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Dynamic Analysis Using Reduced-Order Models
Introduction
High-fidelity finite-element models (FEM) are invaluable for insight during the design and control of a machine or complex mechatronic system. However, they are often too large and computationally intensive to support rapid design iterations, control simulation, or real-time use in a digital twin. We describe a method to transfer a FEM model, developed in a standard FEM software environment such as Ansys, to a reduced-order state-space model for fast and easy analysis in Matlab. This turns large FEM models into compact, control-ready models that preserve input–output behavior and make it straightforward to assess the effects of force inputs, motion profiles, imposed displacements or accelerations, and floor vibrations on complex systems.
Theory: from FEM to State-Space Model
The forced response of a damped FEM model with N degrees of freedom can be written as a system of N coupled second-order differential equations:
[M][\ddot{x}(t)]+[C][\dot{x}(t)]+[K][x(t)]=[F(t)]
with M the mass matrix, C the damping matrix, K the stiffness matrix, x(t) the displacement vector and F(t) the external force vector.
Assuming the system is proportionally damped (i.e. a damping ratio \xi_i is applied to each mode), a modal transformation can be applied to decouple the previous equations. This results in N independent second-order equations in modal space:
\ddot{r_i}(t)+2\xi_i\omega_i\dot{r}(t)+\omega_i^2r_i(t)=f_i(t)
with \xi_i the damping ratio, \omega_i the natural frequency of the i-th mode, r_i(t) the modal coordinate (i.e. displacement in modal space) and f_i(t) the modal force (i.e. the projection of the external force onto the i-th mode shape). The transformation between physical and modal coordinates is given by:
\begin{cases}
[x(t)] = [\phi][r(t)] \\
[F(t)] = [\phi][f(t)] \\
\end{cases}
with \phi the modal matrix of which the columns are the mass normalized mode shapes. The natural frequencies \omega_i and \phi can be exported from most FEM software environments.
From the above N decoupled modal equations a state-space model can be constructed, with A the state matrix, B the input matrix, C the output matrix, D the feedthrough matrix, z(t) the state vector and u(t) the input vector:
[\dot{z}(t)]=[A][z(t)]+[B][u(t)]
\begin{bmatrix}
\dot{r_1} \\
\ddot{r_1} \\
\vdots \\
\dot{r_N} \\
\ddot{r_N} \\
\end{bmatrix}
=
\begin{bmatrix}
0 & 1 & \cdots & 0 & 0 \\
-\omega_1^2 &-2\xi_1\omega_1 & \cdots & 0 & 0 \\
\vdots & \vdots & \ddots & \vdots & \vdots \\
0 & 0 & \cdots & 0 & 1 \\
0 & 0 & \cdots & -\omega_N^2 &-2\xi_N\omega_N \\
\end{bmatrix}
\begin{bmatrix}
r_1 \\
\dot{r_1} \\
\vdots \\
r_N \\
\dot{r_N} \\
\end{bmatrix}
+
\begin{bmatrix}
0 \\
f_1 \\
\vdots \\
0 \\
f_N \\
\end{bmatrix}
[x(t)]=[C][z(t)]+[D][u(t)]
\begin{bmatrix}
x_1 \\
\vdots \\
x_N \\
\end{bmatrix}
=
\begin{bmatrix}
\phi_{1,1} & 0 & \cdots & \phi_{N,1} & 0 \\
\vdots & \vdots & \ddots & \vdots & \vdots \\
\phi_{1,N} & 0 & \cdots & \phi_{N,N} & 0 \\
\end{bmatrix}
\begin{bmatrix}
r_1 \\
\dot{r_1} \\
\vdots \\
r_N \\
\dot{r_N} \\
\end{bmatrix}
+
[0]
In this model, [B][u(t)] is generated by computing [f(t)]=[\phi]^\intercal[F(t)] and then inserting zeros between each row. The D matrix is zero since there is no direct feedthrough from the input to the output.
Reduced Order Modelling
To reduce the computational overhead of the above described state-space model, it is often beneficial to simplify them into a reduced-order state-space model. This is typically done in two steps. The first reduction takes place in FEM by mode selection, where only the first N modes are extracted rather than the full set, based on the frequency range of interest. The second reduction is performed during the state-space construction in Matlab by for example a balanced truncation, where the modes of interest are strategically selected. As a result, the reduced-order state-space model retains sufficient system dynamics for the intended use case, while improving computational efficiency, as demonstrated in the example below.
Example
An aluminum cantilever L-shaped bracket with a central hole is used as a simplified use case. An input impulse force is applied at the free end of the bracket and the displacement of a node located above the loading point is monitored as the output.
The bracket is modeled in Ansys and a modal analysis is performed to determine its natural frequencies and corresponding mode shapes. The animation illustrates the first mode shape, which lies at 469 Hz.
From this analysis, fifty mode shapes and their corresponding eigenfrequencies were extracted and used to construct a reduced-order state-space model in Matlab. The first figure below shows the total system response (sum of all extracted modes) alongside the individual modal contributions. From this figure one can see that not all modes contribute equally: if selection is based only on natural frequency, modes 2 and 3 near 1 kHz might be kept despite their limited effect, while modes 1 and 4 have much greater influence. A better strategy is to rank and select modes by an importance measure (e.g. balanced truncation) rather than frequency alone. This identifies the dominant modes and yields a reduced set that still reproduces the overall response. Based on this reduced selection of modes a reduced state-space model can be constructed. This is shown in the second figure where the responses of three state-space models is compared, i.e. one including all modes, one retaining the three most significant modes, and one retaining only the first eigenmode. It can be seen that the three most significant modes already give a good representation of the full response up to 10 kHz.
Next, a Dirac impulse force is applied to the input node for each of the three state-space models. Since an impulse excites all frequencies, all modes are activated, and the differences between the models should become apparent. It is important to note, however, that a true Dirac impulse (infinite peak force, instantaneous time) is not physically possible.
The figures below show the total impulse response along with a zoomed-in view of the first 1.5 milliseconds.
These results indicate that the overall response is, as expected, primarily governed by the first mode, which explains why all responses appear nearly identical. For all three models, the time it takes for the vibrations to reduce to an amplitude of 10 µm is the same: 0.0798 seconds. This again indicates that the overall response is governed by the first mode. However, a closer examination of the zoomed-in portion reveals that the three-mode model contains higher-frequency content compared to the single-mode model. This is expected, since it retains modes up to 10 kHz, as shown in the previously presented frequency response. Based on these results, it can be concluded that the appropriate number of modes to include depends on the intended use case. For certain applications, it may be sufficient to include only the first mode, for example to extract information about the dominant dynamic behavior of the structure. In contrast, applications requiring higher accuracy or processes that are sensitive to higher-frequency effects necessitate a higher order model.
Another important criterion for selecting the number of modes is computation time. For real-time applications, such as digital twins or model predictive control, the state-space model must be computational small. The figure below shows the normalized computation time of a step response as a function of the number of modes, which increases approximately quadratic with the number of modes. Therefore, an appropriate trade-off must be made between the desired level of computational efficiency and accuracy. If the same analysis were performed directly within the FEM environment, the computation time would likely be significantly higher. By extracting the modes from the FEM model and reducing them to a smaller, yet representative subset, the model becomes computationally more efficient while retaining sufficient accuracy, thereby making it more practically usable.
