D’ALEMBERT FISICO E MATEMATICO

Some mathematical treatises of d’Alembert are presented in in synthetic form. Particular attention is paid to the Traité de dynamique, namely the part concerning the principles on which Mechanics is founded. This includes the formulation of the method later entered introduced in treatises on Mechanics under the name D’Alembert’s Principle. Then some applications to problems in Astronomy are discussed. The content of the note has been presented in a talk given at the Istituto Lombardo Accademia di Scienze e Lettere, in the context of a conference dedicated to the works of D’Alembert owned by the Library.

Virtual Work & d’Alembert’s Principle

This chapter discusses virtual work, returning to the Newtonian framework to derive the central Lagrange equation, using d’Alembert’s principle. It starts off with a discussion of generalised force, applied force and constraint force. Holonomic constraints and non-holonomic constraint equations are then investigated. The corresponding principles of Gauss (Gauss’s least constraint) and Jourdain are also documented and compared to d’Alembert’s approach before being generalised into the Mangeron–Deleanu principle. Kane’s equations are derived from Jourdain’s principle. The chapter closes with a detailed covering of the Gibbs–Appell equations as the most general equations in classical mechanics. Their reduction to Hamilton’s principle is examined and they are used to derive the Euler equations for rigid bodies. The chapter also discusses Hertz’s least curvature, the Gibbs function and Euler equations.

Pre-load on the guiding/stabilizing wheels and the critical lateral force of a straddle-type monorail vehicle

The anti-overturning ability of a straddle-type monorail vehicle is influenced by the contact status between the guiding and stabilizing wheels and the track beam; therefore, an initial pre-load is required for the stabilizing and guiding wheels to enhance the anti-overturning ability of the straddle-type monorail vehicle. Determining the reasonable pre-load for the stabilizing and guiding wheels is a significant problem in the operation of a straddle-type monorail vehicle. D’Alembert’s principle has been adopted to transform the dynamic problem that straddle-type monorail vehicle runs on curve segment to the statics problem. The formula describing the relationship between the critical lateral force of the vehicle and the pre-load of the stabilizing wheels is derived from the lateral roll equation of the straddle monorail vehicle and is verified using the multibody dynamics software UM. Subsequently, the reasonable pre-load for the stabilizing wheels is analyzed from the perspectives of comfort and safety based on the formula of critical lateral force. Finally, the maximum and minimum speed limits on a curve for a straddle-type monorail vehicle are discussed based on the aforementioned analysis.

Substitute Model for Crawler Track Units

Crawlers for mobilisation of working machines in rough terrain realize considerably high propel power. Furthermore, even heavy engineering structures cause only minor ground loading due to the wide contact patch of the crawler track plate. Typically, the driving speed of crawlers is low. Nevertheless, there is a significant speed droop during travel. The resulting vibrations and thus time variant loads eventuates in altered wear of the crawler’s components, as well as in damage of the travel gear drive and the superstructure of the vehicle. In this paper, a substitute model for crawler units is derived based on D’ALEMBERT’s principle. The model includes the most relevant phenomena of longitudinal crawler vehicle dynamics and hence permits to assess the time variant loads on the components thoroughly.

D’Alembert’s Principle

It is explained how the mysterious Principle of Virtual Work in statics is extended to the even more mysterious Principle of d’Alembert’s in dynamics. This is achieved by d’Alembert’s far-sighted stratagem: considering a reversed massy acceleration as an inertial force. A worked example is given (the half-Atwood machine or “black box”). Some counter-intuitive aspects are made intuitive by more examples: the Pluto-Charon system of orbiting planets; Newton’s and then Mach’s explanation of Newton’s bucket. Also, it is demonstrated that the law of the conservation of energy actually follows from d’Alembert’s Principle. The reader is alerted to the astoundingly fundamental nature of d’Alembert’s Principle. It is the cornerstone of classical, relativistic, and quantum mechanics. As Lanczos writes: “All the different principles of mechanics are merely mathematically different formulations of d’Alembert’s Principle”.

A Variationally Consistent Approach to Constrained Motion

A variationally consistent approach to constrained rigid-body motion is presented that extends D'Alembert's principle in a way that has a form similar to Kane's equations. The method results in minimal equations of motion for both holonomic and nonholonomic systems without a priori consideration of preferential coordinates.