<!-- GENERATED by tools/generators/itemDocsEmitter.py from definitions/ - do not edit -->
```{raw} latex
\clearpage
```

(sec-item-nodepointslope23)=
# NodePointSlope23

A 3D point/slope vector node for spatial, shear and cross-section deformable ANCF (absolute nodal coordinate formulation) beam elements, with 3 position and 2 $\times$ 3 slope coordinates, all {ref}`ODE2 <ODE2>`; the slope vectors are the derivatives of the position with respect to the two cross section coordinates $y$ and $z$.

## Interface

- Provides: `Position`, `Orientation`, `PointSlope23`
- Node markers that can be attached: [](#sec-item-markernodeposition), [](#sec-item-markernoderigid), [](#sec-item-markernoderotationcoordinate)
- Objects that take this node: [](#sec-item-objectmasspoint), [](#sec-item-objectancfbeam), [](#sec-item-objectbeamgeometricallyexact)

## Parameters

The parameters of the item; in a dictionary, its type is 'PointSlope23':

| name | type | size | default | description |
|---|---|---|---|---|
| **name** | String |  | '' | node's unique name |
| **referenceCoordinates** | Vector9D | 9 | [0.,0.,0.,0.,1.,0.,0.,0.,1.] | reference coordinates (x-pos,y-pos,z-pos; x-slopey, y-slopey, z-slopey; x-slopez, y-slopez, z-slopez) of node; global position of node without displacement |
| **initialCoordinates** | Vector9D | 9 | [0.,0.,0.,0.,0.,0.,0.,0.,0.] | initial displacement coordinates relative to reference coordinates |
| **initialVelocities** | Vector9D | 9 | [0.,0.,0.,0.,0.,0.,0.,0.,0.] | initial velocity coordinates |
| **visualization** | VNodePointSlope23 |  |  | parameters for visualization of item |

## Visualization parameters

The parameters of `VNodePointSlope23`, given as `visualization`:

| name | type | size | default | description |
|---|---|---|---|---|
| **show** | Bool |  | True | set true, if item is shown in visualization and false if it is not shown |
| **drawSize** | float |  | -1. | drawing size (diameter, dimensions of underlying cube, etc.) for item; size == -1.f means that default size is used |
| **color** | Float4 | 4 | [-1.,-1.,-1.,-1.] | Default RGBA color for nodes; 4th value is alpha-transparency; R=-1.f means, that default color is used |

## Drawing

Drawn as [all nodes](#sec-drawing-nodes); with `nodes.showNodalSlopes`, also its slope vectors.

Settings beyond those of all nodes: [`nodes.showNodalSlopes`](#sec-vsettingsnodes).

## Output variables

Available as `OutputVariableType` in sensors, `Get...Output()` and other functions:

```{tabularcolumns} \Y{0.25}\Y{0.25}\Y{0.5}
```

| output variable | symbol | description |
|---|---|---|
| Position | $\LU{0}{\pv}\cConfig = \LU{0}{[p_0,\, p_1,\, p_2]}\cConfig\tp$ | global 3D position vector of node (=displacement+reference position) |
| Displacement | $\LU{0}{\uv}\cConfig = \LU{0}{[q_0,\, q_1,\, q_2]}\cConfig\tp$ | global 3D displacement vector of node |
| Velocity | $\LU{0}{\vv}\cConfig = \LU{0}{[\dot q_0,\,\dot q_1,\,\dot q_2]}\cConfig\tp$ | global 3D velocity vector of node |
| Acceleration | $\LU{0}{\av}\cConfig = \LU{0}{[\ddot q_0,\,\ddot q_1,\,\ddot q_2]}\cConfig\tp$ | global 3D acceleration vector of node |
| CoordinatesTotal |  | displacement plus reference coordinates of node |
| Coordinates |  | coordinate vector of node (relative to reference configuration) |
| Coordinates_t |  | velocity coordinates vector of node |
| Coordinates_tt |  | acceleration coordinates vector of node |
| RotationMatrix | $[A_{00},\,A_{01},\,A_{02},\,A_{10},\,\ldots,\,A_{21},\,A_{22}]\cConfig\tp$ | vector with 9 components of the rotation matrix $\LU{0b}{\Rot}\cConfig$ in row-major format, in any configuration; the rotation matrix transforms local ($b$) to global (0) coordinates |
| Rotation | $[\varphi_0,\,\varphi_1,\,\varphi_2]\tp\cConfig$ | vector with 3 components of the Euler / Tait-Bryan angles in xyz-sequence |
| AngularVelocity | $\LU{0}{\tomega}\cConfig = \LU{0}{[\omega_0,\,\omega_1,\,\omega_2]}\cConfig\tp$ | global 3D angular velocity vector of node |
| AngularVelocityLocal | $\LU{b}{\tomega}\cConfig = \LU{b}{[\omega_0,\,\omega_1,\,\omega_2]}\cConfig\tp$ | local (body-fixed) 3D angular velocity vector of node |

(description-nodepointslope23)=
## Detailed description

### Coordinates

| index | symbol | kind | meaning | frame |
|---|---|---|---|---|
| 0, 1, 2 | $q_0,\,q_1,\,q_2$ | {ref}`ODE2 <ODE2>` | displacement of the node position $\rv$ | global |
| 3, 4, 5 | $q_3,\,q_4,\,q_5$ | {ref}`ODE2 <ODE2>` | change of the slope vector $\rv_y$ | global |
| 6, 7, 8 | $q_6,\,q_7,\,q_8$ | {ref}`ODE2 <ODE2>` | change of the slope vector $\rv_z$ | global |

### Configuration

$$
\rv = \rv\cRef + [q_0,\,q_1,\,q_2]\tp, \quad
\rv_y = \rv_{y,\mathrm{ref}} + [q_3,\,q_4,\,q_5]\tp, \quad
\rv_z = \rv_{z,\mathrm{ref}} + [q_6,\,q_7,\,q_8]\tp .
$$

### The slope vectors

Unlike the slopes of the cable nodes, the two slope vectors of this node are **not** taken along the
beam axis: they are the derivatives of the position with respect to the two **cross section**
coordinates $y$ and $z$, $\rv_y = \partial \rv / \partial y$ and $\rv_z = \partial \rv / \partial z$,
so that a point of the cross section at $(y,\,z)$ is at $\rv + y\,\rv_y + z\,\rv_z$, as
`ObjectANCFBeam` computes it. The axial direction follows from the positions of the two nodes of the
element. In a beam along the global $x$-axis the slope vectors are $[0,\;1,\;0]\tp$ and
$[0,\;0,\;1]\tp$; they span the cross section, and their lengths and angle carry its deformation -
the element is shear and cross section deformable. The default reference coordinates are those
of a beam along the global $x$-axis; a model gives the reference coordinates of every node.

### Frame and interpretation

All nine coordinates are global (absolute nodal coordinates). The node is used by `ObjectANCFBeam`.

### Rotation

`MarkerNodeRigid` and the output variables of the rotation take the orthonormal frame of the slopes:
$\rv_z$ normalized is its $z$-axis, $\rv_y$ orthogonalized against it and normalized its $y$-axis, and the
$x$-axis is their cross product. The angular velocity $\tomega$ and the rotation Jacobian
$\Jm_{rot} = \partial \tomega / \partial \dot \qv$ are the derivatives of this frame, so joints and torques act
consistently with it (#2763).

### Action on the equations of motion

The nine coordinates lead to nine {ref}`ODE2 <ODE2>` equations, which the element provides; a force at the
node enters the first three.

(miniexample-nodepointslope23)=
## Mini example


```python
#a cantilever of four ANCF beam elements: position and the slopes r_y, r_z of the cross section
L = 1; nElements = 4; F = -0.1
section = exu.BeamSection()
section.stiffnessMatrix = np.diag([1e5, 1e4, 1e4, 100, 100, 100]) #EA, GA_y, GA_z, GJ, EI_y, EI_z
section.massPerLength = 1
section.inertia = 0.01*np.eye(3)
mGround = mbs.AddMarker(MarkerNodeCoordinate(nodeNumber=nGround, coordinate=0))
n0 = mbs.AddNode(NodePointSlope23(referenceCoordinates=[0,0,0, 0,1,0, 0,0,1]))
for i in range(9): #clamped: position and both slopes
    mbs.AddObject(ObjectConnectorCoordinate(markerNumbers=[mGround,
                  mbs.AddMarker(MarkerNodeCoordinate(nodeNumber=n0, coordinate=i))]))
for k in range(nElements):
    n1 = mbs.AddNode(NodePointSlope23(referenceCoordinates=[L*(k+1)/nElements,0,0, 0,1,0, 0,0,1]))
    mbs.AddObject(ObjectANCFBeam(nodeNumbers=[n0,n1], length=L/nElements, sectionData=section))
    n0 = n1
mTip = mbs.AddMarker(MarkerNodeRigid(nodeNumber=n1))
mbs.AddLoad(LoadForceVector(markerNumber=mTip, loadVector=[0,F,0]))

mbs.Assemble()
mbs.SolveStatic()

#converges to F*L^3/(3*EI_z) + F*L/GA_y = -0.3433e-3 as the number of elements grows
exu.sys['testResult'] = mbs.GetNodeOutput(n1, exu.OutputVariableType.Displacement)[1]*1000 #-0.338
```


Relevant Examples (Ex) and TestModels (TM) with weblink to github: [`ANCFBeamEigTest.py`](https://github.com/jgerstmayr/EXUDYN/blob/master/python/TestModels/ANCFBeamEigTest.py) (TM), [`ANCFBeamTest.py`](https://github.com/jgerstmayr/EXUDYN/blob/master/python/TestModels/ANCFBeamTest.py) (TM), [`ANCFCableBeamDampingTest.py`](https://github.com/jgerstmayr/EXUDYN/blob/master/python/TestModels/ANCFCableBeamDampingTest.py) (TM), [`energiesFlexibleBodiesTest.py`](https://github.com/jgerstmayr/EXUDYN/blob/master/python/TestModels/energiesFlexibleBodiesTest.py) (TM), [`geometricallyExactBeamTest.py`](https://github.com/jgerstmayr/EXUDYN/blob/master/python/TestModels/geometricallyExactBeamTest.py) (TM), [`rightAngleFrame.py`](https://github.com/jgerstmayr/EXUDYN/blob/master/python/TestModels/rightAngleFrame.py) (TM)