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 mod-eled in this work. This mathematical model is designed to accuratelydetermine the ballistic performance and maximum internal gas pressureof a specific design at the drawing board stage. The model is based uponbasic principles of physics including, energy analysis principles, Hook’slaw for springs, the ideal gas law, linear and rotational kinetic energy,and Newton’s laws of motion, as well as ballistic quantification of borefriction. Model solution was obtained using MATLAB to numericallysolve a system of linear ordinary differential equations.

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1 Introduction

A spring-air pellet gun is a firearm which fires a bullet or pellet purely fromthe force of compressed air from a spring piston chamber. These firearms areideal for folks who love the shooting sports, but who dont live in the countryside.Spring-air pellet guns are quieter then conventional cartridge firearms. Beingof low power these guns can allow a safe, indoor target range in the privacy ofones own home.

With a pellet gun’s greater safety and much quieter operation, privatelands hunting permission is often easier to obtain. Pellet rifle hunting is a scaleddown version of varmint and small game hunting, normally done with a lightcartridge firearm. Pellet rifles are used instead of cartridge firearms for safetyand perhaps surprisingly, for effectiveness. Because they deliver the same energyon each shot, the trajectory of a spring-air pellet gun is extremely consistent –they are often more accurate then conventional firearms. In addition the factthat they are relatively quite is a great advantage to the hunter. The specificgame which is hunted depends on the power and caliber of the pellet gun inquestion – pellet guns have been successfully used to hunt game as small asmice and rats where conditions exist that prevent effective elimination by othermeans such as traps or poison bait. In addition there are some very large andpowerful spring-air pellet guns which can effectively be used to hunt game aslarge as the various species of North American deer. One such spring-air pelletgun, which is marketed as a survival weapon, is capable of launching a .45-caliber 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 usedto hunt small game up to, and including, fox and is commonly used to huntgophers, rabbits, and squirrels.

Spring-piston guns operate by means of a coiled spring and piston con-tained within a compression chamber. Cocking the gun causes the piston springto be compressed; pulling the trigger releases the piston allowing it to move,compressing the air in the chamber directly behind the pellet. Once the airpressure has risen enough the pellet moves forward, propelled by an expandingcolumn of air. All this takes place in a fraction of a second. Most spring pistonguns are single-shot by nature. Such spring-air pellet guns are able to achievemuzzle velocities near, or in some cases, in excess of the speed of sound from asingle stroke of a cocking lever. The difficulty of the cocking stroke is usuallyrelated to the power of the gun, with higher muzzle velocities requiring greatercocking 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 engi-neering and economic problems. Although quality is a factor, the market sellingprice of the various models of spring-air pellet guns shows a direct relationshipbetween the price and the ballistic performance of the firearm. Spring-air pelletguns with very low power levels sell for as little as twenty dollars. High powersupersonic weapons sell for over five hundred dollars, even in used condition,and often carry four digit price tags when new. It is therefore, very advantagesto be able to quantify the real world muzzle velocity and ballistic energy of aspring-air pellet gun design at the drawing board stage before resources are ex-pended to build a prototype for testing. The primary selling point, and primaryconsideration, of a spring-air pellet gun is the muzzle velocity it is capable ofgenerating with a common weight pellet. However, there is another considera-tion, especially when considering calibers other then .177, this consideration isthe muzzle energy of the pellet. The muzzle energy of the pellet is a function ofits velocity, mass, and gyration spin imparted by the barrels rifling.

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One specific design concern is the maximum internal gas pressure gener-ated within the spring chamber and barrel of the firearm. This maximum gaspressure value has a direct relationship to the necessary strength and quantityof the materials used in the construction of the firearm as well as the specificdesign itself. This will play a significant part in determine the manufacturingcost of a spring-air pellet gun.

The problem of determining these values without an actual working pro-totype in the lab via mathematical analysis of the internal forces at work duringthe firing sequence can be quite daunting. There are two separate moving ob-jects which interact during the firing sequence; the plunger attached to thespring which serves as the mechanism’s power plant and the pellet within thebore. Each of these two objects have their own position, velocity, and accelera-tion vectors and due to the compressed gas which serves as a medium betweenthem the individual values of each of these vectors has an effect on the valueof the other vectors. It appears to be an endless circle where each value is de-pendent upon the other values and there appears to be no solution via normalarithmetic, algebra, or calculus. In order to solve this problem we must goone step higher in the math hierocracy. The only way to solve this particularproblem is through the mathematical science known as ”Ordinary DifferentialEquations” or ODE for short. This system is in affect a system of two seconddegree ODE equations. The compressed gas, which links these two equations,can be used to define the two acceleration terms as functions of the positionterms, thus, simplifying the ODE system from a second order system to a firstorder system. A first order system of two equations is solvable using this math-ematical science.

2 Basic Principles and Equations of Physics

A ground work starting from the most basic principles of physics must belaid in order to build a realistic model of the internal forces at work. First, of allenergy cannot be created or destroyed but merely converted from one form toanother. The total energy input must exactly equal the total energy output, inany system or machine. In the case of a spring-air pellet gun the input energyis the mechanical force imparted by the user in cocking the coil spring withinthe internal workings of the firearm. The energy stored in the spring can bequantified through the integration of Hook’s law which states that the forceexerted by a spring is directly proportional and opposite to the displacement:

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Energy Stored In Spring =∫

(Spring Force)

=∫ xsf

xso

(kx)dx

=12k(xsf

− xso)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 firingsequence begins.

During the firing sequence, the energy released by the spring is transferredto the pellet via the medium of compressed gas in the internal spring plungerchamber and barrel behind the pellet. The pressure of this gas (air) as a functionof its changing volume and temperature, is quantified by the Ideal Gas Law:

PoVo

To=

P (t)V (t)T (t)

The first assumption will now be made in this model. The assumptionis that the temperature of the compressed gas within the internal mechanismremains constant. This assumption is very reasonable when one considers theminuscule mass of gas trapped inside the comparatively massive heat sink of thepellet guns metal components. This massive heat sink should serve to maintaina constant gas temperature during the firing sequence. With this assumptionthe Ideal Gas Law can be simplified for this mathematical model so that thepressure is only a function of the changing volume:

PoVo = P (t)V (t)

P (t) =PoVo

V (t)

Next, since we are only concerned with the pressure above atmospheric pressurewe will subtract the initial (atmospheric) pressure from the equation.

P (t) =PoVo

V (t)− Po

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The compressed gas not only transfers energy but also stores it. Theamount of energy stored in the gas due to its compression at any given time canbe quantified through the integration of the previously determined equation forpressure:

Energy Stored In Compressed Gas =∫ (

PoVo

V (t)− Po

)dx

=∫ Vo

V (t)

(PoVo

V (t)− Po

)dx

= Po

(V 2

o

V (t)− 2Vo + V (t)

)

The pellet during the travel down the bore experiences ballistic frictionforces which result in kinetic energy being converted into heat energy. Thisenergy is not destroyed but wasted in the sense that it is not used to acceleratethe pellet towards its final muzzle velocity. This bore friction can be categorizedinto four separate friction forces which work together to account for the totalenergy wasted as bore friction. The first is simple gravitational friction:

Simple Gravitational Friction = mpgµxp(t)

Where:

mp = Mass of Pelletg = Force of Local Gravitational Fieldµ = Kinetic Coefficient of Friction

xp(t) = Position of Pellet in Barrel

Second, is the rotational rifling friction:

Rotational Rifling Friction =∫ (

Rπcµ(πc2

4 P (t)−mpap(t))sin θ + µ cos θ

)dxp(t)

Where:

R = Twist Rate of Riflingc = Caliber

ap(t) = Acceleration of Pellet in Barrelθ = Angle of Rifling Twist

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Third, is the linear rifling friction:

Linear Rifling Friction =∫ (

4IpRπµap(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 forcedinto the bore. This is an extremely complex statically indeterminant problemfar beyond the scope of this paper. Any sensible engineering analysis in a timeequals money business environment is best done through direct empirical anal-ysis. One must simply quantify the necessary minimum force that is requiredto push a specific pellet through a bore of appropriate caliber at a constant butvery low velocity. This value will remain the same provided the pellet and borematerial remain the same regardless of what other factors change. This can befurther generalized by dividing this force by the coefficient of friction betweenthe pellets material and the barrel material. Thus, this force becomes a uniqueproperty of the specific pellet; allowing quick and easy analysis once this valuehas been determined and tabulated for a variety of common pellet designs. Thegathering of such data can be easily done in any half-way decently equipped labin a workday or less and would continue to be valid for the analysis of futuredesigns for as long as those pellet designs remained popular. Considering thatthe lowly BB pellet design has been around for over a hundred years and thebasic diabolo pellet design has been around for over 40 years – no problemsthere.

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 theprevious mentioned method such that:

FE.C. =Fexp

µexp

Where:

Fexp = Experimentally Determined Minimum Applied Forceµexp = Coefficient of Friction During Experimental Analysis

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The ultimate outcome of the process happening within the internal work-ings of a spring-air pellet gun is to impart a fairly large amount of kinetic energyto the pellet in the form of both linear velocity and rotational velocity. The lin-ear kinetic energy of a projectile’s linear velocity. In this case, the pellet’s linearvelocity is directly proportional to both the projectiles mass and the square ofit’s velocity:

Linear Kinetic Energy of Pellet =12mp(vp(t))2

Where:

vp(t) = Velocity of Pellet in Barrel

The rotational kinetic energy of the pellet is directly proportional to both thepellets 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 kineticenergy and the rotational kinetic energy:

Pellet Energy =12mp(vp(t))2 + 2Ip(Rπvp(t))2

=12mp(vp(t))2 + 2IpR

2π2(vp(t))2

= (12mp + 2IpR

2π2)(vp(t))2

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 equaland 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 therecoil energy, represent the total output energy from the system at any givenmoment during the internal energy conversion process also known as the firingsequence.

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3 Setup Energy Equation in Preparation to Solve

Now that all the basic equations and principles have been gathered to-gether it is time to setup the energy equations in preparation to solve the mod-eling problem. In order to produce a solvable system of equations for the modelwe must end up with two equations such that the velocity of the spring and thevelocity of the pellet are defined as functions of the position of the spring, theposition 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 inputenergy. Followed by input energy is spring energy minus energy sinks, energysinks are energy lost to bore friction, energy stored in the form of compressedgas, and kinetic energy of piston. Output energy is pellet kinetic energy andrecoil kinetic energy, and finally, output energy can be simplified to be twicethe pellet kinetic energy:

Each one of these five terms must be mathematically defined from thepreviously outlined Physics equations and principles and substituted back intothe equation. Due to the massive size and complexity of the resulting equationthis substitution will be in most cases implied rather then actually shown. First-off, it is necessary to derive equations specifically defining the volume of thegas, pressure of the gas, acceleration of the spring piston, and acceleration ofthe pellet as functions of time elapsed during the firing sequence:

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V (t) = Vo − π(rpiston)2xs(t) +π

4c2xp(t)

Where:

V (t) = Volume of Air as a Function of TimeVo = Volume of Air in Cylinder When the Air Gun is Cocked

rpiston = Radius of Spring Pistonxs(t) = Position of Spring Piston as a Function of Timexp(t) = Position of Pellet in Barrel as a Function of Time

———————————————————–

P (t) =PoVo

V (t)− Po

=PoVo

Vo − π(rpiston)2xs(t) + π4 c2xp(t)

− Po

Where:

P (t) = Pressure as a Function of TimePo = Local Atmospheric Pressure

———————————————————–

as(t) =Σ(Force on Piston)

ms

=k(xsf

− xs(t))− π(rpiston)2P (t)ms

Where:

as(t) = Acceleration of Spring Piston as a Function of Time

ms = Mass of Spring Piston + 1/3 Spring Massk = Spring Constant

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ap(t) =P (t)πc2

4mp

=

(PoVo

Vo−π(rpiston)2xs(t)+ π4 c2xp(t) − Po

)πc2

4mp

=PoVoπc2

4mp

(Vo − π(rpiston)2xs(t) + π

4 c2xp(t)) − Poπc2

4mp

Now each of the five terms of the energy equation, Pellet Energy, Spring Energy,Bore Friction, Energy Stored In Compressed Gas, and Piston Kinetic Energywill be mathematically defined as functions of xs(t) and xp(t):

2(Pellet Energy) = 2(12mp + 2IpR

2π2)(vp(t))2

= (mp + 4Ipπ2R2)(vp(t))2

———————————————————–

(Spring Energy) = k

(xsf

xs(t)−12(xs(t))2

)———————————————————–

Due to its extensive length we will define the Bore Friction term to be the sum-mation of its four components, Simple Gravitational Friction,Rotational RiflingFriction,Longitudinal Rifling Friction, and Elastic Compression Friction. Eachof these components will be mathematically defined:

(Simple Gravitational Friction) = mpgµxp(t)

———————————————————–

(Rotational Rifling Friction) =∫ (

Rπcµ(πc2

4 P (t)−mpap(t))sin θ + µ cos θ

)dxp(t)

=∫ (

Rπcµ(mpap(t)−mpap(t))sin θ + µ cos θ

)dxp(t)

=∫ (

Rπcµ(0)sin θ + µ cos θ

)dxp(t)

= 0

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———————————————————–

(Elastic Compression Friction) = µFE.C.xp(t)

———————————————————–

———————————————————–

(Piston Kinetic Energy) =12ms(vs(t))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 energyequation has been defined in terms of vp(t), xp(t), vs(t), and xs(t). When youacknowledge that vp(t) is the derivative of xp(t) you can see that we have thebeginning 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) usingeverything 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 orderdifferential equations:

4 Numerically Solve ODE System

Unfortunately this system is far to complicated to solve via hand calcula-tions (at least in my lifetime). It, therefore, will be necessary to solve this systemvia. numerical analysis methods performed by a computer. The following filewhen 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. Theoutput values will be the muzzle velocity of the pellet, muzzle energy of thepellet, and the maximum gas pressure developed within the internal workingsof spring-air pellet gun during the firing sequence. Necessary Input values aremp, ms, rpiston, c, R, FEC , µ, Ip, Vo, k, xsf

, xso , and the barrel length. Theproper values for g and Po are included within the file but can be changed ifdesired. This would only be done for unique situations such as if the spring-airpellet 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 Valuesglobal 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 materialsI_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)));

figureplot(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

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%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;figureplot(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 Functionsfunction 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 functionfunction 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 ODEsvelocity = zeros(2,1);%system of two first order ODEsvelocity(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) );

---------------------------------------------------------------------------------------------

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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 spring-air pellet gun as a function of the time elapsed during the firing sequence mea-sured in seconds. Please, note that once the pellet reaches the end of the barrelthis graph is no longer valid since the pellet is no longer in a state of internalballistics but rather a state of external ballistics.

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Upon pressing the space bar a second graph will be displayed similar to thisone:

This graph shows the velocity of the pellet within the barrel as a function of thepellets position. Once again, the graph is only valid so long as the xp value hasnot exceeded the barrel length. Worthy of note is the fact that there is a specificideal barrel length where muzzle velocity will be maximized. Any barrel shorteror 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-airpellet 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 occurif the barrel is long enough. Finally the program will display the specific muz-zle velocity, muzzle energy, and maximum internal pressure in the MATLABcommand 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 whichwas built. To this end two of my personal spring-air pellet guns were pressedinto service. These are the two spring-air pellet guns in question:

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Both are sold by the Cummins Tool Corporation and are among the best dealsavailable on the market for the last ten plus years. They are both quality,medium power, spring-air pellet guns capable of generating muzzle velocities inexcess of a hundred meters per second. The rifle is about ten years old and hasbeen 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 theindividual components of the mechanism to wear to fit each other and providevery stable ballistics. The pistol is almost brand new and has been fired only afew hundred times. It is not yet broken in and still displays ”bouncy” ballisticswith a fairly wide velocity spread of ± 20 meters between individual shots.For comparison the rifle usually maintains a spread of less then ± 1 meter persecond with the same box of pellets. Obtaining that small of a velocity spread ina cartridge firearm is practically impossible even with special hand-made matchgrade ammunition.

First, a variety of different pellets of appropriate caliber were obtainedand the specific physical characteristics of each style of pellet was quantifiedand tabulated. The force of elastic compression ( FE.C. ) was determined bydriving each pellet about twenty centimeters deep inside the bore then insertinga cleaning rod with a cup affixed atop it. This cup was then progressively loadedwith lead shot until the pellet moved. The total weight of the pellet, cleaningrod, cup, and lead shot were then divided by the appropriate coefficient of staticfriction. Mass of the pellet was determined with a very precise reloading scaleand the mass moment of inertia was approximated via direct calculation accord-ing to the formula, Ip = kmp(c/2)2. The k value was intelligently estimatedaccording to the individual mass distribution of each pellet:

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The next step involved determining the average specific muzzle velocities ofeach 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 gunwith single digit accuracy. For the rifle a series of ten velocity measurementswere taken about a meter from the muzzle for each pellet and the averagevelocity was calculated and rounded off to the ones place. For the pistol tenvelocity measurements a meter from the barrel were also taken, but the averagevelocity was rounded off to the nearest five meters per second due to the factthat this gun developed such a wide velocity spread from shot to shot. Aspreviously 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 physicalinformation about their internal design was quantified and tabulated:

Finally, the necessary information was input into the mathematical model andthe percent deviation of the model from reality was quantified:

6 Conclusion / Discussion

Ultimately this mathematical model proved to be even more accurate thenI thought it would be. The final mathematical model was found to accuratelypredict real world results within a margin of error of ± 1.8 % and ± 20 % inthe two specific initial test cases involving the spring-air pellet rifle and thespring-air pellet pistol respectively. I suspected that since the model does nottake 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 werein excess of real world performance, however, this does not appear to be thecase. The error values appear to be balanced both slightly above and belowthe 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 gunwith reasonably good seals must be a nominal factor. Although the equationsare complex they are rooted in the most basic principles of physics. With thepossible exception of the bore friction formula it is all senior year, high-schoollevel 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 andvelocity of the spring piston and, thus, properly solve the ODE system. With alittle bit of tweaking the included MATLAB M-file could be built into a graphicalinterface computer program that would serve very well in an industrial designenvironment 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 MachineTool Works, 1958.

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