ObjectJointSliding2D

A specialized sliding joint (without rotation) in 2D between a Cable2D (marker1) and a position-based marker (marker0); the data coordinate x[0] provides the current index in slidingMarkerNumbers, and x[1] the local position in the cable element at the beginning of the timestep.

Additional information for ObjectJointSliding2D:

This Object has/provides the following types = Connector, Constraint
Requested Marker type = _None
Requested Node type = GenericData
Short name for Python = SlidingJoint2D
Short name for Python visualization object = VSlidingJoint2D

The item ObjectJointSliding2D with type = ‘JointSliding2D’ has the following parameters:

name [type = String, default = ‘’]:

constraints’s unique name
markerNumbers [$[m0,m1]\tp$, type = ArrayMarkerIndex, default = [ invalid [-1], invalid [-1] ]]:

marker m0: position or rigid body marker of mass point or rigid body; marker m1: updated marker to Cable2D element, where the sliding joint currently is attached to; must be initialized with an appropriate (global) marker number according to the starting position of the sliding object; this marker changes with time (PostNewtonStep)
slidingMarkerNumbers [$[m_{s0}, \ldots, m_{sn}]\tp$, type = ArrayMarkerIndex, default = []]:

these markers are used to update marker m1, if the sliding position exceeds the current cable’s range; the markers must be sorted such that marker $m_{si}$ at x=cable(i).length is equal to marker(i+1) at x=0 of cable(i+1)
slidingMarkerOffsets [$[d_{s0}, \ldots, d_{sn}]$, type = Vector, default = []]:

this list contains the offsets of every sliding object (given by slidingMarkerNumbers) w.r.t. to the initial position (0): marker m0: offset=0, marker m1: offset=Length(cable0), marker m2: offset=Length(cable0)+Length(cable1), …
nodeNumber [$n_{GD}$, type = NodeIndex, default = invalid (-1)]:

node number of a NodeGenericData for 1 dataCoordinate showing the according marker number which is currently active and the start-of-step (global) sliding position
classicalFormulation [type = Bool, default = True]:

True: uses a formulation with 3 (+1) equations, including the force in sliding direction to be zero; forces in global coordinates, only index 3; False: use local formulation, which only needs 2 (+1) equations and can be used with index 2 formulation
constrainRotation [type = Bool, default = False]:

True: add constraint on rotation of marker m0 relative to slope (if True, marker m0 must be a rigid body marker); False: marker m0 body can rotate freely
axialForce [$f_\mathrm{ax}$, type = Real, default = 0]:

ONLY APPLIES if classicalFormulation==True; axialForce represents an additional sliding force acting between beam and marker m0 body in axial (beam) direction; this force can be used to drive a body on a beam, but can only be changed with user functions.
activeConnector [type = Bool, default = True]:

flag, which determines, if the connector is active; used to deactivate (temporarily) a connector or constraint
visualization [type = VObjectJointSliding2D]:

parameters for visualization of item

The item VObjectJointSliding2D has the following parameters:

show [type = Bool, default = True]:

set true, if item is shown in visualization and false if it is not shown
drawSize [type = float, default = -1.]:

drawing size = radius of revolute joint; size == -1.f means that default connector size is used
color [type = Float4, default = [-1.,-1.,-1.,-1.]]:

RGBA connector color; if R==-1, use default color

DESCRIPTION of ObjectJointSliding2D

The following output variables are available as OutputVariableType in sensors, Get…Output() and other functions:

Position:

position vector of joint given by marker0
Velocity:

velocity vector of joint given by marker0
SlidingCoordinate:

global sliding coordinate along all elements; the maximum sliding coordinate is equivalent to the reference lengths of all sliding elements
Force:

joint force vector (3D)

Definition of quantities


intermediate variables	symbol	description
data node	${\mathbf{x}}=[x_{data0},\,x_{data1}]\tp$	coordinates of node with node number $n_{GD}$
data coordinate 0	$x_{data0}$	the current index in slidingMarkerNumbers
data coordinate 1	$x_{data1}$	the global sliding coordinate (ranging from 0 to the total length of all sliding elements) at start-of-step - beginning of the timestep
marker m0 position	$\LU{0}{{\mathbf{p}}}_{m0}$	current global position which is provided by marker m0
marker m0 velocity	$\LU{0}{{\mathbf{v}}}_{m0}$	current global velocity which is provided by marker m0
marker m0 orientation	$\LU{0,m0}{\Rot}$	current rotation matrix provided by marker m0 (assumed to be rigid body)
marker m0 angular velocity	$\LU{0}{\tomega}_{m0}$	current angular velocity vector provided by marker m0 (assumed to be rigid body)
cable coordinates	${\mathbf{q}}_{ANCF,m1}$	current coordiantes of the ANCF cable element with the current marker $m1$ is referring to
sliding position	$\LUR{0}{{\mathbf{r}}}{ANCF} = {\mathbf{S}}(s_{el}){\mathbf{q}}_{ANCF,m1}$	current global position at the ANCF cable element, evaluated at local sliding position $s_{el}$
sliding position slope	$\LURU{0}{{\mathbf{r}}}{ANCF}{\prime} = {\mathbf{S}}^\prime(s_{el}){\mathbf{q}}_{ANCF,m1} = [r^\prime_0,\,r^\prime_1]\tp$	current global slope vector of the ANCF cable element, evaluated at local sliding position $s_{el}$
sliding velocity	$\LUR{0}{{\mathbf{v}}}{ANCF} = {\mathbf{S}}(s_{el})\dot{\mathbf{q}}_{ANCF,m1}$	current global velocity at the ANCF cable element, evaluated at local sliding position $s_{el}$ ($s_{el}$ not differentiated!!!)
sliding velocity slope	$\LURU{0}{{\mathbf{v}}}{ANCF}{\prime} = {\mathbf{S}}^\prime(s_{el})\dot{\mathbf{q}}_{ANCF,m1}$	current global slope velocity vector of the ANCF cable element, evaluated at local sliding position $s_{el}$
sliding normal vector	$\LU{0}{{\mathbf{n}}} = [-r^\prime_1,\,r^\prime_0]$	2D normal vector computed from slope ${\mathbf{r}}^\prime=\LURU{0}{{\mathbf{r}}}{ANCF}{\prime}$
sliding normal velocity vector	$\LU{0}{\dot{\mathbf{n}}} = [-\dot r^\prime_1,\,\dot r^\prime_0]$	time derivative of 2D normal vector computed from slope velocity $\dot {\mathbf{r}}^\prime=\LURU{0}{\dot {\mathbf{r}}}{ANCF}{\prime}$
algebraic coordinates	${\mathbf{z}}=[\lambda_0,\,\lambda_1,\, s]\tp$	algebraic coordinates composed of Lagrange multipliers $\lambda_0$ and $\lambda_1$ (in local cable coordinates: $\lambda_0$ is in axis direction) and the current sliding coordinate $s$, which is local in the current cable element.
local sliding coordinate	$s$	local incremental sliding coordinate $s$: the (algebraic) sliding coordinate relative to the start-of-step value. Thus, $s$ only contains small local increments.


output variables	symbol	formula
Position	$\LU{0}{{\mathbf{p}}}_{m0}$	current global position of position marker $m0$
Velocity	$\LU{0}{{\mathbf{v}}}_{m0}$	current global velocity of position marker $m0$
SlidingCoordinate	$s_g = s + x_{data1}$	current value of the global sliding coordinate
Force	${\mathbf{f}}$	see below

Geometric relations

Assume we have given the sliding coordinate $s$ (e.g., as a guess of the Newton method or beginning of the time step). The element sliding coordinate (in the local coordinates of the current sliding element) is computed as

\[s_{el} = s + x_{data1} - d_{m1} = s_g - d_{m1}.\]

The vector (=difference; error) between the marker $m0$ and the marker $m1$ (=${\mathbf{r}}_{ANCF}$) positions reads

\[\LU{0}{\Delta{\mathbf{p}}} = \LUR{0}{{\mathbf{r}}}{ANCF} - \LU{0}{{\mathbf{p}}}_{m0}\]

The vector (=difference; error) between the marker $m0$ and the marker $m1$ velocities reads

\[\LU{0}{\Delta{\mathbf{v}}} = \LUR{0}{\dot{\mathbf{r}}}{ANCF} - \LU{0}{{\mathbf{v}}}_{m0}\]

Connector constraint equations (classicalFormulation=True)

The 2D sliding joint is implemented having 3 equations (4 if constrainRotation==True, see below), using the special algebraic coordinates ${\mathbf{z}}$. The algebraic equations read

\[\begin{split}\LU{0}{\Delta{\mathbf{p}}} &=& \Null, \quad \mbox{... two index 3 eqs, ensure sliding body stays at cable}\\ \left[\lambda_0,\lambda_1\right] \cdot \LURU{0}{{\mathbf{r}}}{ANCF}{\prime} - |\LURU{0}{{\mathbf{r}}}{ANCF}{\prime}| \cdot f_\mathrm{ax} &=& 0, \quad \mbox{... one index 1 equ., ensure force in sliding dir.~= 0} \\\end{split}\]

No index 2 case exists, because no time derivative exists for $s_{el}$. The jacobian matrices for algebraic and ODE2 coordinates read

\[{\mathbf{J}}_{AE} = \mr{0}{0}{r^\prime_0} {0}{0}{r^\prime_1} {r^\prime_0}{r^\prime_1}{r^{\prime\prime}_0\lambda_0 + r^{\prime\prime}_1\lambda_1}\]

\[{\mathbf{J}}_{ODE2} = \mp{-J_{pos,m0}}{{\mathbf{S}}(s_{el})} {\Null\tp}{\left[\lambda_0,\,\lambda_1\right]\cdot{\mathbf{S}}^\prime(s_{el}) }\]

if activeConnector = False, the algebraic equations are changed to:

\[\begin{split}\lambda_0 &=& 0, \\ \lambda_1 &=& 0, \\ s &=& 0\end{split}\]

Connector constraint equations (classicalFormulation=False)

The 2D sliding joint is implemented having 3 equations (first equation is dummy and could be eliminated; 4 equations if constrainRotation==True, see below), using the special algebraic coordinates ${\mathbf{z}}$. The algebraic equations read

\[\begin{split}\lambda_0 &=& 0, \quad \mbox{... equation not necessary, but can be used for switching to other modes} \\ \LU{0}{\Delta{\mathbf{p}}\tp} \LU{0}{{\mathbf{n}}} &=& 0, \quad \mbox{... equation ensures that sliding body stays at cable centerline; index3}\\ \LU{0}{\Delta{\mathbf{p}}\tp} \LURU{0}{{\mathbf{r}}}{ANCF}{\prime} &=& 0. \quad \mbox{... resolves the sliding coordinate $s$; index1 equation!}\end{split}\]

In the index 2 case, the second equation reads

\[\LU{0}{\Delta{\mathbf{v}}\tp} \LU{0}{{\mathbf{n}}} + \LU{0}{\Delta{\mathbf{p}}\tp} \LU{0}{\dot{\mathbf{n}}} = 0\]

if activeConnector = False, the algebraic equations are changed to:

\[\begin{split}\lambda_0 &=& 0, \\ \lambda_1 &=& 0, \\ s &=& 0\end{split}\]

In case that constrainRotation = True, an additional constraint is added for the relative rotation between the slope of the cable and the orientation of marker m0 body. Assuming that the orientation of marker m0 is a 2D matrix (taking only $x$ and $y$ coordinates), the constraint reads

\[\LURU{0}{{\mathbf{r}}}{ANCF}{\prime\mathrm{T}} \LU{0,m0}{\Rot} \vp{0}{1} = 0\]

The index 2 case follows straightforward to

\[\LURU{0}{\dot {\mathbf{r}}}{ANCF}{\prime\mathrm{T}} \LU{0,m0}{\Rot} \vp{0}{1} + \LURU{0}{{\mathbf{r}}}{ANCF}{\prime\mathrm{T}} \LU{0,m0}{\Rot} \LU{0}{\tilde \tomega}_{m0} \vp{0}{1} = 0\]

again assuming, that $\LU{0}{\tilde \tomega}_{m0}$ is only a $2 \times 2$ matrix.

Post Newton Step

After the Newton solver has converged, a PostNewtonStep is performed for the element, which updates the marker $m1$ index if necessary.

\[\begin{split}s_{el} < 0 \quad \ra \quad x_{data0}\;-\!\!=1 \nonumber\\ s_{el} > L \quad \ra \quad x_{data0}\;+\!\!=1\end{split}\]

Furthermore, it is checked, if $x_{data0}$ becomes smaller than zero, which raises a warning and keeps $x_{data0}=0$. The same results if $x_{data0}\ge sn$, then $x_{data0} = sn$. Finally, the data coordinate is updated in order to provide the starting value for the next step,

\[x_{data1} \;+\!\!= s.\]

Relevant Examples and TestModels with weblink:

ANCFmovingRigidbody.py (Examples/), ANCFslidingJoint2D.py (Examples/), ANCFslidingJoint2Drigid.py (Examples/), ANCFswitchingSlidingJoint2D.py (Examples/), modelUnitTests.py (TestModels/)

The web version may not be complete. For details, consider also the Exudyn PDF documentation : theDoc.pdf

intermediate variables	symbol	description
data node	\({\mathbf{x}}=[x_{data0},\,x_{data1}]\tp\)	coordinates of node with node number \(n_{GD}\)
data coordinate 0	\(x_{data0}\)	the current index in slidingMarkerNumbers
data coordinate 1	\(x_{data1}\)	the global sliding coordinate (ranging from 0 to the total length of all sliding elements) at start-of-step - beginning of the timestep
marker m0 position	\(\LU{0}{{\mathbf{p}}}_{m0}\)	current global position which is provided by marker m0
marker m0 velocity	\(\LU{0}{{\mathbf{v}}}_{m0}\)	current global velocity which is provided by marker m0
marker m0 orientation	\(\LU{0,m0}{\Rot}\)	current rotation matrix provided by marker m0 (assumed to be rigid body)
marker m0 angular velocity	\(\LU{0}{\tomega}_{m0}\)	current angular velocity vector provided by marker m0 (assumed to be rigid body)
cable coordinates	\({\mathbf{q}}_{ANCF,m1}\)	current coordiantes of the ANCF cable element with the current marker \(m1\) is referring to
sliding position	\(\LUR{0}{{\mathbf{r}}}{ANCF} = {\mathbf{S}}(s_{el}){\mathbf{q}}_{ANCF,m1}\)	current global position at the ANCF cable element, evaluated at local sliding position \(s_{el}\)
sliding position slope	\(\LURU{0}{{\mathbf{r}}}{ANCF}{\prime} = {\mathbf{S}}^\prime(s_{el}){\mathbf{q}}_{ANCF,m1} = [r^\prime_0,\,r^\prime_1]\tp\)	current global slope vector of the ANCF cable element, evaluated at local sliding position \(s_{el}\)
sliding velocity	\(\LUR{0}{{\mathbf{v}}}{ANCF} = {\mathbf{S}}(s_{el})\dot{\mathbf{q}}_{ANCF,m1}\)	current global velocity at the ANCF cable element, evaluated at local sliding position \(s_{el}\) (\(s_{el}\) not differentiated!!!)
sliding velocity slope	\(\LURU{0}{{\mathbf{v}}}{ANCF}{\prime} = {\mathbf{S}}^\prime(s_{el})\dot{\mathbf{q}}_{ANCF,m1}\)	current global slope velocity vector of the ANCF cable element, evaluated at local sliding position \(s_{el}\)
sliding normal vector	\(\LU{0}{{\mathbf{n}}} = [-r^\prime_1,\,r^\prime_0]\)	2D normal vector computed from slope \({\mathbf{r}}^\prime=\LURU{0}{{\mathbf{r}}}{ANCF}{\prime}\)
sliding normal velocity vector	\(\LU{0}{\dot{\mathbf{n}}} = [-\dot r^\prime_1,\,\dot r^\prime_0]\)	time derivative of 2D normal vector computed from slope velocity \(\dot {\mathbf{r}}^\prime=\LURU{0}{\dot {\mathbf{r}}}{ANCF}{\prime}\)
algebraic coordinates	\({\mathbf{z}}=[\lambda_0,\,\lambda_1,\, s]\tp\)	algebraic coordinates composed of Lagrange multipliers \(\lambda_0\) and \(\lambda_1\) (in local cable coordinates: \(\lambda_0\) is in axis direction) and the current sliding coordinate \(s\), which is local in the current cable element.
local sliding coordinate	\(s\)	local incremental sliding coordinate \(s\): the (algebraic) sliding coordinate relative to the start-of-step value. Thus, \(s\) only contains small local increments.