Mathematical Problems in Engineering
Volume 2005 (2005), Issue 4, Pages 393-410

Problems of interaction of vibrating surfaces with processable materials

V. A. Ostapenko

Faculty of Mathematics and Mechanics, National University of Dnepropetrovsk, 13 Nauchnaya Street, Dnepropetrovsk 49050, Ukraine

Received 18 June 2004

Copyright © 2005 V. A. Ostapenko. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We consider the mathematical model of interaction of a vibrating surface with the load placed on it. With the purpose of accounting for influence on behavior of not only system interaction of a blade with a load but also internal interaction of particles of a material, the load is submitted as a finite number of strips with zero thickness. The carrying blade is represented as a vibrating membrane. It is supposed that the weight of the material is comparable to or considerably surpasses the weight of the blade. Therefore, the model takes into account the inertia of the material. In the model with joint movement of the blade and the load, the separation opportunity of the load from the blade is provided. Therefore, there is a phase of separate movement of the blade and the load, with their subsequent connection accompanied with impact. The process of system movement is represented as alternating sequences of joint and separate movements of the load and the blade. The modeling of the process of the interaction of the load and the blade is represented as an initial-boundary value problem. The method of solution is developed and the exact solution of the set problem is obtained in a class of generalized functions.