Vibration of Timoshenko Beams Using Non-classical Elasticity Theories

Santos, J.V. Araújo dos; Reddy, J. N.

doi:10.1155/2012/307806

Cited by 17 publications

(6 citation statements)

References 17 publications

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“…Table 1 shows the resonant frequencies which are rounded off to four decimal digits to compare with the values reported in Thai 30 and Santos and Reddy. 34 In Figure 2, natural frequencies obtained from both classical and non-local theories show that the maximum frequency change happens in short beams (L=h \ 10). Moreover, since for long beams the obtained frequencies converge, the effect of rotary inertia on fundamental frequency can be neglected and the ensuing inaccuracy can be ignored.…”

Section: Governing Equations Of Timoshenko Beam Theorymentioning

confidence: 99%

Non-local vibration of simply supported nano-beams: Higher-order modes

Masoumi

2017

Proceedings of the Institution of Mechanical Engineers, Part N:

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In this article, the effects of some parameters, including rotary inertia, non-local parameter, and length-to-thickness ratio, on natural frequencies are studied for both classical and non-local theories. For Timoshenko beam, the equations of motion and the boundary conditions are derived from Hamilton's principle and then non-local constitutive equations of Eringen are employed to altogether formulate the problem. Afterward, obtained governing equations are used to study the free vibrations of a Timoshenko's simply supported nano-beam. And finally, the effects of above-mentioned parameters on estimated frequencies in classical and non-local elasticity theories are investigated. Results show that the discrepancy between the frequencies of higher-order vibration modes obtained from two theories increases and also significant reductions in natural frequencies occur when the rotary inertia is considered in the computations.

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Section: Governing Equations Of Timoshenko Beam Theorymentioning

confidence: 99%

Non-local vibration of simply supported nano-beams: Higher-order modes

Masoumi

2017

Proceedings of the Institution of Mechanical Engineers, Part N:

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“…Our values for the natural frequencies, in the linear regime, of clamped-clamped and hinged-hinged CNTs and nonlocal beams agreed with the values published in Refs. [12,[16][17][18], and [30][31][32]. Due to space limitations, just one example is shown here, the case of a hinged-hinged beam presented in Ref.…”

Section: Summary Of Convergence and Verification Analysismentioning

confidence: 99%

“…Furthermore, similar values were published in Refs. [18] and [31]. Five shape functions guarantee five digits accuracy, whatever the value of the nonlocal parameter l is.…”

Section: Summary Of Convergence and Verification Analysismentioning

confidence: 99%

Nonlinear Modes of Vibration and Internal Resonances in Nonlocal Beams

Ribeiro

Thomas

2017

Journal of Computational and Nonlinear Dynamics

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A nonlocal Bernoulli–Euler p-version finite-element (p-FE) is developed to investigate nonlinear modes of vibration and to analyze internal resonances of beams with dimensions of a few nanometers. The time domain equations of motion are transformed to the frequency domain via the harmonic balance method (HBM), and then, the equations of motion are solved by an arc-length continuation method. After comparisons with published data on beams with rectangular cross section and on carbon nanotubes (CNTs), the study focuses on the nonlinear modes of vibration of CNTs. It is verified that the p-FE proposed, which keeps the advantageous flexibility of the FEM, leads to accurate discretizations with a small number of degrees-of-freedom. The first three nonlinear modes of vibration are studied and it is found that higher order modes are more influenced by nonlocal effects than the first mode. Several harmonics are considered in the harmonic balance procedure, allowing us to discover modal interactions due to internal resonances. It is shown that the nonlocal effects alter the characteristics of the internal resonances. Furthermore, it is demonstrated that, due to the internal resonances, the nonlocal effects are still noticeable at lengths that are longer than what has been previously found.

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“…Ma et al [17] have developed the static bending and free vibration problems of a Timoshenko beam. dos Santos and Reddy [18] have analyzed classical and nonclassical frequency ratios for a Timoshenko microbeam. Demir et al [19] have examined nonclassical frequencies of carbon nanotubes based on shear deformable beam theory by discrete singular convolution technique.…”

Section: Introductionmentioning

confidence: 99%

“…Similarly, the mass and the stiffness matrices of the second element can be calculated by replacing ( 1 ) by ( 2 ) and ( ) by ( ). Inserting above equations in (18) leads to following eigenvalue problem:…”

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confidence: 99%

Free Vibration Behavior of a Gradient Elastic Beam with Varying Cross Section

Yaylı

2014

Shock and Vibration

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Based on strain gradient elasticity theory, a finite element procedure is proposed for computation of natural frequencies for the microbeams of constant width and linear varying depth. Weak form formulation of the equation of motion is obtained first as in common classical finite element procedure in terms of various kinds of boundary conditions. Gradient elastic shape functions are used for interpolating deflection inside a finite element. Stiffness and mass matrices are then calculated to solve the microbeam eigen value problem. A solution for natural frequencies is obtained using characteristic equation of microbeam in gradient elasticity. The results are given in a series of figures and compared with their classical counterparts. The effect of various slope values on the natural frequencies are examined in some numerical examples. Comparison with the classical elasticity theory is also performed to verify the present study.

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Vibration of Timoshenko Beams Using Non-classical Elasticity Theories

Cited by 17 publications

References 17 publications

Non-local vibration of simply supported nano-beams: Higher-order modes

Non-local vibration of simply supported nano-beams: Higher-order modes

Nonlinear Modes of Vibration and Internal Resonances in Nonlocal Beams

Free Vibration Behavior of a Gradient Elastic Beam with Varying Cross Section

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