A Practical Guide to Numerical Methods in Mechanical Systems: Solving Complex Differential Equations

Abstract

Solving differential equations of the type presented by modern aircraft such as the Airbus A400M and F-16 are fascinating problems. These mechanical systems have high level of nonlinearities that the traditional linear analysis methods are not completely able to explain. In the last ten years computing power has advanced to the point of determining radical changes from the way we approach such complex mathematical problems. Consequently, there has been considerable progress both in the theory and numerical methods for vibration reduction. More specifically, they are of particular importance in aerospace and mechanical engineering where knowledge on natural frequencies and vibration modes is needed for design and certification processes. These include the usage of Dynamic Vibration Absorbers which have had considerable success in attenuating the magnitude of oscillations for different mechanical systems. This very academic guide will teach you how to solve differential equations in mechanical systems in a practical way. We'll explore how these mathematical tools aid us in understanding, and controlling, the complex mechanical behavior of a system. We will be focused on practical applications — how to solve stability analysis, perturbation methods, as well as treating nonlinear systems which are becoming more and more prevalent in modern engineering challenges.