10.1.3. Nodal Equations of Motion

The position of the node \(i\) is written as

(10.7)\[\mathbf{r}_{f_i}=\mathbf{r}_{f}+\mathbf{A}_{f}\mathbf{d}_{f_i}^{'}\]

The velocity of the node \(i\) can be obtained by time differentiation of Eq. (10.8)

(10.8)\[\dot{\mathbf{r}}_{f_i}=\mathbf{A}_{f} ( \dot{\mathbf{r}}_{f}^{'} -{\tilde{\mathbf{d}}}_{f_i}^{'}{\boldsymbol{\omega}}_{f}^{'} +\dot{\mathbf{u}}_{f_i}^{'} )\]

The acceleration of the node \(i\) can be obtained by time differentiation of Eq. (10.9)

(10.9)\[\ddot{\mathbf{r}}_{f_i}=\mathbf{A}_{f}( \ddot{\mathbf{r}}_{f}^{'} -\tilde{\mathbf{d}}_{f_i}^{'}\dot{\boldsymbol{\omega}}_{f}^{'} +\ddot{\mathbf{u}}_{f_i}^{'}+\tilde{\boldsymbol{\omega}}_{f}^{'}\dot{\mathbf{r}}_{f}^{'}+\tilde{\boldsymbol{\omega}}_{f}^{'}\tilde{\boldsymbol{\omega}}_{f}^{'}\mathbf{d}_{f_i}^{'}+2\tilde{\boldsymbol{\omega}}_{f}^{'}\dot{\mathbf{u}}_{f_i}^{'})\]

The nodal equations of motion corresponding the node \(i\) can be derived as

(10.10)\[m_{f_i}\mathbf{A}_{f}( \ddot{\mathbf{r}}_{f}^{'} -\tilde{\mathbf{d}}_{f_i}^{'}\dot{\boldsymbol{\omega}}_{f}^{'} +\ddot{\mathbf{u}}_{f_i}^{'}+\tilde{\boldsymbol{\omega}}_{f}^{'}\dot{\mathbf{r}}_{f}^{'}+\tilde{\boldsymbol{\omega}}_{f}^{'}\tilde{\boldsymbol{\omega}}_{f}^{'}\mathbf{d}_{f_i}^{'}+2\tilde{\boldsymbol{\omega}}_{f}^{'}\dot{\mathbf{u}}_{f_i}^{'})+\mathbf{K}_{f_i}\mathbf{u}_{f_i}^{'}-m_{f_i}\mathbf{g}-\mathbf{f}_{R_i}=0\]

where,

\({m}_{f_i}\) : A lumped nodal mass.

\(\mathbf{K}_{f_i}\) : Sub-stiffness matrix, which is composed of the row vectors corresponding th the node \(i\).

\(\mathbf{A}_{f}\) : The orientation matrix of the RFlex body \(f\).

\(\boldsymbol{\omega}_{f}^{'}\) : The angular velocity of the RFlex body \(f\) measured with respect to the RFlex body reference frame.

\(\mathbf{g}\) : Gravitational acceleration.

\(\mathbf{f}_{R_i}\) : Reaction force vector applied on the node \(i\). When force element or kinematic joints are connected with the node \(i\), the reaction force vector has non-zero value.