Detection of a fatigue crack in a rotor system using full-spectrum based estimation

Sādhanā Vol 4, No 2, February 26, pp 239 25 c Indian Academy of Sciences Detection of a fatigue crack in a rotor system using full-spectrum based estimation C SHRAVANKUMAR and RAJIV TIWARI Department of Mechanical Engineering, Indian Institute of Technology Guwahati, Guwahati 7839, India e-mail: rtiwari@iitgernetin MS received December 24; revised 7 May 25; accepted 8 October 25 Abstract The force due to crack switching has multiple harmonic components of the spin speed These components excite the rotor both in the same and reverse directions of the rotor spin A full-spectrum method using complex Fast Fourier transform equations is developed to obtain force coefficients and displacement coefficients forward and reverse) These coefficients are then used as an input to developed identification algorithms Fault parameters identified are the additive stiffness due to crack, disc eccentricity due to unbalance, and system viscous damping An extended algorithm estimates the crack forces The algorithms are numerically tested Keywords crack forces Switching crack; model-based identification; full-spectrum; fault parameter estimation; switching Introduction 2 Literature review A rotor shaft system during its operation can be subjected to multiple faults; such as unbalance, crack, misalignment, looseness, rub and the bow Fatigue cracks in rotor systems are common but also likely lead to catastrophic faults Accidents due to cracks in turbine rotors have been reported since 95s [] Hence, the online condition monitoring of a rotor shaft for the crack propagation becomes a necessity Condition monitoring literatures available on cracked rotors could be broadly grouped into following methods: the model-based and signal-based methods, the modal testing, and non-traditional methods Sabnavis et al [2] Two identification algorithms are developed in the present work: one for the estimation of rotor crack parameter: ie, additive stiffness in the shaft due to crack, along with disc eccentricity due to unbalance and shaft viscous damping The other algorithm is for estimation of crack forces which contribute to its switching mechanism Both the algorithms use force coefficients and displacement coefficients, directly obtained from full-spectrum as their input Using full-spectrum frequency responses for identification has advantages such as reduced data handling, directly obtaining the forward and reverse displacement coefficients which are otherwise to be estimated), and flexibility to consider the coefficients of higher harmonics of the crack force in identification algorithms For correspondence In this section, a literature review on identification of cracks in rotors, and application of full-spectrum signal-processing for fault diagnostics are presented Crack identification Dimentberg [3] observed that the 2 component in the frequency spectrum is a strong indicator of crack, which occurs because of the shaft crossing two limiting stiffness twice during one revolution The cracked shaft has a critical speed at half the natural frequency of the intact rotor The crack identification from changes in eigen parameters has been studied by Pandey et al [4] Trend analyses of, 2, and3 response amplitudes are expected to increase in direct proportion to the crack size, according to Gasch [5] Dharmaraju et al [6] and Sekhar [7] have studied model-based methods for identifying crack parameters such as the stiffness, location, and depth are used based on an inverse problem approach Darpe et al [8] and Chasalevris & Papadopoulos [9] used coupling effects introduced by the crack for identification Signal-based methods for crack identification have also been used by Liu & Jiang [] Al-Shudeifat & Butcher [] modeled crack based on the time varying area moment of inertia at crack section For each time step, new values of area moment of inertia are calculated, based on which the centroid location and stiffness matrix are updated Shravankumar & Tiwari [2] identified the rotor crack using model-based parameter estimation method Crack was identified by means of the additive stiffness it introduces at the cracked section The additive stiffness is negative or reduction in stiffness 239

24 C Shravankumar and Rajiv Tiwari The main limitation of the work was identification in time domain, which requires handling of large amount of system data, and intermediate steps in the identification algorithm The present work overcomes these limitations of the authors [2] by the application of full-spectrum to model-based identification, which reduces data handling and in addition, the magnitude and phase information of various forward and reverse frequency components harmonics) of crack force and the rotor response can be directly obtained Full-spectrum The concept of full spectrum as forward and backward whirl frequencies has been in use for the development of Campbell diagram However, the extension of Fast Fourier transform technique FFT) to obtain full spectrum is relatively new and has been developed since two decades Now, the use of full-spectrum and related signal processing techniques in fault identification of rotor systems is elaborately reviewed In order to overcome defects in traditional methods of vibration signal processing, Qu et al [3] and Jian [4] introduced the Wigner distribution and Short Period Fourier Transform SPFT) These methods were a precursor to the development of full spectrum Qu et al [5] also introduced the method of holospectrum The full-spectrum was introduced as plus and minus spectrum enhancement by Southwick [6, 7] in the Transient Data Manager2 proprietary software of Bently Nevada Determining the forward and reverse whirl direction of rotors, determining synchronous and nonsynchronous vibration components, degree of orbit ellipticity due to preload and other conditions were discussed 994) Full spectrum cascade plots of a full annular rub with reverse precession were given for both the startup and shutdown conditions Full spectrum plots were also illustrated for the preload and shaft crack conditions Goldman & Muszynska [8] used full spectrum to display the phase correlation between the horizontal and vertical spectral components of a rotor orbit Full spectrum was used to determine whether the rotor orbit is forward or backward in relation to the spin direction Various symptoms of rotor faults that could be detected using full spectrum were highlighted Tuma & Bilos [9] identified the whirl frequency components and the fluid induced instability components using the full spectrum in their study of fluid instability in rotor systems with journal bearings Lee & Han [2] integrated the techniques of Wigner distribution and SPFT with full spectrum and provided the directional Wigner Distribution dwd) in order to study non-stationary rotor vibration They proposed the Shape and Directivity Index SDI) to quantify the shape and direction of instantaneous whirl orbits during the rotor run-up Patel & Darpe [2] used the full spectrum to investigate the directional nature of higher harmonics for detection of rub in cracked rotors Ishida et al [22] studied the non-stationary vibrations of a rotor shaft with nonlinear spring characteristics during acceleration through critical speeds Approximate solutions by means of asymptotic method and amplitude variation curves of individual vibration components were obtained by the complex-dft method They concluded it as the most suitable method for the analysis of non-stationary vibrations Bachschmid et al [23] used the SDI method in order to analyze the ellipticity of filtered orbit and also the amplitude and the inclination angle of the rotor orbit major axis in the study of a large turbo-generator unit subjected to the rotor and stator rubs Shravankumar & Tiwari [24] have applied full-spectrum method for model-based crack identification This work only identifies crack as a reduction in flexibility The present work extends the crack identification to switching crack forces, estimates of which give the actual behavior of a fatigue crack in a rotor system From the literature review, it is clearly observed that applications of full spectrum have been applied chiefly for signal-based fault identification methods The full-spectrum method could be applied for model-based identification methods and it is the main motivation of the present work 3 System modeling of a cracked rotor with transverse crack A Laval rotor with a centrally located rigid disc and a massless elastic shaft is considered, which is supported on rigid bearings figure ) Transverse translation displacements in two orthogonal directions x and y have been considered Gyroscopic effect and shaft elastic coupling are ignored for simplicity A switching crack function based on the hinge model figure 2) is considered The system equations of motion EOMs) along with the time and frequency domain solutions have been elaborately derived in earlier works [2, 24] The final equations are presented briefly for completeness The system EOMs in complex form are obtained by combining the displacements along x and y directions, using a complex variable r = x + j y Combining the EOMs in complex form reduces the number of equations, and applicability of response to full-spectrum which has a pre-requisite of complex response with phase correlation of x and y displacement data For orthogonal directions, the relative phase is 9 The EOMs are obtained as m r + cṙ + kr = 2 w yst) k ξ + exp j 2ωt)) +meω 2 exp jωt + β)) ) Here, m is the disc mass, c is the viscous damping, k is the intact shaft stiffness, r is the complex displacement response, w y is the static deflection of the shaft in y direction, s t) is the switching crack function, k ξ is the additive stiffness due to crack, e is the disc eccentricity due to unbalance, ω is the rotor spin speed, and β is the phase due to

t Crack parameter identification using full-spectrum 24 z o x o x y e x y y a) b) Figure a) A Laval rotor with a cracked shaft b) The unbalanced disc and cracked shaft in inertial and rotational coordinates st) 8 6 4 2 Figure 2 5 5 2 25 3 35 deg) Variation of switching function st) with shaft rotation unbalance The first term on the RHS of Eq ) was further simplified to the following closed-form expression 2 w xst) k 22 + e 2jωt) [ = w x k 22 25 + 39e jωt +6e jωt + 25e j2ωt +6e j3ωt 2e j3ωt 2e j5ωt + 32e j5ωt +32e j7ωt] 2) The first and second terms on the right side of the EOMs Eq )) are the forces due to crack and unbalance faults For a fatigue crack, the crack faces are alternatively opening and closing as the shaft spins Therefore the shaft stiffness changes with time The maximum stiffness is that of the intact shaft The variation or reduction in intact shaft stiffness is periodic and modeled using a switching function st) [5] Hence, this periodic variation of the shaft stiffness multiplied by static deflection of the shaft is considered on the right side of the system EOMs as the crack force The present work aims to identify both these faults The crack force could be further generalized as w y k ξ + i= p i exp j iωt) 3) Here, p i is the force coefficient of i th harmonic component of crack force with direction same as the rotor spin Likewise, p i is the force coefficient in the direction opposite to the rotor spin p ±i which along with w y k ξ contribute the magnitude of the switching crack forces p ±i can be obtained from the full-spectrum coefficients of the switching crack function These results are presented in a later section on numerical illustrations Now, Eq ) is a linear, second order differential equation Its closed-form solution can be obtained as the summation of the particular integrals due to crack and unbalance forces It is given as rt) = + i= p i k ξ w y m iω) 2 exp j iωt) + c jiω) + k + meω2 exp j β) mω 2 exp j ωt) 4) + c jω) + k In practice, the closed-form solutions of the rotor system may not be available Also, in closed-form solutions there is limitation on the number of harmonic components based on the approximation of the Fourier series of the switching function But, in case of time-integrated solution, the switching function can be solved directly Equation ) is written

242 C Shravankumar and Rajiv Tiwari in a state space form, which converts a second order differential equation into two single order equations The single order equations can be solved using fourth order Runge Kutta method of numerical integration The equations in state space form are written as ṙ = v r = m k v c k v + k +meω 2 exp jωt + β)) w x k ξ + ) i= p i exp j iωt) 5) Here, v is the velocity response Both the closed-form and time-integrated responses have been used subsequently for identification The algorithms using closed-form solutions serve to bench mark the algorithm using time-integrated responses These results are discussed in a later section The numerical responses do not contain any noise Hence, in order to mimic a practical rotor response, random noise is generated and added to the numerically generated responses Gaussian white noise is used for the simulation It is defined as a statistical noise that has its probability density function equal to that of the normal distribution It is a random signal with a flat power spectral density, uncorrelated and normally distributed with a mean zero and unit variance The noising response signal could be obtained as r noise t) = r t) + { r t) R 5) N p } 6) Here, R is a random scalar value with mean and standard deviation Numerical responses are added with, 2 and 5 percentages N p /) of its values as the noise In frequency response analyses, the system response is computed at discrete excitation frequencies The forcing is thus a function of frequency It gives a measure of magnitude and phase of the vibration response say, displacement) as a function of frequency, in comparison to the force For a particular harmonic i of the forcing function, the assumed solution is r i ω) exp jiωt) Since, system EOMs are linear, using the principle of superposition, the assumed solution for each harmonic i can be summed up as rt) = n r i ω) exp j iωt) 7) i= n Substituting the assumed solution into the EOMs Eq )), response equations in frequency domain are obtained as follows [{ } ] iω) 2 m + {jiωc) + k } r i = k 22 w x p i + f unb 8) Similar to p i explained above, the r i is the displacement coefficient of the i th harmonic of crack response, obtained using full-spectrum 4 Need for full-spectrum in crack identification problem Conventional FFT gives magnitude spectrum as well as phase spectrum But they do not contain information about the relative phase between two different vibration signals Also, they have no information on the direction of frequency components of crack force or response) with respect to the shaft rotation The rotor orbit is a plot of vibration signals along two transverse directions and contains multiple forward and reverse whirling components Both magnitude and phase information are together required to determine the actual shape of the rotor orbit A full-spectrum plot decomposes the rotor orbit into forward whirl positive) and backward whirl negative) frequency components At a glance, the full-spectrum plot shows whether the orbit of a particular frequency component is forward or reverse with respect to rotor spin direction [8] This is because the full spectrum uses the relative phase correlation of the two vibration signals, which constitute the orbit In the present work, the crack force figure 3) consists of multiple harmonic components of spin speed Some of its harmonics excite the rotor in the same sense of rotation of spin whereas the other harmonics excite the rotor in the reverse sense of spin refer figure 4) The harmonics, which excite the rotor in the same direction of spin, are denoted positive and their magnitude is given by p +i forward force coefficients) The harmonics in the reverse direction as spin are considered negative and their magnitude is given by p i reverse force coefficients) Similarly, the response coefficient magnitudes r +i and r i are forward and reverse respectively It is not possible to directly obtain p i s or r i s reverse force coefficients and reverse displacement coefficients, respectively) using conventional FFT Hence, there is a need for using full-spectrum signal processing in a crack identification problem It can be seen from Eq 4) and Eq 8) that there is a need to know the reverse force coefficients and displacement coefficients in crack problem, as they constitute the time and frequency domain solutions, which will also be used N) force Crack N) force Crack 5 5 5 2 Time s) 5 5 5 5 2 Time s) Figure 3 Variation of the crack force with time in the vertical top) and horizontal bottom) directions at measurement speed of 5 rad/s

p i 3 25 2 5 Crack parameter identification using full-spectrum 243 / t The sampling rate also satisfies the Nyquist Shannon sampling theorem ie, f s 2f max ) where f max is the maximum bandwidth) frequency In a similar manner, response coefficients r i s can be obtained as r d) = N N l= {r l t)} exp 2jdl/N) ) 5 5 5 5 Frequency rad/s) Figure 4 Full-spectrum of the crack force shows force coefficients forward as well as reverse p i s) at measurement speed of 5 rad/s to develop the identification problem Also, in practice, the crack switching function would be unknown, that means a closed-form expression for switching function or crack forcing cannot be assumed Hence, it is more reliable to use the full-spectrum of crack force and responses, and obtain their respective coefficients A full-spectrum requires a complex time domain response in its input For this purpose, system of EOMs in x and y directions are combined in a complex form as mentioned in Eq ), and the full-spectrum of the force function 2 s t) { + exp j2ωt)}, as well as the response r t) are obtained Resultant spectrums will contain magnitude and phase information of both the forward and reverse force and displacement coefficients Obtaining the full-spectrum plot using complex-fft is explained next The full-spectrum force and displacement coefficients can be obtained using standard complex-fft equations [25] Full-spectrum force coefficients p d) which are complex Fourier coefficients) are obtained as N pd)= {sl t)+ exp2jωl t))}exp 2jπkl/N) N l= The full-spectrum converts a time domain complex signal of length N sample length) into a frequency domain complex signal of same length N Its coefficients are complex In Eq 9), the frequency index d length of the full spectrum signal) varies from to N The index d from to N/2 corresponds to p+d) and N/2 to N corresponds to p d); ie, pn/2 +d) = p d)letp i correspond to pd) where 2πd/n = iω, andi =, 2,, l is the number of discrete data in the time history of st) +e j2ωt ) with t as the time interval sampling time) Here, i corresponds to various positive and negative harmonic components of the crack force The acquisition time T)of the time domain signal is N t Samples are acquired at a sampling rate f s ) which is 9) A full-spectrum can be effectively plotted by placing zero frequency component corresponding to DC) at the centre; positive frequencies to the right and negative frequencies to the left The frequency at N/2 corresponds to the Nyquist frequency The maximum frequency or the bandwidth frequency of the full-spectrum depends on the sampling frequency or the sampling time Based on this, the number of harmonics components in the spectrum can be chosen 5 Crack identification algorithm In the authors previous work [2] two identification algorithms have been developed: one for identification of fault parameters: namely, additive stiffness due to crack, and disc eccentricity due to unbalance, along with the system viscous damping parameter An extended algorithm was also developed to identify the switching crack forces, which can be used in real case for obtaining an idea of opening/closing behavior of a fatigue crack The inputs for these identification algorithms, ie, force coefficients were obtained from the closed-form expression of crack force and displacement coefficients were only estimated using displacement responses in time domain This has a limitation on two factors: one is on choosing the higher harmonic components of crack force The closed-form expression of switching function provides the p ±i s limited to the first few harmonic components as in Eq 2) In this closedform expression, crack force is limited to the harmonics i =±, ±3, ±5, +2, +7 On the other hand, full-spectrum can provide p ±i values for any required number of harmonic components of switching function This result is illustrated in a subsequent section and will be particularly useful practical estimation of crack forces Also, instead of a switching function any other time-varying function can also be easily implemented using full-spectrum The other factor is the large amount of time domain data required to estimate the displacement coefficients r ±i This data usage is greatly reduced while directly obtaining r ±i from full-spectrum Now, the generalized form of the identification algorithms developed in the previous work are written, which are suitable for application with full-spectrum Equation 8) which is the system EOM in frequency domain is written in a linear regression form and further used in the identification problem The regression equation is entirely based on frequency domain data ie, force coefficients and

244 C Shravankumar and Rajiv Tiwari displacement coefficients) The generalized identification algorithm is written as follows, for n harmonic components where A 2n+) 3) = b 2n+) ) = x 3 ) = A x = b, ) j nω) r n w y p n j ω) r w y p w y p j ω) r mω 2 e jβ w y p j 2ω) r 2 w y p 2 j nω) r n w y p n n 2 ω 2 ) m k r n ω 2 ) m k r k r ω 2 m k ) r 4ω 2 m k ) r2 n 2 ω 2 ) m k rn c e k 22 2) Bold capital letters are used for matrices and bold small letters are used for vectors Refer Nomenclature for details Unique solution of x can be obtained if the matrix A is square and satisfies the condition for matrix inversion In the general case, A may be any rectangular matrix and cannot be inverted with sufficient conditions Hence, an approximate solution of x or estimates) is found using the least-squares solution, which is a common form of the linear regression x = A T A ) A T b 3) For well-conditioning of the matrices, the identification algorithm is modified and written for multiple measurement speed ranges [26] When the number of regression equations is increased the estimation is also better averaged and the effect of noise is reduced drastically Thus, for accommodating estimation with a set of multiple measurement speeds say ω to ω q, the regression equation Eq 2)) is modified as A ω ) b ω ) A ω 2 ) b ω 2 ) x 9 ) = 4) ) ) A ωq b ωq nq 9) nq ) Thus, the present section algorithm would give some confidence in usage of the full-spectrum in model-based identification However, it has the limitation that the crack force coefficients p ±i are unknown in the real case So, the further idea is to illustrate estimating these coefficients using the cracked rotor response Once by estimating p ±i the periodic crack forcing can also be known Eq 3)) The following section alleviates this difficulty by developing estimation algorithm for the crack force coefficients as well with the help of full-spectrum 6 Identification of switching crack forces The generalized crack force is expressed as w y + k 22 i= p i exp jiωt) in Eq 3) Here, st) or p i defines the crack force completely and it is unknown This implies that by estimating k 22 p i, the profile of the periodic crack force can be obtained at any defined running speed and time interval For this purpose, the identification algorithm, ie Eq 2) is modified into the following estimation equations containing terms k 22 p i in the unknown vector, as shown in Eq 7) Noting that the regression equation contains terms to be estimated for the crack force, iep i and the additive crack stiffness k 22 ; both terms are treated as an unknown, and the equation is rearranged into a new regression equation where p i terms of A are included in the unknown vector x 2 together with k 22 It is given in the following form, Here, A 22n+) 2n+3)) = [ D A 2 x 2 = b 5) D 2 ],where w D 2 = y ; w y w y w y

Crack parameter identification using full-spectrum 245 D = x 22n+3) ) = c e k ξ p n k ξ p k ξ p k ξ p k ξ p 2 k ξ p n ; n 2 ω 2 ) m k r n ω 2 ) m k r ω k r b 2n+) ) = 2 ) m k r 4ω 2 ) m k r2 n 2 ω 2 ) m k rn j nω) r n w y j ω) r w y j ω) r mω 2 e jβ j2ω) r 2 j nω) r n 6) For accommodating estimation with a set of multiple measurement speeds, the regression equation Eq 6)) is modified as A 2 ω ) A 2 ω 2 ) A 2 ωq ) 2n+)q 3 x 23 ) = b ω ) b ω 2 ) b ωq ) 2n+)q 7) From the full-spectrum of crack force thus estimated, p i s can be obtained and used further in Eq 2) estimating crack stiffness, as described in the previous section Now, through numerical simulations identification algorithms will be elaborated in order to illustrate effectiveness 7 Numerical Illustrations In this section, illustrations of identification algorithms have been presented through numerical simulation of a chosen rotor model A Laval rotor, consisting of a simply supported elastic shaft with a disc at the mid-span, is considered for numerical illustration as shown in figure The rotor model data are summarized in table A crack near disc causing variation in stiffness due to switching function st) as shown in figure 2 is considered It should be noted that st) can have any time variation and the present identification algorithm is not restricted by this variation Responses are generated using closed-form expression Eq 4)) as well as by time-integration Eq 5)) The orbit plots of x displacement responses versus y displacement responses are shown in figure 5 The proposed identification algorithms Eq 2) and Eq 6)) are tested with the help of numerical responses obtained The estimates of damping, eccentricity and crack force parameters are obtained and their accuracy is analyzed Obtaining force coefficients Figure 3 and figure 4 show the variation of crack force and its corresponding fullspectrum plot containing p i s, respectively Figure 4 shows the full-spectrum plot with forward and reverse exciting Table Rotor system data for the numerical simulation Parameters Disc mass, m Intact shaft stiffness, k Additive negative) crack stiffness, k ξ Viscous damping in rotor system, c Phase of unbalance, β Shaft deflection due to disc weight, w y Disc eccentricity, e Value 2kg 3235 5 N/m 6 4 N/m 63 N-s/m π/8) rad w y = mg/k = 65 5 m 6 m

246 C Shravankumar and Rajiv Tiwari yt ) displacement m ) x 6 2 2 4 5 a) 5 x 6 5 x 5 x 6 5 5 5 5 b) x 6 x 5 r i x 7 8 6 4 2 5 5 5 c) x 5 xt) displacement m) d) x 5 Figure 5 Orbit plots at a) 8 rad/s, b) 9 rad/s, c) 2 rad/s, and d) 2 rad/s during passage through critical speed due to twice per revolution excitation component crack force coefficients These coefficients p ±i are tabulated for multiple harmonics of crack force in table 2 The p ±i s obtained using full-spectrum are the same as that obtained from closed-form coefficients These are further used as an input for the identification algorithm Eq 2)) to estimate crack and other unknown parameters Obtaining displacement coefficients The displacement coefficients r ±i are obtained from full-spectrum coefficients of the displacement responses of cracked rotor closedform or time integrated), using Eq ) The full-spectrum plot ofr d) against frequency shows various displacement coefficients as in figure 6 7 Simultaneous estimation of crack and rotor parameters using full-spectrum In this section numerical testing of the developed identification algorithm Eq 2)) is performed for various cases and the estimates obtained are compared with the assumed parameters 5 5 Frequency rad/s) Figure 6 The full-spectrum of the cracked rotor response showing displacement coefficients forward and reverse r i s) at measurement speed of 5 rad/s Inverse problem In order to test the algorithms, some of the parameters are treated as unknown grey-box model identification) and are then estimated from the identification algorithm using full-spectrum coefficients This is the inverse problem approach used in the model-based identification The estimates can finally be compared with assumed values in the direct problem) for testing the accuracy of the proposed algorithms In this work both closed-form and time-integrated responses are tested for parameter estimation Numerical responses are also added with different levels of, 2 and 5 percentages of random noise Now different measurement spin-speed ranges are chosen for Eq 4) and Eq 6)) for which the displacement responses are considered for identification Figure 7 and figure 8 show the resonance plots of various frequency components of crack force To avoid measurement error near critical speeds often measurement outside the half-power frequency band is advisable The half-power points are two frequencies on either side of the resonant amplitude They are often referred as side bands and their corresponding amplitudes will be X = 77X res, wherex res is the resonant amplitude and X is the amplitude at half-power points For example, it is noted from figure 7 that the Table 2 Values of p i corresponding to full-spectrum coefficients of crack force I p i i p i i p i i p i 383 25 5 6 5 2 2 25 6 7 3 7 9 3 6 3 22 9 9 9 8 5 22 5 9 2 8 2 7 7 9 7 5 23 7 23 6 9 5 9 32 25 6 25 5 32 22 27 5 27 4 3 22 3 6 29 4 29 3

Crack parameter identification using full-spectrum 247 m) Displacement 4 x 4 2 2 3 4 5 6 4 x 5 2 2 3 4 5 6 5 x 5 2 3 4 5 6 Frequency rad/s) Figure 7 Plot for resonance of a) once per revolution, b)twice per revolution, and c) thrice per revolution harmonics of crack force m) Displacement 4 x 6 2 2 3 4 5 6 2 x 6 2 3 4 5 6 5 x 6 2 3 4 5 6 Frequency rad/s) Figure 8 Plot shows resonance of a) five per revolution, b) seven per revolution, and c) nine per revolution harmonics of crack force half-power point frequencies corresponding to resonance of 3 frequency component are 323 and 336 rad/s From these considerations, the range of frequencies is selected The general identification algorithm Eq 2)) contains harmonics varying from n to n Now, a particular number of harmonics can be chosen for each measurement range by making use of the resonance plots It is observed that the magnitude of higher harmonics is very low of the order of 8 m Also, the sampling time determines the maximum frequency and harmonic in a full-spectrum plot For example, a sampling time of s gives a sampling rate of Hz and a maximum full-spectrum frequency of 5 Hz Based on these, in a given speed range, the harmonic components until the magnitude of 8 m refer figure 7 and figure 8) are chosen and lower magnitudes are neglected The different measurement cases, their corresponding range of frequencies, and the number of harmonics considered in each case are summarized as in table 3 The estimates obtained for different cases of measurement speeds are now discussed For case A, estimates are obtained for a variety of noise levels of, 2, and 5 percentages and are summarized in table 4 The estimates are observed to conform well to the assumed parameters The maximum percentage error in estimates of damping is 4% for 5% noise level; for additive crack stiffness, it is 26% at 5% noise level and there is no error in estimates of the disc eccentricity at any noise level For cases B, C, and D also the estimates are obtained similarly The estimates of 5% noise are summarized in table 5 The maximum percentage error in damping estimates is 9% for case B, 53% for case C, and 22% for case D For eccentricity it is around % for cases B, C, and D for 5% noise level The maximum percentage error in additive crack stiffness is 32% for case B, 3% for case C, and 43% for case D Now, the subsequent section presents results for estimation of crack forces due to switching Table 3 Different speed range and harmonics considered for estimation Cases Range of frequencies rad/s) Harmonics i) considered for algorithm Case A 323 7, 5, 3,,,, 2, 3, 5, 7 Case B 336 984 5, 3,,,, 3, 5 Case C 224 3969 3,,,, 3 Case D 45 6 3,,,, 3 Table 4 Estimates of parameters in the measurement speed range of 3 rad/s Parameters cn-s/m) e 6 m) k ξ 4 N/m) Assumed values 63 6 Noiselevelin % 2 5 2 5 2 5 Estimates closed-form 63 67 62 625 99 99 99 97 6 6 59 57 Estimates time-integrated 67 68 68 68 99 98 6 6 59 57

248 C Shravankumar and Rajiv Tiwari Table 5 Estimates of parameters for measurement cases B, C, and D with 2% and 5% noise in response 35 9 rad/s at 2 39 rad/s at 4 6 rad/s at Measurement speed range rad/s interval rad/s interval rad/s interval Parameters Assumed Noise level 2% 5% 2% 5% 2% 5% value in % cn-s/m) 63 CF 6 599 57 58 595 578 TI 68 67 64 642 637 638 e 6 m) CF 93 9 97 95 2 TI 99 97 99 99 99 97 k ξ 4 N/m) 6 CF 59 56 59 56 58 54 TI 59 57 59 57 59 57 CF Estimates from closed-form responses; TI Estimates from time-integrated responses) 72 Simultaneous estimation of crack forces and rotor parameters using full-spectrum For case A, estimates are obtained for different noise levels of, 2 and 5 percentage The estimates from time-integrated responses are presented in table 6 In order to check the accuracy of the estimates the assumed and estimated switching crack forces are plotted for a convenient spin speed and compared as in figure 9 The comparison is shown for the 2% and 5% noise levels for responses in both the horizontal and vertical directions For cases B, C, and D, the estimates are similarly obtained for the, 2 and 5% noise levels The estimates for 5% noise level in response are presented in table 7 and table 8 Figure 9 shows the comparison of assumed and estimated crack excitation forces at 7 rad/s for 2% and 5% noise levels Similar comparisons are made for cases C and D at spin speeds of 3 rad/s and 5 rad/s for 2% and 5% noise levels as shown in figure and figure From these figures, it is seen that the estimated crack forces conform well to the assumed ones N) force excitation Crack 5 5 5 2 4 5 5 a) c) 5 2 4 5 5 5 5 b) 2 4 d) 2 4 Time s) Figure 9 Comparison of the assumed and estimated crack force at 7 rad/s a) With 2% noise in x response; b) with 2% noise in y response; c) with 5% noise in x response; d) with 5% noise in y response Assumed and Estimated Table 6 Simultaneous parameter estimation in the range 3 rad/s with noise Noise level in % % % 2% 5% Parameters Assumed values CF TI CF TI CF TI CF TI cn-s/m) 63 63 68 67 68 62 68 624 68 e 6 m) 99 99 99 99 97 97 k ξ p N/m) 544 546 55 5838 5896 5529 564 4963 49877 k ξ p 2 N/m) 47 47 472 3993 39972 3969 39772 38969 3973 k ξ p 3 N/m) 748 749 747 6954 6962 6857 6878 6565 6623 k ξ p 5 N/m) 34 34 342 3379 344 3346 3387 3249 3336 k ξ p 7 N/m) 46 462 462 44 453 47 445 35 42 k ξ p N/m) 47 47 469 39954 39969 39737 39769 3988 397 k ξ p 7 N/m) 8 82 84 85 89 87 88 95 8 k ξ p 5 N/m) 46 462 467 388 459 39 452 45 429 k ξ p 3 N/m) 34 34 346 33793 3389 3348 3372 3255 3322 k ξ p N/m) 748 749 748 6965 6964 688 688 6629 6627 CF Estimates from closed-form responses; TI Estimates from time-integrated responses)

Crack parameter identification using full-spectrum 249 Table 7 Simultaneous parameter estimation in the range 35 9 rad/s with noise Noise level in % % % 2% 5% Parameters Assumed values CF TI CF TI CF TI CF TI cn-s/m) 63 63 68 62 64 6 68 594 67 e 6 m) 97 93 99 9 97 k ξ p N/m) 544 546 553 594 597 5736 564 52 49872 k ξ p 2 N/m) 47 47 479 39948 39998 39725 39779 3963 3978 k ξ p 3 N/m) 748 749 76 6964 6939 6878 6889 669 663 k ξ p 5 N/m) 34 34 3429 343 3429 3449 3389 357 3329 k ξ p N/m) 46 47 47 39937 39963 3974 3977 395 3973 k ξ p 5 N/m) 47 462 466 244 464 462 74 455 k ξ p 3 N/m) 8 34 3423 3445 344 348 3385 3585 3329 k ξ p N/m) 46 749 746 6974 696 6899 6874 6679 666 CF Estimates from closed-form responses; TI Estimates from time-integrated responses) Table 8 Simultaneous estimation of damping, eccentricity, and crack force parameters for measurement cases C and D with 2% and 5% noise levels in response 2 39 rad/s 4 6 rad/s Noise level in % Measurement speed range 2% 5% 2% 5% Parameters Assumed values CF TI CF TI CF TI CF TI cn-s/m) 63 57 64 58 642 595 637 578 638 e 6 m) 97 99 95 99 3 99 2 97 k ξ p N/m) 544 5593 5649 49775 4988 587 5892 587 523 k ξ p 2 N/m) 47 3972 39796 3929 3989 39456 39789 3839 3989 k ξ p 3 N/m) 748 6997 6844 697 6569 6647 6979 643 683 k ξ p N/m) 34 39667 39763 38954 3963 39768 39753 3966 3947 k ξ p 3 N/m) 46 358 3432 3669 3443 279 322 32 324 k ξ p N/m) 47 685 6854 6555 66 6833 6942 648 6686 CF Estimates from closed-form responses; TI Estimates from time-integrated responses) N) force excitation Crack 5 5 5 2 3 5 5 a) c) 5 5 5 2 3 5 2 3 2 3 Time s) Figure Comparison of the assumed and estimated crack forces at 3 rad/s a) With 2% noise in x response; b) with 2% noise in y response; c) with 5% noise in x response; d) with 5% noise in y response Assumed and Estimated 5 b) d) N) force excitation Crack 5 5 5 5 5 5 5 a) c) 5 5 5 5 5 5 5 5 5 5 Time s) Figure Comparison of the assumed and estimated crack forces at 5 rad/s a) With 2% noise in x response; b) with 2% noise in y response; c) with 5% noise in x response; d) with 5% noise in y response Assumed and Estimated 5 b) d)

25 C Shravankumar and Rajiv Tiwari It is seen that the estimated crack forces conform very well to the assumed forces It is also observed that the forces in the direction of crack are unsymmetrical about the mean position, while the forces in the orthogonal direction are symmetrical about their mean position figures 9 ) This behavior is due to crack switching 8 General guidelines for crack identification in rotor systems The general guidelines for model-based identification of crack and other unknown system parameters are presented in this section The steps for identification are as follows: The cracked rotor responses, namely, displacement versus time responses, and full-spectrum frequency responses are obtained These responses can be simulated for a chosen data of system and fault parameters In such a case, it constitutes the direct problem The responses can also be obtained directly by means of an experimental set up The second step is the Full-Spectrum analysis Here, displacement responses in two orthogonal directions are combined in a complex form and fed as input to the fullspectrum The displacement coefficients r ±i are obtained from the full-spectrum The third step is the Identification of switching crack forces The input to the identification algorithm consists of displacement coefficients of the previous step discussed above, and known system parameters, such as the mass, intact shaft stiffness, and static deflection of the rotor system The estimated parameters are the terms of switching force k ξ p ±i ), which constitute the crack force magnitude, together with unbalance eccentricity, and viscous damping The plot of the switching crack forces can give the details of opening/closing profile of the fatigue crack in the actual case The fourth step is Full-Spectrum analysis of crack forces From the full-spectrum, the force coefficients p ±i are obtained The fifth step is Identification of crack stiffness Inputs to the identification algorithms include the displacement and force coefficients from the step two and the step four), along with known system parameters The unknown fault parameter, ie additive stiffness due to crack, is estimated for the condition monitoring of crack In addition, the disc eccentricity and the viscous damping are also estimated 9 Conclusions This paper discusses the need and application of fullspectrum for a cracked rotor problem Initially, a Jeffcott rotor with fatigue crack is considered Closed-form solution and numerical responses are obtained for unbalance and crack excitations Effect of measurement noise is considered to simulate experimental data Full-spectrum method is used further to obtain force and displacement coefficients This information is used for crack identification as described follows An identification algorithm has been developed for the simultaneous estimation of additive crack stiffness, viscous damping, and eccentricity An extended algorithm has been developed to estimate switching crack forces, along with damping, and eccentricity The algorithms are completely based on frequency domain full-spectrum) inputs The full spectrum has been employed to obtain force coefficients as well as displacement coefficients, which are then input to the identification algorithm The algorithms have been written in the form of linear regression equations Unknown parameters have been estimated using Least Squares approach As the crack forces have excitation harmonic components both in the forward and reverse directions of spin, full-spectrum is the suitable technique for frequency domain identification The algorithm is tested for different measurement spinspeed ranges and for different noise levels The comparison of estimated and assumed parameters shows robustness of algorithms The estimates of the crack force give an idea of the crack opening/closing function as well The algorithms can accommodate any crack opening/closing function and also can handle required number of coefficients of crack force and response harmonics The use of full-spectrum data as input has the advantage that the algorithms can be used for rotor response with reverse whirl conditions The switching crack model illustrated for the present algorithms makes it applicable for small or moderate cracks, which also means that they can be effectively used for early crack detection Also, as the identification is purely done in frequency domain it helps in data reductions and gives the flexibility to consider higher harmonics of crack force, which can represent the crack breathing function more closely In future, the algorithms can be extended for a multiple crack problem together with location and quantification of crack by finite element modeling of the rotor system The algorithms will be experimentally verified by testing for the developed laboratory and practical rotors with transverse fatigue crack Nomenclature A b c d D, D 2 e exp f s f max FFT i = n,,,,+ n j = k regression matrix regressand or known vector viscous damping real number sub-matrices unbalance eccentricity exponential of sampling rate bandwidth frequency fast Fourier Transform integer showing number of the harmonic component imaginary number intact shaft stiffness

Crack parameter identification using full-spectrum 25 k ξ additive stiffness parameter due to crack along direction ξ m disc mass N sample length of time or frequency response N p percentage of Noise p i crack force coefficients from full-spectrum q number of measurement speeds r t) = x + j y complex displacement r i displacement coefficients from full-spectrum r noise time response with noise s t) periodic, time-varying crack switching function t time t sampling time w x static deflection x translational displacement in x direction y translational displacement in y direction ω rotor spin speed order of frequency β phase angle between crack front direction and unbalance first order time derivative second order time derivative ν state space co-ordinate for velocity References [] Ishida Y 28 Cracked Rotors: Industrial machine case histories and nonlinear effects shown by simple Jeffcott rotor Mech Syst Signal Process 22: 85 87 [2] Sabnavis G, Kirk R G, Kasarda M and Quinn D 24 Cracked shaft detection and diagnostics: A literature review The Shock and Vibration Digest 364): 287 296 [3] Dimentberg F M 96 Flexural vibrations of rotating shafts London, England: Butterworths [4] Pandey A K, Biswas M and Samman M M 99 Damage detection from changes in curvature mode shapes J Sound Vibr 452): 32 332 [5] Gasch R 993 A survey of the dynamic behaviour of a simple rotating shaft with a transverse crack J Sound Vibr 6: 33 332 [6] Dharmaraju N, Tiwari R and Talukdar S 24 Identification of an open crack model in a beam based on force-response measurements Comput Struct 82: 67 79 [7] Sekhar A S 24 Crack identification in a rotor system: a model-based approach J Sound Vibr 27: 887 92 [8] Darpe A K, Gupta K and Chawla A 24 Coupled bending, longitudinal and torsional vibrations of a cracked rotor J Sound Vibr 269: 33 6 [9] Chasalevris A C and Papadopoulos C A 23 Experimental detection of an early developed crack in rotor-bearing system using an AMB Third International Conference of Engineering against Failure, June 26 28, Kos, Greece [] Liu C and Jiang D 23 Experimental study on lateral and torsional vibration of cracked rotor with torsional excitation ASME Turbo Expo 23, June 3 7, 23, Texas, USA [] Al-Shudeifat M A and Butcher E A 2 New breathing functions for the transverse breathing crack of the cracked rotor system: Approach for critical and subcritical harmonic analysis J Sound Vibr 33: 526 544 [2] Shravankumar C and Tiwari R 23a Identification of stiffness and periodic excitation forces of a transverse switching crack in a Laval rotor Fatigue Fracture Eng Mater Struct 363): 254 269 [3] Qu L, Xie A X and Li X 993 Study and performance evaluation of some nonlinear diagnostic methods for large rotating machinery Mech Math Theory 285): 699 73 [4] Jian M 996 Some advanced techniques in fault feature extraction for large rotating machinery PhD dissertation Xian Jiao Tong University [5] Qu L, Liu X and Chen Y 989 Discovering the holospectrum J Noise Vibr Worldwide 58 62 [6] Southwick D 993 Using full spectrum plots Orbit 44): 2 6 [7] Southwick D 994 Using full spectrum plots, Part 2 Orbit 52): 6 [8] Goldman P and Muszynska A 999 Application of full spectrum to rotating machinery diagnostics Orbit 7 2 [9] Tuma J and Bilos J 27 Fluid intability of rotor systems with journal bearings Eng Mech 4: 69 8 [2] Lee C W and Han Y S 998 Use of directional Wigner distributions for identification of the instantaneous whirling orbit in rotating machinery 7 th ISROMAC 22 26 [2] Patel T H and Darpe A K 2 Application of Full spectrum analysis for rotor fault diagnosis IUTAM Symposium on Emerging Trends in Rotor Dynamics : 535 545 [22] Ishida Y, Yasuda K and Murakami S 997 Nonstationary oscillations of a rotating shaft with nonlinear spring characteristics during acceleration through a major critical speed A discussion by the asymptotic method and the complex-fft method) J Vib Acoust 9): 3 36 [23] Bachschmid N, Pennacchi P and Vania A 24 Diagnostic significance of orbit shape analysis and its application to improve machine fault detection J Brazilian Soc Mech Sci Eng 2: 2 28 [24] Shravankumar C and Tiwari R 23b Model-based crack identification using full-spectrum ASME 23 Gas Turbine India Conference December 5 6, 23, Bangalore, Karnataka, India GTINDIA 23-3756) [25] Smith S W 997 The scientist and engineer s guide to digital signal processing California Technical Publishing, San Diego [26] Tiwari R 25 Conditioning of regression matrices for simultaneous estimation of the residual unbalance and bearing dynamic parameters Mech Syst Signal Process 9: 82 95