## 2 Problem Statement
Such a configuration of additional masses to a beam structure is most commonly encountered in pedestrian bridges, where pedestrians either stand still or move at a speed represented by \(c\), with \(x_{0}=ct\). The problems of undamped vibrations, starting from Equation (1), and neglecting the damping forces (second and third terms in Equation (1)), can be described by the following differential equation:  
\[\mu\frac{\partial^{2}\
u(x,t)}{\partial t^{2}}+EI\frac{\partial^{4}\
u(x,t)}{ \partial x^{4}}=p(x,t). \tag{7}\]  
Considering the inertial effect of the additional mass \(\mu_{q}\) as the force \(p(x,t)\), whose position is defined by the Heaviside function:  
\[p(x,t)=-\mu_{q}\frac{\partial^{2}\
u(x,t)}{\partial t^{2}}\bigg{[}H\bigg{(}x- x_{0}+\frac{\lambda}{2}\bigg{)}-H\bigg{(}x-x_{0}-\frac{\lambda}{2}\bigg{)} \bigg{]}, \tag{8}\]  
one can obtain differential equation of undamped free transverse vibrations of a beam with an additional load mass [44]:  
\[\mu\frac{\partial^{2}\
u(x,t)}{\partial t^{2}}+EI\frac{\partial^{4}\
u(x,t)}{ \partial x^{4}}=-\mu_{q}\frac{\partial^{2}\
u(x,t)}{\partial t^{2}}\bigg{[}H \bigg{(}x-x_{0}+\frac{\lambda}{2}\bigg{)}-H\bigg{(}x-x_{0}-\frac{\lambda}{2} \bigg{)}\bigg{]}, \tag{9}\]  
with boundary conditions dependent on the type of support of the beam system, as well as initial conditions:  
\[v(x,0)=v_{0}(x);\;\dot{v}(x,0)=\dot{v}_{0}(x). \tag{10}\]  
Employing modal analysis and introducing an approximation for the inertial effect of the additional masses as following:  
\[\mu_{q}\frac{\partial^{2}\
u(x,t)}{\partial t^{2}}\cong\mu_{q}\ddot{\eta}_{r} (t)V_{r}(x), \tag{11}\]  
differential Equation (9) is reduced to a system of \(r\) independent equations, which can be symbolically represented in the following form:  
\[\mu\ddot{\eta_{r}}(t)V_{r}(x)+EI\eta_{r}(t)\frac{d^{4}V_{r}(x)}{dx^{4}}+\mu_{ q}\ddot{\eta_{r}}(t)V_{r}(x)\bigg{[}H\bigg{(}x-x_{0}+\frac{\lambda}{2}\bigg{)}-H \bigg{(}x-x_{0}-\frac{\lambda}{2}\bigg{)}\bigg{]}=0, \tag{12}\]  
Figure 2: Dynamic model of a continuous beam system with an additional mass \(\mu_{q}\). \[\ddot{\eta_{r}}(t)\left[\mu\int_{0}^{L}{V_{r}}^{2}(x)+\mu_{q}\int_{0}^{L}{V_{r}}^{ 2}(x)\left[H\left(x-x_{0}+\frac{\lambda}{2}\right)-H\left(x-x_{0}-\frac{\lambda} {2}\right)\right]\right] \tag{13}\]  
\[\omega_{dr}(t)=\omega_{r}\sqrt{1-\xi_{r}^{2}}. \tag{6}\]0  
\[\omega_{dr}(t)=\omega_{r}\sqrt{1-\xi_{r}^{2}}. \tag{6}\]1  
where \(\eta_{r}(t)\) denotes a principal coordinate of the \(r\)-th mode and \(\overline{\omega}_{r}\) is the corresponding circular frequency of the \(r\)-th mode of the system with the additional mass:  
\[\omega_{dr}(t)=\omega_{r}\sqrt{1-\xi_{r}^{2}}. \tag{6}\]2  
\[\omega_{dr}(t)=\omega_{r}\sqrt{1-\xi_{r}^{2}}. \tag{6}\]3  
For \(\overline{\omega}_{r}\) in Equation (16), Relation (17) applies as follows:  
\[\omega_{dr}(t)=\omega_{r}\sqrt{1-\xi_{r}^{2}}. \tag{6}\]4  
where:  
\[\omega_{dr}(t)=\omega_{r}\sqrt{1-\xi_{r}^{2}}. \tag{6}\]5  
It should be noted that \(M_{r}\) in Relation (18) represents the generalized mass of the \(r\)-th mode of the system without the additional mass \(\mu_{q}\). Since the contribution of higher modes is typically of lesser significance to the overall vibrations, it is evident that Relation (17) is acceptable for the fundamental mode (\(r=1\)), and for several subsequent modes where deviations become more pronounced, particularly for higher modes when the ratio of additional mass to bridge mass is larger. In the case of pedestrian bridges, it is practically important to understand the change in the frequency of the fundamental mode due to the pedestrian or service vehicle mass, which Relation (17) satisfies, specifically for \(r=1\).