Movement Equations 4: Equilibriums and Small Movements

By Michel Borel and Georges Vénizélos

An important instance of the application of unbuckled solid mechanics is that of its stability and small movements from this situation. The problem expressing goes through the linearization of the movement equations set up in the 3rd volume of this treaty, by their limited development. This book gives and develops the process which leads to the differential linear equations expressing this kind of movement and allowing the study of the equilibrium and the stability of an unbuckled solid.

• Language: English
• Publisher: Wiley
• Release date: Feb 14, 2018
• ISBN: 9781119510642

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Equilibrium, Stationary Movement and Oscillation of a Free Solid

In certain situations, the motion of a free solid, which is only subject to known external efforts that can be expressed and quantified, but not to any links that would introduce onto the motion any unknown strains that would vary with it, can experience situations of equilibrium and oscillations around these. It can even be the motion sought for a particular mechanism. This is why the equations that govern these situations are of great interest to engineers. The approach we develop here is limited to the first order of magnitude (in the sense of limited mathematical developments) of the variations of the situation parameters of a solid; it leads to second order linear differential equations. Far more precise oscillatory motions would require going beyond this first order of magnitude in the limited development of equilibrium equations; this is not the objective of the following presentation, which instead indicates the procedure to do it.

1.1. Expression of the fundamental principle of dynamics for a free solid

In a Galilean frame , the motion of the solid (S), to which is joined the frame , obeys the fundamental principle of dynamics, the torsor expression of which is

where refers to the external efforts torsor acting on (S).

At a point P of the solid, the dynamic torsor has the following reduction elements:

If the motion of the solid (S) is observed from a given frame 〈λ〉, the expression of the fundamental principle of dynamics involves two other torsors that correspond to the existence of the motion of 〈λ〉 in relation to the Galilean frame 〈g〉. This is the inertial drive torsor and the Coriolis drive torsor , the reduction elements of which at a given point Q have the following generic