ObjectConnectorLinearSpringDamper

An linear spring-damper element acting on relative translations along given axis of local joint0 coordinate system; connects to position and orientation-based markers; the linear spring-damper is intended to act within prismatic joints or in situations where only one translational axis is free; if the two markers rotate relative to each other, the spring-damper will always act in the local joint0 coordinate system.

Additional information for ObjectConnectorLinearSpringDamper:

This Object has/provides the following types = Connector
Requested Marker type = Position + Orientation
Short name for Python = LinearSpringDamper
Short name for Python visualization object = VLinearSpringDamper

The item ObjectConnectorLinearSpringDamper with type = ‘ConnectorLinearSpringDamper’ has the following parameters:

name [type = String, default = ‘’]:

connector’s unique name
markerNumbers [\([m0,\, m1]\), type = ArrayMarkerIndex, default = [ invalid [-1], invalid [-1] ]]:

list of markers used in connector
stiffness [\(k\), type = Real, default = 0.]:

torsional stiffness [SI:Nm/rad] against relative rotation
damping [\(d\), type = Real, default = 0.]:

torsional damping [SI:Nm/(rad/s)]
axisMarker0 [\(\LU{m0}{{\mathbf{d}}}\), type = Vector3D, default = [1,0,0]]:

local axis of spring-damper in marker 0 coordinates; this axis will co-move with marker \(m0\); if marker m0 is attached to ground, the spring-damper represents linear equations
offset [\(x_\mathrm{off}\), type = Real, default = 0.]:

translational offset considered in the spring force calculation (this can be used as position control input!)
velocityOffset [\(v_\mathrm{off}\), type = Real, default = 0.]:

velocity offset considered in the damper force calculation (this can be used as velocity control input!)
force [\(f_c\), type = Real, default = 0.]:

additional constant force [SI:Nm] added to spring-damper; this can be used to prescribe a force between the two attached bodies (e.g., for actuation and control)
activeConnector [type = Bool, default = True]:

flag, which determines, if the connector is active; used to deactivate (temporarily) a connector or constraint
springForceUserFunction [\(\mathrm{UF} \in \Rcal\), type = PyFunctionMbsScalarIndexScalar5, default = 0]:

A Python function which computes the scalar force between the two rigid body markers along axisMarker0 in \(m0\) coordinates, if activeConnector=True; see description below
visualization [type = VObjectConnectorLinearSpringDamper]:

parameters for visualization of item

The item VObjectConnectorLinearSpringDamper 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 = diameter of spring; size == -1.f means that default connector size is used
drawAsCylinder [type = Bool, default = False]:

if this flag is True, the spring-damper is represented as cylinder; this may fit better if the spring-damper represents an actuator
color [type = Float4, default = [-1.,-1.,-1.,-1.]]:

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

DESCRIPTION of ObjectConnectorLinearSpringDamper

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

DisplacementLocal: \(\Delta x\)

(scalar) relative displacement of the spring-damper
VelocityLocal: \(\Delta v\)

(scalar) relative velocity of spring-damper
ForceLocal: \(f\)

(scalar) spring-damper force

Definition of quantities


input parameter	symbol	description
markerNumbers[0]	\(m0\)	global marker number m0
markerNumbers[1]	\(m1\)	global marker number m1


intermediate variables	symbol	description
marker m0 orientation	\(\LU{0,m0}{\Rot}\)	current rotation matrix provided by marker m0
marker m0 position	\(\LU{0}{{\mathbf{p}}_{m0}}\)	current position matrix provided by marker m0
marker m1 position	\(\LU{0}{{\mathbf{p}}_{m1}}\)	current position matrix provided by marker m1
marker m0 velocity	\(\LU{0}{{\mathbf{v}}}_{m0}\)	current global velocity vector provided by marker m0
marker m1 velocity	\(\LU{0}{{\mathbf{v}}}_{m1}\)	current global velocity vector provided by marker m1
relative displacement	\(\Delta x = (\LU{0,m0}{\Rot} \LU{m0}{{\mathbf{d}}})\tp (\LU{0}{{\mathbf{p}}_{m1}} - \LU{0}{{\mathbf{p}}_{m0}})\)	scalar relative displacement
relative velocity	\(\Delta v = (\LU{0,m0}{\Rot} \LU{m0}{{\mathbf{d}}})\tp (\LU{0}{{\mathbf{v}}_{m1}} - \LU{0}{{\mathbf{v}}_{m0}})\)	scalar relative velocity; note that this only corresponds to the time derivative of \(\Delta x\) if the markers only move along the axis (in a prismatic joint)

Connector forces

If activeConnector = True, the vector spring force is computed as

\[f_{SD} = k \left(\Delta x - x_\mathrm{off} \right) + d \left(\Delta v - v_\mathrm{off} \right) + f_c\]

if activeConnector = False, \(\tau_{SD}\) is set zero.

If the springForceUserFunction \(\mathrm{UF}\) is defined and activeConnector = True, \(\tau_{SD}\) instead becomes (\(t\) is current time)

\[\tau_{SD} = \mathrm{UF}(mbs, t, i_N, \Delta x, \Delta v, \mathrm{stiffness}, \mathrm{damping}, \mathrm{offset})\]

and iN represents the itemNumber (=objectNumber).

Userfunction: springForceUserFunction(mbs, t, itemNumber, displacement, velocity, stiffness, damping, offset)

A user function, which computes the scalar torque depending on mbs, time, local quantities (relative displacement, relative velocity), which are evaluated at current time. Furthermore, the user function contains object parameters (stiffness, damping, offset). Note that itemNumber represents the index of the object in mbs, which can be used to retrieve additional data from the object through mbs.GetObjectParameter(itemNumber, ...), see the according description of GetObjectParameter.

Detailed description of the arguments and local quantities:


arguments / return	type or size	description
`mbs`	MainSystem	provides MainSystem mbs in which underlying item is defined
`t`	Real	current time in mbs
`itemNumber`	Index	integer number \(i_N\) of the object in mbs, allowing easy access to all object data via mbs.GetObjectParameter(itemNumber, …)
`displacement`	Real	\(\Delta x\)
`velocity`	Real	\(\Delta v\)
`stiffness`	Real	copied from object
`damping`	Real	copied from object
`offset`	Real	copied from object
returnValue	Real	computed force

User function example:

#define simple cubic force for spring-damper:
def UFforce(mbs, t, itemNumber, displacement, velocity, stiffness, damping, offset):
    k = stiffness #passed as list
    return k*displacement + 0.1*k* displacement**3

#markerNumbers and parameters taken from mini example
mbs.AddObject(LinearSpringDamper(markerNumbers = [mGround, mBody],
                                 stiffness = k,
                                 damping = k*0.01,
                                 offset = 0,
                                 springForceUserFunction = UFforce))

MINI EXAMPLE for ObjectConnectorLinearSpringDamper

#example with rigid body at [0,0,0], with torsional load
k=2e3
nBody = mbs.AddNode(RigidRxyz())
oBody = mbs.AddObject(RigidBody(physicsMass=1, physicsInertia=[1,1,1,0,0,0],
                                nodeNumber=nBody))

mBody = mbs.AddMarker(MarkerNodeRigid(nodeNumber=nBody))
mGround = mbs.AddMarker(MarkerBodyRigid(bodyNumber=oGround,
                                        localPosition = [0,0,0]))
mbs.AddObject(PrismaticJointX(markerNumbers = [mGround, mBody])) #motion along ground X-axis
mbs.AddObject(LinearSpringDamper(markerNumbers = [mGround, mBody], axisMarker0=[1,0,0],
                                 stiffness = k, damping = k*0.01, offset = 0))

#force along x-axis; expect approx. Delta x = 1/k=0.0005
mbs.AddLoad(Force(markerNumber = mBody, loadVector=[1,0,0]))

#assemble and solve system for default parameters
mbs.Assemble()
mbs.SolveDynamic(exu.SimulationSettings())

#check result at default integration time
exudynTestGlobals.testResult = mbs.GetNodeOutput(nBody, exu.OutputVariableType.Displacement)[0]

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