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I'm curious about an explanation for why a wind-up spring KERS (Kinetic Energy Recovery System) isn't worth it for a bicycle.

The system I have in mind is instead of braking the rear wheel you engage something that winds up a spring or elastomere as the bike slows. Then some ratchet mechanism stops the elastic from recovering and spinning the wheel in reverse. To use the energy stored in the spring you have a control to release the ratchet and engage some gears that reverse the spin, from reverse to forwards.

Let's assume there are significant losses, say 80%. Given rush-hour start-stop traffic even a 20% energy recovery assist might make sense in terms of the weight/complexity budget.

So how come we don't have such a system already, what is this scenario overlooking?

Leeroy
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    This already exists in regenerative brakes on ebikes. However they're not common. – Criggie Oct 30 '22 at 18:40
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    While a contentious issue, many believe that normal (usually light steel) frames exhibit some amount of storage of spring energy during the power stroke and release it in the dead zone, which can aid in climbing etc. In BQ-influenced times this is often called 'planing,' a term you can Google, but it's been described other ways throughout the history of safety bicycles. – Nathan Knutson Oct 30 '22 at 20:29
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    Here's someone who tried with a flywheel: https://www.youtube.com/watch?v=gahKxbwUcYw – Erty Seidohl Oct 31 '22 at 02:59
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    If I had to do it, it would be with a motor/generator and some big capacitors. – MaplePanda Oct 31 '22 at 07:09

5 Answers5

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Ain't gonna happen.

If you want to store energy of a 100kg cyclist+bike going at 40 km/h (plausible if you want to brake at the end of a downhill), you want to store:

0.5 * 100 * (40/3.6)^2 = 6170 J

Let's think how big spring you need. Elastic energy U is (from Wikipedia):

U = 0.5*V/E * sigma^2 = 0.5*m/(rho*E) * sigma^2

giving us:

m = 2*U*rho*E/sigma^2

...where U = 6170 J, sigma = 1000 MPa = 1000e6 Pa for a good spring steel, rho = 8000 kg/m^3 (approximately) for steel and E = 200 GPa = 200e9 Pa for any kind of steel.

So we need a mass of

m = 2*6170*8000*200e9/1000e6^2 = 19.7 kg

Who would accept an additional weight of 19.7 kg on a bike, when all it can do is to store energy corresponding to braking from 40 km/h to stop -- just once.

Let's consider a 2.6 kg 500 Wh e-bike battery. 500 Wh is 1800000 Joules, or approximately 292 times stopping from 40 km/h to zero.

Which one would you take? 2.6 kg to store energy corresponding to stopping 292 times, or 19.7 kg corresponding to stopping once?

Besides, even those e-bikes that can have 500 Wh battery don't have regenerative brakes.

juhist
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    Good job, actually giving a size estimate for the spring. I had the idea that this spring would need to be rather heavy, but I'm not deep enough into the physics of springs to put numbers to the claim. Just one nitpick: For people in flat areas, storing the energy of coasting at 25 km/h is all that's really needed. That would put the spring mass at (5/8)^2 = 40% of what you calculated, which is a spring of "just" 7.1 kg. However, that's before the weight of the transmission is added... – cmaster - reinstate monica Oct 30 '22 at 20:46
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    @cmaster-reinstatemonica The point of such a mechanical assist isn't to recover all the speed of 25km/h, just to overcome inertia for the first few seconds after braking. Once the bike is rolling accelerating it by pedalling is much more enjoyable. It's those first few strokes from full stop that are the problem. – Leeroy Oct 30 '22 at 22:02
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    @Leeroy May I suggest trying an IGH bike? The first few strokes are a problem if you are not in an adequate gear (= lowest possible gear). Chain shifts have a hard time shifting back to a high gear while you are still crossing the intersection, and chain shifts are impossible to shift down after an emergency brake. Not so Internal Gear Hubs. You can shift from highest to lowest gear at complete stop, and you can shift back up in two to three chunks before you have crossed the intersection. Even a simple, second hand 7 speed IGH will do the trick. Makes you feel like a bullet in a barrel. – cmaster - reinstate monica Oct 30 '22 at 22:13
  • So because such a system would fail the design constraints of weight, complexity and possibly even safety (while the spring is loaded)... The appeal of going for a mechanical system would be that it is less complex than an electric bike. However this scenario really illustrates the outsized advantages of an ebike!

    Very cool answer. Follow-up: what about a hybrid brake-KERS – feathering brake lever loads a small spring to max and beyond that moves on to clamping the brake pads like a regular brake?

     – Leeroy Oct 30 '22 at 22:13
  • @cmaster-reinstatemonica Yep I agree IGH is the way to deal with start-stop traffic situations. But it's not actually recovering any energy. With a KERS you could load up on "free" energy from rolling downhill faster than you need to for example. – Leeroy Oct 30 '22 at 22:18
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    Perhaps compressed air instead of a steel spring. Fails the "wind up" part of the question, but makes the weight and transmission challenges much more tractable. – Perkins Oct 31 '22 at 14:40
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    Why did you decide on this equation? This energy calculation is for a solid isotopic material being deformed. This would be how much mass you need for a solid piece of steel that you stretch to store energy, not a spring which can store more energy with the same lateral movement. Not to say its plausible, but this isnt the right math as far as I can tell. – JMac Oct 31 '22 at 15:12
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    Agree with @JMac, I don't quite follow the equation. The energy stored by a spring is related to how far it stretches. I don't see how you can go directly to a "Joules per kg" value given only the material properties of steel but no information about the spring itself. This seems to imply that all steel springs weighing 1kg can store the same amount of energy, which I am skeptical of. Can you really calculate the energy storage capacity of a spring given only its material and mass? – Nuclear Hoagie Oct 31 '22 at 16:26
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    Yep, the equation may not be 100% accurate for spring, where some parts are in tension and others in compression. However, it gives the right order of magnitude, the amount of elastic energy that can be stored into a material before it fails. – juhist Oct 31 '22 at 16:38
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    @JMac , Nuclear Hoagie, and juhist: Correct, for real springs some of the material will not be stretched/sheared to it's yield point. This is a lower bound of how much steel you'd need at a minimum to store that much energy. For actual springs it might weigh a factor of 2 to 10 more depending on how well the spring maximizes it's specific energy density. – Rick Oct 31 '22 at 16:45
  • @juhist Your link is to a specific type of strain, the axial stretching of a wire (or equivalent). The analysis doesnt work at all for springs. Helical coil springs for example store energy in the twisting of the wire in the spring. This analysis would just be if you had a solid wire and were pulling on both ends to store energy, much less efficient than a coil. – JMac Oct 31 '22 at 17:08
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    @Rick No, this is an overestimate because its assuming a spring can only be as efficient as a solid block of material stressed along one axis in tension/compression. – JMac Oct 31 '22 at 17:09
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    @JMac the second half of your sentence is correct, the first half is incorrect. The maximum possible energy a material can hold is when it's about to yield. A spring that maximizes specific energy storage will get the entire material up to it's yield stress/strain. A simple "spring" that does this is a solid block in compression or tension. A coil spring will only get the surface of the coil up to maximum strain will leaving the material towards the center of the wire at linearly lower strain. Since the outside has more area/volume than the core this is results in a reduction in theoretical... – Rick Oct 31 '22 at 17:35
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    ...energy storage of only about 1/3. So an ideal coil spring has about 2/3 of the specific energy capacity as a solid block in compression. (Of course most of the time a solid block is WAY too stiff so it makes much more sense to use a coil spring, but they are worse springs in terms of specific energy capacity) – Rick Oct 31 '22 at 17:40
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    @JMac oh and if your concern was about the use of the tensile yield stress instead of the shear stress, for most metals it's irrelevant because the failure mode of metals in tension is actually dual sear at 45 degree angles. For materials with a poisons ratio away from 0.3 then you might want/need to specifically look at shear strength vs tensile strength, but for metals it should already be accounted for at least to within 10% or so. – Rick Oct 31 '22 at 17:46
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    @Rick My concern is that an analysis of tension in a solid wire doesnt really make sense to me here. I guess maybe it's true that that is the maximum energy you can store in it, but that analysis still doesnt feel right to me. Comparing an axially stretched piece of metal to a spring doesnt really sit right to me, and definitely isnt how the energy is stored. – JMac Oct 31 '22 at 18:02
  • I was sceptical too at first, thinking something like a thin, long wire spooled around pulleys would surely be more effective than a single solid chunk, so I attacked it with what I knew, and it seems the answer is correct. Maybe this helps: The well known formula for potential spring energy is W=1/2*k*delta_L^2 with spring constant k and elongation delta_L. Let's assume a solid beam with length L and cross section area A. The simple formula for the spring constant of a beam is k=A*E/L with Youngs modulo E. (1/2) – MaxD Nov 01 '22 at 05:31
  • The allowable elongation limited by the material strength sigma is delta_L = L * sigma/E. The mass of the beam is m = A*L*rho, therefore A = m/(L*rho). We get k=m/(L*rho)*E/L=m*E/(rho*L^2). The well known formula for potential spring energy is W=1/2*k*delta_L^2 = 1/2*m*E/(rho*L^2)*L^2*sigma^2/E^2 = 1/2*m*sigma^2/(rho*E) and L cancels out completely. (2/2) – MaxD Nov 01 '22 at 05:32
  • @JMac https://en.wikipedia.org/wiki/Mohr%27s_circle this might help with reconciling how tension and shear store energy in the same way and can in fact be transformed from one to the other through a rotation of axis of the stress/strain tensor. – Rick Nov 01 '22 at 12:20
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    In order to build a stored energy weapon for a battle-bot two decades ago, I compared steel, compressed gas (in a suitable container), and aeroplane elastic as storage media. The elastic won the Joules/kg ratio by a country mile. However, the problem was its endurance at high energy density. Enough for a very short-lived battle bot perhaps, but maybe not for years on a bike. And difficult to weather-proof, even if a lower energy density allowed it a reasonable number of charge/discharge cycles. – Neil_UK Nov 01 '22 at 19:04
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The idea looks nice at first glance, but the deeper you get into the details, the less attractive it becomes:

  • You need to be able to modulate your brakes. As such, you need some transmission between your spring and the wheel that switches gears seamlessly with the action of the brake lever. Did you know that most internal gear hubs have a tiny neutral in between their gears? And with good reason? This is a no-no for a transmission that has a strong, loaded spring on one side.

  • Both the spring and the transmission add weight. Considerable amounts of weight. Many cyclists don't like unnecessary mass.

  • Maintenance with such a strong, loaded spring in place could put you in mortal danger.

  • To be effective, your spring would need to be loaded from the front wheel, but assist acceleration via the rear wheel. Let it act on the wrong wheel, and you are likely to cause the wheel to slip.

  • This could work with an electrical motor, a few supercaps, and some electronic to do the transmission work (= transforming electric voltages). Trying to do it mechanically would be unnecessarily complex.

cmaster - reinstate monica
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    And once you have a motor and a generator (or a motor-generator), you may as well add the battery to have a proper e-bike, instead of the capacitors. Suddenly it is an existing solution. Why it is not more common is another question. – Peter - Reinstate Monica Oct 31 '22 at 08:46
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    @Peter-ReinstateMonica In part, that's what I was alluding at. Nevertheless, such a supercap based system would have benefits as well: It replaces the heavy battery with less than 1kg of supercaps. These can endure any amount of load cycles, and live several times as long as a lithium ion battery. There would be no point in actually charging the caps, it would simply be a feature of the bike that it gives an extra kick with each acceleration after braking. – cmaster - reinstate monica Oct 31 '22 at 11:31
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Part of what makes bicycles attractive is their simplicity. The simple frame-sprockets-chain-wheel with a few ball bearings in between, plus two simplistic brakes, is making the entire system light and cheap. It is an elegant, efficient solution to a transportation demand.

Because of the advent of Lithium ion batteries and because there is a growing segment of the population who can afford to spend thousands of dollars on a bicycle we now have e-bikes which are basically very light, 2-wheeled automobiles.

Your mechanical solution would be similarly heavy, expensive and complicated. But it wouldn't offer the same benefits as an e-bike, which is why it doesn't exist.

The next question may be why normal e-bikes don't use recuperation; the main reason is that the substantial disadvantages in terms of additional gear, weight and sub-optimal direct-drive motors are not worth the 5% gain in battery endurance you would get. Once you have a battery you are good.

Peter - Reinstate Monica
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    Regen braking doesn't really require any extra gear on a direct-drive e-bike, and certainly no extra weight. It's just that both mid-motor and freewheel-geared-hub designs can't do it. But all this about e-bikes and regen is, indeed, another question. – leftaroundabout Oct 31 '22 at 15:24
  • @leftaroundabout Well, some designs (the more desirable ones, with a single brake lever for "both kinds of music") do... but thanks for the link. – Peter - Reinstate Monica Oct 31 '22 at 15:37
  • @leftaroundabout Not so fast. At the very least, you need specialized brake levers. Then the question comes up which wheel has the motor. If it's the rear wheel, regenerative braking may easily make the rear wheel slip, so you need some ABS like modulation of brake power. If it's on the front wheel, you already have the problem on steep ascents, but you also have the problem that regenerative braking power is limited by the motor power. And 250W is no that much compared to the 2.8kW that you want to be able to brake away (a = 4 m/s^2, v0 = 25.2 km/h = 7m/s). – cmaster - reinstate monica Oct 31 '22 at 15:42
  • @cmaster-reinstatemonica you get those levers on any dumb e-scooter. And sure, regen is hardly going to be powerful enough for hard emergency braking – but most braking doesn't need to be that powerful. For the more typical use cases of limiting speed on a road descent or gently slowing down in front of a red traffic light, you get neither to the limits of rear-wheel traction nor motor power. But once again, this is a bit off-topic here and was discussed in the other question. – leftaroundabout Oct 31 '22 at 16:18
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    @leftaroundabout E = 100kg / 2 * (7 m/s)^2 = 2.45 kJ, divide that by the motor power, and you see that T = 2.45 kJ / 250 W = 9.8s. That's an acceleration of a = 7 m/s / 9.8 s = 5/7 m/s^2, which gives a braking distance of s = 5/7 m/s^2 / 2 * (9.8 s)^2 = 34.3 m. That's as early as you would need to brake in order to come to a full stop while recovering the entire energy. And while you can do that for test purposes, regenerative braking would only recover a small fraction of the energy available when riding around the city with realistic braking distances. In short: next to useless. – cmaster - reinstate monica Oct 31 '22 at 16:41
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To summarise the discussion, having this non-electrical KERS doesn't make sense because of its reduced capability.

It only helps with coming to a sudden stop, whereas an (IGH) or an also assists on sustained challenges like climbing hills.

Thus if installed on an IGH bike it's overkill, because you can just switch to a low gear after losing momentum, and get back up to speed without much effort.

On an e-bike the energy you could feasibly recover is insignificant compared to the energy stored in the battery. So it's added weight and complexity on an already heavy and complex bike.

On a single-speed bike you would get more capability from upgrading to an IGH or electric motor. So the cost and complexity budget for such a KERS needs to be significantly lower than an IGH or e-bike upgrade, to account for the reduced capability.

This leaves what appears to be an impossible niche: simpler, lighter and cheaper than an IGH, for mostly urban single-speed (and maaaybe even derailleur) bicycles.

Leeroy
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Has been seriously looked at. World Academy of Science, Engineering and Technology International Journal of Mechanical and Mechatronics Engineering Vol:8, No:4, 2014
5 page study, with a conclusion of

The aim of this system is to improve the fuel efficiency of the vehicle without compromising the vehicle performance by storing the energy which would otherwise be lost during braking in the spiral spring and using it during acceleration. Deep study in this field will result in cost and size reduction of the components.

and in the Indian Journal of Science and Technology Volume 10:36 September 2017
7 page study referencing spring-based KERS in a vehicle

In coming days, spring based regenerative system will gain more attention with the advancement in spring design and spring material. The vehicle with start-stop cycle of driving will be affected the most with the introduction of this technology.

Criggie
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Martin Langley
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    Hi, welcome to bicycles. Link-only answers are discouraged, since a bit of link rot leaves them meaningless. Please include a summary of the links you're including. – DavidW Feb 04 '23 at 21:54
  • Martin - welcome to StackExchange. I've added a precis from each linked PDF, and included the source publication and the year/volume number. You might like to browse the [tour] and see how the Q&A format works too. – Criggie Feb 04 '23 at 23:23
  • Downside of both these reports is that they are primarily targetted at vehicles, referencing fuel saving and stop-go traffic, and driving. OP's original question is more about how this technology might relate to bicycles, and while these papers are of interest, they presume a ~thousand kilogram vehicle with dozens of horsepower compared to a sub-20 kilogram bike with 1/5 to 1/2 horsepower. The car is by far more common, and therefore represents the largest single target so it makes sense. How will this tech trickle down if at all? You can use [edit] to expand your answer futher. – Criggie Feb 04 '23 at 23:26