# Explicit Third-Order Model Reduction Formulas for General Nonlinear Mechanical Systems

## TL;DR

This paper derives explicit third-order reduced models for nonlinear mechanical systems using invariant manifold theories, enabling straightforward analysis of system characteristics without complex coordinate transformations.

## Contribution

It provides closed-form, cubic-order reduction formulas for general nonlinear systems based on LSM and SSM theories, simplifying model analysis and parameter extraction.

## Key findings

- Reduced models accurately predict backbone and forced-response curves.
- Formulas are explicit and depend only on physical and modal parameters.
- Validated on complex systems like a nonlinear Timoshenko beam.

## Abstract

For general nonlinear mechanical systems, we derive closed-form, reduced-order models up to cubic order based on rigorous invariant manifold results. For conservative systems, the reduction is based on Lyapunov Subcenter Manifold (LSM) theory, whereas for damped-forced systems, we use Spectral Submanifold (SSM) theory. To evaluate our explicit formulas for the reduced model, no coordinate changes are required beyond an initial linear one. The reduced-order models we derive are simple and depend only on physical and modal parameters, allowing us to extract fundamental characteristics, such as backbone curves and forced-response curves, of multi-degree-of-freedom mechanical systems. To numerically verify the accuracy of the reduced models, we test the reduction formulas on several mechanical systems, including a higher-dimensional nonlinear Timoshenko beam.