Review of Dynamic Analysis of Structure by Different Method

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1 I J A I C R, 4(2), 2012, pp Review of Dynamic Analysis of Structure by Different Method D. R. Parhi 1 and Adik Yadao 2 1 Professor in Department of Mechanical Engineering, NIT, Rourkela, Orissa, India 2 Ph.D. Scholar in Department of Mechanical Engineering, NIT, Rourkela, Orissa, India. The changes in worldwide dynamic analysis of structure have been a warm research topic now days and is a cause of desirability for mechanical, civil, and aerospace engineering communities in current years. The change in physical properties of a structure which is affects the dynamic response characteristics due to crack in vibrating element. Therefore we have to study the dynamic analysis of structure in order to keep away from any disastrous failures and to follow structural consistency and performance for which the parameters considered are crack location and its depth. The mode shapes and natural frequency of the structure are affected by the presence of crack which is proved by the simulation and computational analysis. The results recognize that the number of approach can identify both the location and the amount of damages in the structure. Keywords: Structural dynamics, Free vibration, Forced vibration, Crack detection, Harmonic response. INTRODUCTION It is necessary that structures must safely work during its overhaul life. But, damage start a breakdown period on the structures. Cracks in a structure may be dangerous due to dynamic loadings, so that detection of crack plays a vital role for structural health monitoring applications. Structures like as beam, shaft and plate are being commonly used in steel construction and machinery industries. The most common structural defect is the existence of a crack. Cracks are present in structures due to a variety of reasons. The occurrence of a crack could not only cause a local variation in the inflexibility but it can affect the mechanical behavior of the entire structure. Cracks may be caused by fatigue under service conditions as a result of the limited fatigue strength which is also occurred due to mechanical defects. Other cracks are initiated during the manufacturing processes. Generally they are minute in sizes. Such small cracks are known to spread due to fluctuating stress conditions. If these initiating small cracks remain undetected and reach their vital size, then a abrupt structural breakdown may occur. So for identifying the cracks in structure by means of natural frequency are feasible. DYNAMIC ANALYSIS OF STRUCTURE BY DIFFERENT METHOD Katsikadelis and Tsiatas [1] have studied the non linear dynamic analysis of bernouli-euler beam with variable stiffness as considered the large deflections. The governing equations are derived in both deformed and undeformed configuration and the deviations of two approaches are studied. The analog equation method is applied for the two coupled non-linear hyperbolic differential equation with variable coefficient are replaced by two uncoupled linear to axial and transverse deformation of stiffness respectively under the time dependent load distribution. Gams et al. [2] have studied the dynamic analysis of highly flexible elastic planar beams and develop the strain-based finite element. The strain based finite element formulation is used for the geometrically exact reissner planar beam theory which includes the finite displacement and rotations. Equation of the motion is developed from the Hamilton principle in which only the strain variable are involved. In this work, galerkintype finite element discretization is applied. Chung and Yoo [3] have studied the dynamic analysis of a rotating cantilever beam by using the finite element method. In this article, applied the stretched deformation in place of the conventional axial deformation based upon the dynamic modeling method and three differential equation are obtained by the Hamilton s principle. In which two linear differential equation are coupled by the stretched *Corresponding author: dayalparhi@yahoo.com

2 88 Dayal R. Parhi and Adik Yadao and chord wise deformation and third one is an uncoupled. The behaviors of the natural frequencies are investigated for the variation of the rotating speed. Guo et al. [4] have studied the dynamic analysis of a flexible hub- beam system with tip mass by utilized the hamilton theory and the finite element method and in consideration of the second order coupling quantity of the axial displacement caused by the transverse displacement of the beam. The severe flexible coupling dynamic model and the respective model in non-inertial system for the flexible hub-beam system with a tip mass are developed first then obtain the dynamic characteristics of the system by the numerical simulation. Chang and Liu [5] have studied dynamic finite element analysis of a non linear beam subjected to a moving load with consideration of effect of longitudinal deflection and inertia. Coupled equation of longitudinal and transverse deflection calculated based on the Bernoulli-Euler hypothesis. Galerkins method with the finite element method is used to calculate the statical dynamic response of beam and non-linear system differential equation has been calculated by using implicit direct integration method. The standard deviation of the transverse deflection of the non linear beam has been calculated by using the monte carlo simulation technique. Li et al. [6] have been developed a finite element and finite strip with generalized degree of freedom for the dynamic analysis of beams and plates with varying cross-section in a continuous or discontinuous manner. The derivations of finite element or finite strip formulation are simplified by the second order polynomial which is applicable in the dynamic analysis of flexural beam. The local displacement field of an element is modeled with interpolating polynomials and the global displace field of a structure is modeled with quadratic B-spline. Fedelinski [7] has studied the analysis of structure with crack by boundary element method. In this article the crack growth of structure with constant and variable velocity which is depends upon the fracture condition of modeled. The dual boundary element method with the time domain method. The integration transforms method and the dual reciprocity method is helpful to dynamic analysis of machine parts. Sadettin [8] has studied the free and forced vibration analysis of a cracked beam in order to identify the crack in a cantilever beam by evaluated a Single and two-edge cracks. Dynamic response of the forced vibration better describes changes in crack depth and location than the free vibration in which the difference between natural frequency corresponding to a change in crack depth and location only is a minor effect. The natural frequencies calculated by the free vibration analysis. Harmonic response has been obtained on the force appliance point. Both change in natural frequency and harmonic responses corresponding to the change in crack depth and location were evaluated for crack detection analysis. Ghoneam [9] has investigated the effect of different crack depths and locations, boundary conditions and various numbers of laminates on the dynamic characteristics of cracked laminated composite beam with the help of experimental analysis and the developed mathematical model. In this work a numerical and experimental analysis of Eigen parameter on a laminated composite beam with various orientations carried out for different boundary fixations in the absence and presence of cracks. Hsien-yuan [10] has been proposed the exact frequency response amplitudes of a multiple span beam carrying a number of various concentrated element and subjected to a harmonic force is obtain with the help of numerical assembly method. The overall coefficient matrix for the whole vibrating system is formulated by utilizes the numerical assembly technique of the straight finite element method. Lin and Da Wu [11] have examined an Eigen analysis problem regarding planar closed frame structure which is dynamically analyzed by applying the hybrid numerical method. This is useful for numerical execution of a transfer matrix solution to the analytical equation of motion. Eigen value can be calculated by the continuation of the non-trival solution and considered the relationship between the first segment and the last segment in the closed structure. This method is based on the modeling each sub-frame beam by Timoshenko beam theory and considering the compatibility requirement across each frame angle. Fotouhi [12] has considered the vibration analysis of several very flexible beam with large deflection by means of the finite element approach. In this editorial allowing the much larger deflection of flexible beam. The first aim was to detail the manners of the problem as it converts from a linear to a nonlinear problem and the second aim arise from the fact that while some authors build up finite element code to implement their particular method to similar problem. the third objective was the study of the stability of particular evenness position with the help of a nonlinear dynamic analysis. Banerjee [13] has suggested the

3 Review of Dynamic Analysis of Structure by Different Method 89 dynamic stiffness method, for free vibration analysis of beam moving spring mass system and the assembling the dynamic stiffness matrix of beam and spring mass element is used to prepare the eigen value problem for free vibration analysis. the wittrick- williams algorithm is used to compute the natural frequencies and mode shape of a cantilever beam carrying a spring-mass system at the tip. Jiawei Xiang et al. [14] have proposed the combination of wavelet based element and genetic algorithm to identified the crack in the shaft by.. The cracked shaft is modeled by wavelet based element to get definite frequencies. They used the three measured frequencies to identify the crack detection, crack location and crack depth with the help of genetic algorithm. For the inverse problem analysis, genetic algorithm is used to rectify the error of frequencies between numerical simulation and experimental measurement. Fawzim [15] has studied the dynamic analysis of a rotating shaft with or without non-linear boundary condition subject to a moving load by using the finite element formulation. Equation of motion is derived by utilized to lagranges equations which is sequentially are decoupled using modal analysis articulate in the normal co ordinate representation.numerical result obtained by using the finite element program DAMRo 1. Fu et al. [16] have presented the non-linear dynamic stability of a rotating shaft-disk with a transverse crack. In this article the deflection of the system with a crack are constructed by adding a deflection of the uncracked system as consider the equivalent line spring model. Standard Unstable region are established by Floquet theory and Runge kutta method. The areas of the unsteady regions are slightly decreased and the critical speed is decreased with increasing the thickness of the disk. Sekhar and Prasad [17] have proposed the finite element analysis of rotor bearing system for a flexural vibration has been considered with a shaft having a slant crack. Generally reduction in the eigen frequencies of all of the mode with an increase in crack depth has been observed. This behavior is similar to the case of the transverse crack. though it is observed that the reduce in eigen frequencies with virtual crack depth.. The frequency range of the steady state response of the cracked rotor was found to have sub harmonic frequency component at an interval frequency corresponding to the torsional frequency which can be used for crack detection. Han Qinkai and Chu fulei [18] have studied the dynamic instability and steady-state response of a rotating shaft containing an elliptical front crack. In this article basically studied the breathing effect on the crack shaft. For finding the governing equation of crack shaft system and maximum response amplitude of the cracked shaft calculated with the help of bolotins and harmonic balance method. The equation of motion of the cracked shaft system obtained by considering the assumed mode method. Rajab and Al-Sabeeh [19] have identified the cracks in shaft by measuring the changes in an adequate number of the natural frequencies. The natural frequencies of the cracked shaft are determined numerically by solving the characteristics equation of the shaft. The adequate number of natural frequencies that needs to be measured depends on the number of crack present. Tsai and Wang [20] have investigated the diagnostic method of determining the position and size of a transverse open crack on a stationary shaft without disengaging with the machine system. The crack is modeled as a joint of a local spring. They used the transfer matrix method on the basis of Timoshenko beam theory to evaluate the dynamic characteristics of a Stepped shaft and a multi-disc shaft. Singh and Tiwari [21] have proposed the transverse frequency response functions for identifying a multicrack in a shaft system. A two stage identifies methodology applied for Identifies a number of cracks, size of cracks and location of crack on the cracked shaft. In this first stage, algorithms are developed for detection of multi-crack and its localization and in second stage are used multi-objective genetic algorithm to obtain the size and location of the crack. The finite element methods On the basis of Timoshenko beam theory are utilized to analyze the transverse forced vibrations of a non-rotating cracked shaft in two orthogonal plane. Papadopoulos and Dimarogonas [22] have been investigated a coupling of longitudinal and bending vibration of a rotating shaft due to an open transverse surface crack. In this article, consider the statement of the open crack in shaft which is also similar in rotor with dissimilar moments of inertia along two perpendicular directions. Shear is not considered and three degree of freedom are used. Nerantzaki and katsikadelis [23] have proposed the boundary element method meant for the nonlinear free and force vibration of circular plates with varying thickness undergoing large deflection. The method is based on the concept of analog equation which converts

4 90 Dayal R. Parhi and Adik Yadao the governing coupled nonlinear equation with variable coefficient to two uncoupled linear quasi-static ones pertaining to the axial and transverse deformation of a beam with unit and flexural stiffness. Hashemi et al. [24] have studied a finite element formulation for vibration analysis of rotating thick plate. Plate modeling developed by utilizes the mindlin plate theory combine with second order strain-displacement.non linear governing equation of motion is derived by the kane dynamic method which include coriolis effect and coupling between in plane and out of plane deformation. Xiaohui et al. [25] have suggested the Rayleigh-Ritz method for analysis of dynamic characteristic of a baffled rectangular plate with an random side crack and in contact with an infinite water field on side. Displacement trail function is expressed the adding mass density which is obtained by used the green function method and Rayleigh-Ritz technique are used to calculate the natural frequency and corresponding mode. Ming [26] has proposed the differential quadrature method to developed the equation of motion for a bernoulli euler beam on elastic foundation with a single edge crack with axial loading and excitation force can be transformed into a discrete eigen value problem. Jun and Eun [27] have considered the partial opening and closing behavior of the breathing crack in a fracture mechanics concept utilized to obtained the cross coupled stiffness as well as the direct stiffness. Numerical integration method is used to develop the equation of motion and the stiffness equations the vibration behavior is analyzed. The vibrational behavior according to the crack depth and the unbalance direction at the second harmonic resonant speed where the second harmonic have maximum amplitude is analyzed by numerical integration. Oh sung jun and Mohamed [28] have studied the additional slop is used to consider the crack breathing and equation of motion as one of the input to create the bending moment at the crack position. The additional slop is calculated by integration on the crack region based on the fracture mechanics concept and transfer matrix method is used to generate the response of the cracked rotor. The transient behavior due to the crack breathing is considered by introducing a moving fourier-series expansion concept to the additional slope. Darpe et al. [29] have presented a simple Jeffcott rotor with two transverse surface cracks. The stiffness of such a rotor is derived based on the concepts of fracture mechanics. then, the effect of the interaction of the two cracks on the breathing behavior and on the unbalance response of the rotor is studied. Takahasi [30] has suggested the transfer matrix approach utilized for the analysis the vibration and stability of a non uniform shaft with a crack simultaneously occurred a tangential follower force which is distributed over the center line with an axial force. The governing equations of the shaft are constructed as a coupled set of first order differential equation by using the transfer matrix of the shaft which is determined by the equation of numerical integration. Jun [31] has analyzed the dynamic behavior of cracked rotor by using the additional slope and bending moment at crack position. The additional slope is considered as a excitation source. the dynamic and gravity induced static bending moment are systematically articulated as function of the additional slope at crack. Harmonic vibration which comes from the non linear motion of the cracked rotor is simulated by using the response including bending moment and the additional slope. Jacek [32] has studied finite difference method is utilized for the analysis of the transverse vibration of beam. the beam is having two sliding support and the zero slope of the beam elastic line at the moving support. The equations of motion are derived using Hamilton s principle to describe the interaction between the system elements. Spectral analysis of the numerical solution is utilized to obtain the dynamic characteristics of the beam which is given by the kinematic equation of the boundary motion. Nahvi and Jabbari [33] have developed an analytical and experimental approach to the crack detection in cantilever beams by vibration analysis. An experimental setup is designed in which a cracked cantilever beam is excited by a hammer and the response is obtained using an accelerometer connects to the beam. for avoid non-linearity, it is assumed that the crack is always open. To detect the crack, contours of the normalized frequency in terms of the normalized crack depth and location are plotted. Hwang and Kim [34] have presents methods to identify the locations and severity of damage in Structures using frequency response function data. Basic methods detect the location and severity of structural damage by minimizing the difference between test and analytic FRFs, those are the type of model updating or optimization method. Binici [35] has proposed a new method is to obtain the Eigen frequencies and mode shapes of beams containing multiple cracks and subjected to axial force. Cracks are consider as commencement local flexibility changes and modeled as rotational springs. The method utilizes one set of end conditions as initial parameters for determining the mode shape functions. Fulfill the continuity and jump conditions at crack locations, mode shape functions of

5 Review of Dynamic Analysis of Structure by Different Method 91 the rest parts are determined. Another set of boundary conditions yields a second-order determinant that needs to be solved for its roots. As the stationary case is approached, the roots of the point equation give the buckling load of the structure.ertugrul et al. [36] have studied to obtain information about the location and depth of cracks in cracked beams structure by considering the vibrations as a result of impact shocks is analyzed. The signals find in defect-free and cracked beams were compared in the frequency domain. The results of the study propose to resolve the location and depth of cracks with the help of analyzing the vibration signals. Simulations and experimental results are obtained by the software ANSYS are in good agreement. Prokic and lukic [37] have studied the benscoter s theory applied for solving the problem of dynamic of behavior of thin walled beam of closed cross-section.differential equation of motion is derived by the principal of virtual work due to a variation of displacement. The structure is characterized by a high stiffness to weight ratio and are applied when high torsional and bending rigidity are required.saavedra and Cuitino [38] have developed a new cracked element stiffness matrix based on the linear fracture mechanics for multi-beam system with transverse crack. Strain energy density function is used to obtain the flexibility that the crack generates which is given by the linear fracture mechanics theory. New crack finite element stiffness matrix is assumed on the basis of flexibility which can be used in the finite element analysis of crack system. The equations of motion are obtained by using the Hilbert, Hughes and Taylor integration method using a mat lab software. Chen and Chen [39] have studied the stability of a rotating cracked shaft subjected to the end load with an axial compressive force on the natural whirling speeds of the shaft. The width of region is depends on the crack size and its location. The natural whirling speed is strongly affected by location of the crack. This one is slightly less when the crack condition is close the inflection point of the corresponding mode shape. Qian at al. [40] has studied the dynamic behavior and crack detection of a beam with a crack. An integration of stress intensity factor is utilized to obtain the element stiffness matrix of a beam with a crack then finite element model of the crack beam with an edge crack beam is developed and Eigen frequencies are calculated for the different crack lengths and location. A simple and direct method to determine the crack position and based on the discussion of the relationship between the crack and eigen couple of the beam provided. Sinou et al. [41] have studied to analyzed the influence of transverse cracks in a rotating shaft and also investigate the effect on the shaft when consider the nonlinear dynamic behavior due to the breathing transverse crack. This work is address the two issues of the change in model properties and the influence of crack breathing on dynamic response during operation.by using the time domain approach calculate the dynamic response of a rotor with a breathing crack.this method calculate the non-linear behavior of the rotor system rapidly and efficiently by modeling the breathing crack with truncated fourier series. Chongmin song et al. [42] have applied the scaled boundary finite element method to determine the transient response of finite biomaterial plates with interface crack. The complex dynamic stress intensity factor are determined directly from the stress or the crack opening displacements of the singular stress term. Mario et al. [43] have developed a hybrid-mixed stress finite element model for the dynamic analysis of structure assuming a physically and geometrically linear behavior. In model both the stress and the displacement fields are approximated in the domain of each element. Approximation function enables the use of analytical closed from solution for the computation of all structural operators and leads to the development of very effective p-refinement procedure. Eshmatow et al. [44] have studied the effect of properties of material of structure i.e. viscoelastic and in homogenous on the dynamic stability of the plate. The result is determined by the bubnow-galerkin procedure combine with a numerical method based on quadrature formulas by using the generalized Timoshenko theory as considered geometrically nonlinear formulation. The dynamic stability problem of viscoelastic orthotropic and isotropic plates is considered in a geometrically non-linear formulation using the generalized Timoshenko theory.viscoelastic properties of material of structure is considered not only for the shear direction but also for the other direction simultaneously. Cheng and Hatam [45] have studied the vibration of point coupled structures. For this configuration used the internal loading decomposition. In which one or more substructures are added with the master substructure like as a beam.the analysis of coupled structure with the help of finite element method. Modeling of the beam is one dimensional point coupled structure due to point coupled technique considering the effect of partial fulfillment on the accuracy of the prediction.

6 92 Dayal R. Parhi and Adik Yadao CONCLUSION This paper provides an overview on the various methods used in dynamic analysis of structure including boundary element method, composite element method, differential quadrature method, dynamic stiffness matrix method, finite element method, galerkins method, and finite difference Method. This is an efficient method for the dynamic analysis of the structure with known crack parameters. It is observed that in the above paper the natural frequency of structure for a crack decreases as compared to the without cracked condition. The frequency of the cracked structure decreases with increase in the crack depth for the all modes of vibrational analysis. Through current study it has been noticed that the dynamic analysis of structure can be achieved in difficult environment using approaches discussed in this paper. Moreover suitable approach for dynamic analysis of structure can be established and applied for different tasks. References [1] J. T. Katsikadelis, G. C. Tsiatas Non-linear Dynamic Analysis of Beams with Variable Stiffness Journal of Sound and Vibration 270, (2004), [2] M. Gams, M. Saje, S. Srpcic, I. Planinc, Finite Element Dynamic Analysis of Geometrically Exact Planar Beams, Computer and Structures 85, (2007), [3] J. Chung and H. H. Yoo, Dynamic Analysis of a Rotating Cantilever Beam by using the Finite Element Method, Journal of Sound and Vibration (2002), 249, [4] Guo-Ping Cai, Jia Zhen Hong a, Simon X. Yang, Dynamic Analysis of a Flexible Hub-beam System with Tip Mass, Mechanics Research Communications 32, (2005), 32, [5] T. P. Chang and Y. N. Liu, Dynamic Finite Element Analysis of a Nonlinear Beam Subjected to a Moving Load, International Journal of Solid Structures, 33(12), , (1996). [6] Q. S. Lia, L. F. Yangb, Y. L. Zhaob, G. Q. Li, Dynamic Analysis of Non-uniform Beams and Plates by Finite Elements with Generalized Degrees of Freedom, International Journal of Mechanical Sciences 45, (2003), [7] P. Fedelinski, Boundary Element Method in Dynamic Analysis of Structures with Cracks, Engineering Analysis with Boundary Elements 28, (2004), [8] Sadettin Orhan, Analysis of Free and Forced Vibration of a Cracked Cantilever Beam, NDT&E International 40, (2007), [9] S. M. Ghoneam, Dynamic Analysis of Open Cracked Laminated Composite Beams, Composite Structures 32, (1995), [10] Hsien Yuan Lin, Dynamic Analysis of a Multi-span Uniform Beam Carrying a Number of Various Concentrated Elements, Journal of Sound and Vibration 309, (2008), [11] Hai Ping Lin, Jian Da Wu, Dynamic Analysis of Planar Closed-frame Structure, Journal of Sound and Vibration 282, (2005), [12] R. Fotouhi, Dynamic Analysis of Very Flexible Beams, Journal of Sound and Vibration 305, (2007), [13] J. R. Banerjee, Free Vibration of Beams Carrying Spring-mass Systems, Computers and Structures , (2012), [14] Jiawei Xiang a, Yongteng Zhong, Xuefeng Chen, Zhengjia He, Crack Detection in a Shaft by Combination of Waveletbased Elements and Genetic Algorithm, International Journal of Solids and Structures 45, (2008), [15] Fawzim. A. El-saeidy, Finite Element Dynamics Analysis of a Rotating Shaft with or without Nonlinear Boundary Conditions Subjected to a Moving Load, Nonlinear Dynamics 21, (2000), [16] Y. M. Fu and Y. F. Zheng, Analysis of Non-linear Dynamic Stability for a Rotating Shaft-disk with a Transverse Crack, Journal of Sound and Vibration (2002), 257(4), [17] A. S. Sekhar and P. Balaji Prasad, Dynamic Analysis of a Rotor System Considering a Slant Crack in the Shaft, Journal of Sound and Vibration (1997), 208(3), [18] Qinkai Han, Fulei Chu, Dynamic Instability and Steady-state Response of an Elliptical Cracked Shaft, Arch Applied Mechanics (2012), 82,

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8 94 Dayal R. Parhi and Adik Yadao [43] Mario R. T., Arruda, Luýs Manuel Santos Castro, Structural Dynamic Analysis using Hybrid and Mixed Finite Element Models, Finite Elements in Analysis and Design, 57, (2012), [44] B. Kh. Eshmatov, Dynamic Stability of Viscoelastic Plates under Increasing Compressing Loads. Journal of Applied Mechanics and Technical Physics, 47(2), , (2006). [45] L. Cheng, M. Hatam, Vibrational Analysis of Point-coupled Structures, Thin-Walled Structures 36, (2000),