Rigid Link
^^^^^^^^^^

This command is used to construct a single MP_Constraint object.

.. function:: rigidLink $type $retainedNodeTag $constrainedNodeTag

.. csv-table:: 
   :header: "Argument", "Type", "Description"
   :widths: 10, 10, 40

   $type, |string|,  "| string-based argument for rigid-link type:
   | **bar** only the translational degree-of-freedom will be constrained to be exactly the same as those at the master node 
   | **beam** both the translational and rotational degrees of freedom are constrained."
   $retainedNodeTag, |integer|, integer tag identifying the retained node
   $constrainedNodeTag, |integer|, integer tag identifying the constrained node

.. note::
   
   1. By retained node, we mean the node who's degrees-of-freedom are retained in the system of equations. The constrained nodes degrees-of-freedom need not appear in the system (depending on the constraint handler). 
   2. For 2d and 3d problems with a **beam** type link, the constraint matrix (that matrix relating the responses at constrained node, :math:`U_c`, to responses at retained node, :math:`U_r`, i.e. :math:`U_c = C_{cr} U_r`, is constructed assuming small rotations. For 3d problems this results in the following constraint matrix:

      .. math::
        :label: rigidConstraintBeam3D

	\begin{bmatrix}
		1 & 0 & 0 & 0 & \Delta Z & -\Delta Y \\
		0 & 1 & 0 & -\Delta Z & 0 & \Delta X \\
		0 & 0 & 1 & \Delta Y & -\Delta X & 0 \\
		0 & 0 & 0 & 1 & 0 & 0 \\
		0 & 0 & 0 & 0 & 1 & 0 \\
		0 & 0 & 0 & 0 & 0 & 1 
	\end{bmatrix}
   

      For 2d problems, the constraint matrix is the following:

      .. math::
        :label: rigidConstraintBeam3D
      
	\begin{bmatrix}
		1 & 0 & -\Delta Y \\
		0 & 1 & \Delta X \\
		0 & 0 & 1
	\end{bmatrix}

      where :math:`\Delta X` is x coordinate of constrained node minus the x coordinate of the retained node. :math:`\Delta Y` and :math:`\Delta Z` being similarily defined for y and z coordinates of the nodes.


   3. For 2d and 3d problems with a **rod** type link the constraint matrix, that which matrix relates the responses at translational degrees-of-freedom at the constrained node to corresponding responses at retained node, is the identity matrix. For 3d problems this results in the following constraint matrix:

   .. math::
        :label: rigidConstraintBeam3D

	\begin{bmatrix}
		1 & 0 & 0  \\
		0 & 1 & 0  \\
		0 & 0 & 1 
	\end{bmatrix}
   

   For 2d problems, the constraint matrix is the following:

      .. math::
        :label: rigidConstraintBeam3D
      
	\begin{bmatrix}
		1 & 0 \\
		0 & 1 \\
	\end{bmatrix}   

   4. The rod constraint can also be generated using the equalDOF command.
      
.. admonition:: Example:

   The following command will constrain node **3** to move rigidly following rules for small rotations to displacements and rotations at node **2** is

   1. **Tcl Code**

   .. code-block:: none

      rigidLink beam 2 3

   2. **Python Code**

   .. code-block:: python

      rigidLink('beam',2,3)