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Journal articles on the topic 'Non-classical Euler-Bernoulli beam'

Author: Grafiati

Published: 6 September 2023

Last updated: 19 July 2025

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Samayoa, Didier, Andriy Kryvko, Gelasio Velázquez, and Helvio Mollinedo. "Fractal Continuum Calculus of Functions on Euler-Bernoulli Beam." Fractal and Fractional 6, no. 10 (2022): 552. http://dx.doi.org/10.3390/fractalfract6100552.

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A new approach for solving the fractal Euler-Bernoulli beam equation is proposed. The mapping of fractal problems in non-differentiable fractals into the corresponding problems for the fractal continuum applying the fractal continuum calculus (FdH3-CC) is carried out. The fractal Euler-Bernoulli beam equation is derived as a generalization using FdH3-CC under analogous assumptions as in the ordinary calculus and then it is solved analytically. To validate the spatial distribution of self-similar beam response, three different classical beams with several fractal parameters are analysed. Some mechanical implications are discussed.

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Yin, Shuohui, Zhibing Xiao, Gongye Zhang, Jingang Liu, and Shuitao Gu. "Size-Dependent Buckling Analysis of Microbeams by an Analytical Solution and Isogeometric Analysis." Crystals 12, no. 9 (2022): 1282. http://dx.doi.org/10.3390/cryst12091282.

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This paper proposes an analytical solution and isogeometric analysis numerical approach for buckling analysis of size-dependent beams based on a reformulated strain gradient elasticity theory (RSGET). The superiority of this method is that it has only one material parameter for couple stress and another material parameter for strain gradient effects. Using the RSGET and the principle of minimum potential energy, both non-classical Euler–Bernoulli and Timoshenko beam buckling models are developed. Moreover, the obtained governing equations are solved by an exact solution and isogeometric analysis approach, which conforms to the requirements of higher continuity in gradient elasticity theory. Numerical results are compared with exact solutions to reveal the accuracy of the current isogeometric analysis approach. The influences of length–scale parameter, length-to-thickness ratio, beam thickness and boundary conditions are investigated. Moreover, the difference between the buckling responses obtained by the Timoshenko and Euler–Bernoulli theories shows that the Euler–Bernoulli theory is suitable for slender beams.

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Stempin, Paulina, and Wojciech Sumelka. "Dynamics of Space-Fractional Euler–Bernoulli and Timoshenko Beams." Materials 14, no. 8 (2021): 1817. http://dx.doi.org/10.3390/ma14081817.

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This paper investigates the dynamics of the beam-like structures whose response manifests a strong scale effect. The space-Fractional Euler–Bernoulli beam (s-FEBB) and space-Fractional Timoshenko beam (s-FTB) models, which are suitable for small-scale slender beams and small-scale thick beams, respectively, have been extended to a dynamic case. The study provides appropriate governing equations, numerical approximation, detailed analysis of free vibration, and experimental validation. The parametric study presents the influence of non-locality parameters on the frequencies and shape of modes delivering a depth insight into a dynamic response of small scale beams. The comparison of the s-FEBB and s-FTB models determines the applicability limit of s-FEBB and indicates that the model (also the classical one) without shear effect and rotational inertia can only be applied to beams significantly slender than in a static case. Furthermore, the validation has confirmed that the fractional beam model exhibits very good agreement with the experimental results existing in the literature—for both the static and the dynamic cases. Moreover, it has been proven that for fractional beams it is possible to establish constant parameters of non-locality related to the material and its microstructure, independent of beam geometry, the boundary conditions, and the type of analysis (with or without inertial forces).

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Ghorbanpourarani, A., M. Mohammadimehr, A. Arefmanesh, and A. Ghasemi. "Transverse vibration of short carbon nanotubes using cylindrical shell and beam models." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 224, no. 3 (2009): 745–56. http://dx.doi.org/10.1243/09544062jmes1659.

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The transverse vibrations of single- and double-walled carbon nanotubes are investigated under axial load by applying the Euler—Bernoulli and Timoshenko beam models and the Donnell shell model. It is concluded that the Euler—Bernoulli beam model and the Donnell shell model predictions have the lowest and highest accuracies, respectively. In order to predict the vibration behaviour of the carbon nanotube more accurately, the current classical models are modified using the non-local theory. The natural frequencies, amplitude coefficient, critical axial load, and strain are obtained for the simply supported boundary conditions.

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Zhang, GY, and X.-L. Gao. "A new Bernoulli–Euler beam model based on a reformulated strain gradient elasticity theory." Mathematics and Mechanics of Solids 25, no. 3 (2019): 630–43. http://dx.doi.org/10.1177/1081286519886003.

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A new non-classical Bernoulli–Euler beam model is developed using a reformulated strain gradient elasticity theory that incorporates both couple stress and strain gradient effects. This reformulated theory is first derived from Form I of Mindlin’s general strain gradient elasticity theory. It is then applied to develop the model for Bernoulli–Euler beams through a variational formulation based on Hamilton’s principle, which leads to the simultaneous determination of the equation of motion and the complete boundary conditions and provides a unified treatment of the strain gradient, couple stress and velocity gradient effects. The new beam model contains one material constant to account for the strain gradient effect, one material length scale parameter to describe the couple stress effect and one coefficient to represent the velocity gradient effect. The current non-classical beam model reduces to its classical elasticity-based counterpart when the strain gradient, couple stress and velocity gradient effects are all suppressed. In addition, the newly developed beam model includes the models considering the strain gradient effect only or the couple stress effect alone as special cases. To illustrate the new model, the static bending and free vibration problems of a simply supported beam are analytically solved by directly applying the general formulas derived. The numerical results reveal that the beam deﬂection predicted by the current model is always smaller than that by the classical model, with the difference being large for very thin beams but diminishing with the increase of the beam thickness. Also, the natural frequency based on the new beam model is found to be always higher than that based on the classical model.

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Ishaquddin, Md, and S. Gopalakrishnan. "Differential quadrature-based solution for non-classical Euler-Bernoulli beam theory." European Journal of Mechanics - A/Solids 86 (March 2021): 104135. http://dx.doi.org/10.1016/j.euromechsol.2020.104135.

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Mei, C. "Vibrations in a spatial K-shaped metallic frame: an exact analytical study with experimental validation." Journal of Vibration and Control 23, no. 19 (2016): 3147–61. http://dx.doi.org/10.1177/1077546315627085.

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In a spatial K-shaped metallic frame, there exist in- and out-of-plane bending, axial, and torsional vibrations. A wave-based vibration analysis approach is applied to obtain free and forced vibration responses in a space frame. In order to validate the analytical approach, a steel K-shaped space frame was built by welding four beam elements of rectangular and square cross-section together. Bending vibrations are modeled using both the classical Euler–Bernoulli theory and the advanced Timoshenko theory. This allows the effects of rotary inertia and shear distortion, which were neglected in the classical Euler–Bernoulli theory, to be studied. In addition, the effect of torsional rigidity adjustment for structures of rotationally non-symmetric cross-section is also examined.

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König, Paul, Patrick Salcher, Christoph Adam, and Benjamin Hirzinger. "Dynamic analysis of railway bridges exposed to high-speed trains considering the vehicle–track–bridge–soil interaction." Acta Mechanica 232, no. 11 (2021): 4583–608. http://dx.doi.org/10.1007/s00707-021-03079-1.

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AbstractA new semi-analytical approach to analyze the dynamic response of railway bridges subjected to high-speed trains is presented. The bridge is modeled as an Euler–Bernoulli beam on viscoelastic supports that account for the flexibility and damping of the underlying soil. The track is represented by an Euler–Bernoulli beam on viscoelastic bedding. Complex modal expansion of the bridge and track models is performed considering non-classical damping, and coupling of the two subsystems is achieved by component mode synthesis (CMS). The resulting system of equations is coupled with a moving mass–spring–damper (MSD) system of the passing train using a discrete substructuring technique (DST). To validate the presented modeling approach, its results are compared with those of a finite element model. In an application, the influence of the soil–structure interaction, the track subsystem, and geometric imperfections due to track irregularities on the dynamic response of an example bridge is demonstrated.

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Zhang, G. Y., X. L. Gao, C. Y. Zheng, and C. W. Mi. "A non-classical Bernoulli-Euler beam model based on a simplified micromorphic elasticity theory." Mechanics of Materials 161 (October 2021): 103967. http://dx.doi.org/10.1016/j.mechmat.2021.103967.

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Yin, Shuohui, Yang Deng, Tiantang Yu, Shuitao Gu, and Gongye Zhang. "Isogeometric analysis for non-classical Bernoulli-Euler beam model incorporating microstructure and surface energy effects." Applied Mathematical Modelling 89 (January 2021): 470–85. http://dx.doi.org/10.1016/j.apm.2020.07.015.

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Zu, J. W. Z., and R. P. S. Han. "Dynamic Response of a Spinning Timoshenko Beam With General Boundary Conditions and Subjected to a Moving Load." Journal of Applied Mechanics 61, no. 1 (1994): 152–60. http://dx.doi.org/10.1115/1.2901390.

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The dynamic response of a spinning Timoshenko beam with general boundary conditions and subjected to a moving load is solved analytically for the first time. Solution of the problem is achieved by formulating the spinning Timoshenko beams as a non-self-adjoint system. To compute the system dynamic response using the modal analysis technique, it is necessary to determine the eigenquantities of both the original and adjoint systems. In order to fix the adjoint eigenvectors relative to the eigenvectors of the original system, the biorthonormality conditions are invoked. Responses for the four classical boundary conditions which do not involve rigidbody motions are illustrated. To ensure the validity of the method, these results are compared with those from Euler-Bernoulli and Rayieigh beam theories. Numerical simulations are performed to study the influence of the four boundary conditions on selected system parameters.

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Akgöz, Bekir, and Ömer Civalek. "Buckling Analysis of Functionally Graded Tapered Microbeams via Rayleigh–Ritz Method." Mathematics 10, no. 23 (2022): 4429. http://dx.doi.org/10.3390/math10234429.

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In the present study, the buckling problem of nonhomogeneous microbeams with a variable cross-section is analyzed. The microcolumn considered in this study is made of functionally graded materials in the longitudinal direction and the cross-section of the microcolumn varies continuously throughout the axial direction. The Bernoulli–Euler beam theory in conjunction with modified strain gradient theory are employed to model the structure by considering the size effect. The Rayleigh–Ritz numerical solution method is used to solve the eigenvalue problem for various conditions. The influences of changes in the cross-section and Young’s modulus, size dependency, and non-classical boundary conditions are examined in detail. It is observed that the size effect becomes more pronounced for smaller sizes and differences between the classical and non-classical buckling loads increase by increasing the taper ratios.

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Yang, Zhenghao, Konstantin Naumenko, Chien-Ching Ma, and Yang Chen. "Closed-Form Analytical Solutions for the Deflection of Elastic Beams in a Peridynamic Framework." Applied Sciences 13, no. 18 (2023): 10025. http://dx.doi.org/10.3390/app131810025.

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Peridynamics is a continuum theory that operates with non-local deformation measures as well as long-range internal force/moment interactions. The resulting equations are of the integral type, in contrast to the classical theory, which deals with differential equations. The aim of this paper is to analyze peridynamic governing equations for elastic beams. To this end, the strain energy density is formulated as a function of the non-local curvature. By applying the Lagrange principle, the peridynamic equations of motion are derived. Examples of non-local boundary conditions, including simple support, clamped edge, roller clamped edge, and free edge, are presented by introducing the interaction domain. Novel closed-form analytical solutions to integral equations are presented for beams with various boundary conditions, including clamped—simply supported, clamped–clamped, simply supported–roller-clamped, and clamped–roller-clamped beams. Furthermore, different types of loadings, including uniformly distributed load, concentrated force, and concentrated moment, are considered. The results are validated by comparing the derived solutions against solutions to the classical Bernoulli–Euler beam theory. A very good agreement between the non-local and the classical theories is observed for the case of the small horizon sizes, which shows the capability of the derived equations of motion and proposed boundary conditions.

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Esfahanian, V., E. Dehdashti, and A. M. Dehrouyeh-Semnani. "Fluid-Structure Interaction in Microchannel Using Lattice Boltzmann Method and Size-Dependent Beam Element." Advances in Applied Mathematics and Mechanics 6, no. 3 (2014): 345–58. http://dx.doi.org/10.4208/aamm.2013.m152.

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AbstractFluid-structure interaction (FSI) problems in microchannels play prominent roles in many engineering applications. The present study is an effort towards the simulation of flow in microchannel considering FSI. Top boundary of the microchannel is assumed to be rigid and the bottom boundary, which is modeled as a Bernoulli-Euler beam, is simulated by size-dependent beam elements for finite element method (FEM) based on a modified couple stress theory. The lattice Boltzmann method (LBM) using D2Q13 LB model is coupled to the FEM in order to solve fluid part of FSI problem. In the present study, the governing equations are non-dimensionalized and the set of dimensionless groups is exhibited to show their effects on micro-beam displacement. The numerical results show that the displacements of the micro-beam predicted by the size-dependent beam element are smaller than those by the classical beam element.

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Fakhreddine, Hatim, Ahmed Adri, Saïd Rifai, and Rhali Benamar. "Nonlinear free and forced vibration of Euler-Bernoulli beams resting on intermediate flexible supports." MATEC Web of Conferences 211 (2018): 02003. http://dx.doi.org/10.1051/matecconf/201821102003.

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This paper deals with the geometrically nonlinear free and forced vibration analysis of a multi-span Euler Bernoulli beam resting on arbitrary number N of flexible supports, denoted as BNIFS, with general end conditions. The generality of the approach is based on use of translational and rotational springs at both ends, allowing examination of all possible combinations of classical beam end conditions, as well as elastic restraints. First, the linear case is examined to obtain the mode shapes used as trial functions in the nonlinear analysis. The beam bending vibration equation is first written in each span. Then, the continuity requirements at each elastic support are stated, in addition to the beam end conditions. This leads to a homogeneous linear system whose determinant must vanish in order to allow nontrivial solutions to be obtained. Numerical results are given to illustrate the effects of the support stiffness and locations on the natural frequencies and mode shapes of the BNIFS. The nonlinear theory is then developed, based on the Hamilton’s principle and spectral analysis. The nonlinear beam transverse displacement function is defined as a linear combination of the linear modes calculated before. The problem is reduced to solution of a non-linear algebraic system using numerical or analytical methods. The nonlinear algebraic system is solved using an explicit method developed previously (second formulation) leading to the amplitude dependent nonlinear fundamental mode of the BNIFS.

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Ziou, Hassina, and Mohamed Guenfoud. "MODAL BEHAVIOUR OF LONGITUDINALLY PERFORATED NANOBEAMS." Advances in Civil and Architectural Engineering 14, no. 27 (2023): 143–59. http://dx.doi.org/10.13167/2023.27.10.

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Nano-electro-mechanical systems (NEMS) require perforated beams for structural integrity. Hole sizes, hole numbers, and scale effects need to be modelled appropriately in their design. This paper presents a new finite element model to investigate the modal behaviour of longitudinally perforated nanobeams (LPNBs) using the classical Euler–Bernoulli beam theory. A symmetric array of holes arranged parallel to the length direction of the beam with equal spacing was assumed for the perforation. The non-local Eringen’s differential form was used to incorporate the nanoscale sizes. The accuracy of the proposed model was verified by comparing the obtained results with the available analytical solutions for fully filled nanobeams. The effects of aspect ratios, non-local parameters, boundary conditions, and perforation characteristics on the modal behaviour of LPNBs were investigated. The non-local parameter reduced the natural frequency owing to a decrease in the stiffness of the structures. However, the perforation filling ratio led to higher values of the fundamental frequency. Furthermore, compared with other boundary conditions, clamped–clamped boundary conditions demonstrated the best performance in terms of the maximum frequency.

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Hosseini, Mohammad, Reza Bahaadini, and Zahra Khalili-Parizi. "Structural instability of non-conservative functionally graded micro-beams tunable with piezoelectric layers." Journal of Intelligent Material Systems and Structures 30, no. 4 (2019): 593–605. http://dx.doi.org/10.1177/1045389x18818769.

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This investigation aims to explore the non-conservative instability of a functionally graded material micro-beam subjected to a subtangential force. The functionally graded material micro-beam is integrated with piezoelectric layers on the lower and upper surfaces. To take size effect into account, the mathematical derivations are expanded in terms of three length scale parameters using the modified strain gradient theory in conjunction with the Euler–Bernoulli beam model. However, the modified strain gradient theory includes modified couple stress theory and classical theory as special cases. Applying extended Hamilton’s principle and Galerkin method, the governing equation and corresponding boundary conditions are obtained and then solved numerically by the eigenvalue analysis, respectively. The results illustrated effects of non-conservative parameter, length scale parameter, different material gradient index, and various values of piezoelectric voltage on the natural frequencies, flutter and divergence instabilities of a cantilever functionally graded material micro-beam. It is found that both the material gradient index and applied piezoelectric voltage have significant influence on the vibrational behaviors, divergence and flutter instability regions. Furthermore, a comparison between the various micro-beam theories on the basis of modified couple stress theory, modified strain gradient theory, and classical theory are presented.

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Tanaka, N., and Y. Kikushima. "Optimal Vibration Feedback Control of an Euler-Bernoulli Beam: Toward Realization of the Active Sink Method." Journal of Vibration and Acoustics 121, no. 2 (1999): 174–82. http://dx.doi.org/10.1115/1.2893961.

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This paper discusses the optimal vibration feedback control of an Euler-Bernoulli beam from a viewpoint of active wave control making all structural modes inactive (more than suppressed). Using a transfer matrix method, the paper derives two kinds of optimal control laws termed “active sink” which inactivates all structural modes; one obtained by eliminating reflected waves and the other by transmitted waves at a control point. Moreover, the characteristic equation of the active sink system is derived, the fundamental properties being investigated. Towards the goal of implementing the optimal control law that is likely to be non-causal, a “classical” velocity feedback control law (Balas, 1979) widely used in a vibration control engineering is applied, revealing a substantial shortcoming. Introduction of a “classical” displacement feedback to the velocity is found to realize the optimal control law in a restricted frequency range. Finally, two kinds of stability verification for closed feedback control systems are presented for distributed parameter structures.

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Bobková, Michaela, and Lukáš Pospíšil. "Numerical Solution of Bending of the Beam with Given Friction." Mathematics 9, no. 8 (2021): 898. http://dx.doi.org/10.3390/math9080898.

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We are interested in a contact problem for a thin fixed beam with an internal point obstacle with possible rotation and shift depending on a given swivel and sliding friction. This problem belongs to the most basic practical problems in, for instance, the contact mechanics in the sustainable building construction design. The analysis and the practical solution plays a crucial role in the process and cannot be ignored. In this paper, we consider the classical Euler–Bernoulli beam model, which we formulate, analyze, and numerically solve. The objective function of the corresponding optimization problem for finding the coefficients in the finite element basis combines a quadratic function and an additional non-differentiable part with absolute values representing the influence of considered friction. We present two basic algorithms for the solution: the regularized primal solution, where the non-differentiable part is approximated, and the dual formulation. We discuss the disadvantages of the methods on the solution of the academic benchmarks.

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Montseny, Gérard. "Diffusive wave-absorbing control: example of the boundary stabilization of a thin flexible beam." Journal of Vibration and Control 18, no. 11 (2011): 1708–21. http://dx.doi.org/10.1177/1077546311419933.

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In this paper we deal with the boundary control of the Euler–Bernoulli beam by means of wave-absorbing feedback. Such controls are based upon the reduction of reflected waves and involve long memory non-rational convolution operators resulting from specific properties of the system. These operators are reformulated under so-called diffusive input–output state-space realizations, which allow us to represent the global closed-loop system under the abstract form d X/d t = A X with A the infinitesimal generator of a continuous semigroup. So, well-posedness and stability of the controlled system result from classical semigroup theory. Finite-dimensional approximations of the diffusive realizations are then studied, with the aim of providing implementable controls close to the ideal ones. Finally, significant numerical simulations are presented.

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Attia, Mohamed A., and Salwa A. Mohamed. "Pull-In Instability of Functionally Graded Cantilever Nanoactuators Incorporating Effects of Microstructure, Surface Energy and Intermolecular Forces." International Journal of Applied Mechanics 10, no. 08 (2018): 1850091. http://dx.doi.org/10.1142/s1758825118500916.

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In this paper, an integrated non-classical continuum model is developed to investigate the pull-in instability of electrostatically actuated functionally graded nanocantilevers. The model accounts for the simultaneous effects of local-microstructure, surface elasticity and surface residual in the presence of fringing field as well as Casimir and van der Waals forces. The modified couple stress and Gurtin–Murdoch surface elasticity theories are employed to conduct the scaling effects of microstructure and surface energy, respectively, in the context of Euler–Bernoulli beam hypothesis. Bulk and surface material properties are varied according to the power-law distribution through the beam thickness. The physical neutral axis position for mentioned FG nanobeams is considered. Hamilton principle is employed to derive the nonlinear size-dependent governing equations and the non-classical boundary conditions. The resulting nonlinear differential equations are solved utilizing the generalized differential quadrature method (GDQM). In addition, the non-classical boundary conditions of nanocantilever beams due to surface residual stress are exactly implemented. After validation of the obtained results by previously available data in the literature, the influences of different geometrical and material parameters on the pull-in instability of the FG nanocantilevers are examined in detail. It is concluded that the pull-in behavior of electrically actuated FG micro/nanocantilevers is significantly influenced by the material distribution, material length scale parameter, surface elasticity constant, surface residual stress, initial gap, slenderness ratio, Casimir, and van der Waals forces. The obtained results can be considered for modeling and analysis of electrically actuated FG nanocantilevers.

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Wakshume, Demeke Girma, and Marek Łukasz Płaczek. "Mathematical Modeling and Finite Element Simulation of the M8514-P2 Composite Piezoelectric Transducer for Energy Harvesting." Sensors 25, no. 10 (2025): 3071. https://doi.org/10.3390/s25103071.

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This paper focuses on the mathematical and numerical modeling of a non-classical macro fiber composite (MFC) piezoelectric transducer, MFC-P2, integrated with an aluminum cantilever beam for energy harvesting applications. It seeks to harness the transverse vibration energy in the environment to power small electronic devices, such as wireless sensors, where conventional power sources are inconvenient. The P2-type macro fiber composites (MFC-P2) are specifically designed for transverse energy harvesting applications. They offer high electric source capacitance and improved electric charge generation due to the strain developed perpendicularly to the voltage produced. The system is modeled analytically using Euler–Bernoulli beam theory and piezoelectric constitutive equations, capturing the electromechanical coupling in the d31 mode. Numerical simulations are conducted using COMSOL Multiphysics 6.29 to reduce the complexity of the mathematical model and analyze the effects of material properties, geometric configurations, and excitation conditions. The theoretical model is based on the transverse vibrations of a cantilevered beam using Euler–Bernoulli theory. The natural frequencies and mode shapes for the first four are determined. Depending on these, the resonance frequency, voltage, and power outputs are evaluated across a 12 kΩ resistive load. The results demonstrate that the energy harvester effectively operates near its fundamental resonant frequency of 10.78 Hz, achieving the highest output voltage of approximately 0.1952 V and a maximum power output of 0.0031 mW. The generated power is sufficient to drive ultra-low-power devices, validating the viability of MFC-based cantilever structures for autonomous energy harvesting systems. The application of piezoelectric phenomena and obtaining electrical energy from mechanical vibrations can be powerful solutions in such systems. The application of piezoelectric phenomena to convert mechanical vibrations into electrical energy presents a promising solution for self-powered mechatronic systems, enabling energy autonomy in embedded sensors, as well as being used for structural health monitoring applications.

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Makkad, Gulshan, Lalsingh Khalsa, and Vinod Varghese. "Fractional thermoviscoelastic damping response in a non-simple micro-beam via DPL and KG nonlocality effect." Scientific Temper 16, no. 04 (2025): 4151–64. https://doi.org/10.58414/scientifictemper.2025.16.4.18.

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This study introduces a novel framework for investigating thermoelastic damping (TED) in viscoelastic micro-scale rectangular beams using the Euler-Bernoulli beam theory (EBBT). A two-temperature thermoviscoelastic model is developed, uniquely integrating non-Fourier effects and fractional-order parameters through a generalized thermoelasticity approach with an internal heat source and the dual-phase-lag (DPL) thermal conduction model. This innovative approach addresses the limitations of classical models by incorporating size-dependent effects and spatially varying thermal properties, providing new insights into thermal-mechanical coupling in micro-scale systems. Explicit formulas are derived for the inverse quality factor and frequency shifts, with the study comprehensively analyzing the influence of parameters such as beam thickness, aspect ratios, end constraints, relaxation constants, and fractional-order parameters. Novel findings reveal critical thickness ranges and phase-lag effects that govern energy dissipation and system dynamics. The results highlight the divergence from existing models, emphasizing the importance of nonlocal and fractional-order frameworks for accurately predicting damping and frequency behavior. Practical applications of the study include the optimization of micro-electromechanical systems (MEMS), nano-scale resonators, sensors, and energy-harvesting devices. By bridging theoretical advancements with tangible engineering solutions, this research provides a robust foundation for future exploration in thermal management and micro-scale system design, marking a significant contribution to the field. Graphical representations illustrate the effects of temperature discrepancy factors on damping and frequency shifts in non-simple micro-beams, offering a comprehensive understanding of the interaction between thermal and mechanical responses.

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Rahmanian, Sasan, Shahrokh Hosseini-Hashemi, and Mohammad-Reza Ghazavi. "Analytical Primary Resonance of Size-Dependent Electrostatically Actuated Nanoresonator Under the Effects of Surface Energy and Casmir Force." International Journal of Applied Mechanics 11, no. 01 (2019): 1950002. http://dx.doi.org/10.1142/s1758825119500029.

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This paper investigates the nonlinear vibration of a size-dependent doubly clamped nanoresonator based on modified indeterminate couple-stress theory and Euler–Bernoulli beam theory. Surface effects, dispersion Casimir force, and fringing field effects are considered in the nonlinear model. The electrostatic actuation is a combination of DC and AC voltages and imposed on the nanobeam through one electrode. The governing differential equation of motion is derived using the extended Hamilton’s principle and discretized to a nonlinear ODE using Galerkin’s procedure. The multiple time scale method is applied to the reduced-order model in order to obtain the nanobeam frequency-response curves analytically under small AC voltage loads. The influences of the mentioned parameters are investigated on the primary resonance characteristics of the nanoresonator. It is shown that the application of non-classical continuum theory results in a softening effect on the dynamic response of the system near primary resonance. Moreover, it is concluded that the influence of surface energy on the system dynamic behavior depends on the value of DC voltage load.

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Limkatanyu, Suchart, Worathep Sae-Long, Jaroon Rungamornrat, et al. "BENDING, BUCKLING AND FREE VIBRATION ANALYSES OF NANOBEAM-SUBSTRATE MEDIUM SYSTEMS." Facta Universitatis, Series: Mechanical Engineering 20, no. 3 (2022): 561. http://dx.doi.org/10.22190/fume220506029l.

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This study presents a newly developed size-dependent beam-substrate medium model for bending, buckling, and free-vibration analyses of nanobeams resting on elastic substrate media. The Euler-Bernoulli beam theory describes the beam-section kinematics and the Winkler-foundation model represents interaction between the beam and its underlying substrate medium. The reformulated strain-gradient elasticity theory possessing three non-classical material constants is employed to address the beam-bulk material small-scale effect. The first and second constants is associated with the strain-gradient and couple-stress effects, respectively while the third constant is related to the velocity-gradient effect. The Gurtin-Murdoch surface elasticity theory is adopted to account for the surface-free energy. To obtain the system governing equation as well as corresponding boundary conditions, Hamilton’s principle is called for. Three numerical simulations are presented to characterize the influences of the material small-scale effect, the surface-energy effect, and the surrounding substrate medium on bending, buckling, and free vibration responses of nanobeam-substrate medium systems. The first simulation focuses on the bending response and shows the ability of the proposed model to eliminate the paradoxical characteristic inherent to nanobeam models proposed in the literature. The second and third simulations perform the sensitivity investigation of the system parameters on the buckling load and the natural frequency, respectively. All analytical results reveal that both material small-scale and surface-energy effects consistently stiffen the system response while the velocity-gradient effect weakens the system response. Furthermore, these sized-scale effects are more pronounced when the underlying substrate medium becomes softer.

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Abouelregal, Ahmed E., Khalil M. Khalil, Wael W. Mohammed, and Doaa Atta. "Thermal vibration in rotating nanobeams with temperature-dependent due to exposure to laser irradiation." AIMS Mathematics 7, no. 4 (2022): 6128–52. http://dx.doi.org/10.3934/math.2022341.

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<abstract> <p>Effective classical representations of heterogeneous systems fail to have an effect on the overall response of components on the spatial scale of heterogeneity. This effect may be critical if the effective continuum subjects' scale differs from the material's microstructure scale and then leads to size-dependent effects and other deviations from conventional theories. This paper is concerned with the thermoelastic behavior of rotating nanoscale beams subjected to thermal loading under mechanical thermal loads based on the non-local strain gradient theory (NSGT). Also, a new mathematical model and governing equations were constructed within the framework of the extended thermoelastic theory with phase delay (DPL) and the Euler-Bernoulli beam theory. In contrast to many problems, it was taken into account that the thermal conductivity and specific heat of the material are variable and linearly dependent on temperature change. A specific operator has been entered to convert the nonlinear heat equation into a linear one. Using the Laplace transform method, the considered problem is solved and the expressions of the studied field variables are obtained. The numerical findings demonstrate that a variety of variables, such as temperature change, Coriolis force due to rotation, angular velocity, material properties, and nonlocal length scale parameters, have a significant influence on the mechanical and thermal waves.</p> </abstract>

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Hosseini-Hashemi, Kamiar, Roohollah Talebitooti, and Shahriar Hosseini-Hashemi. "The exact characteristic equation of frequency and mode shape for transverse vibrations of non-uniform and non-homogeneous Euler Bernoulli beam with general non-classical boundary conditions at both ends." Mechanic of Advanced and Smart Materials 3, no. 1 (2023): 1–20. http://dx.doi.org/10.61186/masm.3.1.1.

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Alimoradzadeh, Mehdi, Mehdi Salehi, and Sattar Mohammadi Esfarjani. "Nonlinear Vibration Analysis of Axially Functionally Graded Microbeams Based on Nonlinear Elastic Foundation Using Modified Couple Stress Theory." Periodica Polytechnica Mechanical Engineering 64, no. 2 (2020): 97–108. http://dx.doi.org/10.3311/ppme.11684.

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In this study, a non-classical approach was developed to analyze nonlinear free and forced vibration of an Axially Functionally Graded (AFG) microbeam by means of modified couple stress theory. The beam is considered as Euler-Bernoulli type supported on a three-layered elastic foundation with Von-Karman geometric nonlinearity. Small size effects included in the analysis by considering the length scale parameter. It is assumed that the mass density and elasticity modulus varies continuously in the axial direction according to the power law form. Hamilton's principle was implemented to derive the nonlinear governing partial differential equation concerning associated boundary conditions. The nonlinear partial differential equation was reduced to some nonlinear ordinary differential equations via Galerkin's discretization technique. He's Variational iteration methods were implemented to obtain approximate analytical expressions for the frequency response as well as the forced vibration response of the microbeam with doubly-clamped end conditions. In this study, some factors influencing the forced vibration response were investigated. Specifically, the influence of the length scale parameter, the length of the microbeam, the power index, the Winkler parameter, the Pasternak parameter, and the nonlinear parameter on the nonlinear natural frequency as well as the amplitude of forced response have been investigated.

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Vatankhah, Ramin, Ali Najafi, Hassan Salarieh, and Aria Alasty. "Exact boundary controllability of vibrating non-classical Euler–Bernoulli micro-scale beams." Journal of Mathematical Analysis and Applications 418, no. 2 (2014): 985–97. http://dx.doi.org/10.1016/j.jmaa.2014.03.012.

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Sinira, B. G., B. B. Özhanb, and J. N. Reddyc. "Buckling configurations and dynamic response of buckled Euler-Bernoulli beams with non-classical supports." Latin American Journal of Solids and Structures 11, no. 14 (2014): 2516–36. http://dx.doi.org/10.1590/s1679-78252014001400010.

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Hrytsyna, Olha. "Local gradient Bernoulli–Euler beam model for dielectrics: effect of local mass displacement on coupled fields." Mathematics and Mechanics of Solids, October 19, 2020, 108128652096337. http://dx.doi.org/10.1177/1081286520963374.

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The size-dependent behaviour of a Bernoulli–Euler nanobeam based on the local gradient theory of dielectrics is investigated. By using the variational principle, the linear stationary governing equations of the local gradient beam model and corresponding boundary conditions are derived. In this set of equations the coupling between the strain, the electric field and the local mass displacement is taken into account. Within the presented theory, the process of local mass displacement is associated with the non-diffusive and non-convective mass flux related to the changes in the material microstructure. The solution to the static problem of an elastic cantilever piezoelectric beam subjected to end-point loading is used to investigate the effect of the local mass displacement on the coupled electromechanical fields. The obtained solution is compared to the corresponding ones provided by the classical theory and strain gradient theory. It is shown that the beam deflection predicted by the local gradient theory is smaller than that by the classical Bernoulli–Euler beam theory when the beam thickness is comparable to the material length-scale parameter. The obtained results also indicate that the piezoelectricity has a significant influence on the electromechanical response in a dielectric nanobeam. The presented mathematical model of the dielectric beam may be useful for the study of electromechanical coupling in small-scale piezoelectric structures.

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Wolfgram, Zachary, and Martin Ostoja-Starzewski. "Plate and beam moment-field formulation." Mathematics and Mechanics of Solids, November 10, 2024. http://dx.doi.org/10.1177/10812865241275552.

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Governing equations of the linear elastic Euler–Bernoulli beam and pure bending Kirchhoff–Love plate are developed in a pure stress form with the evolution of the bending moments fields. In the case of elastodynamics, the governing equations are used to find mode restrictions of vibration within both structures. These vibration restrictions are then compared with the solutions found through the displacement formulation finding both forms to give the same solution under the same boundary conditions. The waves are assumed to be harmonic with only spatial contributions being examined, validating the stress approach. More advanced cases involving anisotropic and non-classical boundary conditions are also explored to demonstrate further implications of following the stress evolution. Elastostatic plates are examined under the compatibility conditions to be met for moment fields to possess uniqueness and to show the CLM (Cherkaev–Lurie–Milton) invariance of the inhomogeneous plane-stress problem.

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Togun, Necla, and Süleyman M. Bağdatli. "Application of Modified Couple-Stress Theory to Nonlinear Vibration Analysis of Nanobeam with Different Boundary Conditions." Journal of Vibration Engineering & Technologies, March 31, 2024. http://dx.doi.org/10.1007/s42417-024-01294-3.

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Abstract Purpose In the present study, the nonlinear vibration analysis of a nanoscale beam with different boundary conditions named as simply supported, clamped-clamped, clamped-simple and clamped-free are investigated numerically. Methods Nanoscale beam is considered as Euler-Bernoulli beam model having size-dependent. This non-classical nanobeam model has a size dependent incorporated with the material length scale parameter. The equation of motion of the system and the related boundary conditions are derived using the modified couple stress theory and employing Hamilton’s principle. Multiple scale method is used to obtain the approximate analytical solution. Result Numerical results by considering the effect of the ratio of beam height to the internal material length scale parameter, h/l and with and without the Poisson effect, υ are graphically presented and tabulated. Conclusion We remark that small size effect and poisson effect have a considerable effect on the linear fundamental frequency and the vibration amplitude. In order to show the accuracy of the results obtained, comparison study is also performed with existing studies in the literature.

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Leslie Darien Pérez Fernández, Douglas Machado da Silva, and Julián Bravo-Castillero. "Asymptotic homogenization of a mechanical equilibrium problem of a functionally-graded Euler-Bernoulli beam with non-periodic microstructure." Ibero-Latin American Congress on Computational Methods in Engineering (CILAMCE), December 2, 2024. https://doi.org/10.55592/cilamce.v6i06.8115.

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To the best of our knowledge, the few classical applications of Keller's two-space method of non-periodic asymptotic homogenization are related to the effective behavior of heterogeneous media in the context of poroelasticity considering fluid flow and saturation. We believe that this is due to the alternative common approach of approximating random or non-periodic microstructures via the periodic replication of a representative volume element, as periodic structures are, generally speaking, much more tractable mathematically and computationally. However, more than 40 years later, a number of preliminary results of applications of the two-space method has arisen on various areas, namely, effective behavior of composite or functionally-graded bars, approximate solution of the electroencephalogram forward problem for neural imaging activity, and modeling of atmospheric pollutant dispersion. These recent applications deal with second-order elliptic or parabolic equations. In this contribution, we present the application of the two-space method to a mechanical equilibrium problem of a functionally-graded Euler-Bernoulli beam with non-periodic microstructure, which relies on a fourth-order elliptic equation. To the best of our knowledge, homogenization of fourth-order equations has been considered only in the periodic case.

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Banerjee, J. Ranjan, Stanislav O. Papkov, Thuc P. Vo, and Isaac Elishakoff. "Dynamic stiffness formulation for a micro beam using Timoshenko–Ehrenfest and modified couple stress theories with applications." Journal of Vibration and Control, October 12, 2021, 107754632110482. http://dx.doi.org/10.1177/10775463211048272.

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Several models within the framework of continuum mechanics have been proposed over the years to solve the free vibration problem of micro beams. Foremost amongst these are those based on non-local elasticity, classical couple stress, gradient elasticity and modified couple stress theories. Many of these models retain the basic features of the Bernoulli–Euler or Timoshenko–Ehrenfest theories, but they introduce one or more material scale length parameters to tackle the problem. The work described in this paper deals with the free vibration problems of micro beams based on the dynamic stiffness method, through the implementation of the modified couple stress theory in conjunction with the Timoshenko–Ehrenfest theory. The main advantage of the modified couple stress theory is that unlike other models, it uses only one material length scale parameter to account for the smallness of the structure. The current research is accomplished first by solving the governing differential equations of motion of a Timoshenko–Ehrenfest micro beam in free vibration in closed analytical form. The dynamic stiffness matrix of the beam is then formulated by relating the amplitudes of the forces to those of the corresponding displacements at the ends of the beam. The theory is applied using the Wittrick–Williams algorithm as solution technique to investigate the free vibration characteristics of Timoshenko–Ehrenfest micro beams. Natural frequencies and mode shapes of several examples are presented, and the effects of the length scale parameter on the free vibration characteristics of Timoshenko–Ehrenfest micro beams are demonstrated.

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Warminski, Jerzy, Lukasz Kloda, Jaroslaw Latalski, Andrzej Mitura, and Marcin Kowalczuk. "Nonlinear vibrations and time delay control of an extensible slowly rotating beam." Nonlinear Dynamics, December 29, 2020. http://dx.doi.org/10.1007/s11071-020-06079-3.

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AbstractNonlinear dynamics of a rotating flexible slender beam with embedded active elements is studied in the paper. Mathematical model of the structure considers possible moderate oscillations thus the motion is governed by the extended Euler–Bernoulli model that incorporates a nonlinear curvature and coupled transversal–longitudinal deformations. The Hamilton’s principle of least action is applied to derive a system of nonlinear coupled partial differential equations (PDEs) of motion. The embedded active elements are used to control or reduce beam oscillations for various dynamical conditions and rotational speed range. The control inputs generated by active elements are represented in boundary conditions as non-homogenous terms. Classical linear proportional (P) control and nonlinear cubic (C) control as well as mixed ($$P-C$$ P - C ) control strategies with time delay are analyzed for vibration reduction. Dynamics of the complete system with time delay is determined analytically solving directly the PDEs by the multiple timescale method. Natural and forced vibrations around the first and the second mode resonances demonstrating hardening and softening phenomena are studied. An impact of time delay linear and nonlinear control methods on vibration reduction for different angular speeds is presented.

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Mohammad, Roslina, Astuty Amrin, and Sallehuddin Muhamad. "THEORETICAL OF DYNAMIC LOADING AGAINST WATER-FILLED SIMPLY SUPPORTED PIPES." Jurnal Teknologi 75, no. 11 (2015). http://dx.doi.org/10.11113/jt.v75.5343.

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The primary aim of this study had been to investigate the effects of water-filled flow on the transient response of a simply supported pipe subjected to dynamically applied loading. The importance of this study is manifested in numerous applications, such as oil and gas transportations, where dynamic loading can be the result of an accident. The classical Bernoulli-Euler beam theory was adopted to describe the dynamic behavior of an elastic pipe and a new governing equation of a long pipe transporting gas or liquid was derived. This governing equation incorporated the effects of inertia, centrifugal, and Coriolis forces due to the flowing water. This equation can be normalized to demonstrate that only two non-dimensional parameters governed the static and the dynamic responses of the system incorporating a pipe and flowing water. The transient response of this system was investigated based on a standard perturbation approach. Moreover, it had been demonstrated that the previous dynamic models, which largely ignored the internal flow effects and interactions between the flow and the structure, normally produced a large error and are inapplicable to the analysis of many practical situations. One interesting effect identified was that at certain flow ratio, the system became dynamically unstable and any, even very small, external perturbation led to a growing unstable dynamic behavior. Such behavior, which is called pipe whip, is well-known to everyone who waters a garden using a flexible long hose.

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Afras, Abderrachid, and Abdelouafi El Ghoulbzouri. "Effects of non-classical boundary conditions on the free vibration response of a cantilever Euler-Bernoulli beams." Diagnostyka, January 4, 2023, 1–13. http://dx.doi.org/10.29354/diag/158075.

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