# When does a Periodic Response Exist in a Periodically Forced Multi-Degree-of-Freedom Mechanical System?

## TL;DR

This paper develops rigorous mathematical criteria to guarantee the existence of periodic responses in periodically forced nonlinear mechanical systems, supporting the validity of numerical detection methods without amplitude restrictions.

## Contribution

It introduces new existence criteria for periodic responses in multi-degree-of-freedom systems, applicable regardless of forcing or response amplitude.

## Key findings

- Existence criteria are essential for confirming periodic responses.
- Conditions are demonstrated to be necessary through examples.
- Results support the use of numerical methods like harmonic balance.

## Abstract

While periodic responses of periodically forced dissipative nonlinear mechanical systems are commonly observed in experiments and numerics, their existence can rarely be concluded in rigorous mathematical terms. This lack of a priori existence criteria for mechanical systems hinders definitive conclusions about periodic orbits from approximate numerical methods, such as harmonic balance. In this work, we establish results guaranteeing the existence of a periodic response without restricting the amplitude of the forcing or the response. Our results provide a priori justification for the use of numerical methods for the detection of periodic responses. We illustrate on examples that each condition of the existence criterion we discuss is essential.