Використання математичного моделювання для оптимального проектування дає можливість перевести процес випробувань реально виготовлених механізмів до випробувань та аналізу комп'ютерної моделі, що значно зберігає матеріальні і часові ресурси на підготовку й упровадження у виробництво сучасних машин або механізмів і гарантує їхню якість і надійність у процесі роботи. Використання такої комп'ютерної моделі можливе при побудові адекватної математичної моделі, яка відображає механічне формоутворення контактуючих поверхонь, процес робочого зачеплення і дозволяє провести порівняльний аналіз к.к.д. передач, які проектуютьсядля трансмісійних пристроїв. Застосування розробленого програмного забезпечення і запропонованих методів побудови моделей механічного формоутворення контактуючих поверхонь і процесу робочого зачеплення дозволяє провести якісний аналіз к.к.д. просторових передач, які розроблюються. При цьому вдається зменшити втрати на тертя в зачепленні і підвищити к.к.д. силових вузлів і редукторів, що в умовах економічної кризи має велике економічне значення.^UThe most important task of mechanical engineering is to constantly improve the quality of machines and mechanisms. A significant role is played by gears and reducers, whichlargely determine the durability, reliability and competitiveness of the mechanisms and machines in which they are used.The use of mathematical modeling for optimal design makes it possible to translate the test process of actually made mechanisms to testing and analysis of a computer model,which significantly saves material and time resources for preparation and implementation of modern machines or mechanisms and guarantees their quality and reliability. The use ofsuch a computer model is possible when building an adequate mathematical model that reflects the mechanical formation of contact surfaces, the process of working gearing andallows for comparative analysis of efficiency. gears that are designed for transmission devices. Therefore, the simulation of wheel engagement remains relevant in the developmentof new types of gears, which is inconceivable without deep knowledge of the basic laws of interaction of interconnected tooth surfaces and requires a comprehensive study of the kinematics and nature of contact of such surfaces to assess the performance of developed gears.The mathematical model of kinematic formation of lateral surfaces of teeth of cylindrical helical gear wheels with the initial contour of the general look is offered in the work. With the help of the considered model the equations of lateral surfaces of geargenerating tools - tool wheels with Novikov gearing are received. The obtained dependences contribute to the study of the parameters of the movement of contact points in the transmission when rolling the spatial wheels with tool surfaces, which are represented by rollers with oblique teeth. During modeling, taking into account the geometry of the initialsurfaces of the spatial transmissions, a mathematical model of the formation of the teeth ofthe spatial wheels with machine gearing was developed. As a result of mathematical modeling the equation of the processed surface of a tooth of a gear wheel and a wheel of spatial transfer is received. Due to mathematical and computer simulation of the working gearof the connected spatial gears, a real-time modeling program of the working gear, instantaneous and total contact areas and projection of the gear curve has been developed. Testsof working gearing of spatial wheels are carried out. The shape and position of the contact area are investigated. Software simulations of the working gear and contact pads confirmed the absence of movement of the instantaneous contact pads along the height of the teeth in the process of their movement along the tooth line from one end plane of the wheels to another. This indicates the fulfillment of the necessary and sufficient criterion of Novikov's engagement for the designed spatial transmissions. The specified method of calculation of efficiency of gearing taking into account contact interaction of lateral surfaces of teeth and speeds of rolling and sliding of contact sites is developed. This allowedthe development of an interactive module for the interaction of the designer of the computer model of transmission with the resources of the simulation software system.Application of the developed software and the offered methods of construction of models of mechanical shaping of contacting surfaces and process of working gearing allows to carry out the qualitative analysis of efficiency. spatial transmissions that are being developed. At the same time it is possible to reduce losses on friction in gearing and to increase efficiency power units and gearboxes, which in times of economic crisis is of great economic importance.The implementation and verification of the adequacy of field experimental research and computer modeling, which was carried out by the considered computational methods.A sample of the gearbox with spatial transmission was designed, which is used for the winch type LSHV / 2Ch-63. The experimental reducer was installed in the mechanism forcontrolling and regulating the change of the angle of rotation of the blades in the rectifier of the fan of the main ventilation of the mine DCD-47U. The use of this reducer allowedto reduce energy losses by 15-20% and by increasing the sliding speed of the lateral surfaces of the teeth to increase the efficiency reducer by 6-8%.The results obtained in the dissertation can be used by design and engineering organizations in the design and manufacture of gears in gearboxes, which are used as high-loadelements of the drives of forging and pressing, concentrating, transport and other types ofequipment.