Comparison of Transfer Mass Matrix Method with Finite Element Method for Modal Analysis of beams

Comparison of Transfer Mass Matrix Method with Finite Element Method for Modal Analysis of beams

Abhishek Kumar Sahu

M.Tech Scholar Department of Civil Engineering National Institute of Technology

Agartala, India

Dr. Surajit Das Assistant Professor

Department of Civil Engineering National Institute of Technology Agartala,India

AbstractIn the present work two different methods for finding natural frequencies of beams having crack are compared. Both the methods are used for modal analysis of cracked beam having transverse cracks. In the present analysis, it is found that the present method of finding the natural frequencies is more accurate than transfer mass matrix method. Kausar H. Barad, in his paper has performed experiment on a cantilever beam having cracks at different locations and obtained natural frequencies corresponding to each crack. The same problem has been analysed by the transfer mass matrix method proposed by D.P Patil and present finite element formulation derived from formulations proposed by A.D Dimarogonas and Uttam Kumar Mishra. The error in natural frequencies obtained by present finite element formulation is less as compared to the transfer mass matrix method. Variation of graph for natural frequencies obtained by present finite element formulation is similar to the experimental one whereas it is not similar for that obtained by transfer mass matrix method.

KeywordsDamage Detection, Structural Health Monitoring, Modal Analysis, Cracks, Beams.

1. INTRODUCTION (Heading 1)

   Structures such as buildings, trusses etc. are subjected to damage due to varying loads. In buildings, beams get damaged earlier than columns. A large number of researchers have developed numerous ways to find out natural frequencies of beams subjected to transverse cracks. Transfer Matrix Method is one of such ways. Patil and Maiti (2003) have developed a method for detection of multiple open cracks in a slender Euler-Bernoulli beam based on frequency measurements. This method is based on approach given by Hu and liang. It has transverse vibration modelling through transfer matrix method and represents crack by rotational spring. The beam is virtually divided into a number of segments and each one is considered to be associated with the damage parameter. These parameters are determined from changes in natural frequencies and are used to determine crack location and crack size. This method eliminates the need of symbolic computation envisaged by Hu and liang to obtain mode shapes of corresponding uncracked beams. Calculations are done for two simultaneous cracks of size 10% and more of section depth. The differences between actual and predicted crack location and sizes are less than 10% and 15% respectively. The no. of segments in which

   beam is divided limits the maximum number of cracks that can be handled. The objective of the work done in paper is to present a method for modelling transverse vibration of beams with multiple open cracks by combining the transfer matrix method and rotational spring based representation of a crack and approximate approach of Hu and liang. This method uses symbolic computation to obtain mode shape of corresponding uncracked beam. Simply supported beam, cantilever beam, beams on elastic foundation and multiple pin supports are examined. Kausar H Barad et.al. (2012) have used the natural frequency to find the crack depth and crack location in the beam like structure. In the present paper, crack in the beam structure is considered as a rotational spring. Characteristic equation based on boundary conditions of the beam is derived. Relationship between crack depth and stiffness is derived and graph is plotted between normalized crack depth and crack location for the first two frequencies. Shen and Taylor (1990) have shown an identification procedure to determine crack location and size from dynamic measurements. The procedure is based on minimization of either mean square or max measure of difference between measurement data (natural frequencies and mode shapes) and corresponding predictions obtained from computational model. Necessary conditions are used for formulation. Method is tested for crack in simply supported Euler- Bernoulli beam. Sensitivity of solution of damage identification to the values of parameters characterizing damage is discussed. Christides and Barr (1983) have derived differential equation and associated boundary conditions for nominally uniform Bernoulli Euler beam containing one or more pairs of symmetric cracks. Reduction of one spatial dimension is achieved using integration over cross-section after plausible stress, strain, displacement. Momentum fields are chosen. Perturbation in stresses induced by crack is incorporated through local function assuming an exponential decay with distance from crack. It includes a parameter evaluated by experimental tests. Experiments on beams containing cuts to simulate cracks are described. Change in first natural frequency with crack depth is matched closely by theoretical predictions. The theory can be extended to beams having non symmetry cracks and has coupling between various forms of motion such as bending and torsion. Such couplings are significant in the regions where both

   frequencies of predominantly bending and predominantly torsional motion and their corresponding

   wavelengths are approximately same. This theory can be

   K P2

   F

   (6)

   applied to beams applied to flexural vibration with crack on one side only In the present paper, natural frequencies are

   II1 bh

   II h

   calculated by transfer mass matrix method and present finite element method for the experimental problem analysed by Kausar H Barad. The results of the two are compared with the

   From definition, the elements of the overall additional flexibility matrix Cij can be

   2

   experimental results and it is observed that the natural

   C i C

   , (I, j=1,2) (7)

   ij P

   PP

   frequencies obtained by former one deviates more from the experimental frequencies in comparison to the latter one.

   1

   1

2. PRESENT FINITE ELEMENT FORMULATION AND

   j i j

   Substituting Eq (4),(5),(6) into Eq (3), then into Eq (1) and Eq (7) subsequently we get,

   TRANSFER MASS MATRIX METHOD

   b 2

   6PL

   6P

   2 P

   2

   Cij

   E ' PP

   1 c

   bp

   F1 h

   1

   bp

   F1 h

   bh

   F1 h

   d

   1. Finite Element Formulation

      i j

      (8)

      When crack is induced in a beam, then its flexibility is

      Substituting i,j (1,2) values, we get

      increased. So, first we calculate the additional flexibility

      induced in it. Then it is added up with the flexibility matrix of

      a a

      2 36L2 h h

      C

      2

      2

      C

      2

      2

      (9)

      intact beam element. The inverse of the overall flexibility

      C11

      E 'b

      p xF1 (x)dx

      xFII (x)dx

      matrix thus obtained is multiplied with the transformation matrix to obtain the required stiffness matrix of the cracked beam element. This stiffness matrix is assembled along with

      0 0

      h

      h

      a

      the stiffness matrices of the intact beam element and

      72 Lc 2

      (10)

      thereafter the natural frequencies are calculated from the equation K 2M=0, where K= Assembled stiffness matrix of the beam, M=Assembled mass matri and = Natural

      C12 E 'bp xF1 (x)dx C21

      0

      0

      a

      frequency (rad/sec). According to Dimarogonas et.al. (1983)

      72 h 2

      (11)

      0

      0

      and Tada et.al. (2000) the additional stain energy due to existence of crack can be expressed as

      C22 E 'bp xF1 (x)dx

      s s 4

      C = GdAC

      (1)

      tan 2 0.923 0.199 1 sin 2

      (12)

      F (s)

      AC

      Where, G = the strain energy release rate, and AC = the effective cracked area.

      I s

      2

      2

      cos s

      2

      2

      2

      2

      2

      2

      G = 1 (2 K

      )2 ( K

      )2 k( K

      )2

      (2)

      FII (s)

      1.122 0.561s 0.085s2 0.180s3

      1 s

      (13)

      E '

      In IIn IIIn

      n1

      n1

      n1

      L3

      e C

      L2

      e C

      Where, E = E for plane stess

      E = E/1-2 for plane strain

      C 3EI

      total L2

      11 2EI 12

      22

      22

      L

      (14)

      k = 1 + =Poissons ratio

      e C

      21

      21

      2EI

      e C

      EI

      E=Youngs Modulus of elasticity.

      KI, KII and KIII = stress intensity factors for sliding, tearing and opening type cracks respectively. Neglecting effect of axial force and for open cracks above equation can be written as

      The stiffness matrix Kcrack of a cracked beam element can be

      tot

      tot

      obtained as Kcrack = LC1 LT , Where, L is the transformation matrix for equilibrium condition

      1 0

      G = 1 (K K )2 K 2

      (3)

      L 1

      L e

      (15)

      E ' I1 I 2

      II1

      1 0

      The expressions for stress intensity factors from earlier studies are given by Uttam Kumar Mishra (2014) as follows

      0 1

      0 1

      Here, equation 1, 2and 3 are coefficients of additional

      KI1

      = 6P1 LC

      bp

      F

      F

      1

      h

      (4)

      flexibility matrix, a= crack depth, h= total depth of the beam, Lc= distance of crack from right node of beam element. E=E/(1-2), where = Poissons ratio, E= modulus of

      KI 2

      6P2

      bp

      F

      1 h

      1 h

      (5)

      elasticity.

   2. TransferMass Matrix Method

   1. P. Patil and S. K. Maiti (2003) have used transfer mass matrix method to find out the natural frequency of a cantilever and simply supported beams. Let there be uniform beam with n cracks located at = x/L = 1, 2, 3, 4,. n. Each crack is represented by a rotational spring with stiffness given by

3. COMPARISON OF PRESENT METHOD OF FINDING NATURAL FREQUENCY WITH THE TRANSFER MASS MATRIX

   METHOD

   Problem Description: A cantilever with crack depth ratio of

   0.5 with cracks located at various locations is taken. It has following properties and is divided into 8 elements for comparison with the natural frequencies obtained by transfer matrix method with cracks considered as rotational spring

   72 f (r )

   72 f (r )

   Ebp

   Ki

   i

   (16)

   model.

   Length, L = 0.78m Breadth, b = 0.04m

   Where,

   Ki is the equivalent spring stiffness for crack i, d is

   Height, h = 0.01m

   depth and b is width of cross section ri(=a/h) is nondimensional crack size and ai is size of the crack. f(ri), is called flexibility function and is given by

   f(ri) = 0.6384(ri)2-1.035(ri)3+3.7201(ri)4- 5.1773(ri)5+7.553(ri)6-7.3324(ri)7+2.4909(ri)8

   (17)

   For an Euler Bernoulli beam the governing equation of motion is

   Mass density, = 7860 kg/m3. Youngs Modulus, E = 210 GPa

   1. onvergence Study

      2

      2 y(x,t)

      2 y(x,t)

      (18)

      x2 EI

      x2

      A

      t2 0

      Through separation of variables y(x,t) = Z(x) cos(t), the mode shape equation is obtained:

      d 4 Z

      2 or d 4 Z 4

      (19)

      EI

      dx4

      Ai Z 0

      • p Z 0

   dx4

   Where =pL, =L1/L, e=2 -1, and K = KL .

   EI

   Equation 13 and 14 can be used to find out natural frequencies for a single crack in a cantilever and simply supported beam.

   Where is mass density (kg/m3), A, cross-sectional area (m2), i , mode i natural frequency (rad/s) , E, modulus of elasticity (N/m2), I, area moment of inertia (m4), and

   Figure 1: Convergence Study of single cracked cantilever beam

   P4 =

   A 2

   EI

   (20)

   B. Comparison of methods

   The general solution of eq (25) can be written as

   Z(x) = C1[cos(px) + cosh(px)] + C2[cos(px) cosh(px)] + C3[sin(px) + sinh (px)] +C4[sin(px) sinh(px)]————(13) Using this relation we calculate displacement Z, slope, bending moment, and shear force at two ends i and i -1 of an arbitrary segment. At crack location there is jump in slope. By forming transfer matrix we derive the following expression for cantilever and simply supported beam as :

   4(1+coshcos)+/K{sinh(cos+cose)- sin(cosh+coshe)+2cosh()sin()-2cos()sinh() 2sin[(1-)]cosh[(1-)] + 2cos[(1-)]sinh[(1-)]}=0

   (21)

   And

   4sin sinh + /K{sinh (cos -cos e)-sin (cosh -cosh e)}=0 (22)

   		Experimental	FEM
   Crack location	Crack depth	1st Frequency	1st Frequency	Error
   No Crack	No Crack	13.45	13.726	-2.052
   0.1	0.5	12.8	12.983	-1.43
   0.2	0.5	13.02	13.188	-1.29
   0.3	0.5	13.15	13.361	-1.6046
   0.4	0.5	13.28	13.5	-1.657
   0.5	0.5	13.37	13.598	-1.705
   0.6	0.5	13.4	13.667	-1.9925
   0.7	0.5	13.43	13.704	-2.04

   		Experimental	FEM
   Crack location	Crack depth	1st Frequency	1st Frequency	Error
   No Crack	No Crack	13.45	13.726	-2.052
   0.1	0.5	12.8	12.983	-1.43
   0.2	0.5	13.02	13.188	-1.29
   0.3	0.5	13.15	13.361	-1.6046
   0.4	0.5	13.28	13.5	-1.657
   0.5	0.5	13.37	13.598	-1.705
   0.6	0.5	13.4	13.667	-1.9925
   0.7	0.5	13.43	13.704	-2.04

   Table 1: Comparison of 1st Natural Frequency (Hz) for single cracked cantilever beam.

   /tr>

   Experimental			FEM		D.P.Patil et.al.
   Crack Location	Crack Depth	2nd frequency	2nd frequency	Error	2nd frequency	Error
   No Crack	No Crack	84.3	86.291	– 2.362	79.84	5.2872
   0.1	0.5	82.89	84.397	– 1.818	81.36	1.842
   0.2	0.5	84.3	85.981	– 1.994	82.7	1.898
   0.3	0.5	83.3	85.398	– 2.519	83.84	– 0.6469
   0.4	0.5	82.87	83.804	– 1.127	84.748	-2.266
   0.5	0.5	81.82	83.64	– 2.224	85.42	-4.4
   0.6	0.5	81.93	83.007	– 1.314	85.86	-4.8
   0.7	0.5	82.5	84.207	-2.07	86.103	-4.367

   Experimental			FEM		D.P.Patil et.al.
   Crack Location	Crack Depth	2nd frequency	2nd frequency	Error	2nd frequency	Error
   No Crack	No Crack	84.3	86.291	– 2.362	79.84	5.2872
   0.1	0.5	82.89	84.397	– 1.818	81.36	1.842
   0.2	0.5	84.3	85.981	– 1.994	82.7	1.898
   0.3	0.5	83.3	85.398	– 2.519	83.84	– 0.6469
   0.4	0.5	82.87	83.804	– 1.127	84.748	-2.266
   0.5	0.5	81.82	83.64	– 2.224	85.42	-4.4
   0.6	0.5	81.93	83.007	– 1.314	85.86	-4.8
   0.7	0.5	82.5	84.207	-2.07	86.103	-4.367

4. RESULTS AND DISCUSSIONS

   From the table 2, it is observed that the errors obtained by transfer mass matrix method of obtaining frequencies as proposed by D.P. Patil is more with respect to experimental frequencies of Kausar H. Barad. It is lesser for frequencies obtained by present finite element method. Also, from figure 3, the variation of natural frequency with crack location for the case of present finite element method is similar to the experimental one, whereas it is not so for the graph of natural frequency variation obtained by transfer mass matrix method. Hence, present method of modal analysis of cracked beams is preferable over transfer mass matrix method of modal analysis.

   Table 2: Comparison of 2ndNatural Frequency (Hz) for single cracked cantilever beam.

5. CONCLUSION

From the results obtained above it can be concluded that

• 1st frequency increases as the crack shifts away from the fixed end of a cantilever beam.

• Due to crack, the natural frequencies of the beam decreases with respect to the undamaged beam.

• Difference between frequencies obtained by transfer mass matrix method is more in comparison to the frequencies obtained by present finite element method.

• Variation of natural frequencies obtained by finite element method is similar to that of experimental one whereas it is not so for those obtained by transfer mass matrix method.

REFERENCES

Figure 2: Comparison of 1st Natural Frequency with experimental results for single cracked cantilever beam.

Figure 3: Comparison of 2nd Natural Frequency obtained by two different methods with experimental results.

1. Uttam Kumar Mishra (2014), Vibration, Buckling and Dynamic Stability of stepped beams with multiple transverse cracks, P.hd thesis, N.I.T Rourkela.

2. A. D. DIMAROGONAS Vibration Engineering. St. Paul, Minnesota: West Publishers. 1976.

3. A.D.Dimarogonas, Vibration of cracked structures: a state of the art review, Engineering Fracture Mechanics, vol. 55, pp. 831-857, 1996.

4. T. CHONDROS M.Sc. Thesis, University of Patras, Greece. Dynamic response of cracked beams. 1977.

5. D. P. Patil and S. K. Maiti (2003), Detection of multiple cracks using frequency measurements, Engineering fracture mechanics 70(2003), 1553-1572.

6. Jialou Hu and Robert Y. Liang (1993), An integrated approach to detection of cracks using vibration characteristics, Journal of franklin institute.

7. Kausar H. Barad, D.S Sharma and Vishal Vyas (2013), Crack detection in cantilever beam by frequency based method, Procedia engineering 51, 770-775.

8. M.H.H. Shen and J.E. Taylor (1991), An identification problem for vibrating cracked beams, Journal of sound and vibration 150(3), 457- 484.

9. S. Christides and A.D.S. Barr (1984), One- Dimensional theory of cracked Bernoulli- euler beams, Int. J. Mech. Sci. Vol. 26, No. 11/12, pp.639-648.