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

WHAT A TUNER DOES.... It is possible to "Tune" a rifle barrel so that where it is pointing at bullet exit time will compensate for small variations of muzzle velocity. In the "tuned" condition the slower bullets are launched at a slightly higher angle than the faster bullets. The difference in launch angle can allow the slower bullets to hit the target at the same elevation as the faster bullets. That is the simple explanation and now the details.

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 flexible barrel's muzzle must launch the two bullets of different velocity at slightly different elevations for there to be no vertical spread at the 50 yard target. With only two place accuracy on the drop values, from 42 yards to 50 yards the round off shows the drop to be the same. With more accuracy, this would not be the case.

RIGIDLY FIX MUZZLE.... If the same two velocity shots were fired from a rigidly fixed muzzle with the exact same launch angle then there would be a vertical spread with the slower bullet striking low by approximately 0.24 inches. This an approximate value because the ballistic program I was using only reports drop values with two place accuracy.

BUILT IN FLEXIBILITY.... The reverse taper barrel design is more flexible than a straight cylindrical barrel. The advantage of the flexible design is that it allows the barrel's muzzle to be pointing at increasing elevations for slower bullets. It is possible to tune a barrel for Positive Compensation to minimize the vertical spread. See the muzzle projection curves below.

FINALLY.... There is useful information in this FEA analysis. But there is no substitute for shooting bullets at a target. 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. Geoffrey Kolbe had performed an experiment on Positive Compensation to minimize vertical spread. Here is his article: Vertical Dispersion with Velocity by Geoffrey Kolbe.

NODES AND ELEMENTS.... The solid elements are made of  6-sided 8-node bricks. The solid brick elements have a node at each corner which define the x, y, & z initial positions of each corner. These nodes are not similar and are not to be confused with the nodes present in vibrating structures.

SUMMARY.... For the reader who doesn't want to wade through all the discussion here is a summary of what a tuner can do to correct for small variations in muzzle velocity from ammo with small variations in muzzle velocity. The Muzzle Projection Curve shows where the muzzle is pointing at a 100 yard target while the rifle is being fired. The most important aspect is the curve is where the muzzle is pointing at the time the bullet exits the muzzle.

Improving accuracy by compensating for small variations in muzzle velocity.

Smaller Groups Left of the Peak - UPWARD SLOPE:
Higher velocity shots exit early while pointing lower at the target but drop less in reaching the target.
Lower velocity shots exit later while pointing higher at the target but drop more in reaching the target.
Counteracting combination. Less vertical spread. Good.

Larger Groups Right of the Peak - DOWNWARD SLOPE:
Higher velocity shots exit early while pointing higher at the target and drop less in reaching the target.
Lower velocity shots exit later while pointing lower at the target and drop more in reaching the target.
Bad additive combination. More vertical spread. Bad.

The additional mass of a Muzzle Tuner slows down the muzzle movements and allows the bullet exit before the peak of the muzzle projection curve during the upward slope without resorting to high pressure loads.

ESTEN'S 22LR RIFLE.... 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 Harrell 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. The reverse taper is effective because it ALLOWS movement.

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 rigidly fixed in space 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 rigidly fixed muzzle. I believe that a rigidly fixed muzzle is impossible with any reasonable rifle configuration.

For each calculated shot's Point of Impact (POI), the following was calculated:

1. Muzzle's projection to the 50 Yd Target
2. Muzzle's vertical velocity transmitted to the bullet at exit
3. Time of Flight (TOF)
4. Drop during the flight to the target (ballistics - 4.345 in for 1035 fps and 4.172 in for 1075 fps)
5. Bullet's vertical displacement from 2. during the TOF
6. Pressure curve scaled for the correct Muzzle Velocity
7. Bullet exit time
8. Gravity of 386 in/sec^2

Note: Not only is the muzzle exit angle changing in time, but the muzzle is also moving in the vertical direction while the bullet is traveling down the barrel. When the bullet exits the muzzle, the bullet will have the same vertical velocity as the muzzle. The bullet's vertical velocity during the TOF will also effect the vertical impact point at the target.

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.

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). 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. The barrel axis points at zero on the 50 yard target. Gravity is applied and the stock deforms as well as the barrel sage. That is why the muzzle starts out pointing approximately 1.35 inches below zero. The muzzle projection includes the vertical motion of the muzzle and the exit angle projected to the 50 yard target. It is as if there were a tiny laser taped to the muzzle and it traces the curves on the target. 

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.... There can be too much of a good thing. 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.

Two-Flats Barrel Contour with Vertical Flexibility and Greater Horizontal Stiffness

THE TWO-FLATS CONTOUR.... The barrel in front of the action is 1.25" in diameter and tapers down to 1.1" in diameter at 4" from the action. From there, the barrel diameter is 1" and the distance between the flats is 0.643". The muzzle is 1" diameter for 2 inches so a tuner could be used. It is 24.75" long. The flats start at 4.5" and extend to 22.25". At each end of the flats, the transition is 0.5" in length. This contour allows for more vertical flexibility in the vertical direction than a 1" full diameter barrel. But it has nearly the same stiffness in the horizontal direction as a 1" diameter barrel.

WIDER RISING PROJECTION WINDOW.... The Muzzle Projection curve shows a wider time period that the muzzle projection is rising that with the Reverse Taper barrel contour. It appears that it would be easier to fine tune if a light weight tuner were used on the muzzle. The sag due to gravity was in the calculations, but subtracted out for the chart so the curves can more easily be compared. There is less muzzle projection sag at the 50 yard target with the Two-Flats barrel than the Reverse Taper barrel. The calculations included gravity of 386 in/sec^2.

TWO-FLATS MOVIE.... Here is the movie clip of the calculated motion for the 1075 muzzle velocity round. The last frame occurs at bullet exit time. The displacements are amplified by 1000X. The stock contacts the simulated sandbag rests with zero friction in the calculation. The rests are fixed in space. 

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 but this is highly unlikely with a single lot of ammo.

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

TUNING & DISTANCE.... Here is a chart that shows the 100 yard trajectory of a 22LR 40 gr bullets with a BC of 0.128 and an average muzzle velocity of 1050 fps. It is assumed that the actual velocity varies from 1040 to 1060 fps with the vertical tuned out. The first case (bottom curves) is where the vertical is tuned out at 50 yards. In this case the vertical at 100 yards is 7.79-7.46 or 0.33”. The rifle is not in tune for 100 yards.

In the second case the vertical is tuned out at 100 yards. But in this case the vertical at 50 yards is 3.89-3.73 or 0.16”.
Anyone can calculate this for their rifle & ammo using free ballistic software. For small variations is muzzle velocity a 22LR rifle in tune at 50 yards is NOT in tune at 100 yards.

LAUNCH ANGLE & DROP.... For the 1040 fps and 1060 fps bullets to strike at the same elevation at the 100 yard target, the launch angle for the 1040 fps bullet has to be elevated 0.2218 degrees above that of the 1060 fps bullet. If the muzzle were somehow fixed in space so that both bullets were launched at exactly the same angle, then the 1040 fps bullet would strike the 100 yard target 0.40" lower than the 1060 fps bullet. It is not rocket science that the slower bullet will drop more in traveling to the target.

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

Last Updated: 07/18/2015
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