10.1.5. Strain Energy

Strain energy is potential energy. For the simple 1 DOF mass-spring-damper system, the strain energy is defined as the following equation.

\(P=\cfrac{1}{2}k\delta^{2}\)

The strain energy of an arbitrary RFlex \(P\) node can be calculated in the same way.

\(\begin{aligned} P_P &=\cfrac{1}{2}\bar{\mathbf{u}}_{P}^{T}\mathbf{K}_{TP}\bar{\mathbf{u}}_{P} + \cfrac{1}{2}\bar{\boldsymbol{\theta}}_{P}^{T}\mathbf{K}_{RP}\bar{\boldsymbol{\theta}}_{P} \\ &= \cfrac{1}{2}(\boldsymbol{\Phi}_{TP}\mathbf{q}_a)^T\mathbf{K}_{TP}\boldsymbol{\Phi}_{TP}\mathbf{q}_a + \cfrac{1}{2}(\boldsymbol{\Phi}_{RP}\mathbf{q}_a)^T\mathbf{K}_{RP}\boldsymbol{\Phi}_{RP}\mathbf{q}_a \\ &= \cfrac{1}{2} \mathbf{q}_a^T\boldsymbol{\Phi}_{TP}^T\mathbf{K}_{TP}\boldsymbol{\Phi}_{TP}\mathbf{q}_a + \cfrac{1}{2}\mathbf{q}_a^T\boldsymbol{\Phi}_{RP}^T\mathbf{K}_{RP}\boldsymbol{\Phi}_{RP}\mathbf{q}_a \end{aligned}\)

where,

\(\mathbf{q}_a\): Modal Coordinate

\(\mathbf{K}_{TP}\): Trans. Stiffness Matrix of \(P\) node

\(\mathbf{K}_{RP}\): Rot. Stiffness Matrix of \(P\) node

\(\bar{\mathbf{u}}_{P}\): Trans. deformation Vector of \(P\) node

\(\bar{\boldsymbol{\theta}}_{P}\): Rot. deformation Vector of \(P\) node}

\(\boldsymbol{\Phi}_{TP}\): Trans. Mode Shape Matrix of \(P\) node

\(\boldsymbol{\Phi}_{RP}\): Rot. Mode Shape Matrix of \(P\) node

Therefore, we can calculate the strain energy of the RFlex body with the summation of all the energy each node. The final formula is represented by Eigen Value and Modal Coordinate.

\(\begin{aligned} P &=\sum\limits_{P=1}^{nNode}\left( \cfrac{1}{2}\mathbf{q}_{a}^{T}\boldsymbol{\Phi}_{TP}^{T}\mathbf{K}_{TP}\boldsymbol{\Phi}_{TP}\mathbf{q}_a + \cfrac{1}{2}\mathbf{q}_{a}^{T}\boldsymbol{\Phi}_{RP}^{T}\mathbf{K}_{RP}\boldsymbol{\Phi}_{RP}\mathbf{q}_a \right) \\ &= \cfrac{1}{2}\mathbf{q}_a^T(\boldsymbol{\Phi}^T\mathbf{K}\boldsymbol{\Phi})\mathbf{q}_a = \cfrac{1}{2}\mathbf{q}_a^T\mathbf{K}_{aa}\mathbf{q}_a \end{aligned}\)

Where,

\({{\mathbf{K}}_{aa}}\) is a diagonal matrix with eigenvalues of RFlex body.