QUICK FACTS
Researchers at Three Rivers Technologies investigated variations that cause bullets
to impact in different locations around their point of aim. Output data consisted of
binary files containing transverse velocity and displacement profiles. These files are
readable by FFT post-processing codes and by PV-WAVE® procedure files, which displayed
animations of the system.
THE PROBLEM
A device that controls
rifle barrel vibration and improves bullet accuracy was evaluated and
designed by computer simulation. A computer program was written by the
authors to solve the transient form of the elastic beam vibration equation
for any given driving function. Output consists of binary files containing
transverse velocity and displacement profiles. These files are readable
by FFT post-processing codes and by PV-WAVE®
procedure files, which display animations of the system. FFT post-processing
was performed to benchmark and verify the simulator with analytical solutions
of mode shapes and frequencies. The simulator then was used to optimize
a design for a set of rifle barrel modifications that alter the vibrational
response to minimize the angular dispersion, or slope, at the muzzle.
Bullets will thus exit the barrel always pointing in the same direction
(parallel to the barrel baseline axis). This minimizes the sensitivity
of precision (measured by group size) to bullet exit time and vibration
initiators. Rifle accuracy is thereby improved for a wide range of loads
for a given rifle.
Barrel vibration is one of the factors affecting the accuracy of rifles. Variations in
loads (propellant charge weights and bullet masses) cause different times-of-flight
from primer ignition to the point in time when the bullet leaves the muzzle.
These variations cause the bullet to impact in different locations around
the point of aim. The size of the bullet dispersion is called group size.
Handloaders have typically reduced bullet group size by "tuning" (or adjusting)
the powder load to the barrel so that minimum bullet group size results.
The goal is to get the bullet to exit the barrel at a point of maximum
barrel deflection, as this represents a point of minimum time-rate-of-change
in barrel slope at the muzzle. This minimizes the sensitivity of bullet
dispersion to statistical muzzle velocity variation. A new approach was
investigated that improves firearm precision by significantly reducing
the magnitude of the angular dispersion of the muzzle over a window of
relevant bullet exit times.
Proposed modifications to the barrel control the shape of the vibrational response so that the
barrel slope at the muzzle is minimized. The first modification (addition
of a mass to the barrel between the breech and the muzzle) isolates the
muzzle from much of the vibrational energy initiated between it and the
chamber through inertial damping (reflection). The second modification
(addition of a flexible cylindrical extension and mass attached to the
muzzle) acts as a cantilever harmonic oscillator, providing a periodic
bending moment to alter the shape of the vibrational response associated
with the transmitted vibrational energy. This flexible extension is designed
so that the barrel slope is minimized at the point where the bullet leaves
physical contact with the barrel, and its flight is determined. The barrel
extension has an inner diameter larger than the bore diameter and gas
release slots to reduce the effect of gas upsetting the bullet after it
has left the muzzle. The design goal is to optimize the positions of the
two masses, the radial dimensions and length of the extension, and the
configuration of lengthwise slots in the extension to minimize the slope
of the muzzle over a window of bullet exit times. This is accomplished
when the barrel and flexible extension forms a resonating segment between
the masses so that any vibrational energy transmitted past the first mass
forms a symmetrical standing wave. The location of zero barrel slope (half
the standing wavelength) is designed to coincide with where the bullet
enters the extension and leaves physical contact with the barrel. The
bullet exits the barrel parallel to the baseline axis for an extended
window of bullet exit times, resulting in significantly less sensitivity
to variations in exit times.
Rifle barrel vibration is initiated by mechanical interaction between the barrel and the bullet
accelerating down the bore, as well as by the severe pressure transient
arising from the burning propellant. Barrel vibrations are considered,
for this application, to be a superposition of transverse vibrational
modes initiated at a continuum of points along the barrel. The short-term
vibrational response includes a particular solution arising from the specific
characteristics of the driving function (the operating deflection shape),
but this response will rapidly transition into the natural modes for the
barrel itself. Although this transition takes longer to occur than the
time required for the bullet to exit the barrel, comparisons with natural
mode shapes and frequencies allow independent verification of the simulator.
THE SOLUTION
To investigate these ideas, a computer program was written to solve the generalized transient
beam vibration equation, which is second order in time and fourth order
in space. Equation (9.23) from the Shock and Vibration Handbook
(Harris and Crede 1976) is reproduced here, and is assumed to be the governing
differential equation for this system:
The coefficients E
and I are Young's modulus (ratio of stress to strain, 
and the area moment for the barrel cross section 
where r is measured from the x-axis and dS
is a differential area over the barrel cross section). The coefficients
c1 and c2 are internal and external
damping constants. If an external rib is attached to the vibrating barrel,
k characterizes its linear restoring force. In the acceleration
term, 
and
S are the material density and the cross-sectional area of the
barrel. F(x,t) is the external driving, or excitation, function.
The steady-state form of this equation reduces to the classic beam flexure
equation. With transient terms set to zero (and neglecting the rib constant,
k), equation (1) reduces to:
where M is the bending moment on the barrel. As another special
case of interest, if the barrel were of zero radial extent, I
would equal zero in equation (1), and one is left with
the classic damped, forced harmonic oscillator for every point along the
barrel. The two terms containing the area moment provide the coupling
between points along the barrel.
Boundary conditions are assumed to be zero deflection and slope at the breech end of the barrel
(where the barrel attaches to the rifle's action), and zero shear and
zero bending moment at the muzzle end of the barrel:
where L is the overall length of the barrel. These boundary conditions
describe the idealized, "clamped-free" beam problem in mechanics, but
do not explicitly model the more complex boundary conditions of mechanical
coupling between rifle barrel and stock along the length of the stock,
coupling to the shooter (whose shoulder is not rigidly fixed), etc. However,
during the 1-2 msec time window of interest, it is believed that these
more complex effects are negligible, and the simple, clamped-free set
of boundary conditions should suffice.
Numerical Methods
Equation
(1) is fourth order in space and second order in time. To solve this
equation, we first split it into two coupled equations that are each first
order in time. If we set 
we can rewrite equation (1):
This can be rearranged as:
Equation (5) is integrated
over time using the "theta-differencing" method, where the integral 
is approximated by a weighted average of future time fj+1
and present time fj (Carnahan et al. 1969):
Using the finite difference formula for a fourth-order derivative,
and integration of equation (5) over time (applying equation
(6) to the 
terms only, while leaving the remaining terms constant over the time step)
results in:
This represents a set of five-point-coupled algebraic equations of the form:
where the coupling coefficients (which are, in general, a function of position)
are defined as follows:
These equations are collected into a five-diagonal matrix, referred to here as a "5 x
N" matrix, where N is the number of mesh intervals used in the barrel
model, which has the following form:
The four boundary conditions are incorporated directly into the 5 x N matrix. The
last two equations in the matrix are reserved for the imposition of the
zero shear and zero bending moment. Using the finite difference formulae
for the second- and third-order derivatives results in:
The first two equations in the matrix impose the zero displacement and slope boundary
conditions. The displacement at x=0, y0,
is zero, and so does not have to be calculated. Therefore, the first equation
in the matrix is for the first point internal to the barrel,
y1. The first equation imposes zero slope at x=0
by requiring reflective symmetry (i.e., by setting y-1
= y1). The second equation in the matrix is a general
internal equation, but takes advantage of the fact that y0
= 0. Formally incorporating these boundary conditions into equation
(11) imposes a zero time derivative on y and the first three
spatial derivatives. However, at t=0, the displacements are all
zero, and so are the three spatial derivatives. Requiring a zero time
derivative keeps the displacement and the spatial derivatives equal to
zero, and so these boundary conditions are equivalent.
These equations are solved by direct inversion of the 5 x N matrix for the transverse velocity
distribution for a single time step. This in turn is integrated once in
time to obtain displacement:
The speed of sound,
is approximately
equal to 5100 m/sec in steel. Any impulse to the barrel will result in
an acoustical pulse traveling down the barrel, exciting small transverse
waves which lag behind the sound pulse. This acoustic speed represents
a small time constant attribute that makes the vibration equation a "stiff"
system, requiring small time steps to resolve the response correctly.
Small time steps and consideration of internal damping are required for
numerical stability. Given the mesh spacing used, the sound pulse requires
about 10-6 sec to traverse one mesh interval, and so 10-7
sec time steps were used (which in turn requires double precision in the
simulation). Animations created from simulated transients display visible
disturbances which appear to travel faster than the primary vibrational
response, but lag behind the longitudinal sound pulse. The initial pulse
reaches the end of the barrel in 0.46 msec (which is about a factor of
3 less than the known speed of sound in steel), but the primary vibrational
response reaches the end of the barrel only after 1.0 msec.
Computer Code Verification
After long times
(several hundreds of milliseconds), very little of the particular solution
is left, and the barrel exhibits only the natural modes. Several simulations
were performed using a step impulse, or "thump," for F(x,t),
and the long-term behavior of the barrel was analyzed using the Fast Fourier
Transform (FFT) (Press et al. 1992) calculation of the power spectrum
and mode reconstruction. The test model was of a barrel of uniform radial
dimensions (not appropriate for most real barrels, but is more amenable
to comparison with analytical solutions for mode shapes and frequencies).
The barrel was modeled with 50-mesh intervals along its 24-inch length,
and the step impulse was applied at mesh interval 17. A transient time
of 307.2 msec was taken, and the comparison was quite good. Figure
1 shows the slope of the barrel evaluated at the muzzle. Figure
2 shows the FFT power spectrum evaluated at mesh interval 35. Figure
3 shows the root-mean-square (always positive) mode shape reconstruction
as a function of frequency. Table 1 compares the frequencies
extracted from the simulator model to those predicted by analytical methods.
Table 2 compares node locations for each of the mode
shapes.
For the first two
harmonics, there are three nodal points, or points of zero deflection
(one for the first harmonic and two for the second). The simulator consistently
uses the cgs metric system of units, and the magnitude of the impulse
was 10 dyne/cm applied over 1-mesh intervals for 0.1 msec. The dyne is
a very small unit of force, and this is consistent with the small calculated
displacements. Equation (1) is linear in
F(x,t), and the magnitude of the response will scale exactly
with the magnitude of the driving function. To compare simulated barrel
vibration with real barrel vibration, normalization to measurement is
required.
Table 1. Frequency
Spectrum Evaluated at Mesh 35.
| Frequency (Hz) |
Simulator |
Analytical |
| Fundamental |
28 |
27.0 |
| First Harmonic |
180 |
169.0 |
| Second Harmonic |
500 |
473.1 |
Table 2. Fourier Mode
Reconstruction. Nodal Points for First Two Harmonics.
| Node Locations |
Simulator |
Analytical |
| First Harmonic |
38/50 = 0.76 |
0.783 |
| Second Harmonic |
25/50 = 0.50 |
0.504 |
| Second Harmonic |
42/50 = 0.84 |
0.868 |
Figure 1. Slope, Evaluated
at the Muzzle, vs. Time. Test Verification Model Problem.
Figure 2. Power Spectral
Density vs. Frequency Evaluated at Mesh Interval 35 of 50.
Figure 3. Mode Reconstruction vs. Frequency.
The points
along the barrel predicted to be nodal points are predicted by theory
and compared to the nodal points predicted by the simulator. This post-processing
of simulator output verifies that the simulator reproduces long-term analytical
solutions (mode shapes as well as their associated frequencies) that agree
reasonably well with analytical descriptions of ideal beams of uniform
cross section.
Optimal Design
Results
The nature of the
driving function is important in the short time of interest. The rifle
model is a .308 Winchester with a 27-inch barrel. The cartridge is a handload,
loaded with 168 grain bullets and 50 grains of Hodgden H4831 powder. The
specific pressure and bullet acceleration curves for this rifle were calculated
using the QuickLOAD software (Brömel 1996). The maximum pressure
was just under 45,000 psi, and the muzzle velocity was 2572 ft/sec. It
was assumed that F(x,t) could be modeled as an impulse traveling
along with the accelerating bullet, and with a relative magnitude proportional
to the pressure curve. Figure 4 shows the slope at
the muzzle as a function of time for both the unmodified barrel and after
modification. The root-mean-square slope at the muzzle,
(defined
from 1.0 to 1.8 msec) for the unmodified case was 9.074 x 10-11
and 7.599 x 10-12 after modification, which indicates a factor
of 12 improvement over the unmodified barrel. This translates potentially
into a factor of 12 reduction in bullet dispersion as different bullet
exit times and vibrational excitations are realized. Of course, different
loads will still exhibit different flight paths to the target, but these
trajectories are a function purely of external ballistics effects and
are, therefore, predictable using these methods (e.g., Brömel 1996;
ADC 1996). With the barrel always pointing in the same direction, no "surprises"
should remain when changing loads. Figure
5 illustrates a comparison between barrel responses with and without
modification. The time shown is the instant when the bullet for the model
load exits the muzzle (1.4 msec). Figure 5a
shows the muzzle pointed significantly away from the baseline axis. In
Figure 5b, squares indicate the positions
of the two masses, and the transition between the barrel and extension
is at mesh interval 108. The small pulses at the locations of the two
masses indicate the dynamic reaction forces exerted on the barrel by the
masses at this point in time. These two figures were taken from animations
of the system, where the total forcing function F(x,t) is superimposed
over the barrel response. In the animations, one can observe the pulse
from the bullet accelerating down the barrel with a magnitude following
the pressure curve, as well as the oscillatory reaction forces from the
two masses.
Figure
4. Slope, Evaluated at the Muzzle, vs. Time. Unmodified Barrel and After
Modification.
Conclusions
A finite difference
computer simulation of rifle barrel vibrations was developed from first
principles. A model of realistic pressure initiators of vibration was
incorporated into the model to give proper weighting of vibration sources.
The model produced good agreement with classical vibrational mode solutions
representing non-transient mode shapes. The transient results of the model
were reasonable and yielded valuable insight into the design of barrel
modifications.
The prototype rifle
and barrel now are being built, which will provide the measurements that
will validate this model. The most significant realization should be small,
tight bullet groups across a range of loads and temperatures (which affects
internal ballistics), with no adjustments required on the part of the
shooter.
Figure
5a. Vibrational Response, No Modification. Snapshot in Time Just as Bullet
Exits Barrel.
Figure 5b. Vibrational Response After Modification. Snapshot in Time Just
as Bullet Exits Barrel.
References
ADC, Inc. 1996. PC
Bullet for Windows, Software for Shooters, software sold and licensed
by ADC, Inc., Scappoose, OR, copyright 1989-96.
Brömel, Hartmut
G. 1996. QuickLOAD Interior Ballistics Program and QuickTARGET
Exterior Ballistics Program, commercial software sold and licensed
by the author, D-63303 Dreieich, Germany, copyright 1996.
Carnahan, Bryce; H.
A. Luther; and James O. Wilkes. 1969. Applied Numerical Methods,
John Wiley & Sons, New York, p. 451.
Harris, Cyril M.,
and C. E. Crede. 1976. Shock and Vibration Handbook, Second Edition,
McGraw-Hill Book Company, New York, p. 9-5.
Press, William H.;
Saul A. Teukolsky; William T. Vetterling; and Brian P. Flannery. 1992.
Numerical Recipes in Fortran 77, Second Edition, The Art of Scientific
Computing, Cambridge University Press, p. 490-551.
Biography
Dr. Schwinkendorf earned his B.S. and M.S. degrees from Oregon State University
in 1981 and 1983, respectively, and his Ph.D. from the University of Washington
in 1996, all in nuclear engineering. His doctoral work was in the field
of severe nuclear reactor accident simulation. Dr. Schwinkendorf has worked
as a reactor physicist and nuclear criticality safety engineer at the
Hanford Site in southeastern Washington state since 1983.
Mr. Roblyer received his B.S. in physics from the University of Oregon
and M.S. in nuclear engineering from the University of Washington, and
has worked in the field of vibration control engineering, specifically
vibration and signature analysis of rotating machinery, at the Hanford
N Reactor.
Both Dr. Schwinkendorf and Mr. Roblyer share an interest in the shooting sports and have jointly
applied for a patent on the rifle barrel vibration control system described
in this paper. Three Rivers Technologies was formed by the authors to
develop this system.
**This paper was reprinted with the permission of Society for Computer Simulation,
and was first presented at the 1998 Advanced Simulation Technologies Conference
(ASTC '98), April 5-9, 1998. SCS supports multiple conferences, or tracks,
at these meetings, and the one we presented to was Military, Government
and Aerospace Simulation. This paper appears on page 66 of the Proceedings.
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