NodePointSlope12

A 3D point/slope vector node for thin ANCF (absolute nodal coordinate formulation) plate 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 in-plane coordinates of the plate.

Interface

Provides: Position, Orientation, PointSlope12
Node markers that can be attached: MarkerNodePosition, MarkerNodeRigid, MarkerNodeRotationCoordinate
Objects that take this node: ObjectMassPoint, ObjectBeamGeometricallyExact, ObjectANCFThinPlate

Parameters

The parameters of the item; in a dictionary, its type is ‘PointSlope12’:

name	type	size	default	description
name	String		‘’	node’s unique name
referenceCoordinates	Vector9D	9	[0.,0.,0.,1.,0.,0.,0.,1.,0.]	reference coordinates (x-pos,y-pos,z-pos; x-slopeX, y-slopeX, z-slopeX; x-slopeY, y-slopeY, z-slopeY) 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	VNodePointSlope12			parameters for visualization of item

Visualization parameters

The parameters of VNodePointSlope12, 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 \(\rv_x^\prime\)	global
6, 7, 8	\(q_6,\,q_7,\,q_8\)	ODE2	change of the slope vector \(\rv_y^\prime\)	global

Configuration

\[ \rv = \rv\cRef + [q_0,\,q_1,\,q_2]\tp, \quad \rv_x^\prime = \rv_{x,\mathrm{ref}}^\prime + [q_3,\,q_4,\,q_5]\tp, \quad \rv_y^\prime = \rv_{y,\mathrm{ref}}^\prime + [q_6,\,q_7,\,q_8]\tp . \]

The slope vectors

The two slope vectors are the derivatives of the position of the mid-surface of a thin plate with respect to its two in-plane coordinates, \(\rv_x^\prime = \partial \rv / \partial x\) and \(\rv_y^\prime = \partial \rv / \partial y\). In a flat plate in the global \(x\)-\(y\) plane they are \([1,\;0,\;0]\tp\) and \([0,\;1,\;0]\tp\). They span the tangent plane of the mid-surface, and their lengths and angle carry its in-plane strains. The default reference coordinates are those of a flat plate in the global \(x\)-\(y\) plane; 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 ObjectANCFThinPlate, whose element scales the slopes by slopesScalingX and slopesScalingY.

Rotation

MarkerNodeRigid and the output variables of the rotation take an orthonormal frame of the slopes: \(\rv_x\) normalized is its \(x\)-axis, \(\rv_y\) orthogonalized against it and normalized its \(y\)-axis, and the \(z\)-axis - the normal of the surface - is their cross product. The angular velocity \(\tomega\) and the rotation Jacobian are not the derivatives of this frame but the least-squares fit \(\tomega = \left(-\tilde\rv_x\tilde\rv_x - \tilde\rv_y\tilde\rv_y\right)^{-1}\left(\tilde\rv_x\dot\rv_x + \tilde\rv_y\dot\rv_y\right)\), symmetric in the two slopes; the two agree for slopes that keep their lengths and their right angle (#2219). The derivative of \(\Jm_{rot}\tp\vv\) with respect to the coordinates is not implemented: a connector whose Jacobian needs it, e.g. a RigidBodySpringDamper at the node with an implicit solver, raises an error; a torque and a joint do not need it.

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 square plate clamped at one edge, from ANCF thin plate elements with position and slopes r_x, r_y
from exudyn.shells import ShellMesh
plate = ShellMesh(vertices=[[0,0,0],[1,0,0],[1,1,0],[0,1,0]], numberOfElementsX=2, numberOfElementsY=2,
                  youngsModulus=2e9, poissonsRatio=0, density=1000, thickness=0.01)
plate.CreateANCFThinPlateElements(mbs) #adds a NodePointSlope12 per mesh point
for node in plate.boundaryNodeNumbers['left']:
    mNode = mbs.AddMarker(MarkerNodeRigid(nodeNumber=node))
    mbs.CreateGenericJoint(itemNumbers=[oGround, mNode]) #clamped: position and orientation
for element in plate.elementNumbers:
    mbs.AddLoad(LoadMassProportional(markerNumber=mbs.AddMarker(MarkerBodyMass(bodyNumber=element)),
                                     loadVector=[0,0,-9.81]))

mbs.Assemble()
mbs.SolveStatic()

#a corner of the free edge; compare q*L^4/(8*D) = 0.0736 of a cantilever strip, D = E*h^3/12
corner = plate.vertexNodeNumbers[1]
exu.sys['testResult'] = mbs.GetNodeOutput(corner, exu.OutputVariableType.Displacement)[2]