coating, additive technology, multibody contact, tilt angle, settlement, surface shape, system of integral equations, rapidly oscillating function.

Various coatings are used in the manufacturing of machine, device, and instrument parts to improve the strength characteristics of the latter in the designed contact regions. Such coatings are applied layer by layer with the use of modern additive technologies [1]. The paper deals with the contact interaction of a coated base with a system of identical rigid dies, the shape of the coating surface being completely identical to that of the die bases. Such a situation occurs, e.g., if the coating is applied by laser cladding or bonding and the rigid die is partly immersed in the still unhardened coating. Depending on the processing technique, the surface shape can be described by rather complicated functions, which are often rapidly oscillating. From the viewpoint of the classification of contact problems, we make one more step in the complication of the periodic contact problem: all dies are identical, but the system is finite (e.g., see [2]–[10]). From the viewpoint of integral equations, we consider a vector integral equation with matrix kernel and with one rapidly oscillating function in the free term [11].

1. Manzhirov Alexander Vladimirovich e-mail: manzh@inbox.ru, Doctor of Sciences (Physics and Mathematics), Professor, Foreign Member of the National Academy of Sciences of the Republic of Armenia, Deputy Director of the Ishlinsky Institute for Problems in Mechanics of the Russian Academy of Sciences, Moscow, Russia; Professor of the Department of Applied Mathematics at the Bauman Moscow State Technical University, Moscow, Russia; Professor of the Department of Higher Mathematics at the National Research Nuclear University (MEPhI), Moscow, Russia. 2. Kurdina Svetlana Pavlovna e-mail: svetlana-ka@yandex.ru, Senior Lecturer of the Department of Applied Mathematics at the Bauman Moscow State Technical University, Moscow, Russia. 3. Kazakov Kirill Evgenievich e-mail: kazakov@ipmnet.ru, Candidate of Sciences (Physics and Mathematics), Associate Professor of the Department of Applied Mathematics at the Bauman Moscow State Technical University, Moscow, Russia. 4. Fedotov Igor Ph.D., Priofessor of Department of Mathematics and Statistics at the Tshwane University of Technology, Pretoria, South Africa.