ObjectConnectorGravity.rst

ObjectConnectorGravity

A connector for additing forces due to gravitational fields beween two bodies, which can be used for aerospace and small-scale astronomical problems; DO NOT USE this connector for adding gravitational forces (loads), which should be using LoadMassProportional, which is acting global and always in the same direction.

Additional information for ObjectConnectorGravity:

• This Object has/provides the following types = Connector
• Requested Marker type = Position
• Short name for Python = ConnectorGravity
• Short name for Python visualization object = VConnectorGravity

The item ObjectConnectorGravity with type = 'ConnectorGravity' has the following parameters:

• name [type = String, default = '']:

  connector's unique name
• markerNumbers [[m0,m1]\tp, type = ArrayMarkerIndex, default = [ invalid [-1], invalid [-1] ]]:

  list of markers used in connector
• gravitationalConstant [G, type = Real, default = 6.67430e-11]:

  gravitational constant [SI:m^3kg^{-1}s^{-2})]; while not recommended, a negative constant gan represent a repulsive force
• mass0 [mass_0, type = UReal, default = 0.]:

  mass [SI:kg] of object attached to marker m0
• mass1 [mass_1, type = UReal, default = 0.]:

  mass [SI:kg] of object attached to marker m1
• minDistanceRegularization [d_{min}, type = UReal, default = 0.]:

  distance [SI:m] at which a regularization is added in order to avoid singularities, if objects come close
• activeConnector [type = Bool, default = True]:

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

  parameters for visualization of item

The item VObjectConnectorGravity has the following parameters:

• show [type = Bool, default = False]:

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

  drawing size = diameter of spring; 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 ObjectConnectorGravity

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

• Distance: L

  distance between both points
• Displacement: \Delta\! \LU{0}{{\mathbf{p}}}

  relative displacement between both points
• Force: {\mathbf{f}}

  gravity force vector, pointing from marker m0 to marker m1

Definition of quantities

intermediate variables	symbol	description
marker m0 position	\LU{0}{{\mathbf{p}}}_{m0}	current global position which is provided by marker m0
marker m1 position	\LU{0}{{\mathbf{p}}}_{m1}
marker m0 velocity	\LU{0}{{\mathbf{v}}}_{m0}	current global velocity which is provided by marker m0
marker m1 velocity	\LU{0}{{\mathbf{v}}}_{m1}

output variables	symbol	formula
Displacement	\Delta\! \LU{0}{{\mathbf{p}}}	\LU{0}{{\mathbf{p}}}_{m1} - \LU{0}{{\mathbf{p}}}_{m0}
Velocity	\Delta\! \LU{0}{{\mathbf{v}}}	\LU{0}{{\mathbf{v}}}_{m1} - \LU{0}{{\mathbf{v}}}_{m0}
Distance	L	|\Delta\! \LU{0}{{\mathbf{p}}}|
Force	{\mathbf{f}}	see below

Connector forces

The unit vector in force direction reads (if L=0, singularity can be avoided using regularization),

{\mathbf{v}}_{f} = \frac{1}{L} \Delta\! \LU{0}{{\mathbf{p}}}

If activeConnector = True, and L>=d_{min} the gravitational force is computed as

f_G = - G \frac{mass_0 \cdot mass_1}{L^2}

If activeConnector = True, and L<d_{min} the gravitational force is computed as

f_G = - G \frac{mass_0 \cdot mass_1}{L^2+(L-d_{min})^2}

which results in a regularization for small distances, which is helpful if there are no restrictions in objects to keep apart. If d_{min}=0 and L=0, there a system error is raised.

The vector of the gravitational force applied at both markers, pointing from marker m0 to marker m1, finally reads

{\mathbf{f}} = f_G {\mathbf{v}}_{f}

The virtual work of the connector force is computed from the virtual displacement

\delta \Delta\! \LU{0}{{\mathbf{p}}} = \delta \LU{0}{{\mathbf{p}}}_{m1} - \delta \LU{0}{{\mathbf{p}}}_{m0} ,

and the virtual work (not the transposed version here, because the resulting generalized forces shall be a column vector,

\delta W_G = {\mathbf{f}} \delta \Delta\! \LU{0}{{\mathbf{p}}} = -\left( - G \frac{mass_0 \cdot mass_1}{L^2} \right) \left(\delta \LU{0}{{\mathbf{p}}}_{m1} - \delta \LU{0}{{\mathbf{p}}}_{m0} \right)\tp {\mathbf{v}}_{f} .

The generalized (elastic) forces thus result from

{\mathbf{Q}}_G = \frac{\partial \LU{0}{{\mathbf{p}}}}{\partial {\mathbf{q}}_G\tp} {\mathbf{f}} ,

and read for the markers m0 and m1,

{\mathbf{Q}}_{G, m0} = -\left( - G \frac{mass_0 \cdot mass_1}{L^2} \right) {\mathbf{J}}_{pos,m0}\tp {\mathbf{v}}_{f} , \quad {\mathbf{Q}}_{G, m1} = \left( - G \frac{mass_0 \cdot mass_1}{L^2} \right) {\mathbf{J}}_{pos,m1}\tp {\mathbf{v}}_{f} ,

where {\mathbf{J}}_{pos,m1} represents the derivative of marker m1 w.r.t.its associated coordinates {\mathbf{q}}_{m1}, analogously {\mathbf{J}}_{pos,m0}.

MINI EXAMPLE for ObjectConnectorGravity

mass0 = 1e25
mass1 = 1e3
r = 1e5
G = 6.6743e-11
vInit = np.sqrt(G*mass0/r)
tEnd = (r*0.5*np.pi)/vInit #quarter period
node0 = mbs.AddNode(NodePoint(referenceCoordinates = [0,0,0])) #star
node1 = mbs.AddNode(NodePoint(referenceCoordinates = [r,0,0],
                              initialVelocities=[0,vInit,0])) #satellite
oMassPoint0 = mbs.AddObject(MassPoint(nodeNumber = node0, physicsMass=mass0))
oMassPoint1 = mbs.AddObject(MassPoint(nodeNumber = node1, physicsMass=mass1))

m0 = mbs.AddMarker(MarkerNodePosition(nodeNumber=node0))
m1 = mbs.AddMarker(MarkerNodePosition(nodeNumber=node1))

mbs.AddObject(ObjectConnectorGravity(markerNumbers=[m0,m1],
                                     mass0 = mass0, mass1=mass1))

#assemble and solve system for default parameters
mbs.Assemble()
sims = exu.SimulationSettings()
sims.timeIntegration.endTime = tEnd
mbs.SolveDynamic(sims, solverType=exu.DynamicSolverType.RK67)

#check result at default integration time
#expect y=x after one period of orbiting (got: 100000.00000000479)
exudynTestGlobals.testResult = mbs.GetNodeOutput(node1, exu.OutputVariableType.Position)[1]/100000

Relevant Examples and TestModels with weblink:

connectorGravityTest.py (TestModels/)

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