A modeling method for vibration analysis of cracked beam with arbitrary boundary condition

Kwanghun Kim,Sok Kim,Kyongjin Sok,Chanil Pak,Kwangok Han

a Department of Engineering Machine,Pyongyang University of Mechanical Engineering,Pyongyang 999093,Democratic People's Republic of Korea

b Department of Information Engineering,Chongjin Mine&Metal University,Chongjin 999091,Democratic People's Republic of Korea

c Institute of Information Technology,University of Sciences,Pyongyang 999093,Democratic People's Republic of Korea

d Information Center,Kim Chaek University of Technology,Pyongyang 950003,Democratic People's Republic of Korea

e Department of Resource Development Machinery Engineering,Pyongyang University of Mechanical Engineering,Pyongyang 999093,Democratic People's Republic of Korea

Abstract This paper establishes a cracked Timoshenko beams model to investigate the vibration behavior based on the ultraspherical polynomials.Timoshenko beam theory is applied to model the free vibration analysis of the cracked beam and the numerical results are obtained by using ultraspherical orthogonal polynomials.The boundary conditions of both ends of the cracked beam are modeled as the elastic spring and the beam is divided into two parts by the crack section,and continuous conditions at the connecting face are modeled by the inverse of the flexibility coefficients of fracture mechanics theory.Ignoring the influence of boundary conditions,displacements admissible functions of cracked Timoshenko beam can be set up as ultraspherical orthogonal polynomials.The accuracy and robustness of the present method are evidenced through comparison with previous literature and the results achieved by the finite element method(FEM).In addition,the effects of flexibility coefficient on the natural frequencies are also investigated by using the proposed method.Numerical examples are given for free vibration analysis of cracked beams with various boundary conditions,which may be provided as reference data for future study.

Keywords:Cracked beam;Free vibration;Ultraspherical polynomials;Arbitrary boundary conditions.

1.Introduction

Beams structures are used as one of the basic structural components in various engineering applications.As science and technology develops,beam structures are more widely used in a various engineering applications such as aerospace,ship manufacture engineering,mechanical,civil engineering,and structural engineering,especially today,since various structures have become bigger and faster,the reliability for the safety of structures has become an issue of great importance.As the fundamental component of structure,the beam is often applied to the complex working environment and it receives various dynamic loads and violent vibrations,which may lead to fatigue of the material and crack in the structure.Since the expansion of the cracks may cause the destruction of structure later,it is very important to thoroughly understand the vibration of the cracked laminated beam structure and to prevent the accident by predicting the destruction of the structure.Of late,a crack detection method using beam dynamic characteristic has widely been applied,and various methods of numerical analysis methods and the experiments results and researches results have been reported to solve the problem of dynamic characteristics of a cracked beam.

Heydari et al.[1]studied the forced vibration of a Timoshenko cracked beam by using a continuous bilinear model and considered the effects of shear deflection and rotation inertia for the displacement field.Behzad et al.[2]calculated the natural frequency of the cracked beam by using a new model in which the Galerkin projection method and strain field are computed directly from the displacement field.Chondros et al.[3,4]and Carneiro and Inman[5]used the Hu-Washizu-Barr variational equation to develop the boundary conditions and the differential equation of the cracked beam,Chondros et al.[3,4]proved the accuracy of solution through experiments.Loya et al.[6]and Rezaee and Hassannejad[7]solved the problem of bending vibrations of cracked Timoshenko beams at simply supported boundary conditions by the perturbation methods.Swamidas et al.[8]used the energy approach to estimate the influence of crack location and size on the natural frequencies of cracked beams,and the problem has been numerically solved by using Galerkin's method.Lin[9]used a transfer matrix method for free vibration analysis of simply supported cracked beam.Khajietal.[10]proposed a new closed-form solutions for vibration analysis of cracked Timoshenko with various classical boundary conditions.Khnaijar and Benamar[11]presented a new discrete model for vibration analysis of the cracked beam,and obtained the natural frequency corresponding to a change in crack location and the depth.Zhao et al.[12]obtained substantial dynamic response of a cracked beam by using Green's function method,and used the transfer matrix method for a multi-cracked beam.

Many researchers analyzed the vibration characteristics of the cracked beam by using Finite Element Method and thus obtained natural frequencies corresponding to a change in crack position and depth[13-17].Ghodke et al.[18]obtained naturalfrequencies and mode shape corresponding to a change in crack location and depth by ANSYS software.Kisa et al.[19,20]presented a novel numerical technique which the finite element and component mode synthesis methods for analyzing the free vibration of the cracked beams.Barad et al.[21]obtained natural frequencies of the cracked beam experimentally,for the detection of crack location and size,and Douka and Hadjileontiadis[22]investigated dynamic behavior of the cracked beam both theoretically and experimentally.Khalkar and Ramachandran[23]investigated the effect of cracks on vibration of the cracked beam by an experimental method.

Most of researchers have studied the free vibration of a cracked beam under classical boundary conditions such as Clamped-Clamped,Free-Free,Clamped-free and simply supported.Ali C a?gri Batihan et al.[24]analyzed the vibration characteristics of the cracked beam supported on Pasternak and generalized elastic foundations.Zhang et al.[25]established cracked elastic-support beam and offset boundary by using ANSYS software,analyzed amplitude-frequency responses of the cracked beam.

Mao[26]investigated impairment inspection and the free vibrations of cracked beams with elastic boundary conditions at both ends.Most of researchers modeled the continuous conditions at the crack section at the by using the inverse of the compliance flexibility coefficients of fracture mechanics theory.

Compliance flexibility coefficients are expressed as a rotational springs[2,3,8,10,24,27]or two springs-a translational spring and a rotationalspring[6,28].Yokoyama etal.[28]investigated the vibration characteristics of the cracked beam using a modified line-spring model.Many researchers used local or total flexibility matrix to model the continuous conditions of crack section[17,29-32].Rakideh[33]considered the cracked section as a local flexibility,namely,a rotational spring for vibration analysis of the cracked Timoshenko beam,and obtained natural frequency data which are to be used to design a neural network for detecting of crack parameters.Neves et al.[34]used the Discrete Element Method in conjunction with local flexibility of a cracked beam.Jalali and Noohi[35]presented a modal-energy based equivalent lumped model in which cracked beam is divided into 3 parts-crack and two beam parts.

Wang etal.[36-47]presented the research results for analysis the vibration characteristics of the various structures(for example,beams,plates and shells)with generalized boundary conditions,and Yegao et al.[48-50]proposed research methods for analyzing the vibration of shells with arbitrary boundary conditions.

As was discussed above,many researchers presented the various methods for analysis the vibration characteristics such as natural frequencies corresponding to a change in crack location and depth in the beams with classic boundary conditions,however,while the study on the vibration analysis for arbitrary boundary conditions is comparatively limited.In addition,for the authors'greatest knowledge,there is little research done for the effects of flexibility coefficient on the natural frequencies.

The purpose of this study is to propose a unified calculation method for free vibration characteristics analysis of cracked beams with arbitrary boundary conditions by using ultraspherical polynomials.

Free vibration analysis model of the cracked beam is taken as befits Timoshenko beams,when taking into account the effects of both the shear deformation and the rotational inertia.The boundary conditions of the beam are set with the use of penalty variables of stiffness attributes which present artificial translational springs and rotational springs.Continuous condition at the crack section is represented by the flexibility coefficients with which two non-cracked beam segments are connected,that is,the crack is set as a translational spring and a rotational spring.The results show its accuracy and robustness by comparing with the existing literature and FEM results,and several numerical examples for the free vibration of a cracked beam with classical boundary conditions and elastic boundary conditions whose results can be provided as a reference for future engineering are presented.In addition,the effects of flexibility coefficient on the natural frequencies will further be investigated.

2.Theoretical formulations

2.1.Energy functions of the Timoshenko beam

The geometric model for the elastically supported cracked Timoshenko beam is shown in Fig.1.The well-known strain energy and the kinetic energy of the Timoshenko beam are expressed as follows[51]:

whereU,Tindicate strain energy and kinetic energy andI,ρ,E,Gare the moment of inertia and mass density per unit length,Young's modulus and shear modulus,A,Lare the area of the cross section and length of the beam,respectively.Andw(x),θ(x)are the transverse displacement and rotation deflection for cracked Timoshenko beam.

In Eq.(1)and(2),the bending momentM(x)and shearing forceQ(x)can be written as[52]

The bending momentM(x)and shearing forceQ(x)of Eq.(3)and(4)are used for vibration analysis of the cracked Timoshenko beam.

2.2.Stiffness coefficient Kc for cracked Timoshenko beam

The additional strain energy due to the existence of the crack can be expressed as[53]

where,Jis the function of strain energy release rate andAis the cross section area of the cracked beam.The function of strain energy release rateJcan be written as

whereE′=Efor plane stress problem,E′=E/(1-ν2)for plane strain problem,kis the shear shape coefficient of the beam section.

Since shearing forces and bending moments exist in Timoshenko beam theory,so the function of the energy release rate can be written as follows;

where KIare the stress intensity factors in simply fracture mode I of deformations due to bending momentMand KIIis the stress intensity factors in simply fracture mode II of deformations due to shearing forceQ.

whereξis the crack depth,FI,FIIare the correction factors for stress intensity factors.The correction factors for stress intensity factorsFI,FIIare expressed as[53]:

Fig.2.Change of flexibility coefficients with the crack depth increase.

In this research,c11is flexibility coefficient corresponding to fracture mode II due to shearing forceQandc22is flexibility coefficient corresponding to fracture mode I due to bending momentM.

It can be seen in Fig.2 that the flexibility coefficients increase as the crack ratioa/hincrease.

Finally,spring stiffness coefficientsKwc,Kθcin the cracked section of the beam can be written as follows:

In this paper,stiffness coefficientsKwc,Kθcof the Eqs.(17)and(18)are adopted as translationalspring stiffness and rotational spring stiffness for the continuity condition of the cracked beam.

2.3.Boundary conditions and continuity conditions

The boundary conditions of the beam are modeled by using artificial translational springs and rotational spring characterized by artificialspring stiffness coefficient.The artificial spring stiffness coefficientcan express various boundary conditions of the cracked beam,accept the flexible choice of the allowable displacement functions.The boundary conditions for an elastically restrained Timoshenko beam are as follows[52]

whereKw0,KwLare the translational spring stiffness atx=0 andL,andKθ0,KθLare the rotational spring stiffness atx=0 andL,respectively.

Eqs.(19)and(20)means a set of boundary conditions.

Fig.3.Dimensions of the cracked beam(a)and model of the cracked beam(b).

From Eqs.(19)and(20),all of the classical boundary conditions can be directly got by setting the spring stiffness values as an exceedingly small or an exceedingly large number.In addition,arbitrary elastic boundary conditions can be obtained depending on how the spring values are set.The potential energy added by boundary conditions can be expressed as

In the presented study,the beam is segregated into two segments by crack section,and the continuity conditions of the separated beam are modeled as the translational spring and rotating spring,which are determined by inverse of the flexibility coefficients of fracture mechanics theory as shown in Fig.3(b).

The strain energy stored in the link springs atthe conjunction between two segments can be described as

whereKwc,Kθcdenote the stiffness of the translational spring and rotating spring between the two non-cracked beam segments,respectively.And,w1,w2and θ1,θ2denote the vertical displacement and rotating displacement,respectively.

2.4.Unified solution and solution procedure

It is very important to select the suitable allowable displacement function for ensuring a stable convergence and accuracy of the solution.The displacement functions of the cracked beam can be flexibly selected by the penalty parameter,fast convergence of the accurate solution can be ensured with the appropriate value of the penalty parameter.On the treatmentof continuous boundary conditions,itmakes the choice of the admissible function flexible to introduce the spring stiffness,which is the penalty parameter in nature[54-59]

An ultrasphericalpolynomialis a polynomialsystem where Chebyshev,Legendre polynomials is generalized,and is an orthogonal polynomial of the special case of Jacobi polynomials.Therefore,the allowable displacement function of the cracked beam is uniformly extended in this paper to the ultraspherical orthogonal polynomials regardless of the element shape and displacement type.The main advantage of this polynomials is to ensure computationally the higher accuracy and robustness.

Thus,the selection of admissible displacement function of the cracked beam is more generalized by the ultraspherical polynomials.

The displacementfunction of each partof the beam divided into two parts by means of the crack section can be expressed as:

where the subscripti(i=1,2)means each parts of the beam divided into two parts by means of the crack section;Wm,i,θm,iare unknown coefficients of the displacement function that we want to obtain.(X)are the ultraspherical polynomials of degreemfor the displacement;ω,tdenote an angular frequency and time,respectively.The nonnegative integersMrepresent the highest degrees taken in the ultraspherical polynomials.

Fig..4.Convergence characteristic of frequency parameters on the boundary spring stiffness.(a)Translational springs and(b)rotational springs.

The ulraspherical polynomials are complete and orthogonal polynomials defined on the interval ofX∈[-1,1].That is why,a linear transformation statute may be introduced for coordinate conversion from intervalx∈[0,Li]of the divided beam to the intervalX(X∈[-1,1])of the ultraspherical polynomials,that is,X=

The total Lagrange energy functions(∏)for the cracked Timoshenko can be written as:

whereTi,Uirepresentkinetic and potentialenergy of divided parts,respectively;Ubounddenotes additional potential energy by spring stiffness in both ends boundary andUcrackrepresent additional potential energy by spring stiffness reflecting the connective conditions in crack section.

The total Lagrange energy function is minimized about unknown coefficients of the Rayleigh-Ritz method.

where M and K are the mass and stiffness matrix of cracked Timoshenko beam,and A representthe coefficient vector.The natural frequencies and the corresponding eigenvectors of the cracked beam can easily be obtained by solving Eq.(29).

3.Numerical results and discussions

In this section,the accuracy and flexibility of the proposed method are confirmed in the way of comparison with the previous literature and FEM results,and the results and discussions of the free vibration of the cracked Timoshenko beam are presented.

The discussions are organized as follows:Firstly,the study of convergence the solution for free vibration analysis of cracked Timoshenko beam has been conducted.Then,the reliability and accuracy of the proposed method are verified and confirmed through the comparison with the results in the precedent literature.Also,based on the reliability and accuracy of the present method,some examples and new results for free vibration analysis of the cracked beam with various classical boundary conditions and arbitrary boundary conditions are accordingly presented in the paper.Lastly,the effects of flexibility coefficient on the natural frequencies in the frequency measurement for detecting of a crack parameters are investigated.

Unless otherwise indicated,the material properties and the dimensionless frequency of the cracked beam is expressed as follows:

3.1.The study of convergence

From the mathematical point of view,the solution of the model may be not converged when the boundary spring stiffness coefficient is defined as a very low values or very high values[55,61].The study on the convergence of boundary parameters is proposed to decide boundary conditions for the cracked beam.To decide the parameters of the translational springs,values of the rotational springs at both ends of the cracked beam are assumed to be smaller than 105and larger than 100.Similarly,those of the translational springs at both ends of the cracked beam are assumed to be smaller than 105and larger than 100to decide the parameters of the rotational springs[52].

Fig.4 shows the convergence characteristic of frequency parameters on the boundary spring stiffness values.The geometric dimensions of the cracked beam used for convergence study are as follows:Beam widthb=0.06,beam height,h=0.02 beam lengthL=1,crack locationLc=0.5,crack ratioa/h=0.2.As can be seen in Fig.4,the frequency parameters rapidly increase as the values of translationalsprings(or rotational springs)increase in the range from 104to 108and beyond this range,there is little variation in the frequency.Therefore,it is assumed in this study that the boundary spring stiffness value needed to characterize the fixed boundary condition is 1015,the stiffness value for characterizing the elastic condition is 107.According to the study,the corresponding spring stiffness for the boundary conditions which are commonly found in practical engineering applications may be defined in terms of spring rigidities as shown in Table 1.

Table 1 The spring stiffness values of the arbitrary boundary conditions.

Table 2 Convergence analysis for Non-dimensional parameters of CC cracked beam.b=0.06,h=0.02,L=1,Lc=0.5,a/h=0.2.

From the theoretical formulation,we can know that the accuracy of the solution relates with a limited number of series in displacement functions in the actual calculation.The case being as such,it is important to appropriately choose the number of series in the ultraspherical polynomials.The convergence results of the non-dimensionalfrequency for cracked Timoshenko beam with classicalclamped boundary are shown in Table 2.

As can be seen in Table 2,stable convergence of the solution is ensured while the number M of the series in the ultraspherical polynomials is larger than 10.Therefore,all of the polynomial series are truncated intoM=10 in the following numerical analysis.The percentage error(?λ-?λ=0)/?λ=0of the solution for the ultraspherical polynomials parameterλ in the cracked Timoshenko beam are shown in Fig.5.From Fig.5,it can be seen that the change of the characteristic parameterλof the ultrasphericalpolynomials does notaffectthe convergence of the solution,and the maximum value of the percentage error does not exceed 0.3×10-4.Therefore,the characteristic parameterλof the ultraspherical polynomials are set intoλ=0 in the following numerical analysis.

3.2.Verifications and numerical examples

In this subsection,the reliability and accuracy of the present method for the analysis of the free vibration characteristic of the cracked Timoshenko beam is demonstrated,based on the convergence study of the current solutions.

Table 3 Comparison of the natural frequencies for non-cracked Timoshenko beam under the different classical boundary conditions.h/L=0.02,b=0.06.

In order to confirm the accuracy of the proposed method,the non-dimensional frequency parameters calculated by the current method for a non-cracked beam and the results of the previous literature are compared.

The cracked beam can be assumed as non-cracked beam,when the stiffness values of the translational spring and rotational spring at crack section are infinite.In this verification,it is assumed that the stiffness valuesKwc.Kθcof the translational spring and rotational spring at crack section are 1015.The comparison of the non-dimensional frequency parameters for non-cracked Timoshenko beam under the different classical boundary conditions is presented in Table 3.

Fig.5.Percentage error of the non-dimensional frequency for the ultraspherical polynomials parameterλin the cracked Timoshenko beam.

From Table 3,we can see that the results obtained with this method are in good agreement with the results of the reference data.

Then,to verify the accuracy and reliability of the proposed method for the cracked Timoshenko beam,the natural frequencies calculated by the current method and the results of the previous literature are compared accordingly.

The comparison of natural frequencies for cracked Timoshenko beam under the differentclassicalboundary conditions such as C-C,C-F,and P-S is shown in Table 4.From the comparison in Table 4,we can see that the results taken by the present method are in excellent accord with those of the references.

From Tables 3 and 4,we can know that the proposed method has the capability to deal not only with the vibration analysis of the non-cracked beam but also with a cracked beam.

Based on the convergence study on the current solution and the verification of accuracy for the presentmethod,to enrich the calculation results of cracked Timoshenko beams as well as improve design efficiency for engineers,some examples of the cracked Timoshenko beam with various classical boundary conditions are numerically provided.Also,new natural frequencies applied for cracked Timoshenko beam with various elastic boundary conditions will be given.In Tables 5-7,the first six natural frequencies for the cracked Timoshenko beams with three types of boundary conditions such as C-C,F-F,and C-F is presented according to the ratio of crack depth on the height of the beam.The results of numerical examples are compared with results obtained with the use of finite element analysis program ABAQUS(in element type-C3D8R,elements-26,856).

Table 4 Comparison of natural frequencies for cracked Timoshenko beam under the various classical boundary conditions.h=0.025,b=0.0125,L=0.1,Lc=0.05.

The result of the comparison shows that there is little significant difference between two results.Table 8 shows the natural frequencies of the cracked Timoshenko beam subject to the elastic boundary condition.As can be seen in Table 8,the changes of the elastic boundary conditions have a conspicuous influence on the natural frequencies.

Effects of crack location on the natural frequency of the cracked beam with E3-E3 elastic boundary condition are shown in Fig.6.

Fig.6 shows the change of the natural frequency of the cracked beam's first four modes according to the location of the cracks.When we are to investigate the change in the natural frequency of the first mode,it is known that the change of the natural frequency is the largest when the crack exists at both ends.And,when cracks are located at0.21 L and 0.79 L of the beam,there is little change in the natural frequency of the beam.

Furthermore,when cracks are located at 0.21 L and 0.79 L of the beam,there is little change in the natural frequency of the beam.When a crack is located on 0.5 L of the beam,there is little change of the natural frequency in the cracked beam.In third mode,when the cracks are located on 0.25 L and 0.75 L of the beam,there is little change in the natural frequency of the beam,and in contrast to the second mode,when the cracks is located on the 0.5Lof the beam,the change of the natural frequencies are the largest.We can see from the fourth mode,when the crack is located on 0.5 L of the beam,there is little change in the natural frequency of the cracked beam as in the second mode.

Table 6 Natural frequencies of CF cracked Timoshenko beam.(h/L=0.05,Lc=0.5).

Table 7 Natural frequencies of FF cracked Timoshenko beam.(h/L=0.05,Lc=0.5).

Table 8 Natural frequencies of the cracked Timoshenko beam with elastic boundary conditions h/L=0.05,Lc=0.5.

Fig.6.Effect of crack location on the natural frequency of the cracked beam with E3-E3 elastic boundary condition.

3.3.Effects of flexibility coefficient on the natural frequencies

The purpose of vibration analysis for cracked beam is to detect the crack parameters such as the locations and depth of a crack.The modes to be measured for detecting the crack in cracked beams may be varied depending on the object being measured,the working conditions,and the surrounding atmosphere.For example,in some cases the second frequency may need to be measured and in some other cases the third frequency may need to be measured.In addition,it is well known thatthe change of the flexibility coefficients due to the crack in cracked beams depends on the locations and depth of the cracks,and the change of the flexibility coefficients affects the natural frequencies.

Thus,in this subsection,the effects of flexibility coefficient on the natural frequencies in the frequency measurement for detecting of a crack parameters in cracked beam with different boundary conditions are investigated.The investigation of effects of flexibility coefficienton the naturalfrequencies will be carried out in crack ratio in 0.2,crack location in 0.5 L.

Many researchers ignored the effects of flexibility coefficient corresponding to translational spring stiffness,this means the value of the translational spring stiffness is infinite(in this paper it equal to 1015).While the value of the translational spring stiffness is constant in 1015,the change of natural frequencies on the change of flexibility coefficient corresponding to rotational spring stiffness is shown in Fig.7 depending on the different classical boundary conditions.

From Fig.7,we can see that the second,fourth and sixth natural frequencies of the cracked beam with C-C,C-F,F-F,and P-P boundary conditions are independent of the changes in flexibility coefficient corresponding to rotational spring stiffness.This means that the cracks cannot be detected by measuring the second,fourth and sixth natural frequencies,and the flexibility coefficientcorresponding to translationalspring stiffness should be considered unconditionally.The change of natural frequencies on the change of flexibility coefficient corresponding to translationalspring stiffness is shown in Fig.8 according to the different classical boundary conditions.Here,the value of the flexibility coefficient corresponding to rotational spring stiffness is assumed as 105on the basis of the analysis of the Fig.8.As can see from Fig.8,the natural frequencies of the cracked beam under all of the classical boundary conditions have a conspicuous changes depending on the values of specific range of the flexibility coefficient corresponding to translational spring stiffness.

Fig.7.Change of natural frequencies as increasing of flexibility coefficients corresponding to rotational spring.

Fig.8.Change of natural frequencies as increasing of flexibility coefficients corresponding to translational spring.

4.Conclusions

In this paper,a method for analyzing the free vibrations of cracked beam under arbitrary boundary conditions by using ultraspherical polynomials is presented.The ultraspherical polynomials are applied to generalize choice of the allowable displacement function and the Rayleigh-Ritz method is used to get the formulation on the basis of the Timoshenko beam theory.The boundary spring technique is adopted to realize the kinematic compatibility and physical compatibility conditions at arbitrary boundary conditions,and continuous conditions at the cracked section are modeled by using inverse of the compliance flexibility coefficients of fracture mechanics theory.The studies of the convergence,accuracy and reliability for the cracked Timoshenko beam with various boundary conditions of the classical,elastic and their combinations are made.In addition,investigated are the effects of flexibility coefficient on the natural frequencies which are significant in the frequency measurement for detecting of a crack parameters are investigated.

The results show good agreement between the present method and the existing literature and finite element analyses.Also,several numerical examples for free vibration of the cracked laminated composite beam with classical boundary conditions and elastic boundary conditions are also conducted.The results of the present paper can be applied as reference data for future researches.

Acknowledgments

The authors would like to thank the anonymous reviewers for their very valuable comments.

The authors also gratefully acknowledge the supports from Pyongyang University of Mechanical Engineering of DPRK.