# Analytic Prediction of Isolated Forced Response Curves from Spectral Submanifolds

## TL;DR

This paper introduces a spectral submanifold-based analytical method to predict isolated forced responses in complex mechanical systems, surpassing numerical techniques especially in high-dimensional cases.

## Contribution

It develops an explicit analytic criterion for isolated forced responses using spectral submanifold theory, applicable to high-dimensional nonlinear vibration problems.

## Key findings

- Accurate predictions in high-dimensional systems
- Explicit criterion for isolated responses
- Refinable predictions without numerical solutions

## Abstract

We show how spectral submanifold theory can be used to provide analytic predictions for the response of periodically forced multi-degree-of-freedom mechanical systems. These predictions include an explicit criterion for the existence of isolated forced responses that will generally be missed by numerical continuation techniques. Our analytic predictions can be refined to arbitrary precision via an algorithm that does not require the numerical solutions of the mechanical system. We illustrate all these results on low- and high-dimensional nonlinear vibration problems. We find that our SSM-based forced-response predictions remain accurate in high-dimensional systems, in which numerical continuation of the periodic response is no longer feasible.