Passive mitigation of low-frequency vibration modes in lightweight structures based on a new generation of acoustic black holes

Topology optimization and generative design are methods used to lighten structures under specific loads. The resulting lightweight components are usually softer in the directions perpendicular to the applied efforts, such as in the case of cranes. This results in the possible excitation of structural modes at very low frequencies, which can lead to premature aging of lightweight structures. In this paper, we present a new design of acoustic black hole (ABH), able to mitigate vibration modes in a test 3D-printed lightweight structure, from 20 Hz upwards. The novel ABH weights less than 10 grams and can be readily screwed on the structure. Its design is based on that of a classic ABH, with additional winglets and/or rods, to increase the vibrating mass without significantly changing the stiffness of the ABH. This results in a strongly reduced ABH cut-on frequency. The winglets and/or rods also result in a greatly increased contact area between the ABH and the viscoelastic dissipative material, with positive effects on the ABH damping efficiency. The new ABHs could successfully be used, first to damp the fundamental vibration mode of aluminum cylinders, and then to reduce the amplitude of the first three structural modes of the test lightweight structure. The new ABH manages to damp modes at frequencies several times lower than the ones found in the literature, which are promising results.


Introduction
Lightweight structures have been omnipresent since the early days of human constructions.The assembly of rigid elements enabled to build the first scaffolds and cranes in the antiquity [1].This method was more recently used for the construction of large structures such as the Montreal Biosphere in Canada, the Nine Bridges Country Club in South Korea, the Eiffel tower in France and roller coasters all around the world (figure 1, first column).
These constructions have a high load to weight ratio that is not only highly beneficial in the building sector but for transportation applications as well, since less mass to move translates into energy savings.Huge efforts are directed towards making the supporting frame of vehicles lighter, especially with the help of numerical methods such as topology optimization and generative design (e.g., for cars [2][3][4] and planes [5][6][7]).Topology optimization is a method aiming at reducing by a given amount the mass of a mechanical component under a specific load, while minimizing the mechanical stress in the component.Generative design, however, is meant to create an as light as possible structure connecting specified locations in space, under well-specified mechanical constrains.In both cases, the result obtained after a few iterations is a tree-like lightweight truss.For the fabrication of components structured in three dimensions, additive manufacturing usually becomes necessary (the shape being too complex), which drastically increases the manufacturing costs (figure 1, second column).However, components with a structure defined in two dimensions only (and extruded into the third one), such as bell cranks, can most often still be fabricated using standard manufacturing processes.
The presence of rotating elements (usually motors) in a lightweight construction can lead to the excitation of low frequency vibration modes.The direction of motion for these modes is usually perpendicular to the load for which the component was optimized in the first place.It is for instance the case of cranes (figure 1, yellow frame).Vibration modes (or resonances) must be avoided (by design) or damped, in order to prevent any early mechanical failure.
Passive dampers have the advantage of being independent from any electronics, resulting in a gain of mass (and thus raw material) and implementation time.These dampers are usually resonators tuned at a given resonance frequency of the structure, in order to mitigate one particular resonance mode [8].This requires fine but also tailored designs for each application.Another option are broadband passive absorbers, such as acoustic black holes (ABHs) .ABHs are additional features mounted on the structure, by one end.They are meant to slow down the incoming mechanical waves in order to trap them.The slowing down of the waves is obtained through the progressive decrease of the stiffness of the element, over its length.This decrease is obtained by a progressive reduction of the thickness of the ABH, following a well defined mathematical law.Consequently, the acoustic energy geometrically localises at the tip of the ABH, where a layer of highly viscous material enables to dissipate the concentrated mechanical energy, as depicted in figure 2. Another way to slow down the incoming mechanical wave is to make the ABH out of a material more and more porous as the wave travels to the tip of the ABH [31].This kind of device is nowadays possible to design and fabricate thanks to programmable porous materials techniques [32].Similarly to regular black holes able to trap electro-magnetic waves, acoustic black holes are supposed to perfectly trap the incoming mechanical waves due to their particular geometry or material properties.However those structures are subjected to physical limitations (minimum achievable thickness), leading to wave reflexion at their tip and, eventually, to resonances.Indeed, being mechanical structures, ABHs have their own resonance modes and the first ones, at low frequencies, are well separated from one another.At higher frequencies however, strong damping causes the resonance peaks to widen and eventually, to overlap.They then form a continuum, which can be interpreted as an absorption band.A so-called cut-on frequency can be defined, above which all frequencies are well damped [33].This cut-on frequency is located around the fundamental mode and should be reduced as much as possible to increase the width of the ABH absorption band.
In this paper, we present a solution, based on a 3D-printed ABH, which should make it possible to strongly reduce the cut-on frequency and therefore, to strongly improve the damping factor at very low frequency.The solution is based on the addition of either winglets or rods on the curved surface of the ABH.Indeed, those structures enable to increase the mass of the ABH.In combination with the progressive decrease in stiffness along the ABH profile, they make it possible to strongly lower the absorption threshold, in frequency domain.We designed and fabricated a first generation of these new ABHs.We tested them first on aluminum cylinders, where we could use the ABHs to damp vibrations around 2 kHz.We then designed a second version of the ABH, to absorb frequencies down to 20 Hz.This last version was used to damp resonance modes of a topologyoptimised and 3D-printed test structure.The vibration measurements were performed using both an accelerometer and a motion amplification system (based on a high-speed and high-resolution camera).All simulations in this paper were performed using the modal analysis module from ANSYS, the Mechanical APDL solver and a program controlled tetrahedral mesh.Only the topology optimization was performed using the static structural module and the topology optimization tool.

Experimental setups
Two test benches ('Setup 1' and 'Setup 2') were designed and used for the purposes of the present study.
Setup 1 consists in a series of suspended aluminum cylinders.They have a diameter of 30 mm, a length of 250 mm and a fundamental resonance frequency close to 2 kHz.A M4 threaded hole was drilled on one of the two ends of the cyclinders, to suspend them from a gantry, using a hook and a chain (figure 3(a)).The chain makes it possible to achieve vibration isolation.Furthermore, four M2 threaded holes were drilled on the other end of the cylinders, to fix the ABH.The modes of the cylinders were then purposely excited, using hammer strikes.The vibration measurements were carried out using a A/128/V1 Miniature Piezo-Tronic IEPE Accelerometer from DJB Instruments (figure 3(b)), located on the side of the cylinders and close to the ABH.Data acquisition and processing was realized using the VibSoft-20 software from Polytec.During signal processing, a 1500 Hz-2 kHz bandpass filter was used, to filter out the DC signal and high frequency noise, and no averaging function was used.
Setup 2 was specifically designed to study the damping effect of the new ABH design on lightweight structures.The structure we have designed and 3D-printed for this study is shown in figure 4(a).To obtain this optimized shape, we started with a full L-shaped structure, fixed on one end and subjected to a particular load on the other end.We then topologically optimized this part for this load, using ANSYS workbench.We then 3Dprinted the part, in PLA (Polylactic Acid), using a Stratasys F170 printer.An unbalanced brushed motor was then mounted on the bottom of the front plate (figure 4(b)).This motor can generate vibrations between 0 and 200 Hz, which is sufficient to excite the first three vibration modes of the structure.
To characterize the vibration behaviour of the lightweight structure, we used the IRIS M system from RDI Technologies (figure 5).The hardware comprises essentially of a high-speed high-resolution camera.The software then uses each of the image pixels as a displacement sensor, and makes it possible to strongly amplify the motion of the vibrating devices under test.Therefore, it enables to visualize vibration modes in one shot and to measure the displacement of any location called region of interest (ROI), in the plane perpendicular to the camera.Filters and fast Fourier transforms (FFT) can also be applied to any ROI.An illustrative video of that system can be found here [34].

ABH design
First, we designed and fabricated a classic version of the ABH, to be mounted on the extremities of the cyclinders of setup 1.The ABH was made of PA12 and is shown in figure 6(a).It has a diameter of 30 mm and a length of 50 mm.The resonance frequencies of the first three resonance modes of the ABH are 1600 Hz, 2200 Hz and 3500 Hz, respectively.The frequency of the first eigen-mode of the cylinder is 2 kHz, which is located between the two first resonance frequencies of the ABH.Because of this mismatch, the vibration energy of the resonance  mode is expected to be only partially absorbed by the ABH.To effectively absorb frequencies around 2 kHz, the cut-on frequency of the ABH must be lowered below this frequency.
Lowering the cut-on frequency of a classic ABH design is doable, by decreasing its stiffness.This stiffness depends on both the length of the ABH and the minimum achievable thickness of the ABH tip, itself limited by the fabrication process.However, in this work, we rather increased the mass of the ABH, to overcome those manufacturing challenges (this also has the effect, as in the case of a simple mass-spring system, of further lowering the resonance frequencies).To this purpose, winglets were added along the ABH profile (figure 6).The winglets were designed and positioned so that the stiffness of the ABH remained roughly unchanged.Indeed, the system can be modeled by a series of high (winglets) and low (gaps between winglets) stiffness values, in which case the lowest ones prevail, since in series, the inverse of the different stiffness add up.The winglets have a thickness of 1 mm and are separated by 1 mm gaps.The frequencies of the three first resonance modes of the new ABH are 380 Hz, 650 Hz and 660 Hz.These frequencies have therefore been reduced by a factor of about 4, compared to the classic ABH configuration, without any significant increase in the space requirement of the ABH.The configuration with winglets has an additional advantage.Here, the contact surface between the viscous material used to dissipate the mechanical energy and the vibrating surface of the ABH is much larger than it is in the classic configuration (the viscous material is located between the winglets).This results in a strongly improved absorption coefficient.
To damp the vibrations of the lightweight structure in setup 2, an even lower cut-on frequency is required.For this purpose, the ABH was lengthened from 50 mm to 100 mm, the winglets were thickened from 1 mm to 3 mm to further increase the ABH mass and they were connected to the ABH body, by a thin 1 mm-wide junction (in order to keep the stiffness as low as possible).The combination of these modifications enabled to lower the first ABH mode down to 22 Hz.Furthermore, the winglets were turned into separated rods (figure 7(a)) in order to provide the ABH with more mechanical degrees of freedom, which resulted notably in the possibility to excite one new mode (namely mode 2, depicted in figure 7(b)).The diameter of the ABH was also reduced to 20 mm, to enable an easier mounting on the structure (figure 7(c)).It was expected that this new ABH (or ABH 3) would make it possible to efficiently dissipate the mechanical energy of modes 1 and 3 of the lightweight structure, but that mode 2 would not be damped (a careful observation of the polarizations of the modes of ABH 3 (Figure 7(b)) shows indeed that none of these modes has the same polarization as mode 2 of the lightweight structure (figure 4(b)).Therefore, they cannot be mechanically coupled).At the end of the design phase, the three ABHs were 3D-printed, using a SLS printer Sinterit Lisa Pro (figure 8).
The profile used to define the winglet/rod heights of ABH 2 and ABH 3 is the inverse of that used to define the ABH thicknesses.We made this choice based on our practical experience with our first ABHs (the aim of the present work was not to achieve the optimum ABH geometry, but to demonstrate that this kind of ABH can have positive effects on vibration damping).According to our practical experience with the first versions of the ABHs, the most important factor to increase damping is that winglets/rods have increasing mass towards the end of the ABH.

Setup 1: cylinders
The impulse response of the cylinder with no damper attached is shown in figure 9.A fit using a logarithmic decay function enables the computation of the damping factor Lambda, using equation (1).Here, A 0 , A 1 , t 0 and t 1 are the amplitude and time of the two points circled in red, in figure 9.The experiments shown in figure 10 were carried out, one after the other.Case number 1 corresponds to the naked cylinder (no damper attached) and case number 3 to a reference measurement, performed using a PA12 cylindrical plug with the same dimensions as ABH 1. Cases 4 and 5 correspond respectively to measurements performed with ABH 1 and 2 attached to the cylinder, but without silicon grease (used as a viscous material to damp the mechanical vibrations).In cases 6 and 7, silicon grease was used.For comparison purposes, measurements were also performed while holding the cylinder in the hand, with as much contact surface as possible between the hand, the fingers, and the cylinder (case 2).
It is observed that the cylinder equipped with ABH 2 produces only an extremely brief noise when hit, which qualitatively demonstrates the efficiency of the greased winglets.This is also confirmed by the experimental results: figure 10 shows indeed, for case 7, a normalized damping rate 25 times higher than the one obtained in case 3 (i.e. using the reference plug), 5 times higher than the one obtained while holding the cylinder by hand (case 2), and 3 times higher than the one obtained using the classic ABH (case 4). Figure 10.Measured damping rates, normalized with the one of a cylinder without ABH (case number 1).For each case, five measurements are performed and both the average value and standard deviation are given.This image has been obtained by the author(s) from the Pixabay website where it was made available under the Pixabay License.It is included within this article on that basis.

Setup 2: lightweight structure
The vibration modes of the lightweight PLA structure were excited, using the 5 mm × 25 mm unbalanced motor and a signal generator, capable of sending a frequency-swept signal to the motor, in the range 0 -200 Hz.The sweep lasted 10 s and was repeated five times in a row.The camera was used to record videos of the vibrating structure from above, with a framerate of 950 frames per second and no averaging.The displacement measurements were subsequently performed on the ROI showed in red, in figure 11 Without grease, the ABH does not provide any damping to the structure (the mechanical energy does get trapped in the ABH, but it is not dissipated).The results obtained with and without ABH are compared in figure 12 (displacements of the ROI in the (x,y) plane).The first and third modes of the structure are clearly damped by the greased ABH, whereas the second mode is not.This is due to the fact that the three ABHs modes The structure weights 400 g.The greased ABH weights 8 g.This represents a mass increase of 2 %.The displacement amplitude of the first and third modes is reduced, in average, by 20 % (figure 12).This corresponds to a reduction factor of 0.8.Assuming a geometrical law, each percent of mass increase results in the multiplication of the displacements by a factor of 0.8 .Finally, since the vibration energy of a harmonic resonator is proportional to the square of the displacement, each percent of added mass results (following our geometrical law assumption) in the multiplication of the vibration energy of the structure by 0.8 2 , that is a reduction of 20 %.This way, a first estimation of the damping efficiency of ABH 3 per percent of added mass is obtained (table 1).

Discussion
In theory, the ABH slows waves down so that incoming mechanical waves never reach its tip.However, in practice as in the present document, the ABH has a finite length.Consequently, waves are always partly reflected at the tip, which results in resonances.It is then possible to achieve better damping around these resonances, provided that the damping material enables to approach the critical regime, i.e.Q factor close to one half (not higher or lower).As we did not check, within this work, whether this condition is fulfilled or not, we cannot conclude that our ABH was in its optimum configuration, which will be part of future work.
Still, our experimental results demonstrate the possibility to use a compact ABH-plug to dissipate vibration energy with high efficiency.The main novelty presented in this work is the improved characteristics of the winglets/rods ABH designs, in comparison to classic ABH.Indeed, the results above all confirm that our ABH structure has damping capabilities largely improved in comparison to the classic ABH and confirm the possibility to attenuate very low frequency modes.
However, the vibration attenuation observed on the lightweight structure using ABH 3 is much lower than that observed on the cylinders, using ABH 1 and ABH 2. This is probably due to the following reasons.ABH 1  and 2, once mounted, constitute a natural geometrical extension of the cylinders.This is not the case for ABH 3. Consequently, a higher fraction of the vibration energy might enter ABH 1 and ABH 2, which results in a better damping efficiency.In addition, the lightweight structure is made of polymer, and not metal.This material damps already the vibrations, which reduces the apparent effect of ABH 3. Finally, at very low frequencies (i.e.below the cut-on frequency), the attenuation of the ABH 3 depends very much on the matching between the frequencies of its eigenmodes and those of the structure.This matching is not perfect in our case, which reduces the damping efficiency of ABH 3.
To address this last issue, it might be interesting to design an ABH shaped like a tuning fork, but with two branches of different length.Each branch should be designed in such a way that the modes of one branch are located in the gaps of the other.Thus, the spectral density of modes would increase at lower frequency, which should result in improved damping properties.

Conclusion
The new ABH designs (ABH 2 and ABH 3) studied in this paper have promising properties.They operate at significantly lower frequencies than conventional ABHs, and dissipate mechanical energy with more efficiency.The new ABHs are also light, compact and easy to mount.The main limitations, which concern the classic ABH as well as the new ones, include the difficulty to obtain a good mechanical coupling between the modes of the structure and those of the ABHs, and the absorption at very low frequency.To solve this last problem, it will be necessary to further reduce the cut-on frequency.An alternative solution could be to increase the spectral density of the resonance modes of the ABH, at low frequency, using a tuning fork type geometry.

Figure 1 .
Figure 1.Lightweight structures classification sketch.This image has been obtained by the author(s) from the Pixabay website where it was made available under the Pixabay License.It is included within this article on that basis.

Figure 5 .
Figure 5. Setup 2. The lightweight structure under test is framed in blue.The camera and computer of the amplification motion system (IRIS M, RDI Technologies) are framed in green.

Figure 9 .
Figure 9. Measured acceleration of a suspended cylinder without ABH, in time domain.The green circle indicates the maximum of the signal.The first red circle is located at 80 % of the maximum amplitude, to ensure that it is located in the free oscillation regime.The second circle is located at 1 % of the maximum amplitude.Both red circles are used to perform the computation of the damping rate, according to equation (1).
(a).The x-and y-axis displacement velocities of the ROI are shown in figure 11(b) as a function of the excitation frequency.The peaks correspond to the resonance modes, whose frequency and polarization (mode shape) correspond well to the predictions of the numerical calculations presented in figure 4(b).Indeed, the first and third modes have a polarization along the x axis, whereas the second mode has a polarization along the y axis.

Figure 12 .
Figure 12.Displacements of the ROI in the (x,y) plane, without and with ABH 3. The amplitude of the resonance modes ranges approximately from 100 μm to 150 μm.ABH 3 attenuates the vibrations due to modes 1 (in red) and 3 (in orange), by about 20 %.The remaining mode, which is left undamped, is mode 2.
a Relative energy dissipation for one percent of mass added to the structure.