## 2 Theoretical Insight
### Bending Behavior of Beam with Mass and Spring at Midspan
A dimensionless parameter \(\gamma=k_{p}/m\omega^{2}\) is introduced to describe the relative importance of the elastic effect compared to inertia. In case the beam is simply supported at both ends, the solution form simplifies as \(\phi_{\omega}(x)=C_{1}\sin(kx)+C_{3}\sinh(kx)\), and the equation of motion together with the zero-slope condition give the following system:  
\[ \begin{cases}2 EIk^{3}[-C_{1}\cos(k^{*})+C_{3}\cosh(k^{*})]\\ \quad=-(m\omega^{2}+k_{p})[C_{1}\sin(k^{*})+C_{3}\sinh(k^{*})],\\ k[C_{1}\cos(k^{*})+C_{3}\cosh(k^{*})]=0\end{cases}  \tag{16}\]  
Solving these equations gives the frequency equation, and rearranging with respect to mass ratio \(m_{b}/m\), it follows that  
\[\frac{m_{b}}{m}=\frac{k^{*}}{2}\left(\tan k^{*}-\tanh k^{*}\right)\left(1+\gamma\right) \tag{17}\]  
with \(m_{b}\) as the weight of the beam, \(k^{*}\) as the dimensionless wavenumber defined as \(kl/2\). The first solutions are illustrated in Figure 5. Expression (17) contains the frequency equation of a beam with support-slide boundary conditions, as well as a corrective term related to the added mass and supporting spring. It shows the rapid decrease in the first root as the added mass increases. As expected, it also shows the shifting of the first root to higher values as the elastic contribution prevails over the mass effects. In case the beam is clamped at one end, the solution form simplifies as \(u(x,\omega)=C_{1}(\sin(kx)-\sinh(kx))+C_{2}(\cos(kx)-\cosh(kx))\), and the equation of motion together with the zero-slope condition give the following system:  
\[ \begin{cases}2 EIk^{3}[C_{1}(-\cos(k^{*})-\cosh(k^{*}))+C_{2}(\sin(k^{*})-\sinh( k^{*}))]\\ \qquad=-(m\omega^{2}+k_{p})[C_{1}(\sin(k^{*})-\sinh(k^{*}))+C_{2}(\cos(k^{*})- \cos(k^{*})))],\\ k[C_{1}(\cos(k^{*})+\cosh(k^{*}))+C_{2}(-\sin(k^{*})-\sinh(k^{*}))]=0\end{cases}  \tag{18}\]  
Solving these equations gives the frequency equation, and rearranging with respect to the mass ratio \(m_{b}/m\), it follows that  
\[\frac{m_{b}}{m}=k^{*}\bigg{(}\frac{1-\cosh k^{*}\cos k^{*}}{\sin k^{*}\cosh k^ {*}+\sinh k^{*}\cos k^{*}}\bigg{)}(1+\gamma) \tag{19}\]  
The first solutions are illustrated in Figure 6. Expression (19) contains the frequency equation of a fixed beam \(1-\cosh k^{*}\cos k^{*}=0\), as well as a term related to the added mass and supporting spring. A comparison of the roots calculated analytically and by a two-dimensional finite element model (by making the frequencies dimensionless with respect to the characteristics of the beam) is presented in Table 2 to show the relevance of the formulation. The finite element model, as used in Section 3.3, is a beam element clamped or supported at both ends. Spring foundation and added mass are specified at mid-length. An eigenfrequency  
Figure 5: Graphical representation of the roots \(k^{*}\) of simply supported beam (17) (**a**) with respect to \(m_{b}/m\) with \(k_{p}=0\), (**b**) with respect to \(\gamma\) with \(m_{b}/m=1\). Figure 6: Graphical representation of the roots \(k^{*}\) of clamped beam (19) (**a**) with respect to \(m_{b}/m\) with \(k_{p}=0\), (**b**) with respect to \(\gamma\) with \(m_{b}/m=1\). study is performed that leads to the characteristic frequencies of the system. Considering the mechanical properties and geometrical parameters, the roots of the characteristic transcendental equation can be obtained. Let us define \(\mu=m/m_{b}\) as the mass ratio between the middle mass and the mass of the beam, and \(\kappa=k_{p}/k_{b}\) the stiffness ratio between the linear spring and the equivalent bending stiffness of the beam. A preliminary parametric study is presented in the following. It allows \(\kappa\) and \(\mu\) to vary respectively within the range [0; 10]. The resonance frequency of the obtained spring-mass system is represented versus these two parameters in Figure 7. Additionally, the fundamental frequency of the clamped beam with mass and spring foundation at mid-length is evaluated from the analytical expression presented above.