Vibration Reduction on Circular Disks with Vibroacoustic Metamaterials

Abstract

Vibroacoustic metamaterials represent an innovative technology developed for broadband vibration reduction. They consist of an array of local resonators and are able to reduce vibrations over a wide frequency range, commonly referred to as a stop band. Vibroacoustic metamaterials may be a promising strategy to reduce out-of-plane vibrations of thin-walled, disk-shaped structures, such as saw blades. However, their behavior in rotating systems has not yet been fully understood. In this study, a vibroacoustic metamaterial integrated into a circular disk for the reduction of out-of-plane vibrations is experimentally investigated in the rotating and non-rotating state. Derived from the predominant frequency range of noise emitted by saw blades, a vibroacoustic metamaterial with a numerically predicted stop band in the frequency range from 2000 Hz to 3000 Hz, suitable for integration into a circular disk, is designed. The resonators of the metamaterial are realized by cutting slots into the disk using a waterjet cutting machine. To experimentally examine the structural dynamic behavior, the disk is excited by an impulse hammer and observed by a stationary optical velocity sensor on a rotor dynamics test stand. The results of the rotating and the non-rotating state are compared. The measurements are carried out at two different radii and at speeds up to 3000 rpm. A distinct stop band characteristic is shown in the desired frequency range from 2000 Hz to 3000 Hz in the rotating and non-rotating state. No significant shift of the stop band frequency range was observed during rotation. However, adjacent modes were observed to propagate into the stop band frequency range. This work contributes to a better understanding of the behavior of vibroacoustic metamaterials in the rotating state and enables future applications of vibroacoustic metamaterials for vibration reduction in rotating, disk-shaped structures such as saw blades, brake disks or gears.

1. Introduction

Metamaterials are artificial materials that are rationally designed and feature effective properties that are not found in nature [1]. They exist in different physical domains such as optics, electromagnetism, mechanics and acoustics [1]. The emergence of metamaterials roots back to Victor Veselago who laid the foundations for understanding metamaterials in 1968 with his theoretical work on materials with negative electrical permittivity and magnetic permeability [2]. Between 1996 and 1999, John B. Pendry et al. [3,4] succeeded in experimentally investigating the metamaterials as described by Veselago and confirming their special properties. Within the field of metamaterials, vibroacoustic metamaterials (VAMMs) with local resonators are an effective approach for vibration and noise reduction. A VAMM consists of periodically arranged resonators that are applied to the target structure. The resonators are all tuned to the same resonance frequency and are placed at intervals smaller than half the wavelength of the frequency that needs to be reduced. The vibrational behavior of these periodically arranged resonators results in a negative effective mass in a certain frequency range [5] for the overall structure. Within this frequency range, a stop band for elastic waves occurs, where no free wave propagation is possible. A schematic VAMM with an exemplary stop band can be seen in Figure 1.

While there are a large variety of works that have focused on the application of VAMMs on non-rotating structures, for example, in aviation and space [6,7], the automotive industry [8] and machinery [9,10], there is limited information regarding their behavior in the rotating state, though thin-walled rotors often face severe vibration problems. For example, the sound emitted by a circular saw blade predominately goes back to the vibrations of the saw blade [11] which are excited by the sawing process. Thin-walled lightweight gears, rotating at large rotational speeds, become excited by the engagement of the teeth. This can lead to extensive gear noise being emitted by the gear body or the housing which is excited by vibrations traveling over the shafts. In addition, there are multiple other examples of rotating structures facing vibration issues, like brake disks, fans in turbo machines and compressors. In rotating structures, centrifugal and gyroscopic forces act on the resonators of the VAMM, and when observed from a stationary observer, modes split into a forward and backward propagating wave. These special effects have to be investigated for a VAMM in the rotating state.

Figure 1. (a) Schematic of a VAMM, (b) schematically depicted stop band in the frequency domain. Source: compare [12].

Figure 1. (a) Schematic of a VAMM, (b) schematically depicted stop band in the frequency domain. Source: compare [12].

While VAMMs have been investigated for the reduction of torsional waves [13,14,15] and bending waves [16,17,18,19] of spinning shafts, there is just a limited amount of works focusing on the reduction of elastic waves in thin-walled rotors. Rosso et al. [20] numerically investigated a so-called metastructure on a thin-walled gear, where the spring element of the resonators is machined out of the gear. However, the investigation was not carried out in the rotating state. Rosso et al. suggested investigating the effects of gyroscopic forces that act on the resonators. In an experimental work by Rieß et al. [12], the authors studied the behavior of a VAMM integrated into a circular saw blade. The spinning saw blade was excited by an impulse hammer and the resulting vibrations were measured with a stationary laser Doppler vibrometer (LDV). The stop band characteristic was validated for a rotational speed of up to 1000 $\mathrm{rpm}$. The work was limited by the signal-to-noise-ratio of the measurement setup. Though the stop band was identified in the rotating state, it was not possible to examine whether the stop band is narrowed down by adjacent modes traveling into the stop band or whether the stop band travels with increasing rotational speed due to a shift in the resonator’s eigenfrequency as a result of centrifugal forces. The presented study has the aim of answering these questions with an experimental investigation. Methodically, it follows the work of Rieß et al. [12] with the improvement of the test setup and the investigation of a circular disk, which is a more general structure than a saw blade. An experimental study is chosen since the numerical simulation of non-axisymmetric systems is subject to limitations [21].

1.1. Vibrations of Rotating Disks

The natural modes of a system consist of natural frequencies and their corresponding mode shapes. Mode shapes represent specific vibration patterns that the structure assumes when excited at the natural frequency associated with that mode. For circular disks, the mode shapes are characterized by the formation of nodal diameters (k) and nodal circles (n) depending on the specific mode. Figure 2b illustrates typical mode shapes of circular disk structures.

For the influence of rotation on natural frequencies, the choice of the reference system is crucial. One option is to measure the vibration from the perspective of a rotating observer. This implies that the measurement system used to capture the structural response has the same rotational speed as the object being measured. Consequently, the position of the measurement point on the object remains constant during the measurement.

The other option involves consideration from the viewpoint of a stationary observer. In this case, the measurement system remains fixed, while the observed measurement point on the object undergoes changes due to rotation.

1.1.1. Rotating Observer

When measuring structural vibrations in simple disk structures using a rotating measurement system at high rotational speeds, gyroscopic effects, centrifugal forces, and shear forces influence the vibration behavior [22,23]. Specifically, centrifugal or centripetal forces result in a stiffening of the structure, leading to an increase in natural frequencies.

Southwell [24] demonstrated that the natural frequencies of a rotating plate can be derived from the natural frequencies of a non-rotating plate. According to Southwell, the frequency associated with a specific mode $f_{\mathrm{e}\omega }$, which is dependent on the rotation speed, can be approximated for modes with exclusively nodal diameters and without nodal circles using the stationary mode frequency $f_{\mathrm{e}}$ and the rotation frequency $f_{\mathrm{n}}$ as follows:

$$f_{\mathrm{e}\omega }^2=f_{\mathrm{e}}^2+\lambda f_{\mathrm{n}}^2$$

(1)

The factor $\lambda$ takes into account the mode shape and can be calculated with the Poisson’s ratio $\nu$ and the number of nodal diameters k: [25]

$$\lambda =\frac{1-\nu }{4}·k^2+\frac{3+\nu }{4}·k$$

(2)

1.1.2. Non-Rotating Observer

When measuring vibrations of circular disks in a stationary reference system, the gyroscopic effect leads to a splitting of the natural frequencies [26,27]. The resulting two waves can be conceptually interpreted as traveling waves moving with or against the direction of rotation of the disk [25]. The wave propagating in the direction of rotation can be attributed to the increasing eigenfrequency $f_1$, while the wave moving against the rotation direction can be associated with the decreasing eigenfrequency $f_2$. Both natural frequencies of the waves traveling with and against the direction of rotation can be calculated using Equation (3) [25].

$$f_{1,2}=f_{\mathrm{e}\omega }\pm k·f_{\mathrm{n}}$$

(3)

Figure 2. Example of a Campbell diagram (a). Solid lines: stationary observer; dashed lines: rotating observer. Mode shapes for circular disks (b). Source: compare [28].

Figure 2. Example of a Campbell diagram (a). Solid lines: stationary observer; dashed lines: rotating observer. Mode shapes for circular disks (b). Source: compare [28].

2. Numerical Design of the Circular Disk with a Vibroacoustic Metamaterial

In order to investigate the behavior of a VAMM for out-of-plane vibration reduction within a spinning disk shaped rotor, a test structure needs to be evaluated. For this, a circular disk with similar dimensions like those of a typical circular saw blade used in woodworking on table saws and chop saws is used. The circular disk features an outer diameter ($d_{\mathrm{o}}$) of 300 $\mathrm{mm}$, an inner diameter ($d_{\mathrm{i}}$) of 30 $\mathrm{mm}$ and a thickness (t) of 2 $\mathrm{mm}$ (see Figure 3). This circular disk is used for integration of a VAMM. The design by numerical methods (finite element method) of the circular disk with an integrated VAMM is presented in the following sections.

2.1. Numerical Model of the Disk

To design the VAMM for integration into the circular disk, an experimentally validated numerical model of the disk is necessary. For this, a finite element (FE) model of the disk is elaborated in Ansys^® Workbench. The model consists of 10,004 hexahedron elements of type SOLSH190 and is depicted in Figure 4a. To determine the material properties as well as the effective clamping diameter, an experimental characterization of the disk clamped at a diameter of 61 $\mathrm{mm}$ is performed. For this, the disk is excited by an impulse hammer at a distance of 137 $\mathrm{mm}$ away from the center of the disk. The resulting vibrations are measured at 116 points with an LDV. The location of measurement points can be seen in Figure 4a where the numerical model is also evaluated. Since this section focuses on the numerical design of the VAMM test structure, the test setup is not shown here. It is described later in Section 3.1 together with the setup for measuring the disk in the rotating state.

Figure 4b shows the experimentally and numerically obtained transfer functions in terms of accelerance, averaged over the 116 evaluation points as shown in Figure 4a. The transfer functions indicate a good agreement between measurement and simulation. To parameterize the model, a Young’s modulus of E = 205 $\mathrm{GPa}$, a density $\rho$ = 7850 $\mathrm{kg}/{\mathrm{m}}^3$ and a Poisson’s ratio of $\nu$ = 0.3 were chosen. Since the clamping of the disk in the experiment with a flange of a diameter of 61 $\mathrm{mm}$ is not ideal, the effective clamping diameter is updated to fit the measurement data. A good correlation is found for a diameter of 58.5 $\mathrm{mm}$. Since the circular disk is made from steel, damping is omitted in the simulation. The width of the peaks in the simulated and measured data matches well. This confirms the assumption of the omitted damping in the simulation.

2.2. Numerical Design of the Unit Cell

With the numerically obtained material parameters of the disk, the unit cell is designed. The design has already been validated for circular saw blades in [12,29,30] and is used as a resonator (see Figure 5). The frequency range at which sawing noise is particularly loud and disturbing is used as the requirement for the stop band frequency range. According to [31,32], this is in between the frequency range of 2000 $\mathrm{Hz}$ and 3000 $\mathrm{Hz}$. Therefore, this frequency range is used as a requirement for the design of the VAMM.

The resonators need to be arranged at a distance smaller than half the wavelength of the vibration that is to be reduced. The wavelength $\lambda$ within the disk can be calculated using Equation (4).

$$\lambda =\sqrt[4]{\frac{{\left(2\pi \right)}^2}{f^2}\frac{t^{2}E}{12\rho (1-{\nu }^2)}}$$

(4)

At a frequency f of 2000 $\mathrm{Hz}$ and with the parameters of the disk as described above, the wavelength $\lambda$ is 98.5 $\mathrm{mm}$. Thus, the resonators need to be smaller than 49.25 $\mathrm{mm}$.

By applying the Bloch theorem [33], the stop band behavior of an infinitely periodic VAMM can be predicted by calculating the dispersion curve of a single unit cell. Figure 6 (right) shows the dispersion curve of the unit cell with parameters a = 24 $\mathrm{mm}$, b = 16 $\mathrm{mm}$, c = 4.5 $\mathrm{mm}$ and d = 1 $\mathrm{mm}$, and a thickness t = 2 $\mathrm{mm}$. Figure 6 (left) shows the resonator in its first eigenmode and the Brillouin contour on which the dispersion curve is calculated. The parameters have been chosen in accordance with [12].

For the infinite periodic VAMM, the dispersion curve indicates a stop band in the frequency range of 2045 $\mathrm{Hz}$ to 2967 $\mathrm{Hz}$ (see Figure 6, right, gray-marked frequency range), where no free wave propagation for elastic waves is possible. The dotted curves on the left and right side of the dispersion curve, which cross the stop band, correspond to in-plane vibrations (longitudinal and shear waves). Since the aim of the work is to reduce the out-of-plane vibrations, these dotted curves can be neglected.

2.3. Numerical Design of Vibroacoustic Metamaterial Compound

The functionality of the designed VAMM array integrated into the circular disk is verified via an FE model. In this model, 68 resonators are integrated into the disk in such a way that there is free space at the circumference and close to the central bore. With this arrangement, the disk can be measured with a stationary laser Doppler vibrometer in these regions without the resonators influencing the measurement. The resonators are rotated by 90° in each quadrant of the circular disk. This ensures that the spring element of the resonators always points in the direction of rotation when the disk rotates clockwise. The model of the disk with incorporated resonators can be seen in Figure 7a.

The resonators are positioned in such a way that there are diameters that are not interfered with by resonators so that a measurement of the vibrations of the disk with a stationary observer is possible. In particular, a portion at the outer diameter of the disk is kept free of resonators so that the disk can be excited there with a stationary impulse hammer without hitting a resonator.

The FE model consists of 17,657 hexahedral elements of type SOLSH190. The material parameters and boundary conditions are the same as for the model of the plain disk. The results of the simulation of the circular disk with and without the VAMM are shown in Figure 7b. Depicted are the transfer functions in terms of accelerance averaged over the 12 evaluation points as indicated in Figure 7a.

Reduced vibration amplitudes are visible within the predicted stop band frequency range of 2045 $\mathrm{Hz}$ to 2967 $\mathrm{Hz}$ as predicted by the unit cell modelling. The stop band is interrupted by three distinct modes at 2448 $\mathrm{Hz}$, 2690 $\mathrm{Hz}$ and 2790 $\mathrm{Hz}$. These frequencies are related to mode shapes with deflections at the circumference of the circular disk. Since no resonators are placed in this area, the vibrations cannot be reduced with the VAMM. Figure 8 depicts these modes.

3. Experimental Characterization

The circular disks, previously designed and investigated in the non-rotating state, are manufactured from 2 $\mathrm{mm}$ galvanized sheet metal using a water jet cutting machine. The manufactured plain disk and disk with the VAMM are shown in Figure 9.

3.1. Measurement Setup

The experimental investigation of vibrations in a rotating system with a stationary observer is a complex measurement task. It necessitates both a suitable excitation mechanism of the rotating system and a preferably non-contact sensor for measuring the resulting system response. Since the stop band is expected in the frequency range between 2000 $\mathrm{Hz}$ and 3000 $\mathrm{Hz}$, an excitation mechanism capable of inducing vibrations up to at least 3000 $\mathrm{Hz}$ is required. Due to rotation, high levels of noise amplitudes occur depending on the measurement method used. Hence, it is necessary to excite sufficiently large amplitudes to measure the system behavior above the noise level.

Preliminary investigations by Schmidt [34] revealed that a measurement setup consisting of an automated impulse hammer for excitation and an LDV for measuring the structural vibrations is most suitable for the underlying measurement problem. The impact force of the impulse hammer is measured with a piezoelectric force sensor at the hammer’s tip. Figure 10 shows the experimental setup together with the utilized rotor dynamics test bench. The circular disk is mounted on an electric motor via a flange with a diameter of 61 $\mathrm{mm}$. The rotational speed of the motor can be adjusted via a frequency converter. The disk is excited with the automated impulse hammer on the side of the disk facing the motor. The automation of the impulse hammer enables a reproducible and remotely controllable excitation. The surface velocities of the disk are measured using an LDV (Polytec PSV-300-H, Germany-76337 Waldbronn). To realize a diffuse reflection behavior, the circular disk is sprayed with chalk spray, which enhances the signal quality.

3.2. Non-Rotating State

The circular disks as shown in Figure 9 are firstly investigated in the non-rotating state using the experimental setup described in Figure 10 to validate the functionality of the VAMM. For this purpose, the circular disks with and without the VAMM are excited with the automatic impact hammer at a radial distance of 137 mm from the center of the disk. The surface velocities are measured at 116 points each according to Figure 11. The measurement points are positioned in such a way that they are situated in between the resonators.

The measurement results can be seen in Figure 12. Here, the transfer function of the accelerance is shown for three groups of measurement points located at the outer edge of the circular disk, in between the resonators, and in the inner area of the circular disk. As expected, the averaged transfer function of the points at the circumference does not exhibit a distinct stop band, as the vibrations excited at the excitation point can propagate freely along the circumference. When considering the middle and inner measurement points, a stop band in the frequency range of approx. 2000 $\mathrm{Hz}$ to 3000 $\mathrm{Hz}$ can be observed. The location of the stop band matches well with the numerical predicted stop band of the finite disk in between 2045 $\mathrm{Hz}$ and 2967 $\mathrm{Hz}$.

As observed in the numerical investigation, distinct peaks occur in the stop band region. The three numerically investigated modes are also present in the measured transfer function at 2452 $\mathrm{Hz}$, 2710 $\mathrm{Hz}$ and 2794 $\mathrm{Hz}$ within the stop band frequency range. These peaks align well with the frequencies of numerically predicted modes and can be attributed to modes with high amplitudes in the circumferential region of the circular disk (see Figure 13). Since there are no resonators placed close to the circumference, these modes cannot be influenced.

3.3. Rotating State

To characterize the circular disks in the rotating state, measurements are conducted at two measurement points with radial distances of $r_{\mathrm{m},\mathrm{i}}$ = 49 $\mathrm{mm}$ and $r_{\mathrm{m},\mathrm{o}}$ = 137 $\mathrm{mm}$ (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 $\mathrm{rpm}$ ranging from 0 $\mathrm{rpm}$ up to 3000 $\mathrm{rpm}$, 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 $G_{\mathrm{XY}}\left(jw\right)$ and the auto-power spectrum $G_{\mathrm{XX}}\left(w\right)$ of the input signal [35] (see Equation (5)).

$$H_1\left(jw\right)=\frac{G_{\mathrm{XY}}\left(jw\right)}{G_{\mathrm{XX}}\left(w\right)}$$

(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 $\mathrm{Hz}$, which requires a sampling frequency of 12,800 $\mathrm{Hz}$. In total, 8192 sampling points are set, 3200 of which can be used. As a result, the frequency resolution is 1.5625 $\mathrm{Hz}$ with a measurement time of 0.64 $\mathrm{s}$. 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 ($r_{\mathrm{m},\mathrm{o}}$ = 137 $\mathrm{mm}$) 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. 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 $\mathrm{Hz}$ and 3000 $\mathrm{Hz}$. 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 $\mathrm{Hz}$ and 3000 $\mathrm{Hz}$ (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 have either no or a negligible influence on the behavior of the VAMM up to the investigated maximum rotational speed of 3000 $\mathrm{rpm}$, 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 $\mathrm{Hz}$. 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 Alsaffar et al. [17] 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 $\mathrm{Hz}$, 214 $\mathrm{Hz}$ and 812 $\mathrm{Hz}$) of the disk with the integrated VAMM at a rotational speed of 3000 $\mathrm{rpm}$. The modes together with their eigenfrequencies and calculated mode-splitting frequencies are presented in

Table 1. Exemplary modes and their analytically calculated frequency splitting at a rotational speed of 3000 rpm with a stationary observer. MP: measurement point.

Number		1	2	3
Mode Shape
Frequency		81 Hz	214 Hz	812 Hz
	Nodal Diameters	1	3	6
MP1	Frequency forward (3)	145 Hz	386 Hz	1129 Hz
	Frequency backward (3)	45 Hz	86 Hz	529 Hz
	Nodal Diameters	1	3	0
MP2	Frequency forward (3)	145 Hz	386 Hz	-
	Frequency backward (3)	45 Hz	86 Hz	-

. These are calculated using Equation (3) from the nodal diameters as derived from the mode shapes.

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). The good agreement of the analytical calculation of the mode splitting with the obtained measurement data confirms the validity of this approach. Thus, modes that are near the stop band region can be analyzed in terms of their nodal diameters to infer the narrowing of the stop band with increasing rotational speed.

4. Conclusions

In the presented work, the behavior of a VAMM for out-of-plane vibration reduction of a thin-walled circular disk has been investigated experimentally in the rotating state with a stationary observer up to 3000 $\mathrm{rpm}$. The leading research questions are whether the stop band still exists under rotation, whether it is subject to a frequency shift or whether it is narrowed down by split modes traveling into the stop band with increasing rotational speed. The study was performed experimentally because a numerical study is bound to limitations since a VAMM is always a non-axisymmetric system. In order to be able to experimentally investigate a rotating disk with an embedded VAMM, such a disk firstly was developed based on numerical simulations in the non-rotating state. The dimensions of the disk were adopted from a circular saw blade usually used in woodworking. The stop band was tuned to a frequency range of approx. 2000 $\mathrm{Hz}$ to 3000 $\mathrm{Hz}$ where circular saw blades are often loudest. For the experimental investigation, a measurement setup consisting of a stationary LDV for measuring the out-of-plane vibrations and an automatic impulse hammer for excitation is suggested. With this setup, Campbell plots were recorded for two different measurement points—one within the resonator array of the VAMM and one on the circumference of the disk. For the measurement point at the circumference, no stop band was obtained since no resonators were placed at the circumference. The inner measurement point indicates a stop band in between approx. 2000 $\mathrm{Hz}$ and 3000 $\mathrm{Hz}$ that does not shift in frequency with increasing rotational speed. Adjoining modes that split up propagate into the stop band region, narrowing it down symmetrically by approx. 200 $\mathrm{Hz}$. As a result, it can be stated that the gyroscopic and centrifugal forces do not have a significant influence on the formation of the stop band. Furthermore, the results show that a VAMM for a thin-walled disk can be numerically designed in the non-rotating state with good accuracy. Adjoining modes that are on the verge of propagating into the stop band can be accounted for with analytical estimations as shown in the last section of this work. For that, the number of nodal diameters needs to be obtained from numerically evaluated mode shapes.

This work contributes to the field of VAMMs for vibration reduction on thin-walled rotors both in terms of the methodology used for the development and in terms of the experimental validation of such a system. The results are applicable to other technical systems where out-of-plane vibrations of thin-walled rotors such as gears, brake disks, saw blades, fans, etc., are to be reduced.