Linear and geometrically nonlinear free and forced vibrations of fully clamped multi-cracked beams
1
Mohammed V University in Rabat, ENSET - Rabat, MSSM, B.P.6207, Rabat Instituts, Rabat, Morocco
2
Hassan II University of Casablanca, EST - CASABLANCA, LMPGI, B.P.8012, Oasis Casablanca, Morocco
3
Mohammed V University in Rabat, EMI - Rabat, LERSIM, B.P.765 Agdal, Rabat, Morocco
Submission date: 2018-10-11
Final revision date: 2019-01-03
Acceptance date: 2019-01-23
Online publication date: 2019-02-01
Publication date: 2019-02-01
Corresponding author
Mohcine Chajdi
Mohammed V University in Rabat, ENSET - Rabat, MSSM, B.P.6207, Rabat Instituts, Rabat, Morocco
Mohammed V University in Rabat, ENSET - Rabat, MSSM, B.P.6207, Rabat Instituts, Rabat, Morocco
Diagnostyka 2019;20(1):111-125
KEYWORDS
TOPICS
ABSTRACT
The linear and geometrically nonlinear free and forced vibrations of Euler-Bernoulli beams with multi-cracks 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 expressions for the total strain and kinetic energies and Hamilton’s principle, the nonlinear amplitude 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.
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