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(sec-item-nodepoint2dslope1)=
# NodePoint2DSlope1

A 2D point/slope vector node for planar Bernoulli-Euler ANCF (absolute nodal coordinate formulation) beam elements. The node has 4 displacement degrees of freedom (2 for displacement of point node and 2 for the slope vector 'slopex'); all coordinates lead to second order differential equations; the slope vector defines the directional derivative w.r.t the local axial (x) coordinate, denoted as $()^\prime$; in straight configuration aligned at the global x-axis, the slope vector reads $\rv^\prime=[r_x^\prime\;\;r_y^\prime]^T=[1\;\;0]^T$.

## Interface

- Python names: `NodePoint2DSlope1` or `Point2DS1`, and `VPoint2DS1` for its visualization
- Provides: `Position2D`, `Orientation2D`, `Point2DSlope1`
- Node markers that can be attached: [](#sec-item-markernodeposition), [](#sec-item-markernoderigid)
- Objects that take this node: [](#sec-item-objectmasspoint2d), [](#sec-item-objectrigidbody2d), [](#sec-item-objectancfcable2d), [](#sec-item-objectbeamgeometricallyexact2d)

## Parameters

The parameters of the item; in a dictionary, its type is 'Point2DSlope1':

| name | type | size | default | description |
|---|---|---|---|---|
| **name** | String |  | '' | node's unique name |
| **referenceCoordinates** | Vector4D | 4 | [0.,0.,1.,0.] | reference coordinates (x-pos,y-pos; x-slopex, y-slopex) of node; global position of node without displacement |
| **initialCoordinates** | Vector4D | 4 | [0.,0.,0.,0.] | initial displacement coordinates: ux, uy and x/y 'displacements' of slopex |
| **initialVelocities** | Vector4D | 4 | [0.,0.,0.,0.] | initial velocity coordinates |
| **visualization** | VNodePoint2DSlope1 |  |  | parameters for visualization of item |

## Visualization parameters

The parameters of `VNodePoint2DSlope1`, 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 vector.

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 | $\LU{0}{\pv}\cConfig = [p_0,\, p_1,\,0]\cConfig\tp$ | global 3D position vector of node (=displacement+reference position) |
| Displacement | $\LU{0}{\uv}\cConfig = [q_0,\, q_1,\,0]\cConfig\tp$ | global 3D displacement vector of node |
| Velocity | $\LU{0}{\vv}\cConfig = [\dot q_0,\,\dot q_1,\,0]\cConfig\tp$ | global 3D velocity vector of node |
| Acceleration | $\LU{0}{\av}\cConfig = [\ddot q_0,\,\ddot q_1,\,0]\cConfig\tp$ | global 3D acceleration vector of node |
| CoordinatesTotal |  | displacement plus reference coordinates of node |
| Coordinates |  | coordinates vector of node (2 displacement coordinates + 2 slope vector coordinates) |
| Coordinates_t |  | velocity coordinates vector of node (derivative of the 2 displacement coordinates + 2 slope vector coordinates) |
| Coordinates_tt |  | acceleration coordinates vector of node (derivative of the 2 displacement coordinates + 2 slope vector coordinates) |

(description-nodepoint2dslope1)=
## Detailed description

### Coordinates

| index | symbol | kind | meaning | frame |
|---|---|---|---|---|
| 0, 1 | $q_0,\,q_1$ | {ref}`ODE2 <ODE2>` | displacement of the node position $\rv$ | global |
| 2, 3 | $q_2,\,q_3$ | {ref}`ODE2 <ODE2>` | change of the slope vector $\rv^\prime$ | global |

### Configuration

Position and slope vector are the sums of reference and current values,

$$
\rv = \rv\cRef + [q_0,\,q_1]\tp, \quad \rv^\prime = \rv^\prime\cRef + [q_2,\,q_3]\tp .
$$

### The slope vector

The slope vector is the derivative of the position of the beam axis with respect to the axial
coordinate $x$ of the element in its reference configuration, $\rv^\prime = \partial \rv / \partial x$.
In a straight beam along the global $x$-axis it is $[1,\;0]\tp$, which is the default of the
reference coordinates. Its **direction** is the tangent of the beam axis, and so the rotation of the
cross section in a Bernoulli-Euler beam; its **length** is one plus the axial strain,
$\varepsilon = \|\rv^\prime\| - 1$, as `ObjectANCFCable2D` computes it. A reference slope of length
other than one therefore describes a pre-strained beam.

### Frame and interpretation

All four coordinates are global, with no rotation parameters: this is the absolute nodal coordinate
formulation. The node is used by the planar ANCF cable elements, `ObjectANCFCable2D` and
`ObjectALEANCFCable2D`, which share the node between neighbouring elements; the beam utilities
(`exudyn.beams`) create them.

### Action on the equations of motion

The four coordinates lead to four {ref}`ODE2 <ODE2>` equations, which the element provides. A force at the
node enters the first two; `MarkerNodeRigid` sees the node as a position with the orientation of the
slope vector.

(miniexample-nodepoint2dslope1)=
## Mini example


```python
#a cantilever of one ANCF cable element: position and slope (r_x) at each node
L = 1; EI = 100; F = -0.1
n0 = mbs.AddNode(NodePoint2DSlope1(referenceCoordinates=[0,0, 1,0])) #position, slope = axis
n1 = mbs.AddNode(NodePoint2DSlope1(referenceCoordinates=[L,0, 1,0]))
mbs.AddObject(ObjectANCFCable2D(nodeNumbers=[n0,n1], length=L, massPerLength=1,
                                bendingStiffness=EI, axialStiffness=1e5))
mGround = mbs.AddMarker(MarkerNodeCoordinate(nodeNumber=nGround, coordinate=0))
for i in [0,1,3]: #clamped: x, y and the y-component of the slope
    mCoord = mbs.AddMarker(MarkerNodeCoordinate(nodeNumber=n0, coordinate=i))
    mbs.AddObject(ObjectConnectorCoordinate(markerNumbers=[mGround, mCoord]))
mTip = mbs.AddMarker(MarkerNodePosition(nodeNumber=n1))
mbs.AddLoad(LoadForceVector(markerNumber=mTip, loadVector=[0,F,0]))

mbs.Assemble()
mbs.SolveStatic()

#the cubic element is exact for a tip load: F*L^3/(3*EI) = -1/3000
exu.sys['testResult'] = mbs.GetNodeOutput(n1, exu.OutputVariableType.Displacement)[1]*1000 #-1/3
```


Relevant Examples (Ex) and TestModels (TM) with weblink to github: [`ALEANCFpipe.py`](https://github.com/jgerstmayr/EXUDYN/blob/master/python/Examples/ALEANCFpipe.py) (Ex), [`ANCFcantileverTestDyn.py`](https://github.com/jgerstmayr/EXUDYN/blob/master/python/Examples/ANCFcantileverTestDyn.py) (Ex), [`ANCFcontactCircle.py`](https://github.com/jgerstmayr/EXUDYN/blob/master/python/Examples/ANCFcontactCircle.py) (Ex), [`ANCFcontactCircle2.py`](https://github.com/jgerstmayr/EXUDYN/blob/master/python/Examples/ANCFcontactCircle2.py) (Ex), [`ANCFmovingRigidbody.py`](https://github.com/jgerstmayr/EXUDYN/blob/master/python/Examples/ANCFmovingRigidbody.py) (Ex), [`ANCFslidingJoint2D.py`](https://github.com/jgerstmayr/EXUDYN/blob/master/python/Examples/ANCFslidingJoint2D.py) (Ex), [`ANCFslidingJoint2Drigid.py`](https://github.com/jgerstmayr/EXUDYN/blob/master/python/Examples/ANCFslidingJoint2Drigid.py) (Ex), [`ANCFswitchingSlidingJoint2D.py`](https://github.com/jgerstmayr/EXUDYN/blob/master/python/Examples/ANCFswitchingSlidingJoint2D.py) (Ex), [`ANCFtestHalfcircle.py`](https://github.com/jgerstmayr/EXUDYN/blob/master/python/Examples/ANCFtestHalfcircle.py) (Ex), [`ANCFtests2.py`](https://github.com/jgerstmayr/EXUDYN/blob/master/python/Examples/ANCFtests2.py) (Ex), [`sliderCrank3DwithANCFbeltDrive.py`](https://github.com/jgerstmayr/EXUDYN/blob/master/python/Examples/sliderCrank3DwithANCFbeltDrive.py) (Ex), [`solverFunctionsTestEigenvalues.py`](https://github.com/jgerstmayr/EXUDYN/blob/master/python/Examples/solverFunctionsTestEigenvalues.py) (Ex), [`createItemNumbersTest.py`](https://github.com/jgerstmayr/EXUDYN/blob/master/python/TestModels/createItemNumbersTest.py) (TM), [`energiesFlexibleBodiesTest.py`](https://github.com/jgerstmayr/EXUDYN/blob/master/python/TestModels/energiesFlexibleBodiesTest.py) (TM), [`energiesTest.py`](https://github.com/jgerstmayr/EXUDYN/blob/master/python/TestModels/energiesTest.py) (TM), ...