Vibration Analysis of a cracked prismatic Beam

IJSRD - International Journal for Scientific Research & Development| Vol. 3, Issue 08, 2015 | ISSN (online): 2321-0613

Vibration Analysis of a Cracked Prismatic Beam Deepak Ranjan Biswal Assistant Professor Department of Mechanical Engineering 1,2 DRIEMS, Odisha, India Abstract—Vibration analysis is one of the fundamental tasks in designing of structural and mechanical system. The effect of vehicle suspension system, vibration absorber on the rotating machineries and the dynamic behaviour of machine tool structures due to excitation are the important information that design engineer wants to obtain. It has been observed that most of the structural members fail due to the presence of cracks. The cracks are developed mainly due to fatigue loading. Existence of crack in a beam not only affects the stiffness but also it affects the mechanical behaviour of the beam. Due to the existence of such cracks the frequencies of natural vibration, amplitudes of forced vibration, and areas of dynamic stability change. In order to identify the magnitude and location of the crack, analysis of these changes is essential. The present article deals with the analysis of a cracked beam with transverse open crack and investigates the mode shape frequencies.

Key words: Cracked Prismatic Beam, Vibration Analysis I. INTRODUCTION Various numerical methods have been adapted for solving vibration problems from several decades. Among them Finite element method (FEM) is the most versatile method for solving such problems. FEM is a numerical technique used for finding approximate solutions of partial differential equation as well as integral equation. This method basically involves of dealing piecewise continuous function for the solution and finding out the parameters of the functions in order to reduce the error. By this method the beams is discretised into finite number of elements and the response for each small elements is calculated. After that all the responses are added together to get global value. Stiffness matrix and mass matrix is calculate for each of the discretized element and at last all have to combine to get the global stiffness matrix and mass matrix. The shape function gives the shape of the beam element at any point along longitudinal direction. This shape function also calculated by finite element method. Both potential and kinetic energy of beam depends upon the shape function. So it can be easily say that potential and kinetic energy depends upon shape function of beam obtained by FEM method.

Beams are considered as one of the commonly used elements in structures and machines, and fatigue cracks are the main cause of beams failure. Existence of damage in structural elements like beams and shafts points to the modification of the vibration modes. The existence of crack in a mechanical member produces local flexibility which would affect the vibration response of the structure. Due to this the dynamic behaviour of the structural elements has been the subject of several investigations for the last few decades. Papadopoulos and Dimarogonas [1] revealed that due to presence of transverse crack in a mechanical member, a local flexibility come into play whose dimension depends on the no of degrees of freedom. It has also been detected that local flexibility is suitable for analysis of cracked beam

when analytical method has been employed for solving differential equations [2]. It is also suitable to use a semianalytical method by using the improved Fourier series [3]– [4], mechanical impedance method [5]. Loyaa [6] proposed direct and perturbation methods for the solution of natural frequencies of the beam subjected to bending vibrations. In his work the beam has been modelled using Timoshenko beam formulation. Viola et al. [7] proposed the dynamic stiffness method for analysis of vibration structures. The free vibration of simply supported beams with crack by using direct and inverse methods has been proposed by Lin [8].

The crack localization and its size in a beam from the free and forced response measurements method is specified by Karthikeyan et al. [9]. Swamidas et al. [10] derived mathematical expressions for identification of crack in beam structures using Timoshenko and Euler beam formulation. Their work especially focused on to estimate the influence of crack size and location on natural frequencies of beam using Timoshenko and Euler beam formulations. Frequency contour method has been used to identify the crack size and location properly. Ali et al. [11] have deliberated the vibration analysis of a cantilever beam.

From their work it has been seen that natural frequency is greatly affected by presence of crack.The magnitude of change in natural frequency specifically depends on depth of crack, number of crack and position of crack.

Lee et al. [12] have established a method to find the natural frequencies of the cracked structure and by using Armon's Rank-ordering method the approximate crack location is obtained using finite element. A technique for shaft crack detection have proposed by Al-Bedoor et al. [13] which formulates the shaft crack detection using an optimization technique by means of finite element method.

In this case they utilized genetic algorithm to search the solution for position of crack.

Owolabi et al. [14] proposed an experimental exploration in order to predict the effects of crack on the structures. The localization of a crack in a non-rotating shaft attached to an elastic foundation, Al-Qaisia et al. [15] proposed a method of reduction of Eigen frequencies and sensitivity analysis. In their proposed work the shaft was modelled and coupled to an experimentally recognized foundation by using FEM. For quantification and localization of damage in beam-like structures Sahin et al.

[16] have introduced a different damage scenario using finite element method. In their work the local thickness of selected elements has been reduced for damage identification. Nahvi et al. [17] have proposed both logical as well as experimental approach for detection of crack in cantilever beams. Behzad et al. [18] have calculated natural frequencies of an open edged crack beam using theoretical and finite element analysis.

Sutar [19] investigated the cracked beam of cantilever type and established the relationship between the calculated modal frequencies with crack depth. The analysis

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Vibration Analysis of a Cracked Prismatic Beam (IJSRD/Vol. 3/Issue 08/2015/128)

is carried out for a single crack placed at different position and different crack depths. For finding out the total flexibility matrix, Zheng [20] has proposed an additional flexibility matrix as a substitute of local flexibility matrix.

The stiffness matrix of the cracked beam has been found out by this overall additional flexibility matrix. It has been observed that considering the additional flexibility matrix, accurate results can be attained than from using the local flexibility matrix. In this case 128-point (1D) or 128 ×128(2D) Gauss quadrature has been employed for computing the parameters. Then best fitted employing least quadrature has been used to get the flexibility parameters.

The best-fitted formulas produce good agreement with the numerical integration results [21]. After getting the cracked beam element stiffness matrix, standard FEM procedure can be followed, to obtain the natural frequency.

Based on the literature survey the present article focussed on the effect of crack on a beam at different positions with different depths. For this purpose Timoshenko beam formulation has been proposed by considering both rotary inertia and shear deformation. A two noded beam element has been used for the investigation of both uncracked and cracked beam.

1 1 w   θ   Eb I b  1   kGAb  θ1  1  (1.2) 2  α 2 α     2

Es 

2

The total strain energy of the beam in matrix form can be written as  θ1  1  α   Eb I b Es     2 0 θ  w1   0 1 α   T

Le

Where Ib is considered as the area moment of inertia about the neutral axis and Ab is considered as the cross sectional area of the beam, Eb is the young’s modulus and, G is the shear modulus of the material of the beam. k is the shear coefficient factor that relays on the material type and geometry of cross-section. The Kinetic energy which is considered as the sum of energy due to velocity (both linear and rotary). Hence by considering rotary inertia the kinetic energy can be written as 2 2 1  w  1  θ  (1.4) Ek  ρb Ab  1   ρb I b  1  2  τ  2  τ  Therefore the total kinetic energy of the beam in matrix form can be written as

 wb  0   τ  dα ρb I b   θb     τ 

 w1  1 Le  τ   ρb Ab Ek     2 0  θ1   0  τ  T

II. MATHEMATICAL MODELING The mathematical modeling consists of formulation of uncracked beam and cracked beam using Timoshenko beam formulations with finite element approach which are illustrated in following sections. A two noded beam with two degrees of freedom per node element has been modeled for the mathematical formulation.

 θ1  (1.3) 0   α  dα kGAb   w1  θ1   α  

(1.5)

Where ρb is considered as the density of the material used for the purpose. The energy equation of the beam can be written by using Hamilton’s principle as

δ   δE s  δEk  δWext dτ  0 (1.6) t2

A. Mathematical Modeling of an uncracked beam Element

t1

Fig.1: Uncracked beam element with two degrees of freedom per node.

The uncracked beam element with two degrees of freedom per node (such as shear and bending) is shown in Fig. 1. The modeling of the beam based on the assumption that it has constant flexural rigidity, mass density and length [15].The displacement functions in the α, β and δ directions can be represented as

 w  u1 α, β , δ , τ   z1θ1 α, τ   z1  1  β1 α  ,  α 

v1 α, β, δ, τ   0 ,

(1.1) w1 α, β, δ, τ   w1 α, τ  Where u1 is considered as axial displacement along the α-axis, v1 is the lateral displacement along β axis which is equal to zero and w1 is considered as time dependent transverse displacement of the centroidal axis. The time dependent rotation of the cross-section about β axis is considered as θ1. The shear slope which has no contribution during finding out the axial displacement at a point at a distance z1 from the centre line is considered as β1(α). The strain energy can be represented as

Here Es, Ek and Wext are represented as strain energy, kinetic energy and external work done. The external work done on the system is zero. τ is represented as the time. Sustituting the above values the governing equation of motion of the beam using Timoshenko beam formulation can be obtained.

Let the displacement w1 can be the represented by a polynomial of degree 3. Hence θ1 can be represented by polynomial of degree 2 for analysis using FEM. (1.7) w1  p1  p2α  p3α 2  p4α 3 Where α is the distance of the finite element node from the fixed end of the beam.p1, p2 p3 p4 are the constants and can be obtained by implementing the structural boundary conditions. Substituting the unknown coefficients and writing them in matrix form ' (1.8) w1 x   S w qw  w1 x   Sθ qw  Where T (1.9) qw   y1θ1 y2θ2  Sw & Sϴ are transverse displacement and bending shape functions, taking the shear (Ø1) into consideration.

The shear (Ø1) is defined as the ratio of the stiffness of bending to the stiffness of shear and is given by mathematical expression as 12Eb I b k yy , 1 (1.10) k υ  1

AbGL2

k

yy

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Vibration Analysis of a Cracked Prismatic Beam (IJSRD/Vol. 3/Issue 08/2015/128)

The elemental mass matrix can be obtained by addition of translational mass and rotational mass and interpreted in matrix form as T L me    SSw   ρb0Ab

0  S w  dα (1.11) ρb I b  S θ  0 θ   Substituting the mode shapes Sw & Sϴ the elemental mass matrix can be obtained. The elemental stiffness matrix which is the combination of the stiffness of bending and stiffness of shear, can be written in matrix form as e

 S θ    S θ   Le    Eb I b  0   α α k e       0 kGA    dα      S  S 0 b S θ   w   S θ   w  α  α   

Where T is represented as the strain energy release rate function and can be written as [22] 1 (1.16)    2   2 .

T Eb

I2

'

Substituting the mode shapes Sw & Sϴ, the elemental stiffness matrix can be found. When Ø1 is deserted, the elemental mass and elemental stiffness matrix condensed to corresponding Euler-Bernoulli beam elemental matrices. The governing equation of motion for the whole structure can be obtained by assembling the elemental mass and stiffness matrices and represented as (1.13) {qw}  []{qw}  (t ) Where [T] and [H] are represented as the global mass and global stiffness matrices for the whole structure and P(t) is the global external force vector. Therefore, the natural frequencies and the mode shapes of the whole system can be interpreted as the solutions of eigenvalue problem (1.14) [T ]   2 [ H ] {υ}  0 2 Where Ω is the natural frequencies of the structure and Φ is the mode shape of the structure for the corresponding natural frequency.





B. Mathematical Modeling of cracked beam Element Fig.2 represents a cracked beam element with two nodes (i, j) having two degrees of freedom per node. A transverse open crack is assumed to be present at a distance lc from the node j. The depth of this crack is a. The cross section of the beam is of rectangular lamina at crack position which is shown in the Fig.2.The width and depth of the cross section is b and h respectively.

II 2

Where Eb΄=Eb for plane stress problem, Eb΄=Eb/1µ2 for plane strain problem. ΠI2, ΠI3, ΠII2 are considered as the stress intensity factors. The values of stress intensity factor can be can be written as  λ (1.17)   σ 3.14α    ni

i

T

(1.12)

I3

n

h

Where σi is the stress due to load Pi., α is the crack depth, Yn(α/h) is considered as the intensity function. By Using Paris equation the displacement can be written as  c . (i,j=2,3) (1.18) i  Pi By using Paris equation the additional flexibility matrix Ψij can be obtained as [24]   2  c (i,j=2,3) (1.19)  ij  i  Pj Pi Pj Substituting (1.17) in (1.16) & (1.15) and then in (1.19) we get 2  6 P L  λ    2 2 c 3.14λ 2     h    bh   2 a b   6 P3  λ  ψ ij  ' 3 .

14 λ        dλ 2 E Pi Pj 0  bh 2 h    2  P2 3.14λ 2  λ   II  b 2 h 2  h  

(i,j=2,3) (1.20) By using the FEM co-ordinate system, each node of the beam element is subjected to shearing force Q and bending moment M. The corresponding transverse displacements and rotations are denoted as ζ and θ. The length of each element is le and flexural rigidity EbIb. Hence the forces and corresponding displacements can be written as   ζ j  ζ i  θi   Q j   (1.21)   θ j  θi

  ovl      M j  

For the intact beam the flexibility matrix can be represented as   ζ j  ζ i  θi   V j   (1.22)     int act   θ  θ M     i  j   j Therefore the total flexibility matrix can be written as (1.23) total  ovl  int act The stiffness matrix kc of the cracked beam element can be represented through equilibrium condition [25] [26] as follows 1 (1.24) kc  total T Where X is the transformation matrix and can be represented as

Fig. 2: Cracked beam element with rectangular cross section Due to the existence of crack additional strain energy developed which can be expressed as [23] [24].

(1.15)  c   dA .

AC

 1 0    1    Le 1 0  0 1 

III. RESULT AND DISCUSSION In order for analysis of the given structure a cantilever beam is taken into consideration. The various geometric and

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Vibration Analysis of a Cracked Prismatic Beam (IJSRD/Vol. 3/Issue 08/2015/128)

physical dimensions of the beam are L=1m, b=0.1m, h=0.01m, E=206Gpa, ρ=7800Kg/m3, µ=0.3.The beam is discretised into number of finite elements. Table 1 represents the convergence result of the beam for different number of finite elements with these given dimensions.

No of elements ω1 2 54.173 4 53.150 6 53.149 8 53.128 10 52.188 12 52.148 14 52.148 Table 1: Convergence study of 1st natural frequency for different finite number of elements.

From Table 1 it is observed that for 12 no of elements the formulation shows good convergence result.

Hence all further calculations have been carried out by considering 12 no of elements. By considering transverse vibration, the natural frequencies of the beam have been found theoretically and are compared with the present FEM results obtained. Ω Theoretical [27] FEM Result % error (rad/sec) Ω1 52.209 52.148 0.116 Ω2 326.304 326.811 -0.155 Ω3 915.134 915.094 0.004 Table 2: Comparison of Frequencies.

Table 2 shows the comparison of natural frequencies between the present finite element method and the theoretical calculated result for a cantilever beam. From the table, good agreement between the two methods has been perceived.

A transverse crack (open type) is considered for the analysis and assumed to be located at different positions of the beam with varying relative crack depths. For this purpose the crack depths (relative) of 0.2, 0.3, 0.4 and 0.5 are considered with relative crack positions of 0.2, 0.4, 0.6 and 0.8.The variations of natural frequencies are shown in the following tables. Ω αc/L a=0.2 a=0.3 a=0.4 a=0.5 Ω1 0.2 51.255 50.613 50.045 49.658 Ω2 0.2 308.287 296.616 287.308 281.475 Ω3 0.2 858.783 828.708 807.505 795.378 Table 3: First three frequencies of cracked Timoshenko beam with αc/L=0.20. Ω αc/L a=0.2 a=0.3 a=0.4 a=0.5 Ω1 0.4 51.665 51.313 50.997 50.781 Ω2 0.4 327.484 328.005 328.495 328.843 Ω3 0.4 888.066 867.117 847.897 834.643 Table 4: First three frequencies of cracked Timoshenko beam with αc/L =0.50. Ω αc/L a=0.2 a=0.3 a=0.4 a=0.5 Ω1 0.6 51.972 51.841 51.722 51.639 Ω2 0.6 324.543 322.846 321.296 320.218 Ω3 0.6 920.574 925.188 929.800 933.245 Table 5: First three frequencies of cracked Timoshenko beam with αc/L =0.60 Ω αc/L a=0.2 a=0.3 a=0.4 a=0.5 Ω1 0.8 52.122 52.102 52.085 52.072

Ω2 0.8 324.187 322.248 320.495 319.284 Ω3 0.8 900.106 889.392 879.977 873.624 Table 6: First three frequencies of cracked Timoshenko beam with αc/L =0.80 Table 3-6 represent the variation of first three frequencies of a cracked Timoshenko beam with different relative position of crack. From the table it is concluded that for a particular crack position, as the relative crack depth increases the natural frequencies decreases. This is due to the fact that there is a reduction of stiffness at the position of crack. As the position of crack changes from clamped end to free end, the natural frequencies increase for a given crack depth. The reason is that there is reduction of stress in the beam element from fixed end to free end. But as the depth of crack increases there is decrease in natural frequencies.

Fig. 3: 1st mode shape of uncracked and cracked cantilever beam (α/h=0.4 αc/L =0.5)

Fig. 4: 2nd mode shape of uncracked and cracked cantilever beam (α/h=0.4 αc/L =0.5).

Fig. 5: 3rd mode shape of uncracked and cracked cantilever beam (α/h=0.4 αc/L =0.5).

Figs.2-4 represents the variation of first three mode shapes of an uncracked and cracked cantilever beam using Timoshenko beam formulations. The relative crack position and crack depth has been considered as 0.5 and 0.4 respectively. It is observed from the figure that there is a reduction of amplitude of cracked beam as compared to uncracked beam and a sudden jump in mode shape at crack

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Vibration Analysis of a Cracked Prismatic Beam (IJSRD/Vol. 3/Issue 08/2015/128)

position which is easily visualise in higher modes of vibration.

IV. CONCLUSION This present article focussed on analysis of a cracked prismatic beam using Timoshenko beam formulation. Two noded beam element has been considered for the analysis.

The equation of motion has been derived by using Hamilton’s principle. A MATLAB code has been developed for finding out the natural frequencies along with the mode shapes for both uncracked and cracked beam. For crack modelling a complete additional flexibility matrix has been used. Based on the result it has been observed that due to the existence of crack in the beam there is a variation in the natural frequency. The change in magnitude of natural frequencies depends on parameters like the change of crack depths, position of the transverse crack.

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