Computational formulation for periodic vibration of geometrically nonlinear structures—part 1: Theoretical background

Lewandowski, Roman

doi:10.1016/s0020-7683(96)00127-8

Cited by 69 publications

(52 citation statements)

References 16 publications

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“…Lau and Cheung proposed the incremental harmonic balance (IHB) method [7]. Chen et al combined the IHB and finite element methods to analyze nonlinear vibrations of plane structures [8] whereas Lewandowski presented a general formulation for computing steady-state vibrations of geometrically nonlinear structures by using a harmonic balance and finite element method [9,10,11]. The classical IHB method is suitable for structural vibration with polynomial nonlinearity, whereas for structural vibration with a more complex nonlinearity, the alternating frequency/time (AFT) domain method proposed by Cameron [12] is widely used.…”

Section: Introductionmentioning

confidence: 99%

Optimization of nonlinear structural resonance using the incremental harmonic balance method

Dou

Jensen

2015

Journal of Sound and Vibration

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We present an optimization procedure for tailoring the nonlinear structural resonant response with time-harmonic loads. A nonlinear finite element method is used for modeling beam structures with a geometric nonlinearity and the incremental harmonic balance method is applied for accurate nonlinear vibration analysis. An optimization procedure based on a gradient-based algorithm is developed and we use the adjoint method for efficient computation of design sensitivities. We consider several examples in which we find optimized beam width distributions that minimize or maximize fundamental or super-harmonic resonant responses.

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Section: Introductionmentioning

confidence: 99%

Optimization of nonlinear structural resonance using the incremental harmonic balance method

Dou

Jensen

2015

Journal of Sound and Vibration

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“…The computation of large oscillations of homogeneous structures by coupling a nonlinear finite element analysis with a direct time integration procedure [27,28] or a harmonic balance method [29,30] is relatively well known today. The finite element modeling of nonlinear vibrations of piezoelectric structures is less common but has been treated in [31] or [32].…”

Section: Introductionmentioning

confidence: 99%

Finite element reduced order models for nonlinear vibrations of piezoelectric layered beams with applications to NEMS

Lazarus

Thomas

Deü

2012

Finite Elements in Analysis and Design

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a b s t r a c tThis article presents a finite element reduced order model for the nonlinear vibrations of piezoelectric layered beams with application to NEMS. In this model, the geometrical nonlinearities are taken into account through a von Ká rmá n nonlinear strain-displacement relationship. The originality of the finite element electromechanical formulation is that the system electrical state is fully described by only a couple of variables per piezoelectric patches, namely the electric charge contained in the electrodes and the voltage between the electrodes. Due to the geometrical nonlinearity, the piezoelectric actuation introduces an original parametric excitation term in the equilibrium equation. The reduced-order formulation of the discretized problem is obtained by expanding the mechanical displacement unknown vector onto the short-circuit eigenmode basis. A particular attention is paid to the computation of the unknown nonlinear stiffness coefficients of the reduced-order model. Due to the particular form of the von Ká rmá n nonlinearities, these coefficients are computed exactly, once for a given geometry, by prescribing relevant nodal displacements in nonlinear static solutions settings. Finally, the low-order model is computed with an original purely harmonic-based continuation method. Our numerical tool is then validated by computing the nonlinear vibrations of a mechanically excited homogeneous beam supported at both ends referenced in the literature. The more difficult case of the nonlinear oscillations of a layered nanobridge piezoelectrically actuated is also studied. Interesting vibratory phenomena such as parametric amplification or patch length dependence of the frequency output response are highlighted in order to help in the design of these nanodevices.

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“…Substitution of time-forms for the entries in (13) and carrying out the integration with respect to time before the space discretisation [17] (14) The components in the last four rows of (14) still require expressing of generalized force amplitudes in terms of generalized displacement amplitudes. To this end the harmonic balance method is used.…”

Section: -3mentioning

confidence: 99%

Influence of elastic supports on non-linear steady-state vibrations of Zener material plates

Litewka¹,

Lewandowski²

2018

AIP Conference Proceedings

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Abstract. The paper reports numerical results of analyses of steady-state harmonic vibrations of von Kármán non-linear plates made from Zener material with various elastic support conditions. Influences of shear deformation and rotary inertia are taken into account, thus the model is able to predict the behaviour of plates with a moderate thickness. The amplitude equation for the plate is obtained using the time-averaged principle of virtual work for the assumed harmonic form of excitation and plate displacements as well as the harmonic balance method for Zener material and non-linear elastic supports. Plates are discretised using 8-noded rectangular plate finite elements. The discretised amplitude equation is solved for the response curves using a path-following method. Results of two numerical examples are presented and the qualitative and quantitative influence of support elastic properties is discussed.

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Computational formulation for periodic vibration of geometrically nonlinear structures—part 1: Theoretical background

Cited by 69 publications

References 16 publications

Optimization of nonlinear structural resonance using the incremental harmonic balance method

Optimization of nonlinear structural resonance using the incremental harmonic balance method

Finite element reduced order models for nonlinear vibrations of piezoelectric layered beams with applications to NEMS

Influence of elastic supports on non-linear steady-state vibrations of Zener material plates

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