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# Transient dynamics, damping, and mode coupling of nonlinear systems with internal resonances

## Abstract

The study of both linear and nonlinear structural vibrations routinely circles the concise yet complex problem of choosing a set of coordinates which yield simple equations of motion. In both experimental and mathematical methods, that choice is a difficult one because of measurement, computational, and interpretation difficulties. Often times, researchers choose to solve their problems in terms of linear, undamped mode shapes because they are easy to obtain; however, this is known to give rise to complicated phenomena such as mode coupling and internal resonance. This work considers the nature of mode coupling and internal resonance in systems containing non-proportional damping, linear detuning, and cubic nonlinearities through the method of multiple scales as well as instantaneous measures of effective damping. The energy decay observed in the structural modes is well approximated by the slow-flow equations in terms of the modal amplitudes, and it is shown how mode coupling enhances the damping observed in the system. Moreover, in the presence of a 3:1 internal resonance between two modes, the nonlinearities not only enhance the dissipation, but can allow for the exchange and transfer of energy between the resonant modes. However, this exchange depends on the resonant phase between the modes andmore »

- Authors:

- Sandia National Lab. (SNL-CA), Livermore, CA (United States)

- Publication Date:

- Research Org.:
- Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)

- Sponsoring Org.:
- USDOE National Nuclear Security Administration (NNSA)

- OSTI Identifier:
- 1559516

- Report Number(s):
- SAND2019-9484J

Journal ID: ISSN 0924-090X; 678450

- Grant/Contract Number:
- AC04-94AL85000

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Nonlinear Dynamics

- Additional Journal Information:
- Journal Name: Nonlinear Dynamics; Journal ID: ISSN 0924-090X

- Publisher:
- Springer

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 42 ENGINEERING

### Citation Formats

```
Mathis, Allen T., and Quinn, D. Dane. Transient dynamics, damping, and mode coupling of nonlinear systems with internal resonances. United States: N. p., 2019.
Web. doi:10.1007/s11071-019-05198-w.
```

```
Mathis, Allen T., & Quinn, D. Dane. Transient dynamics, damping, and mode coupling of nonlinear systems with internal resonances. United States. doi:10.1007/s11071-019-05198-w.
```

```
Mathis, Allen T., and Quinn, D. Dane. Fri .
"Transient dynamics, damping, and mode coupling of nonlinear systems with internal resonances". United States. doi:10.1007/s11071-019-05198-w.
```

```
@article{osti_1559516,
```

title = {Transient dynamics, damping, and mode coupling of nonlinear systems with internal resonances},

author = {Mathis, Allen T. and Quinn, D. Dane},

abstractNote = {The study of both linear and nonlinear structural vibrations routinely circles the concise yet complex problem of choosing a set of coordinates which yield simple equations of motion. In both experimental and mathematical methods, that choice is a difficult one because of measurement, computational, and interpretation difficulties. Often times, researchers choose to solve their problems in terms of linear, undamped mode shapes because they are easy to obtain; however, this is known to give rise to complicated phenomena such as mode coupling and internal resonance. This work considers the nature of mode coupling and internal resonance in systems containing non-proportional damping, linear detuning, and cubic nonlinearities through the method of multiple scales as well as instantaneous measures of effective damping. The energy decay observed in the structural modes is well approximated by the slow-flow equations in terms of the modal amplitudes, and it is shown how mode coupling enhances the damping observed in the system. Moreover, in the presence of a 3:1 internal resonance between two modes, the nonlinearities not only enhance the dissipation, but can allow for the exchange and transfer of energy between the resonant modes. However, this exchange depends on the resonant phase between the modes and is proportional to the energy in the lowest mode. The results of the analysis tie together interpretations used by both experimentalists and theoreticians to study such systems and provide a more concrete way to interpret these phenomena.},

doi = {10.1007/s11071-019-05198-w},

journal = {Nonlinear Dynamics},

number = ,

volume = ,

place = {United States},

year = {2019},

month = {8}

}

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