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1 Introduction

This program calculates the behavior of an automag paintball gun using a simple model of the gun. Please see the automag manual for an explanation of the terminology. If the program is run as is, it will emulate a 0.2 second wait while the gun charges up to about 450 psi, then the trigger is pulled and a ball is ejected at about 290 fps. In order to set up other scenarios, the timing loop in main() and the trigger subroutine UpDate_t() must be modified by the user.

2 Explanation of the Program

At any point in time, the gun is considered to be completely described by specifying a set of pressures, volumes, surface areas, positions and velocities of certain key elements of the gun. In the computer program, the subroutine tstep() takes the set of variables describing the state of the gun, along with the time (t) and a time step (dt), and calculates the new values for these variables at time t+dt. This "marching algorithm" is accurate as long as the time step dt is small enough but if dt is too small, it can take a long time for the program to calculate out to a reasonable amount of time. A good value for dt seems to be about 1 millisecond. A check on whether a small enough time step is being used is to cut the time step by a factor of 3 and see if the results are the same. If the time step is increased by a factor of 3 and the results are about the same, a larger time step can be used.

There are two types of subroutines used by tstep(); flow rate subroutines starting with "Q" and moving part update subroutines starting with "U".

2.1 Flow rate subroutines

There are 4 volumes or chambers in the gun: "s" signifies the source (e.g. a CO2 bottle), "r" is the regulator chamber, "a" is the air chamber, and "b" is the barrel. Thus, for example, PV.Pr is the regulator chamber pressure and PV.Vr is the regulator chamber volume, and Qra() is the subroutine that calculates the rate of flow from the regulator chamber to the air chamber. The regulator valve connects the source and the regulator chamber, the on-off valve connects the regulator chamber and air chamber and the bolt-to-power tube o-ring is the "valve" that connects the air chamber to the barrel. The flow rate subroutines calculate the rate of flow of gas from one chamber to the other according the the crude equation:

Q12 = d(P2*V2)/dt = -d(P1*V1)/dt = (Cl + Z*x)*(P1-P2) (1)where P1 and V1 are the pressure and volume of the upstream chamber, P2 and V2 are the pressure and volume of the downstream chamber, Cl is a leakage flow rate (in order that bad o-rings may be modelled), Z*x is the normal flow rate where x is a measure of how open the valve is and Z is a proportionality constant. (If x becomes negative it is assumed the flow is stopped). Any pressure at time t+dt is calculated (assuming constant volume V) by:

P(t+dt) = P(t) + (Qin-Qout)*dt/V (2)where Qin is the rate into the chamber and Qout is the rate out of the chamber.

2.2 Moving Part Update Subroutines

The moving parts of the gun are moved by pressure differentials applied to various surfaces, by springs, by friction, or by a finger moving the trigger. Generally, the force (F) on a surface of area S due to a pressure difference is given by:

F = S*(P1-P2) (3)where P1 and P2 are the pressures on either side of the surface. The force on an element due to a spring is;

F = K*(x-x0) (4)where x is the length of the spring, x0 is its relaxed length, and K is the spring constant. The force due to friction is taken to be:

F = -R*V (4.5)where V is the velocity of the part and R is a proportionality constant.

If a massive part is being moved, the position (x) and velocity (v) of the part must be known in order to predict its motion. The acceleration of the part is A=F/M where F is the force on the part and M is the mass of the part. The position and velocity of the part at time t+dt is calculated as:

x(t+dt) = x(t) + (v(t) + 0.5*A*dt)*dt (5a) v(t+dt) = v(t) + A*dt (5b)2.25 The Source Pressure

The pressure of the source (PV.Ps) is assumed to derive from the equilibrium vapor pressure over liquid carbon dioxide at temperature PV.T. The vapor pressure data was found in the CRC Handbook of Chemistry and Physics, 74th edition, copyright 1993. The data is given between temperatures of 140 and 300 degrees kelvin. The logarithm of the pressure as a function of temperature was fitted with a third order polynomial and the source pressure is calculated using the four co-efficients of this polynomial.

2.3 The Regulator

It is assumed that a gas source at pressure PV.Ps is applied to the input. The regulator is diagrammed schematically below:

|---------------- | -> to on-off valve | |-| ---------------|- |D|---------- | | E | | |-| | | A |88888| B |---------|C| | | | | | |-| | ---------------|- |D| | |---|-|--| |-| | | | v to sourceA Regulator nut

B Regulator piston

C Regulator valve

D Regulator seal

E Regulator spring pack ("888" represents the spring)

It is assumed that the left end of the regulator valve is always in contact with the regulator piston. (actually the regulator valve spring contacts the right end of the regulator valve and keeps it pushed against the regulator piston) As gas enters from the source, it flows past the regulator valve into the part of the regulator chamber holding the regulator piston. The pressure forces the piston to the left, and the regulator valve moves with it. This process continues until the regulator valve head comes in contact with the regulator seal and stops the flow. Equation (1) is used to calculate the rate of flow of gas past the regulator valve, with:

Qsr = (Zr.Cl + Zr.Z*Zr.x)*(PV.Ps-PV.Pr) (6)

Zr.x distance from the bottom of the regulator valve head to the top of the regulator seal. PV.Ps source pressure PV.Pr regulator chamber pressure Zr.Z flow rate/length for the regulator valve Zr.Cl leakage rate for the regulator valveThe position of the regulator valve is determined by the position of the regulator piston. The regulator piston is pushed to the left by the regulator spring and pushed to the right by the higher pressure in the regulator chamber. The regulator piston can only move so far to the right before it runs into a stop. The force of the regulator spring can be increased by moving the regulator nut to the right, compressing the spring. Lets say Zr.x0 is the value of Zr.x when the regulator piston is against the stop and Zr.T/Zr.H is the distance the regulator nut compresses the spring beyond its relaxed length when the regulator piston is against the stop. Zr.T is the number of turns of the regulator nut (clockwise) and Zr.H is the number of turns per inch of the regulator nut. The position of the regulator piston is found by assuming that the spring force and the pressure force are always equal:

Zr.K*(L0 - L) = Zr.S*(PV.Pr-PV.Px) (7)where L0 is the relaxed length of the regulator spring and L is its compressed length. When the spring is compressed, it is compressed from the left Zr.T/Zr.H inches by the regulator nut and Zr.x0-Zr.x by the pressure force. That means

L0 = L + Zr.T/Zr.H + Zr.x0 - Zr.x (8)The above two equations can be solved for Zr.x

Zr.x = Zr.x0 + Zr.T/Zr.H - (PV.Pr-PV.Px)*Zr.S/Zr.K; (8.5)which is the equation used inUpDate_r();

2.4 The On-Off valve

A schematic diagram of the on-off valve is given below:

------------| to regulator <- | |----o o----| | | -> to air chamber |-----|------ | A--------B---------- SearEquation (1) is used to calculate the flow through the on-off valve with:

Qra = (Zo.Cl + Zo.Z*Zo.x)*(PV.Pr-PV.Pa)

Zo.x distance that the top of the regulator pin is below the on-off o-ring PV.Pr regulator pressure PV.Pa air chamber pressure Zr.Z flow rate/length for the on-off valve Zr.Cl leakage rate for the on-off valveThe position of the on-off pin is controlled by the sear which is in turn controlled by the trigger. It is assumed that the bottom of the on-off pin is connected to the sear but actually the regulator pressure pushes it against the sear. If the trigger is released, the sear is tilted at zero degrees. As the trigger is pulled, The on-off valve position (Zo.x) decreases according to:

Zo.x = -Zt.xs*sin(Zo.B-Zo.Bo)where Zt.xs is the length of the sear arm connecting its pivot point ("B" in the diagram) to the point of contact with the on-off pin ("A" in the diagram). Zo.Bo is the angle at which the on-off pin contacts the teflon on-off o-ring. This equation is implemented in UpDate_o() except it is assumed that the angle Zo.B-Zo.Bo is small enough so that sin(Zo.B-Zo.Bo) can be replaced with Zo.B-Zo.Bo itself.

2.5 The Power Tube

A schematic diagram of the power tube is given below:

----------|-----------|-------------- O |888888888| |---|-----------| Air Chamber | bolt | Barrel |-------|-------| O | ----------|-----|--------------------- | | | | ---------o----------| SearEquation (1) is used to calculate the flow through the power tube valve:

Qab = (Zp.Cl + Zp.Z*Zp.x)*(PV.Pa-PV.Pb) (9)

Zp.x distance that the bolt is to the right of the power tube o-ring ("O" on the diagram) PV.Pa air chamber pressure PV.Pb barrel pressure Zr.Z flow rate/length past the power tube o-ring Zr.Cl leakage rate past the power tube o-ringThe position of the bolt is controlled by the pressure difference between the air chamber and the barrel, the power tube spring ("888" in the diagram), and the sear. If the bolt moves Zp.xc inches to the left of the power tube o-ring, it is latched by the sear, as long as the sear is at an angle of Zt.Bl or greater. If the bolt at the latch point, it is stopped from moving left. If the bolt moves to the right by Zp.xm inches, it strikes a stop in the main body and is stopped from moving right. anywhere between these two points, equations (3) and (4) give the force on the bolt except that if the bolt is in contact with the power tube o-ring, it is assumed that there is an additional frictional force on the bold according to equation (4.5) and equations 5a and 5b then determine the motion. This is implemented in subroutine UpDate_p().

2.6 The Ball and Barrel

The ball is pushed down the barrel by the difference in pressure between the barrel and the atmosphere. A complication here is that the barrel volume (the volume of the barrel between the bolt and the ball) is changing as the ball moves down the barrel. The barrel volume (Vb) is given by:

Vb = 0.25*pi*Zb.D*Zb.D*(Zb.x+0.1) (10)where Vb(t) is calculated before Zb.x is updated and Vb(t+dt) is calculated after Zb.x is updated. The rate equation is then:

P(t+dt)Vb(t+dt) = PV.Pb*Vb(t) + (Qab-Qbx)*dt (11)which can be solved for PV.Pb(t+dt). Equations 3, 5a, and 5b are then used to update the position of the ball. If the ball moves beyond the end of the barrel, the barrel pressure is atmospheric.

I tried worrying about whether the ball was in contact with the foamie and being pushed by the bolt or whether it was being pushed by barrel pressure, but it seemed like it was always being pushed by barrel pressure, so I didn't bother with including this in the model, even though it needs to be included. Also, it shouldn't be too hard to model a vented barrel to determine how much more gas efficient an undrilled barrel is.

2.7 Estimation of Constants

Most of the geometric values were obtained by direct measurement of the gun itself, or were obtained from the automag manual.

2.7.1 The Power tube spring constant

The spring constant of the power tube spring was estimated by placing a 5-pound weight on the spring and noting the deflection.

2.7.2 The Regulator Spring Pack Spring Constant

The manual says that the gun is nominally charged to 450 psi. Not knowing how much the regulator nut compresses the spring pack under these conditions, I assumed it didn't compress it at all. That means the spring constant for the spring pack is found by solving equation (8.5) with Zr.x = 0 and Zr.T=0:

Zr.K = Zr.S*(450-PV.x)/Zr.x0 = 1367.5 (12)This value for Zr.K can probably be improved if I can find out how much the regulator spring pack is compressed when the gun is charged to 450 psi.

2.7.3 Air chamber volume

The manual says that a 7 oz tank of CO2 will provide 400 good shots for the gun. There are probably about 6 usable ounces in the tank so that 70 shots per ounce of CO2 is a reasonable estimate of gas consumption per shot. Assuming that the air chamber fill pressure is is 450 psi, and the temperature is 70 degrees F, we can solve the ideal gas equation for PV.Va:

PV.Pa*PV.Va = n*R*PV.Twhere n = 1/70 oz = 9.2904e-2 mole and R=8.314 J/mole/deg K. This gives an air chamber volume of 0.463 in^3

2.7.4 The Valve Flow Rates/Length

The manual says the gun charges up to 450 psi in about a third of a second. The flow rate constant Zr.Z was found by hit-or-miss until the air chamber filled up to 95 percent of its steady state value (i.e. 430 psi) in about 0.33 seconds. The same value of flow rate/length was used for both the regulator valve and the on-off valve.

2.7.5 Power Tube Flow Rates/Length

When the air chamber is filled up to 450 psi, it is assumed that the ball will be ejected from an 11 inch barrel at 300 fps. The Power Tube Flow Rate/Length was adjusted until this was the case.

2.7.6 Power Tube Friction Coefficient

If the friction coefficient is set to zero, in the "lawn sprinkler" scenario the bolt motion is dominated by the resonant frequency of the bolt and its spring. If the coefficient is much greater than 3.0 all evidence of this natural frequency is lost. I chose a value of 1.0 only because its interesting. The bolt motion has about equal components of 5 and 10 cycles per second. Some thinking has to be done to get the value of this constant correct.

3. Scenarios

A print statement is in the main() program which outputs a number of variables. A few different scenarios are given by the following modifications to the program along with a description of the expected printed output

3.1 Simple Shot - run program as is. The regulator chamber should charge to 449.54 psi and the air chamber should charge to 449.02 psi at 0.005 seconds, then the trigger should go on and a ball should be ejected at 296.69 fps.

3.2 fast stroke: The regulator and air chambers are pre-charged to about 450 psi and the trigger subroutine turns the trigger on to begin with. The first ball comes out at 296.67 fps. The trigger subroutine waits until the air chamber is emptied, then turns the trigger off (0.0176 sec). 0.2 seconds later at 0.2176 sec, the second ball comes out at 263.92 fps, becuase the air chamber could not fully charge in 0.2 seconds.

Zt.B = Zt.Bb; /* Trigger pulled */ PV.Pr = 449.; /* regulator chamber charged */ PV.Pa = 449.; /* regulator chamber charged */ dt = .0002; n = 1150;

----- Insert into UpDate_t() ----- static double delt; double cycle; cycle = 0.2; /* time trigger is off (sec) */ if(Zt.B >= Zt.Bb) { if((Zb.x >= Zb.L) && (PV.Pa <= 1.005*PV.Px)) { Zt.B = 0.0; /* trigger off */ Zp.x = -0.1; /* bolt latched */ Zp.v = 0.0; Zb.x = 0.0; /* ball loaded */ Zb.v = 0.0; /* ball motionless */ delt = t; /* reset timer */ } } else { if(t-delt >= cycle)Zt.B = Zt.Bb; /* Trigger on */ } ret: return(0); ----------------------------------3.3 Lawn sprinkler. A leakage in the on-off valve is put in, and no ball is loaded (This is done by putting the position of the ball Zb.x outside the barrel, i.e. Zb.x >= Zb.L) The trigger is always on, and the regulator and air chamber pressures are set roughly to a steady state value. (They needn't be).

Zb.x = Zb.L; /* no ball */ Zt.B = Zt.Bb; /* trigger down */ Zp.X = 0; /* bolt at zero */ Zo.C = 0.01 /* on-off leak */ PV.Pr = 450; /* steady state */ PV.Pa = 34.7; /* steady state */ n = 3000; dt = .0002; ----- Insert into UpDate_t() ----- { Zt.B=Zt.Bb return(0); } ----------------------------------3.4 Cold Gun - Low temperatures lower the vapor pressure of the CO2 which results in slow charging of the gun and possibly incomplete charging. Set the temperature PV.T to 30 degrees F. and run the short stroke scenario. The second ball will come out at 239.00 fps, which is even less than before, this being due to the lower source gas pressure.

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