Towards a Technique for Nonlinear Modal Reduction
In this paper we discuss an analytical method to enable modal reduction of weakly nonlinear systems with multiple degrees-of-freedom. This is achieved through the analysis of backbone curves—the response of the Hamiltonian equivalent of a system—which can help identify internal resonance within systems. An example system, with two interacting modes, is introduced and the method of second-order normal forms is used to describe its backbone curves with simple, analytical expressions. These expressions allow us to highlight which particular interactions are significant, as well as specify the conditions under which they are important. The descriptions of the backbone curves are validated against the results of continuation analysis, and a comparison is also made with the response of the system under various levels of forcing and damping. Finally, we discuss how this technique may be expanded to systems with a greater number of modes.
KeywordsBackbone curves Second-order normal forms Modal analysis Modal interaction Modal reduction
- 1.Ewins DJ (2000) Modal testing: theory, practice, and application. Research Studies Press, Philadelphia, PAGoogle Scholar
- 4.Kerschen G, Peete rs M, Golinval JC, Vakakis AF (2009) Nonlinear normal modes, Part I: A useful framework for the structural dynamicist. Mech Syst Signal Process 23:170–194Google Scholar
- 6.Doedel EJ, with major contributions from Champneys AR, Fairgrieve TF, Kuznetsov YuA, Dercole F, Oldeman BE, Paffenroth RC, Sandstede B, Wang XJ, Zhang C (2008) AUTO-07P: Continuation and bifurcation software for ordinary differential equations. Concordia University, Montreal. http://cmvl.cs.concordia.ca
- 7.Murdock J (2002) Normal forms and unfoldings for local dynamical systems. Springer, New YorkGoogle Scholar
- 9.Neild SA (2012) Approximate methods for analysing nonlinear structures. Springer, ViennaGoogle Scholar