## 2 Theoretical Insight
### Bending Behavior of Beam with Mass and Spring at Midspan  
The model proposed in this section is based on the bending behavior of a beam of total length \(l=2L\), with added mass and spring foundation located at \(L\). Due to the curved nature of the beams, a preliminary investigation is proposed to identify the model to be used. Let us consider the curved beam as an assembly of two noncollinear beams connected with an angle \(\theta=\arctan(2Q_{0}/L)\), as illustrated in Figure 4, each beam experiencing coupled flexural and longitudinal motions. The longitudinal motion is associated with an axial displacement \(v_{i}\), and axial effort \(F_{i}\), and the bending motion is associated with a transverse displacement \(u_{i}\), section rotation \(u^{\prime}_{i}\), bending moment \(M_{i}\), and shear effort \(V_{i}\), where \(i=1,2\) corresponds to the two segments. The continuity conditions at the connecting point are  
\[v_{1} =v_{2}\cos(\theta)-u_{2}\sin(\theta) \tag{6}\] \[F_{1} =F_{2}\cos(\theta)-V_{2}\sin(\theta)\] (7) \[u_{1} =u_{2}\cos(\theta)+v_{2}\sin(\theta)\] (8) \[u^{\prime}_{1} =u^{\prime}_{2}\] (9) \[M_{1} =M_{2}\] (10) \[V_{1} =F_{2}\sin(\theta)+V_{2}\cos(\theta) \tag{11}\]  
The solution for the bending motion of the \(i\)-th segment is  
\[u_{i}(x,\omega)=C_{1,i}\sin(kx)+C_{2,i}\cos(kx)+C_{3,i}\sinh(kx)+C_{4,i}\cosh(kx) \tag{12}\]  
with \(k\) is the bending wavenumber \(k^{4}=\omega^{2}\rho S/EI\), and the one for the longitudinal motion is  
\[v_{i}(x,\omega)=C_{5,i}\sin(\lambda x)+C_{6,i}\cos(\lambda x) \tag{13}\]  
with \(\lambda\) is the compression wavenumber \(\lambda^{2}=\omega^{2}\rho/E\). Figure 4: Schematic overview of the curved beam approximation and associated notations, for the single curved beam, force projection, and double beam configuration. In order to verify the consistency of this modeling, we consider a straight beam two-times thicker than the curved beam, a pair of curved beams, and a pair of noncollinear beams, together with clamped-free boundary conditions. Stating \(v=u=u^{\prime}=0\), at \(x=0\) and \(v^{\prime}=u^{\prime\prime}=u^{\prime\prime\prime}=0\) at \(x=L\) provides an analytical benchmark case. In such conditions, the first cantilever mode of the straight beam is 455 Hz, the one associated with the pair of curved beams is calculated using a finite element model and yields 450 Hz. The first mode of the pair of noncollinear straight beams is calculated by solving the transcendental equation that is obtained by equating to zero the determinant of the 6 \(\times\) 6 matrix system formed by the six continuity equations. This leads to 453 Hz and is confirmed by the finite element model. This modeling is correct for small values of \(\theta\); otherwise, the length of each noncollinear beam is \(l_{b}=L(2-cos(\theta))/2\). From this macroscopic analysis, the equivalent beam associated to a pair of curved beam is two-times thicker than a single curved beam and can be described by the Euler-Bernoulli beam equation:  
\[EI\frac{\partial^{4}u}{\partial x^{4}}=\rho S\omega^{2}u \tag{14}\]  
from which the solution for the transverse displacement is  
\[F_{1} =F_{2}\cos(\theta)-V_{2}\sin(\theta)\]0  
with \(k\) as the bending wavenumber \(k^{4}=\omega^{2}\rho S/EI\). In the following, the added mass is denoted \(m\), and the stiffness of the vertical spring associated with positive stiffness is denoted \(k_{p}\). Thanks to the symmetry, one half of the beam is considered through a set of four boundary conditions, two at the extremity (supported or clamped), and two at the mid-length, accounting for mass and spring. Specifically, it is assumed that the mass can only move vertically and that the motion of curved beams is symmetric on both sides of it. This point does not experience rotation and is subjected to (i) the difference in shear forces on both sides of the mass, which is related to the mass and its acceleration as \(2 EIu^{\prime\prime\prime}=-m\omega^{2}u\), and (ii) the restoring force from the spring as \(2 EIu^{\prime\prime\prime}=-k_{p}u\).