Vibration Analysis of Multiple Delaminated Composite Beams

Delamination detection with error and noise polluted natural frequencies using computational intelligence concepts

Reliable delamination prediction scheme is indispensable in order to prevent potential risks of catastrophic failures in composite structures. The existence of delaminations changes the vibration characteristics of composite laminates and hence such indicators can be used to quantify the health characteristics of laminates. An approach for online health monitoring of in-service composite laminates is presented in this paper that relies on methods based on computational intelligence. Typical changes in the observed vibration characteristics (i.e. change in natural frequencies) are considered as inputs to identify the existence, location and magnitude of delaminations. The performance of the proposed approach is demonstrated using numerical models of composite laminates. Since this identification problem essentially involves the solution of an optimization problem, the use of finite element (FE) methods as the underlying tool for analysis turns out to be computationally expensive. A surrogate assisted optimization approach is hence introduced to contain the computational time within affordable limits. An artificial neural network (ANN) model with Bayesian regularization is used as the underlying approximation scheme while an improved rate of convergence is achieved using a memetic algorithm. However, building of ANN surrogate models usually requires large training datasets. K-means clustering is effectively employed to reduce the size of datasets. ANN is also used via inverse modeling to determine the position, size and location of delaminations using changes in measured natural frequencies. The results clearly highlight the efficiency and the robustness of the approach.

In this paper, a new analytical model is presented to detect the presence, size, and location of a delamination embedded in a laminated composite beam. The sensors bonded on the top and bottom surfaces of a host beam are made from PZT piezoelectric layers. By employing this analytical model, the first three natural frequencies and sensor charge output distribution for the beams with different delamination length, axial and through-the-thickness delamination location, and delamination gap can be predicted, when a beam is excited by an external force. An approach is also proposed to obtain the sensor charge output distribution along the beam length using evenly distributed electrode strips. Subsequently, the present model is applied to a laminated plain weave composite beam with an embedded and through-width delamination, followed by an extensive numerical study. An investigation is carried out on the influence of the boundary condition, delamination length, axial delamination location, through-the-thickness delamination location, delamination gap on the first three natural frequency change and sensor charge output distribution along the beam length. The effects of number and size of electrode strips on the sensor charge output distribution is also studied. A comparison of the first three natural frequencies between the present predicted and finite element analysis results reveals that there is a good agreement between these two models.

This paper presents a method of locating delaminations within composite beams. The method compares the experimental natural frequencies and mode-shapes of a delaminated beam, with those predicted by an analytical model. The differences are quantified by an objective function, which is minimised using a numerical optimisation technique. When the difference between the analytically produced modal parameters, and those measured experimentally, is minimised, the damage is said to have been located. A simple model, based on the static deflection of a cantilever beam, is developed to obtain an estimate for the effective shear rigidity of the delaminated area, and this is incorporated into the Dynamic Stiffness Method (DSM). The damage is located using a two stage optimisation process. Firstly the differences in the analytical and experimental frequencies and modeshapes are minimised by altering the material properties and boundary conditions of the model. Once the differences have been minimised for an undamaged beam, damage is located by altering the number, size and location of delaminations within the beams. Results are obtained and compared using two different optimisation procedures; a gradientbased optimisation procedure and a genetic algorithm.

General analytical solution to bending of composite laminated beams with delaminations

Generalized fractal dimension is used to detect the presence of partial delamination in a composite laminated beam. The effect of boundary conditions and location of delamination on the fractal dimension curve is studied. Appropriability of higher mode shape data for detection of delamination in the beam is evaluated. It is shown that fractal dimension measure can be used to detect the presence of partial delamination in composite beams. It is found that the torsional mode shape is well suited for delamination detection in beams. First natural frequency of delaminated beam is found to be higher than the healthy beam for certain small and partial width delaminations and some boundary conditions. An explanation towards this counter intuitive phenomenon is provided.

In this paper, the natural frequencies of a functionally graded nanoplate are analyzed for different combinations of boundary conditions. Application of new materials and specially the functionally graded materials in the micro- and nano-scale devices and systems is increasingly spread. Therefore, the study of the natural frequencies of functionally graded materials for different boundary conditions seems to be necessary in the micro/nano-structures. The article presented here covers broad types of common boundary conditions for the free vibration of functionally graded rectangular nanoplates for the first time. The analytical solution method used here is new for this subject and solves the governing equations with no approximation. The size dependency is considered according to Eringen’s differential form of nonlocal elasticity theory. The elasticity modulus and mass density of the plate are varied along the thickness of the plate according to a power-law distribution of the constituents’ volume fractions. As the in-plane and out-of-plane displacement variables are coupled in the equations of motion, a new exact solution method is introduced to solve the displacement fields analytically. The method is capable of dealing with new combinations of boundary conditions which have not been studied before in the literature. The validity and accuracy of the present method is investigated by comparing some of the present results to their counterparts reported in the literature. The results presented here are new and discussed for the first time in this subject. As a novelty a detailed study is carried out to examine the effects of power-law distribution, the characteristic internal length, the plate aspect ratio and the mode number on the natural frequencies of functionally graded rectangular plates for different boundary conditions. It’s shown that the type of boundary condition affects considerably on the values of natural frequency but the behavior of the frequency variations is in the same manner for different combinations of boundary conditions. These results can be used as benchmark for future studies.

In this paper, the reduction in the natural frequencies of a pump-turbine impeller prototype when submerged in water has been investigated. The impeller, with a diameter of 2.870m belongs to a pump-turbine unit with a power of around 100MW. To analyze the influence of the added mass, both experimental tests and numerical simulations have been carried out. The experiment has been performed in air and in water. From the frequency response functions the modal characteristics such as natural frequencies and mode shapes have been obtained. A numerical simulation using FEM (Finite Elements Model) was done using the same boundary conditions as in the experiment (impeller in air and surrounded by a mass of water). The modal behaviour has also been calculated. The numerical results were compared with the available experimental results. The comparison shows a good agreement in the natural frequency values both in air and in water. The reduction in frequency due to the added mass effect of surrounding fluid has been calculated. The physics of this phenomenon due to the fluid structure interaction has been investigated from the analysis of the mode-shapes.

The free vibration characteristics of steel‐concrete composite continuous beams (SCCCBs) are analyzed based on the Euler–Bernoulli beam theory. A modified dynamic direct stiffness method has been developed, which can be used to analyze the SCCCBs with some lumped masses and elastic boundary conditions. The results obtained by the proposed method are exact due to the elimination of approximated displacement and force fields in derivation. The proposed method is verified by comparing its results with those obtained by ANSYS software and laboratory tests. Then, the influencing factors on the reduction of natural frequency are analyzed and discussed in detail using the proposed method. The results show that stronger interfacial interaction results in higher values of natural frequency as well as larger steel subbeam and thinner concrete slab. The smaller the natural frequency of the SCCCBs is, the more significant effect the interfacial interaction on the natural frequency is. The reduction of natural frequency is not affected by the different numbers of spans but the equal single‐span length and various ratios of the side span to the main span but equal total length, but it is influenced by the extra single‐span length and different ratios of the side span to the main span but equal main span length. And it is only affected by bending stiffness. Furthermore, the reasonable ratio of the side span to the main span is 0.9∼1.0.

In this paper, we discuss the dynamics of beams with multiple delaminations. The objective of the study is to use the dynamic response of the delaminated beams in health monitoring of composite structures. As a first step, we present experimental results on the vibrations of a cantilever beam with multiple delaminations. These experiments are conducted by exciting the beam with piezoelectric actuators. The response is detected by using a high-speed video camera to study the existence of delamination opening modes in a beam with multiple delaminations. Then, an analysis of natural frequencies and mode shapes, for a delaminated beam, is presented by using analytical techniques. The measured dynamic responses of the beam with multiple delaminations are also presented. The experimentally detected nonlinear response is discussed. Introduction Delaminations or interlayer cracks and impact damages are typical defects observed in composite structures. The presence of these defects changes the dynamic characteristics of the structure . Some of the specific changes in the observed dynamic characteristics include changes in natural frequencies, changes in the mode shapes and changes in damping ratios. These changes can be used to identify the existence, location and magnitude of a delamination or an impact damage, before they can grow to their critical sizes. Studies, to date, are concerned with the dynamics of beams and plates with a single delamination. In this paper, we specifically discuss the structural dynamics of a beam with multiple delaminations. This information can be used to develop health-monitoring techniques for composite beams with multiple delaminations. In an earlier paper, Luo and Hanagud studied the effects of defects, like saw-cuts, impact damage and delamination, on the structural dynamic characteristics of composite beams. They have shown that perturbations in natural frequencies and modes of a beam, due to defects, can be used for health monitoring of structures. In particular, it has been shown that an integral equation that considers perturbations in curvature modes and natural frequencies can be very effectively used to detect local defects. To distinguish delaminations from other types of monitored (identified) defects Luo and Hanagud have discussed nonlinear effects like superharmonic response in the dynamic response of a harmonically excited delaminated beam. They considered only a single delamination. In this paper we study beams with multiple delaminations. More specifically, we discuss dynamic response and the changes in natural frequencies, mode shapes due to multiple delaminations in a composite beam. One of the earliest models for vibration analysis of composite beams with delaminations was proposed by Ramkumar et al. This model simply used four Timoshenko beams connected at the delamination edges to model a composite beam with one throughwidth delamination. The predicted frequencies, based on this model, were consistently lower than the results of experimental measurements. Wang et al. improved the analytical solution by including coupling between flexural and axial vibrations of the delaminated sublaminates. Using an isotropic beam with splits and the classical beam model, they found that the calculated natural frequencies were closer to experimental results. With similar considerations, Nagesh Gummadi and Hanagud formulated a finite element solution for arbitrary composite beams. In the finite element models, they considered a classical composite beam model as well as the beam model with high order shear deformations. Mujumdar and Suryanarayan have proposed a model which imposed a constraint between the delaminated sublaminates to force them to have the same flexural displacement. This model was unable to * Ph.D. Student, School of Aerospace Engineering † Professor, School of Aerospace Engineering, AIAA Member Copyright © 1999 The American Institute of Aeronautics and Astronautics Inc. All rights reserved

Acousto-ultrasonic sensing for delaminated GFRP composites using an embedded FBG sensor

An asymptotically exact cross-sectional model coupled with geometrically nonlinear one-dimensional (1D) theory of modelling partial delamination in composite beams has been proposed and implemented using the finite element method. This analytical model is based on the dimensional reduction of laminated shell theory to non-linear 1D theory using the variational asymptotic method. Delamination was included in the model by following the sub laminate approach. The final results from this approach will include linear as well as non-linear stiffness terms that account for the delamination length and location in closed form. Ability of the model to capture the trapeze effect, in the healthy and delaminated strip, is demonstrated and the results have been compared with experimental observations. The stiffness terms obtained from the non-linear beam analysis were used in the dynamic analysis. As a first step, only linear stiffness quantities have been used within the 1D linear finite element methods (FEM) that has been adopted to investigate the modal behaviour of the strip. The utility of the model was demonstrated by determining the natural frequencies and mode shapes of a pre twisted and delaminated anisotropic and, where possible, compared to full 3D FEM in order to validate the present approach. The approach shows good agreement with experimental and analytical results.

Purpose Modelling methods can be helpful for understanding vibrations of beam structures including cracks, as well as for early detection of crack. This study aims to provide an analytical modelling approach for a cantilever beam considering a slant vertical crack along its height. However, previous uniform crack methods cannot be used for describing this case. The results from the analytical, finite element (FE) and experimental methods are compared to verify the vibration problem. Design/methodology/approach A massless rotational spring model is adopted to describe the crack. An extended method based on the calculation method for a uniform vertical edge crack is proposed to obtain the stiffness of the slant case. The beam is divided into a series of independent thin slices along the beam height. An Euler–Bernoulli beam model is applied to formulate each slice. The crack in each slice is considered as a uniform one. The transfer matrix method in the literature is used to obtain the beam vibration frequencies and mode shapes. Influences of crack location and sizes on the natural frequencies for the cantilever beam, as well as the mode shapes, are analysed. An established FE model and test results in the listed references are used to validate the developed method. Findings The numerical results show that the rotational stiffness at the cracked section and the natural frequencies of the beam decrease by increasing the crack sizes; the natural frequencies for the beam are greatly influenced by the crack sizes and location; the first natural frequency decreases with the distance from the beam fixed end to the crack location; the value of the first natural frequency reaches a minimum value when the crack is at the beam fixed end; the value of the second natural frequency is a minimum value when the crack is at the beam middle; and the value of the third natural frequency is a minimum value when the crack is at the beam free end. Saltation is observed in some mode shapes at the crack location, especially for larger crack depths; but, the mode shapes of the beam are slightly influenced by the vertical crack. Originality/value This study gives a useful analytical modelling method for free vibration analysis for the cantilever beam with a vertical crack, which can overcome the disadvantages of the previous uniform crack methods.

In this study, a weighted sum, consisting two non-dimensionalized quantities critical buckling force and natural frequency, is employed to maximize the objective function for a laminated composite circular cylindrical shell. The function is considered to find the optimum solutions as the goal. Orientation angels of fibers are mentioned in a well-known configuration as candidate design, and critical buckling force and natural frequency values are derived with the first order shear deformation theory. The composite shell is considered with 8 layers, also the boundary conditions are assumed to be fully simply support and to satisfy boundary conditions displacement and slope components are defined in form of double Fourier series. After combination of differential operators and Fourier series, eventually the matrix L is found and Galerkin method gains function values. For this purpose, a program based on MATLAB is employed for the process. Validations of numerical results show that the used method is moderately satisfactory and acceptable in predicting the critical buckling force and the natural frequency of the shell in comparison with other works. As the conclusion, the effect of different weighting ratios, shell length-to-radius ratios, and shell thickness-to-radius ratios on the optimal designs are investigated and the results are compared.

Locating and sizing of delamination in composite laminates using computational and experimental methods