Theory of barrel twist rate and bullet mass and diameter?

[Bill_Rights]

Yes, thanks 1858. I will see if I can download the Excel spreadsheet and understand it. I am interested!

BTW, what is "SG" again? (I know I knew it before, but old-age CRS syndrome is creeping in.....)

 

[Bill_Rights]

Miller Formula calculator

1858,

I was able to download the Excel spreadsheet. Thanks! (I didn't know we could do this.)

Yeh, so... looking at it, I see that you don't ask for the caliber anywhere, in actual dimensions, such as inches. But the twist and the bullet length are expressed in units of calibers. OK, so I get it. It does not matter what the caliber measurement is, you just make sure to express the other two parameters (twist rate and projectile length) in multiples of caliber diameters.

Cool.

 

[MCMXI]

I see that you don't ask for the caliber anywhere, in actual dimensions, such as inches.

Bullet diameter (in.) is the caliber. You enter bullet diameter and length in inches and the spreadsheet calculates those values in calibers and then uses those values to calculate SG.

SG is a unitless value to indicate gyroscopic stability.

 

[Bill_Rights]

Oh! Excuse me. I was wrong. You corrected me by saying

You enter bullet diameter and length in inches and the spreadsheet calculates those values in calibers

For some reason, I did not see the data entry field for bullet diameter in inches. In addition to CRS, I must be coming down with CSS (can't see sh!#) .

 

[Bill_Rights]

Bullet spinning energy - is it significant?

I said earlier

I assert that any amount of spin, excessive or not, detracts from muzzle velocity. Of course, some spin is necessary, as we've been discussing. I will calculate the bullet energy consumed by spin and post it later...

So, here it is. The needed formulas tying the spin energy to bullet mass, diameter and spin speed rely on the moment of inertia of the bullet. The moment of inertia about the spin axis is a property of the shape of the bullet and the density of the material. I approximated the shape as a cylinder and took the material to be solid lead (density is 11340 kg/m^3 = 11.34 g/cc = 185.83 g/cu.in. = 2867.7 gr/cu. in.). Here are the formulas, in which "W" is the energy or work just to spin up the bullet:

In the formulas, "M" is the mass of the bullet, "r" is the radius of the bullet (half the caliber diameter) and omega is the spin speed, as explained. This all has to be worked out in consistent units. I have a spreadsheet for that, if anybody wants me to upload it.

It turns out that the spin energy carried by the bullet to the target is small, less than 1 %, compared to the linear speed energy, which is technically called the translational energy. Here are three examples sort of covering some extremes.

A 180 grain .308 inch diameter lead bullet travelling at 2620 fps out of a barrel with a twist of 12 inches/turn has:
Bullet translational energy of 3719 Joules = 2743 ft*lbf
Bullet rotational energy of 12.1 Joules = 8.92 ft*lbf.
So the ratio of rotational to translational energy is 0.325 %.
Bullet length, effective (cylinder approximation): 0.842 inches

A 300 grain .452 inch diameter lead bullet travelling at 1500 fps out of a barrel with a twist of 30 inches/turn has:
Bullet translational energy of 2032 Joules = 1499 ft*lbf
Bullet rotational energy of 2.28 Joules = 1.68 ft*lbf.
So the ratio of rotational to translational energy is 0.112 %.
Bullet length, effective (cylinder approximation): 0.652 inches

A 62 grain .223 inch (5.56 mm) diameter lead bullet travelling at 3002 fps out of a barrel with a twist of 7 inches/turn has:
Bullet translational energy of 1682 Joules = 1240 ft*lbf
Bullet rotational energy of 8.42 Joules = 6.21 ft*lbf.
So the ratio of rotational to translational energy is 0.501 %.
Bullet length, effective (cylinder approximation): 0.554 inches

These energies carried by the spin of a bullet do not include the extra friction of the faster spinning bullet lost in the barrel or to the air. Zoogster pointed out these effects earlier. This friction energy is lost as heat, not delivered to the target.

 

[helotaxi]

These energies carried by the spin of a bullet do not include the extra friction of the faster spinning bullet lost in the barrel or to the air. Zoogster pointed out these effects earlier. This friction energy is lost as heat, not delivered to the target.

Those energies have been shown to be somewhere around negligible when compared to the same energies from a straight rifled barrel. Essentially you lose the tiny fraction of a percent used to get the bullet spinning and otherwise the difference is nil.

 

[Bill_Rights]

Since the thread responses indicated the critical importance of bullet length, I added the calculated effective lengths of the bullets used for the sample calculations posted, two posts above. As explained there, for the sake of an easy, text-book calculation of bullet spin moment of inertia, I approximated the true bullet shape by a simple cylinder, with flat, square ends. The true bullet lengths would be longer than these approximate results, for the same weight of bullet.

 

[helotaxi]

But the cylinder would be the worst case scenario with regard to energy required to get up to speed. You also assumed a monolithic lead slug which would be worse than a jacketed bullet.

 

[Bill_Rights]

helotaxi,

Yes, also true. But also the effect of the jacket (being lighter, less dense material) would be somewhere around negligible, unless the jacket was very thick.

For my part, I was interested in terminal ballistics (at or inside the target). How much energy does bullet spin carry to the target and could this play any role on effectiveness at impact or penetration? Since the spin energy is well under 1% of the total bullet kinetic energy, my answer seems to be "no".

You, on the other hand, seem more interested in firing chamber dynamics. If we are going to "dissect" that, we also should consider the energy required to engrave the rifling grooves into the bullet flanks. I am a materials scientist, so I could probably estimate the displacement or deformation energy of the bullet jacket, if I had some idea of the mass of the bullet material that gets moved and how far it gets moved. But this is probably also near to negligible, especially for soft materials such as dead-soft lead.

We might be beating a dead flea on a dead horse .

 

[Bill_Rights]

A contemporary thread touches on barrel twist rate. A link there even touches on spin energy and engraving energy.

Seeking .45/70 barrel twist rate info

http://www.thehighroad.org/showthread.php?t=686216

 

[helotaxi]

If we are going to "dissect" that, we also should consider the energy required to engrave the rifling grooves into the bullet flanks.

Once you factor in the obduration of the bullet with the very high forces invloved, the engraving force is anything but negligible. The rate of onset of that force is one of the determining factors of the initial pressure curve of the cartridge. The leade and freebore are manipulated to modulate that onset rate and are the deterministic differences between the .223 and 5.56x45 chambers as well as the 6.8 SPC and 6.8 SPC II chambers. the reduction in the onset of the engraving force is such that a hotter round can be used while keeping the same max chamber pressure and the longer pressure curve of the hotter load results in higher velocity. In the case of the 6.8 SPC, the change from 1:10 to 1:11 rate of twist was the final adjustment used. Again a rate of onset issue. The peak in linear acceleration is also the peak in rotational acceleration and occurs at peak pressure. Reducing the end result RPM by 10% reduced the required rotational acceleration as well and the energy requirement for that acceleration.

 

[Bill_Rights]

helotaxi,

Could you define "obduration" of the bullet? I looked it up, but it is not clear how the general definition applies:

ob·du·rate
adjective \ˈäb-də-rət, -dyə-; äb-ˈdu̇r-ət, əb-, -ˈdyu̇r-\
1
a) stubbornly persistent in wrongdoing
b) hardened in feelings
2
a) resistant to persuasion or softening influences
— ob·du·rate·ly - adverb
— ob·du·rate·ness - noun

Regardless, I gather that it has to do with deformation of the bullet, which is resisted by the bullet material. It is not unknown to occur that the energy (and of course the force) required to deform a material increases with increasing rate of deformation. And the rate of deformation of a bullet will be FAST.

Considering only the energy, energy is defined as a force acting through a distance (or power expended over a time duration). The distance over which the high engraving forces are active may be only the length of the flanks of the bullet, i.e., short. So the total energy expended in engraving may be small.

Which brings the point (which I have known all along) that energy, itself, may not be the important parameter. In the case of terminal ballistics due to spin, angular momentum may be more important than spin energy, for example. As you point out, force can be critical. Force directly maps to pressure. [pressure] = [force per unit area]

It bears thinking about which of the following quantities are most important to bullet kinematics (kinematics = motion dynamics). Here are the linear quantities, and each has it angular (spin) analog:

position: .................. (self-explanatory)
velocity (or speed): ... [rate of change of position]
acceleration: ............ [rate of change of velocity]
jerk: ........................ [rate of change of acceleration]
momentum: ............... [mass mutiplied by velocity]
force: ...................... [mass mutiplied by acceleration]
pressure: .................. [force applied per unit area OR force divided by area]
energy: .................... [force mutiplied by distance]
power: ..................... [energy expended per unit time OR energy divided by time]

 

[helotaxi]

Basically when the pressure starts to move the bullet forward, the bullet gets "squished" for lack of a better term along its longitudinal axis and expands along its radius. This occurs at pretty much the exact time that the bullet is being squeezed into the rifling.

The squishing of the bullet to fill/seal the bore is what is typically referred to as obduration in this context.

 

[WNTFW]

Hear (Here) you go:
I think you mispelled "obturation"
http://en.wikipedia.org/wiki/Obturation

Sometimes one letter is like a decimal in math.

 

[helotaxi]

Possibly/probably.

 

[WNTFW]

I think "swoll up" would be a better term. Or is it "swole up"?

 

[helotaxi]

That term is generally reserved for results from the gym or telling the wife she looks fat...

 

[murf]

deformation of the bullet base rather than "squish", or "swole up" might be a better way to say it.

obturation is the process of sealing the chamber or barrel from powder gases. bullets obturate barrels.

murf

 

[Bill_Rights]

Well; I learned something today, in spite of my best intentions .

But I will obdurately insist upon obfuscating the difference between obturation and obduration .

Regarding the seething controversy over swole versus swoll, they are both wrong, from a materials point of view. Swelling implies that alien molecules (or atoms) are added to the base material or that molecules of the base material all get farther from each other more or less uniformly. I am 99.9% certain that the total volume of a deformed bullet, after it obturates the bore, is exactly the same as it was before firing, accounting for rub-off of material and swelling due to temperature rise. That is, obturation involves plastic deformation with no change in density of the bullet material.

So, rather than a swole-up bullet, we have a bullet with shock-induced basal-proximate radial expansion deformation. We are now in full obfuscation mode. That and $4.89 will get you a cup of coffee at Kona Krakatoa Koffee Kiosk :banghead:.

 

[WNTFW]

Not to be didactic or pendantic . . . but I think you need to upgrade 'rub off' in your thesis.

Now I want coffee and don't have a KKKK near here.

Re: bullets can't be over stabilized.
I see explanations that say they can be over stabilized. In general it is easy to prove a bullet is under stabilized or if it disintegrates. If it is over stabilized that is harder to prove. It is also insignificant to some degree. I just see some info from very knowledgeable folks saying over stabilization is in fact a real condition.

So are you guys saying it can't be over stabilized? Do you mean that it doesn't matter, don't worry about it. Just looking for some clarification on the statement. That issue seems to be a point of ignition for some people.

I have never had a twist rate problem that I am aware of. Thanks to all the guys that figured it out before I came along. All of my calibers are middle of the road boring.