Forced-response characterization of PBF-LB/AlSi10Mg particle dampers with thin and flat cavities

Powder Bed Fusion (PBF) enables the production of complex geometries which offer the opportunity to manufacture lightweight, stiffness-optimised or integrally designed components. Although these properties are usually advantageous for the performance in many applications, they pose disadvantages under vibration as they lead to low damped components. These are prone to high vibration amplitudes which result in higher sound radiation and a reduced lifetime. Particle damping can counteract these disadvantages. By including cavities during the design process, unmelted powder remains inside the component after its production. This powder dissipates energy under vibration by inelastic impacts and friction in particle-particle or particle-wall-interactions, increasing the damping characteristics of the component. In this work, additively manufactured AlSi10Mg specimens with cavities are investigated with respect to their damping characteristics by experimental modal analysis. The focus of the investigation is on thin and flat cavities that can be easily integrated into components without adapting the external geometry. The damping characteristics in dependence on excitation amplitude and mode are quantified. The extent to which settling effects of the powder during shaking influence the damping is analysed. The vibration of the specimens is forced by an electrodynamic shaker and their response is measured contactlessly via Scanning Laser Doppler Vibrometry (SLDV). A damping effect of up to 564% depending on the mode, excitation amplitude and specimen can be achieved. In addition, a significant settling effect of the powder which hampers the damping effect is identified by CT scans and modal analysis.


Introduction
Over the last decade, the use of additive manufacturing (AM) has gained significance in structural industrial applications due to several benefits AM and especially Laser Powder Bed Fusion (PBF-LB) provides compared to conventional manufacturing processes [1]. Relevant for the present study are its abilities to manufacture integrally designed components on one hand and lightweight structures on the other hand. Integral components are designed to have as few parts as possible, where each part carries multiple functions [2]. This is beneficial from an industrial manufacturing perspective but also from a performance perspective as it increases strength and decreases weight [3]. In general, lightweight design offers the opportunities to realize higher accelerations, increase payload and reduce spent energy, which is especially advantageous in the aerospace, automotive and energy sector [4]. The often-desired high stiffnessper-weight-ratio can be achieved by topology optimization and its geometrically complex solutions are suitable for PBF-LB [5]. Several industrial applications show PBF-LB being used to manufacture lightweight, comparably stiff and integrally designed structural components [6][7][8][9][10][11][12][13]. However, these specific benefits become drawbacks under vibration as lightweight structures are more prone to vibration [14][15][16], stiff structures have a reduced component damping [17] and the lack of friction in joints decreases the damping characteristics of integrally designed components [18]. This leads to a reduced lifespan of the components and a higher sound radiation [19].
One method to counteract these susceptibilities is the use of particle damping. In this passive damping system, the component consists of a solid (primary) mass and a versatile (secondary) particle mass. This secondary mass can be attached to the primary mass as an outside container or can be inserted into the primary mass [20][21][22]. Under vibration, the secondary mass dissipates energy by inelastic collisions or friction at particle-particle or particle-wall interfaces, see Fig. 1 [23]. PBF-LB offers the opportunity to inherently include particle damping by designing cavities inside of the component in which unfused powder remains after manufacturing. Following the "Complexity for free" concept [24], Ehlers et al. [25] have coined the expression "Damping for free" as this technique neither increases component mass nor manufacturing cost compared to the fully fused reference component. However, for now, modelling of conventional as well as AM particle dampers is not reliable as the damping characteristics are highly non-linear and dependent on multiple design parameters like the cavity size, cavity position, excitation amplitude and frequency [26,27]. Nevertheless, some research regarding AM particle dampers has been carried out: Ehlers et al. [25,28] characterized additively manufactured particle dampers of different materials depending on excitation force and frequency for different cavity sizes. One dimension of their slender specimens was up to two orders of magnitude larger than the other two dimensions with a cavity located throughout the centre of the whole specimen. The most significant increase of damping by a factor of 28 for tool steel 1.2709 and a factor of 18 for aluminium alloy AlSi10Mg was observed. Heat treatment of Ti6Al4V specimens prevented any significant damping effect. Schmitz et al. [29][30][31] investigated beam-shaped aluminium specimens with inner cavities of 5 -9 mm diameter and laser beam melted walls out of AISI 316L stainless steel with ~ 20 -60% cavity volume and different internal ribs. At most, the damping was increased by a factor of 225 at the third mode. Künneke and Zimmer [32] studied a more general approach and investigated the damping characteristics of AISI 316L stainless steel particle dampers independently of the stiffness of the vibrating mass using a power measurement technique. In a previous study, Künneke and Zimmer [33] showed the ability to reduce sound radiation by additively manufactured particle damping in an electromagnetic spring-applied brake.
Although these studies show great potential for additively manufactured particle dampers, the integrability into aerodynamic designs like integrally bladed rotors (IBRs) in turbines or other two-dimensional structures can be better achieved by thin and flat cavities. When positioned at the neutral axis, these would also only have a small impact on structural integrity. Particle damping with thin and flat cavities was analysed by Scott-Emuakpor et al. [34][35][36][37][38] with specimens manufactured of Inconel 718 and AISI 316L stainless steel. Their first hypothesis of a densely packed interior pocket and thereby minimized impact behaviour as dissipation mechanism [37] did not hold as CT scans revealed empty spaces in the pocket [38]. Even for low cavity volumes of 1 -4% compared to the previously mentioned studies, the component damping was increased by a factor of up to 10. In the present study, specimens with one thin and flat cavity out of AlSi10Mg powder are analysed, which is a less dense and typically coarser powder compared to the material in previously mentioned studies. The overall specimen behaviour is measured via Scanning Laser Doppler Vibrometry (SLDV) under several different excitation amplitudes to characterise the non-linear particle damping. In addition, settling effects of the powder are depicted which are attributed to the experimental setup and significantly decrease the effectiveness of the particle damping.
inelastic collisions cavity particles (secondary mass) primary mass Fig. 1 Schematic diagram of an additively manufactured particle damper showing system excitation X(t) and system response Y(t) and depicting the two effect mechanisms of energy dissipation; based on [28] 1 3

Specimen design
Two kinds of specimens were designed. The "solid" specimens are fully fused and serve as a baseline for modal parameters. The "cavity" specimens have the same outer geometry as the solid specimens and differ only by an internal cavity. The specimens are of cuboidal geometry with a length of l tot = 120 mm in accordance with the capabilities of the PBF-LB machine employed and to obtain low and, therefore, well measurable resonance frequencies. To investigate thin and flat cavities, the other two dimensions of the specimen are chosen to be w = 24 mm and h = 2 mm. The cavity is positioned in the neutral axis of the specimen and has a thickness of h cav = 0.67 mm. To keep the aim of thin and flat cavities in mind, the other two dimensions are chosen to w cav = l cav = 15 mm. As the specimens are clamped at one side with a clamping length of l BC = 20 mm, the remaining l = l totl BC = 100 mm can oscillate. The positioning of the cavity along the length of the specimen is chosen, so that it would not overlap with a node point in the first three bending modes, as previous studies have shown a larger damping effect for particle dampers being placed in areas of larger displacement for conventional [39] as well as additive particle dampers [35]. The bending mode shapes Φ n for the bending mode n are calculated according to classical beam theory for undamped cantilever beams  Fig. 2 Dimensions of the cavity specimen in mm with the clamped area (red) and the printing direction (z) as well as the bending mode shapes Φ n (x) for n = 1…4 with the respective node points according to classical beam theory with the position x along the length l and the frequency parameter n [40]. The frequency equation shows that the frequency parameter n can take any of a countable set of values, see Table 1. This leads to the first four bending mode shapes according to Fig. 2. The positioning of the outwards boundary of the cavity is chosen to be 25 mm from the free end as this would lead to no overlap between cavity and a node point in the first three bending modes. Nevertheless, a node point in the fourth bending mode and the cavity coincide. (1) sinh n l + sin n l cosh n l + cos n l cosh n x − cos n x (2) 1 + cosh n l cos n l = 0

Laser beam powder bed fusion
The specimens are manufactured out of AlSi10Mg on an SLM 125 PBF-LB machine with process parameters which can be found in Table 2.
The properties of the AlSi10Mg powder in Table 3 are obtained by supplied data sheets from the powder manufacturer as well as previous work on powder characterisation by a Bettersizer S3 Plus for the used powder batch. It shows the significantly lower density and courser particle size compared to the powder in similar studies. The Young's modulus E = 74000MPa for this set of parameters and the Poisson's ratio = 0. 35    Three cavity specimens (labelled #1 -#3) and two solid specimens (labelled #4 & #5) are manufactured. They are positioned on the build platform with the longest dimension in y-direction and the second longest dimension in z-direction, see Fig. 2. This results in an overhang of 0.67 mm in the cavity which is manufacturable without support structures [47]. The specimens are rotated by 15° around the z-axis to avoid recoater crashes and separated by 5.5 mm, see Fig. 3. Due to manufacturing constraints, the specimens have to be manufactured on two different build jobs. It is chosen that the cavity specimens are manufactured on the same build job to ensure their comparability. After the manufacturing processes, the specimens are sawn from the building platforms and cleaned in an ultrasonic bath. The remaining support structures to the building platform are milled off. No subsequent heat treatment nor other mechanical finishing or sandblasting is carried out.

Experimental setup
The specimens are sequentially clamped on an electrodynamic shaker (RMS SW 121) to analyse their modal parameters. Four M4 screws hold the specimen in place and are tightened with a defined torque of 6 Nm. Two dowel pins in the clamp prevent the slipping of the specimen To examine the overall specimen behaviour, 37 points on a specimen are measured sequentially, see Fig. 5. The measurement points around the cavity are positioned more tightly to identify a possible special specimen behaviour at this location. The natural bending frequencies f n * are estimated according to classical beam theory for undamped cantilever beams, compare [48,49].
The results can be seen in Table 4. They are taken as an estimation to define the measured frequency range. The applicability to the cavity specimens with their increased damping characteristics as well as their density and elasticity discontinuity around the cavity is evaluated during the experiments.
The specimens are excited with periodic chirps in the range of f = 50-6000 Hz with a frequency resolution of 500 mHz. This excites the first four bending resonance frequencies n = 1…4, see Table 4. The excitation acceleration X(f) is kept constant over all frequencies within one measurement for a better comparability between the modes. Every measurement is repeated k # a ≥ 4 times to form one measurement batch per specimen # and excitation amplitude a. By this, the dependency of modal parameters on shaking duration is examined. X(f) is then varied after one measurement batch from 100 mm/s 2 , 200 mm/s 2 , 400 mm/s 2 up to 800 mm/s 2 per frequency bin to identify the non-linear damping characteristics of particle damping mentioned previously. The system response is transferred to the frequency domain Y(f), averaged over all 37 measurement points and the frequency response function FRF(f) is calculated, compare [50].

Computed tomography
The distribution of particles inside the cavity before and after the modal analysis is examined by computed tomography (CT) scans. A Bruker microCT SkyScan 1173 with an image pixel size between 10 -15 µm is utilised and several measures during the reconstruction of the 3D-data are taken to achieve calibrated and, therefore, comparable images. The orientation of the specimens during CT scanning is in an upright position, meaning that the longest dimension of the specimen is parallel to the direction of gravity and the specimen is rotated around this axis during the scan. Therefore, the orientation of the specimen is different during CT scanning and modal analysis.

Modal analysis
The FRFs show six distinct peaks, see Fig. 6, which correspond to the first four bending modes, as estimated in Table 4. The other two distinct peaks at about 1400 Hz  Fig. 6 depict the first two torsional modes. The corresponding mode shapes can be seen in Fig. 7. Looking at Fig. 6, it is clearly visible that at 800 mm/s 2 the damping is much higher for the cavity specimens than for the solid specimens. At 100 mm/s 2 the difference in damping is less prominent. A closer look on Fig. 8 Measured resonance frequencies f n of all specimens and calculated natural bending frequencies f n * for the second-to-fourth bending modes n = 2…4 over the excitation amplitude a; the standard deviation s f,kind for all specimens of same kind (black) are depicted Fig. 9 Loss factor η n of all specimens for the second-to-fourth bending modes n = 2…4 over the excitation amplitude a; the standard deviation s η,mb in each measurement batch (color) and the standard deviation s η,kind for all specimens of same kind for the third bending mode at 800 mm/s 2 (black) are depicted exemplarily; the data follows logarithmic trend curves the dependency of modal parameters on excitation amplitudes is taken. The resonance frequencies over the excitation amplitudes for all five specimens as well as the calculated natural frequencies from Table 4 can be seen in Fig. 8. The data from the first bending mode is disregarded as it is often done in similar studies [37], because large tip deflections lead to parasitic friction in the clamped components which influences component damping [28].
Three different characteristics can be highlighted: (1) the standard deviation s f,mb in each measurement batch, (2) the difference of the resonance frequencies over the excitation amplitude and (3) the standard deviation s f,kind for all specimens of same kind at the same excitation amplitude. The results show a very low s f,mb for all measurement batches with a maximum of 0.036% in relation to the respective mean resonance frequency. This is so small that s f,mb is not depicted in Fig. 8 due to visibility reasons, which indicates no relevant effect of shaking duration on the resonance frequencies. The resonance frequencies of each solid specimen decrease over excitation amplitude, see Fig. 8. The resonance frequencies of each cavity specimen also decrease over excitation amplitude for the second bending mode but increase for the third and fourth bending mode. Nevertheless, Fig. 8 shows as well that this effect is not significant when focussing at all specimens of same kind due to the large standard deviation s f,kind .
The loss factor over the excitation amplitudes for all five specimens and all considered bending modes is shown in Fig. 9. Similar to the assessment of the resonance frequencies, three characteristics of the loss factor for every mode are highlighted: (1) The standard deviation s ,mb in each measurement batch is with a maximum of 16% in relation to the respective mean loss factor significantly higher compared to s f,mb , see Fig. 9. The distribution of loss factor in dependence of shaking duration is, therefore, checked in the following paragraph. (2) The difference of the loss factor over the excitation amplitude with 100 mm/s 2 as a baseline is calculated for each specimen and mode, see Table 5. It shows first that the loss factors of the cavity specimens vary a lot more compared to the solid specimens. Second, the change in loss factor for the cavity specimens is by a factor of 4-6 higher in the third bending mode compared to the other modes and thirdly it can be seen that there is a slight decrease in loss factor for the solid specimens. (3) The standard deviation s ,kind for all specimens of same kind at the same excitation amplitude show a much wider spread between cavity specimens than between solid specimens, see Fig. 9. Therefore, the mean of loss factor of the k #1∕#2∕#3 a measurements for each cavity specimen at a specific excitation amplitude a for a specific mode n is put into relation to the mean of loss factor of all k #4∕#5 a measurements for both solid specimens to calculate the damping effect for each cavity specimen 10 Damping effect δη for all cavity specimens for all considered modes at all examined excitation amplitudes a; the highest measured damping effect for each specimen is quantified which expresses the difference of the loss factor of a specific cavity specimen in comparison with the solid specimens for the same mode and at the same amplitude. The damping effect for all cavity specimens for all considered modes can be found in Fig. 10. The bending modes show little to no damping effect for the lower excitation amplitudes. In contrast, the damping effect increases for higher excitation amplitudes. A positive damping effect can be seen for all bending modes but most prominently for the third bending mode. The damping of the torsional modes was also analysed but no significant damping effect was measured.
Because s ,mb shows a significant variance in the damping characteristics per measurement batch, a closer look at each measurement is taken and a decrease in damping over time is identified. To quantify this decrease in loss factor, the difference in loss factor between adjacent measurements within one batch Δ n is calculated, sequentially accumulated, and plotted against shaking duration of the respective specimen, see Fig. 11. As several batches are measured consecutively without removing the specimen from the shaker, their Δ n is offset by the final accumulated difference in damping of the prior batch to visualize the total accumulated difference in loss factor over the entire shaking duration. This shows a low and random variation in loss factor for the solid specimens in all bending modes, but a decrease of loss factor over shaking duration for the cavity specimens, particularly in the third and fourth bending mode. This is especially prominent for (7) #1∕#2∕#3 mode,a = mean #1∕#2∕#3 mean #solid − 1 the specimens #1 and #3, which are also the ones with the significantly higher damping effect, see Fig. 10. The measurement sequence described in Sect. 2.3 was unintentionally altered for specimen #3 by switching the order of the 100 mm/s 2 and 200 mm/s 2 measurement batches. When comparing specimen #3 with specimen #1, this does not seem to have a significant effect. However, comparing the accumulated total difference in loss factor over shaking duration of specimens #1/#3 with specimen #2 another influence factor shows a bigger influence which is discussed later in this study.

Computed tomography
To understand the decrease in loss factor and the particle distribution inside the cavity better, the CT scans of specimens before and after shaking is analysed. As mentioned before, the specimen orientations during CT scanning and modal analysis differ from each other, see Fig. 12. Before shaking, the particles fill up the whole cavity and are evenly distributed. Neither pockets of empty space nor regions with significantly higher packing density are formed. After shaking, a gradient of packing density in the direction of gravity during shaking � ⃗ g shake can be identified as the radiodensity in the cavity after shaking is higher at the bottom and lower at the top as shown in the graph in Fig. 12. In addition, small pockets of empty space at the cavity side against the direction of gravity during shaking can be seen. No gradient of packing density in the direction of gravity during CT scanning � ⃗ g CT or formation of empty space at the cavity side against this direction can be identified.

Discussion
The results in the present study show a considerable damping effect, which is mainly dependent on three parameters: (1) the specific cavity specimen, (2) the mode and (3) the excitation amplitude. The dependency of the damping effect on the specimen (1) can be identified in all examined modes: The damping performance of specimen #2 is always (and sometimes substantially) below the specimens #1 and #3 at comparable modes and excitations. The difference between the specimens of same design is probably based on differences in the manufacturing process or a potentially different handling of the specimens between manufacturing and experiment. The experimental setup is excluded as root cause as the specimens are remounted on the shaker and similar results as in the present study are achieved. The influence of the mode (2) on the damping effect could have two different explanations: The position of the cavity regarding the mode shape as well as the resonance frequency at the mode could be the root cause for the significantly higher damping in the third bending mode. A combination of both explanations is also conceivable and could be investigated by changing the length of the specimen and the position of the cavity in subsequent studies. Scott-Emuakpor et al. [35] proposed a multi-factor model to predict the damping performance, which would need to be validated for specimens out of AlSi10Mg. The dependency on the mode is also linked to the dependency on the specimen as the standard deviation s ,kind for all cavity specimens at the third bending mode is significantly higher compared to the other bending modes, see Fig. 9. The nonlinearity of particle damping, and therefore, the dependency on the excitation amplitude (3) is already mentioned in Sect. 1 of this article as well as in literature [26,27]. Subjected to a higher excitation amplitude, the particles in the cavity move more thereby dissipating more energy via the two effect mechanisms of particle damping depicted in Fig. 1. The well fit logarithmic trend curves for particle damping need to be verified with data for excitation amplitudes between the chosen amplitudes in the present study and beyond that, to ensure if the depicted trend persists. The damping effects measured by Scott-Emuakpor et al. [37] with similar specimen out of denser material were higher but in the same order of magnitude.
The decrease of resonance frequencies of the solid specimens, see Fig. 8, could be influenced by a slight opening of the clamp when the specimens move with larger amplitudes. This would elongate the swinging length according to Eq. (3) by 0.1% which corresponds here to 0.1 mm. The resonance frequencies of the cavity specimens on the other hand increase with excitation amplitude for the third and fourth bending mode, which are the modes with the highest damping effect at 800 mm/s 2 , see Fig. 10. In classical beam theory of damped vibrations, the damped resonance frequency is lower than the undamped natural frequency [40] which contradicts the measured effect. Although the general mechanisms of this effect still need to be investigated, a link between increasing particle damping and increasing is indicated in the scans overhead; the directions of gravity during shaking g shake and during CT scanning g CT are given; the empty space in the cavity after shaking is highlighted resonance frequency with increasing excitation amplitude seems conceivable. In addition, an explanation is still to be found why the resonance frequencies of the cavity specimens in the fourth bending mode are lower than the resonance frequencies of the solid specimens. As mentioned, the movement of powder is necessary to dissipate energy via the described mechanisms. Of special interest is, therefore, the ability of the powder to move, which is assumed to be reduced when the powder is packed more densely. The CT scans in Fig. 12 show areas with higher packing density after shaking, which shows a settling of the powder during shaking in the direction of gravity � ⃗ g shake . This seems to be the root cause for the decrease of loss factor over shaking duration, see Fig. 11. It is additionally assumed that the settled powder gets into a mesostable state, as the powder does not reorientate inside of the cavity in the direction of gravity during CT scanning � ⃗ g CT . This conclusion is drawn from the fact that there is no gradient of packing density in the direction of gravity during CT scanning � ⃗ g CT or formation of empty space at the cavity side against this direction. It is, therefore, deduced that the particles in the present study could have a lower tendency to move in the cavity compared to previous studies, which showed empty spaces existing extensively in bigger cavities [25,28,30] or with powders of smaller particle size [38]. As the specimens in the present study have comparably small cavities filled with coarser powder, a lower tendency to move is assumed which causes the decrease in loss factor. An explanation is still to be found, why the decrease in loss factor is mainly seen in the third and fourth bending mode. The correlation between the decrease in loss factor and the total damping effect of the specific specimen could also be based on the specimen-dependent movability of the powder in the cavity. Specimen #2 experiences a significantly lower damping effect but its loss factor does not decrease as much as specimen #1 and #3, compare Figs. 10, 11. One possible explanation is that the powder in the cavity of specimen #2 is denser packed prior to the experiment due to possible non-regular specimen handling. During shaking, the powder could, therefore, just settle slightly and loose only some of its damping abilities. Another possible explanation is the position of specimen #2 on the build plate during manufacturing between the other specimens, see Fig. 3. This could have resulted in a higher exposure to spatter or heat from the two adjacent specimens, which could have influenced the movability of the powder in the cavity. These hypotheses would need to be tackled with a study on the manufacturing parameter, especially the position and distance to other specimens on the build plate. Special care on specimen handling as well as shaking duration and amplitude would have to be taken care of. That is why only a qualitative analysis of the decrease in loss factor over shaking duration is conducted in the present study, as there are unmeasured times on the shaker due to the experimental procedure, see Fig. 11. Therefore, an additional, unmeasured decrease in loss factor is probable. This makes the settling effect even more significant and opens room for scientific studies to analyse the effect based on at least two parameters: At first, the long-term behaviour due to a constant force (like gravity) at different excitation amplitudes has to be investigated. Second, the external force could be increased to simulate other external forces like the centrifugal force, which is relevant in, e.g., turbine applications. In addition, measures to reactive the damping effect in specimens with densely packed powder have to be analysed.

Conclusion
The present study contributes to a better understanding of the governing influencing factors of AM particle damping with thin and flat cavities. It is shown that a damping effect of up to 564% can be achieved for AlSi10Mg specimens. However, this damping effect is dependent on the mode, excitation amplitude and even specific specimen. The highest damping is found in the third bending mode at the highest excited amplitude of 800 mm/s 2 in case of specimen #1. The two key findings of the present study are (1) the dependency of the damping effect on the specific cavity specimen and (2) the significant settling effect of the powder in the cavity which can be identified in CT scans and has an unfavourable effect on the damping characteristics over shaking duration. This settling effect has the potential to be subject of further scientific studies as well as relevant for future industrial applications.
Funding Open Access funding enabled and organized by Projekt DEAL. All authors certify that they have no affiliations with or involvement in any organization or entity with any financial interest or nonfinancial interest in the subject matter or materials discussed in this manuscript.

Data availability
The data that support the findings of this study are openly available in Zenodo at https:// doi. org/ 10. 5281/ zenodo. 77600 81.
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