[Theory] Flywheel series as a means of breaking the "glass ceiling"

Striving for the ability to "reach out and touch" zombies at very long range has been an interest of mine ever since I started playing HvZ - not only for the tactical utility, but also for the sheer fun of nailing zombies who think that they are out of range. Here, I'd like to talk about an idea that I've been sitting on for a while now: multiple flywheel sets in series. This is a theoretical exploration of the performance potential of such systems. I'd like to build such a system at some point, when I have the materials and opportunity to use it - so probably not any time soon.

In conventional flywheel systems, there is an upper bound on the velocity which a pair of flywheels can produce - the "glass ceiling" - which is reached when the flywheels and dart slip against each other through the entirety of the dart's passage through the flywheels. The coefficient of kinetic friction varies little if at all with flywheel speed and draft effects are not significant for smooth flywheels, so faster flywheels do not result in faster darts. Attempts to increase the acceleration of the dart while between the flywheels by increasing the grip of the flywheels have been of limited success at best. So far, I think that I'm the first to suggest increasing the length of the dart's passage through the flywheels instead, which is what a flywheel series effectively does.

I will assume here that a high exit velocity, perhaps with heavy darts, is desirable. Pain upon impact limits the useable velocity and weight, and for many games a single set of flywheels can already reach this limit; this is not for those games. I'm ignoring the issue of dart stability for now; I'll assume that this problem can be overcome with either modified Elite or similar darts or with stefan-like darts that can work in flywheels.

Using more than two flywheels isn't a completely new idea - several Rayvens with a second set of flywheels installed near the front, called afterburners, can be seen on Nerfhaven, and IIRC someone made a barrel attachment with a Barricade flywheel set which achieves the same effect. In such blasters, there is a length of Nerf pseudobarrel between each set of flywheels, which wastes velocity while contributing nothing to accuracy. Afterburners cannot be placed further back in a Rayven because they would interfere with the trigger/pusher mechanism, but if a blaster other than a Rayven (or a Rayven with a new pusher mechanism) is used, multiple flywheel cages with little or no gap in between them could be installed.

If a secure attachment mechanism could be obtained, it would be straightforward to make a flywheel series front barrel which could go on any blaster. This would allow one to have a single base blaster which could be used for campus HvZ without and more bruise-tolerant games with the front barrel. Likewise, any flywheel blaster could be used as a base for a dedicated flywheel series blaster, making it straightforward to make one with whatever handling characteristics (SA or FA, bullpup or not) are preferred.  

Flywheel series blasters would be relatively safe compared to HPA, although they still carry non-negligible risks. They would not be subject to the risk of launching a dart with injuriously more velocity than intended and cannot explode. In the event of a short, however, the risk and potential intensity of incendiary failure would be greater than for other flywheel systems due to the increased power requirements.

This article has been written primarily about flywheel series blasters made using Nerf flywheels and cages, and homemade systems where flywheels come in pairs (as mounting two flywheels on either side of a tube is much easier than mounting three or four evenly spaced around it). The theoretical work done here is equally applicable to less conventional flywheel systems, though.

The primary advantage of a flywheel series is that the exit velocity can be very high, even if weighted darts are used, while the blaster itself can be compact, simple to build, ergonomic, safer than HPA, and capable of high rates of fire. No existing high-velocity system combines all of these characteristics.

The main disadvantage of a flywheel series is the cost. Really good batteries and many good motors would be needed. Using several blasters just for the flywheel cages would be excessively spendy, if Nerf flywheels are used.

This is where the math starts, so this is as good a place as any for the jump.

    Symbols used

Vi - the dart's initial velocity, before entering the flywheel system
Ve - the dart's velocity upon entering a specific flywheel set
Vl - the dart's velocity upon leaving a specific flywheel set
Vf - the dart's final velocity, upon leaving the flywheel system

L - the total length of the dart's passage through the flywheel system
Li - the length of the dart's passage through one flywheel set
A - the average acceleration of the dart during it's passage through the flywheel system
T - the time that the dart spends in the flywheel system

    Theory

Let L be the length of the dart's passage through the flywheel system, let Vi be the dart's velocity before it enters the flywheel system, let Vf be the dart's velocity as it exists the flywheel system, let A be the dart's acceleration while in the flywheel system (presumed constant for now), and let T be the time that the dart spends in the flywheel system.

L = Vi * T + (A/2) * T^2
T = (Vf -Vi) / A

Substitute,

L = Vi * ( (Vf -Vi) / A) + (A/2) * ( (Vf -Vi) / A)^2

Simplify,

0 =  (1/2) * (Vf -Vi)^2 + Vi (Vf -Vi) - L * A

Apply the quadratic formula and solve for Vf

Vf = sqrt [ Vi^2 + 2 L * A ]

I call this the fundamental flywheel equation.

What if the acceleration is not constant? As it turns out, it does not matter - we can just take the average. A rigorous proof of this would require calculus, but I don't want to scare away the liberal arts majors (and blogger does not offer math formatting). So, here is a less rigorous argument:

Divide the dart's passage through the flywheel system into h non-overlapping segments of equal length (L/h), where (L/h) is small enough that the acceleration is near as makes no difference to constant in each section. Let the acceleration in each section be A1, A2, etc. Let V1, V2, etc. be the velocity of the dart after passage through A1, A2, etc. Applying the fundamental flywheel equation iteratively gives

V1 = sqrt[Vi^2 + 2 (L/h) * A1]

V2 = sqrt[V1^2 + 2 (L/h) * A2] = sqrt[ Vi^2 + 2 (L/h) * (A1 + A2) ]

V3 = sqrt[V2^2 + 2 (L/h) * A3] = sqrt[ Vi^2 + 2 (L/h) * (A1 + A2 + A3) ]

Combining all of the segments gives

Vf = sqrt[ Vi^2 + 2 (L/h) * (A1 + A2 + A3 + A4 . . . ) ]

If we let A be the average acceleration, then  (A1 + A2 + A3 + A4 . . . ) = h*A. Performing this substitution produces the fundamental flywheel equation again:

Vf = sqrt [ Vi^2 + 2 L * A ]

It does not matter whether a dart enters a flywheel set before it leaves the previous one, because the acceleration in the overlap region is equal to the sum of the accelerations which each flywheel set would provide individually, so the term L * A is unaffected. This is good as it is convenient to pack flywheels tightly.

    Performance

The glass ceiling velocity given a certain number of flywheels is equal to the square root of the number of flywheels times the glass ceiling velocity for a single flywheel (if the darts are thrust in with a pusher mechanism which provides negligible velocity, as they probably would for even a high-ROF blaster). This makes calculating the glass ceiling velocity very easy for such blasters.

Since 110 fps = sqrt[2 L * A ], for a single pair of Nerf flywheels operating in the glass ceiling limited regime, we have 2 L * A = 12100 feet squared per second squared. If the dart is thrown with some velocity Vi into the flywheel system with N such flywheel pairs, the final velocity is

Vf = sqrt [ Vi^2 + N * 12100 (fps)^2 ]

This allows for the glass ceiling calculations with various Vi, given in table 1. Of course, the final velocity may be less than this if the flywheel surface speed is insufficient.

Using weighted darts would lower the glass ceiling; more flywheels could be added to compensate.

The calculated glass ceiling velocities do not take into account the energy lost due to the need to crush a Nerf dart as it passes through each flywheel stage. It is unknown how much of this lost energy comes from the dart's and how much comes from the flywheel's kinetic energy. If a significant proportion comes from the dart's energy, then the calculated glass ceiling values will be increasingly inaccurate for higher flywheel counts. This could be avoided by using darts with trimmed heads, as discussed in Design notes.

A flywheel series blaster can be compact even with a large number of flywheels. A system made with Nerf flywheel cages will have to be compact, as only a limited number of such cages are likely to be available. A completely homemade system with 10 flywheel pairs, a generous 0.25 inch gap between flywheels, and flywheels that are 1.25 inches in diameter would be less than 15 inches long, and could fling stock Nerf darts at about 350 fps if my calculations are accurate.

As with any flywheel blaster, the ROF could be as fast as darts can be pushed out of the magazine.

Table 1. The calculated minimum RPM and critical voltage are underestimates as they do not take into account the speed loss during firing.

    Sample power consumption calculations

Let's do some calculations to get a rough idea of the power that would be needed. Let's assume that Blade motors and Nerf flywheels or something similar are used. Blades aren't the best motors for the foremost flywheels in a many flywheel system, for a variety of reasons, but  they serve as a worst case for power consumption. Blades have a free-running speed of 12900 rpm at 3V. Free-running speed is, in theory and to a good approximation in practice, proportional to the voltage. Let's assume that the presence of flywheels has negligible effect on the free-running speed. A Nerf blaster's flywheels are about 1.25 inches in diameter. This gives a surface speed of 23.5 feet per second per volt.

The calculated critical voltages for flywheel cage counts from one to ten are given in table 1. This calculation does not take into account the flywheel speed lost while flinging a dart. However it is nonetheless a decent estimate, as the flywheels towards the front of the blaster, which is where speed is needed, will loose the least speed. Using weighted darts would lower the glass ceiling and thus the critical voltage.

The current required by a multi-stage flywheel system could be problematically large but, even in this worst case, not unfeasibly large. Let's continue to assume the use of Blades, which have a stall current of 3.02 per volt and a free-running current of 0.087 A per volt (assuming that linear extrapolation works here). The calculated stall and free-running currents for various flywheel counts at the corresponding calculated critical voltage are given in table 1. These currents assume that all motors are run in parallel. The free-running current is hefty but manageable for higher numbers of flywheel cages. The stall current for more than one flywheel pair is extremely large. It is highly implausible that any battery pack would be able to meet this demand for current and bring out the full performance potential of these motors, and measures may be necessary to protect the batteries from damage due to the inrush current, which would be similar to a momentary short. A MOSFET with a slow switching time might work.

A better choice of motors would reduce power requirements, perhaps greatly.

    Design notes

Nerf flywheel cages could be used for an easy build, but this would not allow for very many flywheels to be used unless many blasters are used for parts. If a blaster is intended for use in a game which is more bruise-tolerant than most HvZ games but not as bruise-tolerant as a NIC game, then a suitable blaster could be made with a reasonable number of flywheel cages.

It would not be difficult to fabricate a crude but workable system without using parts from multiple Nerf blasters. One could use a pipe or pseudobarrel, with holes cut in it for the flywheels, a block running the length of the pipe, and motors glued to the block. (The motor mounts would have to be strong enough to deal with the crush force of a Nerf dart's head, unless only darts with trimmed heads are used, as discussed later.)

A carefully constructed or 3D printed mount would allow for the flywheels to be packed very densely, and would allow for arrangements where three or four flywheels grip the dart at each stage rather than two. If more than two flywheels grip a dart, the fundamental flywheel equation still applies - three flywheels apply half as much again acceleration as two, etc. There is a limit to how much contact area there can be at each stage, set by the dart's diameter, so a large number of flywheels in one stage would not be advantageous.

The flywheels used, if Nerf flywheels aren't used, is an issue that I leave aside for now as this is largely unexplored territory.

A flywheel series front barrel would offer much more flexibility than a dedicated flywheel series blaster. A flywheel series blaster capable of adjustable muzzle velocity could be made, but it would be tricky to make it perform well in terms of power consumption and avoiding excessive dart wear at each velocity. On the other hand, the ideal performance characteristics for the rearmost flywheel motors in a flywheel series are the same as those for the motors in a single flywheel pair system, so having a flywheel series front barrel and a well-modified flywheel blaster with a secure front barrel connection would allow one blaster to serve as either a low-pain or a long-range primary and to excel in both roles. Also, if one were to play in a game where fully automatic (or any automatic) blasters cannot be used at some point, it would be easier to put a front barrel on a different blaster than to make several separate high-velocity blasters.

However, this combination (low-pain blaster + high-velocity flywheel front barrel) would require that the blaster be capable of firing both low-pain and high-velocity darts, which may be problematic if the high-velocity darts are shorter.

The force required to compress a Nerf dart's head is large, and this could be problematic for homemade systems in several ways. It could damage motors if flywheels are positioned on the motor's axle in a way that produces leverage, and could damage homemade mounts. It also probably causes a significant loss of energy, which may be problematic for many-stage flywheel systems as this much energy is lost at each stage.

Using darts with trimmed heads could help, but this would create other problems if you want to have a low-pain blaster with a high-velocity flywheel front barrel. Darts with trimmed heads would be slightly shorter than stock Nerf darts and this could require that the entire flywheel cage mount (or pusher mechanism) be rebuilt to ensure that the darts reach the flywheels. With stock Nerf darts, the flywheels grip the head of a dart more tightly than the shaft and the contribution to the acceleration provided by the head could plausibly be very large. Putting a straw inside the dart to ensure consistent crush force and reducing the flywheel gap could compensate for this, but would render the blaster unsuitable for use with stock Nerf darts - which is a problem because darts with their heads trimmed would hurt more upon impact.

As was mentioned in Peformance, the energy required to crush a Nerf dart's head is potentially a significant loss of energy, source of uncertainty which impairs optimal design, and limiting factor on a flywheel series' performance. This is another reason why using darts with trimmed heads would be advantageous.

A reasonable compromise would be to have a flywheel blaster with a rebuilt flywheel cage mount to bring the flywheels closer to the magazine, but with the stock flywheel gap maintained. Such a blaster would be able to fling stock Nerf darts as well as any other, and should be able to grip and throw darts with trimmed heads just well enough to push them into a flywheel series front barrel.

If a long pseudobarrel is used for enhancing accuracy, it would be straightforward to compensate for the velocity loss - just add more flywheels. (Whether pseudobarrels do in fact increase accuracy in any way other than decreasing velocity is not an issue that I'll address here. I'll just say that it is plausible that they do on theoretical grounds and that there is anecdotal evidence which supports this.)

As with single flywheel pairs, it would be advantageous to use a higher-than-critical voltage. At the critical voltage, increasing the number of flywheels only increases rapidfire endurance (by increasing total flywheel momentum), whereas increasing the voltage beyond critical increases this as well as windup and recovery times.

Using different motors in the rearmost and foremost flywheels - specifically, high-torque motors in the back and high-speed motors in the front - would allow for performance to be maintained while minimizing dart wear and power consumption. A wise choice of motors is very important as efficiency is a significant concern.

Let's take another look at a flywheel series with Nerf or flywheels or Nerf-like flywheels and with similar motors used throughout. The rearmost motors loose the most speed while firing, by far. To get a rough sense of how much speed each flywheel looses, let's assume that the average acceleration of the dart during its passage through each flywheel is the same as its average acceleration while in the flywheel system and that the velocity imparted by the pusher is negligible. Let's assume that the momentum lost by each flywheel is proportional to the time that the dart spends in the flywheels, which is given by T = (Vl - Ve)/A, where here Ve and Vl are the velocities of a dart upon entering and leaving the individual flywheel in question. Let Li be the length of the dart's passage through the flywheel in question. Let's use nondimensional units with 2 Li*A = 1 and A = 1, and assume that there is no overlap (as this simplifies calculating Ve and Vl). The calculated nondimensional speed losses are given in table 1. With no initial velocity, the rearmost flywheel pair looses 2.4 times as much speed as the second to rear pair, and 3.7 times as much as the fourth from the rear pair!

Now, let's consider a flywheel series front barrel. The dart will be launched into the flywheel system with a non-negligible velocity Vi. We need a way to convert these velocities into our nondimensional units. Note that the glass ceiling velocity is 1 in nondimensional units, so we can convert Vi into these units by using the factor 1 / (110 fps). Using this, we obtain the numbers given in table 1. With even a modest Vi, the relative speed loss of the rearmost flywheels is reduced greatly, but is still significantly larger that that of the rest of the flywheels.

This is not a problem from a performance standpoint - the rearmost flywheels loose the most speed, but they also have the most "extra" speed as the dart travels more slowly while close to the rear of the flywheel system - but this does increase dart wear and power consumption. Dart wear is an increasing function of slippage speed. Therefore it is desirable to have a flywheel surface speed which does not exceed the dart's speed by too much in each flywheel. Running a high-torque motor at high speeds consumes a lot of power. A flywheel series is necessarily going to be problematically power-hungry and increasing efficiency is a significant concern. Simply running the rearmost flywheels at a lower voltage would help to reduce power consumption and dart wear, but would harm performance during rapid firing as the blaster would have low endurance and recovery at the back, which is where it is needed the most. So, it would be advantageous to use high-torque motors in the back and high-speed motors in the front.

The very high RPM required of the motors towards the front of a many-stage flywheel system makes brushless motors suitable. However, brushless motors are expensive - even very small ones cost about $10 each - and, due to the large number of motors that would be needed in a flywheel system, it would be infeasible to use brushless motors throughout. This is another reason why it would be advantageous to allow for the use of different motors at different stages.

There may be problems with flywheels coming off the motors at very high speeds. Flywheels could be attached more securely by using glue and/or motors with long shafts.

    Misc. notes

I'm not entirely confident in the assumption that the speed loss of a flywheel during firing is proportional to the time that the dart spends in the flywheel for stock Nerf darts. If the energy required to compress a dart is a significant contributor to speed loss, then the speed loss will be closer to constant than my calculations indicate, and this would reduce the need for different motors to be used at different stages. This assumption is more reasonable for darts with their head trimmed.

Working with Nerf flywheel systems seems to have made people make some odd assumptions, myself included. Previous work on multi-stage flywheel systems had each flywheel stage engage the dart separately; AFAIK the idea that overlap is not a problem is new. In an earlier draft of this post, I assumed that flywheels would have to come in pairs.

There are some exiting possibilities for well-made homemade systems here. In particular, if darts with a consistent crush force and shaped flywheels which crush a dart evenly around its entire diameter are used, the acceleration provided per flywheel could be increased, perhaps dramatically. 3D printed barrel / motor mounts would allow for modular open-source designs which could be produced by anyone who can commission 3D printed parts, and I suspect that 3D printers will become much more common as time goes on. A modular design would increase the accessibility of the system - in particular, it would make it easy to upgrade an existing blaster by simply adding another flywheel stage the the front. As for the flywheels, they might also be 3D printable - 3D print plastic has a lower tensile strength than injection molded plastic, but it isn't terribly much lower, the surface irregularities that are inevitable on 3D printed products would result in faster dart erosion in the short terms but would ensure that the layer of melted foam which forms on all high-speed flywheels would be less likely to come off in the long term, and while ensuring balance and secure mounting to the motor's shaft may be problematic I am confident that this problem can be overcome.

Credit goes to Toruk for bringing the issue of damage caused by a Nerf dart's head's crush force to my attention, and for pointing out that flywheels don't necessarily come in pairs.