Modal Truncation is a method used for Model Order Reduction. It is used for complex and large numerical models, which require a substantial computational effort and large amounts of memory. Model order reduction methods reduce the size of the numerical models. The reduced models approximate the full system's response characteristics with less computational effort and memory requirements. Good reduced models are able to replace the original model.

For Modal Truncation, the reduction is done using generalized coordinates. The models are transformed from physical space to modal space and the target space is truncated.

Modal Truncation is based on modal superposition: Vibrations of a dynamic system can be expressed using a linear combination of its mode shapes. These mode shapes are acquired using modal analysis. The system is transformed into modal coordinates \mathbf{q}, i.e. coordinates in a subspace spanned by the eigenmodes of the system.

Theory

The transformation to modal space is performed using a transformation matrix \mathbf{V}=\left[ \pmb{\phi}_1,\pmb{\phi}_2,\dots,\pmb{\phi}_r \right] \in\mathbb{R}^{n\times r}, where \pmb{\phi} are the mode shapes of the system. This leads to the reduced system

\begin{align} s^2\mathbf{M}\mathbf{{x}}\left(s\right) + s\mathbf{Cx}(s)+\mathbf{Kx}\left(s\right)=\mathbf{f}\left(s\right)\\ \mathbf{M}\mathbf{V}\mathbf{{q}}\left(s\right) + \mathbf{C}\mathbf{V}\mathbf{{q}}\left(s\right) + \mathbf{KVq}\left(s\right)=\mathbf{f}\left(s\right)\\ s^2\mathbf{V}^H\mathbf{MVq}(s) + s\mathbf{V}^H\mathbf{CVq}(s) + \mathbf{V}^H\mathbf{KVq}(s) = \mathbf{V}^H\mathbf{f}(s) \end{align}

If r=n (i.e. all mode shapes are considered), the system's size is not reduced, but yields the same result as the original system. This strategy is called modal superposition.

If some modes \phi are truncated, the reduced system's size shrinks, but the solution gets less accurate. However, if the significant modes are selected for the reduction basis, the result is sufficiently accurate.

Further reading: (Humar, 2021)

Mode Participation

The choice which mode shapes should be included in the reduction basis \mathbf{V}=\left[ \pmb{\phi}_1,\pmb{\phi}_2,\dots,\pmb{\phi}_r \right] \in\mathbb{R}^{n\times r} depends on the loads acting on the system. The modal participation factor \gamma_i represents the effect of the force on a certain mode shape \pmb{\phi}_i:

\begin{equation} \gamma_{i}=\pmb{\phi}_i^H\mathbf{f} A_i(\Omega), \end{equation}

with the amplification function

\begin{equation} A_i\left(\Omega\right) = \frac{\hat{k}_i}{\sqrt{\left( \hat{k}_i - \hat{m}_i \Omega^2\right)^2 + \left(\hat{c}_i\Omega\right)^2 }} \end{equation}

and the generalized mass, damping, and stiffness \hat{m}_i=\pmb{\phi}_i^H\mathbf{M}\pmb{\phi}_i, \hat{c}_i=\pmb{\phi}_i^H\mathbf{C}\pmb{\phi}_i, and \hat{k}_i=\pmb{\phi}_i^H\mathbf{K}\pmb{\phi}_i.

The modal participation is depending on the driving frequency \Omega . A participation factor of \gamma=0 means that this mode is not excited at all by the given load case.

Examples

Examples and exercises how to apply Modal Truncation can be found here:

References

Humar, J. (2012). Dynamics of structures. CRC press.