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					Nerf engineering

   Ben Trettel
Copyright

This work is c 2009 Ben Trettel and licensed under a Creative Commons
Attribution-Noncommercial-Share Alike 3.0 United States License. For
more information about this license, visit this URL:

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When adapting this work, please retain Ben Trettel as an author.




                                   i
Contents

Preface                                                                                                1

1 Types of Nerf guns                                                                                   2
  1.1 Pneumatic . . . . . . . . . . . . . . . . . . . . . . . . . . . .                                2
  1.2 Spring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                               3

2 Measurement and units                                                                                5
  2.1 Systems of units . . . . . . . . . .     .   .   .   .   .   .   .   .   .   .   .   .   .   .   5
  2.2 Length . . . . . . . . . . . . . . . .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   5
  2.3 Area . . . . . . . . . . . . . . . . .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   5
  2.4 Volume . . . . . . . . . . . . . . .     .   .   .   .   .   .   .   .   .   .   .   .   .   .   5
  2.5 Pressure . . . . . . . . . . . . . . .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   6
  2.6 Velocity . . . . . . . . . . . . . . .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   6
  2.7 Time . . . . . . . . . . . . . . . . .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   6
  2.8 Temperature . . . . . . . . . . . .      .   .   .   .   .   .   .   .   .   .   .   .   .   .   6
  2.9 Weight . . . . . . . . . . . . . . . .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   6
  2.10 Uncertainty and significant figures       .   .   .   .   .   .   .   .   .   .   .   .   .   .   6

3 Parts                                                                                                 7
  3.1 Classification of pipe . . . . . . . . .      .   .   .   .   .   .   .   .   .   .   .   .   .    7
  3.2 Common pipe materials . . . . . . .          .   .   .   .   .   .   .   .   .   .   .   .   .    7
  3.3 Types of fittings . . . . . . . . . . .       .   .   .   .   .   .   .   .   .   .   .   .   .    7
  3.4 Threaded pipe and fittings . . . . . .        .   .   .   .   .   .   .   .   .   .   .   .   .    7
  3.5 Flexible tubing . . . . . . . . . . . .      .   .   .   .   .   .   .   .   .   .   .   .   .    7
  3.6 Special concerns for PVC . . . . . .         .   .   .   .   .   .   .   .   .   .   .   .   .    7
  3.7 Valves . . . . . . . . . . . . . . . . .     .   .   .   .   .   .   .   .   .   .   .   .   .    8
  3.8 Pilot operated valves . . . . . . . . .      .   .   .   .   .   .   .   .   .   .   .   .   .    9
  3.9 Gas reservoirs . . . . . . . . . . . . .     .   .   .   .   .   .   .   .   .   .   .   .   .    9
  3.10 Springs . . . . . . . . . . . . . . . .     .   .   .   .   .   .   .   .   .   .   .   .   .   13
  3.11 Plastic sheet, solid tubes, blocks, etc.    .   .   .   .   .   .   .   .   .   .   .   .   .   13

                                      ii
CONTENTS                                                                                                           iii

   3.12   Nuts, bolts, screws, and threads         .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   13
   3.13   Pumps and compressors . . . .            .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   14
   3.14   Lubricants . . . . . . . . . . . .       .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   14
   3.15   Glues, cements, and putties . .          .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   14
   3.16   Straps . . . . . . . . . . . . . .       .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   15
   3.17   Air cylinders . . . . . . . . . .        .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   15
   3.18   Material selection . . . . . . . .       .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   15
   3.19   Free samples . . . . . . . . . .         .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   15
   3.20   Further reading . . . . . . . . .        .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   15

4 Darts and barrels                                                                                                16
  4.1 Darts . . . . . . . . . . .      . . . . .           .   .   .   .   .   .   .   .   .   .   .   .   .   .   16
  4.2 Barrels . . . . . . . . . . .    . . . . .           .   .   .   .   .   .   .   .   .   .   .   .   .   .   17
  4.3 Combinations of darts and        barrels             .   .   .   .   .   .   .   .   .   .   .   .   .   .   17
  4.4 Further reading . . . . . .      . . . . .           .   .   .   .   .   .   .   .   .   .   .   .   .   .   17

5 Machining and tools                                                                                              18
  5.1 Cutting . . . . . . . . . .      .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   18
  5.2 Drilling holes . . . . . . .     .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   18
  5.3 Taping holes . . . . . . .       .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   18
  5.4 Threading parts together         .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   19
  5.5 Filing and sanding . . . .       .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   19
  5.6 Good practices . . . . . .       .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   19
  5.7 Further reading . . . . . .      .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   19

6 Basic solid mechanics                                                                                            20
  6.1 Stress and strain . . . . .      .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   20
  6.2 Other failure mechanisms         .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   21
  6.3 Pressure vessels . . . . . .     .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   21
  6.4 Springs . . . . . . . . . .      .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   21
  6.5 Bending stress . . . . . .       .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   22
  6.6 Further reading . . . . . .      .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   22

7 Materials                                                                                                        23
  7.1 Plastics . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                                           23
  7.2 Metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                                           23
  7.3 Further reading . . . . . . . . . . . . . . . . . . . . . . . . .                                            23

8 Schematics                                                                                                       24
iv                                                                                          CONTENTS

9 Mechanisms                                                                                                    25
  9.1 Turrets . . . . . . . . . . .     .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   25
  9.2 Magazines . . . . . . . . . .     .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   25
  9.3 RSCB clips . . . . . . . . .      .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   25
  9.4 Rotating turret mechanisms        .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   25
  9.5 Dart-pusher mechanisms . .        .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   25

10 Basic mechanics and dynamics                                                                                 26
   10.1 Basic mechanics and dynamics . .                .   .   .   .   .   .   .   .   .   .   .   .   .   .   26
   10.2 Basic gas laws . . . . . . . . . . . .          .   .   .   .   .   .   .   .   .   .   .   .   .   .   26
   10.3 Conservation laws . . . . . . . . .             .   .   .   .   .   .   .   .   .   .   .   .   .   .   28
   10.4 Basic gas dynamics and fluid flow .               .   .   .   .   .   .   .   .   .   .   .   .   .   .   29
   10.5 Number of shots from a gas source               .   .   .   .   .   .   .   .   .   .   .   .   .   .   29
   10.6 Further reading . . . . . . . . . . .           .   .   .   .   .   .   .   .   .   .   .   .   .   .   33

11 Testing                                                                                                      34
   11.1 Range . . . . . . . . . . . . . .       . . . . .           .   .   .   .   .   .   .   .   .   .   .   34
   11.2 Hang time . . . . . . . . . . . .       . . . . .           .   .   .   .   .   .   .   .   .   .   .   34
   11.3 Muzzle velocity . . . . . . . . .       . . . . .           .   .   .   .   .   .   .   .   .   .   .   34
   11.4 Pressure as a function of time .        . . . . .           .   .   .   .   .   .   .   .   .   .   .   34
   11.5 Projectile position as a function       of time             .   .   .   .   .   .   .   .   .   .   .   34

12 Terminal ballistics                                                                                          35
   12.1 Further reading . . . . . . . . . . . . . . . . . . . . . . . . .                                       35

13 Simulation                                                                                                   36
   13.1 Internal ballistics of a pneumatic gun .                .   .   .   .   .   .   .   .   .   .   .   .   36
   13.2 Internal ballistics of a spring gun . . .               .   .   .   .   .   .   .   .   .   .   .   .   36
   13.3 External ballistics of a dart . . . . . .               .   .   .   .   .   .   .   .   .   .   .   .   36
   13.4 Further reading . . . . . . . . . . . . .               .   .   .   .   .   .   .   .   .   .   .   .   37

14 Optimization and efficiency                                                                                    38
   14.1 Goals of optimization . . . . . . . . . . . . . . .                             .   .   .   .   .   .   38
   14.2 Sources of inefficiency . . . . . . . . . . . . . . .                             .   .   .   .   .   .   39
   14.3 Internal ballistic parameters for pneumatic guns                                .   .   .   .   .   .   39
   14.4 Internal ballistic parameters for spring guns . . .                             .   .   .   .   .   .   40
   14.5 External ballistic parameters . . . . . . . . . . .                             .   .   .   .   .   .   40
   14.6 Optimization of darts . . . . . . . . . . . . . . .                             .   .   .   .   .   .   41
   14.7 Optimization of pneumatic guns . . . . . . . . .                                .   .   .   .   .   .   41
   14.8 Optimization of spring guns . . . . . . . . . . . .                             .   .   .   .   .   .   42
   14.9 Numerical optimization algorithms . . . . . . . .                               .   .   .   .   .   .   42
   14.10Further reading . . . . . . . . . . . . . . . . . . .                           .   .   .   .   .   .   42
CONTENTS                                                                                            v

15 Examples                                                                                        43

16 Putting it all together: FANG                                                                   44

Reference tables and charts                                                                        45
  Material properies . . . . . . . . .     . . . . . . .   .   .   .   .   .   .   .   .   .   .   46
  PVC pipe dimensions and weight .         . . . . . . .   .   .   .   .   .   .   .   .   .   .   49
  PVC fitting dimensions and weight         . . . . . . .   .   .   .   .   .   .   .   .   .   .   49
  Temperature derating of PVC pipe         and fittings     .   .   .   .   .   .   .   .   .   .   49
  Thread data . . . . . . . . . . . . .    . . . . . . .   .   .   .   .   .   .   .   .   .   .   49
  Telescoping brass sizes . . . . . . .    . . . . . . .   .   .   .   .   .   .   .   .   .   .   49
  Tap and drill combinations . . . .       . . . . . . .   .   .   .   .   .   .   .   .   .   .   49
  Valve data . . . . . . . . . . . . . .   . . . . . . .   .   .   .   .   .   .   .   .   .   .   49
  O-ring data . . . . . . . . . . . . .    . . . . . . .   .   .   .   .   .   .   .   .   .   .   49
  Schematic symbols . . . . . . . . .      . . . . . . .   .   .   .   .   .   .   .   .   .   .   49
  Dart optimization table . . . . . .      . . . . . . .   .   .   .   .   .   .   .   .   .   .   49
  Unit conversions . . . . . . . . . .     . . . . . . .   .   .   .   .   .   .   .   .   .   .   49

Glossary                                                                                           49

List of part sources                                                                               50

Index                                                                                              51
Preface

Building Nerf guns is a fun hobby that attracts people with wildly varying
experiences. The hobby requires use of engineering principles to design the
Nerf blasters. Despite that, few of these people are engineers. Relatively
basic explanations of these principles would benefit all in the hobby.
   This book is meant primarily to be used as reference and is not meant to
be read in order. Alternatively, it could be used as a smooth introduction
to many aspects of the mechanical engineering profession.
   The methods, materials, and approaches the book uses are far from the
only methods, materials, and approaches. Creativity is part of what makes
Nerf fun; find new and different ways of doing these things.
   While there is much requisite mathematics knowledge to understand
some of the sections, I try to avoid “heavy lifting” when possible. Sadly, the
physics can not be segregated from the mathematics–other resources should
be used if you lack the necessary mathematics knowledge for a chapter as
explaining the mathematics go beyond the scope of this book.
   Though every attempt to be accurate in this book has been made, I
can not and do not gaurantee the accuracy of what is presented here. This
book is primarily a summary of what I have found helpful in designing Nerf
guns.
   Also, I am a real person and I do appreciate your comments about this
book or questions. Feel free to email me.
   —Ben Trettel
   btrettel@umd.edu




                                      1
Chapter 1

Types of Nerf guns

The primary focus of this book is pneumatic and spring Nerf guns, however,
an overview of some of the options will be enlightening.


1.1       Pneumatic
Some readers might be confused about what a pneumatic gun is. The word
pneumatic refers to pressurized gas–it is a more generalized word to refer to
what often are called air guns. “Air guns” isn’t strictly correct as we’ll see.
Some guns use other gases, however, the vast majority do use air. We’ll
use the words air gun to distinguish guns powered by compressed air from
other pneumatics.
   The simplest pneumatic guns can be represented as a gas chamber and
barrel with projectile separated by a valve. The valve opens; gas expands
though it from the gas chamber into the barrel; and this gas pushes the
projectile until it leaves the barrel.
   An important fact to note at this juncture is that pressure by itself does
not equal a net force on an object. There must be a difference in pressure
on the two sides, a pressure differential, for a net force to result. On a Nerf
dart in the barrel, this pressure differential is the difference between the
barrel pressure and the atmospheric pressure1 .
   The two varieties of pressure, gauge and absolute, help and harm un-
derstanding of this concern. Absolute pressure is just that–absolute. In
absolute pressure, 0 is a vaccuum. Atmospheric pressure is an absolute
    1 Later we’ll discuss total pressure which takes into account the pressure contributions

of a moving gas, so this statement is not completely accurate but is fine to understand
the basics.


                                             2
1.2. SPRING                                                                           3

pressure, and the approximate value2 for most locations and weathers is
about 14.7 psi or 1.01 bar. Gauge pressure is based on the idea that the
force on an object is proportional to pressure differential over it. A gauge
pressure is “referenced” against the current atmospheric pressure is is sim-
ply the difference between the absolute pressure where the gauge is and
atmospheric pressure. For example, if the atmospheric pressure is 14.7 psi
and the absolute pressure in a pipe is 64.7 psi, the gauge will report 50 psi as
this is the difference in pressure between the pipe’s gas and the atmosphere.
    Often with pressure an a or g is added after the units to differentiate be-
tween absolute and gauge pressure. For example, 14.7 psi absolute pressure
is 14.7 psia. This distinguishing point is often used in this book.
    We’ll now examine how in brief pneumatic gun operates. The gas cham-
ber is pressurized and a dart is loaded into the barrel. In these drawings,
the darker the color of the gas, the higher the pressure of the gas.
    The image above shows the gun just before the valve opens. The pres-
sure in the gas chamber is high. There is no pressure differential over the
dart.
    Now the valve opens and some compressed gas is released into the barrel.
Due to friction the projectile has not moved yet.
    The pressure in the barrel builds higher and the projectile begins to
accelerate.
    Gas flows through the valve, eventually making the pressure in the gas
chamber and barrel approximately equal. The flow through the valve is
fairly low now as the pressure differential (which is the primary factor con-
troling flow through the valve) is small. The projectile continues to accel-
erate, eventually leaving the barrel and becoming a ballistic projectile.


1.2      Spring
Spring guns are the most prevalent form of manufactured and homemade
Nerf gun. In spring guns, energy is stored in the compression of a spring
rather than a gas as in pneumatics. The spring is attached to a piston
(commonly called the plunger head or plunger seal ) that seals against a
tube (commonly called the plunger tube). When loaded, the spring is held in
place; the piece that holds the spring in place is called the catch. Typically
a rod leading from the plunger head out of the back of the Nerf gun is used
to cock the gun.
    The photo below shows a catch mechanism from a Lanard MaxShot.
The notches in the plunger rod are for the catch; when pulled back the
   2 Atmospheric   pressure obviously varies with location and weather. More information
on this is in the next chapter on measurements.
4                                     CHAPTER 1. TYPES OF NERF GUNS

rubber band3 pulls the catch into these notches, locking the plunger and
spring into place. When the trigger is pulled, the catch releases the plunger
and air is compressed by the plunger and spring.
    Once the spring is released, the air in the plunger tube and barrel is
compressed, which in turn pushes the dart down the barrel in a superficially
similar way to pneumatic guns.
    The barrel gas pressure in spring guns typically builds slower than that
of penumatics. This is because the gas starts uncompressed. Consequently,
when the projectile begins to move is much more important. If there is little
static friction (i.e. the projectile requires little force to start to move),
the projectile will begin to move down the barrel. Consequently, more
of the total barrel distance will see the projectile moving slowly and the
projectile often will not reach an acceptable speed. High static friction
is therefore desirable in spring guns; the projectile will stay put until the
pressure differential–and consequently the acceleration–is high. This effect
is desirable in pneumatics as well, however, there is no serious need for this
effect.
    See the section titled Desired fit in the Darts and barrels for more basic
information about this pnenomena. The chapter on simulation will also
detail the mechanisms involved in much greater detail.

Air spring guns. A rare variety that is more common in air rifles meant for
hunting. The air spring variety essentially uses air instead of a spring. This
has a number of advantages for air rifles, namely, the fact that the gas will
be heated from adiabatic heating which in turn increases the speed of sound
such that projectiles can travel faster than their pneumatic brethren4 . The
air spring in Nerf could be used to create a gun with a variable maximum
pull force.




    3 Originally, a torsion spring pushed the catch into the notches. The torsion spring

broke and was replaced by a rubber band. Similarly, the unusual lever that pulls the
plunger broke (a common problem for MaxShots) and was replaced by a rope. Later the
front of the plunger tube fractured from repeated slams from the plunger head. With
this track record I am willing to bet MaxShots were not engineered to last long.
    4 See chapter 9 for information about adiabatic heating
Chapter 2

Measurement and units

2.1       Systems of units

2.2       Length

Calipers.

Rulers.

Tape measures.

Measuring wheels.

String.




2.3       Area

2.4       Volume

Geometry.

Weighted water.

                             5
6                     CHAPTER 2. MEASUREMENT AND UNITS

2.5    Pressure
Pressure gauge ratings.

Choosing a pressure gauge

Barometers.


2.6    Velocity
Chronometers.

Coils and magnetic projectiles.




2.7    Time
Stopwatches.

Cameras.


2.8    Temperature
2.9    Weight
2.10    Uncertainty and significant figures
Chapter 3

Parts

Familiarity with a wide variety of parts is part of building and designing
Nerf guns. This chapter will provide basic information about a wide vari-
ety of parts you may encounter while building Nerf guns and some basic
guidelines on how to choose between different parts and materials.


3.1     Classification of pipe

3.2     Common pipe materials

3.3     Types of fittings

3.4     Threaded pipe and fittings

3.5     Flexible tubing

3.6     Special concerns for PVC
Pressure rating.

High temperature derating.

Brittleness at low teperatures.

Weatherability.

                                    7
8                                                     CHAPTER 3. PARTS




3.7      Valves
Ball valves. Ball valves are the most basic form of valve and are ubiqui-
tous. They make poor firing valves due to their slow opening and difficulty
of being triggered. However, they are excellent and convenient for shutting
off flow and can be used in flow lines whenever stopping flow would be
desired.

Pull valves. Pull valves are the same class of valves as those used in
manufactured Nerf guns. Some homemade versions have been made and
existing garden hose nozzles with threads on both the inlet and outlet exist
that can be used.

Pilot operated valves. Pilot operated valves are known by a number
of names.

Pilot valves are completely different and this point often is unclear. Pilot
valves dump the pilot volume. Any valve can be a pilot valve, in fact, even
another pilot actuated valve can be a pilot valve. See the next section title
Pilot actuated valves for more information about pilot actuated and pilot
valves.

Solenoid. Solenoid valves are valves that use a magnetic coil (i.e. a
solenoid) to open or close a valve. Small solenoid valves for devices such as
clothes or dish washers could be used in Nerf as a cheap and simple valve.
Solenoid sprinker valves, often modified for pneumatic operation as pilot
actuated valves1 , can and have been used for Nerf2 but are too large.

Directional control valves. Directional control valves are valves with
flow paths that can change. They often make excellent pilot valves because
the changing flow path can exhaust the pilot volume in one path and fill
the gas chamber in another.

Check valve. Check valves allow flow in only one direction. For this rea-
son they are also known as one-way valves and non-return valves. There
are a few varieties of check valves
    1 http://www.spudfiles.com/forums/viewtopic.php?t=305
    2 http://nerfhaven.com/homemade/cxwq   bamf/
3.8. PILOT OPERATED VALVES                                                               9



Regulators.

Valve flow diagrams for DCVs.

Using provided data.

Performance.


3.8        Pilot operated valves
Piston valves.

Diaphragm valves.

QEVs. Commercially manufactured pilot operated valves are often called
quick exhaust valves. These valves can either be of the piston or diaphragm
variety. They are simple to use and come in a wide variety of sizes, making
them very attractive for Nerf.

DCVs for semi-auto. A DCV can both pilot a pilot actuated valve
and refill the gas chamber.


3.9        Gas reservoirs
Gas reservoirs3 are systems that supply gas to pneumatic Nerf blasters.
There are many varieties, from pressurized gas systems, to ”bladders” and
systems that expand a liquid into a gas.
   If there’s anything you take from this section, make it this: Use gas
reservoirs designed to hold gas. This limits choices to metal LPA and
HPA tanks. There are other options but there are good reasons not to use
them as I will explain.

Safety note. Never exceed pressure ratings. Best practice is to use metal
tanks at pressures below the maximum operating pressure with a pop-safety
valve in place to reduce the likelyhood of overpressurization. HPA tanks
already have a pop-safety valve.

   3 The  title is a little misleading as I include liquid sources like CO2 on this page as
well. Perhaps “propellants” or “gas sources” would be most appropriate.
10                                                       CHAPTER 3. PARTS

Regulators. A regulator maintains the outlet pressure at a certain level.
Ideally the outlet pressure is completely independent of the inlet pressure,
but some regulators are better than others in this respect.
    Pressure regulators are necessary for compressed gas stored in a solid
tank. Compressed liquid and gas stored in a bladder do not require regu-
lators as the pressure is relatively constant (given constant temperature in
liquid systems and minimum expansion in bladder systems), however, if a
lower pressure is desired than the vapor pressure of the gas or the bladder
pressure of the bladder, a regulator can be used.
    There are two types of regulator: relieving and non-relieving. Avoid
relieving regulators–they “relieve” the excess pressure by exhausting it.
This is very wasteful for Nerf.
    For a good small regulator I suggest the Clippard MAR4 series. These
are the smallest (less than 2 1/2 inches long and 3/4 inches wide) regula-
tors I have encountered. A relieving version exists, so be sure to buy the
non-relieving version. Clippard sells them directly, but their shipping is
expensive, so I’d suggest looking on eBay. One disadvantage of this regu-
lator is that the outlet pressure is not completely independent of the inlet
pressure, so some adjustment is necessary. Dual-regulation (using two reg-
ulators in series) is also an option.
    Other regulators exist and are sold from places like McMaster-Carr, but
these regulators are usually designed for air compressors and are very often
too large and/or too heavy.

Aluminum LPA tanks. Low pressure aluminum cylinders are manu-
factured by many companies. Catalina Cylinders5 manufactures a wide
variety of size cylinders with capacities that are perfectly appropriate for
Nerf.
    These LPA (low pressure air, though they can be filled with any non
volatile gas) tanks are very light, not expensive, extremely durable, and very
strong. The 85 cubic inches (ci) tank I received from Catalina Cylinders
is extremely safe at 150 psi, weighs less than half a pound, and was very
cheap.
    Due to these tank’s lack of popularity, they are only available from the
manufacturers directly. Most manufacturers do not sell single units but
are willing to help out someone with an odd project. Until someone buys a
small stockpile of these tanks to resell, contacting the manufacturer directly
is the only way to get one.
    Catalina’s tanks use a custom fitting called the F1 that connects the
uncommon threads of their cylinder to female 1/4 inch NPT.
     4 http://www.clippard.com/store/display.asp?dept   id=163&levels=1
     5 http://www.catalinacylinders.com/lp.html
3.9. GAS RESERVOIRS                                                         11

   • The burst pressure given is just that–the pressure the tank bursts at.
     A safety factor of 2.5 is common for pressure vessels, so the 600 psi
     burst cylinders have a pressure rating of 240 psi.
   • The gasket on the fitting can burst out at high pressure, leaking gas.
     Be sure to tighten the cylinder into the fitting adequately. Also, I have
     successfully used a cable tie tightened around the O-ring to prevent
     bursts.
High pressure air. HPA tanks are 3000 psi or 4500 tanks meant primarily
for paintball. These tanks can easily be used for the similar Nerf game.

CO2.

PET bottles. Many people use soda bottles made from PET plastic
as pressure vessels. This is a dangerous practice that I will not suggest,
especially given cheap and very attractive alternatives like aluminum LPA
tanks.
    PET bottles are not designed to hold gas pressure. They are designed
to hold liquid under pressure with some gas, and the pressure does not
get very high. PET bottles explode very easily, so easily that I never will
suggest them. The remainder of my discussion here are reasons not to use
PET bottles.
    Not all PET bottles are the same. Bottles for non-carbonated beverages
such as water often are not designed to hold pressure, even if they appear
to be otherwise like carbonated bottles. There also are many differences
in bottles between brands. One brand of 2 liter bottle may burst at 100
psi and another may burst at 250 psi. There is the possibility that some
bottles are lemons too even in a good brand. The bottles might not stand
up to repeated pressurization as they are designed to hold pressure for a
certain amount of time.
    A “safety factor”, that is, the number of times below the failure stress
the operating stress is, is very important. Getting useful gas mass from a
PET bottle requires pressures near the burst pressure. Most people do not
seem to have a problem with this, but it is dangerous. The general safety
factor over burst pressure for pressure vessels is 2.5. So if a pressure vessel
bursts at 150 psi, you should operate at a maximum of 60 psi to satisfy the
safety factor. As the burst pressures vary from brand to brand and even
bottle to bottle within brands, no specific recommendation can be made,
but I think 60 psi should be safe for most bottles.
    Temperature is also something to consider. Pumping gas increases it’s
temperature and extremely hot temperatures are easily possible (this is
called adiabatic heating). Temperatures of 600+ degrees Fahrenheit are
12                                                       CHAPTER 3. PARTS

not uncommon if you neglect heat transfer in your analysis. The high
temperatures will be lost relatively quickly to heat transfer. The melting
point of PET is 500 degrees Fahrenheit... of course, the plastic won’t reach
that temperature due to the specific heat of the plastic6 , but it certainly
will be weakened as the heat is transfered. This is an issue soda bottle
manufacturers don’t worry about because they are not pumping gas into
the bottle.
    Even operating at safe pressures, I’d be afraid of sudden stresses on the
bottle. If you fall on your bottle, will it burst? You could do some tests
to figure it out. This is something the water rocket people, who typically
pressurize 2 liters, don’t have to worry about but I think is a very real
possibility for Nerf. Spud gunners don’t have to worry about this either as
they are not playing a game with their creations.

PVC LPA chambers.

Rubber bladders.

Safety. Any pressurized gas container is a bomb if mistreated. Steps
can be taken to virtually eliminate the chance of catastropic failure.

     • Pop-safety and relief valves release gas when the pressure gets
       too high. These are available from essentially any place that sells
       pneumatic components, McMaster-Carr being one of them.

     • Valves to quickly vent chambers can be particularly useful. Often
       a gas charge will be left in a tank after a match so they are a good idea
       to have even not considering safety. Gas chambers, unless specifically
       designed to be like HPA or CO2 tanks, should not be left pressurized
       for extended periods of time because of a phenomena called creep.
       The rapid discharge of a gas can be loud; an exhaust muffler, available
       from McMaster-Carr will reduce the noise level greatly

     • “Safety” valves much in the same vein of mechanical safeties on
       real guns can be useful to prevent gas flow from the gas source to the
       gas chamber. Regulators can be adjusted such that no gas can flow
       through them, making them fill two jobs. Similarly, mechnical stops
       preventing the motion of the trigger exactly like safeties in real guns
       can also be employed.
    6 The specific heat is how much energy is required to raise the temperature of a

unit cube of material one degree. Basically, two cubes with the same temperature but
different specific heats will have different internal energies.
3.10. SPRINGS                                                           13

What doesn’t improve safety. Wrapping duct tape, similar tapes,
denim, etc. around an inadequate chamber will likely make the chamber
less safe. The sleeve will provide no protection in an actual failure. And
while it might increase the pressure at which failure occurs, that just in-
crease the amoung of potential energy to be released, making the explosion
more dangerous.
    Wrapping carbon fiber, a second pipe, or something else much stronger
than the pressure vessel around the vessel probably will improve safety.
However, if you’re going to spend the money on those options, you’d be
better off turning your attention to proper pressure vessels. Also, the in-
crease in weight could be unacceptable.


3.10        Springs
Helical.

Constant force.

Buying springs.

Build your own springs.

See also.


3.11        Plastic sheet, solid tubes, blocks, etc.
Finding cheap plastic sheet.

Telescoping brass tubes.


3.12        Nuts, bolts, screws, and threads
Thread types.

Nuts.
   • Lock-nuts.
Screws and bolts. “Machine screws” are what are used most often in
Nerf.
14                                                CHAPTER 3. PARTS

     • Hex.

     • Phillips.

     • Slotted.

     • Combination.

Washers.

     • Lock-washers.

     • Fender washers.

Spacers.

Making screw threads in materials. See ***** in the machining chap-
ter for details.


3.13       Pumps and compressors
How pumps work.

Ball pumps.

Ball pumps.

Homemade pumps.

Air compressors.

Battery powered compressors


3.14       Lubricants
3.15       Glues, cements, and putties
Very often in design and repairs glues or another bonding material must be
used.

PVC cement. PVC cement is useful for fusing (not just bonding) two
pieces of PVC together.
3.16. STRAPS                15

3.16   Straps
3.17   Air cylinders
3.18   Material selection
3.19   Free samples
3.20   Further reading
Chapter 4

Darts and barrels

4.1     Darts

FBR.

Brands of FBR.

Making Stefans.

Aerodynamics.

Weight distribution and stability.

Holes in the back of darts. I’ve read suggestions to put holes in the
back of darts. Putting a hole in the back of the dart with the intention of
improving performance somehow is the equivalent of doing a voodoo ritual
to improve performance. The only reason to do this would be to improve
balance or mass slightly. Anyone who says differently has no idea of what
they are talking about.

    Simply put, no mechanism for improvement comes to mind and very
rarely are comparisons before and after a change done. If you disagree with
me, do a proper statistical study on the effects to demonstrate there is a
statistical difference.

                                    16
4.2. BARRELS                                                              17

4.2      Barrels
Friction.

Strength.

41 caliber.

50 caliber.

53 caliber.

55 caliber.

66 caliber.

Rifing.

Barrels and accuracy.


4.3      Combinations of darts and barrels
Finding working combinations of barrels and darts is very much luck. The
precise diameter of FBR depends on local pressure and temperature and
thus darts that work in one location may not work in another. The best
advice is to find a consistent high quality foam that is readily available to
you in bulk and then find a barrel that works. Knowing the approximate
size of the FBR (i.e. 53 caliber) can narrow the search to testing just a few
often compatible barrels.
    A good guess for fit can be achieved when the tolerance of the diameter
of the FBR is known. For example McMaster-Carr part number 93295K33
has a diameter of 0.5 inches with a tolerance of ±0.031 inches, which makes
the upper limit in diameter about 0.531 inches. Not surprisingly, the 53
caliber barrels fit this foam well; Aluminum with a diameter of 0.527 inches,
brass with a diameter of 0.5345 inches, and PETG with a diameter of 0.528
inches all fit this foam well.

Desired fit.


4.4      Further reading
Chapter 5

Machining and tools

5.1      Cutting

Pipe and tubes.

Sheet.




5.2      Drilling holes

5.3      Taping holes

A tap puts threads in an already drilled hole. The tap works by carving
threads into the material; the empty spaces fill up with carved off material.
General procedure.

Straight threads.

Tapered threads.

                                    18
5.4. THREADING PARTS TOGETHER                                          19

5.4     Threading parts together
5.5     Filing and sanding
5.6     Good practices
Use lock-nuts or lock-washers when appropriate. Lock-nuts and lock-washers
resist loosening from vibration by holding the screw in tension.
    Don’t glue anything shut that you might want to open it up later.
    Use threaded pipe fittings when possible. Threaded pipe and fittings
are easy to disassemble and allow for changes in design to be made in the
future.
    With PVC pipe, put plenty of pipe between fittings as to allow for cuts
to be made in the future if necessary. These cuts could be necessary if a
part of the blaster breaks or you want to make changes to the design.
    Unions are a little known fitting that can be very useful. If you an-
ticipate unscrewing a threaded pipe would be difficult in the design, use a
union there.


5.7     Further reading
Chapter 6

Basic solid mechanics

6.1       Stress and strain
Stress.

Strain.

Stress strain curves.
   Rubber does not have a linear stress-strain relationship and conse-
quently Hooke’s law does not apply to rubber.

The modulus of elasticity.
    Sometimes distinctions about the modulus are made. When a tensile
test is used the modulus is called the tensile modulus of elasticity. When
a bending test1 is used the modulus is called the flexural modulus of elas-
ticity. These two should be approximately the same for isotropic materials
(i.e. materials where the material properties are the same in all directions).

Yielding.

Ultimate stress.

Hardness. As could be imagined, making a test sample to obtain a stress
strain curve and pull all of the useful points from can be quite and in-
volved procedure. This testing also destroys the specimen, which makes it
unattractive to perform on existing structures that can not be destroyed.
   1 Bending   stress will be described in some detail further in this chapter.


                                            20
6.2. OTHER FAILURE MECHANISMS                                          21

Hardness tests are used sometimes for these reasons.
   There is a relationship between Brinell Hardness and UTS.

Safety factors.



6.2     Other failure mechanisms
Fatigue.

Buckling.

Impact.



6.3     Pressure vessels
Pressure ratings and safety factors.

“Thin-wall” pressure vessels.



6.4     Springs
Hooke’s law for springs.

Energy stored in a spring.

Calculating spring constants from geometry.

Geometric compression limit.

Maximum stress.

Spring material strength.

Fatigue.
   If you are willing to replace springs after a certain number of cycles,
then you do not need to worry about fatigue otherwise.
22                     CHAPTER 6. BASIC SOLID MECHANICS

6.5    Bending stress
Second moment of the area for a tube.

Beam equation.

Deflection.

Useful formulas.


6.6    Further reading
Chapter 7

Materials

Basic knowledge of the materials used in homemade Nerf blasters is essential
to choosing the correct material. Knowing the properties of a material is
not enough; understanding why the material has those properties is more
helpful.


7.1     Plastics
7.2     Metals
7.3     Further reading




                                    23
Chapter 8

Schematics




             24
Chapter 9

Mechanisms

9.1   Turrets
9.2   Magazines
9.3   RSCB clips
9.4   Rotating turret mechanisms
9.5   Dart-pusher mechanisms




                     25
Chapter 10

Basic mechanics and
dynamics

Nerf guns are relatively simple; comprehension of only a few basic principles
are necessary to understand most of the physics propelling the dart.


10.1        Basic mechanics and dynamics
Forces.

Friction.

Pressure.

Energy.

Energy stored in a spring.


10.2        Basic gas laws
Ideal gas law.

Thermodynamic processes.

Boyle’s law.

Isentropic expansion.

                                     26
10.2. BASIC GAS LAWS                                                    27



Potential energy.

Energy stored in an isothermal process. For an isothermal process
P V = constant (this again, is Boyle’s law). Let’s call this constant C.
From this and the work-pressure-volume relationship we can calculate how
much work is done on or by an isothermal process neglecting friction. This
is an ideal case, but to examine efficiency it can be useful.

                                      V2
                              W =          P dV
                                     V1
                             PV = C

   Following this logic we can see that:

                                          V2
                                               dV
                             W =C
                                      V1        V

   Integrating this equation, substituting the limits in and simplifying we
find that:


                                           V2
                              W = C ln
                                           V1
    This form is not too useful by itself. Often we don’t know the starting
or final volumes of the gas. Luckily, we don’t need to; Boyle’s law can find
                                                              P1
the ratio of volumes in terms of the ratio of pressures: V2 = P2 .
                                                         V1


                                           P1
                              W = C ln
                                           P2
    Note the sign convention here. Positive work means that energy is
leaving the system (i.e. the gas), or in other words, that the gas is doing
work on something else (as is the case during expansion of the gas on a
Nerf dart). Negative work means that energy is entering the system, or in
other words, that something else is doing work on the gas (as is the case
when compressing the gas with a pump).
    Now, what C should we choose? The answer depends on what you know.
When the gas is expanding, often you don’t immediately know the final
volume the gas will take; so C = P1 V1 . When the gas is being compressed,
28              CHAPTER 10. BASIC MECHANICS AND DYNAMICS

calculating the initial volume may be time consuming so for simplicity
C = P2 V 2 .
    The most useful expression will be that for the maximum work possible
from expansion of a gas chamber to atmospheric pressure. P is the chamber
operating pressure (an absolute pressure here), V is the chamber volume,
and Patm is the atmospheric pressure.


                                            P
                             W = P V ln                                (10.1)
                                          Patm

    Note that isothermal assumptions are not realistic models of most pro-
cesses in Nerf guns. In reality, things occur rather quickly for an isothermal
process; the temperature won’t be even relatively constant due to adia-
batic heating. Isothermal processes underestimate the energy required to
compress gas into a tank and overestimate the energy available as a tank
expands quickly and will have no chance for heat transfer to equalize the
temperatures. Isothermal assumptions are most reasonable when the com-
pression ratio r = V1 = P1 is near unity. However, the equation is simpler
                   V2
                         P2

than the adiabatic process equation and may be useful for a quick calcula-
tion.

Energy stored in an adiabatic process.




10.3      Conservation laws
Conservation of energy.

Conservation of mass.

Beyond the scope of this book. There are more advanced forms of
these conservation laws; they are far beyond the scope of this book. The
Euler equations describe flow of an inviscid fluid (i.e. a simple fluid with
no viscosity, heat transfer, or diffusion). The Navier-Stokes equations de-
scribe flow of a much more general fluid. Generally these systems of equa-
tions have no analytical solution, so computer simulation (commonly called
“computational fluid dynamics”, abbreviated CFD) is necessary.
    Flow through Nerf guns is relatively simple and the increase in accuracy
offered over the control volume approach taken in this book is negligible
for most situations. What CFD offers is neglegible improvements with
10.4. BASIC GAS DYNAMICS AND FLUID FLOW                                   29

far slower computations than the internal ballistics software described in
the simulation chapter. Consequently we will not discuss these approaches
further.


10.4      Basic gas dynamics and fluid flow
Force of pressure.

Flow through a valve.

Joule-Thomson effect.

Total pressure.

The Reynolds Transport Theorem. The Reynolds Transport Theorem
is a more explicit mathematical statement of the conservation of anything.
As the internal ballistics and
    Earlier the Euler equations (a simplification of the Navier-Stokes equa-
tions) were mentioned as beyond the scope of this book. That is not com-
pletely true; the equations that result from the Reynolds Transport Theo-
rem actually are the “integral” forms of the Euler equations. This again,
goes beyond the scope of this book, but the perspective is helpful.

Control volume analysis.




10.5      Number of shots from a gas source
Questions of the adequacy of a gas reservoir for lasting a Nerf war are com-
mon. With some algebra and logic, the number of shots a gas reservoir will
provide can be reasonably accurately predicted, greatly aiding the design
process.
   For all of these calculations we’ll take the approach that a gas reservoir
becomes useless when its output pressure runs below the operating pres-
sure. In some cases more shots can come from a gas reservoir in this state,
however, they will be of lower pressure and consequently lower power.
   As with most calculations in this section, all pressures are absolute
unless noted otherwise.
   These calculations are simple and may not be representative of reality.
They make a number of assumptions, specifically, that regulators are con-
30               CHAPTER 10. BASIC MECHANICS AND DYNAMICS

sistent and their outlet pressure is not dependent on the inlet pressure, that
the temperature during the use of a liquid gas source is constant, and that
the pressure of a bladder is independent of the bladder’s volume.
    More complicated and accurate formulas can be derived with more re-
alistic assumptions by similar procedures.
    The general approach to find the number of shots one can get from a gas
source is to track mass. The conservation of mass is a powerful statement
and its use here demonstrates that. In brief, figure out how much mass is
removed from the gas reservoir each shot, remove that mass to calculate
the new gas reservoir gas mass, and repeat.

Non-regulated gas sources. Non-regulated gas sources are particularly
complex; the gas flows from the gas reservoir into the gas chamber until the
pressure in the two equalize. This pressure is different for each step; conse-
quently an iterative approach using spreadsheets often is most convenient.
    The time scale of the discharge of gas from a gas reservoir in a Nerf war is
long, therefore an isothermal assumption is useful. To reiterate points made
earlier, isothermal assumptions are good for when there is heat transfer to
ensure that the gas temperature is constant–heat transfer is only possible
with a rather large time scale.
    Pr0 is the starting pressure of the gas reservoir. Vr is the gas reservoir
volume. Vc is the gas chamber volume. Patm is atmospheric pressure. R
is the specific gas constant. T is the temperature of the gas. n is the shot
number. mr (n) is the total gas mass in the gas reservoir after shot number
n. Similarly, Pr (n) is the pressure of the gas reservoir after shot number n.
    Using ideal gas assumptions (specifically, the mass form), note that
            r0 V
mr (0) = PRT r and Pr (0) = Pr0 . ∆mn+1 , the amount of gas mass removed
                                        n
can be found by the ideal gas law as well, but some logic is necessary before
finding it. Gas at atmospheric pressure is left in the gas chamber; this
mass must be subtracted from the total mass of the gas chamber at the
equilibrium pressure (i.e. the common pressure that both the gas reservoir
and gas chamber equalize to).
    That statement is made more explicit like this:


                                    Peq Vr   Patm Vc
                         ∆mn+1 =
                           n               −                             (10.2)
                                     RT       RT

    What is Peq ? It is the pressure the gas reservoir and chamber equalize
at (so Peq = Pr (n+1)). Using the principle of conservation of mass and the
ideal gas laws we can calculate this. As we’ll see, the isothermal assumption
will reduce this to Boyle’s law.
10.5. NUMBER OF SHOTS FROM A GAS SOURCE                                  31

   The gas reservoir has a pressure after the nth shot of Pr (n). The gas
chamber is evacuated after a shot so it has a pressure of Patm .


                         mr + mc = mtotal
                                     Pr (n)Vr
                               mr =
                                       RT
                                     Patm Vc
                              mc =
                                      RT
                                     Peq (Vc + Vr )
                            mtotal =
                                          RT
   By combining these equations one arrives at the following:


                    Pr (n)Vr + Patm Vc = Peq (Vc + Vr )

    Note that the term RT cancels out from each statement–this is because
for an isothermal process P V is proportional to mass–and consequently
Boyle’s law can be seen as simply a statement of the conservation of mass
for isothermal processes.
    Solve for Peq and we end up at:


                                       Pr (n)Vr + Patm Vc
                  Peq = Pr (n + 1) =                                  (10.3)
                                             Vc + Vr

   From here we can derive an explicit formula for ∆mn+1 . Plug Peq into
                                                     n
the equation for ∆mn+1 .
                   n



                           Pr (n)Vr2 + Patm Vc Vr   Patm Vc
                ∆mn+1 =
                  n                               −                   (10.4)
                                RT (Vc + Vr )        RT

    The initial conditions of the mass in the gas reservoir (at “shot zero”)
are:

                                       Pr0 Vr
                              mr (0) =
                                        RT
                              Pr (0) = Pr0

   Unfortunately there does not appear to be any simple equation–even an
implicit one–to find the number of shots one can get from a non-regulated
32                CHAPTER 10. BASIC MECHANICS AND DYNAMICS

gas reservoir. Use of a spreadsheet with the mass difference and the equal-
ization pressure formulas seems ideal. The ugly expressions alone are reason
enough to use a regulator; combine that with the consistency of pressure
and regulators seem extremely attractive.
    To account for the volume of whatever is between the gas reservoir and
the gas chamber (tubing, valves, etc.), figure out approximately what the
volume that is exhausted and replenished is and add that to the gas cham-
ber volume. The volume that is not exhausted must be added to the gas
reservoir volume.

Regulated gas sources. The derivation for regulated gas sources is far
cleaner than that of non-regulated sources as the new equalized pressure of
each shot does nto need to be calculated.
    Like with non-regulated sources we’ll find ∆m, the difference in gas
reservoir mass between shots. Each shot is pressurized to Preg , the reg-
ulated pressure, and after the gas is exhausted what remains is at atmo-
spheric pressure. Note that here the mass removed per shot is independing
of the number of shots taken1 due to the regulator. Following similar logic
to that applied for non-regulated sources we find that:

                                      Vc (Preg − Patm )
                             ∆m =
                                             RT
    Here Preg is the set pressure of the regulator. Using the principle of
conservation of mass (like we did with non-regulated sources again) we can
arrive at the following equation:


                      Pr0 Vr = Pmin Vr + nVc (Preg − Patm )

   Here Pmin is the minimum acceptable pressure of the gas reservoir.
Sometimes you want to refill before the gas reservoir’s pressure drops below
the regulated pressure as a precaution against that–however–most people
are interested in how many shots at the regulated pressure they can get
from a reservoir, so Pmin = Preg . Rewriting the equation we arrive at:


                                     Vr (Pr0 − Preg )
                               n=                                               (10.5)
                                    Vc (Preg − Patm )

     Round this result down as you can’t get a fraction of a shot.
     1 Atleast until gas reservoir pressure drops below the regulated pressure, but our
task is to find where that happens here so we do not worry about that.
10.6. FURTHER READING                                                   33

    Note that this assumes the regulator is perfect and regulates at the
pressure it was set at consistently. Some regulators are better than oth-
ers at this; the small Clippard MAR regulators I have used have an outlet
pressure that seems rather dependent on the inlet pressure. This I do not
consider to be a major problem as adjustments can be easily made on the fly.

Non-regulated liquid sources. Examples of liquid sources are CO2 and
propane.

Regulated liquid sources.

Bladders.


10.6      Further reading
Chapter 11

Testing

Testing can provide a deeper understanding of the dynamics of Nerf guns
and often is necessary to gather empirical data for a theoretical analysis.
This chapter will detail which tests I’ve found useful and how to interpret
and use their data.


11.1      Range
11.2      Hang time
11.3      Muzzle velocity
11.4      Pressure as a function of time
11.5      Projectile position as a function of time




                                    34
Chapter 12

Terminal ballistics

Basic terminal ballistic knowledge is required to understand what makes
projectiles dangerous. This knowledge will make Nerf a safer and more
enjoyable game for all involved.

Kinetic energy density.

Modulus of resilience.

Maximum muzzle velocity for safety. Based on known energy densi-
ties required to cause damage to the eyes and using a safety factor of two,
the table below was constructed to show the maximum safe muzzle velocity
for different dart masses.

Energy the dart absorbs. Softer tipped darts will obviously absorb
more energy than hard tipped darts. Avoid hard objects on the tips. Cap-
tainSlug’s felt tipped darts are one approach.


12.1      Further reading




                                    35
Chapter 13

Simulation

Internal ballistics. Internal ballistics concern how the projectile is ac-
celerated inside of the gun. The internal ballistics of completely subsonic
guns such as Nerf blasters are very well understood, however, they are not
the easiest topic to understand. This chapter will require more knowledge
than the others and I can not distill it more than I will here.

External ballistics. External ballistics are the area of ballistics that con-
cern the projectile after it has left the gun. The dart now is in flight.


13.1         Internal ballistics of a pneumatic gun
In 2009 I wrote my own simulation of the internal ballistics of a pneumatic
gun called BAGS1 .


13.2         Internal ballistics of a spring gun
13.3         External ballistics of a dart
Major forces on darts in flight.

   • Gravity

   • Drag

   • Lift
   1 btrettel’s   air gun simulation, available from http://trettel.org/bags/.


                                            36
13.4. FURTHER READING    37

13.4   Further reading
Chapter 14

Optimization and
efficiency

14.1      Goals of optimization
Minimizing kinetic energy required for a dart to attain a certain
range. Darts that travel further with less energy are good (obviously so).

Maximizing energy efficiency. This essentially means maximizing the
ratio of projectile kinetic energy to the energy put into the system (by
pumping, compressing the spring, etc.).

Maximizing number of shots per tank. Sometimes you don’t care
too much about efficiency but do care about getting the most shots out of
a gas reservoir.
    We’ll see that maximizing energy efficiency and maximizing the number
of shots per tank do peak at different configurations, but there is a great
deal of overlap between these configurations, so both goals can practically
be attained.

Minimizing gas mass used per shot. Often, especially with liquid
sources such as CO2, maximizing the number of shots from a tank involves
essentially minimizing the gas mass used per shot.

Minimizing total energy to achieve a certain distance of range.
Due to the complex dynamics of the problem, the dart configuration that
requires the least kinetic energy to get a certain range isn’t necessarily

                                   38
14.2. SOURCES OF INEFFICIENCY                                                         39

the dart configuration that results in least energy used overall (i.e. poten-
tial energy of the gas in the gas chamber). This is the “best” optimization
goal–we’ll see that this often requires too much computation and some sim-
plifications (i.e. using different efficiency goals) are perfectly acceptable.


14.2        Sources of inefficiency
Slow build up of pressure. If the dart moves down the barrel slowly
from the initial low pressure, a shorter segment of the barrel (an “effective”
barrel length) is used for most of the acceleration.

The pressure the dart exits at is above atmospheric pressure.
This basically means that you’re not extracting all of the energy available
from the gas. Be aware that the ideal length is not necessarily where the
barrel gas static pressure equalizes with the atmosphere1 , but it should be
close.


14.3        Internal ballistic parameters for pneu-
            matic guns
Barrel length. Barrel length plays a major role in optimizing Nerf guns.
A barrel that is too short won’t capture the maximum amount of energy,
and a barrel that is too long will reduce performance due to friction and
with a long enough barrel, pulling a vacuum in the barrel.

Gas chamber volume.

Gas chamber operating pressure.

“Dead” volume. Dead volume isn’t quite useless. In fact, simulation can
show that while too much is bad for performance, often some will increase
performance a small but significant amount over an equivalent blaster with
no dead volume.

Projectile mass. Projectile mass can make a large difference in energy
transfer. Pilot volume. This applies only to pilot operated valves. Less
pilot volume wastes less gas mass. However, too little pilot volume can be
    1 In fact, with valves that close before the dart leaves the barrel, the ideal barrel

length often has considerable pressure in the barrel. See later in this chapter about
HEAR valves.
40                 CHAPTER 14. OPTIMIZATION AND EFFICIENCY

problematic. If the pilot volume is so small such that the movement of the
piston can result in an increase in the pilot pressure, this will result in a
“twitch” that will slow the full opening of the valve. Valve flow coeffi-
cient. A flow coefficient that is too low will cause the pressure to build up
slowly.


14.4      Internal ballistic parameters for spring
          guns
Optimizing spring guns is far more complicated than optimizing pneumatics
as they involve many more variables that significantly affect performance.

Barrel length.

“Dead” volume.

Projectile mass.

Projectile friction. Spring guns build up pressure slowly. Having high
static friction such that the pressure can build high for high acceleration
and low dynamic friction such that the projectile faces little opposition
from the barrel is optimal.

Plunger mass.

Spring stiffness.

Spring length and/or deflection.

Plunger diameter.


14.5      External ballistic parameters
Drag coefficient. Obviously, the less drag affects the dart, the farther it
will travel.

Projectile mass. Like with internal ballistics, projectile mass plays a
major role in efficiency. Too little mass will produce a projectile that is af-
fected too much by drag. Too much mass will simply require more kinetic
energy to go a certain distance simply because the increased mass increase
14.6. OPTIMIZATION OF DARTS                                               41

                                       m·v 2
the kinetic energy (kinetic energy =    2 ).




14.6      Optimization of darts

Dart optimization table. To simplify the optimization of darts, I have
calculated the dart mass and muzzle velocity combinations that get pre-
scribed ranges at specific coefficients of drag. These calculations required
over a week of continuous computing.

Using the dart optimization table. If you desire about 100 feet of
range and you know your darts have a drag coefficient of about 0.6, then go
to the table for a Cd of 0.6 and range of 100 feet. Note where the minimum
required kinetic energy occurs–this is the optimal point. You may not be
able to realistically make a dart with the mass required for this point, so
you can choose an other combination that get similar performance.
   If my darts have a mass of 1.0 grams, then




14.7      Optimization of pneumatic guns

Pilot volume.

HEAR valves.

Heat transfer from adiabatic heating in pneumatic guns. This
is an unavoidable source of inefficiency in all pneumatic guns. Adiabatic
compression increases both the pressure and temperature of the gas–this
temperature difference will equalize with the outside environment eventu-
ally, reducing pressure and consequently the total energy stored. Often
what is mistaken for a leak is in fact heat transfer. Heat transfer is a
significant advantage spring guns have over pneumatics.
   A similar effect that may affect spring guns is hysteresis. An elastic
system with hysteresis’ loading force is higher than the unloading force.
This effect is insignificant in metal springs. Rubber however has pronounced
hysteresis, and for that reason it should be avoided when efficiency is a goal.
The energy lost, like in pneumatic systems, is dissapated as heat.
42          CHAPTER 14. OPTIMIZATION AND EFFICIENCY

14.8    Optimization of spring guns
14.9    Numerical optimization algorithms
14.10    Further reading
Chapter 15

Examples




             43
Chapter 16

Putting it all together:
FANG




             44
Bibliography




               45
Reference tables and
charts

 16.1 Material properties . . . . . . . . . . . . . . . .    .   .   .   .   .   .   .   47
 16.2 PVC pipe dimensions and weight . . . . . . . .         .   .   .   .   .   .   .   48
 16.3 PVC fitting dimensions and weight . . . . . . .         .   .   .   .   .   .   .   48
 16.4 Temperature derating of PVC pipe and fittings           .   .   .   .   .   .   .   48
 16.5 Thread data . . . . . . . . . . . . . . . . . . . .    .   .   .   .   .   .   .   48
 16.6 Telescoping brass sizes . . . . . . . . . . . . . .    .   .   .   .   .   .   .   48
 16.7 Tap and drill combinations . . . . . . . . . . .       .   .   .   .   .   .   .   48
 16.8 Valve data . . . . . . . . . . . . . . . . . . . . .   .   .   .   .   .   .   .   48
 16.9 O-ring data . . . . . . . . . . . . . . . . . . . .    .   .   .   .   .   .   .   48
 16.10Schematic symbols . . . . . . . . . . . . . . . .      .   .   .   .   .   .   .   48
 16.11Dart optimization table . . . . . . . . . . . . .      .   .   .   .   .   .   .   48
 16.12Unit conversions . . . . . . . . . . . . . . . . .     .   .   .   .   .   .   .   48




                                    46
                           Table 16.1: Material properties
This data represent only generic figures found in textbooks and online reference. Use
data provided by manufacturers and suppliers or ask for such data for the most precise
and accurate numbers.




                                                                                              Ultimate strength, σu
                            Elastic modulus, E




                                                                                                                                   Yield strength, σy
                                                                 Density, ρ

       Material
                                                          lb                    g
                   M psi                         GP a    in3                   cm3
                                                                                       ksi                       MP a       ksi                     MP a
                                                          Plastics
            ABS     0.31                         2.1    0.0376                 1.04    —                              —     6.1                         42
         Acrylic    0.43                         2.9    0.0426                 1.18    10.2                           70    9.6                         66
          Delrin    0.35                         2.4    0.0513                 1.42    9.0                            62    —                           —
      Nylon 6/6      0.4                         2.8    0.0412                 1.14    11                             75    6.5                         45
  Polycarbonate     0.35                         2.4    0.0433                 1.20    9.5                            65     9                          62
     Polystyrene    0.45                         3.1    0.0374                 1.03     8                             55    8.0                         55
            PET      0.4                         2.8    0.0494                 1.37    6.6                            46    —                           —
          PETG       0.3                         2.1    0.0459                 1.27    7.7                            53    7.3                         50
          LDPE      0.042                        0.29   0.0332                0.919    1.9                            13    1.4                         9.7
     UHMW-PE        0.12                         0.83   0.0336                0.0929   5.8                            40    3.1                         21
          HDPE      0.23                         1.6    .0347                 0.959    —                              —     4.2                         29
            PVC     0.41                         2.8    0.0513                 1.42    —                              —     7.5                         52
          CPVC      0.36                         2.5    0.0531                 1.47    —                              —     7.3                         50
                                                           Metals
   Al (2024-T3)     10.4                          72    0.100                  2.73    70                             483   50                          345
   Al (6061-T6)     10.4                          72    0.098                  2.70    45                             310   40                          276
   Al (6063-T5)     10.4                          72    0.098                  2.70    27                             186   21                          145
     Brass (260)     16                          110    0.308                  8.53    61.6                           425   52.2                        360




                                                               47
      Table 16.2: PVC pipe dimensions and weight



     Table 16.3: PVC fitting dimensions and weight



Table 16.4: Temperature derating of PVC pipe and fittings



                Table 16.5: Thread data



           Table 16.6: Telescoping brass sizes



         Table 16.7: Tap and drill combinations



                 Table 16.8: Valve data



                Table 16.9: O-ring data



            Table 16.10: Schematic symbols



          Table 16.11: Dart optimization table



             Table 16.12: Unit conversions


                           48
Glossary




           49
List of online part sources




             50
Index




        51
Index

Air gun, 2

Pneumatic Nerf gun, 2




                        52

				
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