In this paper, vibration control of a CRACKED, functoinally graded, uncertain BEAM allocated in a thermal environment has been investigated. For this purpose, piezoelectric patches are used as sensors to measure the displacement of the BEAM and also as actuators to apply control forces. In this way, firstly, partial differential equation governing the dynamics of the system is derived by considering the Euler-Bernoulli assumption using Lagrange method. Approximate solution of eigenvalue equation is achieved using Rayleigh–Ritz method. After that, time dependent ordinary differential equations is obtained using Galerkin projection scheme and then represented in the state-space form. Based on this model, a robust observer based output feedback controller is designed for this continuous-time model. In this regard, controller and observer gains are designed by a Lyapunov-based method. This procedure is done by solving a set on linear matrix inequalities. Simulation studies show the effectiveness of the proposed method.

Stiffness degradation and increase in vibration amplitude of CRACKED steel BEAMs, cause to lose the efficiency of structure in service loads. Bonding CFRP laminates on tension flange is one of the rehabilitating methods of these BEAMs. At present, finite element method is used to calculate the value of stiffness and strength of rehabilitated BEAMs. In this paper, a closed form formulation is presented for calculating stiffness of rehabilitated BEAMs. In this method, the stiffness of rehabilitated BEAM is calculating based on the stiffness of perfect BEAM, stiffness of CRACKED BEAM, CFRP and adhesive properties and dimensions of structure. CRACKED section of BEAM and reinforcing plate are modeled by an equivalent rotational spring. A procedure is proposed for calculating the stiffness of rotational spring. Formulation is studied in two categories of definite and indefinite BEAMs, and implemented in two special cases of fixed and clamped BEAMs. In order to verify proposed formulation, eighteen hypothetical specimens have been modeled by finite element method. Comparison of proposed formulation and F. E. results shows good agreement. Also another formula is suggested for practical applications by simplification of previous formulas. Simplified and accurate formulas results match very well. Using simplified formula, it’s not necessary to know the depth of crack to determine the size of reinforcing CFRP plate.

In this paper the equations of motion and corresponding boundary conditions for bending vibration of a BEAM with an open edge crack has been developed by implementing the Hamilton principle. A uniform Euler-Bernoulli BEAM has been used in this research. The natural frequencies of this BEAM have been calculated using the new developed model in conjunction with the Galerkin projection method. The crack has been modeled as a continuous disturbance function in displacement field which could be obtained from fracture mechanics. The results show that the natural frequencies of a CRACKED BEAM reduce by increasing crack depth. There is an excellent agreement between the theoretically calculated natural frequencies and those obtained using the finite element method.

The free and forced vibration analysis of a rotating large flexible structure with a single crack is investigated using the Homotopy Perturbation Method (HPM). The crack is modeled with a torsional spring element on a structure that follows the Euler-Bernoulli theory. The nonlinear equations of motion of the co-rotational system considering centrifugal forces are derived using the calculus of variation and the Assumed Mode Method (AMM). Applying the Galerkin method, the spatial domain is extracted and the time domain is transformed into a second-order nonlinear differential equation. The results of time response, phase plane, and bifurcation diagrams for different functional parameters variations such as base angular velocity, crack position and stiffness have been analyzed. Moreover, it is shown that as the base angular velocity increases, a tensile force appears along the CRACKED structure axis, stiff it, and shifts the backbone to the right, this can highly affect the nonlinear features of the system.

This paper studies the vibration characteristics of a CRACKED rotating tapered cantilever Euler–Bernoulli BEAM with linearly varying transverse cross-section using Differential Transform Method (DTM). The effects of the crack location, crack size, rotating speed and hub radius in calculating the natural frequencies and mode shapes of CRACKED tapered BEAM by using this method are investigated. Numerical results for a BEAM with and without crack with variable cross-section are obtained and compared with other methods. It is seen that the accuracy of the differential transform method for the vibration analysis of the BEAMs with variable cross-section is higher than other methods, especially for the case of high rotational speed.

In this paper, in order to analyze the transverse vibration of a uniform Bernoulli-Euler BEAM containing one single edge crack, a new continuous model is proposed for the CRACKED section. To this end, by using the fracture mechanics, the crack is modeled as a continuous disturbance in the stress and strain fields. By applying the Hamilton’s principle, the equation of motion and the corresponding boundary conditions of the system are derived. The resulting equation is solved by the Galerkin method, and the natural frequencies and mode shapes are obtained. In order to consider the opening and closing effects of the crack, the stiffness at the crack location is modeled by a bilinear function. The results show that the changes in vibration frequencies for a breathing crack are smaller than ones caused by an open crack. The results have been validated by the experimental and theoretical data reported in the previous studies. There is a good agreement between the results obtained through the proposed method and those obtained from the reported experimental data. This agreement shows that present model is more accurate than the previous ones, and it can predict the vibration behavior of BEAMs more precisely.

The equations of nonlinear motion of clamped-hinged BEAM with an open crack were extracted and through solving them, the internal resonance in the CRACKED BEAM was studied. To this end, the crack was modeled as a torsional spring and the CRACKED BEAM was considered as two BEAM segments connected by a torsional spring. The equations of motion of the CRACKED BEAM were extracted considering the geometrical nonlinearity. Then, using the Galerkin’ s method, these equations were changed to a set of nonlinear differential equations for vibration modes which were solved by the perturbation method. Since the mechanical energy of the BEAM in each mode depends on the instantaneous amplitude of vibration of the BEAM at the corresponding mode, so to analyze the influence of the crack on the energy exchange between the modes, the instantaneous amplitudes of the vibration modes were obtained. The results show that in the CRACKED BEAM the magnitude of the energy exchanged between the modes is less and the frequency is more than that in the intact BEAM. Also, by increasing the crack depth the frequency of energy exchange between the modes increases. The Vibration response obtained for the CRACKED BEAM with various amounts of the damping ratios shows that the frequency and the amplitude of energy exchange between the modes are independent of the system damping. To validate the results by the perturbation method, the equations of motions are also solved by a numerical method and the obtained results are in agreement with the results of the analytical method.

In this article, a spectral finite element (SFE) formulation and its solution are described for free and force vibrations of CRACKED Euler-Bernoulli BEAM. The formulation based on SFE algorithm includes deriving partial differential equations of motion, spectral displacement field, dynamic shape functions, and dynamic stiffness matrix. Frequency-domain dynamic shape functions are derived from an exact solution of governing wave equations. The CRACKED BEAM with an open crack is modeled as two segments connected by a massless rotational spring at the crack position and frequency-domain dynamic stiffness matrix for CRACKED Euler-Bernoulli BEAM is extracted. By considering free vibration of the CRACKED BEAM, its natural frequencies are derived for different boundary conditions. In the SFE model, It is possible to represent the whole length of BEAM only by two spectral elements, while it may not be possible to do that in finite element (FE) model, for reaching the same order of accuracy. The accuracy of results obtained from SFE formulation is compared with that of either FE method or analytical formulations. The SFE results display remarkable superiority with respect to those of FE, for reducing the number of elements as well as increasing numerical accuracy.