It is known that steam locomotives exhibit increased forces to the railway tracks, in comparison to diesel and electrical locomotives. Although mainline operation of steam locomotives ceased in 1960-ties and 1970-ties, many steam locomotives are still preserved, some of them in operational state, and are used for tourist traffic and retro tours on mainline lines, even on those where traffic is shared by high speed trains, for example, in Germany. Imbalanced inertial forces in steam engine are similar to forces in diesel engines, but for the same power all the parts of steam engine are heavier, and connected directly to wheelsets and locomotive mainframe, without any elastic joints. For this reason a steam locomotive in general cannot be a fully balanced engine, and it is the case when the compromise should be found between minimal vertical imbalance (unloading and overloading of the wheel) and acceptable horizontal imbalance (leading to fore-and-aft dynamics in the train consist), as both goals are not achievable. Steam locomotive dynamics has been reviewed in many books, but with many approximations and limitations, to simplify the calculations, but introducing error in the results. A narrow gauge steam locomotive Gr-319 was chosen as the research object, which was being restored in narrow gauge railway Gulbene-Aluksne in Latvia. There were many locomotives built with 0-4-0 wheel formula, having four driving axles and with driving rod connected to the third wheelset, but this particular locomotive has 0-4-0 wheel formula, but with very long and heavy driving rod connected to the fourth wheelset, resulting in bad dynamics, being the case for research.