Model reduction for discrete and elastic structures with inertial quadratic non-linearities

Abstract

This work discusses model reduction for non-linear structural systems under harmonic excitations. Model reduction can be achieved by different techniques, one of the recent techniques being the non-linear normal modes (NNMs) of the system. The dimension reduction achievable depends on the possibilities of internal resonances. A master–slave separation of degrees of freedom is used, and a non-linear relation between the slave and master coordinates is constructed based on the method of multiple timescales. More specifically, three cases involving external resonance of a mode without any internal resonance, and subharmonic as well as superharmonic resonances for systems with 1:2 internal resonances are considered. Reduced-order models based on the ‘Conservative NNMs’ as well as ‘Damped NNMs’ are constructed. The steady-state periodic responses of the reduced models determined by the method of multiple timescales are compared to exact solutions of the system models computed by the bifurcation analysis and parameter continuation software AUTO. The analysis is specifically applied to a spring-mass-pendulum system with external excitation, and to an elastic three-beam-tip-mass structure, which is first reduced to a high-fidelity non-linear discretized model through a Galerkin approximation. Both systems exhibit essential quadratic non-linearities and couplings between the various generalized coordinates. The NNMs of the two systems are used to perform model reductions when excited by harmonic excitations. It is seen that for systems with essential inertial quadratic non-linearities, the technique for model-order reduction through multiple timescales approximation based on NNMs over-predicts the softening non-linear response in each of the cases studied.

Keywords

spring-mass-pendulum system non-linear elastic structures inertial non-linearities internal resonances resonant response non-linear normal modes method of multiple time scales

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References

Nayfeh

A. H.

Younis

M. I.

Abdel-Rahman

E. M.

Reduced-order models for MEMS application. Nonlinear Dyn, 2005, 41, 211–236.

Kerschen

Golinval

J. -C.

Vakakis

A. F.

Bergman

L. A.

The method of proper orthogonal decomposition for dynamical characterization and order reduction of mechanical systems: An overview. Nonlinear Dyn, 2005, 41(1–3), 147–169.

Hung

E. S.

Yang

Y. -J.

Senturia

S. D.

Low-order models for fast dynamical simulation of MEMS microstructures. Proceedings of the International Conference on Solid-state Sensors and Actuators: Transducers, 1997

Chicago, IL

Vol. 2, 1101–1104.

Guyan

R. J.

Reduction of stiffness and mass matrices. AIAA Journal, 1965, 3, 380–380.

Feeny

B. F.

On the physical interpretation of proper orthogonal modes in vibrations. J. Sound Vib, 1998, 211(4), 607–616.

Georgiou

Advanced proper orthogonal decomposition tools: Using reduced order models to identify normal modes of vibration and slow invariant manifolds in the dynamics of planar nonlinear rods. Nonlinear Dyn, 2005, 41, 69–114.

Balachandran

Nayfeh

A. H.

Nonlinear motions of beam-mass structure. Nonlinear Dyn, 1990, 1, 39–61.

Meirovitch

Principles and techniques of vibrations, 1997 (McGraw Hill, New York).

Wang

Bajaj

Nonlinear normal modes in multi-mode models of an inertially coupled elastic structure. Nonlinear Dyn, 2007, 47, 25–47.

10.

Shaw

S. W.

Pierre

Normal modes for nonlinear vibratory systems. J. Sound Vib, 1993, 164, 85–124.

11.

Pesheck

Pierre

Shaw

Accurate reduced-order models for a simple rotor blade model using nonlinear normal modes. Math. Comput. Model, 2001, 33, 1085–1097.

12.

Pecheck

Pierre

Shaw

Modal reduction of a nonlinear rotating beam through nonlinear normal modes. ASME J. Vib. Acoust, 2002, 124, 229–235.

13.

Jezequel

Lamarque

C. H.

Analysis of non-linear dynamical systems by the normal form theory. J. Sound Vib, 1991, 149(3), 429–459.

14.

Shaw

S. W.

Pierre

Normal modes of vibration for nonlinear continuous systems. J. Sound Vib, 1994, 169(3), 319–347.

15.

Vakakis

A. F.

Manevich

L. I.

Mikhlin

Yu. V.

Pilipchuk

V. N.

Zevin

A. A.

Normal modes and localization in non-linear systems, 1996 (Wiley, New York).

16.

Nayfeh

A. H.

Reduced-order models of weakly nonlinear spatially continuous systems. Nonlinear Dyn, 1998, 16, 105–125.

17.

Touze

Thomas

Huberdeau

Asymptotic non-linear normal modes for large-amplitude vibrations of continuous structures. Comput. Struct, 2004, 82, 2671–2682.

18.

Touze

Thomas

Chaigne

Hardening/softening behavior in non-linear oscillations of structural systems using non-linear normal modes. J. Sound Vib, 2004, 273(1–2), 77–101.

19.

Lacarbonara

Rega

Nayfeh

A. H.

Resonant non-linear normal modes. Part I: Analytical treatment for structural one-dimensional systems. Int. J. Nonlinear Mech, 2003, 38, 851–872.

20.

Lacarbonara

Camillacci

Nonlinear normal modes of structural systems via asymptotic approach. Int. J. Solids Struct, 2004, 41, 5565–5594.

21.

Sinha

S. C.

Redhar

Deshmukh

Butcher

E. A.

Order reduction of parametrically excited nonlinear systems: Techniques and applications. Nonlinear Dyn, 2005, 41, 237–273.

22.

Touzè

Amabili

Nonlinear normal modes for damped geometrically nonlinear systems: Application to reduced-order modelling of harmonically forced structures. J. Sound Vib, 2006, 298, 958–981.

23.

Touze

Amabili

Thomas

Reduced-order models for large-amplitude vibrations of shells including in-plane inertia. Comput. Meth. Appl. Mech. Eng, 2008, 197, 2030–2045.

24.

Jiang

Pierre

Shaw

Nonlinear normal modes for vibratory systems under harmonic excitation. J. Sound Vib, 2005, 288, 791–812.

25.

Apiwattanalunggarn

Shaw

Pierre

Jiang

Finite-element-based nonlinear modal reduction of a rotating beam with large-amplitude motion. J. Vib. Control, 2003, 9, 235–263.

26.

Apiwattanalunggarn

Shaw

Pierre

Component mode synthesis using nonlinear normal modes. Nonlinear Dyn, 2005, 41, 17–46.

27.

Soares

M. E. S.

Mazzilli

C. E. N.

Nonlinear normal modes of planar frames discretized by the finite element method. Comput. Struct, 2000, 77, 485–493.

28.

Nayfeh

A. H.

Nayfeh

S. A.

On nonlinear modes of continuous systems. ASME J. Vib. Acoust, 1994, 116, 129–136.

29.

Wang

Bajaj

Kamiya

Nonlinear normal modes and their bifurcations for an inertially coupled nonlinear conservative system. Nonlinear Dyn, 2005, 42, 233–265.

30.

Nayfeh

A. H.

Resolving controversies in the application of the method of multiple scales and the generalized method of averaging. Nonlinear Dyn, 2005, 40, 61–102.

31.

Wang

Bajaj

A. K.

On the equivalence of normal form theory and multiple time scale method. ASME. J. Comput. Nonlinear Dyn, 2009, 4, 021005-1-11–021005-1-11.

32.

Doedel, E. AUTO: Software for continuation and bifurcation problems in ordinary differential equations. Report, Department of Applied Mathematics, California Institute of Technology, Pasadena, CA, 1896.

33.

Crespoda Silva

M. R.

M. A reduced-order analytical model for the nonlinear dynamics of a class of flexible multi-beam structures. Int. J. Solids Struct, 1998, 35, 3299–3315.

34.

Nayfeh

A. H.

Mook

D. T.

Nonlinear oscillations, 1979 (Wiley Interscience, New York).

35.

Wang, F. Nonlinear normal modes and nonlinear model reduction for discrete and multi-component continuous systems. PhD Thesis, Purdue University, West Lafayette, IN, 2008.

36.

Agnes

G. S.

Inman

D. J.

Performance of nonlinear vibration absorbers for multi-degrees-of-freedom systems using nonlinear normal modes. Nonlinear Dyn, 2001, 25, 275–292.