MarkerBodiesRelativeRotationCoordinate

A coordinate-based Marker attached to two rigid bodies or beams which computes the relative rotation between the bodies according to the given axis; this marker can be used together with coordinate-based constraints and connectors (e.g., CoordinateSpringDamper and CoordinateConstraint). NOTE: it is assumed that the two bodies can only rotate about the given axis (e.g., constrained by a revolute joint) – otherwise results may be unexpected. NOTE: this approach is not compatible with FFRF-based flexible bodies and currently requires and intermediate rigid body.

Interface

Provides: Coordinate
Connectors, constraints and loads that can use it: ObjectConnectorCoordinateSpringDamper, ObjectConnectorCoordinateSpringDamperExt, ObjectConnectorCoordinate, ObjectContactCoordinate, LoadCoordinate

Parameters

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

name	type	size	default	description
name	String		‘’	marker’s unique name
bodyNumbers	ArrayObjectIndex		[ invalid (-1), invalid (-1) ]	(symbol: \([b_0,b_1]\tp\)) list of body numbers for which relative coordinate is computed
nodeNumber	NodeIndex		invalid (-1)	node number of NodeGenericData with 1 coordinate which contains previous angle for continuation of angles (initialize accordingly if needed); if node is not supplied, angles will have jump outside \(\pm \pi\)
localPosition0	Vector3D	3	[0.,0.,0.]	(symbol: \(\LU{m_0}{\pv}_0\)) local position on body 0; i.e. local (body-fixed) position where position is measured and force is applied to
localPosition1	Vector3D	3	[0.,0.,0.]	(symbol: \(\LU{m_1}{\pv}_1\)) local position on body 1; i.e. local (body-fixed) position where position is measured and force is applied to
axis0	Vector3D	3	[1.,0.,0.]	(symbol: \(\LU{m_0}{\av}_0\)) axis defined in body 0, along which the relative rotation is measured
offset	Real		0.	(symbol: \(x_\mathrm{off}\)) rotation offset [SI:1] subtracted from the measured rotation; can be used to change the zero rotation
visualization	VMarkerBodiesRelativeRotationCoordinate			parameters for visualization of item

Visualization parameters

The parameters of VMarkerBodiesRelativeRotationCoordinate, 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

Drawing

Drawn as all markers.

Detailed description

The marker consists of two bodies, body \(b_0\) and body \(b_1\) with respective global marker positions \(\LU{0}{\pv}_{m0}\) and \(\LU{0}{\pv}_{m1}\), depending on local positions \(\LU{m_0}{\pv}_0\) and \(\LU{m_1}{\pv}_1\), and marker orientations \(\LU{0,m_0}{\Rot}_{m0}\) and \(\LU{0,m_1}{\Rot}_{m1}\). From the given axis axis0, we compute an orthonormal basis (orthonormal to axis0) relative to marker \(m_0\),

\[ \LU{m_0,b}{\Rot}_b = \left[\LU{m_0}{\xv}_b, \LU{m_0}{\yv}_b, \LU{m_0}{\av}_0\right] \]

The relative rotation marker computes the relative rotation according to

\[ \LU{m_0,m_1}{\Rot}_{rel} = \LU{m_0,0}{\Rot}_{m0} \LU{0,m_1}{\Rot}_{m1} \]

This relative rotation, which represents a rotation about axis \(\LU{m_0}{\av}_0\) is then transformed into the orthonormal basis,

\[ \LU{b}{\Rot}_{rel} = \LU{b,m_0}{\Rot}_b \LU{m_0,m_1}{\Rot}_{rel} \LU{m_0,b}{\Rot}_b \]

and contains the desired rotation about the z-axis, which can be extracted as

\[ \varphi = \mathrm{atan2}(\LU{b}{\Rot}_{rel}[1,0],\LU{b}{\Rot}_{rel}[0,0]) - x_\mathrm{off} \]

The global axis is computed as

\[ \LU{0}{\av}_0 = \LU{0,m_0}{\Rot}_{m0} \LU{m_0}{\av}_0 \]

Using the angular velocities at the two bodies, \(\LU{m_0}{\tomega_0}\) and \(\LU{m_1}{\tomega_1}\), the relative angular velocity, which may be used in coordinate spring-dampers or for velocity-level constraints, is simply computed as

\[ \dot \varphi = \LU{0}{\av}_0\tp \left( \LU{0,m_1}{\Rot}_{m1} \LU{m_1}{\tomega_1} - \LU{0,m_0}{\Rot}_{m0} \LU{m_0}{\tomega_0} \right) \]

Jacobians are computed according to the relative rotation velocity. Using this approach, coordinate constraints can be added to mechanisms to purely add internal drives, not affecting global momenta. Furthermore, coupling to a relative translation can be used to create advanced mechanisms and gears.

It is a coordinate marker: coordinate connectors, coordinate constraints and LoadCoordinate use it. Both bodies must provide position and orientation at a local point, as a rigid body does; Assemble() refuses other bodies.

Mini example

#the rotation of body 1 relative to body 0 about an axis of body 0, held by a coordinate constraint;
#the data node continues the angle beyond +-pi
inertia = InertiaCuboid(density=1000, sideLengths=[0.4,0.2,0.1])
node = mbs.AddNode(NodeRigidBodyRxyz(referenceCoordinates=[0,0,0, 0,0,0]))
body = mbs.AddObject(ObjectRigidBody(nodeNumber=node, mass=inertia.Mass(),
                                     inertia=inertia.GetInertia6D()))
nData = mbs.AddNode(NodeGenericData(numberOfDataCoordinates=1, initialCoordinates=[0]))
mRel = mbs.AddMarker(MarkerBodiesRelativeRotationCoordinate(bodyNumbers=[oGround, body], axis0=[0,0,1],
                                                           nodeNumber=nData))
mGround = mbs.AddMarker(MarkerNodeCoordinate(nodeNumber=nGround, coordinate=0))
oHold = mbs.AddObject(ObjectConnectorCoordinate(markerNumbers=[mGround, mRel]))
mbs.AddLoad(LoadTorqueVector(markerNumber=mbs.AddMarker(MarkerNodeRigid(nodeNumber=node)), loadVector=[0,0,2]))

mbs.Assemble()
mbs.SolveDynamic()

#the constraint holds the relative rotation about z against the torque: its force is the reaction torque
exu.sys['testResult'] = mbs.GetObjectOutput(oHold, exu.OutputVariableType.Force) #2

Relevant Examples (Ex) and TestModels (TM) with weblink to github: involuteGearGraphics.py (Ex), relativeRotationTranslationMechanism.py (TM)