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Journal articles on the topic 'Timoshenko Equation'

Author: Grafiati

Published: 4 June 2021

Last updated: 1 February 2022

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1

Lou, Menglin, Qiuhua Duan, and Genda Chen. "Modal Perturbation Method for the Dynamic Characteristics of Timoshenko Beams." Shock and Vibration 12, no. 6 (2005): 425–34. http://dx.doi.org/10.1155/2005/824616.

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Timoshenko beams have been widely used in structural and mechanical systems. Under dynamic loading, the analytical solution of a Timoshenko beam is often difficult to obtain due to the complexity involved in the equation of motion. In this paper, a modal perturbation method is introduced to approximately determine the dynamic characteristics of a Timoshenko beam. In this approach, the differential equation of motion describing the dynamic behavior of the Timoshenko beam can be transformed into a set of nonlinear algebraic equations. Therefore, the solution process can be simplified significantly for the Timoshenko beam with arbitrary boundaries. Several examples are given to illustrate the application of the proposed method. Numerical results have shown that the modal perturbation method is effective in determining the modal characteristics of Timoshenko beams with high accuracy. The effects of shear distortion and moment of inertia on the natural frequencies of Timoshenko beams are discussed in detail.

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2

Cao, MS, W. Xu, Z. Su, W. Ostachowicz, and N. Xia. "Local coordinate systems-based method to analyze high-order modes of n-step Timoshenko beam." Journal of Vibration and Control 23, no. 1 (August 9, 2016): 89–102. http://dx.doi.org/10.1177/1077546315573919.

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High-frequency transverse vibration of stepped beams has attracted increasing attention in various industrial areas. For an n-step Timoshenko beam, the governing differential equations of transverse vibration have been well established in the literature on the basis of assembling classic Timoshenko beam equations for uniform beam segments. However, solving the governing differential equation has not been resolved well to date, manifested by a computational bottleneck: only the first k modes ( k ≤ 12) are solvable for i-step ( i ≥ 0) Timoshenko beams. This bottleneck diminishes the completeness of stepped Timoshenko beam theory. To address this problem, this study first reveals the root cause of the bottleneck in solving the governing differential equations for high-order modes, and then creates a sophisticated method, based on local coordinate systems, that can overcome the bottleneck to accomplish high-order mode shapes of an n-step Timoshenko beam. The proposed method uses a set of local coordinate systems in place of the conventional global coordinate system to characterize the transverse vibration of an n-step Timoshenko beam. With the method, the local coordinate systems can simplify the frequency equation for the vibration of an n-step Timoshenko beam, making it possible to obtain high-order modes of the beam. The accuracy, capacity, and efficiency of the method based on local coordinate systems in acquiring high-order modes are corroborated using the well-known exact dynamic stiffness method underpinned by the Wittrick-Williams algorithm as a reference. Removal of the bottlenecks in solving the governing differential equations for high-order modes contributes usefully to the completeness of stepped Timoshenko beam theory.

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3

Guo, Shi Wei. "Dual Variables System Analysis for Timoshenko Beam." Applied Mechanics and Materials 423-426 (September 2013): 1473–84. http://dx.doi.org/10.4028/www.scientific.net/amm.423-426.1473.

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Regarding the displacements and internal forces of Timoshenko beams as dual variables, Timoshenko beam problems were included into dual variables system. Corresponding to state transfer solution of Hamiltonian dual equation, transfer form solution of dual variables for Timoshenko beams was presented. Based on transfer form solution, element stiffness equation and the shape functions of Timoshenko beams were deduced, boundary integral equation and the fundamental solution function of Timoshenko beams were obtained, which reveal the intrinsic relationships among the finite element method, the boundary element method and dual variables system of Timoshenko beams. Based on the transfer form solution of Timoshenko beams, transfer matrix method for chain structure of Timoshenko beams was proposed. For chain beam structure problems, transfer matrix method is simple, intuitive, and has the advantages of good boundary adaptability and less calculation in solving the node variables of chain structures with recursive solution. The numerical results demonstrate the feasibility and accuracy of transfer matrix method in complex beam structure problems.

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4

Wang, Jiujiang, Xin Liu, Yuanyu Yu, Yao Li, Ching-Hsiang Cheng, Shuang Zhang, Peng-Un Mak, Mang-I. Vai, and Sio-Hang Pun. "A Review on Analytical Modeling for Collapse Mode Capacitive Micromachined Ultrasonic Transducer of the Collapse Voltage and the Static Membrane Deflections." Micromachines 12, no. 6 (June 18, 2021): 714. http://dx.doi.org/10.3390/mi12060714.

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Analytical modeling of capacitive micromachined ultrasonic transducer (CMUT) is one of the commonly used modeling methods and has the advantages of intuitive understanding of the physics of CMUTs and convergent when modeling of collapse mode CMUT. This review article summarizes analytical modeling of the collapse voltage and shows that the collapse voltage of a CMUT correlates with the effective gap height and the electrode area. There are analytical expressions for the collapse voltage. Modeling of the membrane deflections are characterized by governing equations from Timoshenko, von Kármán equations and the 2D plate equation, and solved by various methods such as Galerkin’s method and perturbation method. Analytical expressions from Timoshenko’s equation can be used for small deflections, while analytical expression from von Kármán equations can be used for both small and large deflections.

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5

White, M. W. D., and G. R. Heppler. "Vibration Modes and Frequencies of Timoshenko Beams With Attached Rigid Bodies." Journal of Applied Mechanics 62, no. 1 (March 1, 1995): 193–99. http://dx.doi.org/10.1115/1.2895902.

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The equations of motion and boundary conditions for a free-free Timoshenko beam with rigid bodies attached at the endpoints are derived. The natural boundary conditions, for an end that has an attached rigid body, that include the effects of the body mass, first moment of mass, and moment of inertia are included. The frequency equation for a free-free Timoshenko beam with rigid bodies attached at its ends which includes all the effects mentioned above is presented and given in terms of the fundamental frequency equations for Timoshenko beams that have no attached rigid bodies. It is shown how any support / rigid-body condition may be easily obtained by inspection from the reported frequency equation. The mode shapes and the orthogonality condition, which include the contribution of the rigid-body masses, first moments, and moments of inertia, are also developed. Finally, the effect of the first moment of the attached rigid bodies is considered in an illustrative example.

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6

Soltani, Masoumeh, and Behrouz Asgarian. "Finite Element Formulation for Linear Stability Analysis of Axially Functionally Graded Nonprismatic Timoshenko Beam." International Journal of Structural Stability and Dynamics 19, no. 02 (February 2019): 1950002. http://dx.doi.org/10.1142/s0219455419500020.

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An improved approach based on the power series expansions is proposed to exactly evaluate the static and buckling stiffness matrices for the linear stability analysis of axially functionally graded (AFG) Timoshenko beams with variable cross-section and fixed–free boundary condition. Based on the Timoshenko beam theory, the equilibrium equations are derived in the context of small displacements, considering the coupling between the transverse deflection and angle of rotation. The system of stability equations is then converted into a single homogeneous differential equation in terms of bending rotation for the cantilever, which is solved numerically with the help of the power series approximation. All the mechanical properties and displacement components are thus expanded in terms of the power series of a known degree. Afterwards, the shape functions are gained by altering the deformation shape of the AFG nonprismatic Timoshenko beam in a power series form. At the end, the elastic and buckling stiffness matrices are exactly determined by the weak form of the governing equation. The precision and competency of the present procedure in stability analysis are assessed through several numerical examples of axially nonhomogeneous and homogeneous Timoshenko beams with clamped-free ends. Comparison is also made with results obtained using ANSYS and other solutions available, which indicates the correctness of the present method.

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7

Lárez, Hanzel, Hugo Leiva, and Darwin Mendoza. "Interior Controllability of a Timoshenko Type Equation." International Journal of Control Science and Engineering 1, no. 1 (August 31, 2012): 15–21. http://dx.doi.org/10.5923/j.control.20110101.03.

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8

Shou-kui, Si. "Backward wellposedness of nonuniform Timoshenko beam equation." Journal of Zhejiang University-SCIENCE A 2, no. 2 (April 2001): 161–64. http://dx.doi.org/10.1631/jzus.2001.0161.

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9

Esquivel-Avila, Jorge Alfredo. "Dynamic Analysis of a Nonlinear Timoshenko Equation." Abstract and Applied Analysis 2011 (2011): 1–36. http://dx.doi.org/10.1155/2011/724815.

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We characterize the global and nonglobal solutions of the Timoshenko equation in a bounded domain. We consider nonlinear dissipation and a nonlinear source term. We prove blowup of solutions as well as convergence to the zero and nonzero equilibria, and we give rates of decay to the zero equilibrium. In particular, we prove instability of the ground state. We show existence of global solutions without a uniform bound in time for the equation with nonlinear damping. We define and use a potential well and positive invariant sets.

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10

AROSIO, A. "ON THE NONLINEAR TIMOSHENKO-KIRCHHOFF BEAM EQUATION." Chinese Annals of Mathematics 20, no. 04 (October 1999): 495–506. http://dx.doi.org/10.1142/s0252959999000564.

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11

Wan, Chunfeng, Huachen Jiang, Liyu Xie, Caiqian Yang, Youliang Ding, Hesheng Tang, and Songtao Xue. "Natural Frequency Characteristics of the Beam with Different Cross Sections Considering the Shear Deformation Induced Rotary Inertia." Applied Sciences 10, no. 15 (July 29, 2020): 5245. http://dx.doi.org/10.3390/app10155245.

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Based on the classical Timoshenko beam theory, the rotary inertia caused by shear deformation is further considered and then the equation of motion of the Timoshenko beam theory is modified. The dynamic characteristics of this new model, named the modified Timoshenko beam, have been discussed, and the distortion of natural frequencies of Timoshenko beam is improved, especially at high-frequency bands. The effects of different cross-sectional types on natural frequencies of the modified Timoshenko beam are studied, and corresponding simulations have been conducted. The results demonstrate that the modified Timoshenko beam can successfully be applied to all beams of three given cross sections, i.e., rectangular, rectangular hollow, and circular cross sections, subjected to different boundary conditions. The consequence verifies the validity and necessity of the modification.

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12

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 multiplied by second derivative of displacement. Damping properties can be treated as additional substitute load in the static case taking into account this substitute axial force. When one considers the Timoshenko beam, the substitute axial force depends additionally on shear properties of rail section, rail bending stiffness, and subgrade stiffness. It is also proved that Timoshenko beam, described by a single equation, from the point of view of solution, is an analogy of the Euler-Bernoulli beam for both constant and variable load. Certain numerical examples are presented and practical interpretation of proved theorems is shown.

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13

Lou, P., G.-L. Dai, and Q.-Y. Zeng. "Finite-Element Analysis for a Timoshenko Beam Subjected to a Moving Mass." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 220, no. 5 (May 1, 2006): 669–78. http://dx.doi.org/10.1243/09544062jmes119.

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This article presents a finite-element formulation of a Timoshenko beam subjected to a moving mass. The beam is discretized into a number of simple elements with four degrees of freedom each. The inertial effects of the moving mass are incorporated into a finite-element model. The equation of motion in matrix form with time-dependent coefficients for a Timoshenko beam subjected to a moving mass is derived from the variational approach. The equation is solved by the direct step-by-step integration method to obtain the dynamic response of a Timoshenko beam and the contact force between the moving mass and the beam. The correctness of this present method is validated by means of comparison with the solution obtained by the assumed-mode method. The present method can be effectively used in computation for a Timoshenko beam with various boundary conditions. Numerical simulations are performed to demonstrate the efficiency of the present method.

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14

Selezov, Igor. "Timoshenko equation, violation of continuity and some applications." Visnyk of Zaporizhzhya National University. Physical and Mathematical Sciences, no. 2 (2019): 150–57. http://dx.doi.org/10.26661/2413-6549-2019-2-17.

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15

Lin, S. M. "Dynamic Analysis of Rotating Nonuniform Timoshenko Beams With an Elastically Restrained Root." Journal of Applied Mechanics 66, no. 3 (September 1, 1999): 742–49. http://dx.doi.org/10.1115/1.2791698.

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A systematic solution procedure for studying the dynamic response of a rotating nonuniform Timoshenko beam with an elastically restrained root is presented. The partial differential equations are transformed into the ordinary differential equations by taking the Laplace transform. The two coupled governing differential equations are uncoupled into two complete fourth-order differential equations with variable coefficients in the flexural displacement and in the angle of rotation due to bending, respectively. The general solution and the generalized Green function of the uncoupled system are derived. They are expressed in terms of the four corresponding linearly independent homogenous solutions, respectively. The shifting relations of the four homogenous solutions of the uncoupled governing differential equation with constant coefficients are revealed. The generalized Green function of an nth order ordinary differential equation can be obtained by using the proposed method. Finally, the influence of the elastic root restraints, the setting angle, and the excitation frequency on the steady response of a beam is investigated.

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16

Tan, Guojin, Wensheng Wang, Yongchun Cheng, Haibin Wei, Zhigang Wei, and Hanbing Liu. "Dynamic Response of a Nonuniform Timoshenko Beam with Elastic Supports, Subjected to a Moving Spring-Mass System." International Journal of Structural Stability and Dynamics 18, no. 05 (May 2018): 1850066. http://dx.doi.org/10.1142/s0219455418500669.

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This paper is concerned with the dynamic response of a nonuniform Timoshenko beam with elastic supports subjected to a moving spring-mass system. The modal orthogonality of nonuniform Timoshenko beams and the corresponding overall matrix of undetermined coefficients are derived. Then the natural frequencies and mode shapes of nonuniform Timoshenko beams are obtained by the Runge–Kutta method and cubic spline interpolation method. By using the Newmark-[Formula: see text] method and the mode summation method, the vibration equation of Timoshenko beams subjected to a moving spring-mass system was established. A comparison of results between the proposed method and finite element method reveals that this method possesses favorable accuracy for dynamic response analysis. In numerical examples, the effects of the support spring and moving spring-mass system on Timoshenko beams have been examined in detail.

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17

Nielsen, Rasmus, and Sergey Sorokin. "The WKB approximation for analysis of wave propagation in curved rods of slowly varying diameter." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 470, no. 2167 (July 8, 2014): 20130718. http://dx.doi.org/10.1098/rspa.2013.0718.

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The Wentzel–Kramers–Brillouin (WKB) approximation is applied for asymptotic analysis of time-harmonic dynamics of corrugated elastic rods. A hierarchy of three models, namely, the Rayleigh and Timoshenko models of a straight beam and the Timoshenko model of a curved rod is considered. In the latter two cases, the WKB approximation is applied for solving systems of two and three linear differential equations with varying coefficients, whereas the former case is concerned with a single equation of the same type. For each model, explicit formulations of wavenumbers and amplitudes are given. The equivalence between the formal derivation of the amplitude and the conservation of energy flux is demonstrated. A criterion of the validity range of the WKB approximation is proposed and its applicability is proved by inspection of eigenfrequencies of beams of finite length with clamped–clamped and clamped-free boundary conditions. It is shown that there is an appreciable overlap between the validity ranges of the Timoshenko beam/rod models and the WKB approximation.

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18

Shen, Ji Yao, Jen-Kuang Huang, and L. W. Taylor. "Timoshenko Beam Modeling for Parameter Estimation of NASA Mini-Mast Truss." Journal of Vibration and Acoustics 115, no. 1 (January 1, 1993): 19–24. http://dx.doi.org/10.1115/1.2930308.

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In this paper a distributed parameter model for the estimation of modal characteristics of NASA Mini-Mast truss is proposed. A closed-form solution of the Timoshenko beam equation, for a uniform cantilevered beam with two concentrated masses, is derived so that the procedure and the computational effort for the estimation of modal characteristics are improved. A maximum likelihood estimator for the Timoshenko beam model is also developed. The resulting estimates from test data by using Timoshenko beam model are found to be comparable to those derived from other approaches.

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19

CAMPELO, A. D. S., D. S. ALMEIDA JÚNIOR, and M. L. SANTOS. "Stability to the dissipative Reissner–Mindlin–Timoshenko acting on displacement equation." European Journal of Applied Mathematics 27, no. 2 (August 26, 2015): 157–93. http://dx.doi.org/10.1017/s0956792515000467.

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In this paper, we show that there exists a critical number that stabilises the Reissner–Mindlin–Timoshenko system with frictional dissipation acting only on the equation for the transverse displacement. We identify that the Reissner–Mindlin–Timoshenko system has two speed characteristics v12 := K/ρ1 and v22 := D/ρ2 and we show that the system is exponentially stable if only if \begin{equation*} v_{1}^{2}=v_{2}^{2}. \end{equation*}In the general case, we prove that the system is polynomially stable with optimal decay rate. Numerical experiments using finite differences are given to confirm our analytical results. Our numerical results are qualitatively in agreement with the corresponding results from dynamical in infinite dimensional.

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20

Lee, Byoung-Koo, Kwang-Kyou Park, and Tae-Eun Lee. "Free Vibrations of Tapered Timoshenko Beam by using 4th Order Ordinary Differential Equation." Journal of the Computational Structural Engineering Institute of Korea 25, no. 3 (June 30, 2012): 185–94. http://dx.doi.org/10.7734/coseik.2012.25.3.185.

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21

Elishakoff, Isaac, Florian Hache, and Noël Challamel. "Critical Comparison of Bresse–Timoshenko Beam Theories for Parametric Instability in the Presence of Pulsating Load." International Journal of Structural Stability and Dynamics 19, no. 02 (February 2019): 1950006. http://dx.doi.org/10.1142/s0219455419500068.

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In this paper, we investigate parametric instability of Bresse–Timoshenko columns subjected to periodic pulsating compressive loads. The results are derived from three theories, namely the Bernoulli–Euler model for thin beams and two versions of the Bresse–Timoshenko model valid for thick beams: The truncated Bresse–Timoshenko model and the Bresse–Timoshenko model based on slope inertia. The truncated Bresse–Timoshenko model has been derived from asymptotic analysis, whereas the Bresse–Timoshenko model based on slope inertia is an alternative shear beam model supported by variational arguments. These models both take into account the rotary inertia and the shear effect. Simple supported boundary conditions are considered, so that the time-dependent deflection solution can be decomposed into trigonometric spatial functions. The instability domain in the load–frequency space is analytically characterized from a Meissner-type parametric equation. For small slenderness ratio, these last two Bresse–Timoshenko models coincide but for much higher slenderness ratio, the parametric instability regions in the load–frequency space shift to the left and widen them as compared to the Bernoulli–Euler model. The importance of these effects differs between the models.

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22

Zhou, D., and Y. K. Cheung. "Vibrations of Tapered Timoshenko Beams in Terms of Static Timoshenko Beam Functions." Journal of Applied Mechanics 68, no. 4 (August 15, 2000): 596–602. http://dx.doi.org/10.1115/1.1357164.

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In this paper, the free vibrations of a wide range of tapered Timoshenko beams are investigated. The cross section of the beam varies continuously and the variation is described by a power function of the coordinate along the neutral axis of the beam. The static Timoshenko beam functions, which are the complete solutions of a tapered Timoshenko beam under a Taylor series of static load, are developed, respectively, as the basis functions of the flexural displacement and the angle of rotation due to bending. The Rayleigh-Ritz method is applied to derive the eigenfrequency equation of the tapered Timoshenko beam. Unlike conventional basis functions which are independent of the cross-sectional variation of the beam, these static Timoshenko beam functions vary in accordance with the cross-sectional variation of the beam so that higher accuracy and more rapid convergence have been obtained. Some numerical results are presented for both truncated and sharp-ended Timoshenko beams. On the basis of convergence study and comparison with available results in the literature it is shown that the first few eigenfrequencies can be given with quite good accuracy by using a small number of terms of the static Timoshenko beam functions. Finally, some valuable results are presented systematically.

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23

Tan, Pei, Lian Wang Chen, and Yu Jiang Li. "Lagrangianlized Nonlinear Dynamic Equation of Timoshenko Beam and Application." Applied Mechanics and Materials 157-158 (February 2012): 1115–20. http://dx.doi.org/10.4028/www.scientific.net/amm.157-158.1115.

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Base on Hamilton's principle, under the continuity conditions of generalized coordinator, generalized force and the deformation, provide nonlinear deformation problem of Timoshenko beam’s Lagrangianlized basic elastic equation and its generalized boundary condition. Lagrangianlized basic elastic equation makes it possible to avoid indirect problem for the variation principle of continuum, which not only simplify the solving of direct problem but also stylize the solving process. The YUQING ancient bridge (in ShangHai) is taken as an example, calculating the kinetics characteristics and making damage diagnosis combined with the pulsation experimental result data.

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24

Arosio, A. "A geometrical nonlinear correction to the Timoshenko beam equation." Nonlinear Analysis: Theory, Methods & Applications 47, no. 2 (August 2001): 729–40. http://dx.doi.org/10.1016/s0362-546x(01)00218-8.

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25

Pirrotta, Antonina, Stefano Cutrona, Salvatore Di Lorenzo, and Alberto Di Matteo. "Fractional visco-elastic Timoshenko beam deflection via single equation." International Journal for Numerical Methods in Engineering 104, no. 9 (June 10, 2015): 869–86. http://dx.doi.org/10.1002/nme.4956.

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26

Santos, J. R., and G. Siciliano. "On a generalized Timoshenko-Kirchhoff equation with sublinear nonlinearities." Journal of Mathematical Analysis and Applications 480, no. 2 (December 2019): 123394. http://dx.doi.org/10.1016/j.jmaa.2019.123394.

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27

Astaburuaga, M. A., C. Fernandez, and G. Perla Menzala. "Local smoothing effects for a nonlinear Timoshenko type equation." Nonlinear Analysis: Theory, Methods & Applications 23, no. 9 (November 1994): 1091–103. http://dx.doi.org/10.1016/0362-546x(94)90094-9.

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28

Berkani, Amirouche, and Nasser-eddine Tatar. "Stabilization of a viscoelastic Timoshenko beam fixed into a moving base." Mathematical Modelling of Natural Phenomena 14, no. 5 (2019): 501. http://dx.doi.org/10.1051/mmnp/2018057.

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In this paper, we are concerned with a cantilevered Timoshenko beam. The beam is viscoelastic and subject to a translational displacement. Consequently, the Timoshenko system is complemented by an ordinary differential equation describing the dynamic of the base to which the beam is attached to. We establish a control force capable of driving the system to the equilibrium state with a certain speed depending on the decay rate of the relaxation function.

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29

White, M. W. D., and G. R. Heppler. "Vibration of a Rotating Timoshenko Beam." Journal of Vibration and Acoustics 118, no. 4 (October 1, 1996): 606–13. http://dx.doi.org/10.1115/1.2888341.

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Timoshenko beam theory is used to model a flexible slewing link with an attached payload using two different rotating frames of reference: pseudo-pinned and pseudo-clamped. The boundary conditions are presented for both formulations; these lead naturally to the frequency equation for the link. The infinite dimensional model of the slewing link is then approximated by a finite dimensional model. Finally, these two formulations are shown to be equivalent through a simple transformation.

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30

Khiem, Nguyen Tien, and Duong The Hung. "A closed-form solution for free vibration of multiple cracked Timoshenko beam and application." Vietnam Journal of Mechanics 39, no. 4 (December 27, 2017): 315–28. http://dx.doi.org/10.15625/0866-7136/9641.

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A closed-form solution for free vibration is constructed and used for obtaining explicit frequency equation and mode shapes of Timoshenko beams with arbitrary number of cracks. The cracks are represented by the rotational springs of stiffness calculated from the crack depth. Using the obtained frequency equation, the sensitivity of natural frequencies to crack of the beams is examined in comparison with the Euler-Bernoulli beams. Numerical results demonstrate that the Timoshenko beam theory is efficiently applicable not only for short or fat beams but also for the long or slender ones. Nevertheless, both the theories are equivalent in sensitivity analysis of fundamental frequency to cracks and they get to be different for higher frequencies.

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31

Zveriaev, E. M. "INTRODUCING OF INTERNAL VISCOUS FRICTION IN EQUATIONS OF BEAM OSCILLATION AS LONG RECTANGULAR STRIP." STRUCTURAL MECHANICS AND ANALYSIS OF CONSTRUCTIONS, no. 1 (February 25, 2021): 54–58. http://dx.doi.org/10.37538/0039-2383.2021.1.54.58.

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Abstract. On the base of the method of simple iterations generalising methods of semi-inverse one of Saint-Venant, Reissner and Timoshenko the one-dimensional theory is constructed using the example of dynamic equations of a plane problem of elasticity theory for a long elastic strip. The resolving equation of that one-dimensional theory coincides with the equation of beam vibrations. The other problems with unknowns are determined without integration by direct calculations. In the initial equations of the theory of elasticity the terms corresponding to the viscous friction in the Navier-Stokes equations are introduced. The asymptotic characteristics of the unknowns obtained by the method of simple iterations allow to search for a solution in the form of expansions of the unknowns into asymptotic series. The resolving equation contains a term that depends on the coefficient of viscous friction.

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32

Bhaskar, Atul. "Elastic waves in Timoshenko beams: the ‘lost and found’ of an eigenmode." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 465, no. 2101 (October 8, 2008): 239–55. http://dx.doi.org/10.1098/rspa.2008.0276.

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This paper considers propagating waves in elastic bars in the spirit of asymptotic analysis and shows that the inclusion of shear deformation amounts to singular perturbation in the Euler–Bernoulli (EB) field equation. We show that Timoshenko, in his classic work of 1921, incorrectly treated the problem as one of regular perturbation and missed out one physically meaningful ‘branch’ of the dispersion curve (spectrum), which is mainly shear-wise polarized. Singular perturbation leads to: (i) Timoshenko's solution and (ii) a singular solution ; ϵ , ω * and k * are the non-dimensional slenderness, frequency and wavenumber, respectively. Asymptotic formulae for dispersion, standing waves and the density of modes are given in terms of ϵ . The second spectrum—in the light of the debate on its existence, or not—is discussed. A previously proposed Lagrangian is shown to be inadmissible in the context. We point out that Lagrangian densities that lead to the same equation(s) of motion may not be equivalent for field problems: careful consideration to the kinetic boundary conditions is important. A Hamiltonian formulation is presented—the conclusions regarding the validity (or not) of Lagrangian densities are confirmed via the constants of motion.

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33

Huyen, Nguyen Ngoc, and Nguyen Tien Khiem. "Modal analysis of functionally graded Timoshenko beam." Vietnam Journal of Mechanics 39, no. 1 (March 30, 2017): 31–50. http://dx.doi.org/10.15625/0866-7136/7582.

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Dynamic analysis of FGM Timoshenko beam is formulated in the frequency domain taking into account the actual position of neutral plane. The problem formulation enables to obtain explicit expressions for frequency equation, natural modes and frequency response of the beam subjected to external load. The representations are straightforward not only to modal analysis and modal testing of FGM Timoshenko beam with general end conditions but also to study coupling of axial and flexural vibration modes. Numerical study is carried out to investigate effect of true neutral axis position and material properties on the modal parameters.

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34

Banerjee, J. R. "Frequency equation and mode shape formulae for composite Timoshenko beams." Composite Structures 51, no. 4 (April 2001): 381–88. http://dx.doi.org/10.1016/s0263-8223(00)00153-7.

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Esquivel-Avila, Jorge. "Global attractor for a nonlinear Timoshenko equation with source terms." Mathematical Sciences 7, no. 1 (2013): 32. http://dx.doi.org/10.1186/2251-7456-7-32.

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Zhang, Chun-Guo, and Ze-Rong He. "Optimality conditions for a Timoshenko beam equation with periodic constraint." Zeitschrift für angewandte Mathematik und Physik 65, no. 2 (April 27, 2013): 315–24. http://dx.doi.org/10.1007/s00033-013-0333-1.

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Can-zhang, Xiao, Ji Yi-zhou, and Chang Bao-ping. "General dynamic equation and dynamical characteristics of viscoelastic Timoshenko beams." Applied Mathematics and Mechanics 11, no. 2 (February 1990): 177–84. http://dx.doi.org/10.1007/bf02014542.

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38

Erofeev, V. I., and A. V. Leontieva. "QUASIHARMONIC BENDING WAVE, DISTRIBUTING IN THE BALK OF TIMOSHENKO, LYING ON A NONLINEAR ELASTIC BASE." Problems of strenght and plasticity 83, no. 1 (2021): 61–75. http://dx.doi.org/10.32326/1814-9146-2021-83-1-61-75.

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In this paper, we consider the modulation instability of a quasiharmonic flexural wave propagating in a homogeneous beam fixed on a nonlinear elastic foundation. The dynamic behavior of the beam is determined by Timoshenko's theory. Timoshenko's model, refining the technical theory of rod bending, assumes that the crosssections remain flat, but not perpendicular to the deformable midline of the rod; normal stresses on sites parallel to the axis are zero; the inertial components associated with the rotation of the cross sections are taken into account. The uniqueness of the model lies in the fact that, allowing a good description of many processes occurring in real structures, it remains quite simple, accessible for analytical research. The system of equations describing the bending vibrations of the beam is reduced to one nonlinear fourthorder equation for the transverse displacements of the beam particles. The nonlinear Schrödinger equation, one of the basic equations of nonlinear wave dynamics, is obtained by the method of many scales. Regions of modulation instability are determined according to the Lighthill criterion. It is shown hot the boundaries of these areas shift when the parameters characterizing the elastic properties of the beam material and the nonlinearity of the base change. Nonlinear stationary envelope waves are considered. An equation that generalizes the Duffing equation, which contains two additional terms in negative powers (first and third), is obtained and qualitatively analyzed. Solutions of the Schr?dinger equation in the form of envelope solitons are found and the dependences of their main parameters (amplitude, width) on the parameters of the system are analyzed. The dynamics of the points of intersection of the amplitudes and widths of "light" solitons in the case of soft nonlinearity of the base is shown within the region of modulation instability.

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Andreew, Vladimir I., Anton S. Chepurnenko, and Batyr M. Yazyev. "Energy Method in the Calculation Stability of Compressed Polymer Rods Considering Creep." Advanced Materials Research 1004-1005 (August 2014): 257–60. http://dx.doi.org/10.4028/www.scientific.net/amr.1004-1005.257.

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The problem of stability of the polymer rod in creeping was solved by energy method in the form of Timoshenko and Ritz. Possible displacements of the points were given in the form of a trigonometric series with undetermined coefficients. Results obtained numerically using complex Matlab under different constraint equations between creep deformations and stresses. There are shown the necessity of taking into account the "junior" component of high-elastic deformation by using the Maxwell-Gurevich equation.

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Ju, S. H., C. C. Leong, and Y. S. Ho. "Theoretical and Numerical Solutions of Maglev Train Induced Vibration." Applied Mechanics and Materials 459 (October 2013): 271–77. http://dx.doi.org/10.4028/www.scientific.net/amm.459.271.

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This paper proposed an efficient method based on theoretical equations to solve the dynamic interaction problem between the Timoshenko beam and maglev vehicles. A systematic PI numerical scheme is developed for the control system of the maglev train. The major advantage is that only one simple equation required in the control calculation, although the original control system is fairly complicated. Numerical simulations indicate that a large time step length can be used in the proposed method to obtain stable and accurate results.

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Bruk, V. M., and V. A. Krys’ko. "Reduction of generalized S. P. Timoshenko equations to a differential operator equation of hyperbolic type." Russian Mathematics 51, no. 2 (February 2007): 68–70. http://dx.doi.org/10.3103/s1066369x07020089.

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Hull, Andrew J., Daniel Perez, and Donald L. Cox. "A High-Frequency Model of a Rectilinear Beam with a T-Shaped Cross Section." Acoustics 1, no. 3 (September 9, 2019): 726–48. http://dx.doi.org/10.3390/acoustics1030043.

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This paper derives an analytical model of a straight beam with a T-shaped cross section for use in the high-frequency range, defined here as approximately 1 to 35 kHz. The web, the right part of the flange, and the left part of the flange of the T-beam are modeled independently with two-dimensional elasticity equations for the in-plane motion and Mindlin flexural plate equation for the out-of-plane motion. The differential equations are solved with unknown wave propagation coefficients multiplied by circular spatial domain functions. These algebraic equations are then solved to yield the wave propagation coefficients and thus produce a solution to the displacement field in all three directions. An example problem is formulated and compared with solutions from fully elastic finite element modeling, a previously derived analytical model, and Timoshenko beam theory. It is shown that the accurate frequency range of this new model is significantly higher than that of the analytical model and the Timoshenko beam model, and, in the frequency range up to 35 kHz, the results compare very favorably to those from finite element analysis.

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Gau, Wei-Hsin, and A. A. Shabana. "Effects of Shear Deformation and Rotary Inertia on the Nonlinear Dynamics of Rotating Curved Beams." Journal of Vibration and Acoustics 112, no. 2 (April 1, 1990): 183–93. http://dx.doi.org/10.1115/1.2930111.

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In this investigation a method for the dynamic analysis of initially curved Timoshenko beams that undergo finite rotations is presented. The combined effect of rotary inertia, shear deformation, and initial curvature is examined. The kinetic energy is first developed for the curved beam and the beam mass matrix is identified. It is shown that the form of the mass matrix as well as the nonlinear inertia terms that represent the coupling between the rigid body motion and the elastic deformation can be expressed in terms of a set of invariants that depend on the assumed displacement field, rotary inertia, shear deformation, and the initial beam curvature. A nonlinear finite element formulation is then developed for Timoshenko beams that undergo finite rotations. The nonlinear formulation presented in this paper is applied to multibody dynamics where mechanical systems consist of an interconnected set of rigid and deformable bodies, each of which may undergo finite rotations. The equations of motion are developed using Lagrange’s equation and nonlinear algebraic constraint equations that mathematically describe mechanical joints and specified trajectories are adjoined to the system differential equations using the vector of Lagrange multipliers.

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Lee, Sen-Yung, and Qian-Zhi Yan. "Exact Static Analysis of In-Plane Curved Timoshenko Beams with Strong Nonlinear Boundary Conditions." Mathematical Problems in Engineering 2015 (2015): 1–12. http://dx.doi.org/10.1155/2015/646391.

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Analytical solutions have been developed for nonlinear boundary problems. In this paper, the shifting function method is applied to develop the static deflection of in-plane curved Timoshenko beams with nonlinear boundary conditions. Three coupled governing differential equations are derived via the Hamilton’s principle. The mathematical modeling of the curved beam system can be decomposed into a complete sixth-order ordinary differential characteristic equation and the associated boundary conditions. It is shown that the proposed method is valid and performs well for problems with strong nonlinearity.

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LIM, TEIK-CHENG. "ANALYSIS OF AUXETIC BEAMS AS RESONANT FREQUENCY BIOSENSORS." Journal of Mechanics in Medicine and Biology 12, no. 05 (December 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, with special emphasis on solids with negative Poisson's ratio, which are also known as auxetic materials. Here, it is shown that the Timoshenko beam theory approaches the Euler–Bernoulli beam theory if the beams are of solid cross-sections and the beam material possess high auxeticity. However, the Timoshenko beam theory is significantly different from the Euler–Bernoulli beam theory for beams in the form of thin-walled tubes regardless of the beam material's Poisson's ratio. It is herein proposed that calculations on beam vibration can be greatly simplified for highly auxetic beams with solid cross-sections due to the small shear correction term in the Timoshenko beam deflection equation.

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Ma, Zhi Yong. "Polynomial Stability for Timoshenko-Type System with Past History." Applied Mechanics and Materials 623 (August 2014): 78–84. http://dx.doi.org/10.4028/www.scientific.net/amm.623.78.

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In this paper, we consider hyperbolic Timoshenko-type vibrating systems that are coupled to a heat equation modeling an expectedly dissipative effect through heat conduction. We use semigroup method to prove the polynomial stability result with assumptions on past history relaxation function exponentially decaying for the nonequal wave-speed case.

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Pişkin, Erhan. "BLOW UP OF SOLUTIONS FOR A TIMOSHENKO EQUATION WITH DAMPING TERMS." Middle East Journal of Science 4, no. 2 (December 27, 2018): 70–80. http://dx.doi.org/10.23884/mejs.2018.4.2.03.

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48

Olsson, Peter, and Gerhard Kristensson. "Wave splitting of the Timoshenko beam equation in the time domain." ZAMP Zeitschrift f�r angewandte Mathematik und Physik 45, no. 6 (November 1994): 866–81. http://dx.doi.org/10.1007/bf00952082.

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49

Ouaoua, Amar, Aya Khaldi, and Messaoud Maouni. "Existence and stability results of a nonlinear Timoshenko equation with damping and source terms." Theoretical and Applied Mechanics 48, no. 1 (2021): 53–66. http://dx.doi.org/10.2298/tam200703002o.

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In this paper, we consider a nonlinear Timoshenko equation. First, we prove the local existence solution by the Faedo?Galerkin method, and, under suitable assumptions with positive initial energy, we prove that the local existence is global in time. Finally, the stability result is established based on Komornik?s integral inequality.

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50

Kharasova, Lilya. "Strength of thin-walled elastic building structures." E3S Web of Conferences 274 (2021): 03019. http://dx.doi.org/10.1051/e3sconf/202127403019.

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The existence theorem is proved within the framework of the shear model by S.P. Timoshenko. The stress-strain state of elastic inhomogeneous isotropic shallow thin-walled shell constructions is studied. The stress-strain state of shell constructions is described by a system of the five equilibrium equations and by the five static boundary conditions with respect to generalized displacements. The aim of the work is to find generalized displacements from a system of equilibrium equations that satisfy given static boundary conditions. The research is based on integral representations for generalized displacements containing arbitrary holomorphic functions. Holomorphic functions are found so that the generalized displacements should satisfy five static boundary conditions. The integral representations constructed this way allow to obtain a nonlinear operator equation. The solvability of the nonlinear equation is established with the use of contraction mappings principle.

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