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A feature of the wheeled vehicles capable of carrying liquid materials is the presence of the external disturbances in the form of interaction with the roadbed. This applies to peculiar transport vehicles, carrying liquid materials considered as non-Newtonian fluid. The internal disturbances in the form of the liquid material free surface interaction is the presence of the external disturbances in the form of interaction with the roadbed. Internal disturbances in the form of the wheeled concrete transport machine working vessel fluid free surface interaction should also be taken into account. The article describes the first part of the task - the interaction of the system with the roadbed. The simplest form of the system motion equations with rolling is in the accompanying coordinate system when written in the form of equations in quasi-coordinates. Based on the apparatus of nonholonomic mechanics, M.V. Keldysh built the theory of the rolling wheels with an elastic tire. Under certain assumptions, the above-mentioned theory in the article is extended to the case of curvilinear motion. These equations are represented as a system of first-order equations convenient for numerical solution.