Nonlinear Vibration with Control by David Wagg & Simon Neild

Author:David Wagg & Simon Neild
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

4.6 Chapter Notes

The aim of this chapter was to provide an introduction to many of the approximate nonlinear analysis techniques. A case study is provided in Chap. 7, in which the dynamics of a single mode of a cable are considered, and the various techniques introduced here are compared. Harmonic balance has long been used as a technique for approximating the response of nonlinear systems. Discussions of the harmonic balance technique and how to apply it can be found, for example, in Worden and Tomlinson (2000), Nayfeh and Mook (1995), Cartmell (1980). The averaging technique is discussed in Verhulst (1996) and Tondl et al. (2000), including details of how to use an amplitude and phase representation rather than the sine and cosine representation that has been used here. In addition, Bakri et al. (2004) compare averaging to the harmonic balance technique for a specific system. Further analysis of cable dynamics using the averaging technique may be found in Gonzalez-Buelga et al. (2008). Perturbation techniques are discussed in detail in Verhulst (1996), Strogatz (2001) and Glendinning (1994). Both Strogatz (2001) and Glendinning (1994) provide interesting discussions on extending the multiple-scales method beyond just a slow and a fast scale. The normal form approach adopted here is a second-order variant in which the equations of motion are used directly. The first-order methods in which the first transform converts the equations of motion to state-space form is discussed in Jezequel and Lamarque (1991), Nayfeh (1993), and Wagg and Neild (2009). The second-order variant is presented in Neild and Wagg (2011) and discussed in more detail in Neild (2012). The case where the forcing is away from resonance is discussed for a two degree of freedom example in Neild and Wagg (2011) and the stability of steady-state solutions in Xin et al. (2013).

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