+\dot{v}_{0r}(u )}{\omega_{dr}}V_{r}(u)du\Bigr{]}e^{-\zeta_{r}\omega_{r}t},\end{array}  \tag{5}\]  
where \(V_{r}(x)\) denotes the eigenfunction of the \(r\)-th mode, \(v_{0r}(x)=v_{r}(x,0)\) and \(\dot{v}_{0r}(x)=\dot{v}_{r}(x,0)\) are the initial conditions, \(\tau\) is the time variable of force \(p\), \(u\) is the position variable, and \(\omega_{dr}\) is the circular frequency of free damped vibrations of the \(r\)-th mode of the system, which is defined as follows:  
Figure 1: Dynamic model for transverse vibrations of a continuous beam system. \[\omega_{dr}(t)=\omega_{r}\sqrt{1-\xi_{r}^{2}}. \tag{6}\]  
The effects of additional load mass on the free transverse vibrations of an elastic oscillatory system (Figure 2) with constant bending stiffness \(EI\) and mass \(\mu\) along span \(L\) are the main focus of this research. The constant mass \(\mu\), per unit length of the beam, is increased by an additional uniformly distributed mass \(\mu_{q}\), per unit length \(\lambda\), located at the position defined by the abscissa \(x_{0}\). It is assumed that the contact between the beam and the additional masses is continuously maintained.