10.1.7. Correction Mode

10.1.7.1. Static Correction Mode

A static correction mode is defined as a static deformation due to imposing unit displacement on the nodal coordinate in the boundary set \(B\) and zero displacements on the remaining nodal coordinates in the boundary set \(B\).

Let a set \(W\) be a complementary set of the set \(B\) as

\(W=U-B\)

where, \(U\) is the set of all nodal coordinates in the finite element model.

Static correction modes, \(\bar{\boldsymbol{\Psi}}_s\) can be obtained as

(10.11)\[\mathbf{K}_{f}\bar{\boldsymbol{\Psi}}_s=\mathbf{F}\]

where,

(10.12)\[\begin{split}\mathbf{K}_{f}={{\left[ \begin{matrix} {\mathbf{K}_{II}} & {\mathbf{K}_{IB}} \\ {\mathbf{K}_{BI}} & {\mathbf{K}_{BB}} \\ \end{matrix} \right]}_{f}}\end{split}\]

(10.13)\[\bar{\boldsymbol{\Psi}}_{s}=\begin{bmatrix} \boldsymbol{\mathrm{\varphi}}_{sI}^{T} & \mathbf{I} \end{bmatrix}^{T}\]

(10.14)\[\mathbf{F}=\begin{bmatrix} 0^T & \mathbf{F}_{B}^{T} \end{bmatrix}^{T}\]

where, \(\mathbf{F}_{B}\) contains the reaction forces due to the \(B\) set constraint. From the first equation in (10.11), the following equation is derived

(10.15)\[\mathbf{K}_{II}\boldsymbol{\varphi}_{sI}+\mathbf{K}_{IB}=0\]

The \(\boldsymbol{\varphi}_{sI}\) can be obtained as

(10.16)\[\boldsymbol{\varphi}_{sI}=-\mathbf{K}_{II}^{-1}\mathbf{K}_{IB}\]

Substitution of Equation. (6) into Equation. (3) gives the static correction mode as

(10.17)\[\begin{split}\bar{\boldsymbol{\Psi}}_{s}=\begin{bmatrix} -\mathbf{K}_{II}^{-1}\mathbf{K}_{IB} \\ \mathbf{I} \end{bmatrix}\end{split}\]

10.1.7.2. Orthonormalization with respect mass matrix

Let \(\bar{\boldsymbol{\varphi}}_c\) be the correction mode vector and \(\boldsymbol{\varphi}_c\) be the orthonormalized correction mode vector with respect to \({\mathbf{M}_{ff}}\), then \({{\bar{\varphi }}_{c}}\) can be represented as

(10.18)\[\boldsymbol{\varphi}_c = \bar{\boldsymbol{\varphi}}_c + \sum\limits_{p=1}^{nmode}{\alpha}_p\boldsymbol{\varphi}_p\]

where \(\boldsymbol{\varphi}_p\) denotes the \(p\)-th vibration normal mode. Gram Schmidt orthogonalization procedure can be used to remove the vibration normal modes from the correction mode. The othonormalized correction mode must satisfy the following equation.

(10.19)\[\boldsymbol{\varphi}_c^T\mathbf{M}_{ff}\boldsymbol{\varphi}_p=\bar{\boldsymbol{\varphi}}_c^T\mathbf{M}_{ff}\boldsymbol{\varphi}_p+\alpha_p=0\]

Since \(\boldsymbol{\varphi}_c\) is orthonomal to \(\boldsymbol{\varphi}_p\), \(\alpha_p\) can be obtained as follows:

(10.20)\[\alpha_p = -\bar{\boldsymbol{\varphi}}_c^T\mathbf{M}_{ff}\boldsymbol{\varphi}_p\]

Substitution of the above equation into Eq. (10.18) gives the orthonormalized correction mode.