## 3 Methods
### Proposed Approach  
The proposed approach uses the same arrangement as the standard described above, but the mass-loaded electrodynamic actuator is now directly coupled to the loading slab. The electrodynamic actuator consists of a coiled wire embedded in a uniform magnetic field so that when an electrical current is flowing, a mechanic force is produced in its moving assembly. By measuring its electrical input impedance, it is possible to capture the loading effect on the transducer vibratory frequency response when coupled to other elements or structures. Consequently, instead of using an accelerometer and an impact hammer, the electrodynamic actuator is used both as an impulse exciter and a vibration sensor simultaneously. On the other hand, by using the extra loading mass, the need to glue the actuator to achieve a stable dynamic behavior thereof was avoided. Moreover, the resonance frequencies of the resulting coupled two-degree-of-freedom system (i.e., the slab-material and the mass-loaded actuator) are closer in the frequency spectrum, thus easing the analysis to be explained next. Let us therefore consider the coupled mechanical system in Figure 1c composed of two masses, \(m_{1}\) and \(m_{2}\) (in kg), and two springs with damping, \(r_{1}\) and \(r_{2}\) (in Ns/m), and stiffness, \(s_{1}\) and \(s_{2}\) (in N/m). Such a system can be linked to the proposed measurement system as follows: \(m_{1}\), \(r_{1}\), and \(s_{1}\) correspond to the mass (including the extra load), damping, and suspension of the loaded electrodynamic actuator, respectively, \(m_{2}\) stands for the mass  
Figure 1: Schematic representation of the mechanical systems (left) and pictures of the corresponding experimental setups (right) used to determine the dynamic stiffness of resilient materials from the ISO 9052-1 standard (top) and the proposed approach (bottom). of the loading slab, and \(r_{2}\) and \(s_{2}\) represents the damping and dynamic stiffness of the resilient material under test, respectively (see Figure 1d). From the free vibration analysis of such a two-degree-of-freedom system, the equations of motion of each mass can be written as  
\[m_{1}\ddot{\xi}_{1}+r_{1}\Big{(}\dot{\xi}_{1}-\dot{\xi}_{2}\Big{)}+s_{1}(\xi_{1 }-\xi_{2})=0, \tag{2}\]  
\[m_{2}\ddot{\xi}_{2}+r_{2}\dot{\xi}_{2}+r_{1}\Big{(}\dot{\xi}_{2}-\dot{\xi}_{1} \Big{)}+s_{2}\xi_{2}+s_{1}(\xi_{2}-\xi_{1})=0, \tag{3}\]  
where \(\xi_{1}\) and \(\xi_{2}\) are the harmonic displacements (in m) from the equilibrium of masses \(m_{1}\) and \(m_{2}\), respectively, whereas \({}^{\bullet}\) and \({}^{\bullet\bullet}\) stand for the first- and second-time derivatives. The above linear differential equations can be written in terms of a matrix product as  
\[ \begin{bmatrix}-m_{1}\omega^{2}+j\omega r_{1}+s_{1}&-j\omega r_{1}-s_{1}\\ -j\omega r_{1}-s_{1}&-m_{2}\omega^{2}+j\omega(r_{1}+r_{2})+(s_{1}+s_{2})\end{bmatrix} \begin{bmatrix}\xi_{1}\\ \xi_{2}\end{bmatrix} = \begin{bmatrix}0\\ 0\end{bmatrix} , \tag{4}\]  
where \(\omega\) is the angular frequency (in rad/s) resulting from the time derivatives. The eigenvalues of that matrix system can be obtained for those values of \(\omega^{2}\) for which the determinant of the matrix is equal to zero from  
\[\Big{(}-m_{1}\omega_{1,2}^{2}+j\omega r_{1}+s_{1}\Big{)}\Big{(}-m_{2}\omega_{1,2}^{2}+j\omega(r_{1}+r_{2})+(s_{1}+s_{2})\Big{)}-(j\omega r_{1}+s_{1})^{2}=0 \tag{5}\]  
The frequencies corresponding to these eigenvalues (i.e., the natural frequencies of vibration, \(\omega_{1,2}=2\pi f_{1,2}\)) correspond to the solutions of the so-called characteristic equation (Equation (5)). These frequencies can be easily obtained by identifying the resonance peak frequencies \(f_{1}\) and \(f_{2}\) (in Hz) on the measured electrical input impedance spectrum of the actuator when an electrical voltage is applied to it (see [12] for a very similar example of the measurement procedure).