# Vibration Reduction on Circular Disks with Vibroacoustic Metamaterials
## 3 Experimental Characterization
### Rotating State  
To characterize the circular disks in the rotating state, measurements are conducted at two measurement points with radial distances of \(r_{\text{m,i}}\) = 49 mm and \(r_{\text{m,o}}\) = 137 mm (see Figure 14). These radii were chosen to ensure that the measurement points are not interrupted by the slots of the resonators during the rotation of the disk. Measurement point 2 (MP2) is rotated by 45\({}^{\circ}\) relative to measurement point 1 (MP1). This is because at this point, a minimum in measurement noise was present. This is likely due to a non-perfect concentric and orthogonal alignment of the LDV with respect to the axis of rotation of the circular disk. Since the measurements are comparative between the circular disk with and without VAMM, the chosen arrangement of the measurement points does not impose any restriction on the significance of the results. The experimental setup developed in Section 3.1 is used for the measurement. The surface velocities excited by the impulse hammer are measured in rotational speed increments of 300 rpm ranging from 0 rpm up to 3000 rpm, which is the maximum reachable speed of the disks in the test bench. Due to imbalances, axial vibrations of the motor axis and the surface roughness of the circular disk sprayed with chalk spray, there is a high level of measurement noise. Because of this, 100 averages per measurement were conducted. In addition to the noise, multiples of the rotational frequency are visible in the measurement signal, which overlay the structural response. For this reason, the transfer function \(H_{1}\) was used for evaluating the measurement data, which is commonly employed in measurement technology when the output measurement quantity is superimposed by noise [35]. \(H_{1}\) results from the quotient of the cross-power spectrum \(G_{\text{XY}}(jw)\) and the auto-power spectrum \(G_{\text{XX}}(w)\) of the input signal [35] (see Equation (5)). Figure 14: Measurement points (MPs, red) on the circular disk without (**a**) and with VAMM (**b**). Figure 13: Modes of the circular disk with VAMM at (**a**) 2452 Hz, (**b**) 2710 Hz and (**c**) 2794 Hz. \[H_{1}\left(jw\right)=\frac{G_{\text{XY}}(jw)}{G_{\text{XX}}(w)} \tag{5}\]  
To be able to investigate the behavior of the rotating disk above the stop band and in order to compare it with the obtained behavior in the non-rotating state, measurements are carried out up to a frequency of 5000 Hz, which requires a sampling frequency of 12,800 Hz. In total, 8192 sampling points are set, 3200 of which can be used. As a result, the frequency resolution is 1.5625 Hz with a measurement time of 0.64 s. The frequency resolution was chosen to differentiate resonances that split as a result of rotation and move towards each other. All rotational speed increments were measured with the same measurement setting, making them comparable in terms of amplitude values. Finding a setting to measure all rotational speed increments is challenging because noise amplitudes increase with increasing rotational speed. At higher rotational speeds, the disk's decaying vibration amplitudes are therefore masked earlier by the noise amplitudes than at lower rotational speeds. Therefore, within one measurement interval, a measurement at high rotational speeds contains less information about the disk's structural dynamic behavior. The settings described above represent a good compromise between measurement time and frequency resolution for the given excitation by the impulse hammer at the different rotational speeds. Figure 15 shows the results of the measurements at measurement point 1 (\(r_{\text{m,o}}\) = 137 mm) as a Campbell plot. Here, the transfer function in terms of admittance is plotted over the rotational speed and the frequency. Clearly visible is the phenomenon of splitting modes, known from Section 1.1.2, that occurs with increasing rotational speed.