Vibration and Variational Methods

This 42-hour course is divided into two identical parts running in parallel. The first part deals with the study of vibration problems in discrete media and in 1D continuous media (rope, beams). The second involves the use of variational formulations to reformulate the problems studied in L3 in RDM and 3D elasticity. This enables us to propose optimized approximate solutions. This part of the course provides a link between RDM, 3D elasticity and the second-semester finite element course.

Model dynamic problems involving discrete and continuous media and analyze the solutions obtained. In connection with the complementary part of this course, know how to use efficient approximate methods (e.g. Ritz method).

Review the L3 courses on RDM and 3D elasticity in order to better link and deepen them. Learn to formulate problems as minimization problems, so as to be able to process them using a finite element code in the second semester.

Completion of a Bachelor's degree in mechanics, and in particular an MMC course and an RDM course.

The course is divided into two parallel parts:

1. Vibrations

1. Vibrations of discrete systems.

Review of basic vibration concepts. Study of a two-degree-of-freedom system with or without damping. Free vibrations. Forced vibrations. Study of the resonance phenomenon. Study of systems with n degrees of freedom. Study of eigenmodes

1. Longitudinal and transverse vibrations in beams.

Free vibration and forced vibration. Normal modes.

1. Applications of variational methods.

1. Variational writing of beam tension and bending equations. Virtual power principle - Energy minimization. Applications: solution approximation, module framing
2. Variational writing of 3D linear elasticity equations. Principle of virtual powers - Energy minimization. Applications: solution approximation, module framing.