This fourth volume in the “Movement Equations” series is positioned in line with the third entry in that it applies the principles established in the third volume, such as the fundamental principle of dynamics as applied to the motion of non-deformable solids, and its various scalar consequences that lead to the movement equations.

During their motion through space, the bodies can encounter situations of equilibrium, static or can be under the form of uniform movements said to be stationary, the stability of which, meaning the ability to maintain itself, must be assessed if the bodies are subject to stresses that tend to move them away from this equilibrium. These stresses induce small movements that can either, after oscillations, bring the body back to its initial state of equilibrium, or amplify and break it. Furthermore, the combination of oscillatory components in a same device can, through the coupling of their respective behaviors, lead to stabilized and stabilizer systems such as the gyroscope.

This volume includes four chapters. Chapter 1, which states the problem, begins with the scalar consequences of the fundamental principle, identifies the conceivable situations of equilibrium, their conditions for acquisition and the values of the situation parameters that correspond to them. It then examines the consequences of infinitesimal variations of the parameters around the values of equilibrium we previously identified and deduces the equations that express the resulting movements. These small movements show the behavior of the mechanical device thus disturbed and give useful indications on its stability.

Chapter 2 is, in a way, a mathematical insert in the book. Studying the behavior of an oscillator requires solving the equations of small movements established in Chapter 1. These are second-order linear differential equations, with constant coefficients that possess an infinite number of solutions; because the motion we want to study is presumably oscillatory in nature, we are looking for solutions of which the form is compatible with this objective, either starting from the vector expression of the differential system to be solved, or by using the Laplace transformation.

Chapter 3 focuses on the study of oscillators. We start with the individual oscillator, the motion of which highlights the different modes of oscillation and the stability of such a device. We then focus on oscillator coupling that demonstrates the reciprocal influence of different mobile components of a system within which they are coupled and their role in the stability of the set. But, in this case, the study of the stability requires processing transfer functions that have no place in this title. However, to illustrate their use, the chapter presents, for example, the use of the Routh criteria, which are used to study the stability of dynamic systems in engineering and electronics, without giving theoretical validation.

Since the gyroscope is a device whose components are coupled, allowing it to display various stabilizer effects in a variety of applications, we dedicate Chapter 4 to it. After expanding on the principle of gyroscopic coupling, the study moves to its use in the case of the gyroscopic pendulum, used in particular for controlling satellites, and the gyro-compass as a navigational instrument that helps maintain true North. The presented problem demonstrates the use of a gyroscope on the swell of a ship.

The “Movement Equations” series will end with a fifth volume, through the study of sets of solids and with an introduction to robotics. The authors will thus have offered the readers a broad panorama on this subject of movement equations, which are at the core of the study of the motion of non-deformable solids, and the use of which is still current in the field of space, even though it is obvious that the ones used are clearly more complex. But the ground work is the same.