Documentation

Solid Inertia

Inertial Properties

Inertia is the resistance of matter to acceleration due to applied forces and torques. The inertial properties of a body include its mass and inertia tensor—a symmetric 3×3 matrix that contains the moments and products of inertia. Mass resists translational acceleration while the moments and products of inertia resist rotational acceleration.

Among the solid properties of a model, the inertial properties have the greatest impact on multibody dynamics. Those that you must specify depend on the type of inertia you are modeling—a point mass or a body with a 3-D mass distribution. They include one or more of the following:

• Mass or density

• Center of mass

• Moments of inertia

• Products of inertia

In a SimMechanics™ model, these properties are time-invariant. Rigid bodies cannot gain or lose mass nor can they deform in response to an applied force or torque. The mass distribution of a body—and therefore its inertia tensor and center of mass—remain constant throughout simulation.

Blocks with Inertia

You can model inertia using the following blocks:

• Solid — Model a complete solid element with geometry, inertia, and color. This block can automatically compute the moments of inertia, products of inertia, and center of mass based on the solid geometry and mass or mass density. During simulation, Mechanics Explorer renders the solid using the geometry and color specified.

• Inertia — Model only the inertial properties of a solid element. You must specify the moments of inertia, products of inertia, and center of mass explicitly. During simulation, Mechanics Explorer identifies the center of mass using the inertia icon .

Inertia in a Model

The Solid and Inertia blocks each contain a reference frame port. To add one such block to your model, connect its reference frame port to another frame entity in the model. Frame entities include frame lines, nodes, and ports. The frame entity to which you connect the block determines the position and orientation of the inertia within the model. See Representing Frames.

As an example, consider the model shown in the figure. The model contains two Solid blocks, labeled Link A and Link B. The reference frame port of Link A connects directly to the World Frame block. Its reference frame is therefore coincident with the World frame.

The reference frame port in Link B connects to the follower frame port in the Rigid Transform block. This block applies a spatial transform between its base frame, coincident with the World frame, and its follower frame, coincident with the Link B reference frame. The Link B reference frame is therefore offset from the World frame.

For examples showing how to position solid elements in a model, see:

Inertia Parameterizations

Once you have connected the Solid or Inertia blocks in a model, you must specify their inertial parameters. These depend on the inertia parameterization that you select. The blocks provide three optional parameterizations:

• `Calculate from Geometry` — Specify mass or density. The Solid block automatically computes the remaining inertial properties based on the solid geometry. Only the Solid block provides this parameterization.

• `Point Mass` — Specify the mass and ignore the remaining inertial parameters. The inertia behaves as point mass with no rotational inertia.

• `Custom` — Manually specify every inertial parameter. You must obtain each parameter through direct calculation or from an external modeling platform.

Custom Inertia

If you select the `Custom Inertia` parameterization, you must specify the moments of inertia, products of inertia, and center of mass explicitly. These parameters depend closely on the reference frame used in their calculations, so you must ensure that frame matches the one used in SimMechanics:

• Moments and products of inertia — Enter with respect to a frame parallel to the reference port frame but with origin at the center of mass.

• Center of mass — Enter with respect to the reference port frame.

Consider the main section of the binary link in Model Binary Link. You model this solid using a single solid block with a `General Extrusion` shape. As described in the Solid block documentation, the reference port frame for a general extrusion has its origin in the XY plane at the [0,0] cross-section coordinate.

The figure shows the solid reference port frame, labeled R. The center-of-mass coordinates must be with respect to this frame. The moments and products of inertia must be with respect to a parallel frame offset so that its origin coincides with the center of mass. This frame is virtual , as it does not correspond to any frame port, line, or node in the model. It is labeled R* in the figure.

Moments and Products of Inertia

You can extract the moments and products of inertia directly from the inertia tensor. This tensor is symmetric: elements with reciprocal indices have the same magnitude. That is:

• ${I}_{xy}={I}_{yx}$

• ${I}_{yz}={I}_{zy}$

• ${I}_{zx}={I}_{xz}$

This symmetry reduces the number of unique tensor elements to six—three moments of inertia and three products of inertia. The complete inertia tensor has the form:

$\left[\begin{array}{ccc}{I}_{xx}& {I}_{xy}& {I}_{zx}\\ {I}_{xy}& {I}_{yy}& {I}_{yz}\\ {I}_{zx}& {I}_{yz}& {I}_{zz}\end{array}\right]$

The moments of inertia are the diagonal elements:

$\left[\begin{array}{ccc}{I}_{xx}& & \\ & {I}_{yy}& \\ & & {I}_{zz}\end{array}\right]$

SimMechanics defines these elements as follows:

• ${I}_{\text{xx}}=\underset{V}{\overset{}{\int }}\left({y}^{2}+{z}^{2}\right)dm$

• ${I}_{\text{yy}}=\underset{V}{\overset{}{\int }}\left({z}^{2}+{x}^{2}\right)dm$

• ${I}_{\text{zz}}=\underset{V}{\overset{}{\int }}\left({x}^{2}+{y}^{2}\right)dm$

The products of inertia are the unique off-diagonal elements, each of which appears in the inertia tensor twice:

$\left[\begin{array}{ccc}& {I}_{xy}& {I}_{zx}\\ {I}_{xy}& & {I}_{yz}\\ {I}_{zx}& {I}_{yz}& \end{array}\right]$

SimMechanics defines these elements as follows:

• ${I}_{\text{yz}}=-\underset{V}{\overset{}{\int }}yz\text{\hspace{0.17em}}dm$

• ${I}_{\text{zx}}=-\underset{V}{\overset{}{\int }}zx\text{\hspace{0.17em}}dm$

• ${I}_{\text{xy}}=-\underset{V}{\overset{}{\int }}xy\text{\hspace{0.17em}}dm$

The inertia tensor is simplest when it is diagonal. Such a tensor provides the moments of inertia about the principal axes of the solid or inertia element—known as the principal moments of inertia. The products of inertia reduce to zero:

$\left[\begin{array}{ccc}{I}_{xx}& 0& 0\\ 0& {I}_{yy}& 0\\ 0& 0& {I}_{zz}\end{array}\right]$

Complex Inertias

Bodies often comprise different materials, have complex shapes, or contain material imperfections that alter their centers of mass and principal axes. One example is an imbalanced automobile wheel after driving through a pothole. You can model complex inertias such as these using two approaches:

• Use a divide-and-conquer approach. Break up the complex solid or inertia into simpler chunks and model each using a separate Solid or Inertia block. The resulting set of Solid and Inertia blocks constitute a compound inertia. You use a similar approach to model complex geometries, such as the binary link geometry in Model Binary Link.

• Manually specify the complete inertial properties using a single Solid or Inertia block with the inertia parameterization set to `Custom`. You must obtain the moments of inertia, products of inertia, and center of mass through direct calculation, from another modeling platform, or from another external source.

For bodies with complex shapes but uniform mass distributions, you can also import a STEP file containing the solid geometry and set the inertia parameterization to `Calculate from Geometry`.