# Node Masses

Node can have mass and inertia. In the normal case the mass matrix is

$\mathbf{M} = \begin{bmatrix} m & 0 & 0 \\ & m & 0 \\ & & m \\ \end{bmatrix}$

but if mass modifiers are included the mass matrix becomes

$\mathbf{M} = \begin{bmatrix} m_{x} & 0 & 0 \\ & m_{y} & 0 \\ & & m_{z} \\ \end{bmatrix}$

Inertia is a tensor quantity with the following terms

$\mathbf{I} = \begin{bmatrix} I_{xx} & I_{xy} & I_{xz} \\ & I_{yy} & I_{yz} \\ & & I_{zz} \\ \end{bmatrix}$

where the terms in the inertia tensor are defined as

\begin{aligned}I_{xx} &= \int_{V}^{}{\rho\left\lbrack \left( y - y_{c} \right)^{2} + \left( z - z_{c} \right)^{2} \right\rbrack}dV\quad & I_{xy} &= I_{yx} = \int_{V}^{}{\rho\left( x - x_{c} \right)\left( y - y_{c} \right)}dV\\ I_{yy} &= \int_{V}^{}{\rho\left\lbrack \left( z - z_{c} \right)^{2} + \left( x - z_{c} \right)^{2} \right\rbrack}dV\quad & I_{yz} &= I_{zy} = \int_{V}^{}{\rho\left( y - y_{c} \right)\left( z - z_{a} \right)}dV\\ I_{zz} &= \int_{V}^{}{\rho\left\lbrack \left( x - x_{c} \right)^{2} + \left( y - y_{c} \right)^{2} \right\rbrack}dV\quad & I_{zx} &= I_{xz} = \int_{V}^{}{\rho\left( z - z_{c} \right)\left( x - x_{c} \right)}dV\end{aligned}

where $\left( x_{a},y_{a},z_{a} \right)$ are the coordinates of the centre of mass. If non-zero values are specified for the off-diagonal terms, it is important that these are consistent with the diagonal terms. If this is not done the principal inertia values can become negative. The inertia matrix is never modified for directions.