Nonlinear Beam Research Articles

Stabilization of nonlinear non-uniform piezoelectric beam with time-varying delay in distributed control input

The stabilization issue for a nonlinear piezoelectric beam system with variable coefficients and time-varying delay in the control input is considered. The system is constituted by two strongly coupled nonlinear hyperbolic equations, in one of which a kind of distributed control is actuated and time-varying delay exists in the control input. The feedback gains are also time-varying ones. We first apply Kato's variable norm technique to show the well-posedness of the system without nonlinear terms and based on which, the well-posedness of the whole nonlinear system is proved by the fixed-point theory. The exponential stability of the nonlinear time-delay coupled PDEs' system is further proved by the energy estimates together with multiplier techniques under certain conditions. The stability of its corresponding electrostatic/quasi-static model is also investigated. Finally, some numerical simulations are given to illuminate the results obtained in this work.

Study on the vibration suppression of beam structures with nonlinear piezoelectric shunt damping

The vibration suppression of the nonlinear beam structure under harmonic excitations by employing the nonlinear piezoelectric shunt damping circuit (PSDC) is researched. The energy approach is utilized to deduce the dynamic equations of the nonlinear beams with nonlinear PSDC. The complexification-averaging approach and pseudo arc-length continuation approach are applied to calculate the frequency domain response of the beam structure. Numerous numerical examples are conducted to verify the present formulations, thereby demonstrating the remarkable efficiency and accuracy. The impacts of several parameters of the proposed PSDC are discussed and the optimal study of the nonlinear PSDC is further performed to obtain the optimal negative and nonlinear capacitances. The comparisons concerning the vibration of the beam structures with and without PSDC are implemented to examine the vibration mitigation performance of the designed PSDC. In the end, the design criterion of the nonlinear PSDC is given.

Enhancing Electrical Generation Efficiency through Parametrical Excitation and Slapping Force in Nonlinear Elastic Beams for Vibration Energy Harvesting.

This study aims to enhance conventional vibration energy harvesting systems (VEHs) by repositioning the piezoelectric patch (PZT) in the middle of a fixed-fixed elastic steel sheet instead of the root, as is commonly the case. The system is subjected to an axial simple harmonic force at one end to induce transversal vibration and deformation. To further improve power conversion, a baffle is strategically installed at the point of maximum deflection, introducing a slapping force to augment electrical energy harvesting. Employing the theory of nonlinear beams, the equation of motion for this nonlinear elastic beam is derived, and the method of multiple scales (MOMS) is used to analyze the phenomenon of parametric excitation. This study demonstrates through experiments and theoretical analysis that the second mode yields better power generation benefits than the first mode. Additionally, the voltage generation benefits of the enhanced system with the added baffle (slapping force) surpass those of traditional VEH systems. Overall, the proposed model proves feasible and holds promising potential for efficient vibration energy harvesting applications in various industrial sectors.

Multiple solutions of a nonlinear elastic beam equation supported at two ends

In this paper, several new existence theorems on pairs of solutions are obtained for the fourth order boundary value problem ${u^{(4)}}(t) = h(t,u(t))$ subject to $u(0) = u(1) = u''(0) = u''(1) = 0$. Further, if $h(t,u) = \mu f(t,u)$, we prove that there exists $ \mu ^ * > 0$ such that there is no solution to the above boundary value problem for $\mu > {\mu ^ * }$, and that there are multiple solutions of the above boundary value problem for $0 < \mu < {\mu^ * }$.

Simplified nonlinear analysis of doubly corrugated cold‐formed steel arches

AbstractDoubly corrugated steel arches are made by pressing transverse corrugations on trapezoidal cold‐formed steel sheets. Several studies show that these transverse corrugations may have a significant detrimental effect on the stiffness and load‐bearing capacity of the original trapezoidal profile. This paper presents an experimental campaign aimed at investigating the effects of transverse corrugations on the behaviour of arches. The campaign includes reduced tests on flat sheets and sheets with a single corrugation, as well as full tests on arches with spans ranging from 10 to 20 m. Subsequently, a simplified design procedure based on nonlinear 2D beam finite element models is proposed. The models account for the transverse corrugation effects by using intermediate rotational springs and equivalent beam section areas, which are calibrated from the results of the reduced experiments. A brief sensitivity study of the finite element results with respect to different modelling factors is conducted, and the different effects of the corrugation assessed. Finally, the simplified finite element models are validated by comparing the experimental and numerical force‐displacement curves, which show reasonably good agreement. However, further research is needed to refine and extend the proposed procedure.

Quasi-periodic solutions for quintic completely resonant derivative beam equations on T2

In the present paper, we consider two dimensional completely resonant, derivative, quintic nonlinear beam equations with reversible structure. Because of this reversible system without external parameters or potentials, Birkhoff normal form reduction is necessary before applying Kolmogorov–Arnold–Moser (KAM) theorem. As application of KAM theorem, the existence of partially hyperbolic, small amplitude, quasi-periodic solutions of the reversible system is proved in this paper.

On the use of frictional dampers for flutter mitigation of a highly flexible wing

The prospect of aeroelastic instability, leading to large-amplitude limit cycle oscillations, is of particular concern for the highly flexible wings of solar high-altitude long-endurance (HALE) aircraft. Among the different strategies of vibration control, passive mitigation through friction dampers stands out as a promising solution in this context. While this concept has been proposed previously, related studies systematically use simple two-degrees-of-freedom airfoil models. In this paper, a more realistic structural model involving a three-dimensional nonlinear beam description of a full wing is considered. A friction damper is installed at a specific location on the wing span, and the response to aerodynamic loads is investigated numerically. Comparisons of critical speed and limit cycle amplitudes with and without the damper suggest improved vibration mitigation with respect to a linear viscous damper.

Effects of Laying Depth and Pipe Arc Length on the Mechanical Performance of Large-Diameter Cold-Water Pipes during Float-and-Sink Installation

The successful operation of a large-diameter cold water pipeline installation is crucial for harnessing the potential of ocean thermal energy conversion. However, there is a shortage of research focused on mechanical performance analysis during installation. This study establishes a pipeline response analysis model based on a nonlinear beam theory to elucidate the underlying mechanical behaviour. Employing the method of singular perturbation, the general solution for the exterior region of the pipeline, the solution at the boundary layer, and the valid solution across the entire domain are derived. A comparison with numerical solutions is conducted to validate the accuracy and effectiveness of the theoretical model. Based on the theoretical analysis, the influence of installation depth and pipeline curvature on the pipeline’s shape, tension, curvature, and stress is discussed. The results indicate that increasing the installation depth leads to intensified pipeline bending and significant deformation, reaching a maximum bending moment of 3.92 MN∙m at a distance of 50~100 m from the bottom of the pipeline. The results also show that, as the pipeline’s arc length increases from 0 to 100 m, the bending curvature, Von Mises stress, and bending stress exhibit a trend of initial growth followed by a decline, peaking at 7.45 MPa, and 6.83 Mpa, respectively, while the actual tension and axial tension decrease initially and then increase, reaching −0.17 MN and −0.17 MPa, respectively, at the maximum arc length. The findings of this study provide valuable insights for practical cold-water pipe installation and laying.

Spatial beam self-cleaning accompanied by self-similar propagation in few-mode graded-index fiber

We numerically investigated a nonlinear Kerr beam self-cleaning (KBSC) dynamics accompanied by self-similar propagation regimes, which leads to single-mode parabolic pulse reshaping and simultaneously high beam quality based on KBSC, for special distributions of initially excited modes in graded-index multimode fiber (GRIN-MMF). We coupled a Gaussian pulse at 1060 nm, with 100 fs duration, into GRIN-MMF supporting 10 modes that fall into four discrete mode groups. As a result, by using initial powers below the KBSC threshold reported in the literature, the output spatial beam evolves from a speckled pattern into a bell-shaped beam; hence, the generated parabolic pulse is mainly carried by the fundamental mode, which is boosted by the KBSC process. We also provide promising indications for KBSC on different higher-order modes.

Inverse design and additive manufacturing of shape-morphing structures based on functionally graded composites

Shape-morphing structures possess the ability to change their shapes from one state to another, and therefore, offer great potential for a broad range of applications. A typical paradigm of morphing is transforming from an initial two-dimensional (2D) flat configuration into a three-dimensional (3D) target structure. One popular fabrication method for these structures involves programming cuts in specific locations of a thin sheet material (i.e. kirigami), forming a desired 3D shape upon application of external mechanical load. By adopting the non-linear beam equation, an inverse design strategy has been proposed to determine the 2D cutting patterns required to achieve an axisymmetric 3D target shape. Specifically, tailoring the localised variation of bending stiffness is the key requirement. In this paper, a novel inverse design strategy is proposed by modifying the bending stiffness via introducing distributed modulus in functionally graded composites (FGCs). To fabricate the FGC-based shape-morphing structures, we use a multi-material 3D printer to print graded composites with voxel-like building blocks. The longitudinal modulus of each cross-sectional slice can be controlled through the rule of mixtures according to the micro-mechanics model, hence matching the required modulus distribution along the elastic strip. Following the proposed framework, a diverse range of structures is obtained with different Gaussian curvatures in both numerical simulations and experiments. A very good agreement is achieved between the measured shapes of morphed structures and the targets. In addition, the compressive rigidity and specific energy absorption during compression of FGC-based hemi-ellipsoidal morphing structures with various aspect ratios were also examined numerically and validated against experiments. By conducting systematical numerical simulations, we also demonstrate the multifunctionality of the modulus-graded shape-morphing composites. For example, they are capable of blending the distinct advantages of two different materials, i.e. one with high thermal (but low electrical) conductivity, and the other is the other way around, to achieve combined effective properties in a single structure made by FGCs. This new inverse design framework provides an opportunity to create shape-morphing structures by utilising modulus-graded composite materials, which can be employed in a variety of applications involving multi-physical environments. Furthermore, this framework underscores the versatility of the approach, enabling precise control over material properties at a local level.

Attenuation of impact waves in a nonlinear acoustic metamaterial beam

Attenuation of impact waves in a nonlinear acoustic metamaterial beam

Robotic check of a subassembly, and its simulation

This paper discusses a quality inspection process of a subassembly of a battery cover, which is performed with an industrial robot equipped with an intelligent end-effector. The subassembly has a plastic pin and a torsion spring part. During intended use of the unit, the necessary force to actuate the pin is determined with measurements and simulations. In the measurement a self-devised intelligent end-effector is equipped with a microcontroller and a beam type load cell in order to measure the spring force. The main aim is to make an automatic decision without operator intervention at the quality check process, which is related to the quality of the subassembly. In the simulation deformation of the spring part is modeled with geometrically nonlinear 2D beam finite elements in the course of assembly and intended use. Co-rotational approach is used to consider the large displacements and rotations with small strains. Penalty method is applied to treat the contact between the spring and pin parts.

Exact solutions of Euler–Bernoulli beams

In numerous real-world applications, transverse vibrations of beams are nonlinear in nature. It is a task to solve nonlinear beam systems due to their substantial dependence on the 4 variables of the system and the boundary conditions. To comprehend the nonlinear vibration characteristics, it is essential to do a precise parametric analysis. This research demonstrates an approximation solution for odd and even nonlinear transverse vibrating beams using the Laplace-based variation iteration method, and the formulation of the beams depends on the Galerkin approximation. For the solution of the nonlinear differential equation, this method is efficient as compared to the existing methods in the literature because the solutions exactly match with the numerical solutions. The Laplace-based variation iteration method has been used for the first time to obtain the solution to this important problem. To demonstrate the applicability and precision of the Laplace-based iteration method, several initial conditions are applied to the governing equation for nonlinearly vibrating transverse beams. The natural frequencies and periodic response curves are computed using Laplace-based VIM and compared with the Runge–Kutta RK4 method. In contrast to the RK4, the results demonstrate that the proposed method yields excellent consensus. The Lagrange multiplier is widely regarded as one of the most essential concepts in variational theory. The result obtained are displayed in the table form. Highlights The highlights of the solution of the Euler–Bernoulli beam equation with quintic nonlinearity using Lagrange multiplier are: 1. Introducing the constraint of the boundary conditions into the equation using Lagrange multipliers. 2. Formulating the equations for the Lagrange multipliers and the deflection of the beam. 3. Solving the resulting system of algebraic equations using numerical methods. 4. Obtaining the deflection of the beam as a function of its length and the applied load. 5. Analyzing the behavior of the beam under different loads and boundary conditions.

Weak form quadrature element analysis of spatial geometrically exact composite beams with torsional warping

Spatial thin-walled laminated composite beams under large deformations with torsional warping effects are studied. A novel two-dimensional cross-section model for composite beams with shear and torsion-warping deformations is proposed wherein the three-dimensional warping and coupling effects caused by inhomogeneous anisotropic materials are considered. A weak form quadrature element formulation is established for nonlinear analysis of geometrically exact composite beams with torsional warping and shear deformations. By combining the two-dimensional cross-section model and the nonlinear one-dimensional beam model, the three-dimensional stress and strain fields can be obtained. Numerical examples are provided to investigate the torsional warping effects and verify the effectiveness of the present formulation. Results of the present beam formulation which are achieved at a significantly lower computational cost are in excellent agreement with those of shell and solid models of commercial codes.

Modular quasi-zero-stiffness isolator based on compliant constant-force mechanisms for low-frequency vibration isolation

To effectively isolate low-frequency vibrations, we present a rigid–flexible coupling quasi-zero-stiffness (QZS) vibration isolator with high-static-low-dynamic stiffness (HSLDS) characteristics. Specifically, the QZS isolator is realized by the development of a compliant constant-force mechanism, formed by parallelly combining a diamond-shape mechanism and a nonlinear bi-stable beam in parallel. To evaluate performance of the QZS isolator, we derived an analytical force–displacement model and dynamic model based on pseudo-rigid body method and Lagrange’s equations. Then, finite element analysis was performed in Workbench to verify theoretical analysis and identify the optimal design parameters. Furthermore, the dynamic responses of the QZS isolator are established with the harmonic balance method. Finally, the relationships among displacement transmissibility and factors including damping, BSB, payload mass, and material property are discussed. The results have shown that our QZS isolator design can effectively isolate vibrations in low frequency.

One-dimensional viscoelastic von Kármán theories derived from nonlinear thin-walled beams

We derive an effective one-dimensional limit from a three-dimensional Kelvin–Voigt model for viscoelastic thin-walled beams, in which the elastic and the viscous stress tensor comply with a frame-indifference principle. The limiting system of equations comprises stretching, bending, and twisting both in the elastic and the viscous stress. It coincides with the model already identified via Friedrich and Kružík (Arch Ration Mech Anal 238:489–540, 2020) and Friedrich and Machill (Nonlinear Differ Equ Appl NoDEA 29, Article number: 11, 2022) by a successive dimension reduction, first from 3D to a 2D theory for von Kármán plates and then from 2D to a 1D theory for ribbons. In the present paper, we complement the previous analysis by showing that the limit can also be obtained by sending the height and width of the beam to zero simultaneously. Our arguments rely on the static Gamma -convergence in Freddi et al. (Math Models Methods Appl Sci 23:743–775, 2013), on the abstract theory of metric gradient flows (Ambrosio et al. in Gradient flows in metric spaces and in the space of probability measures. Lectures mathematics, ETH Zürich, Birkhäuser, Basel, 2005), and on evolutionary Gamma -convergence (Sandier and Serfaty in Commun Pure Appl Math 57:1627–1672, 2004).

Exact structural-property matrices and Timoshenko’s shape functions for nonprismatic frame elements defined by noncentroidal axes

When one is attempting to model framed structures and adopts beam-like finite elements formulated with reference to their centroidal axis—which avoids the interaction between axial and flexural effects—one must, in many practical situations, take into account geometrically complex curved-line finite elements. This fact, which can be quite disadvantageous, is typically called for when solving physically nonlinear problems or modeling nonprismatic beams having cross sections varying asymmetrically around their straight centerlines. This paper proposes then a flexibility-type method based on the principle of virtual forces (PVF) to obtain, according to the Timoshenko beam theory (TBT), exact expressions for the structural properties of nonprismatic frame elements described with reference to noncentroidal axes. The main advantage of this method is that it allows, through the use of straight-line segments, the modeling of complex variable-rigidity frame elements. To interpolate the variable sectional rigidities over the elements, one adopts polynomials of different orders. In addition, this new method allows one to obtain the exact Timoshenko’s shape functions (TSFs) for nonprismatic frame elements with noncentroidal axes. Such shape functions are needed to evaluate deformation-dependent structural properties, such as geometric stiffness. We validate the robustness of the proposed formulation by solving in-plane tapered beams possessing thin-walled monosymmetric cross sections. Using analytical and highly accurate 3D ANSYS responses, this work determines and compares responses described in terms of displacements, rotations, internal forces, and stresses.

Nonlinear generation of hollow beams in tunable plasmonic nanosuspensions

We experimentally demonstrate that a probe beam at one wavelength, although exhibiting a weak nonlinear response on its own, can be modulated and controlled by a pump beam at another wavelength in plasmonic nanosuspensions, leading to ring-shaped pattern generation. In particular, we show that the probe and pump wavelengths can be interchanged, but the hollow beam patterns appear only in the probe beam, thanks to the gold nanosuspensions that exhibit a strong nonlinear response to pump beam illumination at the plasmonic resonant frequencies. Colloidal suspensions consisting of either gold nanospheres or gold nanorods are employed as nonlinear media, which give rise to refractive index changes and cross-phase modulation between the two beams. We perform a series of experiments to examine the dynamics of hollow beam generation at a fixed probe power as the pump power is varied and find that nonlinear beam shaping has a different power threshold in different nanosuspensions. Our results will enhance the understanding of nonlinear light–matter interactions in plasmonic nanosuspensions, which may be useful for applications in controlling light by light and in optical limiting.

Timber-to-steel inclined screws connections with interlayers: Experimental investigation, analytical and finite element modelling

Thanks to the axial resisting contribution, inclined screw connections offer a mechanical advantage over connections with purely transversely loaded fasteners. However, the strong coupling between the axial and the transversal behaviour of the screw makes it challenging to predict the actual connection behaviour. Moreover, the presence of polyurethane interlayers for soundproofing or OSB interlayers of light-frame buildings between main members may increase the bending moment component interacting with the axial force in the screw, thus reducing the connection failure load. This paper experimentally assesses the behaviour of steel-to-timber connections, and a parametric finite element model of Winkler non-linear beam is presented and validated. Finally, a simplified analytical model for capacity prediction is presented. The results of the approach currently provided by the Eurocode 5, of an additive model and the proposed analytical model, are then compared with the results of 500 finite element analysis.

Vibration Suppression for an Elastically Supported Nonlinear Beam Coupled to an Inertial Nonlinear Energy Sink

This paper investigates the vibration suppression of an elastically supported nonlinear cantilever beam attached to an inertial nonlinear energy sink (NES). The nonlinear terms introduced by the NES are transferred as the external excitations acting on the beam. The governing equations of the nonlinear beam with an inertial NES are derived according to the Lagrange equations and the assumed mode method. The linear and nonlinear frequencies of the beam are numerically obtained by the Rayleigh–Ritz method and the direct iteration method, respectively. The frequencies are verified by the results of the finite element analysis and literature. The responses of the beam under shock excitations and harmonic excitations are numerically solved by the fourth-order Runge–Kutta method. The suppression effect of the inertial NES on the transverse vibration of the beam is evaluated through the values of amplitude reduction and energy dissipation. In addition, a parametric analysis of the inertial NES is conducted to improve the vibration reduction effect of the NES on the beam.