The mechanical of the solid is the part of the mechanics which is interested in the objects that one cannot reduce in a material Point. That in particular makes it possible to describe and model rotations of the object on itself.

The object itself is composed of material points, that they are discrete points - for example an assembly of balls connected by rods of negligible mass, each ball being able to be modelled by a material point - or a continuous whole of points. In general, the indeformable solid is supposed; the deformation of the solid raises of the Mécanique of the continuous mediums.

The mechanics of the solid is thus a branch of the mechanics treating of the behavior of the mechanisms made up of rigid parts in general, and sometimes deformable. The main aim being determination of the performances of a system in order to establish a dimensioning adapted to the use considered, or validation of these sizes.

Formal definition

One will call solid indeformable a whole of points as taken two to two, their distance does not vary during time. If the points are discrete, one can note them M_i , and thus

$\backslash \; forall\; (I,\; J),\; \backslash |\; \backslash \; overrightarrow\; \{M\_\; \{I\}\; M\_\; \{J\}\}\; \backslash |\; =\; \backslash \; mathrm\; \{constant\}$.

Points with the solid

The mechanics of the point can apply in each point of the solid, or, in the case of a continuous solid, for each infinitesimal element of volume FD around a point ( X , there , Z ).

Let us consider the Barycentre G of the points of the solid. In the case of a whole of discrete material points M_i and masses m_i , one a:

$(\backslash \; sum\_i\; m\_i)\; \backslash ,\; \backslash \; overrightarrow\; \{OG\}\; =\; \backslash \; sum\_i\; m\_i\; \backslash ,\; \backslash \; overrightarrow\; \{OM\_i\}$,

O being the origin of the Reference frame. In the case of a compact solid occupying a volume V , one can define in each point a Densité ρ, and one has

$x\_G\; then\; =\; \backslash \; int\_V\; \backslash \; rho\; (X,\; there,\; Z)\; \backslash ,\; X\; \backslash ,\; dx\; \backslash ,\; Dy\; \backslash ,\; dz~,$

$y\_G\; =\; \backslash \; int\_V\; \backslash \; rho\; (X,\; there,\; Z)\; \backslash ,\; there\; \backslash ,\; dx\; \backslash ,\; Dy\; \backslash ,\; dz~,$

$z\_G\; =\; \backslash \; int\_V\; \backslash \; rho\; (X,\; there,\; Z)\; \backslash ,\; Z\; \backslash ,\; dx\; \backslash ,\; Dy\; \backslash ,\; dz~.$

By integrating the laws of Newton on the solid, one from of deduced that the movement of the barycentre itself can be described by the mechanics of the point; it is considered that the resultants of the forces of the solid are exerted on the barycentre. For example, if each element of volume is subjected to a weight $\backslash \; rho\; (X,\; there,\; Z)\; \backslash ,\; FD\; \backslash ,\; \backslash \; vec\; \{G\}$, then one can consider that the barycentre is subjected to the weight $m\; \backslash ,\; \backslash \; vec\; \{G\}$ with

$m\; =\; \backslash \; int\_V\; \backslash \; rho\; (X,\; there,\; Z)\; \backslash ,\; dV$.

One can in the same way write the moment in each point of the solid compared to a reference. While integrating this concept, one arrives at the concept of Moment of inertia and kinetic Moment.

One has thus two types of actions to describe, which utilize two models: translations, with and the law center of inertia of Newton, and rotations, with the moments. To synthesize that, one can use a mathematical object called Torseur.

Modeling by the torques

See also: Torque

Relation of Varignon, concept of torque

Is $\backslash \; vec\; \{m\}$ a field of vectors called moment , $\backslash \; vec\; \{R\}$ a vector called resulting and $A,\; B$ two points of the solid , one says that these elements are bound by the relation of Varignon if:

$\backslash \; vec\; \{m\}\; (A)=\; \backslash \; vec\; \{m\}\; (B)+\; \backslash \; overrightarrow\; \{AB\}\; \backslash \; wedge\; \backslash \; vec\; \{R\}$

The vectors $\backslash \; vec\; \{m\}$ and $\backslash \; vec\; \{R\}$ are thus bound, one calls Torseur the couple of these two vectors and one notes it:

$\backslash \; \{T\; \backslash \}\; =\; \backslash \; \{\backslash \; vec\; \{m\}\; (A),\; \backslash \; vec\; \{R\}\; \backslash \}$

Kinematic torque

Are $\backslash \; vec\; \{V\}$ the field of the Flight Path Vectors of the solid $S$ in a reference frame $R$ and $O$ the origin of space. One a:

$\backslash \; vec\; \{V\}\; (has\; \backslash \; in\; S,\; R)\; =\; \backslash \; frac\; \{D\; \backslash \; overrightarrow\; \{OA\}\}\; \{dt\}$

a relation of Chasles gives us then

$\backslash \; vec\; \{V\}\; (has\; \backslash \; in\; S,\; R)\; =\; \backslash \; frac\; \{D\; \backslash \; overrightarrow\; \{OB\}\}\; \{dt\}\; +\; \backslash \; frac\; \{D\; \backslash \; overrightarrow\; \{BA\}\}\; \{dt\}$

however, it is shown that there exists $\backslash \; vec\; \{\backslash \; Omega\}$ such as $\backslash \; frac\; \{D\; \backslash \; overrightarrow\; \{BA\}\}\; \{dt\}\; =\; \backslash \; overrightarrow\; \{AB\}\; \backslash \; wedge\; \backslash \; vec\; \{\backslash \; Omega\}$

then

$\backslash \; vec\; \{V\}\; (has\; \backslash \; in\; S,\; R)\; =\; \backslash \; vec\; \{V\}\; (B\; \backslash \; in\; S,\; R)\; +\; \backslash \; overrightarrow\; \{AB\}\; \backslash \; wedge\; \backslash \; vec\; \{\backslash \; Omega\}$

there is then a relation of Varignon, one can thus define a torque called kinematic torque :

$\backslash \; \{V\; \backslash \}\; =\; \backslash \; \{\backslash \; vec\; \{V\}\; (\backslash \; in\; S,\; R\; has),\; \backslash \; vec\; \{\backslash \; Omega\}\; \backslash \}$

where the field of the moments is the field of the Flight Path Vectors and where the resultant is the vector $\backslash \; vec\; \{\backslash \; Omega\}$ called Flight Path Vector of rotation its standard is the number of instantaneous revolutions of the solid.

Random links:

Ancourteville-sur-Héricourt | Paternò | Verzolet | Gerard Banide | Clicked