Projected Matrices

In order to solve dynamic equilibrium equations for a flexible body, projected mass and stiffness matrices are required.

Local mass matrix $M$ projected on modes defining the finite rigid body motion:
$M_{R} = Φ_{R}^{T} M Φ_{R}$
Local mass matrix $M$ projected on local vibration modes:
$M_{L} = Φ_{L}^{T} M Φ_{L}$
Coupled terms corresponding to the cross projection of the local mass matrix $M$ on the finite rigid body modes and on the local modes, expressed in the global frame:
$M_{C} = {(P Φ_{L})}^{T} M Φ_{R}$
Where, $P Φ_{L}$ is the family of local vibration modes expressed in global coordinates through the rotation matrix $P$ .
The matrix $M_{C}$ is variable with time since the matrix $P$ evolves with the rigid body motion of the flexible body. The former expression is thus split into 9 constant contributions (one for each term of the rotation matrix):
$M_{C} = \sum_{k = 1}^{3} \sum_{l = 1}^{3} P_{k l} M_{C_{k l}}$
Where, $M_{C_{k l}} = {(T_{k l} Φ_{k l})}^{T} M Φ_{R}$
$T_{k l} = [\begin{matrix} δ_{1 k} δ_{1 l} & δ_{1 k} δ_{2 l} & δ_{1 k} δ_{3 l} \\ δ_{2 k} δ_{1 l} & δ_{2 k} δ_{2 l} & δ_{2 k} δ_{3 l} \\ δ_{3 k} δ_{1 l} & δ_{3 k} δ_{2 l} & δ_{3 k} δ_{3 l} \end{matrix}]$
The matrices to input are the 9 $M_{C_{k l}}$ matrices.
Local stiffness matrix $K$ projected on local vibration modes:
$K_{L} = Φ_{L}^{T} K Φ_{L}$
If static modes are present in the local projection basis (refer to Dynamic Analysis in the Radioss Theory Manual), the projected matrix may not be diagonal. However, it may contain a large diagonal block, corresponding to the projection on eigen modes appearing in the basis. The full part and the diagonal part of the matrix are input separately. The shape of the reduced matrix is:
$K_{L} = [\begin{matrix} Φ_{L, s t a t}^{T} K Φ_{L, s t a t} & Φ_{L, s t a t}^{T} K Φ_{L, d y n} \\ s y m & Φ_{L, d y n}^{T} K Φ_{L, d y n} \end{matrix}]$
The full part corresponds to $[\begin{matrix} Φ_{L, s t a t}^{T} K Φ_{L, s t a t} & Φ_{L, s t a t}^{T} K Φ_{L, d y n} \end{matrix}]$ , in which $Φ_{L, s t a t}^{T} K Φ_{L, s t a t}$ is symmetric and $Φ_{L, s t a t}^{T} K Φ_{L, d y n}$ is rectangular. The diagonal part corresponds to $Φ_{L, d y n}^{T} K Φ_{L, d y n}$ .
Coupled terms corresponding to the cross projection of the local stiffness matrix $K$ on the finite rigid body modes expressed in the local frame and on the local modes:
$K_{C} = Φ_{L}^{T} K (P^{T} Φ_{R})$
This expression is again split into 9 contributions:
$K_{C} = \sum_{k = 1}^{3} \sum_{l = 1}^{3} P_{k l} K_{C_{k l}}$
Where, $K_{C_{k l}} = Φ_{L}^{T} K (T_{k l} Φ_{R})$
The matrices to input are now the 9 $M_{C_{k l}}$ matrices.