The influence of correction for shear and rotary inertia in the case of free oscillating physically-nonlinear beams is analysed in the paper. The solution obtained via energetic methods with the ...usage of "moment-curvature" diagram. Various types of materials and transversal shapes are studied. The influence of nonlinearity to the natural frequency is shown and analysed. Obtained solutions can be used in the calculations of building structures.
Nonlinear Deformable-body Dynamics offers a self-contained introduction to the field, and aims to stimulate research in this area. Examples guide readers through the topics and theories discussed, ...making them accessible to a broad, interdisciplinary audience.
In dispersive wave systems with dispersion relations such that the phase speed attains an extremum at a finite wavenumber, a rich variety of solitary waves that feature decaying oscillatory tails is ...known to arise. Here we use the fifth-order Korteweg-de Vries (KdV) equation, a model for small-amplitude gravity-capillary waves on water of finite depth when the Bond number is close to to examine the stability of the two symmetric solitary-wave solution branches that bifurcate at the minimum phase speed. In the vicinity of the bifurcation point, these solitary waves take the form of modulated wave packets with envelopes that can be approximated by the same soliton solution of the nonlinear Schrodinger (NLS) equation, suggesting that both branches would be stable in the small-amplitude limit. It is shown, however, that the branch of the so-called elevation waves is unstable while the branch of depression waves is stable, consistent with numerical results. The coupling between the carrier oscillations and their envelope, an effect beyond all orders of the expansion underlying the NLS equation, is essential to this behaviour: the dimensionless growth rates of the instability modes found for elevation waves are exponentially small with respect to the solitary-wave steepness. The asymptotic procedure followed here would be useful in discussing the stability of solitary waves with decaying oscillatory tails in other settings as well, and details are worked out for a nonlinear beam equation and a modified fifth-order KdV equation.
Topological results are applied to boundary value problems modeling nonlinear beams. A classical continuation theorem is used to prove existence results. Unified proofs,where possible, are given for ...all the physically relevant boundary conditions. A uniqueness result is given for the case of both ends simply supported.
The aim of this paper is to find a computationally efficient and predictive model for the class of systems that we call ‘pantographic structures’. The interest in these materials was increased by the ...possibilities opened by the diffusion of technology of three-dimensional printing. They can be regarded, once choosing a suitable length scale, as families of beams (also called fibres) interconnected to each other by pivots and undergoing large displacements and large deformations. There are, however, relatively few ‘ready-to-use’ results in the literature of nonlinear beam theory. In this paper, we consider a discrete spring model for extensible beams and propose a heuristic homogenization technique of the kind first used by Piola to formulate a continuum fully nonlinear beam model. The homogenized energy which we obtain has some peculiar and interesting features which we start to describe by solving numerically some exemplary deformation problems. Furthermore, we consider pantographic structures, find the corresponding homogenized second gradient deformation energies and study some planar problems. Numerical solutions for these two-dimensional problems are obtained via minimization of energy and are compared with some experimental measurements, in which elongation phenomena cannot be neglected.
We show that the dynamics of high-intensity laser pulses undergoing self-focused propagation in a nonlinear medium can be understood in terms of the topological constraints imposed by the formation ...and evolution of spatiotemporal optical vortices (STOVs). STOVs are born from pointlike phase defects on the sides of the pulse nucleated by spatiotemporal phase shear. These defects grow into closed loops of spatiotemporal vorticity that initially exclude the pulse propagation axis, but then reconnect to form a pair of toroidal vortex rings that wrap around it. STOVs constrain the intrapulse flow of electromagnetic energy, controlling the focusing-defocusing cycles and pulse splitting inherent to nonlinear pulse propagation. We illustrate this in two widely studied but very different regimes, relativistic self-focusing in plasma and nonrelativistic self-focusing in gas, demonstrating that STOVs mediate nonlinear propagation irrespective of the detailed physics.We show that the dynamics of high-intensity laser pulses undergoing self-focused propagation in a nonlinear medium can be understood in terms of the topological constraints imposed by the formation and evolution of spatiotemporal optical vortices (STOVs). STOVs are born from pointlike phase defects on the sides of the pulse nucleated by spatiotemporal phase shear. These defects grow into closed loops of spatiotemporal vorticity that initially exclude the pulse propagation axis, but then reconnect to form a pair of toroidal vortex rings that wrap around it. STOVs constrain the intrapulse flow of electromagnetic energy, controlling the focusing-defocusing cycles and pulse splitting inherent to nonlinear pulse propagation. We illustrate this in two widely studied but very different regimes, relativistic self-focusing in plasma and nonrelativistic self-focusing in gas, demonstrating that STOVs mediate nonlinear propagation irrespective of the detailed physics.
This paper addresses the evaluation of the time-domain response of nonlinear beams endowed with a fractional derivative element crossed by moving loads. Nonlinearities originate from the assumption ...of moderately large displacements of the beam. Following a Galerkin-type solution procedure, beam transversal displacement is represented in terms of the linear modes of vibration and time-dependent generalized coordinates. A novel step-by-step integration scheme, labeled improved pseudo-force method (IPFM), is developed for the numerical solution of the set of coupled nonlinear fractional differential equations governing the time-dependent generalized coordinates. The proposed procedure stems from the extension of a recently developed step-by-step scheme for the dynamic analysis of fractional single-degree-of-freedom systems. The IPFM involves the following main steps: i) to apply the Grünwald–Letnikov approximation of the fractional derivative; ii) to treat terms depending on the unknown values of the response as pseudo-forces; iii) to handle nonlinearities by performing iterations at each time step.
Numerical results are presented to assess the accuracy of the IPFM as well as to investigate the influence of the fractional derivative order and coefficient on nonlinear beam vibrations under moving loads.
•Nonlinear beams with fractional derivative elements under moving loads are analyzed.•A novel method for solving nonlinear fractional differential equations is proposed.•The Grünwald–Letnikov approximation of the fractional derivative is adopted.•Terms depending on the unknown values of the response are treated as pseudo-forces.•Nonlinear terms are handled by performing iterations.
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