Inertia

Inertial properties of a solid or mass disturbance

Body Elements

Description

This block represents the inertial properties of a solid. The solid can be a point mass or a 3-D mass distribution. To represent the inertial properties of a point mass, the dialog box provides a ```Point Mass``` parameterization. Inertial parameters of a point mass include only the total mass of the solid. To represent the inertial properties of a 3-D mass distribution, the dialog box provides a `Custom` parameterization. Inertial parameters of a custom inertia include the total mass of the solid, its center of mass, as well as its moments and products of inertia.

This block can also represent a mass disturbance in a model. The disturbance can have positive or negative inertia. A disturbance with negative inertia reduces the total inertia of the rigid body the block connects to. A disturbance with positive inertia increases the total inertia of the rigid body the block connects to. Use this block to adjust the total inertia of a rigid body.

The visualization pane of Mechanics Explorer identifies the position of an Inertia element with the inertia icon .

Dialog Box and Parameters

Type

Select a method to specify inertia. The default is ```Point Mass```.

TypeDescription
`Point Mass`Treat the inertia as a mass with zero volume. The point mass is located at the reference frame origin.
`Custom`Manually specify all inertial parameters, including mass, center of mass, and moments and products of inertia.

`Point Mass`/`Custom`: Mass

Enter the total mass of the solid. Select a physical unit. The default is `1 Kg`.

`Custom`: Center of Mass

Enter the center of mass coordinates with respect to the solid reference frame in the order [X Y Z]. In a uniform gravitational field, the center of mass coincides with the center of gravity. Select a physical unit. The default is `[0 0 0]`.

`Custom`: Moments of Inertia

Enter the mass moments of inertia of the solid element in the order [Ixx, Iyy, Izz]. Each moment of inertia must refer to a frame whose axes are parallel to the block reference frame axes and whose origin is coincident with the solid center of mass. The moments of inertia are the diagonal elements of the solid inertia tensor,

`$\left(\begin{array}{ccc}{I}_{xx}& & \\ & {I}_{yy}& \\ & & {I}_{zz}\end{array}\right),$`

where:

• ${I}_{xx}=\underset{V}{\int }\left({y}^{2}+{z}^{2}\right)\text{\hspace{0.17em}}dm$

• ${I}_{yy}=\underset{V}{\int }\left({x}^{2}+{z}^{2}\right)\text{\hspace{0.17em}}dm$

• ${I}_{zz}=\underset{V}{\int }\left({x}^{2}+{y}^{2}\right)\text{\hspace{0.17em}}dm$

Select a physical unit. The default is `[1 1 1] kg*m^2`.

`Custom`: Products of Inertia

Enter the mass products of inertia of the solid element in the order [Iyz, Izx, Ixy]. Each product of inertia must refer to a frame whose axes are parallel to the block reference frame axes and whose origin is coincident with the solid center of mass. The products of inertia are the off-diagonal elements of the solid inertia tensor,

`$\left(\begin{array}{ccc}& {I}_{xy}& {I}_{zx}\\ {I}_{xy}& & {I}_{yz}\\ {I}_{zx}& {I}_{yz}& \end{array}\right),$`

where:

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

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

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

Select a physical unit. The default is `[0 0 0] kg*m^2`.

Ports

This block contains frame port R, representing the inertia reference frame.