Barrel Tuner Analysis
Esten's 22LR Rimfire Benchrest Rifle With and Without a Muzzle Tuner
FEA (Finite Element Analysis) of a 22 LR
Rifle Barrel Dynamic Pressure Analysis

Esten's barrel that is modeled is on a Myers Built 10.5# Rimfire Benchrest Rifle. It has a 24.75" Benchmark 2 Groove, Reverse Taper Barrel like currently holds the World Record. It has a Harrel tuner which clamps on to the end 1" of the barrel. The weights screw in the end and are stackable and available in different weights. The Reverse taper barrel has a short 1" long Breach Cylinder of 1.1" Dia. then radius down to .750" From there it tapers UP in dia. to .915" and has a .915 cylinder for the last 2" at the muzzle which is where the tuner clamps on. The details of the tuner and weight locations were not available so a homogeneous tuner mass of same external dimensions was modeled and the density was scaled to give the correct weight.

Table 1: Vertical Spread for 22LR Barrel Configurations with/without A Tuner

22LR AMMO.... In 22LR benchrest, the ammo is fixed and can not be tuned to the barrel. But the barrel can be tuned to the ammo. The only function of tuning is to compensate for small variations in muzzle velocity from shot to shot. The 1035 to 1075 fps range is an extreme spread but gives an indication of what improvements can be made to compensate for velocity variations. The calculations have been performed at these two velocities to determine if variation if muzzle can be compensated for and reduce the vertical spread at the 50 yard target. 
If somehow, the muzzle were fixed in space or magically "stopped" and two rounds were fired, one with a velocity of 1035 fps and the other with 1075 fps there would be a 0.1737 inch vertical spread because of the different amount of drop. See the bottom row of Table 1. A tuner can cause the slower bullet to be launched to a higher point and the faster bullet at a lower point. This can't be done with a "Stopped Muzzle". I believe that a "Stopped Muzzle" is a null set and is impossible with any reasonable rifle configuration.

REVERSE TAPER BARREL.... The vertical spread for the Reverse Taper barrel and No tuner has a steep rising slope at the time of bullet exit. The vertical spread is 0.0910 inches for the two different loads (1035 and 1075 fps) that Calfee recommends for adjusting a tuner. This was the smallest vertical spread of the barrel configurations calculated so far.  The movie clip shows the deformations of Esten's Reverse Taper barrel with a 1075 fps muzzle velocity and no tuner. The last frame of the animation coincides with the bullet exiting the muzzle. The muzzle is not stationary and points to different locations on a 50 yard target as shown by the chart and movie clip. The deformations are amplified by 2000X.

Smaller Groups Left of the Peak (Positive or Upward Slope):
Higher velocity shots exit early (blue curve) while pointing lower at the target but drop less in reaching the target.
Lower velocity shots exit later (red curve) while pointing higher at the target but drop more in reaching the target.
Counteracting combination.

TUNER OF 16.0 OZ.... The vertical spread with the reverse taper barrel and a 16.0 oz tuner is 0.1832 inches. This was the largest vertical spread of the barrel configurations calculated. the bullet exit time is centered on a negative or downward slope of the Muzzle Projection Curve. The movie clip shows the deformations of Esten's Reverse Taper barrel with a 1075 fps muzzle velocity and a 16.0 oz tuner. The last frame of the animation coincides with the bullet exiting the muzzle. The muzzle is not stationary and points to different locations on a 50 yard target. The deformations are amplified by 2000X. The muzzle and 16 oz tuner are not stationary and points to different locations on a 50 yard target as shown by the chart and movie clip. The deformations are amplified by 2000X.

Larger Groups Right of the Peak (Negative or Downward Slope):
Higher velocity shots exit early (blue curve) while pointing higher at the target and drop less in reaching the target.
Lower velocity shots exit later (red curve) while pointing lower at the target and drop more in reaching the target.
Bad additive combination.

For each calculated shot's POI, the following is calculated:
1. Pressure curve scaled for the correct Muzzle Velocity
2. Bullet exit time
3. Drop during the flight to the target. (4.345 in for 1035 fps and 4.172 in for 1075 fps)
4. Muzzle vertical velocity transmitted to the bullet at the time of exit
5. Time of Flight (TOF) (0.151 sec for 1035 fps and 0.146 sec for 1075 fps)
6. Bullet vertical displacement from 4. during the TOF
7. Muzzle's projection to the 50 Yd Target at bullet exit time

The Muzzle Projection Curves include the vertical displacement of the muzzle and the angle of exit, both projected to the 50 yard target. The low Point of Impact (POI) numbers result from the barrel pointing at zero in zero gravity. When gravity is applied, the muzzle sags and its projection points lower on the target. The amount of projected sag can be seen at zero time on the charts. There are times when the muzzle's vertical projection is zero at the top or bottom of each cycle, but the exit angle is not necessarily zero at the same time. 

The Time of Flight (TOF) and the bullet's vertical velocity charts were generated so the additional bullet drop from this contribution could be calculated. The difference in the vertical spread from TOF and the muzzle's vertical velocity is small compared to where the muzzle is pointing but is included in the values listed in Table 1.

There was a lot of work involved in performing these calculations. My motivation is to try to gain an understanding of what is happening when a rifle is fired. The FEA allows one to inspect the very smallest of detail of the event calculated. I am sharing these results in a hope that others might find some useful information in shooting smaller groups.

Table 2: Rifle, Barrel, and Tuner weights

PRESSURE CURVES.... The chamber pressure curves were generated from strain gauge data. The data was scaled for the pressure amplitude to generate the given muzzle velocities for a 40 gr 0.224 caliber bullet in a 24.75 inch barrel. The assumption for these calculations are that higher velocity bullets will exit earlier than lower velocity bullets. For some markedly different pressure curves with early high pressures that drop off to a very low value later, lower velocity bullets could exit earlier that higher velocity bullets.

FLIGHT PATH.... This chart shows the slower bullet actually travels a higher path than the faster bullet for them both to hit the same 50 yard zero. It is clear that the tuner must launch the two bullets of different velocity at slightly different elevations for there to be no vertical spread at the 50 yard target. 

FINALLY.... I think there is useful information in this FEA analysis. But there is no substitute for shooting bullets a target. I think the FEA analysis helps one to understand what is happening when a 22LR is fired and how the barrel moves. The FEA analysis does not capture the urgency and stress of a competition match and is not meant to.

COMPARISON CHARTS.... These charts are for comparing the Muzzle Projection Curves and the Muzzle's Vertical Velocity for each barrel configuration. All charts are at the same scale except for the 0.8" diameter barrel where the scale had to be increased to not clip the curves. The calculated vertical spread is listed after each set of charts. The sag due to gravity is removed from the Muzzle Projection Curves. This would be the case as if the rifle were sighted in at 50 yards zero for each barrel and muzzle tuner configuration.

spread= 0.1108 inches
Note: The scale is increased for only these two charts.

spread = 0.1974 inches

0.1926 inches

0.0910 inches

0.1218 inches

0.1554 inches

  0.1832 inches

Good Hunting... from Varmint Al
For the serious reader: How to Check Another Engineer's Calculation.

Last Updated: 03/26/2008
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