3. Expression of kinematic relationships#

3.1. Obtaining kinematic relationships#

The reasoning is based on the transmission of forces between the master node and the slave nodes. First, all components of the master node and all components of the slave nodes are considered to be involved in the relationship (section § 4 discusses the restriction of relationships to certain components).

Let \({T}_{M}\) be the torsor of effort of the master node that we want to distribute into \({T}_{i}\) torsors of effort on the \(n\) slave nodes. The distribution is made using the following assumptions:

  • torsors \({T}_{i}\) come from the same linear combination \(X{T}_{M}\) from the torsor to the master node, where \(X\) is the matrix associated with the linear combination,

  • this linear combination \(X{T}_{M}\) is distributed over the slave nodes by weighting the transport by the distance between the master node and the slave node on the one hand, and a weighting coefficient set by the user on the other hand.

Each \({T}_{i}\) twister is then written:

(3.1)#\[ {T} _ {i} = {W} _ {i} {i} {S} _ {i} X {T} _ {M}\]

The matrix \(X\) of linear coefficients is determined by ensuring that the torsor applied to the master node is identical to the set of torsors \({T}_{i}\) applied to the \(n\) slave nodes. \(X\) should therefore check:

(3.2)#\[ {T} _ {M} =\ sum _ {i=1} ^ {n} {i=1} ^ {i} {i} {M} =\ sum _ {i=1} ^ {n} {S} {S} _ {i} {S} _ {i} {S} _ {i} {T} {S} _ {T} _ {M}\]

Reframing the problem in a matrix way, it turns out that \(X\) is a solution to the following system:

(3.3)#\[\begin{split} \ left [\ begin {array} {ccc} {S} _ {1} _ {1} ^ {T} &\ cdots & {S} _ {n} ^ {T}\ end {array}\ right]\ left [\ begin {array} {array} {S} {S} _ {W} _ {n} _ {n} _ {n] &\ cdots & 0\\ 0&\ cdots & {W} _ {n}\ end {array}\ right]\ left [\ begin {array} {c} {array} {c} {S} _ {1}\\n\n}\ end {array}\ right]\ left [X\ right]\ left [X\ right] =\ left [X\ right] =\ left [{I} _ {d}\ right]\end{split}\]

Where \({I}_{d}\) is the identity matrix, which is:

(3.4)#\[ X\ mathrm {=} {({S} ^ {T} WS)} ^ {\ mathrm {-} 1}\]

Noting \(S\) is the assembly of the \({S}_{i}\) and \(W\) is the assembly of the \({W}_{i}\).

Note that matrix \({S}^{T}WS\) must be invertible. This is ensured by a relevant choice of master and slave degrees of freedom by the user. For example, if you want to transmit a complete torsor from a three-dimensional model to a beam model, you must ensure that the slave nodes associated with the three-dimensional model are not aligned. If this is the case, since the nodes of a three-dimensional model do not carry the degrees of freedom of rotation, a mechanism is created such that the rotation around the axis formed by the alignment of slave nodes is not blocked.

We now note \({B}_{i}={W}_{i}{S}_{i}X\). From relationship \({T}_{i}={B}_{i}{T}_{M}\), we use the equality of virtual works to generate a constraint on the degrees of freedom from the relationship on the torsors. The work of torsor \({T}_{M}\) on the degrees of freedom of the master node \({d}_{M}\) must be equal to that of the various torsors \({T}_{i}\) on each of the degrees of freedom \({d}_{i}\) of the slave nodes, i.e.:

(3.5)#\[ {T} _ {M} ^ {T} {d} {d} _ {M} =\ sum _ {i=1} ^ {n} {T} _ {i} ^ {d} _ {i} = {T} _ {i} = {T} _ {M} _ {M} _ {M} ^ {M} _ {T} _ {i}\ left (\ sum _ {i=1} ^ {n} {B} _ {B} _ {b} _ {b} _ {i} _ {i} _ {i} {i}} = {T} _ {i} {i}} = {T} _ {i}} = {T} _ {i}} = {T} _ {i} {i}} = {T} _ {i}} = {T} _ {i}} right)\]

The kinematic relationship applied between the master node and the slave nodes is therefore written as:

(3.6)#\[ {d} _ {M} =\ sum _ {i=1} ^ {n} {b} _ {i} ^ {T} {d} _ {i}\]

Interpreting the cutscene:

By using the expression (), we note that we « overturned » the initial problem, since if the initial problem consisted in distributing the torsor of effort at the master node over the slave nodes, in kinematics, this is equivalent to expressing the degrees of freedom \({d}_{M}\) of the master node as a linear combination of the degrees of freedom \({d}_{i}\) of the slave nodes.

By analyzing more closely the kinematics associated with the matrix \(S\), we see that each of the columns corresponds to a rigid body mode (overall movement) of the set of slave nodes. The first three columns are the translational movements (unit displacement in one direction, zero in the other two), and the last three columns are the rotational movements around the master node.

Let us consider the simple case of a uniform weighting \({w}_{i}=1\text{}\forall i\), with a correction factor \({L}_{c}\) assumed to be unitary. Under these conditions, \(W\) is an identity matrix, the relationship () becomes:

(3.7)#\[\begin{split} {d} _ {M} = {({S} ^ {T} S)} ^ {T} S)} ^ {T}}} ^ {T}\ left [\ begin {array} {c} {d} _ {1}\\ ▼} _ {n}\ end {array} {d} _ {n}\ end {array}\ right]\end{split}\]

The relationship () is the least-square solution to a minimization problem that is written as:

(3.8)#\[ {d} _ {M} =\ underset {{d} _ {m}} {\ mathit {argMin}} {\ mathit {argMin}}} {S {d} _ {m} -\ left [\ begin {array} {c} {d} {d} _ {n}\ end {array} _ {m} -\ left [\ begin {array} {array} {m} {m} -\ left [\ begin {array} {{array} {{m} {m} -\ left [\ begin {array} {{array} {{{m}} {c} {d} _ {d} _ {m} -\ left [\ begin {array} {{array} {{m} {\]

In other words, the relationship () seeks to determine the best possible linear combination \({d}_{M}\) of rigid body modes that pass through all the slave degrees of freedom \({d}_{i}\). When the weighting is uniform, the linear constraint constructed by the relationship () therefore consists in imposing the mean rigid body movement of all the slave nodes on the master node. The addition of a \({w}_{i}\) weighting on a node leads to amplifying or reducing the relative influence of this node in the movement of the master node.

The main advantage of the « RBE3 » link is to avoid over-constraining the model. Unlike the case of solid relationships, the slave nodes of a RBE3 relationship can continue to have relative movements.

3.2. Dimensions of matrices#

Note \(\mathrm{NDDLES}\) the total number of slave degrees of freedom. The matrices that are manipulated have the following dimensions:

  • Matrix \(W\) (assembly of \({W}_{i}\)): \(\mathrm{NDDLES}\) rows, \(\mathrm{NDDLES}\) columns;

  • Matrix \(S\) (assembly of \({S}_{i}\)): \(\mathrm{NDDLES}\) rows, six columns;

  • Matrix \({S}^{T}WS\): six rows, six columns;

  • Matrix \(X\): six rows, six columns;

  • Matrix \(B\): \(\mathrm{NDDLES}\) rows, six columns.