Computing Gun Impulse/Recoil Forces

(Created about 2013~)
(Updated 29 June 2016)

General References on Gun Impulses:

Cutting Down the Kick: Understanding and Managing Large Caliber Recoil, April 2015 (2.8~ MB PDF)

The Designer’s Dilemma – Recoil: What to do with it? (RTO-MP-AVT-108) (768~ KB PDF)

A Fire Out of Battery Tank Gun: Theory and Simulation (829~ kb PDF)

Design Tradeoffs For A Very Lightweight 155mm Howitzer For The U.S. Army Light Forces (1.7~ MB PDF)

AMCP 706-251 Engineering Design Handbook Guns Series, Muzzle Devices (May 1968) (5.7~ MB PDF)

Notes on Gun Impulse/Recoil Forces:

Reducing the average recoil force via increasing the recoil distance lowers the stresses on the turret structure and the rest of the vehicle, but it does not reduce the firing impulseand it is the firing impulse that governs whether a gun can be put in a vehicle without suffering:
As a rule of thumb, the comfortable firing impulse for a vehicle mounted weapon is generally held to be between 0.5 to 0.7 kNs/tonne. The Ogorkiewicz Limit is 0.9 kNs/tonne. Anything beyond that will be extremely uncomfortable for the crew; with a high chance of breaking various components – the M551 Sheridan exceeded 1 kNs/tonne when it fired it's 152mm gun with HEAT rounds.
Firing Impulse can be reduced through the use of muzzle brakes, which create an impulse in the opposite direction via deflecting the flow of propellant gases through the muzzle. Unfortunately, the higher efficiency muzzle brakes gain their efficiency at a price – through tremendous overpressure.
In some of the early tests with the Stryker Mobile Gun System, the overpressure from the muzzle brake was enough to shatter the driver's vision blocks. Additionally, infantry providing close support to the vehicle might also not like it when the gun fires and the muzzle brake can kick up a large amount of dust during firing, obscuring the target in your sights, or providing an easy location marker for the enemy.


Overpressure Diagram for 105mm M102 Howitzer
with and without a medium efficiency muzzle brake

Reference: The Designer’s Dilemma – Recoil: What to do with it? (RTO-MP-AVT-108) (768~ KB PDF)

Reference Gun Impulses

Weapon

Weight (kg)

Impulse (N-Sec)

N-sec/kg

155mm M1917 Howitzer

3,750

30,400

8.11

155mm M1918M1 Gun

11,300

49,660

4.39

155mm M1 Gun

12,700

57,630

4.54

155mm M1/M114 Howitzer

5,765

38,170

6.62

155mm M198 Gun-Howitzer
(Medium Efficiency Brake)

7,163

46,260

6.46

155mm M777 Gun-Howitzer

4,218

46,260

10.97

155mm XM282 Howitzer
(M109A6 SPH / High Eff. Brake)
(Loser for M109A6 Upgrade)

59,508 (without brake)
48,139 (with brake)

Reference: The Designer’s Dilemma – Recoil: What to do with it? (RTO-MP-AVT-108) (768~ KB PDF)

Calculating Average Force Exerted on Gun Mounting (Metric)

FRecoil = I2 / (2 * MRecoil * DRecoil)

Where:

FRecoil = Average force on the gun mounting in Newtons (N)
I
= Firing Impulse in Newton-seconds (Ns)
MRecoil = Mass of recoiling parts in kilograms (kg).
DRecoil = Recoil distance in meters (m).
Reference:
Technology of Tanks by Richard M. Ogorkiewicz

EXAMPLE: A modern 105mm gun has a firing impulse of 16,710 Newton-Seconds (Ns), a recoiling mass of 1,350 kg and a recoil distance of 0.280 meters. What is its average recoil force?

16,7102 / (2 * 1,350 * 0.280) = 369,344.05 Newtons (369.34 kN)

Computing the Firing Impulse of Guns with Muzzle Brakes

IBraked = I * SQRT(1 – EBrake)

Where:

IBraked = Firing Impulse on the gun with a muzzle brake in Newton-Seconds (Ns)
I = Firing Impulse on the gun without a muzzle brake in Newton-Seconds (Ns)
EBrake = Efficiency of the Muzzle Brake

Notes: Generally, Muzzle Brake efficiency has been between 25-40%, but purpose built lightweight guns have had even higher efficiencies; with the Cockerill 90mm Mk7 having an efficiency of 55%, while the MECAR 90/46 has an efficiency of 70%.

Reference:
Technology of Tanks by Richard M. Ogorkiewicz

EXAMPLE: Our gun has a firing impulse of 50,000 newtons. If we fit it with a muzzle brake 50% efficient, what would it's firing impulse be?

50,000 * SQRT(1 – 0.50) = 35,355 newtons


Muzzle Brake Efficiency
From: AMCP 706-251
Engineering Design Handbook Guns Series, Muzzle Devices (May 1968) (5.7~ MB PDF)

Calculating Firing Impulse (Metric)

I = (MProj * VProj) + (MCharge * VCharge)

Where

I = Impulse in Newton-Seconds (Ns)
MProj = Mass of projectile in kilograms (kg).
MCharge = mass of propellant charge in kilograms (kg).
VProj = Muzzle velocity of projectile in m/sec.
VCharge = Average velocity of propellant gases in m/sec.
Reference:
Technology of Tanks by Richard M. Ogorkiewicz

EXAMPLE: A modern 105mm gun is firing a projectile massing 5.82 kilograms at a muzzle velocity of 1,500 m/sec, with a charge weight of 5.7 kilograms and a charge velocity of 1,400 m/sec. What is it's impulse force?

(5.82 * 1500) + (5.7 * 1400) = 16,710 Newton-Seconds (16.7 kNs)

Computing Charge Velocity (Experimental)

VCharge = VProj / β

Where

VCharge = Average velocity of propellant gases in m/sec.
VProj = Muzzle velocity of projectile in m/sec.
β = Coefficient. Varies according to muzzle velocity. Known coefficients according to the Rheinmetall Handbook of Weaponry are:
400 m/sec = 3
800 m/sec = 1.5
1,400 m/sec = 1.0

Extrapolating a power-law based on those three points gives us the following equation for β:

β = 576.17 * Vproj-0.877

and the following coefficients:

100 m/sec = 10.15
200 m/sec = 5.52
300 m/sec = 3.87
400 m/sec = 3
500 m/sec = 2.47
600 m/sec = 2.1
700 m/sec = 1.84
800 m/sec = 1.63
900 m/sec = 1.47
1000 m/sec = 1.34
1100 m/sec = 1.23
1200 m/sec = 1.14
1300 m/sec = 1.07
1400 m/sec = 1
1500 m/sec = 0.94
1600 m/sec = 0.89
1700 m/sec = 0.84
1800 m/sec = 0.8
1900 m/sec = 0.76

Calculating Firing Impulse Force, Rheinmetall style (Metric)

I = (MassProj * VelocityMuzzle) + (MassCharge * (VelocityMuzzle * β))

Where

I = Impulse in Newtons
MassProj = Mass of projectile in kilograms
MassCharge = mass of propellant charge in kilograms
VelocityMuzzle = Muzzle velocity of projectile in m/sec.
β = Coefficient. Varies according to muzzle velocity. Typical Coefficients are:
400 m/sec = 3
800 m/sec = 1.5
1,400 m/sec = 1.0
Reference:
Rheinmetall Handbook of Weaponry

EXAMPLE: A modern 120mm gun is firing a projectile massing 7 kilograms at a muzzle velocity of 1,400 m/sec, with a charge weight of 10 kilograms. What is it's impulse force?

(7 * 1400) + (10 * (1400 * 1.0)) = 23,800 newtons