This is machine translation

Translated by Microsoft
Mouseover text to see original. Click the button below to return to the English verison of the page.

Note: This page has been translated by MathWorks. Please click here
To view all translated materals including this page, select Japan from the country navigator on the bottom of this page.

Specify the Inertia of a Solid

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

These properties can be fixed or variable. Inertial properties specified using Inertia and Solid blocks are fixed and cannot change during simulation. Inertial properties specified using the General Variable Mass block can be either fixed or variable, depending on the bock parameter settings.

Blocks with Inertia

You can model the effects of inertia using any of the following blocks;

  • Solid — Complete solid with geometry, inertia, and color. Inertia is fixed but can be computed automatically from solid geometries, including those imported from STEP files. Solid geometries are rendered in detail in the Mechanics Explorer visualization pane, making the Solid block ideal for modeling bodies.

  • Inertia — Mass element with fixed inertia and no geometry. Inertia must be computed or obtained separately and specified explicitly. Inertias are visualized as simple icons. Use this block when only inertia matters—for example, when modeling the mass disturbances due to a geometrical asymmetry caused by physical damage.

  • General Variable Mass — Mass element with fixed or variable inertia and no geometry. Inertia must be computed or obtained separately and specified explicitly—as block parameters if fixed and as physical signals if variable. Inertias are visualized as simple icons, frames, or inertia ellipsoids. Use this block to model the variable mass contents of bodies such as fluid tanks.

Inertia in a Model

To add a General Variable Mass, Inertia or Solid block to a model, simply connect its 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 Working with Frames.

The blocks each provide a reference frame port. The Solid block enables you to create additional frames, exposing a new frame port for each frame added. The figure shows an example. 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 and is the two frames are therefore coincident.

The reference frame port of Link B connects to the follower frame port of the Rigid Transform block. The Rigid Transform block applies a spatial transform between the World frame and the reference frame of the Link B block. The two frames are therefore translated and rotated relative to each other according to the transforms specified in the Rigid Transform block.

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

Inertia Parameterizations

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

  • Calculate from Geometry — Available only in the Solid block. Specify the solid geometry and either the solid mass or mass density and let the block compute the remaining inertial properties.

  • Point Mass — Specify the mass and ignore the remaining inertial parameters. The inertia behaves as point mass with no rotational inertia. Do not connect point masses to joints with rotational degrees of freedom.

  • Custom — Specify all inertial properties, including mass or mass density, center of mass, moments of inertia, and products of inertia. You must calculate or obtain the inertial properties separately.

Inertia Reference Frame

When specifying the inertial properties of a solid using the Custom parameterization, you must ensure that the property values are computed relative to the correct "reference" frames, defined as follows:

  • The moments and products of inertia are defined relative to the inertia reference frame—a frame with its axes parallel to the reference port frame axes but with its origin at the center of mass. This frame does not correspond to any port on the block.

  • The center of mass is defined relative to the reference frame of the solid—that associated with port R.

Specifying custom inertias can be especially challenging for General Extrusion and Revolution shapes. Consider the left hole section of the binary link body shown in Model a Compound Body. As described in that example, you can model the solid geometry as a General Extrusion shape—by specifying the solid cross-section so that it can be extruded along the normal (z) axis.

Left Hole Section of Binary Link Body

However, General Extrusion shapes are defined with their reference frame origins held coincident with the [0,0] cross-section coordinate, which in this case lies outside of the solid boundaries. To correctly specify the inertia of this solid, you must therefore obtain the moments and products of inertia relative to a frame with origin not at the center of mass, nor at center of geometry, but at a seemingly arbitrary location that depends only on the cross-section coordinates that you specify.

Change the [0,0] coordinate and you change the reference frame location itself—and, more importantly, the frame relative to which the moments and products of inertia are defined.

Moments and Products of Inertia

The moments and products of inertia comprise the elements of a 9-element inertia matrix. This matrix is symmetric so that elements with reciprocal indices have the same value. That is:

  • Ixy=Iyx

  • Iyz=Izy

  • Izx=Ixz

This symmetry reduces the number of inertia matrix elements that you must specify from nine to six—three moments of inertia and three products of inertia. The moments of inertia are the diagonal elements:


These elements are defined as:

  • Ixx=V(y2+z2)dm

  • Iyy=V(z2+x2)dm

  • Izz=V(x2+y2)dm

where I is inertia, V is volume, m is mass, and (x, y, z) are the Cartesian coordinates of an infinitesimal mass element relative to the inertia reference frame Inertia Reference Frame. The products of inertia are the off-diagonal elements:


These elements are defined as follows:

  • Iyz=Vyzdm

  • Izx=Vzxdm

  • Ixy=Vxydm

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 in different ways:

  • 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 a Compound Body.

  • 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.

Was this topic helpful?