Free Vibration Analysis of Uniform Beams with Arbitrary Number of Cracks by using Adomian Decomposition Method

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1 World Applied Sciences Journal 19 (1): , 1 ISSN 1818? 95 IDOSI Publications, 1 DOI: 1.589/idosi.was Free Vibration Analysis of Uniform Beams with Arbitrary Number of Cracks by using Adomian Decomposition Method Qibo Mao School of Aircraft Engineering, Nanchang Hang Kong University, 696 South Fenghe Avenue, Nanchang, CN-6, P.R. China Abstract: The Adomian Decomposition Method (ADM) is employed in this paper to investigate the free vibrations of the uniform cracked Euler-Bernoulli beams. The proposed ADM method can be used to analyze the vibration of beams consisting of an arbitrary number of cracks through a recursive way. Based on the ADM and employing some simple mathematical operations, the natural frequencies and the corresponding closed-form series solution of the mode shapes can be obtained by solving a set of algebraic equations with only three unknown parameters. Several numerical examples are presented to demonstrate the feasibility of the proposed method. It is shown that the ADM offers an accurate and effective method of free vibration analysis of beams with arbitrary number of cracks under general boundary conditions. Key words: Adomian decomposition method vibration analysis cracked beam natural frequency mode shape INTRODUCTION Beam-like structures are widely used in various aeronautical, civil and mechanical engineering fields. And racks are the main cause of structural failure. It is possible to predict the location and the depth of a crack based on the changes in vibration parameters. Therefore, the free vibration analysis of cracked beams have been investigated by many researchers and obtained plentiful achievement. No attempt will be made here to present a bibliographical account of previous work in this area. A few selective recent papers are quoted [1-6] which provide further references on the subect. In this paper, a relatively new computed approach called Adomian Decomposition Method (ADM) [7-1] is imposed to analyze the free vibration problem for a beam with arbitrary number of cracks under general boundary conditions. The ADM is useful and powerful method for solving linear and nonlinear differential equations. The goal of the ADM is to find the solution of linear and nonlinear, ordinary or partial differential equation without depending on any small parameter such as the case with the perturbation method. The solution by using the ADM is considered as a sum of an infinite series and rapidly convergence to an accurate solution [7-9]. The main advantages of the ADM are computational simplicity and do not involve any linearization or smallness assumptions. Recently, the ADM has been applied to the problem of vibration of structural and mechanical systems [1-1]. Using the ADM, the governing differential equation for each section of the cracked beam becomes a recursive algebraic equation. The boundary conditions and continuity conditions at crack location become simple algebraic frequency equations which are suitable for symbolic computation. Moreover, after some simple algebraic operations on these frequency equations, we can obtain the natural frequency and corresponding closed-form series solution of mode shape simultaneously. Finally, some numerical examples are given to demonstrate the feasibility of the proposed method. ADM FOR A CRACKED BEAM Consider the free vibration of a uniform Euler-Bernoulli beam of length consisting of J open cracks elastically restrained at both ends, as shown in Fig. 1. It is assumed that the cracks are located at at 1,,, J-1 and < 1< < < J <. The beam is divided into (J+1) sections with the (J+1) mirror systems of reference x ( =, 1,, J). Corresponding Author: Qibo Mao, School of Aircraft Engineering, Nanchang HangKong University, 696 South Fenghe Avenue, Nanchang, CN-6, P.R. China 171

2 World Appl. Sci. J., 19 (1): , 1 w (x, t) w 1 (x, t) w (x, t) w J (x, t) x x 1 x x J k R 1 J k RJ Crack 1 Crack Crack Crack +1 Crack J k T k TJ Fig. 1: The coordinate system for a multiple-cracked beam with elastically restrained at both ends The partial differential equation describing the free vibration in each section is as follows m w x,t w s x,t + =,x x EI t ( =, 1,, J) (1) where subscript denote the beam between the th crack and (+1)th crack. E is Young s modulus. bh I = is the 1 cross-sectional moment of inertia of the beam. m s = ρgh is the mass per unit length. ρ, b and h are the density, width and thickness of the beam, respectively. According to modal analysis approach (for harmonic free vibration), the w (x, t) can be separable in space and time: i t w x,t x e ω =φ () where φ (x ) and ω are the structural mode shape and the natural frequency, respectively. i= 1. Substituting Eq.() into Eq.(1), then separating variable for time t and space x, the ordinary differential equation for each section of the beam can be obtained Eq. () can be rewritten in dimensionless form φ sω dx EI d x m x = () ( ) where X x d ( X ) =, ( X) Ω X =,X R () ( x ) φ msω =, R =, Ω = EI Ω is the dimensionless natural frequency and the nth dimensionless natural frequency is denoted as Ω (n). According to the ADM [7-1], (X ) in Eq.() can be expressed as in terms of an infinite series ( X) = m= 17 [ m ] ( X ) (5)

3 [ m ] World Appl. Sci. J., 19 (1): , 1 where the component function will be determined recurrently. X d Impose a linear operator G =, then the inverse operator of G is therefore a -fold integral operator defined by and G 1 =... (6) 1 d d X d X G G ( X) =( X) X 6 (7) Applying on both sides of Eq.() with G -1, we get Comparison Eq.(7) and Eq.(8), we get [ m ] G G X G X G ( X ) =Ω =Ω m= (8) [ m ] d d X d X 1 ( X) = + X + + +Ω G X 6 m= (9) Finally, by using Eq.(5), the approximated solution of Eq.(9) can be determined by using the following recurrence relation [ ] d d X d X ( X) = + X + + (1) 6 [ m ] 1 [ m1 ] ( ) X =Ω G X m 1 (11) Substitute Eqs.(1), (11) and () into Eq.(5) and approximate the above solution by the truncated series M1 [ ] s M 1 m+ n m d m X X ( X) s m= s= m= = = Ω m n! (1) ( + ) Eq.(1) implies that [ m ] ( X) is negligibly small. The number of the series summation limit M is m= M determined by convergence requirement in practice. s d The unknown parameters (s =, 1,, ) and Ω in Eq.(1) can be determined based on the boundary s condition equations and the continuity conditions of each section of the beam. It will be discussed in next section. NATURA FREQUENCIES AND MODE SHAPES FOR THE B EAM WITH ARBITRARY NUMBER OF CRACKS The boundary conditions at the ends of the beam shown in Fig. 1 can be expressed as 17

4 World Appl. Sci. J., 19 (1): , 1 φ d φ x d x EI k R =, dx dx φ d φ J x J d J x J EI + k RJ =, dx dx J J ( x ) d φ EI k T ( x) dx ( x ) + φ =, at x = (1) d φ J J EI k TJ J( xj) dxj φ =, at x J = J (1) where k and k R are the stiffness of the translational springs and k J and k RJ are the stiffness of the rotational springs at x = and x J = J respectively. Rewriting the boundary condition equations in Eqs.(1) and (1) into dimensionless form, we obtain d d K R =, d + K = (15) T where d J RJ d J RJ d J RJ + K RJ =, J J J kr kt krj KR =, KT =, KRJ =, KTJ EI EI EI = (16) KTJ J RJ ktj EI J = and RJ = Substituting Eq.(15) into Eq.(1), the mode shape function for the first section (X ) can be expressed as a d linear function of () and M 1 m M 1 m+ m X m X ( X) = Ω KT ( m )! Ω ( m+! ) m= m= M 1 m 1 M 1 d m + + m X m X + Ω + KR Ω ( m+ 1! ) ( m+! ) m= m= (17) Due to the localized crack effect, the crack of the beam can be simulated as a massless spring [1]. For each crack between the two sections, conditions can be introduced which impose continuity of displacement, bending moment and shear. Moreover, an additional condition imposes equilibrium between transmitted bending moment and rotation of the spring representing the crack. Consequently, the continuity conditions in dimensionless form are [1-] ( R ) 1 + =, ( ) ( ) d d R d R 1 + = +θ (18) 1 + d d ( R) =, ( ) d d R = (19) where θ is the dimensionless th crack flexibility. the dimensional local compliance function [1, ], given by α θ = 5.6h J h and α is the depth of the th crack. J(α /h) is α J = 1.86r.95r r 7.6r r 16.9r + 17r.97r r h 17 ()

5 where r is the dimensionless depth of the th crack, World Appl. Sci. J., 19 (1): , 1 α r =. h Substituting Eqs.(18) and (19) into Eq.(1), the mode shapes for the section- ( ) can be written as M1 m M1 m 1 m X 1 d R d R + + m X ( X+ 1) =( R) Ω + +θ Ω ( m )! ( m+ 1! ) m= m= M1 m M 1 m X + m X m ( + ) m= + m= d R d R + Ω + Ω m! ( m! ) (1) d Notice that there are only three unknown parameters ( (), and Ω) in Eq.(1) through a recursive way. Substitute Eqs.(17) and (1) into Eq.(16), this boundary condition equation can be expressed as linear functions of d () and, such as d f11( Ω ) + f1 ( Ω ) = () d f1( Ω ) + f ( Ω ) = () From Eqs.() and (), the dimensionless natural frequency Ω can be solved by, N n f11( Ω) f( Ω) f1( Ω) f1( Ω ) = SnΩ = n= () Notice that Eq.() is a polynomial of degree N evaluated at Ω. By using the functions sympoly and roots in MATAB Symbolic Math Toolbox, Eq.() can be directly solved. The next step is to determine the nth mode shape function corresponding to nth dimensionless natural frequency Ω(n). Substituting the solved Ω(n) into Eq.() d or (), the unknown parameter can be expressed as the function of (). ( Ω) ( Ω) d f Ω f Ω f f 11 1 = = 1 (5) Substituting Eq.(5) into Eqs.(18) and (1), the mode shape function for each section can be obtained. The mode shape function for the entire beam can be written as ( X) ( X ) ( X )... ( X ) = By normalizing Eq.(6), the normalized mode shape is defined as 1 1 J J (6) ( X) = 1 ( X) ( X) (7) It should be noted that the mode shape function by using the ADM is a continuous function (closed-form series solution) and not discrete numerical values at knot point by finite element or finite difference methods. 175

6 World Appl. Sci. J., 19 (1): , 1 NUMERICA CACUATIONS In order to verify the proposed method to analyze the free vibration of the multiple cracked beam, several numerical examples with different boundary conditions and cracks will be discussed in this section. As mentioned earlier, the closed-form series solution of mode shape function in Eq. (1) will have to be truncated in numerical calculations. It is important to check how rapidly the dimensionless natural frequencies Ω(n) computed through the ADM converge toward the exact value as the series summation limit M is increased. To examine the convergence of the solution, a beam with only two cracks as in Ref. [1] is considered. The beam under analysis has the following properties: length =.8m, rectangular cross-section with width b =.m and thickness h =.m, Young s modulus E = N/m, density ρ = 78 kg/m. The boundary condition is assumed as clamped-free. In this study, the clamped boundary condition are obtained by setting the stiffness the translational and rotational springs to be extremely large (which is represented by a very large number, 1 1 9, in this paper). Similarly, for free boundary condition, both the stiffness the translational and rotational springs are set to. For simply supported boundary condition, the stiffness the translational and rotational springs are set to and respectively. For comparison purpose, Table 1 shows the natural frequencies f(n) as the function of the series summation limit M and Table 1:The convergence of the natural frequencies f(n) = ω/π (Hz) for a clamped-free beam with crack location R 1 =.15, R =.5 and crack depth r 1 =.1, r =.15 Mode index M Ref.[1] Fig. : The effect of the locations and depths of the second crack on the first natural frequencies of a clamped-free beam (the location and depth of the first crack are R 1 =.15 and r 1 =.1) 176

7 World Appl. Sci. J., 19 (1): , 1 (a) (b) (c) (d) (e) (f) Fig. : The first four mode shapes for two cracked beam with different boundary conditions (Other parameters listed in Table ). (a) K T = 1, K R =, K TJ = 1, K RJ = ; (b) K T = 1, K R =, K TJ = 1, K RJ = ; (c) K T =, K R = 5, K TJ = 6, K RJ = 7; (d) K T = 1, K R = 1, K TJ = 1, K RJ = 1; (e) K T = 1 1 9, K R = 1 1 9, K TJ = 1 1 9, K RJ = ; (f) K T = 1 1 9, K R = 1 1 9, K TJ = 1 1 9, K RJ =

8 World Appl. Sci. J., 19 (1): , 1 Table :The first four natural frequencies f(n) (Hz) for a two-cracks beam under different boundary conditions (crack location R 1 =.15, R =.5; crack depth r 1 =.1, r =.15) Stiffness of springs (Boundary conditions) Mode index K T K R K TJ K RJ e9 1e9 1e e9 1e9 1e9 1e Fig. : The first natural frequency of a clamped-free beam with equally spaced cracks with the same depth ω EI f = =Ω π m s ( Hz) Clearly, the f(n) converges very quickly as the series summation limit M is increased. If M>5 is used, the results based on proposed method are compared with those listed in Ref. [1] and excellent agreement is found. If M = 8 is used, the first fourth natural frequencies can be kept accurate to the tenth decimal place. The excellent numerical stability of the solution can also be found in Table 1. For brief, the series summation limit M in Eq. (1) will be simply truncated to M=1 in all the subsequent calculations. The natural frequencies are kept accurate to the sixth decimal place for comparison purpose. Assume that the location and depth of the first crack location are R 1 =.15 and r 1 =.1, respectively. Figure shows the effect of the locations and depths of the second crack on the first natural frequencies of the clamped-free beam. From Fig., it can be found that the natural frequency is decreased larger when the crack is near the clamped end. This conclusion agrees well with Ref. []. Because the proposed method based on the ADM technique offers a unified and systematic procedure for vibration analysis of the cracked beam with arbitrary boundary conditions. The calculation of the natural frequencies and corresponding mode shapes for different boundary conditions can be very easy. For example, the modification of boundary conditions from one case to another is as simple as changing the values of the stiffness of translational and rotational springs. And it does not involve any changes to the solution procedures or algorithms. Table lists the first four natural frequencies f(n) for the beam with two cracks with different boundary conditions. Figure shows the first four corresponding mode shapes for the cracked beams listed in Table. 178

9 World Appl. Sci. J., 19 (1): , 1 It should be noticed that the ADM can be used to analyze the free vibration of beams consisting of an arbitrary number of cracks through a recursive way. Consequently, the complexity of the vibration is reduced to the same order of a beam without any cracks. The solution can be obtained by solving a set of algebraic equations with only three unknowns. The method can be used to solve the free vibration problems of beams with arbitrary number of cracks. Figure shows the first natural frequency of a clamped-free beam with equally spaced cracks with the same depth. The maximum number of crack in this case is J = 5. From Fig., it can be found that the natural frequencies are decreased when the crack depth is increased, as expected. Second, it is found that the natural frequencies are almost coincided when the number of cracks is larger than J =. It means that the dynamic behavior becomes more and more insensitive when the number of the cracks increases. CONCUSIONS In this paper, the free vibrations of Euler-Bernoulli beams with multiple cracks are analyzed using Adomian decomposition method (ADM). Natural frequencies and corresponding mode shapes with various parameters (such as boundary conditions, locations, depth and number of cracks) are presented. The results using ADM are in excellent agreement with published results. It should be noted that the proposed method can be used to analyze the vibration of beams consisting of an arbitrary number of cracks through a recursive way. And the complexity of the vibration is the same order of a uniform beam without any cracks. The solution can be obtained by solving a set of algebraic equations with only three unknowns The vibration analysis for different boundary conditions and/or number of cracks is as simple as changing the value of corresponding parameters and does not involve any changes to the solution procedures or algorithms. ACKNOWEDGEMENTS This work was sponsored by the National Natural Science Foundation of China (no ), Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry (no. 1-). REFERENCES 1. Shifrin, E.I. and R. Ruotolo, Natural frequencies of a beam with an arbitrary number of cracks. Journal of Sound and Vibration, (): 9-.. in, H.P., S.C. Chang and J.D. Wu,. Beam vibrations with an arbitrary number of cracks. Journal of Sound and Vibration, 58 (5): Ostachowicz, W.M. and M. Krawczuk, Analysis of the effect of cracks on the natural frequencies of a cantilever beam. Journal of Sound and Vibration, 15 (): Orhan, S., 7. Analysis of free and forced vibration of a cracked cantilever beam. NDT&E International, : Caddemi, S. and I. Calio, 9. Exact closed-form solution for the vibration modes of the Euler-Bernoulli beam with multiple open cracks. Journal of Sound and Vibration, 7: Kisa, M.M.A., 7. Gurel. Free vibration analysis of uniform and stepped cracked beams with circular cross sections. International Journal of Engineering Science, 5: Adomian, G., 199. Solving frontier problems of physics: The decomposition method, Kluwer-Academic Publishers. 8. Wazwaz, A.M., 1. Analytic treatment for variable coefficient fourth-order parabolic partial differential equations. Applied Mathematics and Computation, 1: Haddadpour, H., 6. An exact solution for variable coefficients fourth-order wave equation using the Adomian method. Mathematical and Computer Modelling, : ai, H.Y., J.C. Hsu and C.K. Chen, 8. An innovative eigenvalue problem solver for free vibration of Euler-Bernoulli beam by using the Adomian decomposition method. Computers and Mathematics with Applications, 56 (1):

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