Comparing Analytical Approximation Methods with Numerical Results for Nonlinear Systems
Modelling the dynamics of nonlinear systems poses a much more challenging problem than for their linear counterparts; as such, analytical solutions are rarely achievable and numerical or analytical approximations are often necessary to understand the system’s behaviour. While numerical techniques are undoubtedly accurate, it is possible to gain a greater understanding of the processes underpinning the workings of the dynamics. Therefore, it is valuable to investigate the accuracy and practicality of the aforementioned analytical approximation techniques and compare the results with numerical which are known to be accurate. In this paper, the unforced, undamped dynamics (known as backbone curves) of a non-symmetric two-mass oscillator will be calculated using the second-order normals forms (SONF), harmonic balance, and multiple scales techniques. The results of these will then be compared to responses found using numerical continuation. Furthermore, the forced responses will be approximated using the SONF and harmonic balance techniques. In addition, recent work has reported the possibility of using such analytical expressions for parameter estimation from experimental data.
KeywordsBackbone curves Second-order normal forms Harmonic balance Multiple scales Modal analysis
- 1.Neild, S.A., Champneys, A.R.,Wagg, D.J., Hill, T.L., Cammarano, A.: The use of normal forms for analysing nonlinear mechanical vibrations. Phil. Trans. R. Soc. A 373 (2051), 20140404 (2015). http://rsta.royalsocietypublishing.org/content/373/2051/20140404 MathSciNetCrossRefMATHGoogle Scholar
- 7.Hill, T.L., Cammarano, A., Neild, S.A., Wagg, D.J.: Relating backbone curves to the forced responses of nonlinear systems. In: Kerschen, G. (ed.) Proceedings of the 33rd IMAC, A Conference and Exposition on Structural Dynamics, vol. 1, pp. 113–122 (2015)Google Scholar
- 8.Doedel, E.J., Champneys, A.R., Fairgrieve, T.F., Kuznetsov, Y.A., Dercole, F., Oldeman, B.E., Paffenroth, R.C., Sandstede, B., Wang, X.J., Zhang, C.: Auto-07p: Continuation and Bifurcation Software for Ordinary Differential Equations. Concordia University, Montreal (2008)Google Scholar