# Vibration Reduction on Circular Disks with Vibroacoustic Metamaterials
## 3 Experimental Characterization
### Rotating State
As expected from measurements in the non-rotating system, no reduction in amplitude occurs within the stop band frequency range, which is indicated by dashed lines at 2000 Hz and 3000 Hz. The underlying frequency response functions of the Campbell plot can be seen for every step in rotational speed in Figure A1 of the Appendix A. The measurement results at measurement point 2 shown in Figure 16 reveal a stop band in the frequency range between 2000 Hz and 3000 Hz (see also Appendix A, Figure A2). This aligns well with the results from the characterization of the circular disks in the non-rotating state. As discussed in Section 2.2, gyroscopic effects and centrifugal forces seem to  
Figure 15: Measured admittance of the circular disk without VAMM (**a**) and with VAMM (**b**) at measurement point 1.  
have either no or a negligible influence on the behavior of the VAMM up to the investigated maximum rotational speed of 3000 rpm, as the measurement results do not indicate a shift of the stop band towards higher frequencies or its complete disappearance. With increasing rotational speed, the reduction in amplitude in the stop band region appears to be less pronounced. This is likely a result of the increased measurement noise at higher rotational speeds. Amplitudes of low magnitude, as seen in the stop band frequency range, disappear within the measurement noise. This effect can be observed well by examining the respective transfer functions for each rotational speed increment (see Appendix A, Figure A2). In the frequency range below the stop band, the increased mode density is clearly visible in the measurement results. This can also be observed in the measurement data of the circular disk with the VAMM in the non-rotating state (see Figure 12). Individual modes adjoining the stop band propagate into the stop band with increasing rotational speed, causing the stop band region to narrow down symmetrically by approx. 200 Hz. This behavior can be explained by the fact that these modes are outside the stop band in the non-rotating state and are therefore not influenced by the VAMM. In the rotating state, these modes split up into a forward and a backward propagating wave, which are not reduced by the VAMM effect since their eigenfrequency has not effectively changed. A similar phenomenon is known from the work of [PERSON] et al. (2017) on bending vibrations on a shaft, where splitting modes propagate into the stop band region. In order to account for the forward and backwards traveling waves, the analytical approach as described in Section 1.1 can be deployed. With this approach, a VAMM can be designed in the non-rotating state and the adjoining modes narrowing down the stop band frequency range can than be predicted by analytical calculations. By this, the limitations in the numerical simulation using commercially available FE software of a spinning non-axisymmetric system like a VAMM integrated into a circular disk can be avoided. As an example, this approach is validated for three modes (frequencies: 81 Hz, 214 Hz and 812 Hz) of the disk with the integrated VAMM at a rotational speed of 3000 rpm. The modes together with their eigenfrequencies and calculated mode-splitting frequencies are presented in Table 1. These are calculated using Equation (3) from the nodal diameters as derived from the mode shapes. Figure 16: Measured admittance of the circular disk without VAMM (**a**) and with VAMM (**b**) at measurement point 2. The considered modes have a different number of nodal diameters, resulting in a different degree of splitting of the eigenfrequencies according to Equation (3). The eigenmodes at 81 Hz and 214 Hz have an equal number of nodal diameters at measurement points 1 and 2, leading to a uniform splitting of the eigenfrequencies. In the case of the third mode at 812 Hz, there is only a splitting at measurement point 1, as there are no nodal diameters in the region of measurement point 2. To validate the calculations, the split eigenfrequencies are plotted into the Campbell diagram as lines (see Figure 17).