Vibrations and Variational Methods

This 42-hour course is divided into two identical parts that run in parallel. The first part focuses on the study of vibration problems in discrete media and in 1D continuous media (strings, beams). The second part focuses on the use of variational formulations to reformulate the problems studied in L3 in RDM and 3D elasticity.  This allows us to propose optimized approximate solutions. This part of the course establishes a link between RDM, 3D elasticity, and the second-semester course on finite elements.

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

Review the L3 courses on RDM and 3D elasticity in order to better connect and deepen your understanding of them. Learn how to formulate problems as minimization problems so that you can solve them using finite element code in the second semester.

Have completed a bachelor's degree in mechanical engineering, specifically including courses in MMC and RDM.

The course is divided into two parts that run in parallel:

1. Vibrations

1. Vibrations of discrete systems.

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

1. Longitudinal and transverse vibrations in beams.

Free vibrations and forced vibrations. Normal modes.

1. Applications of variational methods.

1. Variational formulation of beam tension and bending equations. Principle of virtual powers - Energy minimization. Applications: solution approximation, modulus bounding.
2. Variational formulation of equations in 3D linear elasticity. Principle of virtual powers - Energy minimization. Applications: approximation of solutions, modulus bounding.