5 th International & 6 th All India Manufacturing Technology, Design and Research Conference (AIMTDR 4) December th 4 th, 4, IIT Guwahati, Assam, India THEORETICAL AND EXPERIMENTAL VERIFICATION OF VIBRATION CHARACTERISTICS OF CRACKED ROTOR SYSTEM IN VISCOUS MEDIUM Adik R. Yadao *, Dayal R.Parhi, Department of mechanical engineering, NIT Rourkela, Orissa 7698 adik.mech@gmail.com, dayalparhi@yahoo.com Abstract This paper contains an attempt to evaluate dynamic behaviors of cantilever cracked shaft with mass attached at the end of the shaft in viscous medium at finite region. In this work theoretical expressions have been developed for finding the fundamental natural frequency and amplitude of the shaft with attached mass using influence coefficient method. External fluid forces are analyzed by the Navier Stoke s equation. Viscosity of fluid and crack depth is taken as main variable parameters. Suitable theoretical expressions are considered, and the results are presented graphically. Further experimental verifications have also done to prove the authenticity of the theory developed. The work leads to the conclusion that, the variation of parameter of cracked shaft system makes an appreciable difference in vibration characteristics of shaft. Keywords:Cracked shaft, Crack depth, Viscous medium, Influence coefficient method. Nomenclature A= Shaft cross-sectional area a = Crack depth L =Length of the shaft D =Diameter of the shaft δ =Whirling radius of the shaft E = Modulus of elasticity of shaft material ɛ = Eccentricity Fx, Fy=Fluid forces on shaft in x and y direction, respectively. β = Relative crack depth (a/d) α = Relative crack position (L/L) I = Section moment of inertia of the shaft L = Total length of the shaft L = Cracked position from left side of shaft. Ms = Mass of the shaft per unit length M = Fluid mass displaced by the shaft per unit length P= Pressure R = Radius of the shaft R = Radius of the cylinder u = Radial flow velocity. v = Tangential flow velocity ν = Coefficient of viscosity µ= Poisson s ratio ƍ = Fluid density ω= Rotating speed ω= Natural angular frequency of the shaft Ω=Angular velocity of whirling 44-Dirn = Direction perpendicular to crack 55-Dirn = Direction along the crack. Introduction Wang et al.()has studied the methodical approach sort out the confines of wind turbine models in analyzing the complex dynamic response of tower blade interaction Mario et al.()has developed a hybrid-mixed stress finite element model for the dynamic analysis of structure assuming a physically and geometrically linear behavior. Nerantzaki et al. (7) have proposed the boundary element method meant for the nonlinear free and force vibration of circular plates with varying thickness undergoing large deflection. Sung Juna(8)have analyzed the dynamic behavior of cracked rotor by using the additional slope and bending moment at crack position. Pennacchi (8)have analyzed the shaft vibrations of a MWfor that proposed a model based diagnostic methodology which is help full to identified a crack in a load coupling of a gas turbine before happening a serious failure problem. Sino et al.(8) have studied the dynamic analysis of an internally damped rotating composite shaft. Natural frequencies and instability thresholds are evaluated with the help of homogenized finite element beam model which is considered the internal damping. Nandi (4) have presented a simple method of reduction for finite element model of non-axisymmetric rotors on no isotropic spring support in a rotating frame. Hashemi et al.(9) have studied a finite element formulation for 46-
THEORETICAL AND EXPERIMENTAL VERIFICATION OF VIBRATION CHARACTERISTICS OF CRACKED ROTOR SYSTEM IN VISCOUS MEDIUM vibration analysis of rotating thick plate. Plate modeling developed by utilizes the mindlin plate theory combine with second order strain-displacement. Zhou et al. (5)have studied the experimental authentication of the theoretical results is required, particularly for the nonlinear dynamic behavior of the cracked rotor. The crack in the rotor was replicated by a real fatigue crack, as a substitute of a narrow slot. Kadyrov et al.()have examined the oscillations of a rigid cylinder in a tubular duct occupied with a viscous incompressible fluid. They used mathematical and the theoretical consequences for eigen frequency exposed to different fluid parameters. In this investigation, a systematic analysis for the vibrational behavior of a cantilever cracked shaft in viscous medium at finite region is obtainable. Damping effect due to viscous fluid is determined with the help of Navier Stoke s equation. Natural frequencyof the shaft used for finding the critical speed of the system is determined using the influence coefficients method.. Theoretical Analysis.. Equation of Motion The Navier- stroke equation for fluid velocity is expressed in the polar coordinate system r-θ as follows, u p u u u u v = + ν + + t ρ r r r r r r θ r θ v p v v v v u = + ν + + + ρ θ θ θ t r r r r r r r Where u and v denote flow velocities in radial and tangential directions, respectively, and p means a pressure. Rewriting the above equation with the help of a stream function ᴪ (r, θ, t) the above equation can be written as, We obtain, ν t ( ψ ) 4 ψ = Equation [] can be divided into two parts i.e. (a) (b) () ψ = ψ + ψ The radial and tangential components of flow velocity at point A in figure are, ua = Rω sin a δωsin( Ωt θ ) va = Rω cos a + δωcos( Ωt θ) When the shaft isimmersed in a fixed circular cylindrical fluid region with radius R, The boundary conditions forr=r are u = v = r= R r= R d F df + F = dr r dr r d F df + + k F = dr r dr r The non-stationary components of flow velocities induced by the whirling motion of a shaft are given as follows R R A + B+ C I ( kr) ud = = jδω e r θ R + D K ( kr ) r ϕ r r j( ωt θ ) R R A + B+ C I ( kr ) + kri ( kr ) + ϕ r r vd = = δω e r R D K ( kr ) krk ( kr ) r.. Analysis of Fluid Forces j( ωt θ ) (3) (4a) (4b) (5) (6a) (6b) (7a) (7b) The flow velocities given by equation (7) to equation (), thenon-stationary component of pressure p can be written as ψ = ψ ψ v t = The solution of above equation can be given by p A p= θ = δρω R + Br e θ r i( ωt θ ) (8) 46-
5 th International & 6 th All India Manufacturing Technology, Design and Research Conference (AIMTDR 4) December th 4 th, 4, IIT Guwahati, Assam, India Only the real parts of equation are meaningful. So and F y after simplification can be express as Fx ( + ε sin ω ) d Y t M M + K Y = F s 55 y (b) x { Re ( ) cos ( ) Im( ) sin ( )} F = mδω H ωt H ωt y { Re ( ) sin ( ) Im( ) cos ( )} F = mδω H ωt H ωt Where H A B C I ( α ) D K ( α ) = and R e ( H ) Im ( H ) denotes the real and imaginary part of H. The coordinates of the center of the shaft are x = cos t and = sin t δ ω ν δ ω d x ( ) ω Im ( ) F = m Re H + m H x d y ( ) ω Im( ) F = mre H + m H y.3. Analysis of Cracked Cantilever Shaft With Mass at Free End In the current analysis a lumped mass at the free end of the cantilever rotating crackedshaftsubmerged in finite fluid region is considered. The ratio of the equivalent lumped mass to the total mass of the shaft in two main direction are given by the expression, α K 44 =, and eq ω M 44 s α e q = dx ω Where M s is the mass of the shaft K 5 5 M dx 5 5 s If a disk with mass M s is attached with the end span of the shaft, a total lumped mass of the shaft is given by the expression M M = M + α M s s eq s M M = M + α M s s eq s (9a) (9b) (a) (b) (a) (b) d ξ * { + M Re( H )} M ω Im ( H ) ( ) cos ( ) + ξ = ε ω ω τ * dξ * d η { + M Re( H )} M ω Im ( H ) ( ) cos( ) + η = ε ω ω τ * dη The steady state solution of the above equation can be obtain in dimensionless form as ξ = δδ cos( ω τ φ ) andη = δδ sin( ω τ φ ) When the 44-direction axis coincides with X axis the amplitude contributes of ɛ in X and Y directions are, * * = cos ( ) and ξ = δ cos ( φ ) ξ δ φ 44 44 The total dimensionless deflection in X and Y direction, when the 44-dir n (perpendicular to crack)and 55-dir n (along the crack) coincide with X-axis and Y- axis respectively is, δ = δ = ξ + ξ alongthe X direction n= 44 44 55 δ = δ = η + η n= 55 44 55 alongthe Y direction In this investigation, consider the dimensionless deflection in 44-dir n (perpendicular to crack). 3. Numerical Results and Discussion (3a) (3b) (4) (5) (6a) (6b) In the current investigation, the cantilever rotor system has the following specification Cantilevercracked shaft with a disc at the free end in viscous fluid at finite region ( + ε cos ω ) d X t M M + K X = F s 44 x (a) ) Material of shaft Mild steel ) Density of material ƍ=783kg/m 3 3) Modulus of elasticity E =.x N/m 46-3
THEORETICAL AND EXPERIMENTAL VERIFICATION OF VIBRATION CHARACTERISTICS OF CRACKED ROTOR SYSTEM IN VISCOUS MEDIUM 4) Length of the shaft L =.m 5) Radius of the shaft R =.m 6) Radius of disc R D =.4m 7) Length of disc L D =.4m 8) Relative crack depth β=.5// 9) Relative crack location α =.65 ) Damping coefficient of viscous fluid ν =.3 /.47 /.633 Stokes. ) Equivalent mass of fluid displaced/corresponding mass of the shaft M * =.58 /.534 /.44 ) Gap ratio q=(r -R )/ R = Illustrations the effect of varying the viscosity of the fluid and crack depth at constant location on the frequency and amplitude of the cantilever crackedshaftwith additional mass which are revolving in the viscous fluid at finite region. In fig., and 3the graph are plotted between dimensionless amplitude ratio and frequency ratio. From fig., and 3it is observed that as the crack depth increase the resonance frequency decreases. It is also found that as the viscosity of the fluid increase the amplitude of vibration decrease, due to increase in crack depth the corresponding amplitude of vibration under same condition decreases. D im e n s io n le s s A m p li tu d e R a ti o 4 8 6 4 v (stokes) M* +++++.3.58 ooooo.47.534 *****.633.44.7.74.76.78.8.8.84.86 Frequency Ratio Figure. Frequency ratio vs. Dimensionless amplitude ratio. Mild steel shaft (R=.m, L=.m, q=, β=.5, α =.65) D i m e n s i o n l e s s A m p l i t u d e R a t i o 8 6 4 D i m e n s i o n l e s s A m p l i t u d e R a t i o v (stokes) M* +++++.3.58 ooooo.47.534 *****.633.44.58.6.6.64.66.68. Frequency Ratio Figure. Frequency ratio vs. Dimensionless amplitude ratio. Mild steel shaft (R=.m, L=.m, q=β=,α=.65) 8 7 6 5 4 3 v (stokes) M* +++++.3.58 ooooo.47.534 *****.633.44.46.47.48.49.5.5.5.53.54.55.56 Frequency Ratio Figure 3. Frequency ratio vs. Dimensionless amplitude ratio. Mild steel shaft (R=.m, L=.m, q=,β=, α =.65) 46-4
5 th International & 6 th All India Manufacturing Technology, Design and Research Conference (AIMTDR 4) December th 4 th, 4, IIT Guwahati, Assam, India 4. Experimental Analysis The Experiment are accompanied by cantilever cracked shaft with additional mass at the free end which is rotating in viscous medium for determining the amplitude of vibration by varying damping coefficient of viscosity of fluid and crack depth of shaft. The speed is controlled by a variac which is connected to the motor shaft from the fixed end of the cantilever shaft.from the free end of shaft the amplitude of the vibration was measured with the help of vibration pick-up devicee and vibration indicator for cantilever cracked shaft rotating in different viscous fluid and with the different crack depth. Table. Influence of crack depth on the amplitude ratio %Error Experimental ν =.633 (stokes) β3 β ν =.47 (stokes) β β3 ν =.633 (stokes) β3 β ν =.47 (stokes) β β3 β3.5.5.5.5 4. 4. 6 4.59 5.6 5.3 4 4.5 4. 4.. 3.7 8.76.64 6.74 8. 9.3 8.36.9 3..7.3 69.7 7.4.39 6..56 9.8 3. 4..9 4. 7.5 89.6 99.76 5.54 9.9 4. Theoretical ν =.633 (stokes) 69.34.8 5.3 ν =.47 (stokes) β β3 3 77.57 7.66 99 5.43 9.7 3 6.6 5.77. 5 4.5.5.9 9.39 67. 4.7 96.67 β 6. 75.46 67.75 5.5 85.6 7.55 86.37 4..7 7.97 35.99 67.8 4. 3.7.9 57.3 74. 9.38.44 8.5 6 3 53 4. 4. 5.4 6.8 4.7.9 4.7 4.7.8 4..64.97 45.88 8.9 9.55.64 56.88 6.7 8.6 65.7 8.8 8.3.6 88.4 8.53 9.87 6.7.65 5.95 6.99 89.79 83. 8.3 4.53 7.46 4.8 7.7 7.5.4 5.33 6.73 7.7 9.43 6.45 6.5 8.8 8. 54 5.3 7.84 Figure 4. Schematic diagram of experimental Setup S. N..5 8 8.9. 3 4.54 4 38.4 5 34 6 9.58 7 7.8 7.6 8 9 46-5
THEORETICAL AND EXPERIMENTAL VERIFICATION OF VIBRATION CHARACTERISTICS OF CRACKED ROTOR SYSTEM IN VISCOUS MEDIUM Table. Influence of varying viscosity of fluid on the amplitude ratio % Error Experimental Theoretical Conclusion V3[stokes] V [stokes] V[stokes] V3[stokes] V [stokes] V[stokes] V3[stokes] V [stokes] V[stokes] S.N..633.47.3.633.47.3.633.47.3 4. 4. 5. 4.59 7 5.6 8 5. 4. 3.7.9 9.3 9.54.9 8.9 9.8 In this paper, vibration characteristics of spinning cantilever cracked shaft with attached mass at the free end in viscous medium at finite span has been analyzed theoretically which have authenticated by the experimentally. From above we determine that as the viscosity of external fluid increases there is a decrease in amplitude of vibration of shaft.this effect can be.7 7.4.56 4.74 67.. 3 3. 4. 5.3 99.76 4. 4.8 96.67 4.54 3.38 4 4. 5.77 4.7.64 38.4 3.8 5 3. 4. 35.99.83.6 34.45 6.9 9.38 3.48 88.4 9.58.96 7 4.7 65.7 8.3.9 6.5 7.8.48 8.9 8.8 4.53 7.89 8.8 7.6 9 8.3 7.46 5.83 8. 7.6 5.64 visualized from figure,and3. Theresult presented in figure,and3.it is establish that as the crack depth increase with constant location the natural frequency and amplitude of vibration of the shaft with crack decrease and the rate of decrease is faster with increase in crack depth. The present study can also be used for rotating shafts in viscous medium such as long rotating shafts used in drilling rigs, high speed centrifugal and high speed turbine rotor etc. References Jianhong Wanga, Datong Qin and Teik C.Lim(), Dynamic analysis of horizontal axis wind turbine by thin-walled beam theory, Journal of Sound and Vibration 39; pp.3565 3586 Mario R.T., Arruda, Luıs Manuel Santos Castro (), Structural dynamic analysis using hybrid and mixed finite element models, Finite Elements in Analysis and Design 57; pp. 43-54. Maria S. Nerantzaki, John T. Katsikadelis (7), Nonlinear dynamic analysis of circular plates with varying thickness, Arch Appl Mech 77; pp.38 39. Oh Sung Juna, Mohamed S. Gadalab (8), Dynamic behavior analysis of cracked rotor, Journal of Sound and Vibration 39; pp. 45. Paolo Pennacchi and Andrea Vania (8), Diagnostics of a crack in a load coupling of a gas turbine using the machine model and the analysis of the shaft vibrations, Mechanical Systems and Signal Processing ; pp.57 78. R. Sino, T.N.Baranger, E. Chatelet and G. Jacquet(8), Dynamic analysis of a rotating composite shaft, Composites Science and Technology 68; pp.337 345. R. Nandi (4), Reduction of finite element equations for a rotor model on non-isotropic spring support in a rotating frame, Finite Elements in Analysis and Design 4; pp.935 95 S.H. Hashemi, S.Farhadi, S.Carra (9), Free vibration analysis of rotating thick plates, Journal of Sound and Vibration 33; pp.366 384. Tong Zhou Zhengce, Sun Jianxu, Xu and Weihua Han (5), Experimental analysis of Cracked Rotor, Journal of Dynamic Systems, Measurement and Control, September 7; 33-3. Van S.G.Kadyrov, J.Wauer and S.V. Sorokin (), A potential technique in the theory of interaction between a structure and a viscous, compressible fluid, Archive of Applied Mechanics 7; pp.45 47. 46-6