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 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: MarkerNodePosition, MarkerNodeRigid, MarkerNodeRotationCoordinate
Objects that take this node: ObjectMassPoint, ObjectANCFBeam, 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; with nodes.showNodalSlopes, also its slope vectors.

Settings beyond those of all nodes: nodes.showNodalSlopes.

Output variables

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

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

Detailed description

Coordinates

index	symbol	kind	meaning	frame
0, 1, 2	\(q_0,\,q_1,\,q_2\)	ODE2	displacement of the node position \(\rv\)	global
3, 4, 5	\(q_3,\,q_4,\,q_5\)	ODE2	change of the slope vector \(\rv_y\)	global
6, 7, 8	\(q_6,\,q_7,\,q_8\)	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 ODE2 equations, which the element provides; a force at the node enters the first three.

Mini example

#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 (TM), ANCFBeamTest.py (TM), ANCFCableBeamDampingTest.py (TM), energiesFlexibleBodiesTest.py (TM), geometricallyExactBeamTest.py (TM), rightAngleFrame.py (TM)