10.9.1. RFlex Modal Force

The user can apply a user-defined load to a RFlex body using RFlex Modal Force. The function of RFlex Modal Force consists of three parts as Modal Load Case, Modal Force and Modal Preload.

Mathematical definitions

A set of nodal force and torque is can be translated to a generalized force with respect to the reference frame of RFlex body as following equation. The 3rd part of equation (10.24)

(10.24)\[\begin{split}\begin{Bmatrix} \mathbf{F}_g \\ \mathbf{T}_g \\ \mathbf{f}_g \end{Bmatrix}= \begin{Bmatrix} \sum\limits_p^{\text{no\_used\_nodes(nun)}} \boldsymbol{\Phi}_p^T{\mathbf{F}}_p \\ \sum\limits_p^{\text{nun}}\tilde{\mathbf{d}}_p^{'} \mathbf{F}_p + \sum\limits_p^{\text{nun}} \mathbf{T}_p \\ \sum\limits_p^{\text{nun}}\boldsymbol{\Phi}_p^{tT} \mathbf{F}_p + \sum\limits_p^{\text{nun}} \boldsymbol{\Phi}_p^{rT} \mathbf{T}_p \end{Bmatrix}\end{split}\]
Where,
\({{\mathbf{F}}_{g}}\) is a generalized force with respect to the reference frame of RFlex body.
\({{\mathbf{T}}_{g}}\) is a generalized torque with respect to the reference frame of RFlex body.
\({{\mathbf{f}}_{g}}\) is a generalized modal force with respect to the reference frame of RFlex body.
\({{\mathbf{F}}_{p}}\) is a force applied on \(p\) node with respect to the reference frame of RFlex body.
\({{\mathbf{T}}_{p}}\) is a torque applied on \(p\) node with respect to the reference frame of RFlex body.
\({{\mathbf{{d}'}}_{p}}\) is a position vector applied on \(p\) node with respect to the reference frame of RFlex body.
\(\mathbf{\Phi }_{p}^{t}\) is a translational mode shape matrix. [3 \(\times\) nmode]
\(\mathbf{\Phi }_{p}^{r}\) is a rotational mode shape matrix. [3 \(\times\) nmode]

If a force and torque a acting on the \(p\) node

The force vector including torque \({{\mathbf{F}}_{p}}\) [6 \(\times\) 1] is acting on the \(p\) node.

../_images/image1544.png

Figure 10.99 Simple example for Modal force

Therefore, we can write (10.25).

(10.25)\[\mathbf{M}_p\ddot{\mathbf{x}}_{p}+\mathbf{K}_p\mathbf{x}_{p}=\mathbf{F}_{p}\]

Where, \({{\mathbf{M}}_{p}}\) is a mass matrix [3 \(\times\) 3] on \(p\) node. \({{\mathbf{K}}_{p}}\) is a stiffness matrix [3 \(\times\) 3] on \(p\) node.

The \({{\mathbf{x}}_{p}}\) is defined as \({{\mathbf{\Phi }}_{p}}\mathbf{a}\). Where, \({{\mathbf{\Phi }}_{p}}\) is a mode shape matrix [6 \(\times\) nmode]. \(\mathbf{a}\) is modal coordinates.

(10.26)\[{{\mathbf{M}}_{p}}(\mathbf{\Phi }_{p}\mathbf{\ddot{a}})+{{\mathbf{K}}_{p}}({{\mathbf{\Phi }}_{p}}\mathbf{a})={{\mathbf{F}}_{p}}\]

Multiply \(\mathbf{\Phi }_{p}^{T}\) to (10.26)

(10.27)\[\mathbf{\Phi }_{p}^{T}{{\mathbf{M}}_{p}}{{\mathbf{\Phi }}_{p}} \mathbf{\ddot{a}}+\mathbf{\Phi }_{p}^{T}{{\mathbf{K}}_{p}} {{\mathbf{\Phi }}_{p}}\mathbf{a}=\mathbf{\Phi }_{p}^{T}{{\mathbf{F}}_{p}}={{\mathbf{f}}_{p}}\]

Finally, we get a mathematical definition of Modal force as follows.

(10.28)\[\mathbf{\hat{M}}\ddot{\mathbf{a}}+\mathbf{\hat{K}}\mathbf{a}=\mathbf{\Phi}^{T}\mathbf{F}=\mathbf{f}\]
(10.29)\[\mathbf{\hat{M}}=\mathbf{\Phi }_{p}^{T}{{\mathbf{M}}_{p}}{{\mathbf{\Phi }}_{p}}=\mathbf{I}\]
(10.30)\[\mathbf{\hat{K}}=\mathbf{\Phi }_{p}^{T}{{\mathbf{K}}_{p}}{{\mathbf{\Phi }}_{p}}= diag(\omega _{1}^{2},\omega _{2}^{2},\cdots ,\omega _{\text{nmode}}^{2})\]