# Multiscale Modeling and Optimization of Aluminum Foam Material Properties Under Dynamic Load
## 2 Materials and Methods
### Methods  
#### 2.3.1 Vibration Test  
An essential phase of this study focused on assessing the dynamic behavior of the tested material by identifying its resonance frequencies within the 10-2000 Hz range. This analysis provides valuable insights into the mechanical response under dynamic conditions, including the measurement of resonance frequencies, amplitude variations, and the quality factor. The results obtained will contribute to the calculation of the material's modal damping coefficients, essential for understanding its vibrational properties. To conduct these tests, an LDS V406 (Bruel & Kjaer, Naerum, Denmark) 196 N electromechanical actuator was used. As illustrated in Figure 2(a), this device efficiently converts electrical energy into mechanical energy, generating a consistent sinusoidal vibration profile. Figure 2: (**a**) Analysis of the internal structure for the B and D density classes of specimens, through CT scan; (**b**) resulting parameters of the measurement. Designed for high-performance applications, the LDS V406 actuator operates within a frequency range of 5 to 9000 Hz, with a nominal peak force of 98 N for sinusoidal waves. However, when combined with an amplifier and cooled via an external fan, its output force increases to 196 N, making it highly effective for precision testing. Furthermore, the actuator features a 38 mm armature diameter and a lightweight moving armature, ensuring stable support for test loads without compromising force capacity. To enhance stability and flexibility, two fiber-laminated flexures are bonded to the armature, providing axial support and improving lateral and rotational degrees of freedom. By leveraging these advanced testing techniques, the study aims to deepen the understanding of the material's mechanical performance under dynamic conditions, paving the way for improved material design and application in engineering fields. For the vibration measurements, a Polytec CLV-3D Compact 3-D, Polytec GmbH, Waldbronn, Germany system, laser vibrometer was used, as shown in Figure 3b. This system operates based on the Doppler effect, detecting frequency shifts in the laser beam reflected from a moving object. The CLV-3D system includes three laser beams, each inclined at 12\({}^{\circ}\) relative to the surface (when using the 160 mm focal length lens) and spaced 120\({}^{\circ}\) apart in the output plane. These beams converge at a single measurement point located precisely 160 mm from the front of the sensor head. To ensure accurate measurements, the sensor must be positioned at this exact distance from the test specimen. Because laser vibrometers require a reflective surface for optimal signal return, small reflective tape markers were placed on the specimens. The system captures vibration data in all three spatial directions (x, y, and z) at the focal point. Figure 4 shows the full setup used during the test procedure. Although the CLV-3D system does not include an automatic calibration routine based on the size or position of the reflective patch, correct placement at the focal point was manually verified. During testing, a 1 g sine excitation was applied via the shaker, and the system's response was cross-checked to ensure signal consistency and measurement reliability. While this procedure does not constitute a formal calibration, it confirmed that the reflective patch was properly positioned and that the recorded vibration data were accurate. To assess the energy dissipation characteristics of the materials, the structural damping coefficient \(\xi\), was also evaluated as part of the vibration test procedure. The structural damping coefficient was calculated using the half-power bandwidth method, based on classical vibration theory [41]. The damping was derived from the resonance behavior of each specimen, which was excited under a broadband sine sweep profile ranging from 10 Hz to 2000 Hz, with a constant acceleration of 1 g and a linear sweep rate of 1 Hz/s. The  
Figure 3: (**a**) LDS V406 electromechanical actuator; (**b**) laser vibrometer.