Need Formula or Data for Ballistic Bore Friction

[turbo1889]

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.

[danski26]

http://www.roymech.co.uk/Useful_Tables/Tribology/co_of_frict.htm

I believe what you are looking for is the "coefficient of friction". The above link lists data, formulas and testing protocol to find the "COF". Otherwise just a Google search of "COF" will give you many hits.

[Duckiller]

Long ago and far away friction was independent of area. Friction is a function of the force the pellet applies radially to the bore. Classic example is a brick sliding on a flat surface. Makes for easier explanations. Friction factor in this example is the force required to move the brick divided by the weight of the brick. To reduce the friction factor you lube the bottom of the brick/flat surface. You need the friction factor for a lead pellet traveling down a steel bore and the amount of the pellet applies to the bore. The pellet must be large enough to seal the air behind the pellet. A pellet slightly less than bore dia may hold back enough air to push the pellet down the bore because the small annular area won't let the air pass freely. Think the subject was discussed in Physics lo many years ago. Emperical data may be obtained by pushing a pellet down bore and measuring force. Bathroom scale pushing rod. You don't want the initial force to get pellet started since that will also include force to swage pellet to bore. Hope this helps, haven't thought about such things for over 40 years. Duckiller

[turbo1889]

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.