Masstran Analysis Interface Module (AIM)
Masstran Analysis Interface Module (AIM)
|
A module in the Computational Aircraft Prototype Syntheses (CAPS) has been developed to compute mass properties using attributions for finite element structural solvers.
An outline of the AIM's inputs, outputs and attributes are provided in AIM Inputs and AIM Outputs and Masstran AIM attributes, respectively.
The mass properties are computed via the formulas:
\begin{eqnarray*} m &=& \sum_i m_i \\ x_{cg} &=& \frac{1}{m}\sum_i m_i x_i \\ y_{cg} &=& \frac{1}{m}\sum_i m_i y_i \\ z_{cg} &=& \frac{1}{m}\sum_i m_i z_i \\ \big(I_{xx}\big)_{cg} &=& \sum_i m_i \big(y_i^2 + z_i^2\big) - m \big(y_{cg}^2 + z_{cg}^2\big) \\ \big(I_{yy}\big)_{cg} &=& \sum_i m_i \big(x_i^2 + z_i^2\big) - m \big(x_{cg}^2 + z_{cg}^2\big) \\ \big(I_{zz}\big)_{cg} &=& \sum_i m_i \big(x_i^2 + y_i^2\big) - m \big(x_{cg}^2 + y_{cg}^2\big) \\ \big(I_{xy}\big)_{cg} &=& \sum_i m_i \big(x_i y_i \big) - m \big(x_{cg} y_{cg} \big) \\ \big(I_{xz}\big)_{cg} &=& \sum_i m_i \big(x_i z_i \big) - m \big(x_{cg} z_{cg} \big) \\ \big(I_{yz}\big)_{cg} &=& \sum_i m_i \big(y_i z_i \big) - m \big(y_{cg} z_{cg} \big), \end{eqnarray*}
where i represents an element index in the mesh, and the mass \(m_i\) is computed from the density, thickness, and area of the element.
The moment of inertias are accessible individually, in vector form as
\[ \vec{I} = \begin{bmatrix} I_{xx} & I_{yy} & I_{zz} & I_{xy} & I_{xz} & I_{yz} \end{bmatrix}, \]
as lower/upper triangular form
\[ \vec{I}_{lower} = \begin{bmatrix} I_{xx} & -I_{xy} & I_{yy} & -I_{xz} & -I_{yz} & I_{zz} \end{bmatrix}, \]
\[ \vec{I}_{upper} = \begin{bmatrix} I_{xx} & -I_{xy} & -I_{xz} & I_{yy} & -I_{yz} & I_{zz} \end{bmatrix}, \]
or in full tensor form as
\[ \bar{\bar{I}} = \begin{bmatrix} I_{xx} & -I_{xy} & -I_{xz} \\ -I_{xy} & I_{yy} & -I_{yz} \\ -I_{xz} & -I_{yz} & I_{zz} \end{bmatrix}. \]
An example problem using the Masstran AIM may be found at Masstran AIM Basic Example.