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# Experimental Random Excitation of Nonlinear Systems with Multiple Internal Resonances

• A. A. Afaneh
• R. A. Ibrahim
Conference paper
Part of the Solid Mechanics and its Applications book series (SMIA, volume 47)

## Abstract

Small nonlinear modal interactions may have considerable effect on the pattern of forced vibration response of certain types of dynamic systems possessing low damping characteristics. The design of these systems may create two or more linear algebraic relationships between the system’s normal mode frequencies. These relationships are referred to as internal (or autoparametric) resonances, and usually take the form £ $$\sum {\rm K}_{\rm j} {\rm \omega}_{\rm j} = 0$$. The constant $${\rm K} = \sum {\Bbb K}_{\rm j} {\rm l}$$ is referred to as the order of internal resonance. where kj are integers. and ωj are the system normal mode frequencies. Third-order internal resonance results from quadratic nonlinear coupling of normal modes. while fourth-order is due to cubic nonlinearities. Cubic nonlinearities can also cause second-order internal resonance (i.e., one-to-one). Simultaneous internal resonances are classified as independent and interacting. Independent internal resonances usually take the form
$$\sum\limits_{\rm j}^{\rm m} {{\rm K}_{\rm j} {\rm \omega }_{\rm j} } = 0,\,\,\,\,\,\,\,\sum\limits_{\ell}^{\rm n} {{\rm K}_{\ell} {\rm \omega }_{\ell} } = 0$$
(1)
such that j=1,2,.., m. and l=m+1. m+2,..., n

## Keywords

Primary Beam Internal Resonance Excitation Level Couple Beam Secondary Beam
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## References

1. 1.
Ibrahim, R. A.: Multiple internal resonance in a structure-liquid system, ASME J. Engineering for Industry 98, 1092-1098, 1976.
2. 2.
Bux, S. L. and Roberts, J. W.: Nonlinear vibrators interactions in systems of coupled beams, J. Sound and Vibration 104(3), 497-520, 1986
3. Cartmell, M. P. and Roberts, J. W.: Simultaneous combination resonances in and autoparametrically resonant system, J. Sound and Vibration 123, 1988, 81–101, 1988.
4. Afaneh, A. A. and Ibrahim, R. A.: Random excitation of coupled oscillators with single and simultaneous internal resonances, submitted for publication.Google Scholar

## Copyright information

© Kluwer Academic Publishers 1996

## Authors and Affiliations

• A. A. Afaneh
• 1
• R. A. Ibrahim
• 1
1. 1.Department of Mechanical EngineeringWayne State UniversityDetroitUSA