Nonlinear dynamics of uniformly loaded Elastica: Experimental and numerical evidence of motion around curled stable equilibrium configurations

Abstract

It has been numerically observed and mathematically proven that for a clamped Euler's Elastica, which is uniformly loaded, there exist, in large deformations, some ‘undocumented’ equilibrium configurations which resemble a curled pending wire. Even if Elastica is one of the most studied model in mathematical physics, we could not find in the literature any description of an equilibrium like the one whose existence was forecast theoretically in [36]. In this paper, we prove that this kind of equilibrium configurations can be actually observed experimentally when using ‘soft’ beams. We mean with soft beams: Elasticae whose ratio between the applied load intensity and the bending stiffness is large enough. Moreover, we prove experimentally that such equilibrium configurations are actually stable, by observing their oscillations around the considered nonstandard equilibrium configuration. To describe theoretically such oscillations we consider, instead of a ‘soft’ Elastica model, directly a Hencky‐type discrete model, i.e. a ‘masses‐springs’ finite dimensional Lagrangian model. In this way we formulate, avoiding the use of an intermediate continuum model, a model for which numerical simulations can be performed without the introduction of any further discretization. In this way, we can also predict quantitatively the motions of soft beams, in the regime of large displacements and deformations. Postponing to future investigations more careful quantitative measurements, we report here that it was possible to get a rather promising qualitative agreement between observed motions and predictive numerical simulations.

Energy-based trajectory tracking and vibration control for multilink highly flexible manipulators

“…where b i = Y i j i /η i is the lumped bending stiffness of the introduced torsional springs, Y i is the Young modulus of constituting material, and j i is the second moment of area of the link's cross-section. We remark that this model is able to take into account also nonlinear elastic behaviors of the system; indeed, it has been shown ] that such a model, in the homogenized limit, converges to that of a nonlinear beam [Turco 2018;Pietraszkiewicz and Eremeyev 2009;Spagnuolo and Andreaus 2019] being shear undeformable, suitable for the description of problems involving large displacements and large deformations (see [Rosi et al 2018;Placidi et al 2017;Baroudi et al 2019] for more details on methods for obtaining material parameters), and whose deformation energy density depends only upon the exact curvature. Note that the linearized form of (9) is simply a quadratic form in the relative angle ϕ i j .…”

“…However, this reformulation of the problem involves a greater complexity of modeling. An alternative approach is based on the isogeometric formulation proposed by [Hughes et al 2005] and further developed by many other research groups (see, e.g., [Greco and Cuomo 2013;Balobanov et al 2016;Weeger et al 2013] and more recently [Greco et al 2017;Yildizdag et al 2018]). The key concept of the isogeometric analysis consists of using B-splines or NURBS curves both for representing the geometry of the system and as interpolating functions in the finite element method.…”

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“…where b i = Y i j i /η i is the lumped bending stiffness of the introduced torsional springs, Y i is the Young modulus of constituting material, and j i is the second moment of area of the link's cross-section. We remark that this model is able to take into account also nonlinear elastic behaviors of the system; indeed, it has been shown ] that such a model, in the homogenized limit, converges to that of a nonlinear beam [Turco 2018;Pietraszkiewicz and Eremeyev 2009;Spagnuolo and Andreaus 2019] being shear undeformable, suitable for the description of problems involving large displacements and large deformations (see [Rosi et al 2018;Placidi et al 2017;Baroudi et al 2019] for more details on methods for obtaining material parameters), and whose deformation energy density depends only upon the exact curvature. Note that the linearized form of (9) is simply a quadratic form in the relative angle ϕ i j .…”

Two-dimensional continua capable of large elastic extension in two independent directions: Asymptotic homogenization, numerical simulations and experimental evidence

“…is expected that such a mechanical behavior may be of great interest in various applications, like in berreinforced materials [33]. Future outlooks include the study of bi-pantographic structures by means of discrete and semi-discrete modelling [27,28,30] and the search for exotic solutions [29] enabled by strong nonlinearities.…”