On the Forced Response of Harmonically and Partially Mistuned Bladed Disks . Part I : Harmonic Mistuning *

This series of two papers focuses on the determination and clarification of the characteristics of the forced response of bladed disks that exhibit a mistuning pattern that is either harmonic or partial. Harmonic mistuning refers to single wavelength variations in structural properties along the disk while partial mistuning is associated with blade characteristics that are random in a specific sector and tuned elsewhere. The results of this analysis demonstrate that many features of the response of these simple systems match not only qualitatively but also quantitatively their counterparts on randomly mistuned bladed disks. Relying on these similarities, simple and reliable approximations of the localization factor and of the mean response are easily derived that exemplify the usefulness of harmonic and partial mistuning patterns. Finally, it is demonstrated both theoretically and by comparison with simulation results that the maximum amplitude of response of a disk closely follows a Weibull-type distribution in all coupling situations, from very weak to very strong.


INTRODUCTION
The investigation of the effects of mistuning, i.e. of blade-to-blade variations in their geometry and structural properties that occur during the manu- facturing process and/or as a consequence of in- service wear, has been ongoing for more than thirty years.These efforts have resulted in a large number of publications (see for example the works of Whitehead, 1966; Ewins, 1969, etc.) in which many qualitative and quantitative aspects of this problem have been clarified.The unpredictability of the exact properties, e.g.stiffnesses, masses, damping coefficients, of the blades of actual turbomachines has often led to the modeling of these structural characteristics as random variables with specified distributions and moments.The assessment of the damaging effects of mistuning then requires the determination of the probabilistic features of the forced response of these random disks, i.e. moments In this light, the present investigation can be considered a dual of earlier perturbation and Monte Carlo simulation studies since it focuses on obtain- ing "the exact solutions of approximate problems".Specifically, the exact forced response will be derived for two types of mistuned bladed disks, i.e. those whose structural properties vary harmo- nically around the disk (harmonic mistuning) and those which are tuned except for a few consecu- tive blades the properties of which vary randomly (partial mistuning).Interestingly, harmonic mistun- ing has already been considered in a few bladed disk analyses (see for example Valero and Bendiksen,  1986, and, for alternate mistuning, Kielb and Kaza,  1984; Kaza and Kielb, 1985), but only as an example of a mistuning pattern, not as a basis to thoroughly elucidate the effects of the localization phenomenon as is intended here.Specifically, both harmonic and partial mistuning and their corresponding forced responses will be related to the more standard case of a randomly mistuning bladed disk to provide new insights into the prediction of their forced response thereby complimenting perturbation and simula- tion analysis results.

STRUCTURAL DYNAMIC MODELING
For simplicity of the analysis, the one-degree-of- freedom per blade model shown in Fig. will be adopted in the sequel to represent the bladed disk.Previous investigations of this system (Sinha, 1986;  Sinha and Chen, 1989; Wei and Pierre, 1988;  Mignolet and Lin, 1993; 1996; Lin and Mignolet,   1997) have shown that this simple model does exhibit the high sensitivity of the forced response with respect to small fluctuations in the blade stiffnesses (k..) that is characteristic of mistuning.
Following previous investigations (Sinha, 1986;   FIGURE  One degree-of-freedom per blade bladed disk model.Sinha and Chen, 1989), the 24 blades will be assumed to have the same masses (m 0.0114 kg), damping coefficients (c#= 1.443Ns/m, approxi- mately 1% of the critical value), but have different stiffnesses.Specifically, the parameters k# are selected to be independent Gaussian random vari- ables with mean k,-430,000 N/m and standard deviation crl=8,000N/m.Further, the aerody- namic and structural coupling between blades is modeled by springs and dashpots of common values kc and Cc.In the present study, different values of kc will be used but the parameter Cc is set to zero in all cases.
Then, the equation of motion of blade j, j 1,2,..., N, can be written in the form m-+ cJ.+ (kj + 2kc) kcX ._ where .(t)represents the force acting on that blade.
Of particular interest in the present investigation is an rth engine order excitation of the form Fj(t) -/>0e i('+(j-)/xw) j 1,2,..., N, (2) where w and Ab are the corresponding natural frequency and interblade phase angle of the tuned system, i.e.
Introducing this assumed solution in Eq. (1) yields the recurrence relation kc eiX'j+l kc e-i/x'./-+(&j + 2kc cos Ab + icoc).--/5"0 j-1,2,...,N, where &#-kj-kt.The solution of this equation must be obtained with the periodicity conditions J0 JN and 2, 2N+1. (7) Before studying some of the characteristics of the response of this mistuned system, it is appropriate to first obtain the basic features of its tuned counterpart, i.e. with &# 0. Of particular impor- tance in the present investigation is its impulse response, i.e. the solution of kc eiA'/j+l kc e-ix'/Tj-1 + (2kc cos Ab / icoc)hj (Sjl where 5#1 denotes the Kronecker symbol and with periodicity conditions similar to Eq. ( 7).The above nonhomogenous equation is equivalent to the homogenous recurrence kc eiA/+l kc e--izx/j-1 with the boundary conditions kc ei/X/2 kc e-iZX/Tu + (2kc cos Ab + icoc)/7 and Noting that the difference equation, Eq. ( 9), is characterized by constant coefficients, its solution can be sought as a linear combination of terms of the form AA j for some parameters A and A.
Proceeding in this manner, it can be shown that where z0 is any of the two complex solutions of the quadratic equation z 2 (2kc cos Ab + icoc) z0 + 0 (13) kc HARMONIC MISTUNING As already observed by Huang (1982), the structure of Eq. ( 6) is similar to the discretized version of the equation of motion of a single-degree-of-freedom with parametric excitation.Specifically, using the finite difference formula d2(0) .+,2j + j- where 0-(2rc/N)(j-1) and A0--2reiN, in Eq. ( 6) yields the approximately equivalent differential equation d22 d2 kcA02 2ikc AOA dO At this point, note that the two roots of Eq. ( 13) are inverse of each other, i.e. z0 and 1/Zo, and that the expression for the impulse response /.
given in Eq. ( 12) is unaffected by the transfor- mation Zo-+ 1/Zo.Thus, one can conventionally assume that the root z0 has a magnitude z01 _< and this condition specifies the appropriate sign in Eq. ( 14).
In the absence of damping, coo-0, it is seen from Eq. ( 14) that Zo e + i/x/ so that z0] 1.However, the presence of dissipation in the system yields a root Zo such that ]Zo] < and the magnitude ]Zo decreases monotonically as the ratio cco/kc increases.Then, when the number of blades, N, is large enough, it may be assumed that ]Zo ev<< and zo[N--j + << 1, j not close to N, so that The importance of Mathieu's equation in the analysis of parametrically excited systems suggests the expansion of the stiffnesses &j in Fourier series, i.e.
N-1 Skj Z/s e-i(2rc/N)s(j-1) and s=O N /s -Z Skj e i(2r/N)s(j-1) j=l (8) /j e-i(i-1)/xbz Combining this relation and the above observa- tions, it is found as expected that damping creates an exponential decay of the propagation of the impulse response around the disk.Further, the exponential decay rate, -ln z0, increases mono- tonically with the parameter cco/kc.
Note from the above equations that the real character of the stiffnesses 5k.implies that/3,*- /3N-, where denotes the operation of complex conjugation.Further, if the random variables are independent of each other, then so are the coefficients/3,, and Combining Eqs. ( 6) and ( 18), yields the governing equation for the response of the system as + 2kc cos/Xga + icoc + /s e-i(2r/N)s(j-1) j s=0 F0. ( To gain insight into the effects of the small Fourier coefficients/3, on the forced response, a straight- forward perturbation analysis has been carried out assuming that ,<<kc, cco.Accordingly, the forced response of the jth blade can be written in the form to first order in the small coefficients /3,.Note from the above relation that the effect of the mistuning harmonic of order s,/3s, is modulated by the inverse of term icco+2kc(cos(2rcr/N)cos(2rr(r-s)/N)).When the coupling stiffness is much larger than the damping term coo, it is seen that the summation is dominated by the terms corresponding to s 0 and s 2r which yield con- tributions of the order of 1/icco.
This result can be explained physically as follows.
The engine order excitation, Eqs. ( 2)-( 4), is in fact a resonant traveling wave which would create on a tuned bladed disk a response of the same form, i.e. given by Eq. ( 5) with .-constant.Then, a perturbation of the mean stiffness of the blades, i.e., /30, alters the natural frequencies of the system so that the forcing excitation becomes off-resonant.This relative shift of the natural and excitation frequencies is thus accompanied by a large decrease in the response of the blades, as observed from Eq. ( 21), since the modal damping ratios are small.
The preeminence of the terms s-2r in Eq. ( 21) is not associated, unlike the s 0 contribution, with the excitation wave but rather with its reflections which are naturally created by the dissimilarities in the blade properties.These reflections (backward wave) are characterized by the same frequency as the excitation (forward wave) and a magnitude that is directly related to the dissimilarities in the blade properties.Thus, in the presence of random mistuning Sk./, Eq. ( 18), the combination of reflec- tions created at each blade-to-blade interface exhibits all negative interblade phase angles when expanded in Fourier series.Of primary effect on the forced response is the component of this backward wave associated with the interblade phase angle A'--(2rc/N)r which induces, as its forward component, Ab-(2rc/N)r, resonance in the system.This excitation is the result of the fluctua- tions in the stiffnesses of the blades and their response to the forward wave, i.e. it is associated with the product 6kjX.(l).Equivalently, the back- ward wave will be generated when &iXj(t)-Skj [ ei(ct+ (J-1)/x?) 0 ei(ct-(/-1)Aq0 ( 22) for some complex number I?0.The above relation yields the critical mistuning pattern as (Skj-/2r e-i2(j-1)A /2re -i(j-1)(2rr/N)(2r) (23) which is exactly of the form of Eq. ( 18) with s 2r thereby confirming the importance of the/2r term in Eq. ( 21).
The above findings have been obtained under the assumption kc >> coo which is required for the first order perturbation result, Eq. ( 21), to hold.To assess the validity of these observations for kc [0,), the exact forced response of the bladed disk was computed in the presence of a single harmonic of mistuning Skj /s e-i(2r/N)s(j-q-/N_s ei(2r/N)s(j-1) ,/3s, cos [2s(j -1) + O.] (24) with, I/3sl (8,000/v/) N/m,/0 N, and r-3 for different values of kc.Interestingly, it was   found that neither the mean amplitude observed over the bladed disk nor the corresponding root mean square value were functions of the phase qSs of the stiffness mistuning distribution.However, their variations with the harmonic number s for three typical Values of the coupling stiffness, kc, are shown in Figs. 2 and 3. Clearly, for values of this parameter much larger than the damping term (cco8,000N/m) and the mistuning strength (lsl-1,633N/m), i.e. for kc=45,430N/m in Figs. 2 and 3, both the mean and root mean square values of the amplitude exhibit sharp peaks at the expected values of the harmonic number, that is at s 0, s 2r, and s N-2r.
As the blade-to-blade coupling is reduced, keeping the damping level and the mistuning strength constant, it is seen that the peaks have decreased substantially in magnitude and have broadened to include not only the values s=2r and s=N-2r but also the neighborhooding harmonic numbers, see Figs. 2 and 3 for kc-8,000N/m.This result is still consistent with the first order perturbation results given by Eq. ( 21).
Reducing further the coupling stiffness to 1,000 N/m leads to a rather different situation, as seen in Figs. 2 and 3. Specifically, it is seen that the mean value of the amplitude of response is almost independent of the harmonic order s > 0 except for s=N/2 (=12) value at which a sharp peak is encountered.A similar, but less dramatic, situation can be observed in the plot of the root mean square of the amplitude.At the contrary of the previous observations, this finding cannot be ex- plained by investigating Eq. ( 21) since this equation   would provide a similar effect of all harmonic numbers s.This situation is in fact not surprising since the validity of Eq. ( 21) is limited to the cases where kc >> c,/3,,, which is clearly not true here since , , , (= 1,633 N/m) and c (= 8,868.38N/m) are both larger than ke (: 1,000N/m).Nevertheless, the preeminence of the harmonic orders s N/2 can be explained on physical grounds; it is symptomatic of a strong localization of vibration in the system.(Wei and Pierre, 1988; Castanier and Pierre, 1993).
That is, the reflected waves shed at each blade-to- blade interface with dissimilar properties rapidly decay as they propagate along the disk.Then, the effects of fluctuations in blade characteristics with long wavelengths which consist of slowly varying changes in the amplitude of vibration of the blades will be damped before they produce a large variability in the response.Thus, it is the short wavelength changes in stiffness, i.e. those associated with harmonic numbers s close to N/2, that will have the largest effect on the dynamic behavior of the system.Additional physical under- standing of this situation can be obtained by relying on a recent analysis (Castanier and Pierre, 1993) of the localization phenomenon in damped nearly periodic structures.Although the present investi- gation focuses on cyclic assemblies, the strong localization results derived therein are applicable in the present context since localization prevents the flow of "information" around the disk so that the cyclic character of the system only has a minor influence on its response.
The study of Castanier and Pierre (1993), has revealed the critical role of the two parameters o-/R and 5/R where (25) so that cr / + o kc (26) Specifically, it was demonstrated in that study that strong localization occurs when either /R >> or air >> 1.With the values/3 1,633 N/m and ca;= 8,868.38N/m used in obtaining the results shown in Figs. 2 and 3, it is found that a/R= 1.63 and /R 8.87 for kc 1,000 N/m and the strong local- ization character of the corresponding response is confirmed.The other cases investigated in Figs. 2 and 3, i.e. kc= 8,000 N/m and 45,430 N/m, corre- spond to the values (a/R, /R) (0.204, 1.114) and (0.036,0.201), respectively, which can both be classified as weak to moderate localization situa- tions.Thus, to obtain a complete picture of the effects of localization in harmonically mistuned bladed disks, it is necessary to also investigate the cases in which one of the ratios 6/R or air is much larger than unity but the other one is not.
To this end, the coupling stiffness was kept as kc= 1,000N/m and the mistuning strength /3s was reduced to a tenth of its previous value, i.e.
/3. 163.3 N/m, so that air 0.163 < 1. Shown in Figs. 4 and 5 are the mean and root mean square amplitudes of the response for the values ca;= 8,868.38N/m, /R 8.868 (curve labeled (2)), and cc=886.84N/m,/R=0.887 (curve labeled (3)).Also displayed in these figures and labeled as (1) are the curves corresponding to kc 1,000 N/m shown in Figs. 2 and 3. Comparing the curves (1) and ( 2), it is seen that the decrease in mistuning strength in this system strongly localized by damping has not changed the physical characteristics of the forced response, i.e. the preeminence of the harmonics, sN/2, it has only changed the magnitudes of their effects.
Reducing the damping term ca; to 886.84 N/m (0.1% of critical) yields the curves labeled (3) which are very similar those seen in Figs. 2 and 3 in connection with kc=8,000N/m.This result is expected since the corresponding values of c,/R and 5/R are 0.163 and 0.887, respectively, which are representative of a mildly localized disk.
To investigate the dominance of the effects of the localization associated with mistuning, the coupling stiffness was again set as kc= 1,000N/m but the mistuning strength was selected to be /3.= 8,000N/m, so that cr/R 8.00.Shown in Figs. 6 and 7 are the mean and root mean square amplitudes of the response as functions of the mistuning harmonic number for the values cco=8,868.38N/m(curve labeled (4)), and coo= 886.84N/m (curve labeled (5)).In the first case, for which 6/R 8.868 so that (5/R a/R, it is found that the overall features of the mean and root mean square values of the amplitude of response are very similar to those shown in Figs. 2 and 3, kc 1,000 N/m case.However, there are some important differences between these sets of curves.Specifically, although the increase in the strength of mistuning has increased the localization in the system, it is seen that the response exhibits an increased sensitivity with respect to the harmonics s 4, 6, and 8 of the mistuning pattern.These com- ponents which have period of 3, 4, and 6 blades correspond to longer range interactions than their s N/2 12 counterpart which was found domi- nant when 5/R > cr/R and 5/R >> 1.In fact, increas- ing further the strength of mistuning leads to even FIGURE 6 Mean amplitude of response as a function of harmonic number.Case (4) (right axis): kc= 1,000 N/m, 113,,1-8,000 N/m, c= 1.443 Ns/m; Case (5) (left axis): kc= 1,000N/m, ]fl,.l 8,000N/m, c=0.1443Ns/m.higher peaks at s 4, 6, and 8 (N/6, N/4, and N/3) and a reduced one at s--12 (N/2).
From the above sets of observations, it is con- cluded that the strong localization due to damping leads to a response which is most sensitive to the shortest wavelength fluctuations in the structural properties.Strong localization that is mostly generated by mistuning leads to a special sensitivity of the response with respect to the discrete set of wavelengths that are integer divisors of the number of blades in the disk.Moreover, the sensitivity of the response is largest for the smallest divisors.
This observed evolution, with changing coupling stiffness, of the characteristics (harmonics) of the mistuning pattern that are most dominant in the forced response of the disk closely parallels the findings of a recent investigation by Lin and   Mignolet (1997).Specifically, it was demonstrated in that study that the largest modal contributions in the response of the system are associated with the modes of the tuned system whose natural frequencies are closest to the excitation frequency when the coupling stiffness is large.Further, as this param- eter decreases, the number of modes that have a strong presence in the distribution of the blade amplitudes increases.Finally, similar observations were also found to hold in the weakly coupling limit kc << cw provided that the modes of the uncoupled system are used.
In this earlier paper, these findings were then used to formulate a novel adaptive perturbation strategy in which the dominant modal contribu- tions were accurately computed while the remain- ing ones were approximated.The excellent results obtained with this approach and the similarities between this study and the present investigation suggest reformulating the governing equation for the response of the system, Eq. ( 6), in the form kc ei/X/j+l kc e-i/x/ ,/-- MISTUNED BLADED DISKS 39 (s) Root mean square amplitude of response as a function of harmonic number.Case (4) (right axis): kc 1,000N/m, ]/3.,.] 8,000 N/m, c= 1.443 Ns/m; Case (5) (left axis): kc-1,000 N/m, Iris 8,000 N/m, c=0.1443Ns/m.
with s 2r.When the ratio u N/s is an integer, it is seen that the above equation describes the response of a bladed disk which is composed of s perfectly identical sets of u blades.In this case, the f,(0) j.(0) periodicity of this system implies that and the response of the bladed disk requires only the determination of __jJ)!) j_ 2, u.This reduced set of unknowns produces a computa- tional savings that can be dramatic especially for small values of u.
When u=N/s is not an integer, the forced response of the disk is periodic only with period N but still can be expressed as S where lcm(N, s) is the least common multiple of N and s.Introducing Eq. ( 32) in Eq. (31) yields a system of equations for the parameters 0) in the form (-kc ei/xc kc e-i/xfc-n +/0 + 2kc cos A + icc) (0) +/3N-2(0) +/3 (0) n+l n-1 --/0 5,,0 n 0, 1,2,... ,m, ( where =e i(2r/N)s.Note the extreme similarity between this equation and Eq. ( 6) with the fol- lowing important differences: (i) The variables 0) denote the Fourier coeffi- cients of the response not the response itself as in Eq. ( 6).(ii) The roles of the coupling and mistuning are reversed; the response parameters 0) and are related to each other through the mis- tuning terms/3, and tiN--s, not kc as in Eq. ( 6), while the variations of the parameters of the above difference equation are associated with the coupling stiffness kc, not with/3, and u-,.
The above findings demonstrate that there exists a deep rooted duality between mistuning and coup- ling: the connection between the response of neighborhooding blades is provided by the structural coupling while mistuning links their Fourier com- ponents, or equivalently the modes of the tuned system.This observation provides new insight into the successes achieved with the adaptive perturbation technique (Lin and Mignolet, 1997).That is, when the coupling is strong, the response is localized to a few modes of the tuned system and a very reliable approximation of the amplitudes can be obtained by including a few modal components.On the contrary, when the coupling is weak, the response is localized to a few blades and the modes of the decoupled system clearly represent the dynamics of the system.

SUMMARY
In this first paper, the exact characteristics of harmonically mistuned bladed disks have been investigated to provide phenomenological insights into the effects of full random mistuning on the response of these systems.The contributions of this part of the investigation can be summarized as follows.
(1) The features of the mistuning pattern that most seriously affect the resonant forced response have been highlighted and their dependency on the level of blade-to-blade coupling has been elucidated.In particular, it was demonstrated that strongly coupled bladed disks are espe- cially sensitive to harmonic fluctuations in the structural properties of the blades that cor- respond to a wave propagating in the tuned system at the same frequency as the excitation but in the reverse direction (backward wave).As the level of coupling is reduced, the preemi- nence of this Fourier component is reduced and the short wavelength harmonics of, the mistuning pattern produce increasing fluctua- tions of the response of the disk.
(2) The presence of strong localization effects created by a small level of structural damping (1% of critical) has been demonstrated for the first time and the differences between the forced responses of bladed disks strongly localized primarily by damping or by mistuning have been presented.Specifically, it was concluded that the strong localization due to damping leads to a response which is most sensitive to the shortest wavelength fluctuations in the struc- tural properties.Strong localization that is mostly generated by mistuning leads to a special sensitivity of the response with respect to the discrete set of wavelengths that are integer divisors of the number of blades in the disk.
Moreover, the sensitivity of the response is largest for the smallest divisors.
(3) The level of blade-to-blade coupling and the strength of mistuning were shown to play dual roles of each other.Specifically, the connec- tion between the response of neighborhooding blades is provided by the structural coupling while mistuning links their Fourier compo- nents, or equivalently the modes of the tuned system.This observation indicates that when the coupling is strong, the response is localized to a few modes of the tuned system and a very reliable approximation of the amplitudes can be obtained by including a few modal compo- nents.On the contrary, when the coupling is weak, the response is localized to a few blades and the modes of the decoupled system clearly represent the dynamics of the system.

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FIGURE 2
FIGURE 2 Mean amplitude of response as a function of harmonic number for different coupling stiffnesses, ,1= 1,633 N/m, c 1.443 Ns/m.

FIGURE 3
FIGURE 3 Root mean square amplitude of response as a function of harmonic number for different coupling stiffnesses, /3,.1,633 N/m, c 1.443 Ns/m.

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