The intention of this course is to provide a solid extension of the classical notions of analytical mechanics to a set of generalized functions (distributions) as well as to consider differential equations of motion with discontinuous right handside (and extend them to differential inclusions). In doing so non-smooth analysis that deals with approximation of sets and maps will be applied to problems with impact and dry friction. This course, addresses the student to understand these extensions in different problems posing and problem solving tasks, by means of recognizing, identifying and formulating appropriate models and by choosing the appropriate either numerical or analytical solving procedures.

At the end of the course students will be expected to have ability: to apply this new knowledge in engineering disciplines involving non-smooth mechanics; to recognize various motion of the real life systems and the effects of both regular in impulsive forces and torques; to analyze friction and dissipation of energy with respect to constraints that follow from inequality for entropy; to use computer tools in prediction of various motions by means of appropriate models; to communicate with other engineers within a team work; This course prepares the student for further learning as well as for practice, hard work, creative thinking, further development of skills in design of new solutions of engineering problems.

This course covers the following topics. Introduction to theory of impacts. Distributional derivatives. The generalized Euler-Lagrange equations of the second kind. Changes of kinetic energy during collisions. Regularizations - the Hertz-type impact theories. The Zener model and restrictions that follow from Clausius-Duhem inequality. The Fremond approach. The Hertz-Signorini-Moreau law of unilateral contact. Linear complementarity problems. Generalized derivative, generalized differential. Dry friction models. Differential inclusions. The Filippov theorem. Mechanical systems with set-valued force laws. Non-smooth potentials. The augmented Lagrangian method. The Gauss principle. Numerical integration methods. Examples of non-smooth mechanical systems in engineering. Bifurcations of non-smooth systems.

Lectures, auditory exercises, demonstration of computer tools. Homeworks, as a check of understanding and usage of the introduced notions that can be done within groups. Either a practical examination part -- two problems done by them own -- or seminar work based on a real problem presented in periodicals. Individual work with each of the groups which extends the knowledge and skills in mechanics, mathematical analysis and computer tools, as well as the foreign language the student use. The examination ends with a final talk on the introduced notions and skills.