Math in Railroading, one & two generations ago--
Some time back I had commented that raising HP by adding cylinders faced problems because the crankshaft being heavier would have critical vibration frequencies (in torsion) moved to lower RPMs. HOT WATER then commented that often some of the run points would be placed between critical vibration RPM. I was curious to see how well this might have worked, and much later I found some EMD calculations (for 8 cylinders): some vibration but apparently not to runaway amplitudes if the time between run settings was not too slow. What stood out was that these calculations had been done by hand. Don't recall how I knew, but I had some experience:
In the time frame 1959-60 school year, I chose to look into matrix computations, having heard this would be the coming thing in engineering math. My new school did not lack for quality of its instructor, to the point where, having divided us into teams of two, and my teammate promptly quitting, this instructor saw this as a trivial problem. Perhaps I was compensated by getting first choice of the several methods of solution that the teams would compare.
The class problem was a 6x6 matrix, for a 4-cylinder gasoline engine, plus rear flywheel and front vibration damper; there are 6 critical rpm's for vibration. Perhaps of interest is that the largest matrix (in the 70's) that the really good hand-held engineering calculators could handle was 6x6. The only computer on campus was the bookkeepers' IBM 650, to be programmed in machine language, and loaded by hand-punched cards, on an evening loan once a week. (I actually got output even though it was extemely difficult to program. Well, it was only an electronic Frieden.)
My recollection is that EMD actually further reduced the burden of manual calculation by quartering an 8 by 8 matrix as their results displayed less that the full eight singularities. Using a 4x4 matrix & assuming the number of operations as n^4 (n to the 4th) the effort will be cut by a factor of 256. The problem is called the "Eigenvalue" problem, to extract these from a matrix. I think setting the input up per the "tri-diagonalization" method is most suited for reduction to extracting a 4x4 corner for the actual calculation work.
As a measure of the advance of capabilities in matrix work, in 1975, a mattrix of 276x276 was used for a very acute angle (30 deg) crossing of a curved bridge over 5 electrified tracks of the PRR, by then PC. Run in STRESS or STRUDEL, there was failure to converge to a solution. A suggested renumbering of nodes placed the most significant input data as close to the diagonal as possible, as above. It worked, using a 4 MHz GA (similar to a 1 MHz IBM 1130, but with 17k core). Five years passed before such a difficult design was finally repeated in this country.
I can't recall exactly the calendar dates that higher horsepower engines were sought, but the end of the 50's can't be far off. What I am thinking is that the vibration analysis of 12-cylinder engines was a lot less needed calculation, and a lot more risky guesswork, as I look at the history of what could be done at the time and what it would have cost to have done more.