Algebraic Firing Solution

[TorpX]

An alternate method for torpedo firing solution:
I made the computations for this method some time ago when I was playing SHCE. I've used it quite a
lot in that game, but also in SH 3 and SH 4. It will work with any SH game and does not rely on any game
exploit. I became interested in this after reading the S class boats did not have any TDC, and had to make
use of other methods for obtaining a torpedo firing solution. I know others have made use of similar
solutions, so I thought I'd post my version.


For those who have only used the in-game TDC, this type of method is fundamentally different. It
involves calculating a firing solution for the target as it reaches a discrete point, whereas the TDC
allows continuous updates of the firing solution. This means with the USN TDC, one can launch at any point;
the sub can change course and wait until a more favorable track is obtained, increase speed, etc., etc.
None of this "breaks" the solution. Not so with this method. It is like the difference between methods of
lead in duck hunting; continuous tracking allows the shot to be made any time after proper lead is
established, whereas "trapping" (aiming at a point ahead) requires the shot to be made at the moment the
target reaches the calculated point.


So, if this method is more limited, why use it? This is a fair question. The main reason is
realism. The S class did not have any TDC and had to use some sort of calculation method, such as the Mk
VIII angle solver. Unfortunately, we don't have this in the game. [It should be noted here that the
difficulties involved in using the Mk VIII angle solver, lead to the development of the TDC, and was a big
step forward in fire control.] The other reason I use this method is just because it is both challenging
and interesting (at least to me), to do it this way. It gives one a deeper appreciation for the Torpedo Data
Computer.


This method requires the gyro mechanism be set to fire straight ahead (or behind), and a
longitudinal spread be used, if any.

The method is simply four equations for calculating the following:

1. Lead Angle
2. Time to Position
3. Track Range
4. Torpedo Run Time

Some would consider only the lead angle to be necessary, but I feel that the track range and
torpedo run time should be known in any proper method. The time to position is not necessary, but is
helpful in practice. Knowing this allows one to lower the periscope, instead of leaving it up waiting for
the target to reach the firing bearing. The track range not only provides a run time for the torpedos, but
also gives one an idea of the quality of the set-up, and allows a change in plan to obtain a better range,
if that is indicated. Of course, the torpedo run time provides an indication of whether the torpedos hit
the target, or just exploded somewhere out there. (It also tells you how good your plot and data were.)


Below are the equations used:




To utilize the method, you must have all the usual elements of firing data, the range, the bearing
angle, the track angle, the target speed, and torpedo speed. Many will notice I didn't mention the angle on
the bow. This is because it is implicit in the track angle. If you know the track angle, you can get the
AoB. If you know the AoB, you can get the track angle. They are really two halves of the same coin. I
originally derived the equations in terms of the AoB, on the theory that I might want to make an attack
without plotting, in an emergency, but in practice, I did not use them this way. I played around with them,
trying to put them into a more intuitive or simpler form. Finally I altered them to use the track angle,
since USN documents use this more often. One form is not really better than the other, but I think using
the track angle is a little less confusing than using the AoB.


It doesn't matter how you obtain the data, as long as you have it. Naturally, the more accurate it
is, the better your firing solution will be. Garbage In Garbage Out, as they say.


Here is an example to illustrate:
[Note that the diagrams are not to scale, and are just to help illustrate the concept.]




In this example, the final observation is made with the sub at U0 and the target at T0. The
calculations are made with the Range (T0 to U0), the Track Angle (Angle T3_T2_U0) and the Bearing Angle
(Angle T2_U0_T0). The torpedo(s) are launched when the sub is at U1 and the target at T1. They impact with
the target at T2.


Let's say you have been tracking a target and have good estimates of it's speed and course. You
have turned on to your attack approach course and do not intend to change speed. It is important to
understand that the calculated firing solution is rendered invalid if you change course or speed before you
launch your torpedos. The same goes, if the target should happen to change course or speed. You get from
your final observation:

Torpedo Track Angle__________________________ 135 deg.
Bearing Angle*________________________________ 94 deg.
Range to Target____________________________ 2,600 yds.
Target Speed___________________________________ 9.5 kts.
Sub Speed______________________________________ 2.0 kts.
Torpedo Speed_________________________________ 30.0 kts.

*I deliberately used the term Bearing Angle instead of relative target bearing. Relative Target
Bearing, 266 deg., Angle T2_U0_T0 (measured clockwise) cannot be used, instead the Bearing Angle, Angle
T2_U0_T0, (measured counter-clockwise) must be used, (i.e. 360 - 266 = 94). If in the above example, the
target was to starboard (right), the Relative Bearing 94 deg. could be used without alteration. But, when
the target is to port (left)
, the Relative Bearing of 266 deg. will not give correct answers. This is
because of the sine function. In this case the bearing angle is 360 - Relative Bearing, and the Firing
Bearing will be 360 - 16.1 deg. instead of 16.1 deg.

Doing the calculations yields:
Lead Angle____________________________________ 16.1 deg.
Time to Position_____________________________ 582 sec.
Track Range________________________________ 1,757 yds.
Torpedo Run Time_____________________________ 104 sec.

Using this solution for an attack, one would turn the scope to 343.9 or 344 deg. bearing and launch
on the target when it reaches that point. Spread the torpedos by launching on different sections of the
ship as they pass by. The time to position tells you it will take 582 sec. to reach that bearing. In
theory, you could submerge and fire at that time, without any visual observation, but I have never tried to
do it this way. Using the scope with a visual reference reduces the effects of errors which are likely to
be present in the data. If the time to position is significant, as in this case, one can make use of the
time by refining the estimates of speed and course, or plotting out alternate attack approaches, rechecking
calculations, etc. If the target reaches the firing bearing too early, or late, it is an indication that
either the calculations or firing data are in error.


Back to our example. Let's say you don't like the set-up and want to get closer. Using all the
above data, except the sub speed, which we increase to 11 kts., and doing the same calculations we get:

Lead Angle____________________________________ 16.1 deg.
Time to Position_____________________________ 858 sec.
Track Range________________________________-2,897 yds.
Torpedo Run Time_____________________________-172 sec.

What! Negative range? Negative time? I know what your thinking. What kind of torpedo voodoo is
this!? Before you turn off your computer in disgust, let me explain. It turns out that if the sub in
question goes on this track at 11 kts., it will cross the target's track in front of the target. In other
words, there is no bow shot in this set up. (If you were plotting this out in the game, you would probably
notice this before you did the calculations. If your just pulling numbers out of your hat, you probably
would not. Don't ask me how much time I wasted because of this.) However, this brings up another question.
What if we want to make a stern shot? Can the equations be used to do this? The answer is yes, but we must
make a few changes in our procedure.


First, we must cross the target's track before we can get the proper data. The geometry changes
when we cross the track, so the initial data will not give us the correct answers. We can make our final
observation the moment we cross the track, if we want to, or any time after. As a side note, I should point
out that the collision course speed in our example is 6.25 kts. Below this speed, the target will cross in
front of the sub, above this speed our sub will cross in front of the target. We should choose a speed
significantly above or below this value.


Collision Course Speed = Vt * [sin (alpha - theta)/ sin theta]


This is what the set will look like for a stern shot:




This is the same original course and target data, but with the tracks extended.


So, if we do the math, we find with our first example that at 11 kts we will cross the track after
390 seconds. As I said we can make our final observation now or anytime after. At 490 sec. after our
initial position we find:

Torpedo Track Angle____________________________ 45.0 deg.
Bearing Angle*_________________________________ 28.6 deg.
Range to Target_____________________________ 1,549.6 yds.
Sub Speed______________________________________ 11.0 kts.
*Relative target bearing is 208.6 deg., we must use 28.6, (208.6 - 180 = 28.6).
All other data is the same as before.

Here are the changes we must make in our procedure:

We must consider the geometry from the stern of the boat. (Pretend the stern is the bow here.)
1. torpedo track angle changes as we cross the track. The geometry is such that the track angle for
the stern shot is 45 deg.(angle T3-T2-U0). This is the supplementary angle to the original track angle,
(i.e. 180 - 135 = 45).

2. The bearing angle as taken from the stern. (Angle T2_U0_T0).
3. We must use the negative value of sub speed, - 11.0 kts. (We are moving away from the target's
track now.)
4. The lead angle obtained is also from the stern. (Angle T2_U1_T1, firing bearing = 180 +/- lead
angle.)

Here is the data for our calculations:

Torpedo Track Angle___________________________ 45.0 deg.
Bearing Angle*________________________________ 28.6 deg.
Range to Target____________________________ 1,549.6 yds.
Target Speed___________________________________ 9.5 kts.
Sub Speed_____________________________________-11.0 kts.
Torpedo Speed_________________________________ 30.0 kts.

Crunching the numbers, we obtain:
Lead Angle____________________________________ 10.4 deg.
Time to Position_____________________________ 117 sec.
Track Range________________________________ 1,342 yds.
Torpedo Run Time______________________________ 79 sec.

So, with this we rotate our scope or TBT to 190.4 deg (180 + 10.4) and wait for the target to cut
across our sight.


I tried to use precise figures in the examples so anyone who wishes to can make a scale drawing or
otherwise check the math. If there are any doubters (I suspect there will be a few), the easiest way to
check is to consider the length of the relevant triangles and use the Law of Sines to calculate the
numbers.


Some might consider a method which requires calculations or this sort to be cheating. But since we
have no Mk VIII angle solver, we are left with the choice of either using the game TDC (obviously not
ideal), or some method of our own. This method brings limitations which are, at least, somewhat comparable,
to what one would have using a Mk VIII angle solver.

[Claves_Mortis]

So gorgeous!

A good explanation with neatly arranged pictures, thanks for your afford and time to share.

[jcope]

This is great. I enjoy playing this way as well. I have done all the math and created tracking spreadsheets in my iPad Numbers app to do similar things.

The first I have is a target tracking spreadsheet into which I feed observations on target and own sub position, and it tells me the target's course and speed. Only plotting own sub position is required.

The second spreadsheet I have calculates optimal (minimum time) intercept course for the target or any point I choose relative to the target, using only results calculated by the first spreadsheet and my chosen speed to intercept. It also tells me if an intercept is not possible at a given speed.

Using the two, you could actually intercept and sink a target in the least amount of time without ever plotting the target's position. I've never tried to intercept "blind" like that, but you could. Maybe I'll try it and record a video.

And with your equations you could also set up your firing solution.

I could try to figure out a way to share the spreadsheets, or even just the math. I could also put your equations into one of the tabs.

What would be extremely cool is to find a way to auto-plot the data entered into the first spreadsheet. I find the plotting to be tedious and it adds a fair bit of error into the solution.

Also: in a third spreadsheet I've attempted the math to solve for target's course using the "three bearing method", but with the advantage that you don't have to take the bearings at fixed time intervals which gives you the chance to be more precise and get a better solution. You do have to be stationary though, which I find is rarely practical. I'd like to do the "moving sub" version, and also add the fourth bearing to determine target position and speed as well, but I don't know the math for those yet.

[Dorjun Driver]

Very nice.

[Onkel Neal]

The time to position is not necessary, but is
helpful in practice. Knowing this allows one to lower the periscope, instead of leaving it up waiting for
the target to reach the firing bearing. The track range not only provides a run time for the torpedos, but
also gives one an idea of the quality of the set-up, and allows a change in plan to obtain a better range,
if that is indicated.

That's supremely important, and I always read accounts where the skipper would determine the initial setup and on subsequent observations, determine how true the solution was, and make adjustments to fine tune it in.

Very well done, I rated this thread as "Excellent"

[TorpX]

Thank-you for the compliments.

To be honest, I wasn't sure if this would interest anybody, or if it would be in the catagory of "we already knew this".

[msumpsi]

That is a beautifull piece of mathematical work done there, and sure that these were the kind of things the guys that worked on the TDC were doing.

But honestly, do you think the skippers had those equations in mind in the middle of a real patrol and playing with their lives? To start with, you need a calculator to solve the angle equations, which were non existence, or the TDC which was their calculator. It you want to be realistic and not use the TDC in the pigboats, then just use stadimeter, and guestimate angle on the bow and speed, or use the clock, the stadimeter and the nav map to get the measures (assuming playing with map updates off).

In any case, is a very good work of maths...hats off. I do appreciate your effort and mathematical skills.

[Dorjun Driver]

Originally Posted by msumpsi

...

But honestly, do you think the skippers had those equations in mind in the middle of a real patrol and playing with their lives?

For some of us, the "equations" simply play out in our minds eye. No numbers, just pictures.

A bit of a drag, actually.

[jcope]

Originally Posted by msumpsi

That is a beautifull piece of mathematical work done there, and sure that these were the kind of things the guys that worked on the TDC were doing.

But honestly, do you think the skippers had those equations in mind in the middle of a real patrol and playing with their lives? To start with, you need a calculator to solve the angle equations, which were non existence, or the TDC which was their calculator. It you want to be realistic and not use the TDC in the pigboats, then just use stadimeter, and guestimate angle on the bow and speed, or use the clock, the stadimeter and the nav map to get the measures (assuming playing with map updates off).

In any case, is a very good work of maths...hats off. I do appreciate your effort and mathematical skills.

I would assume there were tables based on the math for quick lookup. Because it would be impossible to solve the equations by hand quickly enough to provide useful results.

There were also circular and linear slide rules for solving math problems, not as quickly as a calculator can. But in the hands of an expert not a lot slower.

[TorpX]

Originally Posted by msumpsi

But honestly, do you think the skippers had those equations in mind in the middle of a real patrol and playing with their lives? To start with, you need a calculator to solve the angle equations, which were non existence, or the TDC which was their calculator.

No, I certainly didn't mean to imply that RL skippers were doing geometry problems like this in the middle of an approach. But before they had the TDC, they had the Torpedo Angle Solver Mk VIII. (see link below) I'm sure the USN had instructors or mathematicians at the Naval Academy, who worked this out and used the knowledge to formulate attack procedures. I'm pretty sure they knew all this, and more, before the Mk VIII was developed. Otherwise, how could they have designed it?

Mk VIII Angle Solver:
http://www.hnsa.org/doc/banjo/index.htm

I don't know nearly as much about the details of USN torpedo attack procedures as I would like to, but I would guess that once they had the necessary data, they could calculate the lead angle and such in 1 to 2 minutes. Of course, if the target zigged in the meantime, they would have to start over, with no guarantee of success. With the TDC, the procedure would be much faster (obviously).

from jcope:

I would assume there were tables based on the math for quick lookup. Because it would be impossible to solve the equations by hand quickly enough to provide useful results.

I don't know what might have been used before the Torpedo Angle Solver Mk VIII. They might have used some sort of tables or the like. But you make a good point; the doctrine and procedures would have to allow an attack to be delivered in a reasonable time frame to be of any use. The limitations (or shortcomings of) the Mk VIII led to the development of the TDC.

[troopie]

It'd be interesting to know what methods were used in WWI.

Edit: Have copy 'n' pasted your OP to a word document for future personal referance, hope u don't mind. Thanks for the great post!

[irish1958]

Originally Posted by troopie

It'd be interesting to know what methods were used in WWI.

Edit: Have copy 'n' pasted your OP to a word document for future personal referance, hope u don't mind. Thanks for the great post!

Check out: http://www.subsim.com/radioroom/showthread.php?t=156161
I think they just pointed the boat at the target, got as close as they could and hoped for the best.

[TorpX]

Originally Posted by troopie

It'd be interesting to know what methods were used in WWI.

Edit: Have copy 'n' pasted your OP to a word document for future personal referance, hope u don't mind. Thanks for the great post!

Anyone is welcome to copy, save, and make full use of all this.

As far as WWI (or earlier) fire control, I don't really know. I looked at some of the documents at Hansa.org about early USN torpedos. The first torpedo that had gyro guidance listed was the Whitehead Mk 3 which entered service ca. 1898. IMO, gyro guidance implies a certain level of sophistication in fire control. Really, the building of any auto-motive torpedo is something of an engineering feat. They were very complex pieces of ordnance. The Mk 10 could turn 90 degrees. I don't think they would've bothered with this if they had no way to compute a firing solution.

[I will note here that in the later 1800's, sophisticated methods were developed for controlling coast artillery, and computing trajectories of artillery in general. It was a period of rapid technical progress.]

[Singed]

Beautiful, going to play with this and see if I can add it to my bag of tricks.

[twm47099]

Originally Posted by msumpsi

... To start with, you need a calculator to solve the angle equations, which were non existence, or the TDC which was their calculator.

An awful lot of scientific and engineering math and trig was done before calculators and computers.

Slide rules were invented in the 17th century. One of the first things we were taught in engineering school (1960's) was how to use a slide rule and how to develop nomographs and other graphical methods for specialized calculations. When I started work after high school as a CO-OP at a Navy lab much of what I worked on were graphical aids. Most of the older engineers working their used custom graphical aids instead of slide rules or mechanical calculators for their routine calculations. Specialized applications (like artillery) had special slide rules. After a few months using a slide rule, complex equations could be performed very quickly. Way-back-when there were some demonstrations where a very experienced user would perform a series of complex calculations faster than an experienced calculator user. His answer was only accurate to a few decimal places, while the calculator was accurate to 10 or more, but the slide rule accuracy was more than sufficient for a targeting solution. Tables of logs, and trig functions were common place and used for higher precision calculations where time wasn't as critical. One of the jobs of the Navy engineers and scientists was developing these mechanical/graphical "calculators" for their weapons systems.


The 'banjo' and is-was are examples of slide rule/graphical aid for doing vector analysis that worked, although if the setup wasn't steady state the results could be inaccurate. The TDC and PK allowed the automatic input of own-ship course and speed which in turn allowed accurate solutions while maneuvering (as long as the target remained steady state.)

A few years ago I bought a used copy of the Navy Basic Fire Control Mechanisms tech manual from 1944. The way the different calculator mechanisms (+-*/, integration and differentiation, trig functions, range rate, etc) were implemented is fascinating. One method for generating trig (and other) functions was a groove in the shape of the function cut into the surface of a disk. As the disk was rotated to the proper "x" value, a follower riding in the grove moved a rod proportional to the "y" value that was then used by some other components, essentially just replacing slide rules and dials with very precision mechanics and multiple simultaneous inputs.

One thing that does surprise me is that these worked in submarine operations, particularly after a depth charging. For example, some of the integrators are very complex with motions being transferred through balls rolling on other balls, rolling on a flat plate and a cylinder. It seems like one good shock would bend something or misalign something very easily and one spray of salt water would start corrosion that would put the precision components out of action pretty quickly. I assume maintenance of the TDC was a time consuming priority.

Tom