WOLFRAM SYSTEMMODELER

Body

Rigid body with mass, inertia tensor and one frame connector (12 potential states)

Wolfram Language

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SystemModel["Modelica.Mechanics.MultiBody.Parts.Body"]
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Information

This information is part of the Modelica Standard Library maintained by the Modelica Association.

Rigid body with mass and inertia tensor. All parameter vectors have to be resolved in frame_a. The inertia tensor has to be defined with respect to a coordinate system that is parallel to frame_a with the origin at the center of mass of the body.

By default, this component is visualized by a cylinder located between frame_a and the center of mass and by a sphere that has its center at the center of mass. If the cylinder length is smaller as the radius of the sphere, e.g., since frame_a is located at the center of mass, the cylinder is not displayed. Note, that the animation may be switched off via parameter animation = false.

Parts.Body

States of Body Components

Every body has potential states. If possible a tool will select the states of joints and not the states of bodies because this is usually the most efficient choice. In this case the position, orientation, velocity and angular velocity of frame_a of the body will be computed by the component that is connected to frame_a. However, if a body is moving freely in space, variables of the body have to be used as states. The potential states of the body are:

  • The position vector frame_a.r_0 from the origin of the world frame to the origin of frame_a of the body, resolved in the world frame and the absolute velocity v_0 of the origin of frame_a, resolved in the world frame (= der(frame_a.r_0)).
  • If parameter useQuaternions in the "Advanced" menu is true (this is the default), then 4 quaternions are potential states. Additionally, the coordinates of the absolute angular velocity vector of the body are 3 potential states.
    If useQuaternions in the "Advanced" menu is false, then 3 angles and the derivatives of these angles are potential states. The orientation of frame_a is computed by rotating the world frame along the axes defined in parameter vector "sequence_angleStates" (default = {1,2,3}, i.e., the Cardan angle sequence) around the angles used as potential states. For example, the default is to rotate the x-axis of the world frame around angles[1], the new y-axis around angles[2] and the new z-axis around angles[3], arriving at frame_a.

The quaternions have the slight disadvantage that there is a non-linear constraint equation between the 4 quaternions. Therefore, at least one non-linear equation has to be solved during simulation. A tool might, however, analytically solve this simple constraint equation. Using the 3 angles as states has the disadvantage that there is a singular configuration in which a division by zero will occur. If it is possible to determine in advance for an application class that this singular configuration is outside of the operating region, the 3 angles might be used as potential states by setting useQuaternions = false.

In text books about 3-dimensional mechanics often 3 angles and the angular velocity are used as states. This is not the case here, since 3 angles and their derivatives are used as potential states (if useQuaternions = false). The reason is that for real-time simulation the discretization formula of the integrator might be "inlined" and solved together with the body equations. By appropriate symbolic transformation the performance is drastically increased if angles and their derivatives are used as states, instead of angles and the angular velocity.

Whether or not variables of the body are used as states is usually automatically selected by the Modelica translator. If parameter enforceStates is set to true in the "Advanced" menu, then body variables are forced to be used as states according to the setting of parameters "useQuaternions" and "sequence_angleStates".

Connectors (1)

frame_a

Type: Frame_a

Description: Coordinate system fixed at body

Parameters (26)

animation

Value: true

Type: Boolean

Description: = true, if animation shall be enabled (show cylinder and sphere)

r_CM

Value:

Type: Position[3] (m)

Description: Vector from frame_a to center of mass, resolved in frame_a

m

Value:

Type: Mass (kg)

Description: Mass of rigid body

I_11

Value: 0.001

Type: Inertia (kg·m²)

Description: (1,1) element of inertia tensor

I_22

Value: 0.001

Type: Inertia (kg·m²)

Description: (2,2) element of inertia tensor

I_33

Value: 0.001

Type: Inertia (kg·m²)

Description: (3,3) element of inertia tensor

I_21

Value: 0

Type: Inertia (kg·m²)

Description: (2,1) element of inertia tensor

I_31

Value: 0

Type: Inertia (kg·m²)

Description: (3,1) element of inertia tensor

I_32

Value: 0

Type: Inertia (kg·m²)

Description: (3,2) element of inertia tensor

angles_fixed

Value: false

Type: Boolean

Description: = true, if angles_start are used as initial values, else as guess values

angles_start

Value: {0, 0, 0}

Type: Angle[3] (rad)

Description: Initial values of angles to rotate frame_a around 'sequence_start' axes into frame_b

sequence_start

Value: {1, 2, 3}

Type: RotationSequence

Description: Sequence of rotations to rotate frame_a into frame_b at initial time

w_0_fixed

Value: false

Type: Boolean

Description: = true, if w_0_start are used as initial values, else as guess values

w_0_start

Value: {0, 0, 0}

Type: AngularVelocity[3] (rad/s)

Description: Initial or guess values of angular velocity of frame_a resolved in world frame

z_0_fixed

Value: false

Type: Boolean

Description: = true, if z_0_start are used as initial values, else as guess values

z_0_start

Value: {0, 0, 0}

Type: AngularAcceleration[3] (rad/s²)

Description: Initial values of angular acceleration z_0 = der(w_0)

sphereDiameter

Value: world.defaultBodyDiameter

Type: Diameter (m)

Description: Diameter of sphere

cylinderDiameter

Value: sphereDiameter / Types.Defaults.BodyCylinderDiameterFraction

Type: Diameter (m)

Description: Diameter of cylinder

enforceStates

Value: false

Type: Boolean

Description: = true, if absolute variables of body object shall be used as states (StateSelect.always)

useQuaternions

Value: true

Type: Boolean

Description: = true, if quaternions shall be used as potential states otherwise use 3 angles as potential states

sequence_angleStates

Value: {1, 2, 3}

Type: RotationSequence

Description: Sequence of rotations to rotate world frame into frame_a around the 3 angles used as potential states

I

Value: [I_11, I_21, I_31; I_21, I_22, I_32; I_31, I_32, I_33]

Type: Inertia[3,3] (kg·m²)

Description: inertia tensor

R_start

Value: Modelica.Mechanics.MultiBody.Frames.axesRotations(sequence_start, angles_start, zeros(3))

Type: Orientation

Description: Orientation object from world frame to frame_a at initial time

z_a_start

Value: Frames.resolve2(R_start, z_0_start)

Type: AngularAcceleration[3] (rad/s²)

Description: Initial values of angular acceleration z_a = der(w_a), i.e., time derivative of angular velocity resolved in frame_a

Q_start

Value: Frames.to_Q(R_start)

Type: Orientation

Description: Quaternion orientation object from world frame to frame_a at initial time

phi_start

Value: if sequence_start[1] == sequence_angleStates[1] and sequence_start[2] == sequence_angleStates[2] and sequence_start[3] == sequence_angleStates[3] then angles_start else Frames.axesRotationsAngles(R_start, sequence_angleStates)

Type: Angle[3] (rad)

Description: Potential angle states at initial time

Components (4)

R_start

Type: Orientation

Description: Orientation object from world frame to frame_a at initial time

world

Type: World

Description:

cylinder

Type: Shape

Description:

sphere

Type: Shape

Description:

Used in Examples (13)

LineForceWithTwoMasses

Demonstrate line force with two point masses using a JointUPS and alternatively a LineForceWithTwoMasses component

Pendulum

Simple pendulum with one revolute joint and one body

PendulumWithSpringDamper

Simple spring/damper/mass system

PointGravity

Two point masses in a point gravity field

SpringDamperSystem

Simple spring/damper/mass system

SpringMassSystem

Mass attached with a spring to the world frame

SpringWithMass

Point mass hanging on a spring

ThreeSprings

3-dim. springs in series and parallel connection

RollingWheelSetDriving

Rolling wheel set that is driven by torques driving the wheels

RollingWheelSetPulling

Rolling wheel set that is pulled by a force

HeatLosses

Demonstrate the modeling of heat losses

UserDefinedGravityField

Demonstrate the modeling of a user-defined gravity field

PlanarLoops_analytic

Mechanism with three planar kinematic loops and one degree-of-freedom with analytic loop handling (with JointRRR joints)

Used in Components (5)

BodyShape

Rigid body with mass, inertia tensor, different shapes for animation, and two frame connectors (12 potential states)

BodyBox

Rigid body with box shape. Mass and animation properties are computed from box data and density (12 potential states)

BodyCylinder

Rigid body with cylinder shape. Mass and animation properties are computed from cylinder data and density (12 potential states)

RollingWheel

Ideal rolling wheel on flat surface z=0 (5 positional, 3 velocity degrees of freedom)

RollingWheelSet

Ideal rolling wheel set consisting of two ideal rolling wheels connected together by an axis