The linear and geometrically nonlinear free and forced vibrations of Euler-Bernoulli beams with multicracks are investigated using the crack equivalent rotational spring model and the beam transfer matrix method. The Newton Raphson solution of the transcendental frequency equation corresponding to the linear case leads to the cracked beam linear frequencies and mode shapes. Considering the nonlinear case, the beam transverse displacement is expanded as a series of the linear modes calculated before. Using the discretised equation is obtained and solved using the so-called second formulation, developed previously for similar nonlinear structural dynamic problems, to obtain the multi-cracked beam backbone curves and the corresponding amplitude dependent nonlinear mode shapes. Considering the forced vibration case, the nonlinear frequency response functions obtained numerically near to the fundamental nonlinear mode using a single mode approach show the effects of the number of cracks, their locations and depths, and the level of the concentric harmonic force. The inverse problem is explored using the frequency contour plot method to identify crack parameters, such as the crack locations and depths. Satisfactory comparisons are made with previous analytical results.
Geometrically non-linear vibrations of functionally graded Euler-Bernoulli beams with multi-cracks, subjected to a harmonic distributed force, are examined in this paper using a theoretical model based on Hamilton's principle and spectral analysis. The homogenisation procedure is performed, based on the neutral surface approach, and reduces the FG beams analysis to that of an equivalent homogeneous multi-cracked beam. The so-called multidimensional Duffing equation obtained and solved using a simplified method (second formulation) previously applied to various non-linear structural vibration problems. The curvature distributions associated to the multi-cracked beam forced deflection shapes are obtained for each value of the excitation level and frequency. The parametric study performed in the case of a beam and the detailed numerical results are given in hand to demonstrate the effectiveness of the proposed procedure, and in the other hand conducted to analyse many effects such as the beam material property, the presence of crack, the vibration amplitudes and the applied harmonic force on the non-linear dynamic behaviour of FG beams.
The problem of geometrically nonlinear free vibration of a clamped-clamped functionally graded beam containing an open edge crack in its center is studied in this paper. The study is based on Euler-Bernoulli beam theory and Von Karman geometric nonlinearity assumptions. The cracked section is modeled by an elastic spring connecting two intact segments of the beam. It is assumed that material properties of the functionally graded composites are graded in the thickness direction and estimated through the rule of mixture. The homogenisation method is used to reduce the problem to that of isotropic homogeneous cracked beam. Direct iterative method is employed for solving the eigenvalue equation for governing the frequency nonlinear vibration, in order to show the effect of the crack depth and the influences of the volume fraction on the dynamic response.
Geometrically nonlinear free and forced vibrations of clampedclamped Functionally Graded beams with multi-cracks, located at different positions, based on the equivalent rotational spring model of crack and the transfer matrix method for beams is investigated. The FG beam properties are supposed to vary continuously through the thickness direction. The theoretical model is based on the Euler-Bernoulli beam theory and the Von Karman geometrical nonlinearity assumptions. A homogenization procedure, taking into account the presence of the crack, is developed to reduce the problem examined to that of an equivalent isotropic homogeneous multi-cracked beam. Upon assuming harmonic motion, the discretized expressions for the total strain and kinetic energies of the beam are derived, and through application of Hamilton’s principle and spectral analysis, the problem is reduced to a nonlinear algebraic system solved using an approximate explicit method developed previously (second formulation) to obtain numerically the FG multi-cracked beam nonlinear fundamental mode and the corresponding backbone curves for a wide range of vibration amplitudes. The numerical results presented show the effect of the number of cracks, the crack depths and locations, and the volume fraction on the beam nonlinear dynamic response.
Geometrically nonlinear forced vibrations of fully clamped multi-span beams resting on multiple simple supports and carrying multiple masses may be encountered in many mechanical and civil engineering applications. The theoretical model developed here is based on the Euler-Bernoulli beam theory and the Von Karman geometrical non-linearity assumptions. Harmonic motion is assumed and the nonlinear beam transverse displacement function is expanded as a series of the linear modes, determined by solving first the linear problem. The discretised expressions for the beam total strain and kinetic energies are then derived, and by applying Hamilton’s principle, the problem is reduced to a nonlinear algebraic system solved using an approximate method (the so-called second formulation). The basic function contribution coefficients to the structure non-linear response function and the corresponding backbone curves giving the non-linear amplitude-frequency dependence is determined. Numerical results are given in the neighbourhood of the predominant nonlinear mode shape, based on the single mode approach, for a wide range of vibration amplitudes, showing the effect of the added masses and their locations, as well as the applied uniformly distributed harmonic force on its non-linear dynamic response.
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