Vibration Analysis Of Cantilever Shaft With Transverse Cracks R.K Behera, D.R.K. Parhi, S.K. Pradhan, and Seelam Naveen Kumar Dept. of Mech Engg. N.I.T., Rourkela,7698 Dept. of Mech. Engg Dept. of Mech. Engg, P.C.E., Rourkela Rourkela, 7698 N.I.T, Rourkela, 7698 ABSTRACT It has been observed since long that, the dynamic behavior of a structure changes due to presence of crack. Scientific analysis of such phenomena can be utilized for fault diagnosis and detection of cracks in structures. In this paper attempts have been made to detect the cracks of a mild steel cantilever shaft. Theoretical expressions have been developed for determining natural frequencies and mode shapes for elastic cantilever shaft having 'two' cracks using flexibility influence coefficients and local stiffness matrix. The numerical results for the beams having no crack, 'single' crack and two cracks are compared. Mode shapes have been plotted for relative crack depth.5 for 'single' crack and.5 each for 'two' cracks respectively. It is observed from the numerical results that, there are appreciable changes in vibration characteristics of the cantilever shaft with and without cracks which can be utilized for multi crack identification of structures. INTRODUCTION Since the dynamic behavior of structure changes due to presence of crack, identification and location of cracks are required in structural design. The frequencies of natural vibrations, amplitude of forced vibrations and areas of dynamic stability change due to existence of such cracks [-6].An analysis of these changes make it possible to identify the
magnitude and location of the crack. This information enables us to determine the degree of sustainability of the structural element and the whole structures. LOCAL FLEXIBILITY OF A CRACKED SHAFT UNDER BENDING AND AXIAL LOADING The presence of transverse surface cracks of depth a and a at a distance L and L respectively from the fixed end on a shaft of diameter D introduces a local flexibility Fig., which can be defined in a matrix form. The geometry of the cracked section is shown in Fig.. The cantilever shaft is subjected to axial force P and bending moment P which gives coupling with the longitudinal and transverse vibration motion. y U U U 3 a dξ η L Y L Y Y 3 d -b o b R h ξ L Fig. : Beam Model ξ Fig. : Geometry of cracked section Using the available expressions for stress intensity factors, Castigliano s theorem and strain energy release rate the compliance matrix can be obtained. The local stiffness matrix can be obtained by taking the inversion of compliance matrix. The stiffness matrices for relative crack position β and γ are obtained as:
k k C C K = = and k k C C k K = k k k C = C C C ANALYSIS OF VIBRATION CHARACTERISTICS OF A CRACKED SHAFT A cantilever shaft of length L and radius 'R' with two crack depths at a distance L and crack depth a at a distance L from the fixed end is considered (Fig.).If T is the period of vibration, by substituting x = x / L, U = U / L, Y = Y / L, t = t / T and β = L / L, γ = L / L the system can be derived with the help of equations for longitudinal and transverse vibration in non dimensional form []. U i x = U i c u t and 4 4 Y i x = Y i c y t () where i = for x β, i = for β x γ, i = 3 for γ x, c u = c T / L, and c y = c T / L, u y The normal function for the system can be defined as u (x) = A cos(k u x) A sin(k u x) () + u (x) = A cos(k u x) A sin(k u x) (3) 3 + 4 u (x) = A cos(k u x) A sin(k u x) (4) 3 5 + 6 y (x) = A cosh(k y x) + A sinh(k y x) + A cos(k y x) A sin(k y x) (5) 7 8 9 + y (x) = A cosh(k y x) + A sinh(k y x) + A cos(k y x) A sin(k y x) (6) 3 + y (x) = A cosh(k y x) + A sinh(k y x) + A cos(k y x) A sin(k y x) (7) 3 5 6 7 + Where x = x / L, u = u / L, y = y / L, t = t / T k u u u y y = ωl / c,c = (E / ρ), k = ( ωl / c ) y,c = (EI / µ ), µ = Aρ A i,(i =,8) are the constants to be determined from boundary conditions[ 7] 4 8 a
NUMERICAL ANALYSIS The mode shapes for no crack, single crack and two cracks are plotted. They are compared in order to observe the change in mode shapes. For small relative crack depth it is difficult to notice the change in mode shape. However for large crack depth the change in mode shapes are quite substantial. The mode shapes for relative crack depth of.5 are shown in Fig. 3 Fig..5 3.5.5 Amplitu d e 4 6 8 - - -3 4 6 8 Fig. 3: First mode of transverse vibration a/d=.5, L / L =.5 cracked, uncracked Fig. 5: Third mode of transverse vibration a/d =.5, L / L =.5 cracked, uncracked
Amplitu d e - 4 6 8 - -3 B eam P osition Fig. 4: Second mode of transverse vibration a/d =.5, L / L =. 5 cracked uncracked.5.5.5 -.5 -.5 -.5 4 6 8.5.5.5 4 6 8 Fig. 6: First mode of transverse vibration / D =.5, a / D =.5, L / L =.5, L / L.5 cracked, uncracked a =.5.5 -.5 4 6 8 - -.5 Fig. 7: Second mode of transverse vibration a / D =.5, a / D =.5, L / L =.5, L / L cracked uncracked =.5 Fig. : Second mode of longitudinal vibration a / D=.5,a / D=.5,L / L=.5,L / L=.5 cracked, uncracked
3 - - -3 4 6 8.5.5 -.5 - -.5 4 6 8 Fig.8: Third mode of transverse vibration a / D =.5, a / D =.5, L / L =.5, L / L =. 5 cracked, uncracked..8.6.4. 4 6 8 Fig. 9: First mode of longitudinal vibration a / D =.5, a / D =.5, L / L =.5, L / L cracked, uncracked =. 5 Fig. : Second mode of longitudinal vibration a /D=.5,a /D=.5,L /L=.5,L /L=.5 cracked, uncracked
CONCLUSION It is observed from the numerical results that, there are appreciable changes in vibration characteristics of the cantilever shaft with and without cracks which can be utilized for multi crack identification of structures NOMENCLATURE A= cross-sectional area of shaft A i, i =,8 = unknown co-efficients of matrix A a, a =depth of crack b = half the width of the crack B = vector of exciting motion C ij = elements of the compliance matrix R(D/)=Radius of shaft E = young s modulus of elasticity F i,i =, = experimentally determined function h=height of rectangular strip I = moment of inertia of shaft section i,j =variable J= strain energy release rate k =Local flexibility matrix element ij L = length of shaft L, L = location of first and second crack from fixed end P i,i =, = axial force (i=), bending moment (i= ) u i,i =, = normal functions (longitudinal) (x) U i,i =, =longitudinal vibration, U i (x, t) x, y = co-ordinate of the shaft Y = amplitude of the exciting vibration Y i,i =, = normal functions (transverse) y i (x) Greek symbols ω = natural circular frequency β, γ = relative crack locations λ= πη/ h µ = A ρ u i
ν =poison ratio ξ = coordinate at the cracked surface ρ =mass density of the shaft ξ = = ξ a D = relative crack depth a D = relative crack depth REFERENCES. Papadopoulos, C.A. and Dimarogonas, A.D., Stability of cracked rotors in the coupled vibration mode. Transactions of the ASME, Journal of Vibration, Acoustics, Stress, and Reliability in Design, 356-359. 988.. Anifantis, N. and Dimarogonas, A.D.,Stability of columns with a single crack subjected to follower and vertical loads. International Journal of Solids and Structures 9(3), 8-9.983. 3. Gounaris, G.and Dimarogonas, A.D., A finite element of a cracked prismatic beam for structural analysis, Computer and Structure 8(3), 39-33. 988. 4. Chen, L-W. and Chen, C Lu., Vibration and stability of cracked thick rotating blades. Computer and Structure 8(), 67-74.988. 5. Dimarogonas, A.D. and Papadopoulos, C.A., Vibration of cracked shaft in bending. Journal of Sound and Vibration 9(4), 583-593.983. 6. Cawley, P., and Adams, R.D.,The Location of Defects in structures from Measurements of Natural Frequencies. Journal of Strain Analysis, 4(), 49-57.979. 7. Parhi D.R.K., Behera A.K., and Behera R.K., Dynamic characteristics of cantilever beam with transverse crack. Journal of Aeronautical Society of India.Vol.47.3-43.995.