Developments of structural dynamics

This course is devoted to the dynamic implementation of continuous structural elements vs discrete models. The matrix representation and implicit solution of Lagrange’s equation are at the heart of this approach, in the framework of conservative structural systems, with Gaussian modes. The prototype of the continuous element being the prismatic beam - as an illustration, but with general value - and the implicit model/solution leads to the major place of natural eigenshapes in vibration and shocks theory and general dynamic representation of structures and foundations of the dynamic tests and certification. In fact, this course is a perspective on the double vision of any dynamic structure, with the continuous point of view and a digital model in perspective. At the end of this course, the physicist, the mechanical engineer, the specialist in control will have a general overview of the dynamic representation and implicit solution applicable in structural analysis and control of general dynamic systems. In a certain way, it is founded on the famous discussions between Albert Einstein and Niels Bohr. Recommendation: we suggest having a look, at the course "Foundation of structural dynamics: from static to discrete dynamics". This course proposes an original point of view of the foundations of dynamics. And could highlight the present course.

• Analytical dynamics
• We will discover the power of Lagrange formalism able to generate the equation of any discrete dynamic system. This seems miraculous because, from energies, we directly extract the complete dynamic equations.
• Linear structures
• Linearity is an important domain. Not only because it corresponds to powerfull mathematic tools but also because it's a nominal way to move for a structure. In this topic, we master the modeling of any digital linear system.
• Discrete eigenshapes
• We solve the digital dynamic system by Gaussian diagonalization, thus we discover the concept of natural shapes (or eigenshapes), fundamental for the dynamic behavior of any structure. This opens the world towards resonances and implicit analysis. In fact, the modes represent the actual dynamic DNA of the structure.
• Dynamics of beam
• Real structures are made with continuous beams and shells. Beams are the ideal prototype for the demonstration of continuous modes showing clearly that there is no fundamental difference between discrete and continuous dynamic shapes. We solve the essential problem of bended dynamic beam representing for instance a bridge, a wing, a javelin in flight, etc.
• General assessment