Theory and computations about calculating recoil in firearms

 Theory and computations about calculating recoil in firearms

Abstract:

               Recoil can be expressed mathematically by the physical law of the conservation of momentum, the law of the conservation of momentum states that for two or more bodies in an isolated system action upon each other, their total momentum remains constant unless an external force is applied. In simply it is stated, the law says that for every action there is an equal and opposite action also the energy can neither be created nor destroyed.

Recoiling of a gun:

               Recoil is the backward momentum of a gun when it is discharged. The recoil caused by the gun exactly balances the forward momentum of the projectile and exhaust gases according to the Newton’s third law. The theory behind the recoil is when the gun is fired the projectile starts moving inside the gun barrel by the pressure created by the expanding gases, in this time the projectile moves forward and the case that holds the chemical gun powder travels the rearward by the expanding gas pressure, why it travels rearward because after ignition of the primer the burning take place inside the cartridge case and the expanding pressure expands inside the chamber in upward, downward direction and forward and rearward direction. There is no gap or openings provided in the upward and downward direction, only it can go either rearward or forward direction, so the expanding gas pushes the projectile through the opening provided in forward direction also it pushes the spent case through the opening provided in the back of the barrel. The backward pushing energy is transferred from the fire arm to shooter. In simple way after the firing the projectile goes forward and the gun comes the backward by the pressure of the expanding gas, (1) free recoiling of firearm is equal and opposite in direction to the momentum of the bullet (or shot charge/slug and wad column) and the propellant gases. (2) Felt recoil (also referred to as “Perceived” recoil) can also vary by modifying the rate of application of force by devices such as recoil pads or damping devices and the influence of action type. (3) Because the propellant gases are extremely difficult to weigh, for purposes of this application, the propellant gas weight will be equated to the powder charge weight. Generally, we can use the recoil force to cycle the weapon automatically until the ammo exist. (4) Recoil-operated weapons are those weapons that rely on recoil activity to operate the bolt and 2 related parts. (5) The bolt, locked to the barrel during firing, is released during recoil after the chamber pressure has become safe. The recoil force varies on the power of the ammunition and the mass of the firearm.

Free Recoil energy is also called as kinetic energy and it can be expressed as - FRE = ½MV2.

Where:
 M = Mass (or the weight of the firearm, including all attachments) 
 V = Velocity of the recoiling firearm

To find out the velocity we can use the mentioned formula:
                                   
Where:
 WE = Weight (in grains) of the projectile
 VE = Velocity of projectile in feed per second 
WPG = Weight of the propellent gases in grains 
VPG = Velocity of the propellent gases in feet per second (fps) 
7000 = Conversion factor for grain to pounds 
WF = Weight of firearm in pounds

If, therefore:
                                             
Then,
                                                               
    
Finally, we got,
               


The weight of the propellent gases WPG is equal to the propellent charge weight WPC. The formula given below is used to find out the recoil energy of the firearm.

                                                          
                                                                  




The weights of the gun, of the projectile and of the propellent gases or powder charge are easily determined. The effective velocity of the propellent gas, as much more difficult measurement, varies in general with muzzle pressure and projectile velocity. (6)


The following VPG should be used:
 VPG = VEf

High powered rifles VPG = 1.75VE 
Shotguns (average length) VPG = 1.50VE 
Shotguns (long barrel) VPG = 1.25VE 
Pistol & revolvers VPG = 1.50VE

If the calculations are made in metric system, the below formula can be used

                                          

Here mass to weight conversion factor is 2 x 9.8

Let us see a sample problem about calculating recoil, 

Problem no 1

How much Free Recoil Energy would be developed by an INSAS rifle weighing 4.6 Kilograms firing a 5.56 x 45 mm with bullet weighing 4.16 grams and loaded with 27 grains of powder and the velocity of the projectile is 890 mps.

Solution, 

Given data 

Weight of the fire arm in kilograms Wf = 4.6 kilograms 
Weight of the projectile in grams WE = 4.16 grams 
Weight of the propellent charge in grams WPC = 1.81 grams 
Velocity of the projectile in mpsVE = 890 mps 

The following velocity of the propellent gas VPG should be used VPG = VEF 
where for 

The f for high powered rifles is VPG = 1.75 VE 

Apply the given data into the given formula, 

why we choose metric formula because the data's are given in kilograms
                                         
                                             


FRE = (4.6 ÷ 2 x 9.8) x (4.16 x 890 + 1.81x 890 x 1.75 / 4.6) 2 
FRE = (4.6 ÷ 1.96) x (4.16 x 890 + 1.81 x 890 x 1.75 / 4.6)2 
FRE = (23.46) x (37024 + 28190 / 4.6) 2 
FRE = (23.46) x (65214 / 4.6 )2
FRE = (23.46) x (14.17) 2 
FRE = (23.46) x (200.7) 
FRE = 4.70 Kg mps The free recoil would be given in kilo-meters.


References: 
1.) Gun recoil technical SAAMI standards (sporting arms and ammunition manufacturing institute) 
2.) Gun recoil technical SAAMI standards (sporting arms and ammunition manufacturing institute) 
3.) Gun recoil technical SAAMI standards (sporting arms and ammunition manufacturing institute)
4.) Engineering design handbook automatic weapons, 1970 US Army material command. 5.) Engineering design handbook automatic weapons, 1970 US Army material command. 6.) Gun recoil technical SAAMI standards (sporting arms and ammunition manufacturing institute) 
7.) Bullet specifications are taken from TECHNOLOGY IN FOCUS paper about small arms from DRDO


 


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