A A A Modelling and visualization of surround buckling in electro-dynamic audio transducers Mattia Cobianchi 1 and Chris Spear 2 Bowers & Wilkins / Sound United B&W Group Ltd - Lennox Wood – Southwater RH139JJ, UK ABSTRACT The surround (or front-suspension) of electrodynamic transducers typically used in loudspeakers and headphones is a device that provides the axial restoring force for the diaphragm movement and re- strains the lateral and tilting movement. A non-linear phenomenon typical of surrounds is pressure- induced buckling, a sudden change in the shape of the surround under load. The question addressed by this paper is how to predict the conditions under which a surround will buckle, and how to measure and visualize it in physical prototypes. The modelling methodology was based on structural finite- element analysis, while the measurement of transducer displacement and video recording with a high- speed camera allowed the experimental verification. This methodology has been applied and tested on semi-circular rubber surrounds. The impact of working temperature and manufacturing toler- ances have also been explored. It was found that it’s possible to predict the pressure threshold trig- gering a specific buckling mode within a 10% error. Full 3D modelling is advisable to assess the buckling pressure of non-axisymmetric modes common in real transducers. At the same time, 2D modelling has been proven enough for the evaluation of the worst-case scenario / lowest buckling pressure. ‘Surround or front suspension spider 1. INTRODUCTION The surround of a transducer, also called “front suspension” (Figure 1), has multiple functions. Figure 1 - Transducer cross section with parts - Adapted from Svjo [CC BY-SA 3.0 (https://creativecommons.org/licenses/by-sa/3.0)] 1 mattia.cobianchi@soundunited.com 2 chris.spear@soundunited.com 21-24 auGUST Scortsi event Cas La inter.noise | ee 2022 It provides some restoring force for centralising the cone in the axial as well as the radial directions (and complements the spider in this role), the acoustic sealing to avoid short circuits between the front and rear radiation from the membrane, and a mechanical termination for the membrane vibration modes. The surround is contributing to the moving mass, thus should be as light as possible, but also have enough area to allow for the desired excursion. Some common materials used in the speaker industry are polyurethane foam, rubbers, cotton fabric or other treated fabrics and polymeric films (mainly for tweeters and compression drivers). Suspension parts are critical for the transducer behaviour, performance and reliability, and the geom- etry of the part is normally the most important aspect to consider in the design. The material and the manufacturing process are also important. Design work thus involves mainly optimizing the geometry and selecting the right material for the application, rather than “hunting” for exotic and expensive engineering materials. The surround main design parameter is the stiffness as a function of the transducer displacement, but some other phenomena need to be taken into account: mechanical vibration modes in the surround causing irregularities in the acoustic response; the acoustic pressure inside the box causing a sudden flipping over of the surround (also known as inversion, roll collapse or buckling). Buckling is a form of structural instability where the progressive increase in the applied load will result at some point in the bifurcation of the deflection. The load at which this happens is called critical load (shortened in CL from now on). Many factors like material inhomogeneities, geometrical imperfections, eccentricity of the applied load etc. can make the exact prediction of the CL very difficult. There are two different types of surround buckling: one is caused by the internal stresses in the ma- terial when the surround geometry is deformed, as shown in (1), the other happens only when a trans- ducer is put inside a box, and a pressure difference between the external ambient pressure and the internal box pressure builds up following the box volume change for high excursions. This mecha- nism has been analysed with structural finite element analysis by Steve Mowry in (2) and other arti- cles written by him for the industry magazine “Voice Coil”. No literature is known to the authors that shows buckling modes in physical samples and a verification of their critical loads with structural finite element analysis. This paper contributes in filling this gap by reporting some numerical modelling and empirical results towards a practical engineering solution to predict dynamic buckling loads of surrounds at the design stage. Buckling theory and the physics involved will not be treated here but is extensively covered in textbooks like (3) and (4) for buckling in structural mechanics and in (1) for static buckling of transducer components. While buckling does not pose a risk to the structural integrity of the part if this is made of rubber, it can create wrinkles and stress accumulation areas in fabrics and polymeric films and thus pose a danger to the part reliability. In both cases though, it has a detrimental effect on the sound quality since it will produce audible artefacts. This, depending on the type of signal that has triggered the buckling and its modal shape, can for example transform a bass note into a motorboating noise, or a kick drum into a clapping noise. These artefacts are produced by a combination of two factors: the sudden deformation of the geometry radiating sound, and the non-linear distortion generated by the discontinuity in the surround stiffness versus displacement curve that the buckling produces. As for simple arches, the CL depends only on the geometry, Poisson ratio and Young modulus, but not on density (1). While design considerations about how to optimize the geometry to be buckling resistant are beyond the scope of this paper, it’s worth mentioning that Mowry has reported in (2) that “as a guideline in uniform thickness, buckling resistance goes as a quantity between the thickness squared and the thickness cubed”. Some companies have also introduced elaborated surround geometries to increase the buckling stiff- ness without penalizing the weight by increasing the thickness, such as in the Purifi long-throw trans- ducers or the Kef subwoofer surround disclosed in the patent (5). 2. MODELLING Finite element analysis was performed in the software Comsol Multiphysics 6.0 running on a laptop with CPU Intel Core i7-7600 @2.8Ghz 2 cores, 16 GB ram and Win10 Pro OS. 2.1. Geometries The geometries analysed where all of the half roll type. A first round of simulation took place to find the best modelling strategy and test for mesh convergence. In this round, a semi-circular roll with a 5.5mm radius, 0.5mm thickness and 144mm diameter (6.5” nominal) was modelled in 2D axial sym- metry and full 3D with both shell and solid formulation. A second round of simulation studied instead four surround roll geometries imported from 3D CAD data of physical samples with roll thickness ranging from 0.5mm to 1.5mm and diameters from 6.5” to 12” that we had available to test the accuracy of the predictions against measurements. 2.2. Mesh In solid 2D, triangular and rectangular elements were tried, both providing good results, but given the very short computation time of about 30s to solve for 2000 degrees of freedom (DOF), the addi- tional time required to set up a correct mapped mesh is probably not justified, and triangular elements are thus recommended for 2D problems. The number of mesh elements per thickness was not critical and one single triangular element per thickness was enough to get a CL very close to the asymptotic value reached for the maximum number of DOF. In Figure 2, the calculated CL versus the number of DOF is plotted for the solid 2D showing that as little as 2E3 DOF are enough to converge. In shell 2D, using a number of elements to achieve the same “radial” resolution of the roll used in the 2D solid, the computation time was a little bit higher, but the CL value the same as solid 2D within 5%. In Figure 3 for shell 3D models with a free triangular mesh, the convergence happens at 1.1E5 DOF, which were solved in 46s. In solid 3D, free tetrahedral meshing was tested against a swept mesh using a mapped mesh on the boundary acting as the source face. The convergence results in Figure 4 show that the free tetrahedral mesh allows to converge from about 2E5 DOF (taking about 90s of compu- tation time), while in Figure 5 the convergence “seems” to happen around 5E5 DOF if looking only at the CL value. 5300 5300 4300 4300 3300 3300 2300 2300 1300 1300 0 5000 10000 15000 20000 0 500000 1000000 Figure 2 - Solid 2D - CL in Pa versus number DOF Figure 3- Shell 3D - CL in Pa versus number DOF 5300 5300 4300 4300 3300 3300 2300 2300 1300 1300 0 500000 1000000 0 500000 1000000 1500000 2000000 Figure 5- Solid 3D swept mapped mesh - CL in Pa versus num- Figure 4- Solid 3D free tetrahedral mesh - CL in Pa versus ber DOF number DOF What is not visible from this figure though, is that while the free tetrahedral mesh leads to physical buckling modes even from the lowest number of DOF, the swept mapped mesh although elegant and practical in allowing the user to control finely the mesh size in the three spatial directions, leads to unphysical buckling modes as in Figure 6 and Figure 7 until about 5E5 DOF, and the first result that shows both a correct CL and a physical mode is the one in Figure 8 at 6.26E5 DOF. In shell 3D the same issue of late convergence and unphysical modes with a mapped mesh on the boundary was found, although the results are omitted here for lack of space. Figure 6 - Unphysical buckling mode for Figure 7- Unphysical buckling mode for Figure 8- Physical buckling mode for 2.36E5 DOF 4.72E5 DOF 6.26E5 DOF According to these results, a free tetrahedral mesh even with one single tetrahedron per thickness is the recommended meshing strategy for buckling problems in solid 3D, and a triangular mesh in shell 3D. The converged CL value difference between 3D solid and shell formulation is within 5%, thus the shell formulation is a viable and time-efficient option taking about half the time of the solid for- mulation problem to solve. 2.3. Physics and boundary conditions The Structural Mechanics module was used testing both the solid and shell formulation. Shell has some advantages: it allows to draw only lines in 2D or surfaces in 3D, making the geometry prepara- tion quicker, it´s easier to mesh and generally results in problems with a lower number of degrees of freedom that are quicker to run. It may prove hard though to faithfully reproduce in shell some com- plex ribbing patterns or variable thickness profiles. Common boundary conditions imposed were a rigid connector with a spring foundation to represent the spider stiffness on the surround lip where it is glued to the cone while preventing any deformation of that surface since it’s glued to a stiff cone, a boundary load on the inside of the roll of 1 Pascal to represent the loading by the pressure difference between the inside and outside of the box, and a fixed constrain on the surround landing where it is glued to the chassis. These are summarized in Figure 9, using the 2D geometry for clarity. Common physics settings were the use of a linear elastic formula- tion for the material model assuming also homogeneous and isotropic properties. The discretization settings were set to use quadratic elements. The simplification of a rigid connection boundary condition is a hypothesis that should be tested when there is the suspect that the cone can partially deform, as in the case of thin paper cones, polypropyl- ene cones etc. oy Figure 9 - Boundary conditions in 2D For the solid formulation both in 2D and 3D, it's worth mentioning that since the analysis was con- ducted for rubber, a material that is quasi-incompressible and with a Poisson ratio of 0.499, a “pres- sure formulation” in the material model node had to be selected - see (6, 7 page 414). 2.4. Material properties The type of material used in all of the studies was rubber, and this is also the most common surround material used in hi-fi loudspeakers. In the range of strain that it has to withstand in a speaker surround it can be considered a perfectly elastic material, isotropic and homogeneous [2, 6, section 33.5]. The material properties relevant in a buckling analysis are then the Young modulus and Poisson ratio. Rubber comes in many different types of compounds, of which SBR (styrene butadiene rubber), NBR (nitrile butadiene rubber), or a mix of the two are the most common options. The shore hardness (usually on the scale A) is the other specification required, which determines its Young modulus. Typical shore hardness values used range thus between SHA 40 and SHA 60 and seldom go above this. The Poisson ratio of rubber, as a quasi-incompressible material, was set for all to 0.499. The four transducers analysed used three different types of rubber, with a density ranging from 1200 to 1300kg/m^3. For two of them, laboratory measurements were available in the form of DMA (Dy- namic Mechanical Analysis) data for samples provided by the authors. The value at the lowest fre- quency, the closest to the static one, was used in the modelling. For the third material of SHA 50, an empirical relationship that relates shore hardness with the Young modulus was used (9,10). In Table 1 the Young modulus for all the physical samples modelled is reported. Table 1 - Young modulus value used for the surroun d rubber of physical samples Transducer model Young Modulus MPa 6.5" woofer 3.05 6.5" bass-mid 3.40 10" woofer 4.00 12" subwoofer 3.05 “Prescribed displacement equal t00 along rin 2D axialsymmetryand along x and yin3D spring foundation to represent ‘Boundary load equal to-1 aja nti Pascal of uniformpressure 2.5. Type of study A linearized buckling analysis (7 page 302, 11) was performed, using the pre-defined study in the Structural Mechanics module of Comsol called “Linear Buckling” and setting to 10 the number of buckling modes to calculate. This is the simplest type of analysis, which basically computes the in- ternal stress state of the structure from the applied static load in its undeformed state, and then solves for an eigenvalue problem to determine the critical loads. This is not completely representative of the loading conditions used in the experiment, since the load in the latter case was applied in sinusoidal regime at low frequencies (5-10 Hz), but it proved to be a good approximation. 2.6. Modelling Results All the results using colour scales have been plotted using the colour-deficiency optimized Cividis colourmap (12). These results show the buckling mode shape as a deformed shape with the colour mapping the local displacement. As it is an eigenvalue problem, the deformation amplification was chosen to ease the mode identification, while the displacement values and scale are arbitrary. The relevant results to assess the performance of the design are the critical load value (CL), which represents the multiplication coefficient of the boundary load set as a boundary condition, and the mode shape. Since the boundary load was set to 1 Pascal for all models, all CLs below are also in Pascal. The actual buckling shape of a structure for a given load can be different from one of the eigenmodes calculated in the eigenvalue problem because it is the result of the non-linear post-buckling evolution of the linear mode, thus the observed buckling shape in a real surround may not be immediately recognizable as one of the modes of the basis. And as described in the previous section, since in the experiment the application of the load in sinusoidal regime was also deforming the surround roll, this would result in a slightly different geometry from the one around which the problem was linearized. For annular plates with both inner and outer edges clamped, Bloom and Coffin state that “the axisym- metric buckled state is not always necessarily associated with the lowest critical load.” (13) This has been for example verified numerically when analysing a very shallow geometry, finding that indeed the first buckling mode with the lowest CL is not axisymmetric, Figure 10 and Figure 11. Figure 10 - Shallow large surround buckling mode Figure 11 - Shallow large surround buckling mode / side view In this case it’s recommended that the model is always run in full 3D. For what concerns the geometries analysed in this paper, which can all be considered as a section of a toroidal shell, we found that the axial symmetric mode was always the lowest in CL, and it is fol- lowed by higher order modes which are closely spaced in CL value within 15% of the axisymmetric CL – see Table 2, where the buckling mode observed in the test samples is highlighted in bold. 9512 Surtece: Otaplecement magnitude (mm) Table 2 - First ten modelled buckling modes critical loads of the four test samples Buckling mode number Critical load - Pascal 6.5" woofer 6.5" bass-mid 10" woofer 12" subwoofer 1 2911.0 2243.2 1364.1 3349.8 2 2998.1 2385.5 1408.5 3469.7 3 2999.7 2393.8 1408.7 3478.8 4 3062.9 2402.2 1428.0 3510 5 3064.1 2409.7 1428.0 3523 6 3182.2 2458.6 1431.1 3546.3 7 3182.9 2460.5 1431.8 3549 8 3320.3 2530.6 1435.4 3569.7 9 3322.6 2531.4 1435.5 3575.6 10 3388.9 2570.6 1443.9 3622.6 This suggests that a good engineering approach to modelling buckling would be to always start with the 2D problem. This would provide the axisymmetric buckling mode CL in a very short computation time and from the above results this is also likely to be the worst case scenario. The exact mode that will be triggered in a physical sample is a function of many factors, some controllable, some not. Among these, material imperfections (very hard to spot), asymmetries in the gluing of the parts that can cause both mass and stiffness imbalances leading to rocking modes in the moving assembly dur- ing the excursion, and asymmetries in the back pressure on the cone and surround due to standing waves in the enclosure and/or asymmetric box construction. 3. MEASUREMENTS AND HIGH SPEED VIDEO RECORDING 3.1. Stimulus playback to trigger dynamic buckling The transducer needs to be mounted inside the enclosure it will be used in, ideally, but for the purpose of verifying the CL value, a small sealed test box of known volume can also be used. Sandbags or bricks can be put inside the box to reduce the volume if needed. The box volume can be calculated from the 3D drawing or it can be inferred by comparing the free air resonance frequency of the trans- ducer with the in-box resonance as long as no absorbing material is used inside the box to increase its apparent volume. An audio amplifier connected to a signal generator can be used, and a sinusoidal tone played-back while stepping up the voltage until buckling is visible in the high speed video recording or its artefacts become audible. The frequency of the tone can be chosen according to different criteria: around the system resonance so that the moving parts velocity is maximum and forced con- vection cooling efficiency is also at its maximum to keep the voice coil temperature low; at a frequency that is low enough to allow the camera to acquire enough frames per oscillation cycle; if the transducer is tested in the final product enclosure, at the frequency where the internal box pressure is maximum. For our measurements we used a Klippel Distortion Analyser system using the TRF module to gen- erate tone bursts of increasing peak voltage at frequencies between 5 and 10 Hz. The advantage of this technique is that the high crest factor of the tone burst allows high peak displacement with limited heating of the voice coil, and the system automatically acquires the displacement measured with a triangulation laser whose purpose is described in the next section. 3.2. Indirect measurement of box pressure through volume displacement The direct measurement of the internal box pressure in typical loudspeakers requires the use of spe- cialised 1/4 or 1/8 inch high pressure microphones since the values can easily reach 165 to 170 dB SPL, and the drilling of a pass through hole for the cable routing. A practical indirect measurement can instead be obtained, if the internal net volume of the box is known, by measuring the cone displacement to extract the volume displacement and thus the relative volume modulation. Triangulation laser displacement sensors are common in transducer R&D labor- atories and this was the preferred method used for this investigation. The pressure difference 𝑃 𝑒 be- tween the outside ambient pressure and the inside pressure in a sealed enclosure can be calculated from the adiabatic gas equation (14) as Equation 1 −𝑘 ∆𝑉 −1൨ (1) 𝑃 𝑒 = 𝑃 0 ቀ1 − 𝑉 ቁ Where 𝑃 0 is the static pressure (about 100000 Pascal), k is the coefficient of adiabatic process (about 1.4 for air, and often indicated with γ), V is the enclosure volume, ∆𝑉 is the volume modulation (equal to the transducer surface area multiplied by the displacement). 3.3. High speed video recording High speed video recording (HSVR) has become very accessible in the past few years thanks to the advances in imaging sensors and processing power on consumer cameras and smart phones. Current mid-range smartphones can shoot up to 980 frames per second at 720p, and flagship models up to 1920 frames per second at 720p resolution. For our recordings we used an inexpensive Sony DSC- RX100M5 consumer camera that can record video up to 960fps (800x270) and an old Fujifilm Finepix HS10 used at 120fps, 640x480 pixel resolution. Both provide full manual control over fo- cus and aperture. At such high fps, the amount of light necessary for a correct exposure requires the use of powerful external photographic lights. To avoid flickering, if an incandescent bulb is used, it must be of suffi- cient power and thus size and mass to avoid a slight dimming when the AC current passes through 0. If LED lights are used, these must use a non-pulsed driver, or a pulsed driver at a frequency above the camera fps. Since rubber, as mentioned above, would soften if heated up, one or two LED lights providing 5000 lm each were used. A picture of the setup is visible in Figure 12. Figure 12 - High speed video recording setup 4. RESULTS AND DISCUSSION In Table 3, a summary of the modelling and experimental results for the CL of the observed buckling modes is available. All CLs modelled in Comsol are within a 10% error from the measured pressure. Table 3 – Comparison of modelled and measured buckling critical loads Transducer Young Modulus Measured pressure Comsol critical load for observed Error model of buckling mode bm order MPa Pa Pa % 6.5" woofer 3.05 3159 2911 7.9 6.5" bass-mid 3.40 2394 2459 2.7 10" woofer 4.00 1388 1364 1.7 12" subwoofer 3.05 3300 3623 9.8 It’s important to note that the visual evaluation of HSVR frames to assess at which displacement/pres- sure buckling is happening is not a straightforward process and adds some uncertainty in the estima- tion of the CL pressure. In Figure 13, four different frames from four different forward peak displacement values for one of the transducers under test show how the visibility of buckling progressively increases, and for each the computed box pressure is reported in the top left corner. The determination of which is the exact pressure from which the structure can be considered buckled is somewhat arbitrary. For comparing modelling results with displacement measurements and HSVR, the choice was to consider as CL the lowest pressure at which a buckling mode starts to be visible. In the case below this was 3159 Pa, and the area circled in red was the one monitored. In Figure 14 the modelled 1 st buckling mode and its CL are in good agreement with the experimental mode and load value of Figure 13. Figure 14- 6.5" woofer 1st modelled buckling mode Figure 13- 6.5" woofer HSVR frames of forward peak dis- placement at four driving voltages In Figure 15, the observed buckling mode of the 12” subwoofer is non-axisymmetric and shows 10 radial lines well captured in the modelled buckling shown in Figure 16. Please refer to section 2.6 for an explanation of the differences between the modelled and observed mode. Critical load factor=-2911 Surface: Displacement magnitude (m) Critical load factor=3622.6 Surface: Total displacement (mm) A6a6x10 x10 oy vo Figure 16- 12" subwoofer 10th modelled buckling mode Figure 15- 12" subwoofer triggered buckling mode Because of limited space, for the other two transducers tested only the buckling modes modelled are reported in Figure 17 and Figure 18 with their relative critical pressures, to be compared with the measured ones availabe in Table 3. Figure 17- 10" woofer 1st modelled buckling mode Figure 18- 6.5" bass-mid 6th modelled buckling mode 5. PRACTICAL ENGINEERING CONSIDERATIONS AND LIMITATIONS In a product development environment, there are always trade-offs to negotiate. For a surround opti- mised to avoid any buckling, the worst case scenario towards which to design would be the bottom tolerance for the shore hardness of the rubber and for the roll thickness, and the maximum continuous operating temperature. Typically the tolerance on a shore A hardness specification is ± 5 degrees, thus the bottom tolerance to use in the simulation is 5 degrees lower than nominal. If we refer to the shore hardness to Young modulus relationship available in (9,10) it's easy to conclude that using a lower shore hardness makes the design more robust in this regard, at the expenses of the surround thickness and thus weight and the transduction efficiency. The tolerance on the thickness depends on the supplier process capabilities, where 10% of the thickness would be a common tolerance. This translates in modelling a geometry that is 10% thinner than nominal. Since rubber is a viscoelastic material whose properties depend on temperature, with a softening for an increase in temperature, it's important to consider what will be a reasonable maximum operating Critical load factor=-1364.1 Surface: Displacement magnitude (rm) 2458.6 Surface: Total displacement (mm) A 207,30" x10 20 ie 16 a 32 ho 2 6 4 2 0 vo temperature. Typical causes for an increase in temperature are the conversion of mechanical energy to heat because of the intrinsic loss / mechanical damping of the material, or the increase in temper- ature of the air in the box because of the heating of other parts inside it (transducer voice coil, ampli- fier, power supply etc.). For an SBR (Styrene Butadiene Rubber), an increase from 20 to 80 Celsius could for example result in halving the Young modulus. Some limitations of this investigation include: the lack of an analytical solution to use as a benchmark for the buckling numerical analysis; a limited number and variety of geometries tested that restricts the generalizability of the findings; some uncertainty in the material properties and the lack of control on potential imperfections that can have an important impact on the value of the critical loads; the approximation of a dynamic buckling condition with a static linearized buckling model; the absence of temperature monitoring of the surround material itself with an infrared camera to monitor if the rubber experienced any temperature rise that could have caused its softening; the uncertainty on the internal box pressure measurement due to the indirect measurement through the transducer displacement; some uncertainty in the estimation of the CL pressure by visual inspection of HSVR frames. 6. CONCLUSIONS It was found that it’s possible to predict the pressure threshold triggering a specific buckling mode within a 10% error within the limited population of the samples studied. It was not possible to tell which one of the many modelled buckling modes that can be triggered was actually triggered in real samples, as this would require a more complex non-linear post buckling analysis. It may well be the case that this is not possible at all since many of the factors involved in dictating which mode the structure will collapse to (material imperfections, assembly tolerances, asymmetrical pressure loading from asymmetric box construction etc.) cannot be known a priori. Full 3D modelling is advisable to assess the buckling pressure of non-axisymmetric modes common in real transducers, but 2D modelling has been proven enough for the evaluation of the worst-case scenario / lowest buckling pressure. High speed video recording is an accessible yet powerful technique to capture vibrating structures deflection in transducers that has allowed to document the shape of the different buckling modes triggered. Three main areas can be explored for future work: more advanced modelling, an investigation into the material model impact on the buckling, and the objective and subjective evaluation of the acoustic artefacts. For what regards modelling, it would be useful to perform a linear buckling analysis on the pre- deformed surround shape at different displacements considering the internal stress status at those displacement to test the assumption by Mowry (2) that the rest position presents the worst case sce- nario in terms of critical load and/or a non-linear post buckling analysis with the addition of typical imperfections to try to predict the exact buckling mode that will be triggered in reality. The viscoelastic properties of rubber (entailing frequency-dependent Young modulus and damping) and their impact on the critical load and modes have not been explored in this study, but can play an important role in real products where the maximum box pressure is at frequencies far from those used in our experiments. Finally, the sound radiation by buckling modes with acoustic measurements and subjective evaluation is an important object of study to verify if an inevitable buckling behaviour can be made less audible by designing the surround geometry in order to support specific buckling modes. 7. AUTHORS CONTRIBUTIONS Conceptualization MC and CS, numerical modelling MC, measurements MC and CS, analysis of results and discussion MC. All authors have read and agreed to the published version of the manu- script. 8. ACKNOWLEDGEMENTS We thank our colleagues Ali Salehzadeh Nobari for his contribution at the modelling stage and some verification measurements, and Roberto Magalotti for his help in reviewing the manuscript. 9. REFERENCES 1. Bolaños F. Stress Analysis on Moving Assemblies and Suspensions of Loudspeakers. In: journal of the audio engineering society [Internet]. San Francisco, Ca, USA; 2006. p. 20. Available from: http://www.aes.org/e-lib/browse.cfm?elib=13773 2. Mowry S. Searching for the Ideal Surround. Voice Coil. 2009 Oct;21(12). 3. Jerath S. Structural stability theory and practice: buckling of columns, beams, plates, and shells. Hoboken: Wiley; 2021. 4. Lindberg HE. Little Book of Dynamic Buckling [Internet]. 2003. 39 p. Available from: http://www.lindberglce.com/tech/LittleBook.PDF 5. Allan James Skellett, Jack Anthony Oclee - Brown. LOUDSPEAKER DRIVER SURROUND. US 2018 / 0242086 A1. 6. Sönnerlind H. Modeling Linear Elastic Materials – How Difficult Can It Be? [Internet]. 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Optimizing colormaps with consideration for color vi- sion deficiency to enable accurate interpretation of scientific data. PLOS ONE. 2018 Aug 1;13(7):e0199239. 13. Bloom F, Coffin D. Handbook of thin plate buckling and postbuckling. Boca Raton, FL: Chap- man & Hall/CRC; 2000. 786 p. 14. Zoltogorski B. non-linear distortions of loudspeaker radiators in closed enclosures. In: journal of the audio engineering society [Internet]. Munich, Germany; 1999. Available from: http://www.aes.org/e-lib/browse.cfm?elib=8286 Previous Paper 17 of 769 Next