10.1.6. Kinetic Energy

Kinetic energy is defined as the following for a lumped mass.

\(T=\cfrac{1}{2}mv^2\)

The kinetic energy can be calculated in the same way.

\(\begin{aligned} T &= \cfrac{1}{2} \begin{bmatrix} \dot{\mathbf{r}}_f^T & \boldsymbol{\omega}_f^T & \dot{\mathbf{a}}_f^T \end{bmatrix} \begin{bmatrix} \mathbf{M}_{rr} & \mathbf{M}_{rw} & \mathbf{M}_{ra} \\ & \mathbf{M}_{ww} & \mathbf{M}_{wa}\\ \textbf{Sym} & & \mathbf{M}_{aa} \end{bmatrix} \begin{bmatrix} \dot{\mathbf{r}}_f \\ \boldsymbol{\omega}_f \\ \dot{\mathbf{a}}_f \end{bmatrix} \\ & =\cfrac{1}{2}\dot{\mathbf{r}}_f^T\mathbf{M}_{rr}\dot{\mathbf{r}}_f +\cfrac{1}{2}\boldsymbol{\omega}_f^T\mathbf{M}_{ww}\boldsymbol{\omega}_f +\cfrac{1}{2}\dot{\mathbf{a}}_f^T\mathbf{M}_{aa}\dot{\mathbf{a}}_f \\ & \qquad\qquad\qquad +\dot{\mathbf{r}}_f^T\mathbf{M}_{rw}\boldsymbol{\omega}_f +\dot{\mathbf{r}}_f^T\mathbf{M}_{ra}\dot{\mathbf{a}}_f +\boldsymbol{\omega}_f^T\mathbf{M}_{wa}\dot{\mathbf{a}}_f \end{aligned}\)

Where,

\(\dot{\mathbf{r}}_f\): Velocity of RFlex Body w.r.t. RFlex Body Ref. Frame

\(\boldsymbol{\omega}_f\): Angular Velocity of RFlex Body

\(\dot{\mathbf{a}}_f\): Modal Velocity of RFlex Body

If ignoring the RFlex rigid motion, then the Kinetic Energy is calculated with following equation. The final formula is represented by Modal Velocity.

\(\begin{aligned} T &= \cfrac{1}{2}\dot{\mathbf{a}}_f^T\mathbf{M}_{aa}\dot{\mathbf{a}}_f \\ & = \cfrac{1}{2}\dot{\mathbf{a}}_f^T\left\{\sum(\boldsymbol{\Phi}_{Tp}^T\mathbf{m}_p\boldsymbol{\Phi}_{Tp}+\boldsymbol{\Phi}_{Rp}^T\mathbf{I}_p^{'}\boldsymbol{\Phi}_{Rp})\right\}\dot{\mathbf{a}}_f \\ & = \cfrac{1}{2}\dot{\mathbf{a}}_f^T\left\{\sum(\boldsymbol{\Phi}_{p}^T\mathbf{M}_p\boldsymbol{\Phi}_{p})\right\}\dot{\mathbf{a}}_f \\ & = \cfrac{1}{2}\dot{\mathbf{a}}_f^T\dot{\mathbf{a}}_f\qquad\because\sum(\boldsymbol{\Phi}_{p}^T\mathbf{M}_p\boldsymbol{\Phi}_{p})=\mathbf{I} \end{aligned}\)