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Journal articles on the topic 'Euler-Bernoulli beam equation'

Author: Grafiati

Published: 4 June 2021

Last updated: 18 February 2022

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1

LIM, TEIK-CHENG. "ANALYSIS OF AUXETIC BEAMS AS RESONANT FREQUENCY BIOSENSORS." Journal of Mechanics in Medicine and Biology 12, no. 05 (2012): 1240027. http://dx.doi.org/10.1142/s0219519412400271.

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The mechanics of beam vibration is of fundamental importance in understanding the shift of resonant frequency of microcantilever and nanocantilever sensors. Unlike the simpler Euler–Bernoulli beam theory, the Timoshenko beam theory takes into consideration rotational inertia and shear deformation. For the case of microcantilevers and nanocantilevers, the minute size, and hence low mass, means that the topmost deviation from the Euler–Bernoulli beam theory to be expected is shear deformation. This paper considers the extent of shear deformation for varying Poisson's ratio of the beam material,

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2

Naz, R., and F. M. Mahomed. "Dynamic Euler-Bernoulli Beam Equation: Classification and Reductions." Mathematical Problems in Engineering 2015 (2015): 1–7. http://dx.doi.org/10.1155/2015/520491.

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We study a dynamic fourth-order Euler-Bernoulli partial differential equation having a constant elastic modulus and area moment of inertia, a variable lineal mass densityg(x), and the applied load denoted byf(u), a function of transverse displacementu(t,x). The complete Lie group classification is obtained for different forms of the variable lineal mass densityg(x)and applied loadf(u). The equivalence transformations are constructed to simplify the determining equations for the symmetries. The principal algebra is one-dimensional and it extends to two- and three-dimensional algebras for an arb

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3

Czyczula, Wlodzimierz, Piotr Koziol, and Dorota Blaszkiewicz. "On the Equivalence between Static and Dynamic Railway Track Response and on the Euler-Bernoulli and Timoshenko Beams Analogy." Shock and Vibration 2017 (2017): 1–13. http://dx.doi.org/10.1155/2017/2701715.

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The paper tries to clarify the problem of solution and interpretation of railway track dynamics equations for linear models. Set of theorems is introduced in the paper describing two types of equivalence: between static and dynamic track response under moving load and between the dynamic response of track described by both the Euler-Bernoulli and Timoshenko beams. The equivalence is clarified in terms of mathematical method of solution. It is shown that inertia element of rail equation for the Euler-Bernoulli beam and constant distributed load can be considered as a substitute axial force mult

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4

Fatima, Aeeman, Fazal M. Mahomed, and Chaudry Masood Khalique. "Noether symmetries and exact solutions of an Euler–Bernoulli beam model." International Journal of Modern Physics B 30, no. 28n29 (2016): 1640011. http://dx.doi.org/10.1142/s0217979216400117.

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In this paper, a Noether symmetry analysis is carried out for an Euler–Bernoulli beam equation via the standard Lagrangian of its reduced scalar second-order equation which arises from the standard Lagrangian of the fourth-order beam equation via its Noether integrals. The Noether symmetries corresponding to the reduced equation is shown to be the inherited Noether symmetries of the standard Lagrangian of the beam equation. The corresponding Noether integrals of the reduced Euler–Lagrange equations are deduced which remarkably allows for three families of new exact solutions of the static beam

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5

Zhang, Jin-lun, Liao-jun Zhang, Ren-yu Ge, Li Yang, and Jun-wu Xia. "Study on Natural Frequencies of Transverse Free Vibration of Functionally Graded Axis Beams by the Differential Quadrature Method." Acta Acustica united with Acustica 105, no. 6 (2019): 1095–104. http://dx.doi.org/10.3813/aaa.919388.

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Functionally gradient materials with special mechanical characteristics are more and more widely used in engineering. The functionally graded beam is one of the commonly used components to bear forces in the structure. Accurate analysis of the dynamic characteristics of the axially functionally graded (AFG) beam plays a vital role in the design and safe operation of the whole structure. Based on the Euler-Bernoulli beam theory (EBT), the characteristic equation of transverse free vibration for the AFG Euler-Bernoulli beam with variable cross-section is obtained in the present work, and the gov

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6

Tekin, Ibrahim, and He Yang. "INVERSE PROBLEM FOR THE TIME-FRACTIONAL EULER-BERNOULLI BEAM EQUATION." Mathematical Modelling and Analysis 26, no. 3 (2021): 503–18. http://dx.doi.org/10.3846/mma.2021.13289.

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In this paper, the classical Euler-Bernoulli beam equation is considered by utilizing fractional calculus. Such an equation is called the time-fractional EulerBernoulli beam equation. The problem of determining the time-dependent coefficient for the fractional Euler-Bernoulli beam equation with homogeneous boundary conditions and an additional measurement is considered, and the existence and uniqueness theorem of the solution is proved by means of the contraction principle on a sufficiently small time interval. Numerical experiments are also provided to verify the theoretical findings.

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7

SHEN, HUI-SHEN. "A NOVEL TECHNIQUE FOR NONLINEAR ANALYSIS OF BEAMS ON TWO-PARAMETER ELASTIC FOUNDATIONS." International Journal of Structural Stability and Dynamics 11, no. 06 (2011): 999–1014. http://dx.doi.org/10.1142/s0219455411004440.

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Postbuckling, nonlinear bending, and nonlinear vibration analyses are presented for a simply supported Euler–Bernoulli beam resting on a two-parameter elastic foundation. The nonlinear model is introduced by using the exact expression of the curvature. Two kinds of end conditions, namely movable and immovable, are considered. The nonlinear equation of motion, including beam–foundation interaction, is derived separately for these two kinds of end conditions. The analysis uses a two-step perturbation technique to determine the postbuckling equilibrium paths of an axially loaded beam, the static

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8

Tang, Shao-Qiang, and Eduard G. Karpov. "Artificial boundary conditions for Euler-Bernoulli beam equation." Acta Mechanica Sinica 30, no. 5 (2014): 687–92. http://dx.doi.org/10.1007/s10409-014-0089-7.

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9

Baysal, Onur, and Alemdar Hasanov. "Solvability of the clamped Euler–Bernoulli beam equation." Applied Mathematics Letters 93 (July 2019): 85–90. http://dx.doi.org/10.1016/j.aml.2019.02.006.

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10

Khaldi, Rabah, and Assia Guezane-Lakoud. "On generalized nonlinear Euler-Bernoulli Beam type equations." Acta Universitatis Sapientiae, Mathematica 10, no. 1 (2018): 90–100. http://dx.doi.org/10.2478/ausm-2018-0008.

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Abstract This paper is devoted to the study of a nonlinear Euler-Bernoulli Beam type equation involving both left and right Caputo fractional derivatives. Differently from the approaches of the other papers where they established the existence of solution for the linear Euler-Bernoulli Beam type equation numerically, we use the lower and upper solutions method with some new results on the monotonicity of the right Caputo derivative. Furthermore, we give the explicit expression of the upper and lower solutions. A numerical example is given to illustrate the obtained results.

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11

Adair, Desmond, and Martin Jaeger. "A power series solution for rotating nonuniform Euler–Bernoulli cantilever beams." Journal of Vibration and Control 24, no. 17 (2017): 3855–64. http://dx.doi.org/10.1177/1077546317714183.

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A systematic procedure is developed for studying the dynamic response of a rotating nonuniform Euler–Bernoulli beam with an elastically restrained root. To find the solution, a novel approach is used in that the fourth-order differential equation describing the vibration problem is first written as a first-order matrix differential equation, which is then solved using the power series method. The method can be used to obtain an approximate solution of vibration problems for nonuniform Euler–Bernoulli beams. Specifically, numerical examples are presented here to demonstrate the usefulness of th

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12

Torabi, K., H. Afshari, and E. Zafari. "Transverse Vibration of Non-Uniform Euler-Bernoulli Beam, Using Differential Transform Method (DTM)." Applied Mechanics and Materials 110-116 (October 2011): 2400–2405. http://dx.doi.org/10.4028/www.scientific.net/amm.110-116.2400.

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Analysis of transverse vibration of beams is presented in this paper. Unfortunately, complexities which appear in solving differential equation of transverse vibration of non-uniform beams, limit analytical solution to some special cases, so that the numerical method is presented. DTM is a numerical method for solving linear and some non-linear, ordinary and partial differential equations. In this paper, this technique has been applied for solving differential equation of transverse vibration of conical Euler-Bernoulli beam. Natural circular frequencies and mode shapes have been calculated. Co

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13

Copetti, Rosemaira Dalcin, Julio C. R. Claeyssen, and Teresa Tsukazan. "Modal Formulation of Segmented Euler-Bernoulli Beams." Mathematical Problems in Engineering 2007 (2007): 1–18. http://dx.doi.org/10.1155/2007/36261.

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We consider the obtention of modes and frequencies of segmented Euler-Bernoulli beams with internal damping and external viscous damping at the discontinuities of the sections. This is done by following a Newtonian approach in terms of a fundamental response of stationary beams subject to both types of damping. The use of a basis generated by the fundamental solution of a differential equation of fourth-order allows to formulate the eigenvalue problem and to write the modes shapes in a compact manner. For this, we consider a block matrix that carries the boundary conditions and intermediate co

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14

Johnpillai, A. G., K. S. Mahomed, C. Harley, and F. M. Mahomed. "Noether Symmetry Analysis of the Dynamic Euler-Bernoulli Beam Equation." Zeitschrift für Naturforschung A 71, no. 5 (2016): 447–56. http://dx.doi.org/10.1515/zna-2015-0292.

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AbstractWe study the fourth-order dynamic Euler-Bernoulli beam equation from the Noether symmetry viewpoint. This was earlier considered for the Lie symmetry classification. We obtain the Noether symmetry classification of the equation with respect to the applied load, which is a function of the dependent variable of the underlying equation. We find that the principal Noether symmetry algebra is two-dimensional when the load function is arbitrary and extends for linear and power law cases. For all cases, for each of the Noether symmetries associated with the usual Lagrangian, we construct cons

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15

Baglan, Irem. "Fourier Method for Inverse Coefficient Euler-Bernoulli Beam Equation." Communications Faculty Of Science University of Ankara Series A1Mathematics and Statistics 68, no. 1 (2018): 514–27. http://dx.doi.org/10.31801/cfsuasmas.431883.

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16

Zamorska, Izabela. "Solution of differential equation for the Euler-Bernoulli beam." Journal of Applied Mathematics and Computational Mechanics 13, no. 4 (2014): 157–62. http://dx.doi.org/10.17512/jamcm.2014.4.21.

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17

Ávalos, G. Gómez, J. Mun͂oz Rivera, and Z. Liu. "Gevrey Class of Locally Dissipative Euler--Bernoulli Beam Equation." SIAM Journal on Control and Optimization 59, no. 3 (2021): 2174–94. http://dx.doi.org/10.1137/20m1312800.

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18

Shimpi, Rameshchandra P., Rajesh A. Shetty, and Anirban Guha. "A simple single variable shear deformation theory for a rectangular beam." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 231, no. 24 (2016): 4576–91. http://dx.doi.org/10.1177/0954406216670682.

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This paper proposes a simple single variable shear deformation theory for an isotropic beam of rectangular cross-section. The theory involves only one fourth-order governing differential equation. For beam bending problems, the governing equation and the expressions for the bending moment and shear force of the theory are strikingly similar to those of Euler–Bernoulli beam theory. For vibration and buckling problems, the Euler–Bernoulli beam theory governing equation comes out as a special case when terms pertaining to the effects of shear deformation are ignored from the governing equation of

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19

Sulbhewar, Litesh N., and P. Raveendranath. "Geometric Effects on the Accuracy of Euler-Bernoulli Piezoelectric Smart Beam Finite Elements." Advanced Materials Research 984-985 (July 2014): 1063–73. http://dx.doi.org/10.4028/www.scientific.net/amr.984-985.1063.

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Numerical analysis of piezoelectric smart beams plays an important role in the design of smart beam based control systems. In general, smart beams are thin and Euler-Bernoulli piezoelectric beam element is widely used for their structural analysis. Accuracy of Euler-Bernoulli piezoelectric beam element depends on the appropriate assumptions for electric potential involved in the formulation. Most of the Euler-Bernoulli piezoelectric beam finite elements available in the literature assume linear through-thickness potential distribution. It is shown that the accuracy of these conventional formul

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20

Adhikari, S., M. I. Friswell, and Y. Lei. "Modal Analysis of Nonviscously Damped Beams." Journal of Applied Mechanics 74, no. 5 (2006): 1026–30. http://dx.doi.org/10.1115/1.2712315.

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Linear dynamics of Euler–Bernoulli beams with nonviscous nonlocal damping is considered. It is assumed that the damping force at a given point in the beam depends on the past history of velocities at different points via convolution integrals over exponentially decaying kernel functions. Conventional viscous and viscoelastic damping models can be obtained as special cases of this general damping model. The equation of motion of the beam with such a general damping model results in a linear partial integro-differential equation. Exact closed-form equations of the natural frequencies and mode sh

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21

Abassi, Wafik, Adil El Baroudi, and Fulgence Razafimahery. "Vibration Analysis of Euler-Bernoulli Beams Partially Immersed in a Viscous Fluid." Physics Research International 2016 (February 21, 2016): 1–14. http://dx.doi.org/10.1155/2016/6761372.

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The vibrational characteristics of a microbeam are well known to strongly depend on the fluid in which the beam is immersed. In this paper, we present a detailed theoretical study of the modal analysis of microbeams partially immersed in a viscous fluid. A fixed-free microbeam vibrating in a viscous fluid is modeled using the Euler-Bernoulli equation for the beams. The unsteady Stokes equations are solved using a Helmholtz decomposition technique in a two-dimensional plane containing the microbeams cross sections. The symbolic software Mathematica is used in order to find the coupled vibration

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22

Ali Nojoumian, Mohammad, Ramin Vatankhah, and Hassan Salarieh. "Adaptive boundary control of the size-dependent behavior of Euler–Bernoulli micro-beams with unknown parameters and varying disturbance." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 231, no. 10 (2015): 1777–90. http://dx.doi.org/10.1177/0954406215622651.

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In this paper, modeling and vibration control of a strain-gradient clamped-free Euler–Bernoulli micro-beam exposed to varying disturbance is studied. A strain-gradient model of the Euler–Bernoulli micro-beam is represented in this paper and consisted of one partial differential equation and six ordinary equations as governing motion equation and boundary conditions, respectively. A boundary controller is proposed to suppress the system’s vibration. The controller is derived based on the direct Lyapunov method. An adaptation law is devised to assure system’s stability under parametric uncertain

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23

Foroudastan, S., John Peddieson, and M. C. Holman. "Application of a Unified Viscoplastic Model to Simulation of Autoclave Age Forming." Journal of Engineering Materials and Technology 114, no. 1 (1992): 71–76. http://dx.doi.org/10.1115/1.2904143.

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A unified viscoplastic constitutive equation is combined with Bernoulli/Euler beam equations to create a model of autoclave age forming of beam specimens. Predictions based on the model are obtained numerically and compared with experimental data. Encouraging agreement is found between simulations and observations. Issues related to extensions of the methodology to more complicated geometries are discussed.

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24

Song, Jiang Yong. "An Elliptic Integral Solution to the Multiple Inflections Large Deflection Beams in Compliant Mechanisms." Advanced Materials Research 971-973 (June 2014): 349–52. http://dx.doi.org/10.4028/www.scientific.net/amr.971-973.349.

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In this paper, a solution based on the elliptic integrals is proposed for solving multiples inflection points large deflection. Application of the Bernoulli Euler equations of compliant mechanisms with large deflection equation of beam is obtained ,there is no inflection point and inflection points in two cases respectively. The elliptic integral solution which is the most accurate method at present for analyzing large deflections of cantilever beams in compliant mechanisms.

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25

Xia, Yong Jun, and Qian Miao. "Large Deformation Geometric Nonlinear Beam Element Based on U.L. Format." Advanced Materials Research 446-449 (January 2012): 3596–603. http://dx.doi.org/10.4028/www.scientific.net/amr.446-449.3596.

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Based on the geometric deformation of the Euler-Bernoulli beam element, the geometric nonlinear Euler-Bernoulli beam element based on U.L. formulation is derived. The element’s transverse first-order displacement field is constructed using the cubic Hermite interpolation polynomial, and the first-order Lagrange interpolation polynomial is used for the axial displacement field. Then the additional displacements induced from the rotation of the elemental are included into the transverse and longitudinal displacement fields, so those displacement fields are expressed as the quadratic function of

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26

Fan, Tao. "Variational Principle of Carbon Nanotubes with Temperature Changes Based on Nonlocal Euler-Bernoulli Beam Model." Key Engineering Materials 452-453 (November 2010): 785–88. http://dx.doi.org/10.4028/www.scientific.net/kem.452-453.785.

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In this paper, the CNTS are considered as the Euler-Bernoulli beams which have been used in many references about the CNTS. Taken the thermal-mechanical coupling and small scale effect into account, the variational principle for the CNTS is presented by the variational integral method. With the derivation of the varitional principle, the vibration governing equation is illustrated, which will be benefit for the numerical simulation with finite element method in further investigations. From the stationary value conditions deduced by the variational principle, the influences of the temperature c

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27

Babaei, Alireza, and Masoud Arabghahestani. "Free Vibration Analysis of Rotating Beams Based on the Modified Couple Stress Theory and Coupled Displacement Field." Applied Mechanics 2, no. 2 (2021): 226–38. http://dx.doi.org/10.3390/applmech2020014.

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In this paper, transverse vibration analysis of rotating micro-beam is investigated based on the modified couple stress theory. The simply-supported micro-beam is modeled utilizing Euler-Bernoulli and Timoshenko beam theories. The system is rotating around a fixed axis perpendicular to the axial direction of the beam. For the first time, displacement filed is introduced as a coupled field to the translational field. In other words, the mentioned rotational displacement field is expressed as a proportional function of translational displacement field using first (axial), second (lateral), and t

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28

Mao, Qibo. "Application of Adomian Modified Decomposition Method to Free Vibration Analysis of Rotating Beams." Mathematical Problems in Engineering 2013 (2013): 1–10. http://dx.doi.org/10.1155/2013/284720.

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The Adomian modified decomposition method (AMDM) is employed in this paper for dynamic analysis of a rotating Euler-Bernoulli beam under various boundary conditions. Based on AMDM, the governing differential equation for the rotating beam becomes a recursive algebraic equation. By using the boundary condition equations, the dimensionless natural frequencies and corresponding mode shapes can be easily obtained simultaneously. The computed results for different boundary conditions as well as different offset length and rotational speeds are presented. The accuracy is assured from the convergence

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29

Halder, Amlan Kanti, Andronikos Paliathanasis, and Peter Gavin Lawrence Leach. "SIMILARITY SOLUTIONS AND CONSERVATION LAWS FOR THE BEAM EQUATIONS: A COMPLETE STUDY." Acta Polytechnica 60, no. 2 (2020): 98–110. http://dx.doi.org/10.14311/ap.2020.60.0098.

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We study the similarity solutions and we determine the conservation laws of various forms of beam equations, such as Euler-Bernoulli, Rayleigh and Timoshenko-Prescott. The travelling-wave reduction leads to solvable fourth-order odes for all the forms. In addition, the reduction based on the scaling symmetry for the Euler-Bernoulli form leads to certain odes for which there exists zero symmetries. Therefore, we conduct the singularity analysis to ascertain the integrability. We study two reduced odes of second and third orders. The reduced second-order ode is a perturbed form of Painlevé-Ince

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30

Ren, Yan-Ming, and Hai Qing. "Bending and Buckling Analysis of Functionally Graded Euler–Bernoulli Beam Using Stress-Driven Nonlocal Integral Model with Bi-Helmholtz Kernel." International Journal of Applied Mechanics 13, no. 04 (2021): 2150041. http://dx.doi.org/10.1142/s1758825121500411.

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Static bending and elastic buckling of Euler–Bernoulli beam made of functionally graded (FG) materials along thickness direction is studied theoretically using stress-driven integral model with bi-Helmholtz kernel, where the relation between nonlocal stress and strain is expressed as Fredholm type integral equation of the first kind. The differential governing equation and corresponding boundary conditions are derived with the principle of minimum potential energy. Several nominal variables are introduced to simplify differential governing equation, integral constitutive equation and boundary

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31

Zhou, Cui Lian. "Riesz Basis Generation of the Euler-Bernoulli Beam Equation with Boundary Energy Dissipation." Advanced Materials Research 433-440 (January 2012): 123–27. http://dx.doi.org/10.4028/www.scientific.net/amr.433-440.123.

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In this paper, the Riesz basis generation of the Euler-Bernoulli beam equation with with boundary energy dissipation is studied. Using the regular property of the boundary conditions, it is shown that the Riesz basis property holds

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32

El baroudi, Adil, and Fulgence Razafimahery. "Transverse vibration analysis of Euler-Bernoulli beam carrying point masse submerged in fluid media." International Journal of Engineering & Technology 4, no. 2 (2015): 369. http://dx.doi.org/10.14419/ijet.v4i2.4570.

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In the present paper, an analytical method is developed to investigate the effects of added mass on natural frequencies and mode shapes of Euler-Bernoulli beams carrying concentrated masse at arbitrary position submerged in a fluid media. A fixed-fixed beams carrying concentrated masse vibrating in a fluid is modeled using the Bernoulli-Euler equation for the beams and the acoustic equation for the fluid. The symbolic software Mathematica is used in order to find the coupled vibration frequencies of a beams with two portions. The frequency equation is deduced and analytically solved. The finit

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33

Mirghaffari, Ali, and Behrooz Rahmani. "Active vibration control of carbon nanotube reinforced composite beams." Transactions of the Institute of Measurement and Control 39, no. 12 (2016): 1851–63. http://dx.doi.org/10.1177/0142331216649431.

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In this paper, active vibration control of carbon nanotube reinforced composite beams subjected to a temperature rise is studied. For this purpose, piezoelectric patches are used as sensors to measure the displacement of the beam and as actuators to implement control forces. The governing equation of motion of this beam is derived from the Euler–Bernoulli theory and Hamilton’s principle. Galerkin’s method is utilized to obtain the temporal ordinary differential equations. An optimal observer-based output feedback controller is designed by the linear quadratic regulator ( LQR) methodology to en

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34

Bassuony, M. A. "A Legendre-Laguerre-Galerkin Method for Uniform Euler-Bernoulli Beam Equation." East Asian Journal on Applied Mathematics 8, no. 2 (2018): 280–95. http://dx.doi.org/10.4208/eajam.060717.140118a.

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35

Marín, Francisco J., Jesús Martínez-Frutos, and Francisco Periago. "Robust Averaged Control of Vibrations for the Bernoulli–Euler Beam Equation." Journal of Optimization Theory and Applications 174, no. 2 (2017): 428–54. http://dx.doi.org/10.1007/s10957-017-1128-x.

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36

Blaszczyk, Tomasz. "Analytical and numerical solution of the fractional Euler–Bernoulli beam equation." Journal of Mechanics of Materials and Structures 12, no. 1 (2017): 23–34. http://dx.doi.org/10.2140/jomms.2017.12.23.

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37

Bokhari, Ashfaque H., F. M. Mahomed, and F. D. Zaman. "Symmetries and integrability of a fourth-order Euler–Bernoulli beam equation." Journal of Mathematical Physics 51, no. 5 (2010): 053517. http://dx.doi.org/10.1063/1.3377045.

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38

Khatami, Iman, Mohsen Zahedi, Abolfazl Zahedi, and Mohammad Yaghoub Abdollahzadeh Jamalabadi. "Akbari–Ganji Method for Solving Equations of Euler–Bernoulli Beam with Quintic Nonlinearity." Acoustics 3, no. 2 (2021): 337–53. http://dx.doi.org/10.3390/acoustics3020023.

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In many real word applications, beam has nonlinear transversely vibrations. Solving nonlinear beam systems is complicated because of the high dependency of the system variables and boundary conditions. It is important to have an accurate parametric analysis for understanding the nonlinear vibration characteristics. This paper presents an approximate solution of a nonlinear transversely vibrating beam with odd and even nonlinear terms using the Akbari–Ganji Method (AGM). This method is an effective approach to solve nonlinear differential equations. AGM is already used in the heat transfer scie

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39

Kim, Cheol Woong, Sang Heon Lee, and Kee Joo Kim. "Modified Bernoulli-Euler Laminate Beam Theory Using Total Effective Moment in LIPCA." Key Engineering Materials 334-335 (March 2007): 413–16. http://dx.doi.org/10.4028/www.scientific.net/kem.334-335.413.

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The applicable bending moment equation of LIPCA is necessary even though the fiber layer ply orientation is changed. The aim of this research is to evaluate the relationship between the total effective moment (ME) and Bernoulli-Euler bending moment (M) when the ply orientations of unidirectional CFRP, which is one of the various laminate configurations in LIPCA, are changed. Since the related previous equation between the performance stroke range (h) and the radius of curvature (ρ) was just applicable to the CFRP ply orientation [0], it will be modified using these results. The related modifie

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40

Diyaroglu, Cagan, Erkan Oterkus, and Selda Oterkus. "An Euler–Bernoulli beam formulation in an ordinary state-based peridynamic framework." Mathematics and Mechanics of Solids 24, no. 2 (2017): 361–76. http://dx.doi.org/10.1177/1081286517728424.

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Every object in the world has a three-dimensional geometrical shape and it is usually possible to model structures in a three-dimensional fashion, although this approach can be computationally expensive. In order to reduce computational time, the three-dimensional geometry can be simplified as a beam, plate or shell type of structure depending on the geometry and loading. This simplification should also be accurately reflected in the formulation that is used for the analysis. In this study, such an approach is presented by developing an Euler–Bernoulli beam formulation within ordinary state-ba

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41

Lu, Nian Li, and Li Xia Meng. "The Element Stiffness Matrix of a Tapered Beam with Effects of Shear Deformation and its Stability Application." Advanced Materials Research 308-310 (August 2011): 1383–88. http://dx.doi.org/10.4028/www.scientific.net/amr.308-310.1383.

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Starting from second-order effect, the governing differential equation of a tapered beam considering effects of axial force and shear deformation is established, the exact element stiffness matrix of a tapered beam with effects of shear deformation is proposed, and whose inertia moment is quadratic along the longitudinal axis. When the effect of shear deformation is ignored, the proposed stiffness matrix will degenerate into the Bernoulli-Euler ones. By using of the presented stiffness matrix, the stability and nonlinear of structures which contain tapered elements can be analyzed. Finally, th

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42

Sun, Yuxin, Shoubin Liu, Zhangheng Rao, Yuhang Li, and Jialing Yang. "Thermodynamic Response of Beams on Winkler Foundation Irradiated by Moving Laser Pulses." Symmetry 10, no. 8 (2018): 328. http://dx.doi.org/10.3390/sym10080328.

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In this paper, the exact analytical solutions are developed for the thermodynamic behavior of an Euler-Bernoulli beam resting on an elastic foundation and exposed to a time decaying laser pulse that scans over the beam with a uniform velocity. The governing equations, namely the heat conduction equation and the vibration equation are solved using the Green’s function approach. The temporal and special distributions of temperature, deflection, strain, and the energy absorbed by the elastic foundation are calculated. The effects of the laser motion speed, the modulus of elastic foundation reacti

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Mao, Qibo. "Free Vibrations of Spinning Beams Under Nonclassical Boundary Conditions Using Adomian Modified Decomposition Method." International Journal of Structural Stability and Dynamics 14, no. 07 (2014): 1450027. http://dx.doi.org/10.1142/s0219455414500278.

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This study employs the Adomian modified decomposition method (AMDM) for the dynamic analysis of Euler–Bernoulli beams spinning about their longitudinal axes under various boundary conditions. Based on the AMDM, the governing differential equations for the spinning beam become a recursive algebraic equation system. By using the boundary condition equations, the natural frequencies can be readily obtained. The computed results under different classical and nonclassical boundary conditions as well as spinning speeds are presented. The accuracy is assured from comparison with published results. It

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Streator, J. L., and D. B. Bogy. "Accounting for Transducer Dynamics in the Measurement of Friction." Journal of Tribology 114, no. 1 (1992): 86–94. http://dx.doi.org/10.1115/1.2920873.

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In many studies of sliding interfaces, the measurement of friction is complicated by quasi-harmonic vibrations of the transducer system. An analytical technique is introduced which accounts for the dynamic characteristics of a force transducer under periodic excitation, and is used to compute the forcing function in the sliding interface. The force transducer is modeled as an elastic cantilever-beam with an attached rigid mass. The forcing function is obtained by solving the time-dependent, fourth-order partial differential equation of Euler-Bernoulli beam theory. The solution is facilitated b

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Xue, Yan Xia, and Zhen Chao Su. "Dynamical Analysis of a Cantilever Column with a Tip Mass Subjected to Subtangential Follower Force." Applied Mechanics and Materials 427-429 (September 2013): 346–49. http://dx.doi.org/10.4028/www.scientific.net/amm.427-429.346.

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Based on the theory of Bernoulli-Euler beam and d Alembert principle, the differential equation of a cantilever column with a tip mass subjected to a subtangential follower force is built, the solution of the differential equation under the specific boundary conditions is found, frequency equation is formed for computing the system frequencies, several cases of this system is investigated.

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Jafari, Mohammad, Harijono Djojodihardjo, and Kamarul Arifin Ahmad. "Vibration Analysis of a Cantilevered Beam with Spring Loading at the Tip as a Generic Elastic Structure." Applied Mechanics and Materials 629 (October 2014): 407–13. http://dx.doi.org/10.4028/www.scientific.net/amm.629.407.

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Although fundamental, vibration of a cantilevered Euler-Bernoulli beam with spring attached at the tip is not found in literatures and is here solved analytically and numerically using finite element approach. The equation of motion of the beam is obtained by using Hamilton’s principle. Finite element method is utilized to write in-house program for the free vibration of the beam. Results show plausible agreements.

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Khang, Nguyen Van, Nguyen Phong Dien, and Nguyen Thi Van Huong. "On the compression softening effect of prestressed beams." Vietnam Journal of Mechanics 28, no. 3 (2006): 145–54. http://dx.doi.org/10.15625/0866-7136/28/3/5574.

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The main objective of the present paper is to study the transverse vibration of the prestressed beams. The differential equation of the transverse vibration of the Euler-Bernoulli beam is developed, in which the initial axial strain in every cross section of the beam is taken into account, so that the initial normal stress is not equal to zero. We have proposed some formulae to determine the natural frequencies of the prestressed beam. The forced transverse vibration of the beam with a moving external force has been considered. From this it follows compression softening effect of prestressed b

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Zhang, Guo-Dong, and Bao-Zhu Guo. "On the spectrum of Euler–Bernoulli beam equation with Kelvin–Voigt damping." Journal of Mathematical Analysis and Applications 374, no. 1 (2011): 210–29. http://dx.doi.org/10.1016/j.jmaa.2010.08.070.

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Hörmann, Günther, and Ljubica Oparnica. "Distributional solution concepts for the Euler–Bernoulli beam equation with discontinuous coefficients." Applicable Analysis 86, no. 11 (2007): 1347–63. http://dx.doi.org/10.1080/00036810701595944.

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Achouri, Zineb, Nour Eddine Amroun, and Abbes Benaissa. "The Euler-Bernoulli beam equation with boundary dissipation of fractional derivative type." Mathematical Methods in the Applied Sciences 40, no. 11 (2016): 3837–54. http://dx.doi.org/10.1002/mma.4267.

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