1. Comparative Modal Analysis of Rifle Barrels with Different Cross Sections using Finite Elements BY NICHOLAS MCDERMOTT

2. Problem Description The first objective is an exploration of natural frequencies of different cross sectional shapes of barrels to determine how much the cross sectional shape of the barrel affects its mode shapes. The second objective is to determine which cross sectional shape of barrel is best for accuracy and weight constraints by comparing the frequency achieved (higher means a more rigid barrel) and the weight of the barrel. The third objective is to determine which barrel cross section, if any, is more influenced by outside boundary conditions, such as a barrel band.

3. Methodology

4. Results – Circular Cross Section The modal shapes of the circular cross section are shown. The natural frequencies are listed below. ModeFrequency [Hz] 22 in Barrel24 in Barrel26 in Barrel Bending175.29702163.30078853.951065 2466.624149392.743609335.264724 31282.160251082.1296926.040251 42448.811752074.845021780.07859 53925.646853341.056112875.72297 Torsion11435.167641315.613871214.41432 Extension12315.829072123.42481959.8014

5. Results – Octagonal Cross Section The modal shapes of the octagonal cross section are shown. The natural frequencies are listed below. ModeFrequency [Hz] 22 in Barrel24 in Barrel26 in Barrel Bending1 71.93273360.46007151.388016 2 445.750366375.262678319.340097 3 1225.929611035.059506883.114005 4 2352.446041992.1532691701.238133 5 3775.1044853210.4967242752.415918 Torsion1 1434.0028251314.3529291207.357164 Extension1 2316.3475042123.8723421960.121095

6. Results -Triangular Cross Section The modal shapes of the triangular cross section are shown. The natural frequencies are listed below. ModeFrequency [Hz] 22 in Barrel24 in Barrel26 in Barrel Bending1 70.57477259.26179850.506246 2 437.382987368.022336314.17715 3 1203.7521351015.814662868.705844 4 2303.322461950.4397191672.827526 5 3701.2902083145.6185972706.350299 Torsion1 1369.0761681254.8952451158.29561 Extension1 2316.0663582122.1998021958.976653

7. Barrel Band Boundary Conditions and Results A barrel band was applied at mid length of the barrel and it restrained the barrel in the all directions, except the longitudinal direction of the barrel. All three cross sections had the same mode shapes shown.

8. Weight Percent Reduction Over Frequency Percent Reduction Ratio The octagonal cross section full barrels are 10.58% lighter than the circular barrel. The same process was done for the 1 st frequency, giving a frequency reduction of 4.5%. Once the percent reduction in weight and frequency is known the weight reduction was divided by the frequency reduction to obtain the ratio of percent weight reduction to percent frequency reduction of 2.36. The triangular cross section full barrels reduce the weight of the barrels by 16.56%. The reduction in frequency is more significant at approximately 6.35%. However, the weight to frequency reduction is an average of 2.61 for the triangular cross sections compared to 2.32 for the octagonal cross section. Making the triangular barrel the more efficient for weight reduction.

9. BarrelWeight [lb]1st Frequency Weight Reduction Frequency Reduction Weight / Frequency Reduction Ratio 22 Inch Circular Cross Section Full Barrel7.2249675.297021N/A 24 Inch Circular Cross Section Full Barrel7.8838463.300788N/A 26 Inch Circular Cross Section Full Barrel8.5398853.951065N/A 22 Inch Circular Cross Section Tapered Barrel3.63236103.268716N/A 24 Inch Circular Cross Section Tapered Barrel3.961886.861819N/A 26 Inch Circular Cross Section Tapered Barrel4.2912473.862583N/A 22 Inch Octagonal Cross Section Full Barrel6.46171.92979410.57%4.47%2.364506827 24 Inch Octagonal Cross Section Full Barrel7.0488860.46007110.59%4.49%2.359983762 26 Inch Octagonal Cross Section Full Barrel7.6367651.38801610.58%4.75%2.226059688 22 Inch Octagonal Cross Section Tapered Barrel3.58408101.2288991.33%1.98%0.672908396 24 Inch Octagonal Cross Section Tapered Barrel3.9106885.0706361.29%2.06%0.625730405 26 Inch Octagonal Cross Section Tapered Barrel4.2372872.4716411.26%1.88%0.667735716 22 Inch Triangular Cross Section Full Barrel6.0293270.57477216.55%6.27%2.638723592 24 Inch Triangular Cross Section Full Barrel6.5774459.26179816.57%6.38%2.597016497 26 Inch Triangular Cross Section Full Barrel7.1255650.50624616.56%6.39%2.59375845 22 Inch Triangular Cross Section Tapered Barrel3.3313298.0361088.29%5.07%1.635633124 24 Inch Triangular Cross Section Tapered Barrel3.635282.373428.24%5.17%1.595368804 26 Inch Triangular Cross Section Tapered Barrel3.9390870.0012438.21%5.23%1.569797626 Weight vs. Frequency Reduction Comparison

10. Conclusions The expected outcomes of the study were met. It is evident that rifle barrels, regardless of their cross section, will have the same modal shapes. The variation in natural frequency was a result of the changes in area moment of inertia and the weight of the barrels. The triangular cross section barrels have the highest weight percent reduction over frequency percent reduction ratios. Regardless of the cross section all barrels the same modal shapes when the additional barrel band boundary condition was used.