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In the numerical procedures involving the force of dry friction some type of numerical approximation for this investigated phenomenon is used in the process of calculations. Elliptic approximations for the local transition region are proposed, and numerical results are presented in this paper. The smoothness and continuity of this approximation is demonstrated by several computational experiments.

Abstract. In the numerical investigation of vibration problems with dry friction it is accepted to use some type of approximation to this phenomenon. Often linear variation of the force of friction in a region around zero velocity is assumed. In this paper trigonometric variation is proposed and comparison of numerical results is performed. From the presented results higher precision of this approximation is observed.

The article presents the results of studies of energy, environmental and fuel efficiency indicators of a passenger car with a gasoline injection system. The engine is equipped with a device additionally supplying liquefied petroleum gas (LPG) and it has an ignition timing variator (ITV). As a result of research, an improvement in the fuel efficiency of a car in thermal equivalent was found when the engine was running on LPG and a slight increase in energy performance was observed. Emissions of pollutants with exhaust gases change in different ways: emissions of carbon monoxide practically do not change, emissions of hydrocarbons and nitrogen oxides are growing, carbon dioxide emissions from LPG operation are reduced, socio-economic damage when the engine is running on LPG is also reduced.

The purpose of this research is to perform comparative analysis of the effect of fuel additive SO‐2E on the economical and ecological parameters of a direct‐injection Diesel engine, operating on Diesel fuel and shale oil alternately. It was proved that multifunctional fuel additive SO‐2E applied in proportion 0,2 vol % is more effective for improving combustion of shale oil than Diesel fuel. At light operation range the treated shale oil savings based upon fuel energy content throughout wide speed range 1400–2000 min−1 reduce from 14,6–12,3MJ/kWh to 11,6–11,8 MJ/kWh or by 20,5–4,1 %. Maximum NO emission for treated Diesel fuel was reduced by 7,8–11,8 %, whereas NO2 simultaneously increased by 3,8–7,4 %. In the case of treated shale oil both harmful pollutants were reduced by 22,9–28,6 % and by 41,6–13,4 %, respectively. The exhaust gas opacity and CO emissions at the rated performance regime for both fuels were obtained a bit higher, whereas HC emission for treated shale oil increases 1,9 times and for Diesel fuel remains on the same level.