5. 2DB and 3DB case processing#

There are four scenarios for the cloud of « solidified » nodes:

  • Volumic: there are at least 4 non-coplanar nodes (within one epsilon);

  • Plan: there are at least 3 non-aligned nodes (within one epsilon);

  • Segment: there are at least 2 nodes that are not combined (within one epsilon);

  • Punctual: all the nodes are geometrically combined (to the nearest epsilon)

The routine that determines the case also returns the 1, 2, 3 or 4 nodes that allow you to « define » the solid. The kinematic relationships that we write depend (quite slightly) on the case.

Let’s take the example of the « Plan » case in 3D.

We have three nodes \(A\), \(B\), \(C\) that are not aligned. We write that the square of the three distances \(\mathit{AB}\), \(\mathit{AC}\), and \(\mathit{BC}\) remains constant during movement. These three relationships are non-linear. They are quadratic and can easily be derived to obtain the tangent linearized problem.

For each node (\(M\)) different from \(A\), \(B\), \(C\), the barycentric coordinates \(\alpha\), \(\beta\) and \(\gamma\) are calculated such that:

\[\]

: label: eq-4

M=alpha A+beta B+gamma C

Then, we write the three linear relationships:

\[\]

: label: eq-5

U (M) =alpha U (A) +beta U (B) +gamma U (C)

In total, if the cloud has \(n\ge 3\) nodes, we write:

  • Three quadratic relationships (easily linearized);

  • \(3(n-3)\) linear relationships.

The cloud had \(\mathrm{3n}\) degrees of freedom. We wrote \(3n-6\) independent relationships. It has six degrees of freedom left, which corresponds to the number of possible movements for a 3D solid.

Notes:

  • For the « segment » case in 3D, for example, we write \(\mathrm{3n}-5\) relationships, which means that the solid has only five possible movements, which is normal because the rotation of the solid around the line is indeterminate;

  • Each solid generates only a few non-linear relationships (a maximum of six).