Modelica.Mechanics.MultiBody.Parts

Rigid components such as bodies with mass and inertia and massless rods

Information

Package Parts contains rigid components of a multi-body system. These components may be used to build up more complicated structures. For example, a part may be built up of a "Body" and of several "FixedTranslation" components.

Content

Model	Description
Fixed	Frame fixed in world frame at a given position. It is visualized with a shape, see shapeType below (the frames on the two sides do not belong to the component):
FixedTranslation	Fixed translation of frame_b with respect to frame_a. It is visualized with a shape, see shapeType below (the frames on the two sides do not belong to the component):
FixedRotation	Fixed translation and fixed rotation of frame_b with respect to frame_a It is visualized with a shape, see shapeType below (the frames on the two sides do not belong to the component):
Body	Rigid body with mass, inertia tensor and one frame connector. It is visualized with a cylinder and a sphere at the center of mass:
BodyShape	Rigid body with mass, inertia tensor, different shapes (see shapeType below) for animation, and two frame connectors:
Fixed BodyBox	Rigid body with box shape (mass and animation properties are computed from box data and from density):
BodyCylinder	Rigid body with cylinder shape (mass and animation properties are computed from cylinder data and from density):
PointMass	Rigid body where inertia tensor and rotation is neglected:
Mounting1D	Propagate 1-dim. support torque to 3-dim. system
Rotor1D	1D inertia attachable on 3-dim. bodies (without neglecting dynamic effects)
BevelGear1D	1D gearbox with arbitrary shaft directions (3D bearing frame)

Components Fixed, FixedTranslation, FixedRotation and BodyShape are visualized according to parameter shapeType, that may have the following values (e.g., shapeType = "box"):
 

All the details of the visualization shape parameters are given in Visualizers.FixedShape

Colors in all animation parts are defined via parameter color. This is an Integer vector with 3 elements, {r, g, b}, and specifies the color of the shape. {r,g,b} are the "red", "green" and "blue" color parts, given in the ranges 0 .. 255, respectively. The predefined type MultiBody.Types.Color contains a menu definition of the colors used in the MultiBody library (this will be replaced by a color editor).

Extends from Modelica.Icons.Package (Icon for standard packages).

Package Content

Name	Description
Fixed	Frame fixed in the world frame at a given position
FixedTranslation	Fixed translation of frame_b with respect to frame_a
FixedRotation	Fixed translation followed by a fixed rotation of frame_b with respect to frame_a
Body	Rigid body with mass, inertia tensor and one frame connector (12 potential states)
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)
PointMass	Rigid body where body rotation and inertia tensor is neglected (6 potential states)
Mounting1D	Propagate 1-dim. support torque to 3-dim. system (provided world.driveTrainMechanics3D=true)
Rotor1D	1D inertia attachable on 3-dim. bodies (3D dynamic effects are taken into account if world.driveTrainMechanics3D=true)
BevelGear1D	1D gearbox with arbitrary shaft directions and 3-dim. bearing frame (3D dynamic effects are taken into account provided world.driveTrainMechanics3D=true)
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

Modelica.Mechanics.MultiBody.Parts.Fixed

Frame fixed in the world frame at a given position

Information

Element consisting of a frame (frame_b) that is fixed in the world frame at a given position defined by parameter vector r (vector from origin of world frame to frame_b, resolved in the world frame).

By default, this component is visualized by a cylinder connecting the world frame and frame_b of this components, as shown in the figure below. Note, that the visualized world frame on the left side and Fixed.frame_b on the right side are not part of the component animation and that the animation may be switched off via parameter animation = false.

Parameters

Connectors

Modelica.Mechanics.MultiBody.Parts.FixedTranslation

Fixed translation of frame_b with respect to frame_a

Information

Component for a fixed translation of frame_b with respect to frame_a, i.e., the relationship between connectors frame_a and frame_b remains constant and frame_a is always parallel to frame_b.

By default, this component is visualized by a cylinder connecting frame_a and frame_b, as shown in the figure below. Note, that the two visualized frames are not part of the component animation and that the animation may be switched off via parameter animation = false.

Parameters

Connectors

Modelica.Mechanics.MultiBody.Parts.FixedRotation

Fixed translation followed by a fixed rotation of frame_b with respect to frame_a

Information

Component for a fixed translation and fixed rotation of frame_b with respect to frame_a, i.e., the relationship between connectors frame_a and frame_b remains constant. There are several possibilities to define the orientation of frame_b with respect to frame_a:

• Planar rotation along axis 'n' (that is fixed and resolved in frame_a) with a fixed angle 'angle'.
• Vectors n_x and n_y that are directed along the corresponding axes direction of frame_b and are resolved in frame_a (if n_y is not orthogonal to n_x, the y-axis of frame_b is selected such that it is orthogonal to n_x and in the plane of n_x and n_y).
• Sequence of three planar axes rotations. For example, "sequence = {1,2,3}" and "angles = {90, 45, -90}" means to rotate frame_a around the x axis with 90 degrees, around the new y axis with 45 degrees and around the new z axis around -90 degrees to arrive at frame_b. Note, that sequence={1,2,3} is the Cardan angle sequence and sequence = {3,1,3} is the Euler angle sequence.

By default, this component is visualized by a cylinder connecting frame_a and frame_b, as shown in the figure below. In this figure frame_b is rotated along the z-axis of frame_a with 60 degree. Note, that the two visualized frames are not part of the component animation and that the animation may be switched off via parameter animation = false.

Parameters

Connectors

Modelica.Mechanics.MultiBody.Parts.Body

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

Information

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.

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".

Parameters

Connectors

Modelica.Mechanics.MultiBody.Parts.BodyShape

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

Information

Rigid body with mass and inertia tensor and two frame connectors. 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. The coordinate system frame_b is always parallel to frame_a.

By default, this component is visualized by any shape that can be defined with Modelica.Mechanics.MultiBody.Visualizers.FixedShape. This shape is placed between frame_a and frame_b (default: length(shape) = Frames.length(r)). Additionally a sphere may be visualized that has its center at the center of mass. Note, that the animation may be switched off via parameter animation = false.

The following shapes can be defined via parameter shapeType, e.g., shapeType="cone":

A BodyShape component has potential states. For details of these states and of the "Advanced" menu parameters, see model MultiBody.Parts.Body.

Parameters

Connectors

Modelica.Mechanics.MultiBody.Parts.BodyBox

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

Information

Rigid body with box shape. The mass properties of the body (mass, center of mass, inertia tensor) are computed from the box data. Optionally, the box may be hollow. The (outer) box shape is by default used in the animation. The hollow part is not shown in the animation. The two connector frames frame_a and frame_b are always parallel to each other. Example of component animation (note, that the animation may be switched off via parameter animation = false):

A BodyBox component has potential states. For details of these states and of the "Advanced" menu parameters, see model MultiBody.Parts.Body.

Parameters

Connectors

Modelica.Mechanics.MultiBody.Parts.BodyCylinder

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

Information

Rigid body with cylinder shape. The mass properties of the body (mass, center of mass, inertia tensor) are computed from the cylinder data. Optionally, the cylinder may be hollow. The cylinder shape is by default used in the animation. The two connector frames frame_a and frame_b are always parallel to each other. Example of component animation (note, that the animation may be switched off via parameter animation = false):

A BodyCylinder component has potential states. For details of these states and of the "Advanced" menu parameters, see model MultiBody.Parts.Body.

Parameters

Connectors

Modelica.Mechanics.MultiBody.Parts.PointMass

Rigid body where body rotation and inertia tensor is neglected (6 potential states)

Information

Rigid body where the inertia tensor is neglected. This body is solely defined by its mass. By default, this component is visualized by a sphere that has its center at frame_a. Note, that the animation may be switched off via parameter animation = false.

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

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 PointMass variables frame_a.r_0 and der(frame_a.r_0) are forced to be used as states.

Parameters

Connectors

Modelica.Mechanics.MultiBody.Parts.Mounting1D

Propagate 1-dim. support torque to 3-dim. system (provided world.driveTrainMechanics3D=true)

Information

This component is used to acquire support torques from a 1-dim.-rotational mechanical system (e.g., components from Modelica.Mechanics.Rotational) and to propagate them to a carrier body.

The 1-dim. support torque at flange_b is transformed into 3-dim. space under consideration of the rotation axis, parameter n, which has to be given in the local coordinate system of frame_a.

All components of a 1-dim.-rotational mechanical system that are connected to a common Mounting1D element need to have the same axis of rotation along parameter vector n. This means that, e.g., bevel gears where the axis of rotation of flange_a and flange_b are different cannot be described properly by connecting to the Mounting1D component. In this case, a combination of several Mounting1D components or the component BevelGear1D should be used.

Reference
Schweiger, Christian ; Otter, Martin: Modelling 3D Mechanical Effects of 1-dim. Powertrains. In: Proceedings of the 3rd International Modelica Conference. Linköping : The Modelica Association and Linköping University, November 3-4, 2003, pp. 149-158

Parameters

Connectors

Modelica.Mechanics.MultiBody.Parts.Rotor1D

1D inertia attachable on 3-dim. bodies (3D dynamic effects are taken into account if world.driveTrainMechanics3D=true)

Information

This component is used to model the gyroscopic torques exerted by a 1-dim. inertia (so called rotor) on its 3-dim. carrier body. Gyroscopic torques appear, if the vector of the carrier body's angular velocity is not parallel to the vector of the rotor's axis. The axis of rotation of the rotor is defined by the parameter n, which has to be given in the local coordinate system of frame_a. The default animation of this component is shown in the figure below.

This component is a replacement for Modelica.Mechanics.Rotational.Components.Inertia for the case, that a 1-dim.-rotational mechanical system should be attached with a 3-dim. carrier body.

The Boolean parameter exact was introduced due to performance reasons. If exact is set to false, the influence of the carrier body motion on the angular velocity of the rotor is neglected. This influence is usually negligible if the 1-dim.-rotational mechanical system accelerates much faster as the base body (this is, e.g., the case in vehicle powertrains). The essential advantage is that an algebraic loop is removed since then there is only an action on acceleration level from the powertrain to the base body but not vice versa.

Reference
Schweiger, Christian ; Otter, Martin: Modelling 3D Mechanical Effects of 1-dim. Powertrains. In: Proceedings of the 3rd International Modelica Conference. Linköping : The Modelica Association and Linköping University, November 3-4, 2003, pp. 149-158

Parameters

Connectors

Modelica.Mechanics.MultiBody.Parts.BevelGear1D

1D gearbox with arbitrary shaft directions and 3-dim. bearing frame (3D dynamic effects are taken into account provided world.driveTrainMechanics3D=true)

Information

This component is used to model a 1-dim. gearbox with non-parallel axes (defined by parameters n_a, n_b). A 3-dim. bearing frame is necessary to reflect the correct support torque, as the axes of rotation of flange_a and flange_b and the direction of the support torque vector are different in general.

Note: The name BevelGear1D is kept only for simplicity. Regardless, this component could be used to model any kind of gearbox with non-parallel axes. For a usage example, see Examples.Rotational3DEffects.BevelGear1D.

Reference
Schweiger, Christian ; Otter, Martin: Modelling 3D Mechanical Effects of 1-dim. Powertrains. In: Proceedings of the 3rd International Modelica Conference. Linköping : The Modelica Association and Linköping University, November 3-4, 2003, pp. 149-158

Extends from Modelica.Mechanics.Rotational.Interfaces.PartialTwoFlanges (Partial model for a component with two rotational 1-dim. shaft flanges).

Parameters

Connectors

Modelica.Mechanics.MultiBody.Parts.RollingWheel

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

Information

A wheel rolling on the x-y plane of the world frame including wheel mass and simple visualization. The rolling contact is considered being ideal, i.e. there is no slip between the wheel and the ground. The wheel can not take off but it can incline toward the ground. The frame frame_a is placed in the wheel center point and rotates with the wheel itself. Therefore, a rotational joint with rotation axis n={0,1,0} should be used to connect the wheel to a carrier.

Note

To work properly, the gravity acceleration vector g of the world must point in the negative z-axis, i.e.

inner Modelica.Mechanics.MultiBody.World world(n={0,0,-1});

Parameters

Connectors

Modelica.Mechanics.MultiBody.Parts.RollingWheelSet

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

Information

Two wheels are connected by an axis and can rotate around this axis. The wheels are rolling on the x-y plane of the world frame. The coordinate system attached to the center of the wheel axis (frameMiddle) is constrained so that it is always parallel to the x-y plane. If all generalized coordinates are zero, frameMiddle is parallel to the world frame.

Note

To work properly, the gravity acceleration vector g of the world must point in the negative z-axis, i.e.

inner Modelica.Mechanics.MultiBody.World world(n={0,0,-1});

Parameters

Connectors