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 (Brmel 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., Br mel 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.

<< PREVIOUS

MORE >>