Euler-Bernoulli Model Research Articles

Photothermoelastic Interactions in Silicon Microbeams Resting on Linear Pasternak Foundation Based on DPL Model

In this work, the photothermal interactions in semiconductor microbeams during the photo-thermo-elastic process have been investigated using the generalized photothermal theory. The proposed mathematical model is constructed based on the Euler–Bernoulli model, the heat equation with two-phase lag and coupled plasma wave equation that indicates the prediction of thermal, elastic and photovoltaic effects in the microbeam resonators. Based on the introduced model, the dynamic influence of thermoelastic photovoltaic microbeam resting on an elastic foundation medium with two parameters has been studied. The Winkler foundation parameter is one of these parameters while the second is the shear foundation parameter. In the field of Laplace transform, the governing equations have been solved while the inverse transforms are found numerically using a tried-and-true approximation technique based on Fourier transform series. The numerical calculations of thermophysical field variables have been discussed and graphically presented. The effects of the magnetic field, Winkler and shear foundation parameters, and lifetime of photogenerated electron have been investigated and studied in detail. Comparisons have been made between the proposed model and previous models that have been derived as special cases from the presented results.

Parametric resonances of nonlinear piezoelectric beams exploiting in-plane actuation

A nonlinear Euler–Bernoulli model of slender piezoelectric beams is employed to investigate parametric resonance motions driven by a pulsating voltage with a DC component. The beam model is based on 3D electric charge conservation and 1D reduction of the momentum balance laws assuming as space coordinate the arclength along the beam base line as space coordinate. The 3D constitutive relationships for a piezoelectric material are specialized according to the Euler–Bernoulli ansatz of transverse unshearability. The nonlinear piezoelastic problem of the parametrically excited beam is directly attacked by the method of multiple scales up to the fifth nonlinear order overcoming modal projection drawbacks and severe restrictions on the oscillation amplitude range. The transition curves, separating regions of stable and unstable trivial solutions in the voltage frequency–amplitude plane, are obtained for various modes of a PVDF beam. The onset of parametric resonances and the post-critical large motion are investigated upon variations of meaningful parameters such as the prestress DC voltage amplitude. The analytical outcomes are compared with those provided by the finite element code ABAQUS which yields the numerical solution of the full 3D nonlinear piezoelectric problem. Moreover, the frequency–response curves to within fifth order are computed applying the pseudo-arclength method to the algebraic frequency–response equation. The frequency–response curves of the lowest few modes reveal new aspects of the parametric resonance motions which can be exploited in applications such as dynamic morphing of thin surfaces.

Nonlinear vibration analysis of a supercritical fluid-conveying pipe made of functionally graded material with initial curvature

In the view of the dynamic behaviors of fluid-conveying pipes become more complex in the supercritical case that it is quite common in engineering, we concentrate on the supercritical fluid-conveying pipe composed of functionally graded material and study the nonlinear vibration for the first time to provide an insight reference for the possible applications of the practice pipe. Firstly, the nonlinear dynamic equations are deduced by Hamiltonian’s principle based on the Euler-Bernoulli and Timoshenko theories. Then the non-trivial equilibrium expressions and the critical fluid velocities are derived analytically for the straight pipe and pipe with initial curvature. For the straight FGM pipe, we conclude that the critical fluid velocity of the Euler-Bernoulli model is larger than that of the Timoshenko model. For the FGM pipe with initial curvature, it is shown that the equilibrium deformations of Euler–Bernoulli model are larger than those of Timoshenko model. For a larger initial curvature, the critical fluid velocity becomes smaller, and the bifurcation occurs earlier for both pipe models. Moreover, Galerkin method is used to analysis the frequencies of the supercritical fluid-conveying FGM Timoshenko pipes based on the buckling configuration. We obtain that the frequencies increase as the power-law exponent and initial tension increase.

Multi-fidelity analysis and uncertainty quantification of beam vibration using co-kriging interpolation method

In this paper, a multi-fidelity surrogate model is created using co-kriging methodology to determine the natural frequencies of beams by combining the fidelities of Euler-Bernoulli (low-fidelity) and Timoshenko (high-fidelity) beam finite element models. This study of free vibration of beams involves uncertainties in material properties. The sampling space for the co-kriging surrogate model is created using an optimal latin hypercube sampling technique. It is shown that the co-kriging surrogate model predicts the natural frequencies with high computational efficiency and accuracy, in the entire design space. The computational efficiency and utility of the multi-fidelity model is demonstrated through its application to a problem of quantifying uncertainties in the natural frequencies of a tapered beam using Monte Carlo Simulation.

Existence of a weak solution to a steady 2D fluid-1D elastic structure interaction problem with Tresca slip boundary condition

We study a steady state fluid–structure interaction problem between an incompressible viscous Newtonian fluid and an elastic structure using a nonlinear boundary condition of friction type on the fluid–structure interface. This condition, also known as Tresca slip boundary condition, allows the fluid to slip on the interface when the tangential component of the fluid shear stress attains a certain threshold function. The governing equations are the 2D Stokes equations for the fluid, written in an unknown domain depending on the structure displacement, and the 1D Euler–Bernoulli model for the structure. We prove that there exists a weak solution of this nonlinear coupled problem by designing a proof based on the Schauder fixed-point theorem. The theoretical result will be illustrated with a numerical example.

Optimization of Piezoelectric Sensor-Actuator for Plate Vibration Control Using Evolutionary Computation: Modeling, Simulation and Experimentation

The development of lightweight, stronger, and more flexible structures has received the utmost interest from many researchers. For this reason, piezoelectric materials, with their inherent electromechanical coupling, have been widely incorporated in the development of such structures to attenuate their vibrations. However, one of the main challenges is to find the optimal control and sensor-actuator placement. This paper presents an active vibration control for flexible structures, whereby a simply supported plate is taken as the benchmark model. A feedback controller with a collocated sensor-actuator configuration is used. Both disturbance and control signal acting on the plate is created by using piezoelectric (PZT) patches. The analytical model is derived based on the Euler-Bernoulli model. The Optimal location of the collocated sensor-actuator, as well as PID controller gains, are determined using Ant Colony Optimization (ACO) technique, then compared with the Genetic Algorithm (GA) and enumerative method (EM). Optimization in this paper is based on minimizing frequency average energy. The optimal performance value of piezoelectric patch sensor-actuator position and PID controller gains are verified experimentally. It was found that PID controller gains and collocated sensor-actuator location optimizations using ACO, GA and enumerative methods give similar results, which implies the effectiveness of ACO as an optimization technique. More than 20 % of attenuation achieved using the available hardware setup.

A size-dependent elastic theory for magneto-electro-elastic materials

A size-dependent elastic theory for magneto-electro-elastic (MEE) nano-materials is proposed. The theory features not only the inclusion of the classical parameters such as piezoelectric and piezomagnetic constants, the magneto-electro, dielectric and magnetic permeability coefficients, but also the nonlocal and strain gradient parameters and their induced high-order MEE parameters. The governing equations and the boundary conditions are derived with the aid of the variational principle. To illustrate the theory, the general solutions of the complete boundary value problems of a one-dimensional beam problem is formulated. It is found that the equation of motion of the present beam model is two orders higher than that of the classical Euler—Bernoulli model. Therefore, it indicates that the general solutions presented in this paper may be served as benchmark theoretical results for the future study.

Effect of Surface and Interfacial Tension on the Resonance Frequency of Microfluidic Channel Cantilever.

The bending resonance of micro-sized resonators has been utilized to study adsorption of analyte molecules in complex fluids of picogram quantity. Traditionally, the analysis to characterize the resonance frequency has focused solely on the mass change, whereas the effect of interfacial tension of the fluid has been largely neglected. By observing forced vibrations of a microfluidic cantilever filled with a series of alkanes using a laser Doppler vibrometer (LDV), we studied the effect of surface and interfacial tension on the resonance frequency. Here, we incorporated the Young–Laplace equation into the Euler–Bernoulli beam theory to consider extra stress that surface and interface tension exerts on the vibration of the cantilever. Based on the hypothesis that the near-surface region of a continuum is subject to the extra stress, thin surface and interface layers are introduced to our model. The thin layer is subject to an axial force exerted by the extra stress, which in turn affects the transverse vibration of the cantilever. We tested the analytical model by varying the interfacial tension between the silicon nitride microchannel cantilever and the filled alkanes, whose interfacial tension varies with chain length. Compared with the conventional Euler–Bernoulli model, our enhanced model provides a better agreement to the experimental results, shedding light on precision measurements using micro-sized cantilever resonators.

Variation of fraction in FOPID controller for vibration control of Euler–Bernoulli beam

Fractional-order derivatives and integrals have gained serious attention by the researchers in past few decades. It has been observed that fractional calculus is more suitable in modeling real life systems. In this paper, Fractional-Order PID (FOPID) controller is studied to suppress the vibration of cantilever beam modeled by Euler–Bernoulli model. Finite difference method is applied to analyze the behavior of the system at different fractional orders of PID controller, and the governing equations are simulated using MATLAB. The response of the system is evaluated at different fractional-order control points, under constant PID parameters ( $$K_{p} ,K_{i} ,{\text{and}} K_{d}$$ ). Furthermore, the variation of frequency in transient response and $$5\%$$ settling time is observed. Simulation results reveal that, though system with fractional order controller is more flexible now, the classical model of PID controller can also be properly tuned through PID constants to achieve the resultant response of the system. So, due to more simplified modelling, the classical model has an edge over the fractional order in the vibration control of cantilever beam.

Characterization and Optimization of Piezoelectric Bimorph Cantilever Structure for Ambient Vibration-Based Energy Harvesting Application

Piezoelectric energy harvesting (PEH) provides an essential mode for the transformation of mechanical vibration to electrical form for providing continuous power to wireless sensor nodes in today’s connected world. This work describes an analytical model based on the Euler-Bernoulli model. The proposed model is in agreement with the FEM simulation model, with results differing by 6%. Variation in geometric parameter, the effect of resonant frequency, output power and stability of the design provides a constraint domain for structure optimization have been described along with the effect of damping. The designed harvester generates a power of 1.31 mW at 90 Hz resonant frequency.

Modeling the pure bending of plates with analog circuit models

Previous work created a way to model flexural bending of Euler-Bernoulli beams as an analog circuit model [C. J. Childs and S. C. Thompson, J. Acoust. Soc. Am. 145, 1895 (2019)]. Further work has been completed to expand the modeling technique from one-dimension into two-dimensions. This allows for enhanced accuracy in modeling physical structures, including a Timoshenko bar or pure bending of plates. The modeling technique will be presented to show bending of plates with comparable accuracy and computational resource advantages, as seen with the Euler-Bernoulli model.

The Vibration of Simply Supported Non-Uniform Cross Sectional Pipe Conveying Fluid Resting on Viscoelastic Foundation

The induced flexural vibration of slender pipe systems with continuous non uniform cross sectional area containing laminar flowing fluid lying on extended Winkler viscoelastic foundation is considered. The Euler Bernoulli model of the pipe has hinged ends. The inlet flow is considered constant steady that interacts with the wall of the pipe. The mathematical model is developed and its corresponding solution is obtained. The influence of the combination of variation of cross section, foundation stiffness and damping on the critical velocities, complex natural frequencies and stabilization of the system is presented.

Buckling and Nonlinear Vibration of Size-Dependent Nanobeam based on the Non-Local Strain Gradient Theory

Based on the nonlocal strain gradient theory, a Euler-Bernoulli nanobeam model subjected to the compressive axial force and resting on the Winkler-Pasternak layer is developed to study buckling and free nonlinear vibration problems. Critical buckling force and nonlinear frequency of simply supported nanobeam are analytically derived. Comparison of obtained analytical solutions with published and numerical ones shows accuracy of the present solutions. Effects of the scale factor, the aspect ratio, the Winkler parameter and the Pasternak parameter on the critical buckling force ratio and the vibration response of the nanobeam are studied in this work.

Comparative Study of Beam Theories on the Effect of Span-Depth Ratio for Symmetric and Un-symmetric Loadings

In the present work, a comparative study is done between Shear-indeformable (Euler-Bernoulli) model and Shear-deformable (Timoshenko) model for two different types of beams (simply supported and fixed beam) under uniformly distributed loading condition and analysed for different Span-Depth ratio (L/D). For un-symmetric loading, a simply supported beam subjected to external point moment is also analysed for different L/D ratios to check the behaviour of beam. Transverse Displacement is taken as a yardstick. Shape functions for Euler-Bernoulli and Timoshenko beam models are developed in MATLAB software by using the Finite Element Method for calculating deflection at different points on the beam. It was observed that for large L/D ratio, the deflections are same for both the beam theory models but as L/D ratio decreases, the deflections for both the beam theories vary from each other. For small L/D ratio, Timoshenko beam model gives more accurate results, since the Timoshenko beam theory is a higher order beam theory than the Euler-Bernoulli beam theory, it is known to be superior in predicting the response of the deep beam.

A Locking-Free DPG Scheme for Timoshenko Beams

Abstract We develop a discontinuous Petrov–Galerkin scheme with optimal test functions (DPG method) for the Timoshenko beam bending model with various boundary conditions, combining clamped, simply supported, and free ends. Our scheme approximates the transverse deflection and bending moment. It converges quasi-optimally in L 2 {L_{2}} and is locking free. In particular, it behaves well (converges quasi-optimally) in the limit case of the Euler–Bernoulli model. Several numerical results illustrate the performance of our method.

Transient analysis of a fixed-head pile group in multi-layered transversely isotropic media due to horizontal loadings

This study presents a FEM-BEM coupled method to research the dynamic response of a fixed-head pile group in multi-layered transversely isotropic media subjected to transient horizontal loadings. Piles are discretized into finite elements on the basis of Euler-Bernoulli model. Fundamental transient solutions of the media derived by the analytical element method are taken as kernel boundary element functions. Then pile-soil interaction solutions are obtained using the FEM-BEM coupled method. Numerical examples are carried out to confirm the correctness of the theoretical derivation and to discuss further the influence of length-diameter ratio, transversely isotropic characteristics and stratification of the media on the transient horizontal response of the pile group.

Parametric resonances of Timoshenko pipes conveying pulsating high-speed fluids

Parametric resonances of pipes caused by pulsating fluids have received much attention. However, the main concern is the pulsation of subcritical flow. Moreover, it is usually based on the Euler-Bernoulli pipe model. This paper focuses on revealing the characteristics of parametric resonances of the Timoshenko pipe with pulsation of supercritical high-speed fluids. Supercritical flow causes the linear instability of the pipe. The coupled partial differential equations with varying parameters are derived for governing the vibration of the pipe around the non-trivial static equilibrium configuration. A direct multi-scale method is developed to analytically obtain parametric resonance responses from coupled partial differential equations with varying parameters. For the first time, the nonlinear parametric response of the Timoshenko pipe is verified by using the finite difference method (FDM). Some interesting phenomena are demonstrated. For example, unlike parametric resonance at subcritical speeds, the steady-state response of velocity pulsation excitations at supercritical speeds is not monotonic with changes in some physical parameters. For another example, when pulsation occurs at supercritical speeds, the smaller steady-state response and the instability region can be obtained through the Timoshenko pipe model. In addition, the relationship between the instability threshold of the velocity pulsation amplitude and the slenderness ratio is non-monotonic, suggesting that the Euler-Bernoulli pipe model may simplify some vibration characteristics. Due to these different phenomena, the necessity of studying the velocity pulsation in the supercritical high-speed range and the necessity of the Timoshenko model are demonstrated.

Geometrically exact planar Euler-Bernoulli beam and time integration procedure for multibody dynamics

A new formulation of geometrically exact planar Euler-Bernoulli beam in multi-body dynamics is proposed. For many applications, the use of the Euler-Bernoulli model is sufficient and has the advantage of being a nodal displacement-only formulation avoiding the integration of rotational degrees of freedom. In this paper, an energy momentum method is proposed for the nonlinear in-plane dynamics of flexible multi-body systems, including the effects of revolute joints with or without torsional springs. Large rotational angles of the joints are accurately calculated. Several numerical examples demonstrate the accuracy and the capabilities of the new formulation.

Predictive Models for Elastic Bending Behavior of a Wood Composite Sandwich Panel

Strands produced from small-diameter timbers of lodgepole and ponderosa pine were used to fabricate a composite sandwich structure as a replacement for traditional building envelope materials, such as roofing. It is beneficial to develop models that are verified to predict the behavior of these sandwich structures under typical service loads. When used for building envelopes, these structural panels are subjected to bending due to wind, snow, live, and dead loads during their service life. The objective of this study was to develop a theoretical and a finite element (FE) model to evaluate the elastic bending behavior of the wood-strand composite sandwich panel with a biaxial corrugated core. The effect of shear deformation was shown to be negligible by applying two theoretical models, the Euler–Bernoulli and Timoshenko beam theories. Tensile tests were conducted to obtain the material properties as inputs into the models. Predicted bending stiffness of the sandwich panels using Euler-Bernoulli, Timoshenko, and FE models differed from the experimental results by 3.6%, 5.2%, and 6.5%, respectively. Using FE and theoretical models, a sensitivity analysis was conducted to explore the effect of change in bending stiffness due to intrinsic variation in material properties of the wood composite material.

Thin-walled composite beam cross-sectional analysis using the mechanics of structure genome

Quick and accurate predictions of cross-sectional properties are required in the preliminary design of thin-walled composite beams. Existing literature provides accurate analytical solutions to the problem, albeit limited to open and single cell sections with stress continuity constraints imposed. Mechanics of structure genome has been used to develop an asymptotically correct model for thin-walled open, closed, single and multi-cell cross sections consisting of any arbitrary anisotropic material without imposing any additional conditions. In this paper, Euler-Bernoulli thin-walled beam cross-sectional model is developed in a finite element framework and demonstrated using several examples, followed by a comparison with existing approaches.