10.1.2. Invariant Variables
RecurDyn RFlex has the 9 invariant variables as follows.
Invariant variable M1: \({\zeta}_{1}=\sum\limits_{p=1}^{nNode} {m_p}\) (1x1)
Invariant variable M2: \({\zeta}_{2}=\sum\limits_{p=1}^{nNode} {m_p}{{\mathbf{d}}^{'}_{0p}}\) (3x1)
Invariant variable M3: \({\zeta}_{3}=\sum\limits_{p=1}^{nNode} ({m_p} \tilde{{\mathbf{d}}}^{'}_{0p}\tilde{{\mathbf{d}}}^{'}_{0p}-\mathbf{I}^{'}_{p})\) (3x3)
Invariant variable N1: \({\zeta}_{4}=\sum\limits_{p=1}^{nNode} {m_p} \boldsymbol{\Phi}^{j}_{T_p}\) j=1, nM ((3x1)x nM)
Invariant variable N2: \({\zeta}_{5}=\sum\limits_{p=1}^{nNode} {m_p} \tilde{{\mathbf{d}}}^{'}_{0p} \tilde{\boldsymbol{\Phi}}^{j}_{T_p}\) j=1, nM ((3x3)x nM)
Invariant variable N3: \({\zeta}_{5}^{T}=\sum\limits_{p=1}^{nNode} {m_p} \tilde{\boldsymbol{\Phi}}^{j}_{T_p} \tilde{{\mathbf{d}}}^{'}_{0p}\) j=1, nM ((3x3)x nM)
Invariant variable N4: \({\zeta}_{6}=\sum\limits_{p=1}^{nNode} {m_p} \tilde{\boldsymbol{\Phi}}^{j}_{T_p} \tilde{\boldsymbol{\Phi}}^{k}_{T_p}\) j=1, nM ((3x3)x nM x nM)
Invariant variable N5: \({\zeta}_{7}=\sum\limits_{p=1}^{nNode} ({m_p} \tilde{{\mathbf{d}}}^{'}_{0p} \boldsymbol{\Phi}^{j}_{T_p}+\mathbf{I}^{'}_{p}{\boldsymbol{\Phi}}_{Rp}^{j})\) j=1, nM ((3x1)x nM)
Invariant variable N6: \({\zeta}_{8}=\sum\limits_{p=1}^{nNode}{{{m}_{p}}\mathbf{\tilde{\Phi }}_{Tp}^{j}\mathbf{\Phi }_{Rp}^{k}}\) j,k=1, nM ((3x1)x nM x nM)
- Where,
- \(m_p\) is a lumped mass of \(p\) node.\(\mathbf{d}_{0p}^{'}\) is a position vector of \(p\) node with respect to RFlex body reference frame.\(\mathbf{I}_{p}^{'}\) is a moment of inertia of \(p\) node with respect to RFlex body reference frame.\(\mathbf{\Phi}_{Tp}^{j}\) and \(\mathbf{\Phi}_{Rp}^{j}\) are translation and rotational mode shape with \(j\)-th mode of \(p\) node.
nM is number of modes.
\(\mathbf{\tilde{a}}\equiv \left[ \begin{matrix} 0 & -{{a}_{3}} & {{a}_{2}} \\ {{a}_{3}} & 0 & -{{a}_{1}} \\ -{{a}_{2}} & {{a}_{1}} & 0 \\ \end{matrix} \right]\)
Mass matrix for RFlex body is defined as follows using the RD/RFlex Invariant variables. (\(r\) : translational DOF, \(w\) : rotational DOF, \(a\) : modal coordinates)
\(\mathbf{M}=\left[ \begin{matrix} {{\mathbf{M}}_{rr}} & {{\mathbf{M}}_{rw}} & {{\mathbf{M}}_{ra}} \\ \mathbf{M}_{rw}^{T} & {{\mathbf{M}}_{ww}} & {{\mathbf{M}}_{wa}} \\ \mathbf{M}_{ra}^{T} & \mathbf{M}_{wa}^{T} & {{\mathbf{M}}_{aa}} \\ \end{matrix} \right]\)\({\mathbf{M}_{rr}}=\mathbf{1}{{\zeta }_{1}}\)\({\mathbf{M}_{rw}}=-({{{\mathbf{\tilde{\zeta }}}}_{2}}+{{{\mathbf{\tilde{\zeta }}}}_{4}}\mathbf{a})\approx -{{{\mathbf{\tilde{\zeta }}}}_{2}}\)\({\mathbf{M}_{ra}}={{\mathbf{\zeta }}_{4}}\)\({\mathbf{M}_{ww}}=-({{\mathbf{\zeta }}_{3}}+{{\mathbf{\zeta }}_{5}}{{\mathbf{a}}_{i}}+\mathbf{\zeta }_{5}^{T}{{\mathbf{a}}_{i}}+{{\mathbf{\zeta }}_{6}}{{\mathbf{a}}_{i}}{{\mathbf{a}}_{k}})\approx -({{\mathbf{\zeta }}_{3}}+{{\mathbf{\zeta }}_{5}}{{\mathbf{a}}_{i}}+\mathbf{\zeta }_{5}^{T}{{\mathbf{a}}_{i}})\)\({\mathbf{M}_{wa}}={{\mathbf{\zeta }}_{7}}+{{\mathbf{\zeta }}_{8}}{{\mathbf{a}}_{j}}\)\({\mathbf{M}_{aa}}=\mathbf{1}\)Mass center position of RFlex body
\({{\mathbf{r}}_{cm}}=\cfrac{\sum\limits_{p=1}^{nNode}{{{m}_{p}}{{\mathbf{r}}_{p}}}}{\sum\limits_{p=1}^{nNode}{{{m}_{p}}}}={{\mathbf{r}}_{f}}+{{\mathbf{A}}_{f}}\cfrac{{{\zeta }_{2}}+{{\zeta }_{4}}\mathbf{a}}{{{\zeta }_{1}}}\)
- Where,