Eulerian Mechanics of Belts

The proposal is motivated by the scientifically challenging and practically important problem of the analysis of nonlinear belt drive dynamics. Belt particles permanently enter and leave the zones of contact with the pulleys. This causes inherent difficulties at the mechanical modeling. For instance, a common finite element model of the belt may suffer from numerical excitation of vibrations as new nodes are coming and leaving the contact zone, and analytical treatment so far is mainly restricted to steady regimes. Moreover, certain effects related to inertia and extensibility of the belt are often left out of scope. Hence, important questions of transient dynamics and stability did not yet obtain comprehensive answers in the literature. The project will focus upon breaking new grounds in modeling the non-stationary dynamics of belt drives by means of a proper and consistent transformation of the equations of motion of the belt to a description, which can be classified as Eulerian (spatial). Established string and rod models will be applied for classical flat or modern multi-groove belts. After the transformation of the equations, mechanical fields become functions of a coordinate along the undeformed contour of the belt. An idealized contact model between the belt and the pulleys will allow avoiding unnecessary details. Coupled transverse and longitudinal vibrations of the belt will be considered using jump conditions at the boundaries of the contact zones. Delay equations for the transport of the solution by the pulleys and expressions for the reactive forces on the pulleys will follow accordingly. Analytical treatment will feature perturbation techniques for small transverse vibrations of the belt and will allow studying transient dynamics, frequency responses and stability of steady regimes. Non-trivial coupling effects and resonances in the system will be considered by taking into account the transport of the strain of the belt by the pulleys between the free spans and the traveling of the boundary points of the free spans along the contour of the pulleys when transverse vibrations occur. Results will be validated and extended with the numerical method of finite differences. Small eccentricities of the pulleys, flexibility of the shafts and bending stiffness of the belt will additionally be considered along with the important questions of structural stability. The research will be concluded with a novel efficient finite element scheme in a non-material description, which accounts for the material flow between the elements. The proposed cooperation of the members of the Russian and Austrian research groups has already started with joint publications on the basic analysis of stationary motion and preliminary results concerning transient dynamics and longitudinal vibrations of the belt. The financial volume applied for from the Austrian side is 258.510 , which includes the salary for one post-doctoral researcher for 2 years and one doctoral student for 3 years, plus travel expenses. Planned duration is 36 months (01.2015 till 12.2017). The project will be carried out at the Institute of Technical Mechanics of the Johannes Kepler University Linz.

The international project "Eulerian Mechanics of Belts" studies the questions how to simulate the nonlinear dynamics accurately and how to use non-linear models of elastic longitudinal structures for this purpose. This simulation must include both elastic strings and elastic rods: under different conditions can either one of the two models be applicable. The project is a fundamental research project in which, however, the practical applicability for industrial goals is obvious.