TIMING.... The approximate time that it takes a 3300 fpsmuzzle velocity bullet to exit the barrel, assuming a constant acceleration, is0.0011 seconds. Actual exit times would be longer since the bullet is not underconstant acceleration and the time from the firing pin first hitting the primer,ignition starts, is not included.
The view is a typical pressure vs. time trace from RecreationalSoftware, Inc. Note that the zero time starts at a 12,000 psi pressure andbullet exit is somewhere near the + signs. The primer ignition was at an earliertime. The velocity of sound in 416 stainless steel is 14,900 fps and a stresswave has time to propagate up and back the full length of the barrel 4 or moretimes after ignition and while the bullet is traveling within the barrel. Themuzzle end of the barrel has ample time to 'know' that something isgoing on at the breach end before the bullet exits.

VIBRATION PERIOD.... As an example, assume a mode 1frequency of approximately 100 Hz that has a period of vibration of 0.010seconds. Therefore the time it takes the barrel to make its first upward swingis a fourth of a period or about 0.0025 seconds. This is the approximate timethe bullet exits. The mode 2 vibration has a period of approximately 0.0002seconds and could be at the top of its thirteenth upward swing, at the 12.25thperiod swing, or about 0.0025 seconds. These two modes could add and amplify theexit angle of the muzzle near the peak of its upward swing, just as the bulletexits. The tuner could be adjusted to slightly change the two frequencies sothat they reinforce each other at the time of bullet exit.

FINDING THE SWEET SPOT.... When tuning a load to aparticular rifle, possibly the sweet spot is not when the bullet exits thebarrel at the maximum of the vibration's upward swing but, slightly beforethe maximum height or exit angle of the upward swing.
UPWARD SWING.... The vertical amplitude of vibration is more heavilyexcited than the horizontal vibration because the center of gravity of the rifleis located below the barrel's centerline and the bullet's travel down the barrelcauses a vertical turning moment about the rifles center of gravity. Thevertical vibration is most important. Also, the barrel is initially slightlydeflected downward due to gravity. When the round is fired, the pressure alsotends to straighten the barrel like a bourdon tube in a pressure gage. As thebarrel straightens, it over shoot in the upward direction and this adds to theexcitation of the Mode 1 vibration. As a side note, the axial extensionvibration mode is also probably heavily excited. This is the mode where thebarrel extends and shortens axially. But, this axial mode should only have anegligible affect accuracy.
AVERAGE VELOCITY BULLET.... Possibly the sweet spot occurs when thebullet, with the average velocity, for a particular load, exits the barrel justbefore the peak of its upward swing.
FASTER BULLET.... A faster bullet will exit the barrel earlier and exitslightly before the average velocity bullet and the angle of the upward swing ofthe barrel will be slightly less. So the bullet's launch angle is slightly less,but the bullet is going faster and drops less.
SLOWER BULLET.... A slower bullet will exit later and the barrel'svertical swing will be higher and at a steeper angle when the slower bulletexits. The bullet is launched at a higher angle but is slower and will dropslightly more.
This combination, within limits, would print the bullet at about the samevertical location on the target for the normal variations in velocity from roundto round.

Mode 1 -80.304 Hz

Mode 2 -439.22 Hz

Mode3 - 919.5 Hz

Mode4 - 1790 Hz

Nodes

Mode5 - 2374 Hz

FIRST FIVE MODE SHAPES.... These are the first fivevibration Mode shapes, in the vertical plane, that the barrel and tuner undergowhen excited. The displacements are shown here greatly amplified so the shapechange can be easily seen. The actual displacements are very small and evensmaller at the higher frequency Modes.

Good Hunting... from Varmint Al
For the serious reader: How toCheck Another Engineer's Calculation.
More info on ModalAnalysis.

Last Updated: 02/08/2013
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