Generalized Steady State (GSS)

This repository collects MATLAB examples demonstrating Generalized Steady State (GSS) prediction for nonlinear mechanical systems under aperiodic (earthquake, chirp, noise, quasi-periodic, chaotic) forcing. Each example computes a unique GSS compares it against a full-order finite-element modal time integration, and in some cases against Galerkin-ROM and data-driven autoencoder+LSTM models [1].

Reference

R. S. Kaundinya, I. Thiel, B. Kaszás, S. Jain, G. Haller, "Predicting Generalized Steady States in Aperiodically Forced Mechanical Systems," Journal of Sound and Vibration (2026). https://www.sciencedirect.com/science/article/pii/S0022460X26003871

Installation

Run the install.m file in the main folder to add this package and its external dependencies (see External packages below) to the MATLAB path.

Each example folder follows the same layout: the main, runnable scripts sit at the top level, and the helper functions they call live in a private/ subfolder.

To reproduce the results/figures for a given example, make sure MATLAB's Current Folder is set to that example's own directory, then run the file named *_GSS.m (the prefix varies by example, as listed below).

.
├── vonKarmanBeam_GSS/
├── AxialMovingBeam_GSS/
├── OscillatorChain_GSS/
├── vonKarmanShell_GSS/
└── Partial data-driven GSS/
    ├── Oscillator Chain/
    └── vonKarman beam/

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Equation-based GSS examples

vonKarmanBeam_GSS/

A cantilevered von Karman beam (finite element model) under recorded earthquake ground motion forcing.

• vonKarmanBeam_AnchorTool.m — computes the GSS (piecewise-exact and Newmark solves) and compares it against a full-order Newmark time integration.
• VkBeamCluster.m — the same GSS-vs-full-order comparison, run as a mesh-refinement sweep in parallel on a cluster.
• rw_calculation_vonKarmanBeam_AnchorTool.m — computes the forcing weakness ratio r_w for this example.

AxialMovingBeam_GSS/

An axially moving beam with gyroscopic and nonlinear (viscoelastic) damping forces, exhibiting 1:3 internal resonance between its first two bending modes, subject to chirp base excitation.

• AxialMovingBeam_chirp_GSS.m — computes the GSS under the chirp forcing, compares it to a full-order time integration, and plots the beam's spatial deflection w(x,t) as a movie.
• rw_calculation_AxialMovingBeam_chirp.m — computes r_w for the chirp forcing example.

OscillatorChain_GSS/

A chain of coupled oscillators with cubic and quadratic nonlinearities, under three forcing scenarios.

• OcillatorChain_largerforcing.m and Oscillator_chain_LTSM_compare_GSS.m — filtered-noise/recorded forcing; compare the GSS (at increasing truncation orders) against a full-order time integration and an autoencoder+LSTM model [1] trained on the same data.
• Osc_chain_periodic.m — periodic forcing; sweeps the forcing frequency and compares the maximum response amplitude across the full-order model, the exact linear solution, and the GSS under both periodic and aperiodic solve modes.
• rw_calculation_OscillatorChain.m — computes r_w for the aperiodic forcing example.

vonKarmanShell_GSS/

A von Karman shell-based shallow curved panel, under two forcing scenarios.

• Plate_QP_GSS.m / Plate_QP_GSS_Galerkin.m — quasi-periodic wind pressure fluctuations, the _Galerkin file adds a Galerkin-projected ROM comparison (on a truncated modal basis) alongside the comparisons with the full order model.
• Plate_chaotic_GSS.m / Plate_chaotic_GSS_Galerkin.m — chaotic (Rössler) wind pressure fluctuation.
• rw_calculation_QP.m / rw_calculation_chaotic.m — r_w for the quasi-periodic and chaotic cases respectively.

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Partial data-driven GSS examples

These two examples combine a data-driven SSM identification step (learned from simulated or recorded trajectories via SSMLearn/fastSSM) with a GSS correction term that accounts for the external forcing. These files include comparisons with autoencoder+LSTM models from [1].

Partial data-driven GSS/Oscillator Chain/

• Oscillator_Chain_SSM_GSS.m — learns a 2D data-driven SSM from free trajectories of a stochastically-forced oscillator chain, adds a GSS correction to account for the forcing, and compares the resulting prediction against a full-order time integration and an autoencoder+LSTM model.

Partial data-driven GSS/vonKarman beam copy/

• vonKarman_beam_SSM_GSS.m — the same data-driven SSM + GSS computation now applied to a cantilevered von Karman beam. Constructs a vector Padé approximant of the GSS to extend its convergence domain, and compares the result against a full-order run and autoencoder+LSTM predictions (6D and 11D latent-dimension models).
• rw_calculation_vonKarman_beam.m — computes r_w for this example.

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Notes

• Every main script assumes the SSMTool-based toolbox classes (DynamicalSystem, SSM, ImplicitNewmark, etc.) are on the MATLAB path via run ../../install.m, relative to each example's own folder.
• A few scripts depend on external data files (recorded forcing, autoencoder+LSTM weights/predictions) present within in each folder.

External packages

This package uses the following external open-source packages:

1. Sandia tensor toolbox: https://gitlab.com/tensors/tensor_toolbox
2. Combinator: https://www.mathworks.com/matlabcentral/fileexchange/24325-combinator-combinations-and-permutations
3. YetAnotherFECode: Zenodo, http://doi.org/10.5281/zenodo.4011281

References

[1] Simpson, T., Dervilis, N., & Chatzi, E. (2021). Machine learning approach to model order reduction of nonlinear systems via autoencoder and LSTM networks. Journal of Engineering Mechanics, 147(10), Article 04021061. https://doi.org/10.1061/(ASCE)EM.1943-7889.0001971

Contact

Please report any issues/bugs to Roshan S. Kaundinya (roshan.kaundinya@mavt.ethz.ch)