Free Vibration of an Euler–Bernoulli Beam with Arbitrary Nonuniformities and Discontinuities

AIAA Journal ◽  
2021 ◽  
pp. 1-4
Author(s):  
Alok Sinha
Keyword(s):  
Free Vibration ◽  
Bernoulli Beam ◽  
Euler Bernoulli Beam

scholarly journals Free Vibration Analysis of DWCNTs Using CDM and Rayleigh-Schmidt Based on Nonlocal Euler-Bernoulli Beam Theory

2014 ◽  
Vol 2014 ◽  
pp. 1-13 ◽  
Author(s):  
Maria Anna De Rosa ◽  
Maria Lippiello
Keyword(s):  
Carbon Nanotubes ◽  
Free Vibration ◽  
Van Der Waals ◽  
Beam Theory ◽  
Van Der Waals Forces ◽  
Bernoulli Beam ◽  
Bernoulli Beam Theory ◽  
Double Walled Carbon Nanotubes ◽  
Walled Carbon Nanotubes ◽  
Euler Bernoulli Beam

The free vibration response of double-walled carbon nanotubes (DWCNTs) is investigated. The DWCNTs are modelled as two beams, interacting between them through the van der Waals forces, and the nonlocal Euler-Bernoulli beam theory is used. The governing equations of motion are derived using a variational approach and the free frequencies of vibrations are obtained employing two different approaches. In the first method, the two double-walled carbon nanotubes are discretized by means of the so-called “cell discretization method” (CDM) in which each nanotube is reduced to a set of rigid bars linked together by elastic cells. The resulting discrete system takes into account nonlocal effects, constraint elasticities, and the van der Waals forces. The second proposed approach, belonging to the semianalytical methods, is an optimized version of the classical Rayleigh quotient, as proposed originally by Schmidt. The resulting conditions are solved numerically. Numerical examples end the paper, in which the two approaches give lower-upper bounds to the true values, and some comparisons with existing results are offered. Comparisons of the present numerical results with those from the open literature show an excellent agreement.

Large Amplitude Free Vibration of Nanobeams Subjected to Magnetic Field Based on Nonlocal Elasticity Theory

2015 ◽  
Vol 764-765 ◽  
pp. 1199-1203 ◽  
Author(s):  
Tai Ping Chang
Keyword(s):  
Magnetic Field ◽  
Free Vibration ◽  
Nonlocal Elasticity ◽  
Beam Theory ◽  
Nonlinear Frequency ◽  
Bernoulli Beam ◽  
Nonlinear Free Vibration ◽  
Vibration Behavior ◽  
Bernoulli Beam Theory ◽  
Euler Bernoulli Beam

In the present study, nonlinear free vibration behavior of nanobeam subjected to magnetic field is investigated based on Eringen's nonlocal elasticity and Euler–Bernoulli beam theory. The Hamilton's principle is adopted to derive the governing equations together with Euler–Bernoulli beam theory and the von-Kármán's nonlinear strain–displacement relationships. An approximate analytical solution is obtained for the nonlinear frequency of the nanobeam under magnetic field by using the Galerkin method and He's variational method. In the numerical results, the ratio of nonlinear frequency to linear frequency is presented. The effect of nonlocal parameter on the nonlinear frequency ratio is studied; furthermore, the effect of magnetic field on the nonlinear free vibration behavior of nanobeam is investigated.

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