Internal Ballistics of a Spring

Internal Ballistics of a Spring-Air Pellet Gun
Stephen J. Compton
05/18/2007
Abstract
The internal workings of the common spring-air pellet gun is modeled in this work. This mathematical model is designed to accurately
determine the ballistic performance and maximum internal gas pressure
of a specific design at the drawing board stage. The model is based upon
basic principles of physics including, energy analysis principles, Hook’s
law for springs, the ideal gas law, linear and rotational kinetic energy,
and Newton’s laws of motion, as well as ballistic quantification of bore
friction. Model solution was obtained using MATLAB to numerically
solve a system of linear ordinary differential equations.
1
1
Introduction
A spring-air pellet gun is a firearm which fires a bullet or pellet purely from
the force of compressed air from a spring piston chamber. These firearms are
ideal for folks who love the shooting sports, but who dont live in the countryside.
Spring-air pellet guns are quieter then conventional cartridge firearms. Being
of low power these guns can allow a safe, indoor target range in the privacy of
ones own home.
With a pellet gun’s greater safety and much quieter operation, private
lands hunting permission is often easier to obtain. Pellet rifle hunting is a scaled
down version of varmint and small game hunting, normally done with a light
cartridge firearm. Pellet rifles are used instead of cartridge firearms for safety
and perhaps surprisingly, for effectiveness. Because they deliver the same energy
on each shot, the trajectory of a spring-air pellet gun is extremely consistent –
they are often more accurate then conventional firearms. In addition the fact
that they are relatively quite is a great advantage to the hunter. The specific
game which is hunted depends on the power and caliber of the pellet gun in
question – pellet guns have been successfully used to hunt game as small as
mice and rats where conditions exist that prevent effective elimination by other
means such as traps or poison bait. In addition there are some very large and
powerful spring-air pellet guns which can effectively be used to hunt game as
large as the various species of North American deer. One such spring-air pellet
gun, which is marketed as a survival weapon, is capable of launching a .45caliber 180 grain pellet at velocities approaching the speed of sound ( Speed
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of Sound ≈ 340 m/s ). The most common caliber in the United States is the
.177 caliber. The more powerful versions are capable of effectively being used
to hunt small game up to, and including, fox and is commonly used to hunt
gophers, rabbits, and squirrels.
Spring-piston guns operate by means of a coiled spring and piston contained within a compression chamber. Cocking the gun causes the piston spring
to be compressed; pulling the trigger releases the piston allowing it to move,
compressing the air in the chamber directly behind the pellet. Once the air
pressure has risen enough the pellet moves forward, propelled by an expanding
column of air. All this takes place in a fraction of a second. Most spring piston
guns are single-shot by nature. Such spring-air pellet guns are able to achieve
muzzle velocities near, or in some cases, in excess of the speed of sound from a
single stroke of a cocking lever. The difficulty of the cocking stroke is usually
related to the power of the gun, with higher muzzle velocities requiring greater
cocking effort. The better quality spring-air guns can have long service lives,
often exceeding thirty years.
The design and manufacture of spring-air pellet guns offer a set of engineering and economic problems. Although quality is a factor, the market selling
price of the various models of spring-air pellet guns shows a direct relationship
between the price and the ballistic performance of the firearm. Spring-air pellet
guns with very low power levels sell for as little as twenty dollars. High power
supersonic weapons sell for over five hundred dollars, even in used condition,
and often carry four digit price tags when new. It is therefore, very advantages
to be able to quantify the real world muzzle velocity and ballistic energy of a
spring-air pellet gun design at the drawing board stage before resources are expended to build a prototype for testing. The primary selling point, and primary
consideration, of a spring-air pellet gun is the muzzle velocity it is capable of
generating with a common weight pellet. However, there is another consideration, especially when considering calibers other then .177, this consideration is
the muzzle energy of the pellet. The muzzle energy of the pellet is a function of
its velocity, mass, and gyration spin imparted by the barrels rifling.
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One specific design concern is the maximum internal gas pressure generated within the spring chamber and barrel of the firearm. This maximum gas
pressure value has a direct relationship to the necessary strength and quantity
of the materials used in the construction of the firearm as well as the specific
design itself. This will play a significant part in determine the manufacturing
cost of a spring-air pellet gun.
The problem of determining these values without an actual working prototype in the lab via mathematical analysis of the internal forces at work during
the firing sequence can be quite daunting. There are two separate moving objects which interact during the firing sequence; the plunger attached to the
spring which serves as the mechanism’s power plant and the pellet within the
bore. Each of these two objects have their own position, velocity, and acceleration vectors and due to the compressed gas which serves as a medium between
them the individual values of each of these vectors has an effect on the value
of the other vectors. It appears to be an endless circle where each value is dependent upon the other values and there appears to be no solution via normal
arithmetic, algebra, or calculus. In order to solve this problem we must go
one step higher in the math hierocracy. The only way to solve this particular
problem is through the mathematical science known as ”Ordinary Differential
Equations” or ODE for short. This system is in affect a system of two second
degree ODE equations. The compressed gas, which links these two equations,
can be used to define the two acceleration terms as functions of the position
terms, thus, simplifying the ODE system from a second order system to a first
order system. A first order system of two equations is solvable using this mathematical science.
2
Basic Principles and Equations of Physics
A ground work starting from the most basic principles of physics must be
laid in order to build a realistic model of the internal forces at work. First, of all
energy cannot be created or destroyed but merely converted from one form to
another. The total energy input must exactly equal the total energy output, in
any system or machine. In the case of a spring-air pellet gun the input energy
is the mechanical force imparted by the user in cocking the coil spring within
the internal workings of the firearm. The energy stored in the spring can be
quantified through the integration of Hook’s law which states that the force
exerted by a spring is directly proportional and opposite to the displacement:
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Z
Energy Stored In Spring =
(Spring Force)
Z
xsf
(kx)dx
=
xso
=
1
k(xsf − xso )2
2
Where:
xso = Length of Spring Compressed Before Cocking
xsf = Length of Spring Compressed After Cocking
This equation represents the total energy stored in the spring before the firing
sequence begins.
During the firing sequence, the energy released by the spring is transferred
to the pellet via the medium of compressed gas in the internal spring plunger
chamber and barrel behind the pellet. The pressure of this gas (air) as a function
of its changing volume and temperature, is quantified by the Ideal Gas Law:
Po Vo
P (t)V (t)
=
To
T (t)
The first assumption will now be made in this model. The assumption
is that the temperature of the compressed gas within the internal mechanism
remains constant. This assumption is very reasonable when one considers the
minuscule mass of gas trapped inside the comparatively massive heat sink of the
pellet guns metal components. This massive heat sink should serve to maintain
a constant gas temperature during the firing sequence. With this assumption
the Ideal Gas Law can be simplified for this mathematical model so that the
pressure is only a function of the changing volume:
Po Vo = P (t)V (t)
Po Vo
P (t) =
V (t)
Next, since we are only concerned with the pressure above atmospheric pressure
we will subtract the initial (atmospheric) pressure from the equation.
P (t) =
Po Vo
− Po
V (t)
5
The compressed gas not only transfers energy but also stores it. The
amount of energy stored in the gas due to its compression at any given time can
be quantified through the integration of the previously determined equation for
pressure:
Po Vo
− Po dx
V (t)
Z Vo Po Vo
=
− Po dx
V (t)
V (t)
2
Vo
− 2Vo + V (t)
= Po
V (t)
Z Energy Stored In Compressed Gas =
The pellet during the travel down the bore experiences ballistic friction
forces which result in kinetic energy being converted into heat energy. This
energy is not destroyed but wasted in the sense that it is not used to accelerate
the pellet towards its final muzzle velocity. This bore friction can be categorized
into four separate friction forces which work together to account for the total
energy wasted as bore friction. The first is simple gravitational friction:
Simple Gravitational Friction = mp gµxp (t)
Where:
mp = Mass of Pellet
g = Force of Local Gravitational Field
µ = Kinetic Coefficient of Friction
xp (t) = Position of Pellet in Barrel
Second, is the rotational rifling friction:
Z
Rotational Rifling Friction =
2
Rπcµ( πc4 P (t) − mp ap (t))
sin θ + µ cos θ
Where:
R = Twist Rate of Rifling
c = Caliber
ap (t) = Acceleration of Pellet in Barrel
θ = Angle of Rifling Twist
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!
dxp (t)
Third, is the linear rifling friction:
Z Linear Rifling Friction =
4Ip Rπµap (t)
c(cos θ − µ sin θ)
dxp (t)
Where:
Ip = Mass Moment Inertia of Pellet
Fourth, is the friction due to the elastic compression of the pellet being forced
into the bore. This is an extremely complex statically indeterminant problem
far beyond the scope of this paper. Any sensible engineering analysis in a time
equals money business environment is best done through direct empirical analysis. One must simply quantify the necessary minimum force that is required
to push a specific pellet through a bore of appropriate caliber at a constant but
very low velocity. This value will remain the same provided the pellet and bore
material remain the same regardless of what other factors change. This can be
further generalized by dividing this force by the coefficient of friction between
the pellets material and the barrel material. Thus, this force becomes a unique
property of the specific pellet; allowing quick and easy analysis once this value
has been determined and tabulated for a variety of common pellet designs. The
gathering of such data can be easily done in any half-way decently equipped lab
in a workday or less and would continue to be valid for the analysis of future
designs for as long as those pellet designs remained popular. Considering that
the lowly BB pellet design has been around for over a hundred years and the
basic diabolo pellet design has been around for over 40 years – no problems
there.
Elastic Compression Friction = µFE.C. xp (t)
Where:
FE.C. = Elastic Compression Force of Pellet
The elastic compression force of the pellet is empirical data obtained via the
previous mentioned method such that:
FE.C. =
Fexp
µexp
Where:
Fexp = Experimentally Determined Minimum Applied Force
µexp = Coefficient of Friction During Experimental Analysis
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The ultimate outcome of the process happening within the internal workings of a spring-air pellet gun is to impart a fairly large amount of kinetic energy
to the pellet in the form of both linear velocity and rotational velocity. The linear kinetic energy of a projectile’s linear velocity. In this case, the pellet’s linear
velocity is directly proportional to both the projectiles mass and the square of
it’s velocity:
Linear Kinetic Energy of Pellet =
1
mp (vp (t))2
2
Where:
vp (t) = Velocity of Pellet in Barrel
The rotational kinetic energy of the pellet is directly proportional to both the
pellets mass moment of inertia and the square of its rotational velocity:
Rotational Kinetic Energy of Pellet = 2Ip (Rπvp (t))2
Thus, the total kinetic energy of the pellet is the summation of the linear kinetic
energy and the rotational kinetic energy:
1
mp (vp (t))2 + 2Ip (Rπvp (t))2
2
1
= mp (vp (t))2 + 2Ip R2 π 2 (vp (t))2
2
1
= ( mp + 2Ip R2 π 2 )(vp (t))2
2
Pellet Energy =
Finally, only half of the output energy will actually by imparted to the pellet.
Newton’s third law of motion clearly state that for every action there is an equal
and opposite reaction – in the case of firearms this is commonly known as recoil.
The summation of these two equal energy outputs, the pellet energy and the
recoil energy, represent the total output energy from the system at any given
moment during the internal energy conversion process also known as the firing
sequence.
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3
Setup Energy Equation in Preparation to Solve
Now that all the basic equations and principles have been gathered together it is time to setup the energy equations in preparation to solve the modeling problem. In order to produce a solvable system of equations for the model
we must end up with two equations such that the velocity of the spring and the
velocity of the pellet are defined as functions of the position of the spring, the
position of the pellet in the barrel, and time elapsed during the firing sequence.
First let’s begin with the fact that the output energy must equal the input
energy. Followed by input energy is spring energy minus energy sinks, energy
sinks are energy lost to bore friction, energy stored in the form of compressed
gas, and kinetic energy of piston. Output energy is pellet kinetic energy and
recoil kinetic energy, and finally, output energy can be simplified to be twice
the pellet kinetic energy:
Each one of these five terms must be mathematically defined from the
previously outlined Physics equations and principles and substituted back into
the equation. Due to the massive size and complexity of the resulting equation
this substitution will be in most cases implied rather then actually shown. Firstoff, it is necessary to derive equations specifically defining the volume of the
gas, pressure of the gas, acceleration of the spring piston, and acceleration of
the pellet as functions of time elapsed during the firing sequence:
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V (t) = Vo − π(rpiston )2 xs (t) +
π 2
c xp (t)
4
Where:
V (t) = Volume of Air as a Function of Time
Vo = Volume of Air in Cylinder When the Air Gun is Cocked
rpiston = Radius of Spring Piston
xs (t) = Position of Spring Piston as a Function of Time
xp (t) = Position of Pellet in Barrel as a Function of Time
———————————————————–
P (t) =
=
Po Vo
− Po
V (t)
Po Vo
− Po
Vo − π(rpiston )2 xs (t) + π4 c2 xp (t)
Where:
P (t) = Pressure as a Function of Time
Po = Local Atmospheric Pressure
———————————————————–
Σ(Force on Piston)
ms
k(xsf − xs (t)) − π(rpiston )2 P (t)
=
ms
as (t) =
Where:
as (t) = Acceleration of Spring Piston as a Function of Time
ms = Mass of Spring Piston + 1 /3 Spring Mass
k = Spring Constant
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ap (t) =
=
=
P (t)πc2
4mp
Po Vo
2
Vo −π(rpiston )2 xs (t)+ π
4 c xp (t)
− Po πc2
4mp
4mp
Po πc2
Po Vo πc2
−
4mp
Vo − π(rpiston )2 xs (t) + π4 c2 xp (t)
Now each of the five terms of the energy equation, Pellet Energy, Spring Energy,
Bore Friction, Energy Stored In Compressed Gas, and Piston Kinetic Energy
will be mathematically defined as functions of xs (t) and xp (t):
1
2(Pellet Energy) = 2( mp + 2Ip R2 π 2 )(vp (t))2
2
= (mp + 4Ip π 2 R2 )(vp (t))2
———————————————————–
1
(Spring Energy) = k xsf xs (t) − (xs (t))2
2
———————————————————–
Due to its extensive length we will define the Bore Friction term to be the summation of its four components, Simple Gravitational Friction,Rotational Rifling
Friction,Longitudinal Rifling Friction, and Elastic Compression Friction. Each
of these components will be mathematically defined:
(Simple Gravitational Friction) = mp gµxp (t)
———————————————————–
2
Rπcµ( πc4 P (t) − mp ap (t))
sin θ + µ cos θ
!
Rπcµ(mp ap (t) − mp ap (t))
=
sin θ + µ cos θ
Z Rπcµ(0)
=
dxp (t)
sin θ + µ cos θ
Z
(Rotational Rifling Friction) =
Z =0
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dxp (t)
dxp (t)
———————————————————–
(Elastic Compression Friction) = µFE.C. xp (t)
———————————————————–
———————————————————–
(Piston Kinetic Energy) =
1
ms (vs (t))2
2
Where:
vs (t) = Velocity of Spring Piston as a Function of Time
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Finally, after all that pushing symbols around every terms of the energy
equation has been defined in terms of vp (t), xp (t), vs (t), and xs (t). When you
acknowledge that vp (t) is the derivative of xp (t) you can see that we have the
beginning of a first order system of ODEs. There is a little hitch, however,
we still need a second equation defining vs (t) which is the derivative of xs (t),
both of which are terms in our energy equation. Solving directly for vs (t) using
everything that has been learned so far is the most direct route:
Now with a little bit of arithmetic we have a completed system of two first order
differential equations:
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Numerically Solve ODE System
Unfortunately this system is far to complicated to solve via hand calculations (at least in my lifetime). It, therefore, will be necessary to solve this system
via. numerical analysis methods performed by a computer. The following file
when entered into MATLAB as a function M-file will preform these
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calculations and also display a number of conceptual graphs along the way. The
output values will be the muzzle velocity of the pellet, muzzle energy of the
pellet, and the maximum gas pressure developed within the internal workings
of spring-air pellet gun during the firing sequence. Necessary Input values are
mp , ms , rpiston , c, R, FEC , µ, Ip , Vo , k, xsf , xso , and the barrel length. The
proper values for g and Po are included within the file but can be changed if
desired. This would only be done for unique situations such as if the spring-air
pellet gun were fired at high altitude or on Mars instead of Earth.
--------------------------------------------------------------------------------------------function y = Project %Allows M-file to be called as a function in MATLAB
%Imput Specific Design Values
global m_p m_s r_piston c R f_e mu I_p g P_o V_o k x_s_f x_s_o R_angle
m_p = ****;
%mass of pellet ( kg )
m_s = ****;
%mass of spring piston + 1/3 spring mass ( kg )
r_piston = ****;
%radius of spring piston in air gun ( m )
c = ****;
%caliber ( m )
R = ****;
%twist rate of rifling ( 1/m )
f_e = ****;
%experimentally quantified elastic compression friction force for pellet-style ( N )
mu = ****;
%kinetic coeficient of friction between pellet and barrel materials
I_p = ****;
%mass moment of inertia of pellet along its lognitudinal central axis ( kg*m^2 )
g = 9.801;
%force of local gravitational feild (use standard earth value for analysis) ( m/(kg*s^2) )
P_o = 1.013*10^5;
%local atmospheric pressure (normally set at see level for analysis) ( Pa )
V_o = ****;
%Volume of air in cylinder when the air gun is "cocked" ( m^3 )
k = ****;
%spring constant ( N/m )
x_s_f = ****;
%compression of spring assembly after cocking ( m )
x_s_o = ****;
%compression of spring assembly before cocking ( m )
barrel_length = ****;
% ( m )
R_angle = atan(R*(pi*c)); %Twist angle of Rifling Formula -- Don’t Change Formula !!!
%First solve the initial value problem using "ode45"
tspan=[0,0.01];x0=[0;0];
%Time Span ( tspan ) may have to be increased or decreased for different models
[t,x] = ode45(@problem_func,tspan,x0);
x(:,1) = real((x(:,1) + ((x_s_f-x_s_o)-x(:,1)).*(x(:,1) >= (x_s_f-x_s_o))));x(:,2) = real(x(:,2));
%Make a plot for (t,x_p)
plot(t,x(:,2))
xlabel(’t’)
ylabel(’x_p’)
title(’Position of Pellet in Barrel as a Function of Time’)
shg
pause
%Next make a plot for (x_p,v_p)
vel_s = real((x(:,1)<(x_s_f-x_s_o)).*(k.*(x_s_f - x(:,1)) - ( (pi.*r_piston^2.*P_o.*V_o)
./(V_o - pi.*r_piston.^2.*x(:,1) + (pi./4).*c.^2.*x(:,2)) ) + pi.*r_piston.^2.*P_o).*((t)./(m_s)));
vel_p = real(sqrt( (SpringEnergy(x(:,1)) - BoreFriction(x(:,1),x(:,2))
- EnergyStoredInCompressedGas(x(:,1),x(:,2)) - (1./2).*m_s.*(max(vel_s)).^2 )
./(m_p + 4.*I_p.*pi.^2.*R.^2)));
figure
plot(x(:,2),vel_p)
xlabel(’x_p’)
ylabel(’v_p’)
title(’Velocity of Pellet in Barrel as a Function of Pellet Position in Barrel’)
shg
pause
14
%Next make a plot for (x_p,P(x_p))
Pres = ( (P_o.*V_o)./(V_o - pi.*r_piston.^2.*x(:,1) + (pi./4).*c.^2.*x(:,2)) ) - P_o;
figure
plot(x(:,2),Pres)
xlabel(’x_p’)
ylabel(’P(x_p)’)
title(’Internal Pressure Within Spring-Air Pellet Gun as a Function of Pellet Position in Barrel’)
shg
%Find and display the muzzle velocity ( m/s )
[p8,p8s,p8mu] = polyfit(x(:,2),vel_p,8);
muzzle_velocity = polyval(p8,barrel_length,p8s,p8mu)
%Find and display the muzzle energy ( J )
muzzle_energy = (m_p + 4.*I_p.*pi.^2.*R.^2).*(muzzle_velocity).^2
%Find and display the Maximum Internal Pressure Within Spring-Air Pellet Gun ( Pa )
max_pressure = max(abs(Pres))
%sub-function Library for this file
%Preliminary Functions
function f1 = SpringEnergy(x_s)
global m_p m_s r_piston c R f_e mu I_p g P_o V_o k x_s_f x_s_o R_angle;
f1 = k.*(x_s_f.*x_s - (1./2).*x_s.^2);
function f2 = SimpleGravitationalFriction(x_p)
global m_p m_s r_piston c R f_e mu I_p g P_o V_o k x_s_f x_s_o R_angle;
f2 = m_p.*g.*mu.*x_p;
function f4 = LognitudinalRiflingFriction(x_s,x_p)
global m_p m_s r_piston c R f_e mu I_p g P_o V_o k x_s_f x_s_o R_angle;
f4 = (( (4.*I_p.*R.*pi.*mu.*P_o.*V_o)./(m_p.*c.*(cos(R_angle) + mu.*sin(R_angle))) )
.*log(V_o - pi.*r_piston.^2.*x_s + (1./4).*pi.*c.^2.*x_p) - ( (I_p.*R.*pi.^2.*mu.*c.*P_o.*x_p)
./(m_p.*(cos(R_angle) + mu*sin(R_angle))) ));
function f5 = ElasticCompressionFriction(x_p)
global m_p m_s r_piston c R f_e mu I_p g P_o V_o k x_s_f x_s_o R_angle;
f5 = f_e.*mu.*x_p;
function f6 = BoreFriction(x_s,x_p)
global m_p m_s r_piston c R f_e mu I_p g P_o V_o k x_s_f x_s_o R_angle;
f6 = SimpleGravitationalFriction(x_p) + ElasticCompressionFriction(x_p) + LognitudinalRiflingFriction(x_s,x_p);
function f7 = EnergyStoredInCompressedGas(x_s,x_p)
global m_p m_s r_piston c R f_e mu I_p g P_o V_o k x_s_f x_s_o R_angle;
f7 = P_o.*(( (V_o.^2)./(V_o - pi.*r_piston.^2.*x_s + (pi./4).*c.^2.*x_p) ) - V_o - pi.*r_piston.^2.*x_s
+ (pi./4).*c.^2.*x_p);
%Primary ODE system of two linear equations function
function velocity = problem_func(t,x )
global m_p m_s r_piston c R f_e mu I_p g P_o V_o k x_s_f x_s_o R_angle;
%Initialize for a system of ODEs
velocity = zeros(2,1);
%system of two first order ODEs
velocity(1)=(x(1)<(x_s_f-x_s_o)).*(k.*(x_s_f - (x(1) + ((x_s_f-x_s_o)-x(1)).*(x(1) >= (x_s_f-x_s_o))))
- ( (pi.*r_piston^2.*P_o.*V_o)./(V_o - pi.*r_piston.^2.*(x(1) + ((x_s_f-x_s_o)-x(1)).*(x(1)
>= (x_s_f-x_s_o))) + (pi./4).*c.^2.*x(2)) ) + pi.*r_piston.^2.*P_o).*((t)./(m_s));
velocity(2)=sqrt( (SpringEnergy((x(1) + ((x_s_f-x_s_o)-x(1)).*(x(1) >= (x_s_f-x_s_o)))) - BoreFriction((x(1)
+ ((x_s_f-x_s_o)-x(1)).*(x(1) >= (x_s_f-x_s_o))),x(2)) - EnergyStoredInCompressedGas((x(1)
+ ((x_s_f-x_s_o)-x(1)).*(x(1) >= (x_s_f-x_s_o))),x(2)) - (1./2).*m_s.*(max(velocity(1))).^2 )
./(m_p + 4.*I_p.*pi.^2.*R.^2) );
---------------------------------------------------------------------------------------------
15
When run the program will display a graph similar to this one:
This graphically shows the position of the pellet within the barrel of the springair pellet gun as a function of the time elapsed during the firing sequence measured in seconds. Please, note that once the pellet reaches the end of the barrel
this graph is no longer valid since the pellet is no longer in a state of internal
ballistics but rather a state of external ballistics.
16
Upon pressing the space bar a second graph will be displayed similar to this
one:
This graph shows the velocity of the pellet within the barrel as a function of the
pellets position. Once again, the graph is only valid so long as the xp value has
not exceeded the barrel length. Worthy of note is the fact that there is a specific
ideal barrel length where muzzle velocity will be maximized. Any barrel shorter
or longer then this value will result in less then optimum muzzle velocity.
17
Pressing the space bar once more will display the final graph:
This clearly displays the internal gas pressure developed within the spring-air
pellet gun as a function of the pellets position as it travels down the barrel –
a point where the maximum pressure ”spikes” is clearly visible and will occur
if the barrel is long enough. Finally the program will display the specific muzzle velocity, muzzle energy, and maximum internal pressure in the MATLAB
command window and the program will terminate.
5
Compare Models Predictions to Real Life Data
It is necessary to confirm the accuracy of the mathematical model which
was built. To this end two of my personal spring-air pellet guns were pressed
into service. These are the two spring-air pellet guns in question:
18
Both are sold by the Cummins Tool Corporation and are among the best deals
available on the market for the last ten plus years. They are both quality,
medium power, spring-air pellet guns capable of generating muzzle velocities in
excess of a hundred meters per second. The rifle is about ten years old and has
been fired literally thousands of times – it is not worn out but rather worn in.
Most quality spring-air pellet guns get more accurate with use – this allows the
individual components of the mechanism to wear to fit each other and provide
very stable ballistics. The pistol is almost brand new and has been fired only a
few hundred times. It is not yet broken in and still displays ”bouncy” ballistics
with a fairly wide velocity spread of ± 20 meters between individual shots.
For comparison the rifle usually maintains a spread of less then ± 1 meter per
second with the same box of pellets. Obtaining that small of a velocity spread in
a cartridge firearm is practically impossible even with special hand-made match
grade ammunition.
First, a variety of different pellets of appropriate caliber were obtained
and the specific physical characteristics of each style of pellet was quantified
and tabulated. The force of elastic compression ( FE.C. ) was determined by
driving each pellet about twenty centimeters deep inside the bore then inserting
a cleaning rod with a cup affixed atop it. This cup was then progressively loaded
with lead shot until the pellet moved. The total weight of the pellet, cleaning
rod, cup, and lead shot were then divided by the appropriate coefficient of static
friction. Mass of the pellet was determined with a very precise reloading scale
and the mass moment of inertia was approximated via direct calculation according to the formula, Ip = kmp (c/2)2 . The k value was intelligently estimated
according to the individual mass distribution of each pellet:
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The next step involved determining the average specific muzzle velocities of
each pellet/gun combination through the use of a tool know as a ”Chronograph”
which measures the velocity of a speeding bullet coming out the barrel of a gun
with single digit accuracy. For the rifle a series of ten velocity measurements
were taken about a meter from the muzzle for each pellet and the average
velocity was calculated and rounded off to the ones place. For the pistol ten
velocity measurements a meter from the barrel were also taken, but the average
velocity was rounded off to the nearest five meters per second due to the fact
that this gun developed such a wide velocity spread from shot to shot. As
previously outlined this is due to the fact that it has not yet been broken in.
Next, the pellet guns were completely disassembled and the necessary physical
information about their internal design was quantified and tabulated:
Finally, the necessary information was input into the mathematical model and
the percent deviation of the model from reality was quantified:
6
Conclusion / Discussion
Ultimately this mathematical model proved to be even more accurate then
I thought it would be. The final mathematical model was found to accurately
predict real world results within a margin of error of ± 1.8 % and ± 20 % in
the two specific initial test cases involving the spring-air pellet rifle and the
spring-air pellet pistol respectively. I suspected that since the model does not
take into account compressed air leakage from the internal workings during
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the firing sequence that the model would consistently yield results that were
in excess of real world performance, however, this does not appear to be the
case. The error values appear to be balanced both slightly above and below
the actual real world values. Due to how quickly the firing sequence takes place
(less then a hundredth of a second) any such leakage from a spring-air pellet gun
with reasonably good seals must be a nominal factor. Although the equations
are complex they are rooted in the most basic principles of physics. With the
possible exception of the bore friction formula it is all senior year, high-school
level physics. Calculating a solution, however, takes some pretty strong math.
Not only is a working knowledge of Ordinary Differential Equations required,
but math logic analysis is necessary to properly limit both the position and
velocity of the spring piston and, thus, properly solve the ODE system. With a
little bit of tweaking the included MATLAB M-file could be built into a graphical
interface computer program that would serve very well in an industrial design
environment for this specific sector of the firearms industry.
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7
Bibliography
Beichner, Robert J., Jewett, John W. Jr., and Serway, Raymond A.
Physics For Scientists and Engineers with Modern Physics. 5th ed.
Orlando, FL: Saunders College Publishing, 2000.
Pytel, Andrew and Kiusalaas, Jaan. Engineering Mechanics Dynamics. 2nd ed.
Pacific Grove, CA: Brooks/Cole Publishing Company 1999.
Oerlikon Pocket Book. Oerlikon-Zrich, Switzerland: Oerlikon Machine
Tool Works, 1958.
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