1
Set of Solids with Neither Loops Nor Branches

The direct application of the fundamental principle of dynamics to the movement of a chain of solids, as developed in the previous volumes of this series, assumes that each solid is considered as non-deformable. Yet, when the arrangements of these solids form loops or branches, during movement these are generally subject to structural deformation like tension, compression, bending, torsion, etc. The study of such movements requires recourse to theories on mechanical structures, reaching beyond the scope of this series.

1.1. Identifying a chain of solids with neither loops nor branches

In a Galilean frame , we consider a series of n solids (S1), (S2), . (Si), .(Sn) linked to one another, forming a chain with neither loops nor branches if

(S1) is linked to and (S2) only;

(S2) is linked to (S1) and (S3) only;

.

(S1) is linked to (Si-1) and (Sj+1) only;

.

(Sn) is linked to (Sn-1) only.

Let:

• - "upstream of (Si)" refer to the subset of solids (Sj) such that 1 = j = i;
• - "downstream of (Si)" refer to the subset of solids (Sj) such that i = j = n;
• - "(Sj-1)" refer to the previous solid (Sj);
• - "(Sj+1)" refer to the next solid (Sj).

The solid (S1), with which the frame <1> is associated, is located with respect to the reference frame <g> using k1(k1 < 6) parameters forming the group .

Each of the solids in the chain is framed and located analogously with respect to that preceding it. Thus, the solid (Si), with which the frame <i> is associated, is located with respect to solid (Si-1), with which the frame <i - 1> is associated, using the ki (ki < 6) parameters forming the group .

1.2. Applying the fundamental principles of mechanics

The fundamental principles governing mechanics take on particular meaning and importance in the case of chains of solids and they are worth explaining and commenting on. The case of a set (D) composed of two solids (S1) and (S2) articulated to each other at a common point provides an appropriate illustration suitable for highlighting their application.

Figure 1.1. Set of two disjoint solids

Let us begin by recalling the two basic principles that are applied when studying chains of solids.

1.2.1. Principle of effort generators

Two effort generators S´ and S?, considered disjoint, i.e. such that S´ n S? = Ø, act jointly on the effort receiver R according to the diagram shown in Figure 1.2.

Figure 1.2. Effort generation diagram

The efforts exerted by S´ on R are represented by the torsor {S R}, those exerted by S? on R are represented by the torsor {S R?}. and those exerted by uniting two effort generators S´ and S? on the same effort receiver R are represented by the torsor {S´ ? S? R} such that:

1.2.2. Principle of effort receivers

The effort generator acts simultaneously on the two effort receivers R´ and R?, disjoint (R´ n R? = Ø) according to the diagram shown in Figure 1.3.

Figure 1.3. Effort reception diagrams

The efforts exerted by S on R´ are represented by the torsor {S R´}, those exerted by S on R? are represented by the torsor {S R?} and those that S exerts jointly on the two effort receivers R´ and R? are represented by the torsor {S R´ ? R?} such that:

1.2.3. Applying the fundamental principle of dynamics

In the Galilean frame < g >, consider the set (D) = (S1) ? (S2) To simplify the expression of the torsor equations that result from applying the fundamental principle of dynamics to the different moving bodies that may be considered, the two solids will be noted 1 and 2, respectively. These equations also involve the external environment of these bodies: represents the external environment of (D), that is, everything within its field of evolution, with the exception of those constituting (D), strictly excluding (S1) and (S2), which are also disjoint.

We must also consider that the reduction elements of a dynamic torsor are summations achieved using the particles M of a solid, which makes it possible to write, insofar as the two solids are disjoint,

1.2.3.1. Applying the fundamental principle to solid 1

As the frame is Galilean, the fundamental principle of dynamics can be applied to the movement of solid 1 with the equation

Note that is the outside of 1, composed of all that is not (S1), and we have

hence, according to the principle of effort generators

The fundamental principle of dynamics applied to the movement of (S1) is therefore written as

By separating the actions expressed by each of the two torsors into known efforts and links, we obtain an exploitable expression of the fundamental principle of dynamics applied to the movement of (S1)

1.2.3.2. Applying the fundamental principle to solid 2

As the frame <g> is Galilean, the fundamental principle of dynamics can be applied to the movement of solid 2 with the equation

Note that , is the outside of 2, composed of all that is not (S2), and we have

Hence, according to the principle of effort generators

The fundamental principle of dynamics applied to the movement of (S2) is therefore written as

Thus, by separating each of the two torsors according to the known efforts and the links, we obtain:

1.2.3.3. Fundamental principle applied to the set (D)

As it is assumed, in mechanics, that effort torsors may be separated according to the physical laws (gravitation, electromagnetism and contact forces), to which a mechanical set is subjected, the following expression expresses in short the application of the fundamental principle of dynamics to the movement of (D) in the Galilean frame <g>

where D = 1?2.

According to the principle of effort receivers

that is, after separating into known efforts and links:

1.2.4. Theorem of mutual actions

The sum of the two equations obtained by application of the fundamental principle of dynamics to each solid (S1) and (S2), respectively, gives us:

Consequently

With, in addition:

The combination of these different equations results in the expression of the theorem of mutual actions that two solids exert on one another

If the separation into known efforts and links, conducted previously, is applied to this relation, this expression is then written as

In principle, whatever the physical law involved in the given or calculable actions exerted by solid (S2) on solid (S1), we can write:

We therefore deduce reciprocity in the link efforts that are exerted between the two solids

1.2.5. Summary of equations obtained

For the set (D) = (S1)?(S2), to recap:

• - two independent torsor equations, i.e. 12 scalar equations;
• - three independent link torsors, unknown, , , representing 18 scalar unknowns;
• - p location parameters.

The 12 scalar equations identified above include 18+p unknowns. As a result, in a system corresponding to a set of two solids, there is a deficit of

18+p-12=6+p equations.

In the specific scenario where one of the three link torsors is null, the deficit is reduced to p equations.

1.3. Study of the movement of a chain of solids (case of three solids)

1.3.1. Applying the fundamental principle of dynamics

In order to illustrate the principles that apply to the study of the movement of several solids interconnected by means of an articulated device according to the rule presented above, we will explore further the case of three solids constituting a chain comprising neither loops nor branches, therefore meeting the following conditions:

The torsor equations translating the movement of each solid in the pseudo-Galilean frame < ? > are written as

The three solids are disjoint: given the composition of the chain and applying what was shown in section 1.2, according to the theorem of mutual actions, we can write that

We can furthermore deduce the torsor equation of the fundamental principle of dynamics by applying it to the mechanical set (D)

The external and...