A A A Combination of Acoustic Black Holes with point masses Ste ff en Ho ff mann 1 Sebastian Rothe 2 Sabine C. Langer 3 Technische Universität Braunschweig, Institute for Acoustics Langer Kamp 19, 38106 Braunschweig, Germany ABSTRACT Passive acoustic measures are suitable for the targeted improvement of the vibration behaviour of technical products. In this context, Acoustic Black Holes (ABH) as innovative measures have shown great e ff ect for an e ffi cient damping of vibrations and a simultaneous mass reduction. The vibration-reducing e ff ect, however, is primarily apparent in the higher frequency range. The thickness reduction results in a mass reduction of the system, which could worsen the vibroacoustic behaviour in the lower frequency range. The idea of the approach here is to use the stinted material in form of point masses to counteract this disadvantage at low frequencies by a targeted placement or direct consideration in the manufacturing process. The result is a mass-neutral acoustic design measure. In order to investigate the e ff ectiveness of the ABH point mass combination, finite element models of generic beam and plate structures with ABH are used in this paper. In addition, the influence of point masses in terms of their position and number on the vibroacoustic behaviour is simulated. One focus is the assessment of the advantage by combining ABH with point masses. To validate the results, experimental investigations are carried out and compared with the numerical results. 1. INTRODUCTION AND MOTIVATION In mechanical engineering, the importance of audible and perceptible vibrations for product quality has increased significantly in recent years. Continuously improving calculation methods enable the acoustics-oriented design and the direct integration of acoustic measures into new technical products. There are numerous possibilities to develop those products acoustic-oriented by applying sti ff eners, mass or damping treatments. Special passive damping measures are Acoustic Black Holes (ABH), which are known since 2000 [1]. Due to targeted local dissipation, they can e ffi ciently reduce structural vibrations, especially in the higher frequency range. In this paper, ABH are combined with point masses to improve their positive acoustic e ff ect also in the lower frequency range. The motivation of this paper can be introduced with the numerical simulation results in Figure 1. It shows three frequency response functions (FRF) of the mean squared admittance level L a of three 1 ste ff en.ho ff mann@tu-braunschweig.de 2 sebastian.rothe@tu-braunschweig.de 3 s.langer@tu-braunschweig.de a slaty. inter.noise 21-24 AUGUST SCOTTISH EVENT CAMPUS O ¥, ? GLASGOW aluminium beams with the same damping treatment but di ff erently shaped ends. All beams are treated with the same constrained layer damping (CLD) patch and have the same lengths, thicknesses and boundary conditions. They only di ff er in the design of the beam ends where the damping patch is applied. In addition to the reference beam (REF) with no special shaped end, one beam is designed with a step shape (STP) whose minimum thickness is optimally adapted to its applied damping patch by using the analytical approach of Ross , Kerwin and Ungar [4]. The end of the third beam (ABH) is designed according to the ABH shape function by Mironov [2]. 172 REF STP ABH L a [dB] 111 49 REF 12 2 54 107 159 L a [dB] STP ABH 0.5 1.0 1.5 2.0 2.5 3.0 f [kHz] 0.00 0.02 0.04 0.06 η Figure 1: Comparison of numerical results of three beams with di ff erently shaped damping measures. Although the material thickness of STP variant is optimally adapted to the damping patch over the length of the measure, the measure generates less system damping η in higher frequency ranges than the ABH beam, whose thickness profile is not optimised according to the damping foil (see η curves in Figure 1 (bottom)). On the one hand, this shows how important the smooth impedance matching is at the transition to the measure. On the other hand, the first two resonance peaks are less damped for the ABH beam compared to the other. Here, targeted point masses in combination with the ABH could improve the structure-borne sound behaviour and thus finally the damping e ff ect in the low frequency range. A sensible placement and design of the point masses is investigated by numerical variations and validated by experimental studies on beams. The collected knowledge is subsequently transferred to plate structures. 2. POINT MASS VARIATION ON BEAM WITH ABH As already mentioned in the introduction, an ABH beam serves as generic structure for the investigations. The dimensions of the aluminium beam are shown in Figure 2. The end of the beam is shaped according to the polynomial ABH shape function (see [2]) with a polynomial degree of n = 2. The ABH area is covered on the flat side with a CLD patch ( 3M Damping foil 2552 [6]). The excitation of the structure is done at one of the free beam ends to make sure that many resonant modes are excited. The numerical model is preprocessed and solved using Abaqus . All structures are discretised with 8-node shell elements (S8R) with quadratic ansatz function and reduced integration. The base material (aluminum) is modeled linear elastic with a sti ff ness of 69 GPa and a density of 2700 kg / m 3 . The excitation is done by a point force of F = 1 N and free boundary conditions. The intermediate layer of the damping foil is meshed with 20-node volume elements with quadratic ansatz function and reduced integration (C3D20R and at least 5 nodes over thickness). Here, the shell and volume elements sharing the nodes and no contact constraints are used. The material behaviour of the damping layer is excitation top layer constrained layer 5 0.13 0.5 99 30 15 619 0.25 Figure 2: Principal beam set up with ABH and CLD patch (dimensions in mm). considered viscoelastic and is based on preliminary studies with the corresponding damping foil [8]. The top layer (aluminium) is modeled in the same way as the base structure (shell elements and same material parameters). Experiments are carried out to validate the numerical model. Due to the low mass of the structure and strong influences of boundary conditions on the dynamic behaviour on the system, the beam is supported quasi-free and excited without contact according to DIN EN ISO 6721-3 [5]. The velocity is measured at the other free end of the beam with an one point laser vibrometer. For the reason of contactless excitation, no force can be measured and accordingly only the velocity level L v can be plotted (see Figure 3, top). In contrast, for the simulation the admittance level L a is plotted (according to the y-axis on the right). So only the resonance frequencies (representing the sti ff ness) and the system damping of the measured and simulated frequency response functions (FRF) can be compared. To do so, the resonance frequency deviation ∆ f r between simulation and measurement is shown below the frequency responses in Figure 3. The experimentally and numerically determined flexural loss factor is plotted at the bottom diagram of the figure. Since the resonance frequency deviation remains below 1 . 5 % up to 2 . 8 kHz (see Figure 3, center) and the experimentally determined damping fits well with the simulated one (compare Figure 3, bottom), the numerical model is assumed to be validated for the e ff ects studied in this paper. L v (exp.) L a (sim.) 164 119 L v [dB] L a [dB] 85 87 6 55 2 ∆ f r [%] 1 ABH beam with applied CLD 0 0.08 0.04 η f 0.00 0.5 1.0 1.5 2.0 2.5 f [kHz] Figure 3: Validation of the simulated admittance L a with measured velocity curve at the end of the beam by comparing resonance positions and loss factors. With the validated numerical model, the position variations of the point masses can be performed. The point mass is considered as concentrated mass (translational inertia) at one node in the finite element model. No rotary inertia is taken into account. The point mass (12 g ≈ 5 % of inital ABH beam mass) is applied on the center line in longitudinal direction x (origin is opposite of ABH beam end) and on the flat side of the beam. The ABH beam with point mass (PM) thus still has a mass advantage compared to the reference beam of ∆ m = − 3% ( m REF = 250 g, m ABH = 230 g). A total of 268 positions are examined. The mean squared admittance sum level L a,s (area under the admittance curve) is plotted over the corresponding position of the point mass in Figure 4 (top, left). 174 236 reference beam x PM = 0 . 540 m x PM = 0 . 619 m 113 218 L a,s [dB] L a [dB] 200 52 ABH beam 182 8 L a (bands) η (damping) 152 0.080 0.050 102 0.053 0.036 L a [dB] η η 0.021 52 0.027 0.007 1 0.000 0.0 0.2 0.4 0.6 x PM [m] 0.5 1.0 1.5 2.0 2.5 3.0 f [kHz] Figure 4: Point mass position variation on ABH beam (numerical results). On the right hand side of Figure 4, the admittance curves of the PM position with the lowest and highest sum level as well as the admittance of the reference beam (depicted in Figure 3) are compared. As can be seen in Figure 4 on the left, the sum level is lowest and the averaged loss factor η is highest when the point mass is placed at the end of the beam at the thinnest point of the ABH. On the right- hand side of Figure 4, it can be seen in the admittance curves (blue and grey curve) that a reduction in the admittance level is particularly noticeable at low frequencies (first two resonances). To validate this e ff ect, a corresponding point mass is applied at the end of the ABH on the beam in the experimental setup (see Figure 5). In contrast to the measurement related to Figure 3, a force sensor is used to measure the input force and an electrodynamic shaker is used for excitation. The strong influence of the shaker on the dynamic behaviour of the specimens is clearly visible in the comparison between Figure 3 and Figure 5 (ABH beam REF). 60 ABH beam (reference) ABH beam with PM L a [dB] 41 21 2 40 L a [dB] 25 9 0.13 0.06 η 0.00 0.5 1.0 1.5 2.0 2.5 f [kHz] 0.04 0.15 0.26 f [kHz] Figure 5: Measured mean square admittance level L a and loss factors η of the ABH beam plate with and without PM ( 12 g ≈ 5% m ABH ) at thinnest point of ABH. However, since no force can be determined with the first measurement method used for the experimental investigations of Figure 3 (no quantitative level comparison possible), the experimental data in Figure 5 is used for validation of the e ff ect in Figure 4. As in the simulation, an admittance level reduction and a damping increase in the low frequency range are evident in the curves of Figure 5 despite the influences of the shaker. In addition to the pure increase in damping in the first but also higher resonances, the second resonance shows a tuned mass damper (TMD) e ff ect. Since the e ff ect is not clearly visible in the numerical results, only the low-frequency increased damping e ff ect can be validated (compare Figure 4). With the knowledge gained from the variation studies on the beams the method is transferred to plates with ABH in the following section where similar e ff ects should be seen. 3. POINT MASS VARIATION ON PLATE WITH ABH For the point mass variation on a plate with ABH, a 5 mm thick aluminium rectangular plate is considered. The ABH with polynomial degree of n = 5, treated with round damping patch in the middle of the ABH, is placed according to the sketch in Figure 6. A photo of the manufactured plate is shown in Figure 7 (right). The dimensions of the layer thicknesses and important geometric parameters of the structure are given in Figure 6. 99 136 5 suspension points top layer constrained layer 0.25 r = 77 395 100 0.13 excitation 139 117 495 0.5 Figure 6: Principal plate set up with ABH and CLD patch (dimensions in mm). The finite element (FE) model of the base plate is set up with 8-node shell elements (quadratic ansatz functions and reduced integration). A validation with solid elements was performed but is not discussed here. The element size is estimated using the minimum bending wavelength for an infinitely extended plate. At least 14 nodes per bending wavelength are considered for the FE mesh. Regarding the material thickness reduction in the ABH, the FE mesh is adapted to the reduced sti ff ness and thus changing bending wavelengths. Linear elastic and isotropic material with a Young’s modulus of 70 GPa and a density of 2700 kg / m 3 is assumed for the base structure. The mass and sti ff ness proportional damping ( Rayleigh damping) is slightly adapted to the plate test stand ( α 2 = 0 . 256 and β 2 = 4 . 62e − 7). In the area of the applied CLD patch, the intermediate and top layers of the patch are not homogenised. The full modeling of all layers of the damping foil allows the consideration of frequency-dependent e ff ects due to the consideration of interacting e ff ect between the layers. The intermediate layer is modeled with quadratic solid elements. To avoid high computational cost, the damping layer is assumed to be a linear elastic material ( E 2 = 14 . 61 MPa, η 2 = 1 . 1 , ρ 2 = 1192 kg / m 3 ) despite its true viscoelastic behaviour. For the top layer, the assumptions of the base structure apply. The force is applied by a concentrated point force according to Figure 6. The modal based direct solution is done at the resonance frequencies and 15 non-linear distributed frequency steps between two eigenfrequencies. There are no boundary conditions considered in the model. After solving the systems of equations, the required quantities for the assessment of the individual variation steps are determined. In this paper, the averaged surface velocity is considered as assessment quantity because it is the scaling quantity for the radiated sound power of components. For this purpose, all nodal velocities of an element are averaged and weighted with its surface. To enable the comparison to the measurement, the mean squared velocity is divided by the force (mean squared admittance). In addition to di ff erent numerical convergence studies (not further discussed here) a comparison with experimental results to validate the numerical model is done (see Figure 7). For this purpose, the manufactured plate is quasi-free suspended with two textile threads, excited with a shaker (with force sensor) at one point and the surface velocity is measured with a laser scanning vibrometer. The comparison of the mean square admittance levels L a from measurement and simulation is shown in Figure 7. This shows a high degree of matching of the curves, which is su ffi ciently accurate for the investigations carried out here. measurement simulation 110 excitation L a [dB] 80 50 ABH L a [dB] 75 50 0.5 1.0 1.5 2.0 2.5 3.0 f [kHz] Figure 7: Comparison of the mean square admittance levels from measurement and simulation. The modeling of the point masses is done in the same way as for the beams (see Section 2). One PM with a mass of 12 g is applied. This corresponds to only 0 . 5 % of the mass of the ABH plate. The combination of the measures thus still has a mass advantage of ∆ m = − 10 % compared to the reference plate without ABH. In Figure 8 the results of the numerical PM position variation are shown. On the right hand side the FRFs ( L a ) for the best and worst PM position are plotted. Below this diagram the same values are averaged in third-octave bands. The sum level of the mean squared admittance level L a,s is used as the assessment quantity for the position rating. On the left side the resulting L a,s for the respective PM position are shown as a contour plot. Blue values correspond to low and red values to high sum levels. Linear interpolation is performed between the 400 positions. The area of the ABH is highlighted in grey in the contour plot. For the variation in Figure 8 (left), the mean squared admittance sum levels in the frequency range from 0 − 1 . 5 kHz are used to assess the position of the point mass. Comparing the one-third octave bands of the best and worst position, clear di ff erences in the frequency range become apparent. Here, a di ff erence of up to 43 dB can be seen, especially at low frequencies, and an average di ff erence of 10 dB in the frequency range under consideration. While the unfavorable positions for the point mass are mainly located at the edges of the plate, the best positions are in the center of the ABH. Thus, for the studied plate, it can be stated that for additional vibration reduction at low frequencies, a point mass should be placed in the center of the ABH. This could be taken into account directly in production, for example during injection molding or die casting. To validate the e ff ects of the PM on the plate (Figure 7, right), the numerical results are investigated experimentally. For this purpose, point masses in the form of magnets are placed in the center of the PM worst PM best 211 223 L a [dB] 300 139 222 y PM [mm] L a,s [dB] 200 67 221 183 100 L a [dB] 148 220 0 112 0 200 400 x PM [mm] 0.5 1.0 1.5 2.0 2.5 3.0 f [kHz] Figure 8: Point mass positions variation on ABH plate – left : point mass positions rating according to sum level ( 0 − 1 . 5 kHz) right : FRFs for the best and worst PM position. ABH according to Figure 8 and the structure is measured as already explained for the validation of the numerical model. The resonant frequency highlighted in Figure 9 (right), whose modal loss factor is more than 4 times higher by placing the PM, is investigated in more detail in the next step. For this purpose, the numerically and experimentally determined deflection shapes of this resonance peak of the plate with and without PM are shown in Figure 10. As can be seen, the numerically determined deflection shapes and their resonant frequencies hardly di ff er from the experimental ones. The loss factors show deviations which, however, seem to be su ffi ciently accurate with respect to the measurement method. 48 ABH plate (reference ) ABH plate with PM L a [dB] 31 14 3 20 L a [dB] 11 3 0.030 0.015 η 0.000 0.5 1.0 1.5 2.0 2.5 f [kHz] 0.20 0.29 0.38 f [kHz] Figure 9: Measured mean square admittance level L a and loss factors η of the reference ABH plate with and without PM ( 12 g ≈ 0 . 5% m ABH ) in center of the ABH. Both the numerical and the experimental data show an increased L a peak in the region of the ABH after placing the point mass in the center of the ABH. On the one hand side, due to the higher deflections of the structure, the damping patch is much more deformed. The resulting high shear deformation in the damping layer leads to the significant increase in damping (loss factor η ). On the other hand, a TMD e ff ect as for the beam studies cannot be seen. Since the measure with regard to lower resonant deflection shapes is mostly located near nodal lines, the positive e ff ect of the PM compared to the beam example (see Section 2) is much smaller, but still clearly visible. plate with PM in ABH center ABH plate without PM simulation experiment max f = 294 Hz, η = 0 . 0142 f = 291 Hz, η = 0 . 0030 L a min f = 298 Hz; η = 0 . 0108 f = 292 Hz, η = 0 . 0038 Figure 10: Comparison of the numerically and experimentally determined deflection shapes of the ABH plate with and without a PM in the center of the ABH. 4. CONCLUSION AND OUTLOOK The focus of the paper was to investigate the combination of point masses with ABH. The aim was to use some of the stinted lost through the thickness reduction when inserting the ABH in order to improve the vibration reduction e ff ect of the ABH. The ABH increase the system damping especially in the higher frequency range, while ABH structures lose performance in the lower frequency range. In this range, additional mass in the form of point masses can provide benefit. The positive acoustic e ff ect to be expected from the combination of point masses with ABH could be proven with the studies carried out here on a generic beam and a rectangular plate. The same structures can be damped significantly more (especially at low frequencies) by placing a point mass at the end of the ABH shape (beam) or in the middle of a 2D ABH (plate). Depending on the considered frequency range and the occurring deflection shapes, the modal loss factors can be increased by a factor of three to five. In higher frequency ranges, however, there is hardly any increase in damping. It should be emphasised that through the combination of measures (ABH and PM), especially for the plate structure, there is still a mass advantage compared to the initial structure without ABH and PM. Even the application of small PM (less than the mass removed by the integration of the ABH) at a targeted position can lead to considerable advantages in the resulting damping. In further studies, more suitable positions should be chosen for the two-dimensional ABH with PM in order to get them better activated already by the first modes. In addition, further variations in the number and mass of the PM to be placed can be carried out. Transferring the methodology to ABH in more complex realistic structures would be an interesting step as well. REFERENCES [1] Krylov, V. V. and Shuvalov, A. L. Propagation of localised flexural vibrations along plate edges described by a power law. Proceedings of the Institute of Acoustics, 22, pp. 263—270, 2000. [2] Mironov, M. A.: Propagation of a flexural wave in a plate whose thickness decreases smoothly to zero in a finite interval. Soviet Physics Acoustics-USSR, 34 (3), pp. 318—319, 1988. [3] Pelat, A., et al.: The acoustic black hole: A review of theory and applications. Journal of Sound and Vibration, 476, 115316, 2020. [4] Ross, D., Ungar, E. E. und Kerwin, E. M.:Damping of Plate Flexural Vibrations by Means of Viscoelastic Laminae, Bolt Beranek and Newmon lnc, Cambridge, Massachusetts, 1959 [5] DIN EN ISO 6721-3: Kunststo ff e - Bestimmung dynamisch-mechanischer Eigenschaften - Part 3, Deutsches Institut für Normung, Beuth Verlag, 2021. [6] 3M Industrial Adhesives and Tapes Division. Technical data - damping foil 2552 (november 2017). https: // multimedia.3m.com / mws / media / 1566784O / iatd-damping-foil-2552- tech-data-sheet.pdf. Accessed: 2022-04-26. [7] Rothe, S., Gha ff ari Mejlej, V., Vietor, T., and Langer, S. C.: Optimal adaptation of acoustic black holes by evolutionary optimization algorithms. PAMM 16 (1), 625–626, 2016. [8] Ho ff mann, S., Rothe, S. and Langer. S. C.: Vorgehen zur Charakterisierung und Modellierung von Mehrschichtbelägen, Conference proceedings of DAGA, Stuttgart, 2022. Previous Paper 85 of 769 Next