# Iterative techniques for analyzing nonlinear vibrating dynamical systems in the frequency domain

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## Abstract

The nonlinear response of prototypical structures experiencing harmonic excitation is studied using novel techniques called iterative harmonic analysis (IHA) and iterative modal analysis (IMA). First, a simple damped oscillator with a cubic hardening stiffness nonlinearity is studied, and IHA is used in this single-degree-of-freedom system. In this first section, a high-order harmonic balance is applied, and IHA is employed in order to find the amplitude coefficients for different harmonics and their codependence. Additionally, a set of nested sums are identified that describe the harmonic coupling explicitly. Secondly, a pinned–pinned nonlinear beam of rectangular cross-section is studied, and IMA is applied to find the amplitude coefficients for different modes and their codependence. The nonlinearity is introduced through the membrane effect, where axial strain due to transverse deflection becomes a significant contributor to the system behavior. Typical frequency-domain methods cannot be easily applied to these systems as the solutions of the differential equations lead to intricate coupling between coefficients of the solution, and no analytical expression exists for those coefficients. Based upon these examples, other nonlinear systems may also be considered in future work using either a modal-based or finite element model. Finally, the advantages of the new method (reduced computational cost) as well as the limitations (effectively the same as those of an *n*th-order harmonic balance) are emphasized.

## Keywords

Nonlinear vibrations Structures Frequency domain Modal analysis Iterative methods Harmonic balance## References

- 1.Amer, Y.A., El-Sayed, A.T., Kotb, A.A.: Nonlinear vibration of the Duffing oscillator to parametric excitation with time delay feedback. Nonlinear Dyn.
**85**, 2497–2505 (2016)MathSciNetCrossRefGoogle Scholar - 2.Culver, D.R., Dowell, E.H.: High frequency response of a plate carrying a concentrated nonlinear spring-mass system. J. Sound Vib. (2016). https://doi.org/10.1016/j.jsv.2016.05.048i Google Scholar
- 3.Domokos, G., Holmes, P., Royce, B.: Constrained euler buckling. J. Nonlinear Sci.
**7**, 281–314 (1997)MathSciNetCrossRefMATHGoogle Scholar - 4.Dowell, E.H., Tang, D.: High frequency response of a plate carrying a concentrated spring–mass system. J. Sound Vib.
**213**, 843–863 (1998)CrossRefGoogle Scholar - 5.Dowell, E.H., Tang, D.: Dynamics of Very High Dimensional Systems. World Scientific, Singapore (2003)CrossRefGoogle Scholar
- 6.Eisley, J.G.: Nonlinear vibration of beams and rectangular plates. J. Appl. Mech.
**15**, 167–175 (1964)MathSciNetMATHGoogle Scholar - 7.Herbert, R.E.: Random vibrations of a nonlinear elastic beam. J. Acoust. Soc. Am.
**36**, 2090–2094 (1964)CrossRefGoogle Scholar - 8.Iyengar, N.G.R., Murthy, P.N.: Non-linear free vibration of a simply-supported beam by programming techniques. J. Sound Vib.
**20**, 277–286 (1972)CrossRefMATHGoogle Scholar - 9.Lewandowski, R., Wielentejczyk, P.: Nonlinear vibration of viscoelastic beams described using fractional order derivatives. J. Sound Vib.
**399**, 228–243 (2017)CrossRefGoogle Scholar - 10.Nayfeh, Ali, Mook, Dean: Nonlinear Oscillations. Wiley, Hoboken (1995)CrossRefMATHGoogle Scholar
- 11.Saito, H., Mori, K.: Vibrations of a beam with non-linear elastic constraints. J. Sound Vib.
**66**, 1–8 (1979)CrossRefMATHGoogle Scholar - 12.Takahashi, K.: A method of stability analysis for non-linear vibration of beams. J. Sound Vib.
**67**, 43–54 (1979)CrossRefMATHGoogle Scholar - 13.Takahashi, K.: Non-linear free vibrations of inextensible beams. J. Sound Vib.
**64**, 31–34 (1979)CrossRefGoogle Scholar - 14.Weeger, O., Wever, U., Simeon, B.: Isogeometric analysis of nonlinear Euler–Bernoulli beam vibrations. Nonlinear Dyn.
**72**, 813–835 (2013)MathSciNetCrossRefMATHGoogle Scholar - 15.Weeger, O., Wever, U., Simeon, B.: Nonlinear frequency response analysis of structural vibrations. Comput. Mech.
**54**, 1477–1495 (2014)Google Scholar - 16.Yoshimura, T., Hino, J., Ananthanarayana, N.: Vibration analysis of a non-linear beam subjected to moving loads by using the Galerkin method. J. Sound Vib.
**104**, 179–186 (1986)CrossRefGoogle Scholar