In this paper, flexural vibrations of cracked micro-and nanobeams are studied. The model is based on the theory of nonlocal elasticity applied to Euler-Bernouilli beams. The cracked-beam model is established using a proper modification of the classical cracked-beam theory consisting of dividing the cracked element into two segments connected by a rotational spring located at the cracked section. This model promotes a discontinuity in bending slope, which is proportional to the second derivative of the displacements. Frequency equations of cracked nanobeams with some typical boundary conditions are derived and the natural frequencies for different crack positions, crack lengths, and nonlocal length parameters are calculated. The results are compared with those corresponding to the classical local model, emphasizing the differences occurring when the nonlocal effects are significant.
The natural frequencies for bending vibrations of Timoshenko cracked beams with simple boundary conditions have been obtained. The beam is modelled as two segments connected by two massless springs (one extensional and another one rotational). This model promotes discontinuities in both vertical displacement and rotation due to bending, which are proportional to shear force and bending moment transmitted by the cracked section, respectively. The differential equations for the free bending vibrations are established and then solved individually for each segment with the corresponding boundary conditions and the appropriated compatibility conditions at the cracked section. The problem is also solved by the perturbation method and both procedures are applied to the case of a simply supported cracked beam. The results show that the perturbation method provides simple expressions for the natural frequencies of cracked beams and it gives good results for shallow cracks. E mail address: ppfer@ing.uc3m.es (J. Ferna´ndez Sa´ez).
This paper is focused on the behavior of boring bars with a passive dynamic vibration absorber (DVA) for chatter suppression. The boring bar was modeled as a cantilever Euler-Bernoulli beam and only its first mode of vibration was considered. The stability of the two-degree-of-freedom model was analyzed constructing the stability diagram, dependent on the bar characteristics and on the absorber parameters (mass, stiffness, damping, and position). Two analytical approaches for tuning the absorber parameters were compared. The selection criterion consisted on the maximization of the minimum values of the stability-lobes diagram. Subsequent analysis performed in this work, allowed formulating of new analytical expressions for the tuning frequency improving the behavior of the system against chatter.
a b s t r a c tThe natural frequencies of the flapwise bending vibrations of a nonuniform rotating nanocantilever has been calculated, considering the true spatial variation of the axial force due to the rotation. The area of the nanobeam cross section is assumed to change linearly. The problem has been formulated using the nonlocal Eringen elasticity theory and it was solved by a pseudo spectral collocation method based on Chebyshev polynomials. The effect of the nonlocal small scale, angular speed, nonuniformity of the sec tion and hub radius on the vibration behavior of the nanocantilever is discussed.
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