In 1888 Krylov joined the department of ship construction of St Petersburg Maritime Academy. There he was taught advanced mathematics by Aleksandr Nikolaevich Korkin, a student of Chebyshev, who was an expert in partial differential equations. Korkin always put in extra effort for the students who he recognised as having exceptional abilities, giving them personal tuition and posing them very difficult and challenging problems. Krylov benefited greatly from Korkin's tuition and he graduated First Class in 1890 after only two years of study. Korkin then persuaded Krylov to stay at the Maritime Academy and take over teaching his courses. This Krylov did, remaining there to teach for almost 50 years :-
He taught various theoretical and engineering sciences for about fifty years at this military-maritime institute, creating from among his students a large school of shipbuilders who were both engineers and scientists.In 1898 Krylov received a Gold Medal from the Royal Institution of Naval Architects for his outstanding contributions to the theory of oscillating motion of a ship.
The Naval Administration Towing Tank was set up in 1894 to perform ship model tests to find engine power requirements for specified speeds and hull lines which required the least power. It was the first experimental basin to test ship design in Russia and the sixth the world. The Superintendent of the Tank was A A Grekhnev but after he left, Captain of the Admiralty A N Krylov was appointed 'the Acting Superintendent of the Tank' on 3 January 1900. Krylov proposed establishing a scientific institution that would include the Towing Tank, testing and physical-chemical laboratories for research on ship construction materials, a mechanical laboratory and an electrical engineering laboratory. On 3 April 1902 Emperor Nicolas II visited the Towing Tank and:-
... deigned to be entirely pleased with both the performed experiments and the observed order of things ...expressing his Imperial gratitude to Krylov as the Superintendent of the Tank. Krylov's work at the Towing Tank covered :-
... theories of buoyancy, stability, rolling and pitching, vibration, and performance, and compass theories.These were precisely the topics that Euler called 'naval science'. But Krylov went even further, joining a voyage on board the 1-rank cruiser Askold in 1902.
In January 1908, Krylov, by this time a General, was appointed Chief Naval Ship Construction Inspector and President of the Maritime Engineering Committee :-
His courage and integrity led to conflicts with officials of the Maritime Ministry and his refusal to do further work for them.in  Botchev gives examples which illustrate Krylov's struggle against officials:-
To promote his innovations, he often had to fight against stagnation and rigid views of the top officials. Once, giving a speech at an important meeting before a large audience, Krylov addressed naval officers asking them for their support in his "fight against the rut in shipbuilding". Being a naval officer at that time, he received an official reprimand for this speech. Another incident also quite remarkably illustrates Krylov's personality: on a sitting of a high-rank technical committee Krylov once took with him several technicians directly from the ships so that they could support his opinion in the debates.Krylov gave up his posts at the Maritime Ministry in 1910. In 1914 he was awarded an honorary doctorate in applied mathematics from Moscow University. In the same year he was elected a member of the Russian Academy of Sciences, becoming a full member in 1916. After the October Revolution of 1917 Krylov sided with the Soviet Government but tried to influence their attitude towards science. In 1919, along with others, he organised the Russian Association of Physicists. In February 1919 they set up a committee of four, Ioffe, Krylov, Anri and Lazarev, to coordinate the receipt of foreign literature, instruments and equipment, and to restart foreign research travel. They called for government support for new institutes, the reestablishment of ties between physicists in Russia and abroad, and the resumption of publication of scientific journals. Indeed Krylov was one of the first scientists allowed to travel to the West after the Revolution, travelling to London in 1921 to re-establish contacts between Russia and the West.
From 1927 until 1932 he was director of the Physics-Mathematics Institute of the USSR Academy of Sciences. He became an honoured scientist and engineer of the Russian Soviet Federated Socialist Republic in 1939 and, in 1943, was awarded the State Prize for his work compass theory and made a "hero of socialist labour".
Krylov made many mathematical advances in his applications of mathematics to shipbuilding. In hydrodynamics, among many advances, he made significant contributions to the theory of ships moving in shallow water. In 1904 he constructed a mechanical integrator to solve ordinary differential equations, being the first in Russia to make such an instrument. He improved Fourier's method for solving boundary value problems in a 1905 paper and gave many applications :-
While using mathematics and mechanics to work out his theory of ships, Krylov simultaneously improved the methods of both disciplines. In a paper on forced vibrations of fixed-section pivots (1905), he presented an original development of Fourier's method for solving boundary value problems, pointing out its applicability to a series of important questions: for example, the theory of steam-driven machine indicators, the measurement of gas pressure in the conduit of an instrument, and the twisting vibrations of a roller with a flywheel on its end.He studied the acceleration of convergence of Fourier series in a paper in 1912, and studied the approximate solutions to differential equations in a paper published in 1917.
In 1931 he found a new method of solving the secular equation determining the frequency of vibrations in mechanical systems which is better than methods due to Lagrange, Laplace, Jacobi and Le Verrier. This paper On the numerical solution of the equation by which, in technical matters, frequencies of small oscillations of material systems are determined deals with eigenvalue problems. Krylov writes in the paper (see for example ):-
It is clear that, if for k = 2 and k = 3 it is easy to compose this [secular] equation, then for k = 4 the laying-out becomes cumbersome, and for values k more than 5 this is completely unrealisable in a direct way. Therefore one should use methods where the full development of the determinant is avoided. The aim of the paper ... is to present simple methods of composition of the secular equation in the developed form, after which, its solution, i.e. numerical computation of its roots, does not present any difficulty. Before we describe these methods, it is good to return a bit back and consider how the first creators of these methods, Lagrange and Laplace, and then such a great astronomer as Le Verrier and such a great mathematician as Jacobi proceeded ...This obvious interest in the history of mathematics and mechanics come over in Krylov's work in a number of ways :-
Krylov's practical interests were combined with a deep understanding of the ideas and methods of classical mathematics and mechanics of the seventeenth, eighteenth, and nineteenth centuries; and in the world of Newton, Euler, and Gauss, he found forgotten methods that were applicable to the solution of contemporary problems.In fact Krylov published the first Russian translation from Latin of Newton's Philosophiae Naturalis Principia Mathematica in 1915. Three editions were published during Krylov's lifetime, then in 1989 a facsimile reproduction of the third edition was published. On 5 October 1933 Krylov gave a speech in Moscow during the celebration at the USSR Academy of Sciences of the 150th anniversary of the death of Euler. This speech was published in 1935, and a Bulgarian translation of the speech was published in 1983 on the occasion of the 200th anniversary of Euler's death.
Finally let us mention some of the classic texts which Krylov published. The first edition of Lectures on Approximate Calculations appeared in 1911, the second edition in 1932, the third in 1935, and the fourth in Krylov's collected works. The first edition of On Some Differential Equations of Mathematical Physics Having Application to Technical Problems appeared in 1913, the second edition in 1932, and the fourth appeared in 1948 as part of Krylov's collected works. Vibration of ships was first published in 1936 as a textbook in shipbuilding for Technical Schools. In 1943 he published Thoughts and materials on teaching mechanics. The ideas on teaching he presented in this text are considered in detail in .
Krylov married Elisaveta Dmitrievna Dranitsyna; their daughter Anna, later became Anna Kapitsa after marrying the famous physicist Pyotr Leonidovich Kapitsa who the Nobel Prize for Physics in 1978 for his research in magnetism and low-temperature physics. Krylov died in Leningrad (now St Petersburg) and is buried in the Volkovo cemetery.
Article by: J J O'Connor and E F Robertson