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turbo1889
03-25-2007, 07:01 AM
Hello, I'm working on a project for my ODE (Math-222m) class involving the mathematical modeling of the internal ballistics of the common Spring/Air pellet rifle. I've got basically all the necessary physics formulas down for building the model (Spring Energy Equation / Ideal Gas Law / Linear Kinetic Energy / Rotational Kinetic Energy / etc.) except for one. I need a formula for the bore friction during the pellets trip down the barrel. I've done a lot of searching but so far have yet to dig up any hard math or even imperial data -- All I've found so far is general references. Example, "Our pellets; due to their special . . . . coating reduce bore friction significantly and can result in some cases of a velocity increase of up to 20%" That don't help me at all I need the hard stuff, not this watered down junk. So far Ive learned that there seems to be a direct relationship between the magnitude of the contact surface between the pellet and the barrel and the magnitude of the bore friction force opposing the pressure force induced by the compressed gas. This contact surface could be easily built into the model as a property of the individual pellets but I need a way to calculate the magnitude of the resulting friction force. If any of you out there have information on this subject please jump in -- I have yet to find an engineering book or gun book that has this illusive tid-bit of information I am looking for. All help is appreciated.

kg42
03-26-2007, 10:36 PM
Look for coefficients of friction for metals, or something similar (steel/steel, steel/brass, etc). I saw a table last week while searching for something else.
It was said to be very theoretical.

kg

turbo1889
03-26-2007, 11:38 PM
Well I was able to get a response on an Engineering forum from a full fledged Ballistic Engineer on this subject and thought I should post the results here for those that are interested:

The input values for the following formulas:
P = force due to gas pressure on base of bullet
alpha = angle of twist
F_x = bore friction acting against the bullets travel down the bore
F_y = bore friction acting against the bullets twist in the bore
D = The gun's caliber
J = bullets moment of inertia with respect to its central axis parallel to the bore
u = the coefficient of friction between the bullet material and the barrel (Examples: Lead/Blued-Steel, Copper/White-Steel, Jacket-Brass/Stainless-Steel, etc.)
v’ = instantaneous derivative of the velocity of the bullet down the bore
w’ = instantaneous derivative of the rotational velocity of the bullet in the bore
m = mass of the bullet

The formula for finding alpha if rate of twist is known (Example: (1 twist) / (in 12") ):

alpha = arctan( (pi)*D*( rate of twist ) )

The formula for finding a bullets moment of inertia where k is a constant unique to the bullets shape (or) alternately finding the bullets moment of inertia when i the bullets radius of gyration is known:

J = k*m*( D/2 )^2 = m*i^2

The formula for the instantaneous friction force apposing bullet travel down the bore:

F_x = ( ( 2*u*J*w’ ) / ( D*( cos(alpha) - u*sin(alpha) ) ) )

The formula for the instantaneous friction force apposing the bullets twist in the bore:

F_y= ( u*( P - m*v’ ) ) / ( sin(alpha) + u*cos(alpha) )

In order to obtain the net energy lost due to bore friction it is necessary to sum the integration of these two formulas for the full length of the barrel. Explicit formulas for P, v’, and w’ must be calculated with respect to the bullets position within the barrel. For my air-gun analysis this won't be too difficult, but for a gun that runs off gun powder rather then compressed air things get way, way, complicated.

Buy the way these formulas are built-up from those put forth on page 119 of the "Oerlikon Pocket Book", published by the Oerlikon Company of cannon fame -- 1981 edition.