Relevant bibliographies by topics / Timoshenko Equation

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Academic literature on the topic 'Timoshenko Equation'

Author: Grafiati
Published: 4 June 2021
Last updated: 1 February 2022

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Journal articles on the topic "Timoshenko Equation"

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 (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, et al. "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 (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 (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 (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 (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 (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 (1999): 495–506. http://dx.doi.org/10.1142/s0252959999000564.

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