Nonlinear Elastic Beam Research Articles

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

Analysis of a Wind-Driven Power Generation System with Root Slapping Mechanism

This study introduces a groundbreaking slap-type Vibration Energy Harvesting (VEH) system, leveraging a rotating shaft with magnets to induce vibrations in an adjacent elastic steel sheet through magnetic repulsion. This unique design causes the elastic sheet to vibrate, initiating the oscillation of a seesaw-type rigid plate lever. The lever then slaps a piezoelectric patch (PZT) at the elastic steel sheet’s root, converting vibrations into electrical energy. Notably, the design enables the PZT to withstand deformation and flapping forces simultaneously, enhancing power conversion efficiency. The driving force for the rotating shaft is harnessed from the downstream flow field generated by moving objects like rotorcraft, fixed-wing aircraft, motorcycles, and bicycles. Beyond conventional vibration energy harvesting, this design taps into additional electric energy generated by the PZT’s slapping force. This study includes mathematical modeling of nonlinear elastic beams, utilizing the Method of Multiple Scales (MOMS) for in-depth vibration mode analysis. Experimental validation ensures the convergence of theory and practice, confirming the feasibility and superior voltage generation efficiency of this slap-type VEH concept compared to traditional VEH systems.

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 $0 < \mu < {\mu^ * }$.

Определение прогибов физически нелинейной балки на упругом основании

The article deals with the issues of determining the deflections of a beam lying on an elastic foundation with a constant coefficient of rigidity, while the beam is made of a material that has a non-linear relationship between stresses and deformations. The physical non-linearity of the beam material is taken into account by approximating the relationship between stresses and strains with a cubic parabola; such an approximation well describes the deformation curves of a non-linear elastic body with the same diagram of the work of the material in tension and compression. As an example, the deflections of a non-linear elastic beam of rectangular cross section, which lies on the Winkler base and carries a uniformly distributed load along the entire length, are considered for three cases of support fastenings at the edges: with two hinged supports, with two terminations, and with termination and hinged support. The method of successive approximations is used to obtain a solution to the non-linear equation for deflections, which depends on dimensionless parameters that take into account the influence of an elastic foundation, the physical non-linearity of the material, and a uniformly distributed load. The dependence of the change in the value of the maximum relative deflection on the coefficient of the bed and the distributed load, obtained by calculation, taking into account the influence of the physical non-linearity of the material for three cases of support fastenings, is given. The results of the calculations showed that the presence of additional bonds reduces the influence of the physical non-linearity of the beam material on its deflections.

Triple longitudinal fracture of inhomogeneous beam under four-point bending an analytical study by using a non-linear elastic material model

The present paper is devoted to fracture analysis of an inhomogeneous non-linear elastic beam configuration with three parallel longitudinal cracks. The material is continuously inhomogeneous along the beam height. The beam is subjected to four-point bending. The longitudinal fracture behavior is studied by applying the J-integral approach. Solutions to the J-integral are obtained for the three cracks. For this purpose, the curvature and the coordinates of neutral axes of the crack arms are determined by using the equations for equilibrium of the elementary forces in the cross-sections of different portions of the beam. The solutions to the J-integral are valid for arbitrary locations of the cracks along the beam height. Thus, the solutions are very useful for evaluating of the effects of locations of cracks on the fracture. The longitudinal fracture behavior is studied also in terms of the strain energy release rate in order to verify the solutions to the J-integral. Solutions to the strain energy release rate are derived for the three cracks by considering the complementary strain energy in the beam. The solutions to the J-integral are applied to evaluate the effects of locations of the cracks, the material inhomogeneity, the sizes of the beam cross-section, and the coordinates of the applications points of the external forces on the longitudinal fracture behavior of the beam.

Relation-theoretic almost $ \phi $-contractions with an application to elastic beam equations

&lt;abstract&gt;&lt;p&gt;In this article, we prove some results on existence and uniqueness of fixed points for an almost $ \phi $-contraction mapping defined on a metric space endowed with an amorphous relation. Our results generalize and improve several well known fixed point theorems of the existing literature. To substantiate the credibility of our results, we construct some examples. We also apply our results to determine a unique solution of a boundary value problem associated with nonlinear elastic beam equations.&lt;/p&gt;&lt;/abstract&gt;

Vibration characteristic analysis of flexible rotor-bearing system subjected to external combined loads

This paper proposed a general five degrees of freedom dynamic model for investigating the vibration characteristics of flexible rotor-bearing system based on nonlinear elastic Hertz contact theory and Timoshenko beam theory by employing differential quadrature finite element and improved Newton-Raphson methods. The flexible rotor-bearing system is made up of stepped shaft and angular contact ball bearing (ACBB). First, the validation of the established model is verified by the existing literatures and finite element software ANSYS 18.2. Then, the dynamic behaviors of ACBB are discussed systematically for the investigations of the vibration behaviors of flexible rotor-bearing system provide reference. Finally, the vibration characteristics of flexible rotor-bearing system subjected to external combined loads are investigated from different perspectives thoroughly. The results show that the preload, external force, radial force and torque have important influence on the vibration behaviors of flexible rotor-bearing system.

Analysis of Double Elastic Steel Wind Driven Magneto-Electric Vibration Energy Harvesting System

This research proposes an energy harvesting system that collects the downward airflow from a helicopter or a multi-axis unmanned rotary-wing aircraft and uses this wind force to drive the magnet to rotate, generating repulsive force, which causes the double elastic steel system to slap each other and vibrate periodically in order to generate more electricity than the traditional energy harvesting system. The design concept of the vibration mechanism in this study is to allow the elastic steel carrying the magnet to slap another elastic steel carrying the piezoelectric patch to form a set of double elastic steel vibration energy harvesting (DES VEH) systems. The theoretical DES VEH mechanism of this research is composed of a pair of cantilever beams, with magnets attached to the free end of one beam, and PZT attached to the other beam. This study analyzes the single beam system first. The MOMS method is applied to analyze the frequency response of this nonlinear system theoretically, then combines the piezoelectric patch and the magneto-electric coupling device with this nonlinear elastic beam to analyze the benefits of the system’s converted electrical energy. In the theoretical study of the DES VEH system, the slapping force between the two elastic beams was considered as a concentrated load on each of the beams. Furthermore, both SES and DES VEH systems are studied and correlated. Finally, the experimental data and theoretical results are compared to verify the feasibility and correctness of the theory. It is proven that this DES VEH system can not only obtain the electric energy from the traditional SES VEH system but also obtain the extra electric energy of the steel vibration subjected to the slapping force, which generates optimal power to the greatest extent.

A corotational finite element approach coupled to a discrete forcing method to solve hyperelastic deformation induced by two-phase flow

Fluid–structure interaction is an ever occurring phenomenon in engineering. In power plants for instance, fuel rods, tubes in steam generator or cooling towers may deform under the extreme load caused by the fluid flowing around them. It may also be valuable to study this kind of interaction during pipelines installation or for wind generating structures. If the deformation is sufficiently important as to influence the fluid behaviour, the problem must be tackled as a whole. Therefore, fluids and structures cannot be processed separately. In this work, a fluid–structure interaction method taking into account the large deformation of simple geometry is developed in neptune_cfd, a 3D multiphase flow code based on Eulerian–Eulerian approach with a single pressure. The solid structures are geometrically non-linear elastic beams in plane strain. The coupling interface is tracked with a discrete forcing method based on a time and space dependent porosity method. The latter is updated several times for each time steps in order to ensure convergence.The present formulation is checked on a semi-analytical problem and validated for a dam break over a flexible plate with experimental confrontation and comparison.

Transverse Vibration Energy Harvesting of Double Elastic Steel

This study uses the piezoelectric technology to collect vibration energy from the fixed-fixed nonlinear elastic beams attached with the piezo-patch between the two ends. Both single elastic steel sheet (SESS) and double elastic steel sheet (DESS) systems are investigated and correlated. To simulate the power generation of the vibration energy harvester (VEH) of both the SESS and the DESS in different engineering elements, the simple harmonic external force generated by a shaker at the location of the piezo-patch is used as the source. With this, more vibration converted electric energy is derived from the transverse deformation and flapping from the DESS than the SESS beam. The equation of a nonlinear Euler–Bernoulli beam is coupled with the electric energy equation of the piezo-patch to simulate the SESS VEH system. The flapping force from the DESS VEH system can be considered the concentrated external load applied on the SESS beam model. The method of multiple scales (MOMS) is employed to analyze this nonlinear problem. The fixed points plots and the numerical results confirm this theory presented for the two beam systems, which can be used for evaluating similar engineering systems. Experiments are also performed in this study. The Taguchi method is used to analyze the optimum locations of the shaker and piezo-patch, as well as the confidence level of the factors. The method of nonlinear analysis presented in this study demonstrates its accuracy compared with the linear case. The transverse DESS VEH model proposed is proved to be feasible and more effective than the SESS system.

Longitudinal Fracture Analysis of Inhomogeneous Beams with a Continuously Varying Height

Longitudinal fracture behavior of nonlinear elastic inhomogeneous beam configurations with a continuously varying height along its length is analyzed. The general solution of the strain energy release rate problem is obtained by considering the energy balance. Beam configurations with the longitudinal crack located arbitrarily along the height are analyzed. The beams exhibit smooth material inhomogeneity in the height direction. The solution derived is valid within the arbitrary law of continuous beam height variations. Longitudinal fracture in a simply supported nonlinear elastic inhomogeneous beam with a height that varies in a parabolic law is investigated by applying the general solution of the strain energy release rate problem. The J-integral approach is used to verify this solution. The effect of various geometrical and material parameters on the longitudinal fracture of the simply supported beam configuration is evaluated and discussed in detail.

Exponential Stability for a Nonlinear Timoshenko System with Distributed Delay

This paper is concerned with a nonlinear Timoshenko system modeling clamped thin elastic beams with distributed delay time. The distributed delay is defined on feedback term associated to the equation for rotation angle. Under suitable assumptions on the data, we establish the exponential stability of the system under the usual equal wave speeds assumption.

Theoretical analysis of fracture in non-linear elastic functionally graded beam of linearly changing thickness with three longitudinal cracks

The present paper is focussed on fracture analysis of a functionally graded nonlinear elastic cantilever beam structure with three longitudinal cracks. The cross-section of the beam is a rectangle. The beam height increases linearly towards the clamped end. The material is functionally graded along the thickness of the beam. The J-integral approach is applied in the fracture analysis. The solutions to the J-integral for the three cracks are verified by deriving of the strain energy release rate. For this purpose, the complementary strain energy cumulated in the beam is differentiated with respect to the areas of the three cracks. A parametric investigation of the longitudinal fracture behaviour of the beam is carried-out by applying the solutions to the J-integral. The effects of the beam geometry, the crack length and the material gradient along the beam thickness on the longitudinal fracture behaviour are evaluated and discussed.

New fixed point theorems for the sum of two mixed monotone operators of Meir–Keeler type and their applications to nonlinear elastic beam equations

In this paper, we present a new fixed point theorem for the sum of two mixed monotone operators of Meir–Keeler type on ordered Banach spaces through projective metric, which extends the existing corresponding results. As applications, we utilize the results obtained in this paper to study the existence and uniqueness of positive solutions for nonlinear singular fourth-order elastic beam equations with nonlinear boundary conditions.

Nonlinear isogeometric multiscale simulation for design and fabrication of functionally graded knitted textiles

We present a nonlinear multiscale modeling and simulation framework for the mechanical design of machine-knitted textiles with functionally graded microstructures. The framework operates on the mesoscale (stitch level), where yarns intermesh into stitch patterns, and the macroscale (fabric level), where these repetitive stitch patterns are composed into a fabric. On the mesoscale, representative unit cells consisting of single interlocked yarn loops, modeled as geometrically exact, nonlinear elastic 3D beams, are homogenized to compute their effective mechanical properties. From this data, a B-Spline response surface model is generated to represent the nonlinear orthotropic constitutive behavior on the macroscale, where the fabric is modeled by a nonlinear Kirchhoff–Love shell formulation and discretized using isogeometric finite elements. These functionally graded textiles with locally varying properties can be designed and analyzed by parameterizing the stitch value, i.e., the loop length of a single jersey stitch, and the knitting direction as mesoscopic design variables of the macroscopic response surface constitutive model. To validate the multiscale simulation and design approach, numerical results are compared against physical experiments of different tensile loading cases for various grading scenarios. Furthermore, the versatility of the method for the design of functionally graded textiles is demonstrated.

Longitudinal fracture analysis of inhomogeneous beams with continuously changing radius of cross-section along the beam length

This paper focusses on deriving a general solution to the strain energy release rate for a longitudinal circular cylindrical crack in non-linear elastic inhomogeneous beams with circular cross-section. The radius of the cross-section changes continuo

Analyses of delaminations in non-linear elastic multilayered inhomogeneous beam configurations

This paper presents investigation of delamination fracture behavior of multilayered non-linear elastic beam configurations by using the Ramberg-Osgood stress-strain relation. It is assumed that each layer exhibits continuous material inhomogeneity along the width as well as along thickness of the layer. An approach for determination of the strain energy release rate is developed for a delamination crack located arbitrary along the multilayered beam height. The approach can be applied for multilayered beams of arbitrary cross-section under combination of axial force and bending moments. The layers may have different thickness and material properties. The number of layers is arbitrary. The approach is applied for analyzing the delamination fracture behavior of a multilayered beam configuration subjected to four-point bending. The beam has a rectangular cross-section. The delamination crack is located symmetrically with respect to the beam midspan. The strain energy release rate is derived assuming that the modulus of elasticity varies continuously in the cross-section of each layer according to a hyperbolic law. In order to verify the solution to the strain energy release rate, the delamination fracture behavior of the multilayered non-linear elastic four-point bending beam configuration is studied also by applying the method of the J-integral. The solution to the strain energy release rate derived in the present paper is used in order to perform a parametric study of delamination.

A high density piezoelectric energy harvesting device from highway traffic – Design analysis and laboratory validation

This paper introduces an innovative piezoelectric energy harvesting device with a high density of the energy harvested from highway traffic. The piezoelectric energy harvesting device has a compression-to-compression force amplification mechanism provided by clamped-clamped nonlinear elastic beams. The amplification mechanism enables the device to fully explore the power conversion potential of the piezoelectric material and can deliver the harvested electricity far more than that generated by the same piezoelectric material under direct compressive loading without amplification. For example, when the amplification factor is equal to 10, the generated electricity is nearly 100 times bigger. A multi-objective optimal design problem of the piezoelectric energy harvesting device is formulated. A set of multi-objective optimal designs has been found numerically in the parameter space. The paper presents the analysis of the nonlinear beams and a finite element model of multi-layer piezoelectric ceramic stacks. Impulsive time dynamic loadings from the passing vehicles are considered. The dynamic response of the piezoelectric energy harvesting device to the impulsive traffic loadings is investigated. The laboratory test results are presented to validate the mathematical model and mechanical design of the device. In a quasi-static load cycle of 1333N, a preloaded piezoelectric energy harvesting device prototype (142×42×84mm3) is able to generate a voltage of 128V and a potential electric energy of 120mJ while the sinking displacement is 2.54mm. Numerical results of the dynamic response show that the piezoelectric energy harvesting device performs well over a wide range of vehicle speed from 8.05 to 128.75 km/h. High speed seems beneficial to energy harvesting.

Optimal thickness distribution of stepped nonlinear Gao beam

We present an optimal shape design problem of a transversely and axially loaded nonlinear stepped elastic beam with piecewise constant thickness distribution. The nonlinear beam model proposed by D.Y. Gao in 1996 is considered for modelling the state problem. The objective of the optimization is to find the thickness distribution to maximize the stiffness of the beam subjected to geometrical constraints, namely the preservation of volume is imposed. The stiffness is characterized by the compliance cost functional. The existence of solutions of the optimization problem and convergence properties of discretization using finite element approach is analysed. The theory is illustrated by numerical results.