3.3.4. Mass
This allows the user to measure the total mass and the moment of inertia for the selected bodies or subsystems. If the user does not specify a reference marker for the Inertia Position, the moments of inertia are calculated from the mass center of selected bodies.
Figure 3.12 Definition of Mass Properties
Mass: Shows the total mass of selected bodies or subsystems. The total mass can be computed from the following equation.
\({{m}_{t}}=\sum\limits_{i=1}^{n}{{{m}_{i}}}\)
where \({{m}_{i}}\) is the mass of each body.
Mass Center
\({{r}_{cm}}=\frac{\sum\limits_{i=1}^{n}{{{m}_{i}}{{r}_{i}}}}{{{m}_{t}}}\)
where \({{r}_{i}}\) is the position vector of the mass center of each body.
Inertia Moment
If the inertia moment \({{J}_{i}}^{\prime \prime }\) which is defined on the inertia marker of a body is transformed into the center marker of the body, the transformed inertia moment \({{J}_{i}}^{\prime }\) can be computed as follows.
\({{J}_{i}}^{\prime }={{C}_{i}}{{J}_{i}}^{\prime \prime }{{C}_{i}}^{T}-{{m}_{i}}({{s}_{im}}{{^{\prime }}^{T}}{{s}_{im}}^{\prime }I-{{s}_{im}}^{\prime }{{s}_{im}}{{^{\prime }}^{T}})\)
where \({{C}_{i}}\) and \(I\) are the orientation matrices of the inertia marker with respect to the center marker and the identity matrix, respectively. \({{s}_{im}}^{\prime }\) is the position vector from the inertia marker to the center marker with respect to the center marker reference frame.
If the inertia moment \(J{{_{i}^{{}}}^{\prime }}\) which is defined on the center marker of the body is transformed into the target marker, the transformed inertia moment \({}^{t}J{{_{i}^{{}}}^{\prime \prime }}\) can be computed as follows.
\({}^{t}{{J}_{i}}^{\prime \prime }={{({{A}_{t}}{{C}_{t}})}^{T}}{{A}_{i}}{{J}_{i}}^{\prime }{{A}_{i}}^{T}{{A}_{t}}{{C}_{t}}+{{m}_{i}}({{d}_{ii}}{{^{\prime \prime }}^{T}}{{d}_{ii}}^{\prime \prime }I-{{d}_{ii}}^{\prime \prime }{{d}_{ii}}{{^{\prime \prime }}^{T}})\)
where \({{A}_{i}}\) is the orientation matrix of the center marker of i body. \({{A}_{t}}\) and \({{C}_{t}}\) are the orientation matrices of the center marker of the target body and the target marker, respectively. \({{d}_{ii}}^{\prime \prime }\) is the distance vector from the target marker to the center marker of i body with respect to the target marker reference frame. This can be computed as follows.
\({{d}_{ii}}^{\prime \prime }={{({{A}_{t}}{{C}_{t}})}^{T}}({{r}_{i}}-{{r}_{t}}-{{A}_{t}}{{s}_{t}}^{\prime })\)
If the target is the position of the mass center of the selected bodies, both \({{A}_{t}}\) and \({{C}_{t}}\) are the identity matrix and \({{s}_{t}}^{\prime }\) is the zero vector. Thus, the distance vector can be defined as follows.
\({{d}_{ii}}^{\prime \prime }={{r}_{i}}-{{r}_{cm}}\)
Finally, the sum of inertia moments is determined by the following equation.
\({}^{t}{J}''=\sum\limits_{i=1}^{n}{{}^{t}{{J}_{i}}^{\prime \prime }}\)
Figure 3.13 Mass Properties dialog box
Name: Displays the names of all available bodies or subsystems.
Add/Remove: Select bodies to add or remove from the list.
Add: Add a row to the list.
Delete: Delete the current row from the list.
All Body: Add all bodies and subsystems in a model.
Inertia Position: Selects Mass Center or reference Marker.
Preview: Show Mass Center on the working window.
Figure 3.14 Preview Mass Center on the working window
Calculate with Body Principal Axis: Calculates the Moments of Inertia by the body’s principal axis. If not using this option, it is calculated by the orientation of the center marker.
Result
Mass: Displays the total mass of the selected bodies.
Moments of Inertia (Ixx, Iyy, Izz, Ixy, Iyz, and Izx): Displays the components of the moments of inertia.
Position of Mass Center: Displays the position of the mass center with respect to Ground.InertiaMarker.
Orientation of Mass Center: Displays the orientation of the Mass Center when the option Calculate with Body Principal Axis is on.
Step to Calculate Mass Property
Add rows to the list by clicking Add.
Select the desired bodies or subsystems. (If the user wants to select more bodies or subsystems, repeat steps 1 and 2).
Select Inertia Position type. If the user selects Marker, the user should navigate a marker to select the reference marker for Inertia Position.
Click Calculate.