Divergence, Hopf and Double-Zero Bifurcations of a Nonlinear Planar Beam

Luongo, A.; Egidio, A. Di

doi:10.4203/ccp.79.78

Cited by 4 publications

(11 citation statements)

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“…The influence of driving frequency, spring rigidity and strut frequency on the system response was investigated in order to determine the regions of periodic solutions and routes to chaos, with reference to the behaviors of the load-bearing element (beam), of the non-structural element (truss) and of the overall system. A system of bifurcations is derived, and further analyses could be done on the basis of multiple scale analysis [44,46,47]. In more detail, for fixed spring rigidity (k imp = 10 −3 ) and strut frequency (ω a 1 = 2.68), and for variable forcing frequency (ω = 0.8 − 4.0), the bifurcation of a one-cycle periodic solution to a two-cycle periodic solution was evidenced in the beam response (ω = 3.35) by the impact velocity diagram, as well as nonlinear resonances for beam (ω = 1.25, 1.35, 2.5) and truss (ω = 1.25, 1.40, 2.78) by the displacement amplitude curves.…”

Section: Resultsmentioning

confidence: 99%

“…and from Eqs. (39) and (44). The non-dimensional horizontal mass density is derived from (58) 2 , (44) and (68)…”

Section: Numerical Simulationsmentioning

confidence: 99%

“…, and the non-dimensional axial frequencies from (53) 2 , (39), (44) and (68), The non-dimensional data are therefore subsumed in Table 3, and the number of both flexural and axial modes we take into account in the numerical simulations is 3. Finally, in this paper we use the following Rayleigh attenuation coefficients:α = 0 andβ = 0.01.…”

Section: Numerical Simulationsmentioning

confidence: 99%

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Soft-impact dynamics of deformable bodies

Andreaus

Chiaia

Placidi

2012

Continuum Mech. Thermodyn.

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Systems constituted by impacting beams and rods of non-negligible mass are often encountered in many applications of engineering practice. The impact between two rigid bodies is an intrinsically indeterminate problem due to the arbitrariness of the velocities after the instantaneous impact and implicates an infinite value of the contact force. The arbitrariness of after-impact velocities is solved by releasing the impenetrability condition as an internal constraint of the bodies and by allowing for elastic deformations at contact during an impact of finite duration. In this paper, the latter goal is achieved by interposing a concentrate spring between a beam and a rod at their contact point, simulating the deformability of impacting bodies at the interaction zones. A reliable and convenient method for determining impact forces is also presented. An example of engineering interest is carried out: a flexible beam that impacts on an axially deformable strut. The solution of motion under a harmonic excitation of the beam built-in base is found in terms of transverse and axial displacements of the beam and rod, respectively, by superimposition of a finite number of modal contributions. Numerical investigations are performed in order to examine the influence of the rigidity of the contact spring and of the ratio between the first natural frequencies of the beam and the rod, respectively, on the system response, namely impact velocity, maximum displacement, spring stretching and contact force. Impact velocity diagrams, nonlinear resonance curves and phase portraits are presented to determine regions of periodic motion with impacts and the appearance of chaotic solutions, and parameter ranges where the functionality of the non-structural element is at risk

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Section: Resultsmentioning

confidence: 99%

“…and from Eqs. (39) and (44). The non-dimensional horizontal mass density is derived from (58) 2 , (44) and (68)…”

Section: Numerical Simulationsmentioning

confidence: 99%

Section: Numerical Simulationsmentioning

confidence: 99%

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Soft-impact dynamics of deformable bodies

Andreaus

Chiaia

Placidi

2012

Continuum Mech. Thermodyn.

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“…The present formulation applies to a generic incremental step if the geometric stiffness matrix is disregarded. The general formulation for non linear Kirchhoff rods can be found in [2] (see also [43,44] for planar beams).…”

Section: The Kirchhoff Rod Modelmentioning

confidence: 99%

An isogeometric implicit G1 mixed finite element for Kirchhoff space rods

Greco

Cuomo

2016

Computer Methods in Applied Mechanics and Engineering

118

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configuration space is necessary for modelling the Kirchhoff rod (the centroid curve position vector and the torsional angle) [1,2]. In the literature this kind of manifold is also known as ribbon, see [3].Kirchhoff rod models have the advantage, with respect to shear deformable models, that shear locking is automatically avoided. Furthermore, it is used for modelling particular spatial structures, like cables with bending and torsional stiffness; for instance, in [4] the authors present a bending-stabilized cable model, while in [5,6] the authors consider the effect of torsional stiffness in aeroelastic analysis.In [1] a geometrically exact formulation of space rods based on a Lagrangian description was presented, that does not depend on the particular geometry of the centroid curve (differently from what has been done for instance in [7][8][9][10]). An orthogonal frame was introduced on the rod axis, that can be different from the natural frame (Bishop frame) used by Langer and Singer [11]. A pull-back of the strain along the directors was used to define the deformation of the cross section, analogously to what is done in non local and second gradient theory, i.e. [12][13][14][15][16][17].Since high continuity is required for the interpolation of the displacements in a Kirchhoff-Love rod model, the B-spline interpolation used in isogeometric analysis appears to be a natural choice for the development of numerical approximations of thin structural models. A first example of isogeometric interpolation for non polar rods can be found in [18] in which the authors have considered the polar formulation of rods developed in [19]. Many others numerical isogeometric formulations for rods have been proposed since (see, e.g., [20][21][22][23][24][25]).When multiple elements are used for discretizing the model, C 0 continuity for the end rotations of the element is needed, that for the Kirchhoff rod model means a G 1 continuity constraint on the deformation of the centroid curve, i.e., the unit tangent vectors have to coincide at the ends of adjacent elements; from an incremental point of view this means that the velocity of rotation at the ends of adjacent elements must be equal.In [26,27] G 1 continuity for space rods was obtained by means of a change of basis, generalizing Hermite's interpolation. The required geometric continuity was thus achieved without introducing Lagrangian or penalty terms in the formulation, as done for instance by the bending strip method [28]. An alternative strategy of a multi-patch approach for non polar shells can be found in [29].Although it has been claimed that sufficiently high degrees of interpolation avoid locking phenomena, isogeometric models, like all displacement based formulations, suffer from this pathology, that arises from the coupling terms appearing in the strain energy. Membrane and shear locking were observed in plane beam models [30][31][32][33] and membrane and flexural locking in space rod models [10,[34][35][36]. In [1,36] severe locking due to these interactions was fo...

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“…However, all the previous papers concern linear systems, while very few contributions deal with the effect of the paradox in the nonlinear regime, notwithstanding that a large number of researchers have been attracted by this topic. Some relevant contributions can be found, for example in [21–29], with reference to discrete [21–23] or continuous [24–29] systems, respectively. The post-critical scenario of general discrete systems, in the presence of linear damping, is studied in [21] by applying the normal form theory; there, an example concerning Ziegler’s column is discussed.…”

Section: Introductionmentioning

confidence: 99%

Nonlinear hysteretic damping effects on the post-critical behaviour of the visco-elastic Beck’s beam

Luongo

D’Annibale

2016

Mathematics and Mechanics of Solids

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The effects of nonlinear hysteretic damping on the post-critical behaviour of the visco-elastic Beck's beam are discussed in this paper. The model consists of an inextensible and shear-undeformable cantilever beam, internally and externally damped, loaded at the free end by a follower force. Equations are derived in finite kinematics by taking as the configuration variable the rotation θ of the cross-section, instead of the usual deflection v of the beam axis, thus making the description of the contact internal actions simpler. The linear stability analysis is carried out and results of the literature are re-obtained. Then, a multiple-scale approach is developed in order to evaluate the nature, super-or subcritical, and the amplitude of the limit-cycle, occurring close to the Hopf critical load. The effects of the nonlinear damping on the amplitude of the limit-cycle are finally discussed for different linear damping coefficients.

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Divergence, Hopf and Double-Zero Bifurcations of a Nonlinear Planar Beam

Cited by 4 publications

References 0 publications

Soft-impact dynamics of deformable bodies

Soft-impact dynamics of deformable bodies

An isogeometric implicit G1 mixed finite element for Kirchhoff space rods

Nonlinear hysteretic damping effects on the post-critical behaviour of the visco-elastic Beck’s beam

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