pyyeti.ode.make_A¶
- pyyeti.ode.make_A(M, B, K)[source]¶
Setup the state-space matrix from mass, damping and stiffness.
- Parameters:
M (1d or 2d ndarray or None) – Mass; vector (of diagonal), or full; if None, mass is assumed identity
B (1d or 2d ndarray) – Damping; vector (of diagonal), or full
K (1d or 2d ndarray) – Stiffness; vector (of diagonal), or full
- Returns:
A (2d ndarray) – The state-space matrix defined as shown below
Notes
The typical 2nd order ODE is:
\[M \ddot{q} + B \dot{q} + K q = F\]The 2nd order ODE set of equations are transformed into the 1st order ODE:
\[\begin{split}\left\{ \begin{array}{c} \ddot{q} \\ \dot{q} \end{array} \right\} - \left[ \begin{array}{cc} -M^{-1} B & -M^{-1} K \\ I & 0 \end{array} \right] \left\{ \begin{array}{c} \dot{q} \\ q \end{array} \right\} = \left\{ \begin{array}{c} M^{-1} F \\ 0 \end{array} \right\}\end{split}\]or:
\[\dot{y} - A y = w\]When the M, B and K are assembled into the A matrix, they must not contain any rigid-body modes. See
eigss().