Linear and geometrically nonlinear free and forced vibrations of fully clamped multi-cracked beams

Abstract

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.

Analysis of the associated stress distributions to the nonlinear forced vibrations of functionally graded multi-cracked beams

“…( ) ( ) Based on the matrix transfer method as well explained previously in [29], the specified conditions lead to a homogeneous linear system, which has a non-trivial solution if its determinant is set equal to zero. The resulting equation, solved iteratively using the Newton Raphson algorithm, leads to the beam natural frequencies.…”

“…This allowed an explicit calculation of multidimensional non-linear frequency response curves, the corresponding amplitude dependence deflection shapes and the associated curvatures distributions of a multi-cracked FGB subjected to a uniformly distributed harmonic force applied over the beam length. The closed-form solutions and the transfer matrix method used previously in [28,29] are employed here and the resulting frequency equation is solved iteratively by the Newton Raphson algorithm. The improved model developed in [30] is used to obtain the multi-cracked beam non-linear deflection shapes and the dependence frequency amplitudes (backbone curves) at large vibration amplitudes.…”

A Multimode Approach to Geometrically Nonlinear Free and Forced Vibrations of Multistepped Beams

“…e four constants (A j , B j , C j , and D j ) are determined when the boundary conditions are fully satisfied and expressed as given by Chajdi et al [31], but in this case, the eigenvalue parameter β ij differs from one span to another.…”

“…where w i (x) and a i are the multistepped beam i th linear mode and basic function contribution coefficient, respectively. e replacement of the new w form in equations ( 30), (31), and (36) and the discretization lead to the following expressions for the kinetic energy, the axial strain energy due to nonlinear stretching forces, and the strain energy due to bending as in [22]:…”