Mass matrix in global coordinate frame

Transformation of a vector $\boldsymbol{a}$ from local coordinate frame $\boldsymbol{e}_i$ into global coordinate frame $\hat{\boldsymbol{e}}_i$ yields

$\displaystyle \hat{a}_i \hat{\boldsymbol{e}}_i$	$\displaystyle =$	$\displaystyle a_j \boldsymbol{e}_j \quad \vert \cdot \hat{\boldsymbol{e}}_k$	(40)
$\displaystyle \hat{a}_i \delta_{ik}$	$\displaystyle =$	$\displaystyle a_j \boldsymbol{e}_j \cdot \hat{\boldsymbol{e}}_k$	(41)
$\displaystyle \hat{a}_k$	$\displaystyle =$	$\displaystyle T_{jk} a_j = (T^T)_{kj} a_j$	(42)

$\displaystyle \boldsymbol{a} = \boldsymbol{T} \boldsymbol{\hat{a}},$

(43)

with the transformation matrix

$\displaystyle \boldsymbol{T} = \left[ \begin{array}{ccc} - & \boldsymbol{e}_1 & - \\ - & \boldsymbol{e}_2 & - \\ - & \boldsymbol{e}_3 & - \end{array} \right]$

(44)

Therefore, the variation of the kinetic energy becomes

$\displaystyle \delta \boldsymbol{d}^T \boldsymbol{M} \boldsymbol{d} = \delta \b... ...{\hat{d}}^T \boldsymbol{T}^T \boldsymbol{M} \boldsymbol{T} \boldsymbol{\hat{d}}$

(45)

and thus the final mass matrix in global coordinate frame $\hat{\boldsymbol{e}}_i$ yields

$\displaystyle \hat{\boldsymbol{M}} = \boldsymbol{T}^T \boldsymbol{M} \boldsymbol{T}$

(46)