A perturbation method for evaluating nonlinear normal modes of a piecewise linear two-degrees-of-freedom system

Vestroni, Fabrizio; Luongo, Angelo; Paolone, Achille

doi:10.1007/s11071-008-9337-3

Cited by 53 publications

(29 citation statements)

References 32 publications

(42 reference statements)

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“…The nonlinear dynamics of elastic or suspended cables, shear indeformable beams, tethered satellite systems, orbiting strings, chains of sliding beams and pendulums has been dealt with in the papers [14][15][16][17][18][19][20][21][22][23][24][25][26][27]. Very often making use of perturbation methods on reduced systems of ordinary differential equations relevant to the different structures involved in the papers, original contributions can be summarized in the following points.…”

Section: Nonlinear Dynamicsmentioning

confidence: 99%

“…6. The detection of analytical expressions for nonlinear free-vibration frequencies and mode shapes in case of a breathing-cracked beam, modeled as a two DOF system, is dealt with in [26] proposing a modification of the classical Lindstedt-Poincaré method, able to tackle non-smooth piecewise linear systems. Differently for the classic approach, valid for smooth systems, the complementary solution of the passive coordinate equations must be taken into account in order to satisfy continuity and periodicity of motion; accordingly, both passive and active frequencies contribute to the motion although they are incommensurable.…”

Section: Nonlinear Dynamicsmentioning

confidence: 99%

See 1 more Smart Citation

On the contribution of Angelo Luongo to Mechanics: in honor of his 60th birthday

Piccardo

D’Annibale

Zulli

2014

Continuum Mech. Thermodyn.

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This paper intends to summarize the scientific production of Angelo Luongo on the occasion of his Sixtieth Birthday, focusing on his main contributions in the field of Mechanics. The task will not be easy because of the breadth of his scientific production, only apparently attributable to a restricted number of keywords. In fact, even when the work seems purely algorithmic, speculation on the physical and mechanical aspects of the problem is always present, providing new interpretations and innovative openings to a careful reader. Similarly, also the works, which apparently seem to be high-level applications, always reserve methodological aspects that are not negligible. The editorial choice to divide his papers through a small number of keywords is certainly simplistic, but offers the possibility to better highlight all the connections among his variegated production. The most original contributions of Angelo Luongo in the context of perturbation methods, linear and nonlinear dynamics and control, elastic buckling and structural analysis, bifurcation and stability of non-conservative systems, are discussed in detail. Finally, the Angelo Luongo's central role in the creation and development of activities of the international research center M&MoCS is pointed out.

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Section: Nonlinear Dynamicsmentioning

confidence: 99%

Section: Nonlinear Dynamicsmentioning

confidence: 99%

On the contribution of Angelo Luongo to Mechanics: in honor of his 60th birthday

Piccardo

D’Annibale

Zulli

2014

Continuum Mech. Thermodyn.

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“…In Ref. [18], the classical Lindstedt-Poincare method for evaluating nonlinear normal modes of a piecewise linear two-degreeof-freedom system is adapted to analyze the nonlinear normal modes of a simple piecewise linear twodegree-of-freedom system representing a beam with a breathing crack. In this study, numerical results obtained by a Poincare map approach show the existence of superabundant normal modes which arise in the unstable interval of the first mode and can be predicted.…”

Section: Introductionmentioning

confidence: 99%

Dynamic analysis of a simply supported beam resting on a nonlinear elastic foundation under compressive axial load using nonlinear normal modes techniques under three-to-one internal resonance condition

2012

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In this paper, the Nonlinear Normal Modes (NNMs) analysis for the case of three-to-one (3:1) internal resonance of a slender simply supported beam in presence of compressive axial load resting on a nonlinear elastic foundation is studied. Using the EulerBernoulli beam model, the governing nonlinear PDE of the beam's transverse vibration and also its associated boundary conditions are extracted. These nonlinear motion equation and boundary condition relations are solved simultaneously using four different approximate-analytical solution techniques, namely the method of Multiple Time Scales, the method of Normal Forms, the method of Shaw and Pierre, and the method of King and Vakakis. The obtained results at this stage using four different methods which are all in time-space domain are compared and it is concluded that all the methods result in a similar answer for the amplitude part of the transverse vibration. At the next step, the nonlinear normal modes are

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“…Much effort in science and engineering has focused on this kind of problems [3]; typical mechanical applications include: oscillators colliding with a deformable or rigid stop [4,5], beams with breathing cracks [6][7][8][9], stick-slip mechanical systems with friction [10][11][12].…”

Section: Introductionmentioning

confidence: 99%

“…NNM is any periodic motion of the undamped autonomous system in which all generalized coordinates vibrate without necessarily passing through the zeros simultaneously [9,13,14].…”

Section: Introductionmentioning

confidence: 99%

Experimental evidence of bifurcating nonlinear normal modes in piecewise linear systems

Giannini¹,

Casini²,

Vestroni³

2010

Nonlinear Dyn

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Abstract.A system with piecewise linear restoring forces, typical of damaged beams with a breathing crack, exhibits bifurcations characterized by the onset of superabundant modes in internal resonance with significantly different shape than that of modes on fundamental branch.A 2-DOF frame with piecewise linear stiffness is analyzed by means of an experimental investigation; the frame is forced by an harmonic base excitation and the operative modal shapes as well as the response amplitude are directly measured; the results are compared with numerical outcomes for different damping. This study shows that the shapes and the frequencies of certain Nonlinear Normal Modes (NNM) of the related autonomous system strongly affect the forced response both in the numerical and the experimental environment. Therefore, it is possible to match the NNM with the forced response of the system, leading to the prospective of identifying severity and position of the damage from experimental tests.

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A perturbation method for evaluating nonlinear normal modes of a piecewise linear two-degrees-of-freedom system

Cited by 53 publications

References 32 publications

On the contribution of Angelo Luongo to Mechanics: in honor of his 60th birthday

On the contribution of Angelo Luongo to Mechanics: in honor of his 60th birthday

Dynamic analysis of a simply supported beam resting on a nonlinear elastic foundation under compressive axial load using nonlinear normal modes techniques under three-to-one internal resonance condition

Experimental evidence of bifurcating nonlinear normal modes in piecewise linear systems

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