Nonlinear Beam Research Articles

Nonlinear vibration and parametric excitation of magneto‐thermo elastic embedded nanobeam using homotopy perturbation technique

AbstractThe present study focuses on the modeling and analyzing the nonlinear vibration patterns and parametric excitation of embedded Euler–Bernoulli nanobeams subjected to thermo‐magneto‐mechanical loads. The Euler–Bernoulli nanobeam is developed with external parametric excitation. The governing equation of motion is derived by utilizing nonlocal continuum theory and nonlinear von Karman beam theory. Subsequently, the homotopy perturbation technique is employed to determine the vibration frequencies. Finally, the modulation equation of Euler–Bernoulli nanobeams is derived for simply supported boundary condition. In order to validate our findings, we conduct a comparative analysis against existing literature, thereby underscoring the effectiveness and robustness of our proposed methodology. The influence of stress, magnetic potential, temperature, damping coefficient, Winkler coefficient, and nonlocal parameters are tested numerically on nonlinear frequency‐amplitude and parametric excitation–amplitude responses. The numerical examples indicate the significant impact of physical variables on the nonlinear frequency and parametric excitation. The primary objective of this study is to investigate the effects of external physical variables on the dynamic behavior of nanobeams, particularly in nonlinear regimes. This study provides insights into the design and control of nanostructures under complex load conditions, contributing to developing advanced materials and nanosystems.

An optimal penalty method for the joint stiffening in beam models of additively manufactured lattice structures

Additive manufacturing is revolutionizing structural design, with lattice structures becoming increasingly prominent due to their superior mechanical properties. However, simulating these structures quickly and accurately using the finite element method (FEM) remains challenging. Recent research has highlighted beam element simulation within FEM as a more efficient alternative to traditional solid FE simulations, achieving similar accuracy with reduced computational resources. However, a significant challenge is managing the lack of rigidity at nodes and the prevalence of low aspect ratio beams. While various methodologies have been proposed to address these issues, there is still a gap in the comprehensive evaluation of their limitations. An optimal node penalization methodology is required to expand the limited range of accurately represented lattice behavior. A preliminary study investigates lattice geometries through comparative analysis of solid and beam FE simulations. Built on this, we developed a methodology suitable to linear, dynamics and nonlinear beam FE simulations, contributing to enhanced computational speed and accuracy. Several lattice structures were printed using material jetting and quasi-static compressive tests were conducted to validate the methodology’s accuracy. The numerical results reveal a good accuracy between the proposed beam FE methodology and the experimental data, offering a better alternative to conventional FEM for energy absorption in terms of computing time.

Strong–strong simulations of combined beam–beam and wakefield effects in the Electron–Ion-Collider

Collective wakefield and beam–beam effects play an important role in accelerator design and operation. These effects can cause beam instability, emittance growth, and luminosity degradation, and warrant careful study during accelerator design. In this paper, we studied the combined wakefield and beam–beam effects in an Electron Ion Collider design using strong–strong simulations. The simulation results show that the nonlinear beam–beam effects help suppress wakefield driven instability in the nominal working tune regime. In other tune regimes, the coherent beam–beam modes interact with the wakefields and cause a beam instability. The simulation results also show the importance of maintaining nominal crab cavity voltage. If the crab cavity voltage drops significantly the beam can become unstable.

Dose increment in ultrashort particle irradiations via coherent stopping power

We present a relativistic self-consistent theory of the coherent stopping power (CSP) of ultradense charged particle beams propagating in a dense matter. CSP corresponds to a collective inelastic collision which adds to the ordinary stopping power of individual particles. Unlike the latter, which depends only on the particles' energy, CSP depends upon many more parameters such as the total charge of the ensemble and its charge density and shape. This paves the way for a broad variety of novel methods to tailor particle absorption and penetration in a dense matter. CSP losses can be explained both by collective excitations of single or multiple molecules and by the emission of coherent Cherenkov radiation. Here we demonstrate that the former mechanism is more relevant than the latter. We find that the coherent energy absorption of subpicosecond particle bunches in water occurs exciting a broad Debye process in the GHz range and an intermolecular stretching vibration mode in the THz region. We generalize the Bethe-Bloch stopping power formula to coherent effects including self-forces, dielectric screening, and absorption by the dense matter. For the sake of a self-consistent dynamical theory including phase space evolution, we have also generalized the Fermi-Eyges theory of particle diffusion to the presence of forces. Nonlinear dynamics is demonstrated, inducing nonlinear dose release, beam self-focusing, and self-enhancement of coherent losses. Given the advent of ultraintense particle sources and their use for biomedical and other relevant applications, our results may be of paramount importance for contemporary and future developments of science and technology. Published by the American Physical Society 2024

Discrete element method simulation of high-speed vehicle collisions with road barrier systems

AbstractThe behavior of road or perimeter protection barriers under vehicle impact are usually investigated based on crash tests and finite element (FE) numerical approaches, which are ether expensive or time-consuming. Several studies have proposed to reduce the computation time of the numerical analysis by substituting the complex FE models of vehicles using simplified mass–spring–damper system models. However, these models have drawbacks since consideration of different vehicle impact angles is difficult and they are unable to correctly simulate the risk of high-speed vehicle collision running over the barrier. In this paper, a new approach is proposed to simulate the collision of vehicles with barriers based on the discrete element method (DEM). Here, to save computation time only a handful of 3D non-spherical particles are used to represent the barrier and vehicle. These particles are generated based on the super-quadric function, which is capable of generating a variety of shapes needed for the model. The contact detection and evaluation are carried out based on discrete function representation of the particles with uniform sampling. The bond between two discrete elements is defined using a nonlinear cohesive beam model since the distance between the elements is relatively large. The simulation results obtained based on this approach are more accurate and complete than the simplified mass–spring models and computationally more efficient than the FE model.

A hyperelastic beam model for the photo-induced response of nematic liquid crystal elastomers

Liquid crystal elastomers (LCEs) are a novel class of materials created by combining polymeric solids with stiff, rod-like molecules known as nematic mesogens. These materials exhibit large, reversible deformations under mechanical, thermal, and optical stimuli. In this work, we develop a nonlinear beam formulation for analyzing the finite elastic deformation of beam-like structures made of LCEs under photo-actuation. This formulation applies to ideal, non-ideal, isotropic genesis, and nematic genesis LCEs. We establish the variational form of the problem based on the principle of virtual work. To solve numerical examples, we also develop a nonlinear finite element formulation based on B-spline functions. Several numerical examples are presented to demonstrate the applicability of the proposed formulation.

Nonlinear parametric generation and optical vortex transfer in graphene ensemble under Landau quantization

We explore the dynamics of nonlinear parametric generation and light beam propagation in a Landau-quantized graphene structure with three energy levels interacting with two laser pulses, utilizing the Maxwell-Bloch equations. By applying a laser field to one transition of the graphene sample while keeping the second beam initially absent, the distinctive preparation of the graphene sample, coupled with its weak interaction with laser radiation, results in the parametric generation of a new laser beam in a different transition. We investigate the influence of diverse system parameters on both the efficiency of the generated beam and the propagation dynamics of both beams. Our findings reveal that manipulating these parameters can induce oscillations in the intensity of propagated beams, mitigate absorption losses during propagation allowing for earlier relaxation, and enhance the efficiency of energy transfer from the initial to the generated beam. Additionally, we demonstrate the transfer of optical vortices within the graphene ensemble by introducing an optical vortex to the initial beam. This scheme holds promise for applications in high-dimensional quantum information processing.

The Impact of the Nonlinear Integral Positive Position Feedback (NIPPF) Controller on the Forced and Self-Excited Nonlinear Beam Flutter Phenomenon

This article presents a novel approach to impact regulation of nonlinear vibrational responses in a beam flutter system subjected to harmonic excitation. This study introduces the use of a Nonlinear Integral Positive Position Feedback (NIPPF) controller for this purpose. This technique models the system as a three-degree-of-freedom nonlinear system representing the beam flutter, coupled with a first-order and a second-order filter representing the NIPPF controller. By applying perturbation analysis to the linearized system model, the authors obtain analytical solutions for the autonomous system with the controller. This study aims to reduce vibration amplitudes in a nonlinear dynamic system, specifically when 1:1 internal resonance occurs. The Routh–Hurwitz criterion is utilized to evaluate the system’s stability. Furthermore, the frequency–response curves (FRCs) exhibit symmetry across a range of parameter values. The findings highlight that the effectiveness of vibration suppression is directly related to the product of the NIPPF control signal after comparing with different controllers. Numerical simulations, conducted using the fourth-order Runge–Kutta method, validate the analytical solutions and demonstrate the system’s amplitude response. The strong correlation between the analytical and numerical results highlights the accuracy and dependability of the proposed method.

Prediction of sectional collapse of thin-walled structure under pure bending by nonlinear composite beam theory

Brazier (1927) found that when one dimension of the beam cross-section was relatively smaller than the others, large in-plane displacements over the cross-section might occur, even though the strains could remain very small. Under this circumstance, the so-called Brazier effect refers to the cross-sectional ovalization, which leads to nonlinear bending buckling and collapses. This paper studies the Brazier effect by the nonlinear Variational Asymptotic Beam Sectional Analysis (VABS) theory which considers finite cross-sectional deformations. Nonlinear VABS reduces three-dimensional (3D) continuum to a one-dimensional (1D) beam analysis and a two-dimensional (2D) cross-sectional analysis featuring both geometric and material nonlinearities without unnecessary kinematic assumptions. The present theory is implemented using the finite element method (FEM) in the VABS code, a general-purpose beam cross-sectional analysis tool. An iterative method is applied to solve the finite warping field for the classical-type model using the Euler–Bernoulli beam theory. The deformation gradient tensor is directly used to deal with finite deformation, various strain definitions, and several types of material laws. Numerical examples demonstrate the capabilities of VABS to predict the sectional collapse of thin-walled structures under pure bending.

Exponential time propagators for elastodynamics

We propose a computationally efficient and systematically convergent approach for elastodynamics simulations. We recast the second-order dynamical equation of elastodynamics into an equivalent first-order system of coupled equations, so as to express the solution in the form of a Magnus expansion. With any spatial discretization, it entails computing the exponential of a matrix acting upon a vector. We employ an adaptive Krylov subspace approach to inexpensively and accurately evaluate the action of the exponential matrix on a vector. In particular, we use an apriori error estimate to predict the optimal Krylov subspace size required for each time-step size. We show that the Magnus expansion truncated after its first term provides quadratic and superquadratic convergence in the time-step for nonlinear and linear elastodynamics, respectively. We demonstrate the accuracy and efficiency of the proposed method for one linear (linear cantilever beam) and three nonlinear (nonlinear cantilever beam, soft tissue elastomer, and hyperelastic rubber) benchmark systems. For a desired accuracy in energy, displacement, and velocity, our method allows for 10−100× larger time-steps than conventional time-marching schemes such as Newmark-β method. Computationally, it translates to a ∼1000× and ∼10−100× speed-up over conventional time-marching schemes for linear and nonlinear elastodynamics, respectively.

Asymptotic analysis of geometrically nonlinear beam vibrations: Kirchhoff and Bolotin equations

AbstractThe paper analyzes various approximate models of geometrically nonlinear vibrations of a beam. In practice, simplified equations are often based on the quasi‐static Kirchhoff hypothesis—neglecting axial inertia. This hypothesis is justified with the prescribed end‐displacements of the beam in the axial direction. Under dead loading, quasi‐static Kirchhoff hypothesis results in a linear equation. The corresponding approximate equations obtained in this paper are based on the asymptotic procedure. The ratio of bending stiffness to reduced tensile/compressive stiffness is taken as a small parameter. Axial inertia is taken into account in the equation of the first approximation. Introduced by V.V. Bolotin concept “nonlinear inertia” is discussed. The most common errors in using the quasi‐static Kirchhoff hypothesis are analyzed.

Non-Linear Elastic Beam Deformations with Four-Parameter Timoshenko Beam Element Considering Through-the-Thickness Stretch Parameter and Reduced Integration

Non-Linear Elastic Beam Deformations with Four-Parameter Timoshenko Beam Element Considering Through-the-Thickness Stretch Parameter and Reduced Integration

The influence of piezoelectric on the nonlinear stochastic vibration of BN nanoresonator

The outstanding physical properties of hexagonal boron nitride (h-BN) make it highly valuable for use in nanoelectromechanical systems. We investigate the nonlinear stochastic vibration of h-BN nanowire affected by piezoelectric. The nonlinear beam model considering the impact of piezoelectric effect excited by random force is established. Molecular dynamic simulations were utilized to determine the potential energy of h-BN nanowires under varying amplitudes affected by an external electric field. The findings suggest that an increase in the intensity of the electric field can result in buckling behavior, leading to the appearance of two stable points. The cases of pre-buckling and post-buckling of nonlinear dynamic behavior of h-BN nanowire induced by piezoelectric effect is discussed in this paper. Furthermore, the impact of the intensity of random force on the nonlinear stochastic vibration characteristics of h-BN nanowire is also examined.

Vibration isolation of nonlinear honeycomb sandwich beams

Vibration isolation of nonlinear honeycomb sandwich beams

Experimental bifurcation analysis of a clamped beam with designed mechanical nonlinearity

We use control-based continuation (CBC) to perform an experimental bifurcation study of a periodically forced dual-beam. The nonlinearity is of geometric nature, provided by a thin, clamped beam. The overall system exhibits hysteresis and bistability in its open-loop frequency response due to a hardening, Duffing-like nonlinear stiffness, which can be designed or adjusted by choosing the properties of the thin beam. We employ local stabilising feedback control to implement CBC and track stable periodic solutions past the fold points. Thus obtained continuous solution branches are used to generate the solution surface over the plane of excitation amplitude and frequency. This surface features two curves of fold bifurcations that meet at a cusp point, and they delimit the experimentally observed bistability range of this nonlinear beam.

Size-dependent nonlinear bending of microbeams based on a third-order shear deformation theory

In this paper, the size-dependent nonlinear bending of microbeams subjected to mechanical loading is studied using a finite element formulation. Based on the von Kármán nonlinear relationship and the third-order shear deformation theory, a size-dependent nonlinear beam element is derived by using the modified couple stress theory (MCST) to capture the microstructural size effect. The element with explicit expressions for the element vector of internal forces and tangent stiffness matrix is derived by employing the transverse shear rotation as a variable. Nonlinear bending of microbeams under different mechanical loading is predicted with the aid of Newton–Raphson iterative method. Numerical investigation shows that the derived element is efficient, and it is capable of giving accurate results by several elements. The obtained results reveal the importance of the micro-size effect on the nonlinear behavior of the microbeams, and the deflections are overestimated when the microstructural effect is ignored. The effects of the material length scale parameter, boundary conditions and loading type on the bending response of the microbeams are studied and highlighted.

On the maximization of entropy in the process of thermalization of highly multimode nonlinear beams.

We present a direct experimental confirmation of the maximization of entropy which accompanies the thermalization of a highly multimode light beam, upon its nonlinear propagation in standard graded-index (GRIN) optical fibers.

Design of hyperbolic quasi-zero stiffness metastructures coupled with nonlinear stiffness for low-frequency vibration isolation

Quasi-zero stiffness (QZS) metastructures with simple structures exhibit a high low-frequency vibration isolation performance, but the bearing capacity and working regions are limited due to the linear positive stiffness elements. In this study, a novel hyperbolic QZS metastructure coupled with nonlinear stiffness is designed by combining a pair of cosine-shaped beams and arc-shaped beams for low-frequency vibration isolation. Finite element models (FEM) and experimental analysis are utilized to examine the static characteristics of the metastructure, revealing a wide QZS region and significant load-bearing capacity. The metastructure can achieve effective vibration isolation properties through reasonably designing the unit cells. The influence of geometrical parameters on the QZS region is investigated to optimize and realize desirable QZS features. The dynamic equation and response of the vibration isolation system are determined using the harmonic balance method. Moreover, the corresponding displacement transmissibility is confirmed by numerical simulation. The results of the dynamic sweep frequency experiment further verify the validity of the theoretical analysis model. The hyperbolic QZS metastructure system possesses good vibration isolation performance due to the introduction and compensation of a nonlinear positive stiffness arc beam element. The proposed hyperbolic QZS metastructure provides a significant route for the stiffness compensation of metastructures.

Nurbs-based Timoshenko formulation of a geometrically nonlinear planar beam

Nurbs-based Timoshenko formulation of a geometrically nonlinear planar beam

Nonlinear dynamics and forced vibrations of simply-supported fractional viscoelastic microbeams using a fractional differential quadrature method

In this paper, free and forced nonlinear vibrations of fractional viscoelastic microbeams are modeled based on Euler-Bernoulli theory, the Von Karman’s nonlinear strain relations, the modified couple stress theory (MCST), and the fractional Kelvin-Voigt viscoelastic model. In the present work, the nonlinear-fractional order governing equations are discretized in the space domain by Galerkin’s method. Two different approaches are introduced to solve the resulting nonlinear fractional-order Duffing equation in the time domain. In the first approach, a time marching fractional finite difference method is presented to compute the transient time response starting from the initial conditions. This approach is time consuming if the frequency-amplitude curves of steady state response are required since it provides just one point on the frequency-amplitude curve. More importantly, this approach usually converges only for the stable branch with smallest amplitude of frequency-amplitude curve. In the second approach, we introduce a novel fractional differential quadrature method (FDQM) to discretize the fractional Duffing equation and apply a pseudo-arc length algorithm to directly construct the frequency-amplitude curves. Effects of the fractional-order, linear and nonlinear viscoelasticity coefficients, viscous damping parameter, microstructure parameters, and the micro-beam thickness on the nonlinear dynamics of the viscoelastic micro-beam are analyzed numerically. Numerical results show that each of these parameters can change natural frequency and/or the damping behavior of the structure. The present model can be used for designing and analyzing the nonlinear microstructure viscoelastic beam under dynamic loads.