43RD ANNUAL ARMAMENT SYSTEMS: GUN & MISSILE SYSTEMS CONFERENCE Gun-Barrel Vibrations of Rapid-Fire Medium Caliber Guns Prof Tom Dawson U. S. Naval Academy April 23, 2008 with support from NAVSEA Naval Gunnery Project Office and Naval Surface Warfare Center Dahlgren I. Background Barrel Vibrations Barrel vibrations can affect accuracy of both slow firing and rapid firing guns by causing positive or negative “muzzle jump”during projectile launch. Vibrations can arise from gravity-caused barrel droop or barrel curvature from manufacturing… or both. Possible enhanced effect with rapid-fire guns where vibrations from one round continue to exist and can be reinforced by subsequent rounds. Present Interest … in rapid-fire medium-caliber gun mounts and the effects of barrel vibrations on their accuracy. Motivation …generated by evaluation and selection of gun mounts for the new Littoral Combat Ships and for other similar Navy needs Early Navy Work Near-muzzle barrel vibrations of 3-in/70 Gun Mount during travel of projectile through the barrel -Single Firings Dahlgren Report of 1951 B. M. Gurley & S.E. Hedden Novel use of optical reflections and “Fastax” camera to measure rotations (slopes) of barrel section near muzzle during projectile launch …and Early Record of Barrel Vibrations Section Rotation Δφ = φ1−φ2 φ2 φ1 Early Army Work Dispersion of machine-gun fire as influenced by firing rate. Increased dispersion measured when firing rate near fundamental barrel vibration frequency - or twice that frequency. Basis for design criterion that: barrel frequency in cycles/sec should generally be 4 (or more) times the firing rate in rounds/sec. 1955 Report on Barrel Vibrations D. E. Wente R. L. Schoenberger B. E. Quinn First study of dynamic amplification of barrel vibrations from “tuned” firing rates Dynamic Amplification of Barrel (from 1955 Army Report) Dispersion Avg R Circle Barrel curvature from manufacture Previous Numerical Work on Barrel Vibrations Numerical studies were carried out at the Army’s Watervliet Arsenal during 1970’s (and onward) with attention restricted to barrel vibrations before projectile exit: No multiple firings. II. Computer Model Lumped-Mass Model & Mechanic 16 mass model mass mn Pn-1 Pn V’s & M’s depend on v’s at n-1, n, n+1 & projectile load P (if If P between If P between between n-1 n-1 and n n and n+1 and n+1) Pn = F Pn-1 = F Otherwise 0 Otherwise 0 Projectile Forces on Barrel Centrifugal Forces Simple Example Actual Case Friction Forces Typical Coeff of Friction μ ≈ 0.20 to 0.30 III. “Generic” 3in /60 Gun Mount (Firing 14 lb Projectiles) Generic Barrel Barrel (No attachments) 5’ 10’ Flexible section divided into 16 Rigid Flexible lumped masses. Accuracy checked Assumed Fixed by increasing Flexible Length Inside Dia = 3 in number to 32 and Outside Dia = 4 in then to 48 … vibration frequencies as shown in the ω = 62.0 rad/sec = 9.87 cycles/sec 1 following ω2 = 387 rad/sec = 61.6 cycles/sec etc Assumed Projectile Velocity in Barrel 3500 3000 S 2500 Muzzle (15 ft) 2000 1500 From Leduc Formula VELOCITY (F/S) assuming 50% of aS 1000 muzzle velocity V = 2.25 ft achieved at first 15% + bS 500 of barrel length 0 0246810121416 DISTANCE ALONG BARREL S (ft) Demonstration of Adequacy of Lumped-Mass Model - Idealized Case- Negligible Friction between Spinning Projectile and Barrel Convergence with number of mass elements 0.014 0.012 VERTICAL DISPLACEMENTS 0.01 (Relative to static values) 0.008 0.006 Distance X 0.004 0.002 32 Mass Model 16 Mass Model Displacemnet (in). Displacemnet 0 48 Mass Model -0.002 Friction Coeff μ = 0 -0.004 Projectile Location at X=5’ -0.006 01234567891011 Distance X along Barrel (ft) Adequacy of Computer Solution 0.014 P 0.012 VT 0.01 0.008 Computer Simplified Case Program Constant Projectile Force 0.006 P=2000 lbs Constant Projectile Velocity 0.004 From V= 2500 ft/sec Analytical Friction Coeff μ =0 0.002 (Exact) Theory Vertical Deflection (in) . (in) Deflection Vertical 0 Instantaneous Projectile Position -0.002 at VT=5' -0.004 -0.006 024681012 Distance along Barrel (ft) Detailed Results -Actual Case- First-Round Barrel Response Free Vibrations following First Round Response Characteristics after Multiple Rounds First-Round Barrel Response Muzzle Deflections vs. Time 0.02 Muzzle Deflections (relative to static values) 0.01 Friction First Round μ = 0.3 0 FV Friction -0.01 μ =0.2 FH Vertical Displacement (in). Displacement Vertical -0.02 f= μ FH -0.03 00.511.522.533.54 Time (msec) Dynamic Section Rotation Comparison with 1951 Dahlgren Data 0.002 0.0015 Theory (μ = 0.2) 0.001 Data Round D 0.0005 0 -0.0005 Theory (μ = 0) (solid line) Optical Measurements -0.001 B. M. Gurley & S. E. Hedden Time Normalized to Generic NPG Report 804 3" / 60 -0.0015 Relative Section Rotation (rad) Dahlgren (1951) Rotations as Measured -0.002 00.511.522.533.5 Time (msec) Free Vibrations following First Round Free Vibration of Muzzle (note check of analytical solution & damping value) 0.6 Muzzle Vibration about Static Position 0.4 after First-Round Projectile Exit 0.2 0 Displacement (in) -0.2 Moderate Damping -0.4 Free-Vibration Analytical Solution K = 0.03 Computer Solution -0.6 0 100 200 300 400 500 Time (msec) Response Characteristics after Multiple Rounds Dynamic Amplification 1.5 Muzzle Deflection after 4 Round Burst 113 1 Relative to Static Values 124 0.5 81 71 0 -0.5 Damping K = 0.03 Friction Coefficient 0.30 -1 Relative Displacement (in). Displacement Relative 3''/60 Generic 126 -1.5 60 70 80 90 100 110 120 130 140 Firing Rate (rds/min) Barrel Deflections (note continuing input of energy 1 Actual Deflections 0.8 Firing Rate 113 rds/min 0.6 Moderate Damping Begin Round 5 K = 0.03 0.4 Friction Factor μ = 0.30 Begin Round 4 0.2 Begin Round 3 Deflection (in) Deflection 0 -0.2 Initial Static Droop -0.4 024681012 Distance along Barrel (ft) IV. Application to USN Mark 75 (80 rds/min) Mark 75 3"/62 (80 rds/ min) USS Curts FFG 38 Barrel Details –Mark 75 (3in/62) Gun Mount Muzzle Break 10 ft (effective) Barrel & Water Jacket Bore Evacuator & Lock Nut Mark 75 - Idealized Barrel Description Barrel (with attachments) 62’’ 120’’ Rigid Flexible 97 lbs 56 lbs Gas Evac & Lock Muzzle Break Nut & ½ Al Cover & ½ Al Cover Modal Frequencies Flexible Length ω1 = 41 rad/sec (6.5 cycles/sec) Inside Dia = 3 in ω2 = 276 rad/sec (44 cycles/sec) Outside Dia = 4 in etc Leduc formula for projectile velocity Variable Firing Rates Normal (Bell-Shaped) Distribution 0.12 . 0.1 Average 80 rds/min 0.08 Std Dev = 4 rds/min Normal Distribution 68 % 0.06 ±4 rds/min 0.04 0.02 95 % Probabiliy Density (1/rds/min) Density Probabiliy ±8 rds/min 0 -14-12-10-8-6-4-20 2 4 6 8101214 Differrence between Firing Rate & Average Firing Rate (rds/min) Dispersion: Theory vs. Measurement 8 +8´ 2020 Round Round Bursts ModerateDamping Damping (K =0.03) (K =0.03) Target Distance Distance 1500' 1500' 6 FrictionFriction Coeff (μ = 0.20)(μ = 0.20) Std Dev 4 R(Theory) = 2.91’ USN R(Data) = 2.93’ 2 .Mark 75 RR = = 2.93' 2.93' =2=2mrad mrad …3”/ 62 Gun Mount 0 0 -8 -6 -4 -2 0 2 4 6 8 ….firing ….14 lb -2 Projectiles -4 Mean Firing Rate TwoTwo Bursts Burst --Theory Theory 80 rds/min Std Dev 4 rds/min TwoTwo Bursts- Burst –Dahlgren Dahlgren Data-6 Data Freq Ratio = 4.9 -8´ -8 -8´ 0 +8´ V. Application to Study of Oto Melara 76 mm/62 SR (120 rds/min) Oto Melara 76mm/62 SR (Super Rapid Gun Mount Mark 75 vs 76mm SR Mark 75 (∼1970) SR(∼2000) Firing Out of Battery In Battery Mark 75 Avg Firing rate 80 rds/min 120 rds/min Accuracy (10-rd burst) ∼1.9 mrad 0.30 mrad* *Reported on web page 76mm SR with standard shield Extracted From Web Page: Italian 76mm/62 (3″)… The SR is an improved faster-firing version of the Mark 75…. Accuracy improved partly by reducing the weights of the moving parts. Claims are that these changes have reduced the radial-error standard deviation values to less than 0.3 mrad for 10-round burst Examination with Theory What if firing rate of Mark 75 is increased 50% to 120 rds/min? See table. Dispersion increased from about 1.9 mrad at 80 rds/sec to about 6.5 mrad at 120 rd/min (for 10-Rd Bursts) Mark 75 (modified) Firing Rate Radial Dispersion* (rds/min) (Std Dev in mrad) 80 1.9 mrad 120 6.5 mrad * 10 Rd Bursts (avg of 4) What if weights of gas evacuator & muzzle break are then reduced by 50%? (Avg of data from generic and Mark 75) Dispersion reduced from about 6.5 mrad to about 4.5 mrad (for 10-rd bursts) Conclusion: Cannot achieve reported accuracy for 120 mm SR with only a reduction of add-on weights of Mark 75 when modified for 120 rds/min What if increased damping of barrel vibrations? See graph below. Dispersion (dashed line) reduced from about 4.5 mrad to about 0.5 mrad for 200% increase in damping. 7 Mark 75 6 10 Round Bursts (Avg 120 rds/min) 5 Avg of 4 Bursts Average Curve for (circles) 4 (50% reduction in add-on Dispersion (mrad). Dispersion weights) 3 Generic 3''/60 ( 120 rds/min) 2 0.03 Avg present 1 Mark 75 Std Dev of Radial 0 0.02 0.04 0.06 0.08 0.1 0.12 Damping Coefficient VI.