Nonlinearity induced bifurcations and chaos in loudspeakers

[Zapper]

Hi. I skimmed the second article you linked. I did an MS thesis on chaotic dynamics in electronic circuits many years ago, so I get the general drift of the paper. Chaotic responses generally entail multiple stable or quasi-stable states. An example is an umbrella - it is normally concave downwards, but a gust of wind can snap in concave upwards. When driven by a large alternating gust, like in front of the monster subs you want to make, it may oscillate chaotically between the two states.

The paper didn't describe the physical mechanism behind the chaos but I suspect it's due to cone breakup. In fact, it's very similar to the umbrella I described - the cone can flex so much that the mechanical restoring force resisting bending starts to decrease and may even go to zero and reverse direction (in which case the cone could get stuck in the deformed condition). The large amplitude signal is forcing the cone to to flex in breakup modes with highly nonlinear restoring forces, and this generates the chaotic response. This has actually been used by guitarists seeking a very grungy sound, driving a speaker deep into breakup, or even cutting the speaker cone to assist breakup. Maybe the chaotic response is the source of the unique sound of speaker breakup.

In any event, the THD of the speaker will be very large before any chaotic response occurs, so the chaotic response will occur in some region between "sounds like cr*p" and "oh sh*t the speaker's on fire".

 

[kemmler3D]

These music styles don't require clean neutral subwoofers, they need to be very loud and go low (40Hz minium, mostly lower) and distortion is an inherent part of the sound, even in the music itself.

Not sure if that's true in this case. Did you listen to the samples in post #12? I wouldn't say they're really 240 BPM or anything, more like ~120-140ish with a lot of extra kick drums. In effect you have almost a droning bass with a lot of transients as you go. In order to differentiate each drum hit in a live venue you're actually going to need really clean response and a reasonably accurate impulse response, at least not afflicted with a lot of group delay.

For this particular genre of music I'd prioritize phase response and flat FR a lot more than I would for most dance genres, because otherwise the fast rolling kick drum effect will be spoiled. If you can still manage high SPL at low frequencies, then it would probably help but I don't get the impression it's the top priority here.

Nice, nothing wrong with a bit of old gabber but sadly it doesn't get much play time at the parties i usually go too

Breakcore is the odd one out of the genres i mentioned and not quite as fast as the others.

Haha yep i am indeed old enough to remember the dubstep era, i actually still quite enjoy it. More-so the chunkier UK style rather than the squeaky skrillex stuff but it do still hear it played out somewhat regularly here.

I'll link a few assorted tracks to clarify some of the styles that i've found issues with, its often common in these genres the bass line will be kick-bass-bass-bass but at 200bpm, the in between bass bits can really start to blur together especially at the lower frequencies where there is often high group delay on some box types.

https://soundcloud.com/wachumarecords%2F12-mirror-me-unalloyed-gold

https://soundcloud.com/bilycore%2Fbilycore-kromacore

https://soundcloud.com/theongoingarcanum%2Flaw-of-the-jungle

https://soundcloud.com/jiazihulive%2Fjiazihu-vallak-macabre-tale-release-soon-in-va-cyberroots

Weirdly, not the same psycore as the old psycore, a whole new branch came out from the psytrance scene about 6-7 years ago as a darker faster version of darkpsy+hitech and has basically no connection to the previous genre by the same name from the hardcore electronic scene. But yeah breakcore ain't new.


Yep i think that's the only real conclusion to come for unless i try to rebuild the simulation formulas with my own inputs.
Luckily if the hornresp sims are anything to go by the displacement is actually fairly low in the 30-100hz region where most of the music is.

Yes, this is quite different than the sort of music I had in mind. Honestly my reaction was not super positive, (dark ambient sounds with 1/16th note kick drums? okay)... some was OK and reminded me of Autechre if they fell in love with fast bass drums. That said, it would make a very different impression if I had had some LSD or a lot of shrooms in the past couple hours. So, probably fit for purpose?

 

[Waxx]

Not sure if that's true in this case. Did you listen to the samples in post #12? I wouldn't say they're really 240 BPM or anything, more like ~120-140ish with a lot of extra kick drums. In effect you have almost a droning bass with a lot of transients as you go. In order to differentiate each drum hit in a live venue you're actually going to need really clean response and a reasonably accurate impulse response, at least not afflicted with a lot of group delay.

For this particular genre of music I'd prioritize phase response and flat FR a lot more than I would for most dance genres, because otherwise the fast rolling kick drum effect will be spoiled. If you can still manage high SPL at low frequencies, then it would probably help but I don't get the impression it's the top priority here.

Yes, this is quite different than the sort of music I had in mind. Honestly my reaction was not super positive, (dark ambient sounds with 1/16th note kick drums? okay)... some was OK and reminded me of Autechre if they fell in love with fast bass drums. That said, it would make a very different impression if I had had some LSD or a lot of shrooms in the past couple hours. So, probably fit for purpose?

The thing is, the people does not care about that, it's the vibe of the music in combination with the drugs they take that is what they want. Accuracy is not a factor there. I don't think you've ever been to a rave with this kind of music if you ask such questions to be honest. When i was sound engineer i was probally the only guy in the room that was not totally wasted from dope (i only smoked weed, even no alcohol). That is how it was and still is today. Most are on coke, speed (amphetamines) and/or ket (ketamine), or combinations with or variations of those drugs. They want the energetic vibes and going party all night, not an accurate sound to listen to.

 

[kemmler3D]

The thing is, the people does not care about that, it's the vibe of the music in combination with the drugs they take that is what they want. Accuracy is not a factor there. I don't think you've ever been to a rave with this kind of music if you ask such questions to be honest. When i was sound engineer i was probally the only guy in the room that was not totally wasted from dope (i only smoked weed, even no alcohol). That is how it was and still is today. Most are on coke, speed (amphetamines) and/or ket (ketamine), or combinations with or variations of those drugs. They want the energetic vibes and going party all night, not an accurate sound to listen to.

I've been to my share of raves... did you listen to the samples?? Obviously you need a certain amount of bass to get the party going, but going for a TL / horn design and sacrificing transient response is not the way forward for this particular genre. If I am guessing right, these folks are not primarily on coke, speed or ket, they're on psychedelics. Totally different vibe. But I guess OP can enlighten us as to the chemical tendency of his audience.

 

[Waxx]

I don't know what raves you go to, but the breakcore and psycore raves i went to most people were on ket and or speed, also in recent times. I like the music, but i'm a bit fed up with the druggy crowd (I may be a bit old for it also), so i don't go that often anymore. Psychidelics is more for goa, tribal or psytrance, not for the harder styles that go above 150bpm. Even on acid raves, speed and ket is way more common than psychidelics.

 

[kemmler3D]

I don't know what raves you go to, but the breakcore and psycore raves i went to most people were on ket and or speed, also in recent times. I like the music, but i'm a bit fed up with the druggy crowd (I may be a bit old for it also), so i don't go that often anymore. Psychidelics is more for goa, tribal or psytrance, not for the harder styles that go above 150bpm. Even on acid raves, speed and ket is way more common than psychidelics.

Fair point, but listening to OP's examples of this new stuff, I couldn't really imagine (people) listening to it on anything but psychedelics. My hot take is it's a bunch of dark ambient with constant 1/16 note kicks and kick rolls over it. Describing it as high BPM is sort of misleading IMO, I would personally struggle to dance to it properly. But I'm more of a disco house guy so maybe don't listen to me.

 

[Zink]

Some thoughts in case you haven't already considered them as well.
You can reduce lowest mode inside by having the transducer midway of long dimension, or in case of multiple transducers put their acoustic center mid of the longest dimension. Non-parallel walls just randomize modes, and internal divider panels can make even longer paths and lower modes and so on, not really addressing any of it. They could be effective on short wavelengths, but also any damping material is. Perhaps there is something to exploit like 1/4 wave resonators, but in general the lowest mode would be loudest and hardest to tame so most problematic one. Anything that fits inside the box has quite little effect to it due to huge wavelength, hence putting transducer midway of long dimension is a good start for a good sounding closed box. If it is a problem in the first place.

If you imagine such box where woofer is middle of long dimension, for a given volume it could be relatively slim in depth and have relatively big square baffle with 1-4 woofers on it so that their acoustic center is center of the baffle. But then there is the utility, like standard truck dimensions, ply sheet sizes and so on. On the other hand, borrowing from big horn systems you might be able to get some extra output by maximizing baffle area and assemble a huge wall with the boxes. Not sure if there is enough benefit on such low frequencies though in any reasonable number of boxes and with big wall you might end up interference issues higher up in frequency requiring more processing. You'd have to make manifold kind of thing to house the driver magnet out but not sticking beyond the "baffle" robbing the volume inside, only few panels sizes and no angled panels for easy assembly though. And so on.

Thanks i will keep this in mind some useful ideas. If the driver was embedded too far inside the box is there chance it would start to cause a bandpass effect?

Hi. I skimmed the second article you linked. I did an MS thesis on chaotic dynamics in electronic circuits many years ago, so I get the general drift of the paper. Chaotic responses generally entail multiple stable or quasi-stable states. An example is an umbrella - it is normally concave downwards, but a gust of wind can snap in concave upwards. When driven by a large alternating gust, like in front of the monster subs you want to make, it may oscillate chaotically between the two states.

The paper didn't describe the physical mechanism behind the chaos but I suspect it's due to cone breakup. In fact, it's very similar to the umbrella I described - the cone can flex so much that the mechanical restoring force resisting bending starts to decrease and may even go to zero and reverse direction (in which case the cone could get stuck in the deformed condition). The large amplitude signal is forcing the cone to to flex in breakup modes with highly nonlinear restoring forces, and this generates the chaotic response. This has actually been used by guitarists seeking a very grungy sound, driving a speaker deep into breakup, or even cutting the speaker cone to assist breakup. Maybe the chaotic response is the source of the unique sound of speaker breakup.

In any event, the THD of the speaker will be very large before any chaotic response occurs, so the chaotic response will occur in some region between "sounds like cr*p" and "oh sh*t the speaker's on fire".

Cone breakup would indeed be a good explanation for the chaotic modes. The 2021 paper linked in the OP makes speculations regarding the overall stiffness of the driver in a given box but unclear to me if cone breakup modes are part of the given formula and seems to imply the occurrence is related to an imbalance between the air and nonlinearity in the suspension Kms(x). Some quotes from the article:

The experimental results suggest that the considered loudspeakers shows bifurcations and chaotic regimes, depending on the amplitude of the current. The theoretical reason for that was the nonlinear effective stiffness Keff(x), probably a consequence of the viscoelastic properties of moving parts

It is interesting to note that the theoretical explanations for the bifurcations and chaos in woofers reported in the literature, are not well established. Therefore, the adopted expression for Keff(x) in Eq. (11), even not being free from criticism, is inside that scope. But the fact is that, whatever the explanations or models, the new term due to the coupling with the sealed enclosure raises the effective stiffness of the woofer + enclosure system in relation to Keff , being now a decreasing function of V0. Therefore, the system becomes stiffer for smaller sealed enclosures, acting against the appearance of bifurcations and chaos. Moreover, beating frequencies involving f0 and its harmonics with woofer + enclosure resonance frequency f res are excited toward chaos for larger volumes and toward regular dynamics for small volumes. These conclusions, even far of being complete, are in the correct direction.

The reach of the model (3) should be evaluated noticing that the enclosure acts as a pressure reacting agent mostly affecting system’s stiffness. That is, it results in terms of the form A(x)x more than in terms of the forms B(x)dx/dt or C(x)d^2x/dt. In that sense, any other type of nonlinearity out the form A(x)x is not directly influenced by the presence of the enclosure itself. Even then, the proposed procedure for estimating the ideal volume can be promptly adapted to wider situations, taking into consideration other relevant types of nonlinear effects.

Our study also permits some predictions about what to expect in the case of vented enclosures. With effect, in this case the restoring force due to V 0 tends to be less effective when compared to the sealed enclosure. Therefore, the situation is shifted to a situation closer to that where V 0 ¼ 1, where less compression is expected but also with more distortions. In this sense, maybe our study points to an explanation for the common popular opinion that woofers installed in sealed boxes are generally more precise and musical than when in vented ones, which tend to be more efficient.

Seems that perhaps if a transducer had a completely flat Kms(x) would play a large part in reducing the phenomenon.

Yes, this is quite different than the sort of music I had in mind. Honestly my reaction was not super positive, (dark ambient sounds with 1/16th note kick drums? okay)... some was OK and reminded me of Autechre if they fell in love with fast bass drums. That said, it would make a very different impression if I had had some LSD or a lot of shrooms in the past couple hours. So, probably fit for purpose?

I'm not going to deny it's a very acquired taste and not super popular but there is a dedicated little bunch of people who can't seem to get enough of it!

I don't know what raves you go to, but the breakcore and psycore raves i went to most people were on ket and or speed, also in recent times. I like the music, but i'm a bit fed up with the druggy crowd (I may be a bit old for it also), so i don't go that often anymore. Psychidelics is more for goa, tribal or psytrance, not for the harder styles that go above 150bpm. Even on acid raves, speed and ket is way more common than psychidelics.

Fair point, but listening to OP's examples of this new stuff, I couldn't really imagine (people) listening to it on anything but psychedelics. My hot take is it's a bunch of dark ambient with constant 1/16 note kicks and kick rolls over it. Describing it as high BPM is sort of misleading IMO, I would personally struggle to dance to it properly. But I'm more of a disco house guy so maybe don't listen to me.

You are spot on with this it's mostly a psychedelic lsd+ketamine crowd, hardcore genres don't really get played at all at these parties no hardtek, raggatek, hardstyle, hardcore or hard techno stuff at all really it's mostly psytrance and this technical fast-ish dark ambient sound psycore, DNB does make an occurrence at these events too.

Having been to a bunch of these parties and having a dozen close friends who are producers in the genre we regularly critique the sound systems and the consensus for sure is for a accurate easy listening sound with dynamic clean bass effects at high volume. The dancing style is generally more of a groovy sway rather than a jump around rave type situation.

 

[tehas]

@Zink did you end up building out this system? Interested to hear your thoughts on the 18nls4000, its one of the few pro woofers that is suitable for home theater LFE duties in a small sealed enclosure.

 

[Zink]

@Zink did you end up building out this system? Interested to hear your thoughts on the 18nls4000, its one of the few pro woofers that is suitable for home theater LFE duties in a small sealed enclosure.

Not yet but will be getting into it soon, hopefully placing an order for drivers in coming months. Since i wrote this post too 18sound released the 18NTLS5000 which as far as i can see notably exceeds the 18NLS4000 in pretty much every aspect so i will be using those instead. The NTLS5000 is an absolute monster of a driver honestly there is basically nothing else on the market that even compares it has a very unusual mixture of TS parameters they have balanced out. Basically stuck the biggest possibly motor on it and then to compensate made the cone very heavy to bring the fs back down resulting in low sensitivity but extremely low vas/box size requirement for sealed in particular.

Having a search on loudspeaker database for drivers > 18" with VAS <100L and Qts < 0.4 leaves only 8 results (NTLS not listed there yet). link most of which are just extremely expensive car audio drivers and one other IPAL driver from 18sound which is very similar in many ways but obvious a pain to use a driver with a RE that drops as low as 1.5ohm. and a pair of PD drivers that do look good, but have basically half of the motor efficiency factor (BL^2 / RE) and xmax of the 18NTLS5000. The NTLS5000 has vas of 75L and Qts of 0.33 making it possible to have a Qtc 0.5 driver in only a ~70L box. Considering the drawing i did on the first post was quite large changing to this driver can potentially half the overall box size and be even higher quality due to the tetracoil benefits too. I believe without doubt it was intended to be aimed at high end cinema but using the modern engineering of high end PA drivers but it perfectly suits what i want them for so glad i didn't get the NLS4000 already!

 

[Axo1989]

... Yes, this is quite different than the sort of music I had in mind. Honestly my reaction was not super positive, (dark ambient sounds with 1/16th note kick drums? okay)... some was OK and reminded me of Autechre if they fell in love with fast bass drums. That said, it would make a very different impression if I had had some LSD or a lot of shrooms in the past couple hours. So, probably fit for purpose?

Important to consider the physio-chemical reproduction system in this case.

 

[Axo1989]

I'll link a few assorted tracks to clarify some of the styles that i've found issues with, its often common in these genres the bass line will be kick-bass-bass-bass but at 200bpm, the in between bass bits can really start to blur together especially at the lower frequencies where there is often high group delay on some box types.

https://soundcloud.com/wachumarecords%2F12-mirror-me-unalloyed-gold

https://soundcloud.com/bilycore%2Fbilycore-kromacore

https://soundcloud.com/theongoingarcanum%2Flaw-of-the-jungle

https://soundcloud.com/jiazihulive%2Fjiazihu-vallak-macabre-tale-release-soon-in-va-cyberroots

Weirdly, not the same psycore as the old psycore, a whole new branch came out from the psytrance scene about 6-7 years ago as a darker faster version of darkpsy+hitech and has basically no connection to the previous genre by the same name from the hardcore electronic scene. But yeah breakcore ain't new.

Nice selection from my PoV. Autechre on different drugs works for me. Incidentally and not what you are after but I enjoyed these tracks on my Audio Physic Codex (which have relatively modest 10" bass drivers mounted inside the cabinet, venting internally and front and back through ceramic foam). You need more for performance a large space full of people of course.

You could consider a stupidly large RiPol stack to keep reverberation under control. Or a DBA or Trinnov wave forming installation.

Edit: I missed "outdoor shows" so none of that.

 

[tehas]

Not yet but will be getting into it soon, hopefully placing an order for drivers in coming months. Since i wrote this post too 18sound released the 18NTLS5000 which as far as i can see notably exceeds the 18NLS4000 in pretty much every aspect so i will be using those instead. The NTLS5000 is an absolute monster of a driver honestly there is basically nothing else on the market that even compares it has a very unusual mixture of TS parameters they have balanced out. Basically stuck the biggest possibly motor on it and then to compensate made the cone very heavy to bring the fs back down resulting in low sensitivity but extremely low vas/box size requirement for sealed in particular.

Having a search on loudspeaker database for drivers > 18" with VAS <100L and Qts < 0.4 leaves only 8 results (NTLS not listed there yet). link most of which are just extremely expensive car audio drivers and one other IPAL driver from 18sound which is very similar in many ways but obvious a pain to use a driver with a RE that drops as low as 1.5ohm. and a pair of PD drivers that do look good, but have basically half of the motor efficiency factor (BL^2 / RE) and xmax of the 18NTLS5000. The NTLS5000 has vas of 75L and Qts of 0.33 making it possible to have a Qtc 0.5 driver in only a ~70L box. Considering the drawing i did on the first post was quite large changing to this driver can potentially half the overall box size and be even higher quality due to the tetracoil benefits too. I believe without doubt it was intended to be aimed at high end cinema but using the modern engineering of high end PA drivers but it perfectly suits what i want them for so glad i didn't get the NLS4000 already!

Makes sense, that ntls5000 looks spectacular particularly with the tetracoil motor. You should ask 18s for the Klippel LSI results, they were more than happy to share the raw db for the nls4000. The main reason I'll probably still stick with the nls4000 is the xvar - there's about 5 more mm of xvar which will be just shy of 3dB more output in the excursion limited domain below Fb, in particular for my application in 10-25Hz.

 

[Zink]

Makes sense, that ntls5000 looks spectacular particularly with the tetracoil motor. You should ask 18s for the Klippel LSI results, they were more than happy to share the raw db for the nls4000. The main reason I'll probably still stick with the nls4000 is the xvar - there's about 5 more mm of xvar which will be just shy of 3dB more output in the excursion limited domain below Fb, in particular for my application in 10-25Hz.

Already got in contact with 18s and asked for the LSI a few weeks ago, however they only sent back the Bl(x) and Kms(x) images and not the full document but from those alone i can say it looks as good as one can expect from a driver that size, up with the very best. Unclear if it's ok to share those images around online. Essentially Bl is near-completely flat until around the 8-10mm where it starts to pull down a bit at the corners but even past there the curve is not particularity steep. Kms also very good i would say flat up to 5mm and at around 10mm it has maybe 1-1.5 N/mm change so also no concerns. I'd love to see some pro-sound size drivers with the purifi surround on it one day

Personally, while the nls4000 does indeed have more raw xmax, for my intended pathway of attempting to maintain very high bass definition at higher volumes i plan to try and keep the drivers playing within that highly linear region <10mm xmax. So exceeding that significantly appears counterproductive as it just brings back well known nonlinear distortions and It's been my strong opinion for a while now that when you start pushing any subwoofer beyond it's halfway xmax point you are just trying to squeeze out the last drops of diminishing returns, you don't need a driver with more xmax you just need more drivers/subs!

 

[tehas]

apologies, was away from asr for the last month. I agree with your position on running drivers in their most linear range, but thats not always possible in a home environment where space is constrained. Hence the search for drivers that minimize nonlinear distortions all the way to xmax

 

[Zink]

Will be cnc cutting real boxes soon but i got impatient and converted a few old sub boxes into temporary testers. Still sealed ~85L volume. Did some very rough eq an sweeps in my lounge room and was very happy with results, then got them outdoors on the weekend for a bit more of a flex, and wow was very blown away somehow even with the much lower volume output vs other subs (ported and such) the actual perceived loudness and impact was MUCH more than anticipated Sadly was busy partying and did not get any outdoor sweeps.

This sweep was one box indoors, not sensitivity normalized.

 

[SoundsGood64]

Howdy, recently i've been interested in the idea of building large sealed subwoofers (18" drivers, ~200L) but recently came across some papers which i haven't seen discussed anywhere online showing some complex effects of nonlinear distortions, particular in larger boxes when user with higher input power. I've had a bit of a read over the papers but the implications still are unclear to me how much of an issue this is and finding what balance of box and driver parameters are ideal to avoid it.

The papers basically show at higher power levels a rapid breakdown of speaker behavior particular in larger boxes due to nonlinearity in effective stiffness between the motors suspension and the compliance of the air in the box. I've seen a handful of old comments online mentioning nonlinearity of air being an issue but often not giving good explanations why. It seems most transducers are generally only linear within a few mm of xmax, the newest paper gives a full complex mathematical formula for showing these effects using wave digital filters but the math is a bit above my level and i'm unsure how i could substitute in my own values for transducer/box to find the limitations of my own designs and where the limits lie. Has anyone got more information on this topic or done any of their own research or have thoughts? It mentioned while the graphs look concerning in theory higher order box designs (BR, bandpass ect) should have even worse issues than sealed :O

Here are the papers and a borrowed image from one.

https://www.sciencedirect.com/science/article/abs/pii/S0003682X21000682

First and newest paper is not public so you will have to track down your own copy...
but the older three are free to read:

https://www.researchgate.net/publication/267797837_Confirmation_of_chaos_in_a_loudspeaker_system_using_time_series_analysis

https://www.researchgate.net/publication/23638269_Verification_of_chaotic_behavior_in_an_experimental_loudspeaker

https://www.researchgate.net/publication/5325548_A_route_to_chaotic_state_on_an_electrodynamic_loudspeaker

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Wow, just came across this thread. I don't know about these physics, but bifurcation is neat. I wonder what happens to the bifurcation if there's just a small change, perhaps due to something not exact, randomness, or error. These are likely deterministic but maybe a really small change could impact the system in a big way.