Mechanical Actions

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Chapter 10 Mechanical Actions

In this chapter, we seek to model the mechanical actions exerted on a material system \(\Sigma\text{.}\) We define a mechanical action as any cause capable of either maintaining a material system in equilibrium, or of modifying its motion or shape. On the simplest level, recall that the action of a force \(\bF_A\) exerted on a system \(\Sigma\) is properly defined by indicating both the value of vector \(\bF_A\) and its line of action (or simply a point on this line of action): hence this mechanical action is defined by the bound vector \((A, \bF_A)\) or in the language of screws by the slider denoted as

\begin{equation*} \left\{ \begin{array}{c} \bF_A \\ \bze \end{array} \right\}_A \end{equation*}

Screws are useful tools to account for the sum of discrete forces, or for the sum of forces distributed over \(\Sigma\) (or over a subset of \(\Si\)). For instance, the action of force \(\bF_A\) acting through point \(A\) and force \(\bF_B\) acting through point \(B\) results in the screw (see Figure 10.0.1)

\begin{equation*} \left\{ \begin{array}{c} \bF_A \\ \bze \end{array} \right\}_A + \left\{ \begin{array}{c} \bF_B \\ \bze \end{array} \right\}_B = \left\{ \begin{array}{c} \bF_A + \bF_B \\ \bF_A \times\br_{AQ} +\bF_B \times\br_{BQ} \end{array} \right\}_Q \end{equation*}

More generally, in the case of distributed forces, we consider two types of interactions between material systems \(\Sigma_1\) and \(\Sigma_2\text{:}\) action at-a-distance (such as gravitational actions) exerted at every point within the volume of each systems, and contact actions exerted on all or part of their boundary. From a local description of elementary forces modeled as bound vectors (or sliders) acting on infinitesimal elements of volume or surface of \(\Sigma_2\text{,}\) we obtain a global description of the action of \(\Sigma_1\) on \(\Sigma_2\) in term of a screw called action screw, and denoted as \(\{ \cA_{\Sigma_1 \to \Sigma_2} \}\text{.}\) The global effect of all mechanical actions exerted on a material system will be defined in terms of the total external action screw.

Figure 10.0.1.