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An Estimation of the Recoil of the Main Centerline Gun on the Galley
Reexamination of the Recoil in order not to Kill the Sixteenth Century Galley Crew
The maximum muzzle velocity


An Estimation of the Recoil of the Main Centerline Gun
on the Galley

The main centerline gun of the galley is on the bow where is so small space. This is 52-55 pounder weighing 5,500 pounds and the length is longer than 4 m.
I doubt whether such big gun could be safely operated by the gunners in the small space.
There is little knowledge about the main centerline gun. However, it is known that the gun barrel is installed in the wooden box and the box is in the track. The recoil is restrained by the friction between this box and the track and the tension of the breeching rope.
I estimated the main centerline gun how to move after shooting.

Movement of the Gun without Breeching


According to action-reaction low
equation 1

equation 2

m, v0 : The mass and the initial velocity of the cannonball
M, V0 : The mass and the initial velocity of the gun barrel

On the otherhand, the velocity V(x) of the gun in the wooden box at x accelarated with accelaration coefficient a is as follows.
equation 3

Since the gun in the wooden box is being decelarated by friction between the wooden box and track, a is described as follows.
equation 4

myu: the coefficient of kinetic friction
g : gravitational accelaration coefficient


Therefore, the velocity V(x) of the gun barrel is as follows.
equation 5

equation 6

Then, the energy of the gun carrage is described as next equation.
equation 7


The Calculated Results

The velocity and the energy of the gun barrel are calculated using equation 6 and 7. The results are shown in the following figures where m = 55 pounds, v0 = 1,200 ft/sec and 1,800 ft/sec6), and M = 5,500 pounds. The coefficient of kinetic friction of wood on wood myu is assumed to be 0.2 1).
velocityenergy

These figures indicate that the recoil distance without breeching is 3.4 m for v0 =1,200 ft/sec and 7.7 m for 1,800 ft/sec, respectively.
The initial kinetic energy of the gun barrel is 17,000 J at v0 = 1,200 ft/sec and 37,500 J at 1,800 ft/s.


Restraint by the Breeching

Since the distance from 3.4 m to 7.7 m is too long in the small space of the galley, the recoil had to be restrained by a breeching rope.

When a force F(x) is applied by the recoil of the gun to a rope and the rope stretches along x axis from x = 0 to l , the energy E is described as follows.
equation 8

Therefore, if we know the relation between the tension of the rope and stretch length, the energy can be calculated.
At the age of the Battle of Lepanto, hemp or Manila ropes were used.
There is little data about these ropes. But I found the data about a 20mm diameter Manila rope4) as summarized in the following table.

force(kg) force(N) stretch(%)
0 0 0
150 1470 5
1000 9800 15
2750 26950 break


As shown in this table, the stretch length is not linearly proportional to the force. This means that the stretched rope longer than several percent is not in the elastic region but in the plastic region. (Young's modulus of the hemp rope is 32 GPa 5). This datum indicates that the elastic region of the hemp rope is calculated to be shorter than 0.3% stretch.)
So, I made a following second order approximation equation using these numerical data.
equation 9

The curve made by this equation is shown in the following figure. In this figure, the stretch at the breaking point is an estimated value.
rope20mm

Since the equation 9 is for the 20 mm diameter rope, the equation for ropes with the diameter d mm is as follows.
equation 10

Therefore, using equation 8, the energy of the rope is as follows.
equation 11


The Calculated Result for a 5m-long Rope


According to "The Arming and Fitting of English Ships of War 1600-1815", "Normally a breech rope was three times the length of the gun." "Each end of this thick piece of rope was fitted by means of an eye-splice to a ringbolt on the side of the ship, either side of the gun. The rope passed through eyebolts on each side of the gun, and was spliced to the button of the cascable."3) Therefore, the rope length of each side of the gun is 1.5 times of the length of the gun. I think this length would be determined by the requirement to move the gun into the inside of the gunports and make the reloading easier.
In the case of the galley, there is no reason to recoil the gun for reloading.
Therefore, the length of the breeching rope is not required to be so long. I think that the shortest breeching rope is the most reasonable in order to operate the gun on the small space deck on the galley. Because the distance between the trunnions and the breech is about 2 m, the shortest rope length would be about 5 m.

I calculated the relation between the energy and the stretch length of the ropes with various diameter from 20 mm to 100 mm using equation 11. The calculated result is shown in the following figure.
rope_energy

This figure shows that the thicker rope is shorter stretch length.

The next figure is the dependency of the stretch length on the diameter of the rope where the cannonball velocities are 1,800 ft/s ( E =37,500 J ) and 1,200 ft/s ( E =17,000 J ).
diameter

This result indicates that 15% stretch ( 40% of breaking strain ) requires diameter of 50-70 mm (circumference of 6-9 inches).

Conclusion

The recoil distance of the 55-pounder gun weighing 5,500 pounds on the track without breeching rope is 3-8 m.
According to "The Arming and Fitting of English Ships of War 1600-1815", "it was said that a 32-pounder gun (without breeching) with a normal charge on a level platform would recoil 11 ft." 2)
Of course, the 32-pouder gun is on the carrage with trucks. So, my calculation results are a little longer. This means that the coefficient of kinetic friction of 0.2 I used might be too small.
In order to restrain this recoil action, a very thick rope with the circumference of 6-9 inches is needed. By using these thick ropes, the recoil distance restrained shorter than 40 cm (= 5 m x 0.15 / 2). However, the stretch length of 15% is equivalent to the 40 % of breaking strain and is not in the elastic region but in the plastic region.
Therefore, the several times of firing might break the breeching rope.

I think that the operation of the gun on the galley must have been very dangerous!!

But then, neither Robert Hook nor Isaac Newton were born in 16 th century. Such dangerous situation might be not surprising.

Reference

1)Frictional Forces:http://www.physics.unc.edu/labs/content/frictionalforces.html
2)Brian Lavery: The Arming and Fitting of English Ships of War 1600-1815, p139 (2000) Conway Maritime Press
3)ibid p141
4)Sailing Smacks-Rope:http://www.alberta-ck318.freeserve.co.uk/rope.htm#manila20
5)Ropes:http://www-materials.eng.cam.ac.uk/mpsite/short/OCR/ropes/default.html
6)Private comminucation from Prof. Dr. John Francis Guilmartin, Jr.




Reexamination of the Recoil in order not to Kill
the Sixteenth Century Galley Crew

Above estimation showed that the operation of the main centrline gun was very dangerous.
I used several assumed values in the above estimation. However, these values lack sufficient basis. Therefore, I will try to modify them in order not to kill the sixteenth century galley crew.

The initial velocity of the cannonball

The initial velocity of the cannonball has been assumed to be between 1,200 ft/s and 1,800 ft/s.
The shooting range R is calculated in the next equation where air resistance is neglected.
equation k1
h : the height of the gun above the sea-level
v0 : the initial velocity of the cannonball


The height of the gun would be reasonable to be about 1.5 m.
The calculated results are shown in the following table.
velocity (ft/s) range (m)
1,800 299
1,200 199
These ranges are reasonable.

The dash speed of the galleys is about 7 knots. When the galley is oared with full speed, it takes about a minute for 200-300 m. It means that there is no time for reloading the gun. Therefore the extremely strong rope might have not been needed.


The coefficient of kinetic friction of wood on wood

I assumed that the coefficient of kinetic friction of wood on wood was 0.2. There are many data about the coefficient of static friction that is 0.5. But there is a little datum about the coefficient of kinetic friction. Although, it is known that the coefficient of kinetic friction depends on the velocity and weight, there is no numerical datum about wood on wood. Therefore, the basis of this value of 0.2 is flimsy.
I calculated the dependency of the recoil distance on the coefficient of kinetic friction.
The calculated result is shown in the following figure.
the coefficient of kinetic friction vs stopping distance


I think that the recoil distance of the main centerline gun without breeching would be around 1 m from the recoil of the British 32-pounder2). In the case of the initial velocity of the cannonball of 1,800 ft/s, the coefficient of kinetic friction must exceed 1.
Because the coefficient of static friction is less than the coefficient of static friction (Amontons-Coulomb's Low), the coefficient of kinetic friction must be less than 0.5. So, the initial velocity of 1,200 ft/s, recoil distance of 1.4 m and the coefficient of kinetic friction of 0.5 are considered to be reasonable.
In these conditions, the velocity and the energy of the gun barrel is shown in the following figure.
the coefficient of kinetic friction of 0.5the coefficient of kinetic friction of 0.5


Restraint by the breeching

Because the frictional force, in the case of the coefficient of kinetic friction of 0.2, is far less than the tension of the rope, I neglected the restraint by the friction in the above calculation. But in high coefficient of kinetic friction, the restraining energy must be the sum of the frictional force and the tension of the rope. Therefore the total energy Etotal is described as the following equation from equation 7 and 11.
equation k2
Efriction : the restraining energy by the friction
Erope : the restraining energy by the tension of the rope

I calculated the energies in the case of the rope length of 5 m (Case A) and in another case of the rope length of 6 m (Case B). In the Case B, the gun barrel is restrained in the first stage up to 50 cm of the recoil only by the frictional force, then reatrained by the sum of the frictional force and the tensile stress of the rope.
The calculated result is shown in the following figures.
Case A Case B
Case A : rope length = 5 m Case B : rope length = 6 m
Up to 50 cm restrained only by friction and
then restrained by the friction
and rope tension.

The energy at the strech=0 in the Case B is the energy caused by the frictional force for the initial sliding distance of 50cm. That is 16,700-10,600=6,100 J .
The relations between the rope stretch and the rope diameter which are obtained from above two figures are shown in the following figure.

The safety factor of the rope is normally selected between 12:1(8% of breaking force) and 6:1(16%) in present day4). These values are the stretches of 6.5% and 9.5%, respectively.
If we select the safety factor of 6:1, the rope diameter of 80 mm is required in the Case A, and the diameter of 50 mm in the Case B. Although, the 80 mm diameter is too thick at this age, 50 mm (circumference of 6.2 inches) would be reasonable.
In the Case B using 50 mm diameter rope, the recoil distance is 0.5+6x0.09/2 = 0.77 m.
The kinetic energy of the gun barrel in this case is shown in the following figure.
The diameter of the breeching rope: 50 mm
The length of the breeching rope: 6 m
The energy decreases up to 50cm by the friction of wood on wood.
Longer than 50 cm, the energy is absorbed
by the sum of the friction and the tension of the breeching.


Conclusion: The condition not to kill the galley crew

Supposing that the initial velocity of the cannonball was 1,200 ft/s and the coefficient of kinetic friction of wood on wood was 0.5, the recoil distance of the main centerline gun without breeching would be 1.4 m. This value is consistent with the datum of the British 32-pounder.

If they used a 6 m long rope and up to 50 cm recoil was restrained only by the frictional force then restrained by the sum of the frictional force and the tensile stress of the rope, the 6.2 inches circumference rope could be used in the safety factor 6:1. In this condition, the repetitive firing would be possible and the recoil distance would be only 77 cm.

No trouble !!

The maximum muzzle velocity

From above reexamination, it has been found that the muzzle velocity of 1,200 ft/s is resonable and the recoil distance is 77 cm. This means that the track length of 2.8 m is enough.
On the otherhand, the track length is considered to be nearly equal to the length of the raised fighting platform above the guns. This length is about ten feets7). So, the track length might be about 3 m. In the case of track length of 3m, the maximum recoil distance must be 1 m.
I decided to search for the maximum muzzle velocity in the condition using 50 mm diameter rope, with the safety factor of 6:1 under the recoil distance of 1 m.
The kinetic energy of the gun garrel in the case of the cannonball velocity of 1,600 ft/s is shown in the next figure.

The orange curve shows the energy where the rope length is 6 m and up to 50 cm recoil is restrained only by the frictional force then restrained by the sum of the frictional force and the tensile stress of the rope.
The tensile stress of the rope oversteps the safety factor 6:1 at the recoil distance of 80 cm.
The green curve shows the energy where the rope length is 6.4 m and up to 70 cm recoil is restrained only by the frictional force. In this condition, the gun stops at the distance of just 1 m and the tesile stress is not beyond the safety factor.

Therefore I conclude that the maximum muzzle velocity of the main centerline gun is 1,600 ft/s.
The shooting range becomes about 270 m.

Reference

7)John Francis Guilmartin, Jr.:Gunpowder & Galleys, p223 (2003) Conway Maritime Press

Acknowledgement

I wish to thank Prof. Dr. John Francis Guilmartin, Jr., Ohaio State University, for his helpful discussions and encouragement.