Qualitative Analysis of Forced Response of Blisks With Friction Ring Dampers
Denis Laxalde, Fabrice Thouverez, Jean-Jacques Sinou, Jean-Pierre Lombard
aa r X i v : . [ phy s i c s . g e n - ph ] J a n Qualitative Analysis of Forced Response of BlisksWith Friction Ring Dampers
D. Laxalde a,b , F. Thouverez a , J.-J. Sinou a , J.-P. Lombard b ( a ) Laboratoire de Tribologie et Dynamique des Syst`emes (UMR CNRS 5513)´Ecole Centrale de Lyon, 36 avenue Guy de Collongue, 69134 Ecully Cedex, France ( b ) Snecma – Safran group, 77550 Moissy-Cramayel, France
Abstract
A damping strategy for blisks (integrally bladed disks) of turbomachinery involving afriction ring is investigated. These rings, located in grooves underside the wheel of theblisks, are held in contact by centrifugal loads and the energy is dissipated when relativemotions between the ring and the disk occur. A representative lumped parameter model ofthe system is introduced and the steady-state nonlinear response is derived using a multi-harmonic balance method combined with an
AFT procedure where the friction force iscalculated in the time domain. Numerical simulations are presented for several dampercharacteristics and several excitation configurations. From these results, the performance ofthis damping strategy is discussed and some design guidelines are given.
Keywords:
Bladed-disk, friction damping, nonlinear dynamics, harmonic balance method
Turbomachinery bladed disks operate in severe environments in terms of aerodynamic loads andmay experience high vibratory stresses due to resonance or flutter. Their ability to withstandthese stress levels, which may cause high cycle fatigue, mainly depends on the external inputsof damping. In bladed disk assemblies, friction in interfaces is the most widely used source ofexternal damping; this includes shroud contact in blades or platform dampers. These deviceswere widely studied in the past and numerous methods of design and analysis are found inthe literature. The interest reader can find examples based on lumped parameter or equivalentmodels in works of Griffin (1990), Sinha and Griffin (1983), Ferri and Heck (1998) or Csaba(1998) and, more recently, extended to more complex structures in works by Guillen and Pierre(1999), Nacivet et al. (2003) or Petrov and Ewins (2003) among others. The contact interfacecould be one dimensional, two dimensional or three dimensional (e.g. Sanliturk and Ewins, 1996;Chen et al., 2000; Nacivet et al., 2003).Beside, numerous experimental works were presented both in contact kinematic description(see Menq et al. (1986); Yang and Menq (1998) for example) and dynamic response prediction(e.g. Berthillier et al., 1998; Yang and Menq, 1998; Sanliturk et al., 2001).However, in blisks (integrally bladed disks where disk and blades are a single piece), inherentdamping is very low and mechanical joints no longer exist leading to the disappearance of nearlyevery sources of energy dissipation. Consequently, new methods and devices need to be found1or the damping of blisks. Among others, friction damping using a ring is a solution which isfound, experimentally, to be efficient on single piece rotating structures (such as labyrinth sealsfor example). These rings are located in dedicated grooves underside wheels and contact ispermanently maintained by centrifugal load due to engine rotation; friction and slipping occurduring the differential motion of the two bodies in contact.As opposed to under-platform dampers in bladed disk assemblies, these devices were seldomstudied; Ziegert and Niemotka (1993) have presented methods of analysis and optimal designof split ring dampers for seals were the motion is studied using a quasi-static beam-like de-scription (strength of materials). This quasi-static description lead to a good understanding ofthe phenomenon and give some qualitative results. However, a nonlinear dynamics analysis isrequired to raise satisfying design rules.In this paper, methods for predicting nonlinear steady-state response of blisks with frictionring dampers are presented along with some qualitative results based on a representative lumpedparameter modelling. The dynamical analysis is performed using the multi-Harmonic BalanceMethod (
HBM ). This method is found to be efficient and was widely used, in the past, to studyfriction dampers in bladed disks assemblies or shroud contacts. Some examples have beenpresented by Wang and Chen (1993), Sanliturk et al. (1997) and Yang and Menq (1997) formono-harmonic vibrations and by Pierre et al. (1985), Petrov and Ewins (2003), Nacivet et al.(2003), and Guillen and Pierre (1999) for multi-harmonic vibrations.Results from numerical simulations (using this nonlinear analysis) are then presented anddescribed. Several parametric studies are presented in terms damper characteristics, contactcharacteristics, excitation configuration and mistuning are presented and some design guidelinesare raised.
Lumped parameter models have been widely used to study friction. Beside from being sim-pler to study and compute (particularly for parametric studies), they can provide some inter-esting and valuable qualitative results if correctly designed. In turbomachinery applications,several models have been proposed. Among others, we can mention Sinha and Griffin (1983)or Wang and Chen (1993) for friction damping applications and Griffin and Sinha (1985) andLin and Mignolet (1996) for mistuning applications.In this paper, we present a new lumped parameter model. It consists in an elementary sectorof a blisk and its associated friction ring element and it is represented in figure 1.As this system is rotationally periodic (or cyclic), its modes are such that the deflectionof a given elementary sector (described in figure 1) is the same as, but with a constant phasedifference from, the preceding (or following) sector. If N is the number of elementary sectors ofthe structure, the phase angle is 2 πn/N where n is called the nodal diameter number . There are N possible values of n (from 0 to N −
1) but only N/ N/ N is evenor ( N − / N − / N is odd, represent different mode shapes; theothers being orthogonal to them with the same eigenfrequency. Consequently, except for n = 0or n = N/ N is even, the natural frequencies are repeated. Further information on thissubject can be found in the literature (e.g. Thomas, 1979; Wildheim, 1981). As an example,figure 2 displays the natural frequencies versus nodal diameter number of the (blisk) lumpedparameter model with N = 24 sectors.The blisk lumped-parameter model was designed so that it stands as much as possible fora real blisk, particularly in terms of coupling between the blades and the disk. There are threefamilies of modes and regions of various modal density. This modal density is directly linked tothe coupling between the blades and the disk in the particular mode. A general rule, which for2 Sfrag replacements k k m m b m b k b k b c b c b c G k G k t m r k r k r c r c r Blisk sectorRing sector
Figure 1: Lumped-parameter model N a t u r a l fr e qu e n c y [ H z ] Figure 2: Frequency / diameter diagram3hat matter applies to the present example, is that the blade/disk coupling can be important islow frequency/low nodal diameter and high frequency/high nodal diameter regions whereas inlow frequency/high nodal diameter or high frequency/low nodal diameter regions this couplingis generally weak. For example, in the two first families the coupling between the blades and thedisk is weak for high nodal diameters whereas in small nodal diameters the coupling is stronger.Also note an important coupling appearing in the 2 nd and 3 rd modes with nodal diameternumber 2. Recalling that, in operating conditions, the external forcing generally acts on theblades this coupling parameter is an important factor regarding the efficiency of the friction ringdamping since it determines the disk participation in the global motion. The energy dissipationdue to friction occurs when the relative displacement between the disk and the ring is enough;this is achieved when the blisk is excited in areas of strong coupling where both the blades andthe disk move. The numerical values of the lumped-parameter are shown in table 1.Table 1: Numerical values of the model m k k G m b k b m b k b ξ . N/m 6 . N/m 350 g 1 . N/m 250 g 2 . N/m 1 ‡ Finally, concerning the ring, its mass and stiffness are chosen so that its lumped-parametermodel represents a circular beam in extension motion and its stiffness k r represents the longi-tudinal rigidity of the ring.An examination of the local behaviour of the friction area can help understanding the fric-tion dissipation mechanism. With reference to figure 3(a), both the disk and the ring experiencebending motion (bending stress σ b ) and are held in contact (slipping or not) due to the cen-trifugal load P . The friction forces in the interface ( T ≤ µP ) between the two elements willgenerate tension or compression stresses ( σ t/c ) mainly in the ring which ensure the continuityof stresses at the interface as shown in figure 3(a). As a consequence the friction load is directlylinked to the tension stresses and, assuming a linear elastic deformation, to the axial rigidity ofthe ring. PSfrag replacements TP Disk Ring σ b σ b σ t/c (a) Local behaviour in the frictional region (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) PSfrag replacements
GrooveRing (b) Friction ring in its groove
Figure 3: Physical model and local behaviourFigure 3(b) shows a physical model of the friction ring in its groove. In order to be assembledin these groove, underside the wheel of the disk, the ring needs to be split; the influence of thissplit on the dynamics of the system will be discussed later.The system of equations of motion can be represented as: M ¨ x ( t ) + C ˙ x ( t ) + Kx ( t ) = f ext ( t ) + f nl ( t ) (1)where x = [ x d , x r , x b , x b ] T is the physical displacements vector which gathers displacements4f the disk, the blades and the ring degrees-of-freedom. The matrices M , C and K are thestructural matrices of mass, damping and stiffness respectively. The vector f ext is an externaldistribution of rotating forces of Engine Order type, so that, considering a bladed disk with Nsector, the j th sector’s forcing term is: f ( j ) ext ( t ) = f (1) ext exp (cid:20) i (cid:18) ωt − ( j −
1) 2 πpN (cid:19)(cid:21) , (2)where p is engine order.The vector of nonlinear forces f nl due to friction is expressed, according to the Coulomb law,as: f nl ( t ) = − µP sign( ˙ u ( t )) , (3)where u ( t ) = x d ( t ) − x r ( t ) is the relative displacements vector between the ring and the disk, µ is the constant friction coefficient, and P the normal contact load, which can be expressed forrotating structures as: P = m r R Ω (4)where m r is the ring’s mass, R is the distance from the axe of rotation and the ring and Ω isthe engine rotation speed. Due to the high contact load resulting from the engine rotation, isassumed that the ring and blisk remain in contact at all times. The aim of the nonlinear analysis performed here is to calculate the steady-state response of thesystem under a periodic external forcing. Among the variety of method available to achieve thisgoal, the Harmonic Balance Method (Nayfeh and Mook, 1979; Szempli´nska-Stupnicka, 1990)presents several advantages in terms of efficiency and accuracy with respect to its computationalcosts.The idea is to express the time-dependent variables of the nonlinear equation (1) in termsof Fourier series (truncated in practice); the response x ( t ) is then: x ( t ) = X + N h X n =1 { X n,c cos( nωt ) + X n,s sin( nωt ) } . (5)The new unknowns of the problem are the Fourier components X n,c and X n,s which are balanced(using a Galerkin procedure) and the equation of motion can be rewritten using this formalismfor each terms of equation (1) in the frequency domain as: ΛX = F ext + F nl (6)with: X = (cid:2) X , X ,c , X ,s , . . . , X N h ,s (cid:3) T , (7a) F nl = h F nl , F ,cnl , F ,snl , . . . , F N h ,snl i T , (7b) F ext = h F ext , F ,cext , F ,sext , . . . , F N h ,sext i T . (7c)5he matrix Λ is block-diagonal, Λ = diag ( K , Λ , . . . , Λ N h ), and: Λ k = (cid:18) − ( kω ) M + K kω C − kω C − ( kω ) M + K (cid:19) (8)The problem (6) is nonlinear and is usually solved iteratively using a Newton-like method.One of the issue of the HBM is that the Fourier components of the nonlinear forces F nl cannot, in most cases, be derived straightforward. In particular, in problems involving friction,the nonlinear forces are only known (using equation (3)) in the time domain and the expansionin the frequency domain is not obvious. When few harmonics (no more than 3 in practice)are retained an analytical derivation of these frequency domain components is possible (e.g.Nayfeh and Mook, 1979; Wang and Chen, 1993). To overcome this problem when more harmon-ics are considered, the use of an Alternating Frequency Time method (e.g. Cameron and Griffin,1989) can be a solution. This method will be detailed in the next section.
The aim is, in the Newton procedure, to derive the Fourier components of the nonlinear forcesas a function of the displacements Fourier components. In computational applications, the ideaof the
AFT method is to use Discrete Fourier Transformation (DFT) to derive the Fouriercomponents of the nonlinear forces for given displacements in the frequency domain.Let’s start from X n,c and X n,s , the harmonic components of the response, predicted by agiven iteration of the Newton-like method. Using, an IDFT ( Inverse Discrete Fourier Trans-form ) procedure, one can express the associated displacements x ( t ) and velocities ˙ x ( t ) in thetime-domain. Then using a nonlinear operator the nonlinear force in time-domain is derived(here this step is achieved using equation (3) as described in the next section). Finally, usinga DFT ( Discrete Fourier Transform ) algorithm one expresses the harmonic components of thenonlinear force in the frequency-domain. The following diagram illustrates this procedure: X n,c , X n,s IDFT −−−−→ x ( t ) , ˙ x ( t ) y F n,cnl , F n,snl ←−−−− DFT f nl ( x , ˙ x , t ) At a given iteration of the Newton scheme and during the
AFT procedure, the friction forcesneed to calculated, in the time domain, in accordance with the velocity and displacement given(by the Newton method). This is generally done using the Coulomb law or, more usually, usinga regularized Coulomb law and several strategies can be used. Some authors have proposeda regularized expression of the sign function (e.g. Berthillier et al., 1998) or arctangent or hy-perbolic tangent function (e.g. Petrov and Ewins, 2002), some others have used the theory ofdistributions (e.g. Pierre et al., 1985), and some have used an iterative procedure in the timedomain (e.g. Poudou and Pierre, 2003; Guillen and Pierre, 1999; Petrov and Ewins, 2003). Forthe latter strategy, it is convenient to add a tangent stiffness to each friction element; doing so,the Coulomb law can be used straightforward to compute at each time step the equilibrium ofthe contact point. In previous studies on under-platform dampers, this stiffness was includedin the damper model. However, in general contact problem, this tangent stiffness appears asan equivalent penalty stiffness and can then account for the elasto-plastic shear deformationsof the contact asperities of the bodies in contact.Figure 4 shows a friction element defined in relative displacements, with a tangent stiffness:6 u is the relative between two points in contact, u = x d − x r in our example. ˆ z is the relative displacement of the contact point. k (cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1) zu t Figure 4: Friction element (relative displacements)The nonlinear forces vector given by equation (3) becomes: f nl ( t ) = ( − k t ( u ( t ) − z ( t )) if k t | u ( t ) − z ( t ) | ≤ µP, − µP sign ( ˙ z ( t )) if k t | u ( t ) − z ( t ) | ≥ µP. (9)In the time domain of the AFT procedure, velocities and displacements are given in oneperiod of the motion. These are entries for the computation of the associated nonlinear forcesin one period of the motion. This computation has to be performed time-iteratively (like anumerical integration) because of the history dependency of the friction law. For example,given the relative displacements and velocities at time t i , u ( t i ) and ˙ u ( t i ) and the displacementof the contact point z ( t i ), the friction force can be predicted (using equation (9)) for time t i assuming a sticking state ( z ( t i ) = z ( t i − )) for the contact point: f prenl = − k t ( u ( t i ) − z ( t i − )) . (10)The Coulomb law is then used to correct the friction force and the displacement of the contactpoint: f nl ( t i ) = f prenl if | f prenl | < µP − µP · f prenl | f prenl | if | f prenl | ≥ µP, (11a) z ( t i ) = u ( t i ) − f nl ( t i ) k t . (11b)Using this iterative scheme, the periodic friction force can be derived. In this section, several aspects of friction ring damping are investigated through numericalsimulations and discussed. All nonlinear numerical simulations were performed using the multi-harmonic balance procedure and a third order Lagrange predictor was used for the continuationof the frequency responses. A d th order Lagrange predictor, see Stoer and Bulirsch (1980) forexample, requires the last d + 1 points to predict the next one ˆ X p +1 and an interpolation isbuild on the arc-length s (which is here the excitation frequency) as:ˆ X p +1 = d X i =0 L i · X p − d + i with L i = d Y j =0 j =1 s ( p +1) − s ( p − d + j ) s ( p − d + i ) − s ( p − d + j ) . (12)The step ∆ s is adjusted a posteriori as a function of the (Newton-like) corrector’s performance(number of iterations) at the previous step. 7ll frequency response plots represent the blade response (degree-of-freedom x b in figure1), with, in dashed line, the linear response (without ring) and in solid lines, the nonlinearresponses (with friction ring). Parametric studies on the type of excitation (or blade/diskcoupling as mentioned earlier), on contact parameters, damper characteristics are presentedand the efficiency of this damping strategy is discussed. Then the influence of ring’s splitdiscussed. And finally, the impact mistuning and friction ring damping is investigated. In this first part, the influences of the ring damper mass and of the excitation configuration areinvestigated. The rotation speed is kept constant is these simulations; however, note that asthe ring’s mass changes, the contact normal load also change with respect to equation (4).
Strongly blade/disk coupling
With reference to the frequency/nodal diameter map offigure 2, we first focus on a small engine order excitation ( p = 2 for example in equation (2))with frequency range around the first modal family (similar behaviour would appears in othermodal family however).Several frequency responses are plotted in figure 5 for different damper mass. From these
209 210 211 212 213 214 215 216 217 218 2190.511.522.533.5 x 10 −3 Excitation frequency [Hz] V i b r a ti on a m p lit ud e [ m ] Figure 5: Frequency responses with strong blade/disk coupling; (- - -) : linear response, (—–) :nonlinear responses at different damper mass (g)response, we can first notice that the damping is efficient in this configuration when the ring issufficiently thick and that an optimal value appears ( m ring ≈ g ). At this optimal value, thereduction of vibration amplitude is nearly 4. Also note the progressive frequency shift of theresonance peaks as the ring’s mass increase, this is due to two facts combined. First, consideringthe associated linear problem, as the ring’s thickness increases the global system becomes stifferwhich account for the frequency shift to the higher frequency. And second, the nonlinear effect8f friction has usually a softening effect on system’s dynamic and, with respect to figure 5,as the ring’s mass decreases the contact normal load also decreases (assuming a fixed rotationspeed) and more and more slipping occurs which account for the frequency shift to the lowerfrequency as the damper’s mass decreases. Weakly blade/disk coupling
A weaker blade/disk coupling can be found in higher nodaldiameter in figure 2. As an example, let’s consider a 6 engine order excitation on the secondmodal family and the associated frequency responses plotted in figure 6. In this example, the
615 620 625 630 6350.20.40.60.811.21.41.61.8 x 10 −4 Excitation frequency [Hz] V i b r a ti on a m p lit ud e [ m ] Figure 6: Frequency responses with weak blade/disk coupling; (- - -) : linear response, (—–) :nonlinear responses at different damper mass (g)friction ring damper is clearly less efficient; even if an optimal mass appears (around 5g), thereduction of vibration’s level is not as important as in strong coupling configuration. In effect,in the configuration, there is not enough motion in the disk as the blades are excited and as aconsequence no relative motion between the disk and the ring can occur.
In this section, we focus on the influence of the normal contact load (which is, in our applica-tion, related to the engine rotation speed) on the friction ring damping efficiency. To do so,the damper mass is fixed as well as the excitation parameters (Engine Order 2) and severalsimulations were performed for different values of the engine rotation speed.In figure 7, the results are depicted and the first remarks is that the existence of an optimalrotation speed/normal load is clear around 3000 rpm. Moreover, one can see that that in aquite large vicinity of this optimal value (from 2000 to 6000 rpm), the friction ring damping isstill efficient.This is a key feature since a turbomachinery often operates at different rotation speeds(take-off, cruise, landing, . . . ). As a consequence, the rings can, for example, be optimally9
09 210 211 212 213 214 215 216 2170.511.522.533.5 x 10 −3 Excitation frequency [Hz] V i b r a ti on a m p lit ud e [ m ] Figure 7: Frequency responses at various rotation speed; (- - -) : linear response, (—–) :nonlinear responses at different rotation speeds (rpm)designed for the cruise speed and be also efficient at take-off or landing speeds.
As mentioned earlier, in order to be assembled the ring need to be split. However, consideringthe model described in figure 1, a split in the ring would necessary spread over one elementarysector and generate an exaggerated gap compared to the technological reality. As a consequence,we built a refined model, presented in figure 8, in which the disk mass, m , is refined into threemasses, m ′ . This allows the split to spread only over a portion of one sector.The influence of the ring’s split was then investigated by comparing the dynamic response ofthe system with or without a split. First, the ring is assumed to be fully stuck (no sliding occurs)and the system is then linear. The frequency response plots resulting from these simulationsare displayed in figure 9.We focused on the first resonance peak of the 2 engine order excitation and comparing theblade’s resonant responses with (figure 9(b)) or without (figure 9(a)) ring’s split, it appearsthat each blades have a different response and that the maximum level is more important whenthe ring is split. This symmetry breaking, which corresponds to the separation of the initialtwin modes (two-fold degeneracy), is similar to the consequence of mistuning and can lead tolocalization phenomena (see Wei and Pierre (1988a,b) for example).When the nonlinear effect are introduced (figure 10), the same phenomena of symmetrybreaking and increasing of resonant responses appear. However, the response amplification islimited by the nonlinear effect and then smaller than in the linear case.These results show the influence of the split in the ring damper on the global dynamics ofa bladed disks. However, these observations have to be put in perspective with respect to theinfluence of the inherent mistuning (discussed next) of a real bladed disks which may generally10 Sfrag replacements k ′ k ′ m ′ m ′ m ′ m b m b k ′ G k ′ G k ′ G N th sector Figure 8: Fine lumped-parameter model
210 212 214 216 218 22000.511.522.533.54 x 10 −3 Excitation frequency [Hz] V i b r a ti on a m p lit ud e [ m ] (a) Continuous ring
210 212 214 216 218 22000.511.522.533.54 x 10 −3 Excitation frequency [Hz] V i b r a ti on a m p lit ud e [ m ] (b) Split ring Figure 9: Linear frequency responses with and without ring’s split; (- - -) : without a ring,(—–) : with ring (split or not). 11
10 212 214 216 218 2200.511.522.53 x 10 −3 Excitation frequency [Hz] V i b r a ti on a m p lit ud e [ m ] (a) Continuous ring
210 212 214 216 218 2200.511.522.53 x 10 −3 Excitation frequency [Hz] V i b r a ti on a m p lit ud e [ m ] (b) Split ring Figure 10: Nonlinear frequency responses with and without ring’s split; (- - -) : without a ring,(—–) : nonlinear responses.be more important.
To conclude on the numerical results, the influence of mistuning was investigated. Mistun-ing refers to small variations in the properties of a system originally designed to be cyclicallysymmetric. These variations are mainly due to manufacturing tolerances or material inhomo-geneity and are inevitable. Mistuning can lead to significant changes in dynamical behaviourwith respect to the tuned case as presented by Wei and Pierre (1988a,b); vibratory amplitudesin forced response are increased and impact on high cycle fatigue can be considerable. Theseeffects are even more important in Integrally Bladed Disks since their ”natural” damping isfar smaller than for bladed disks assemblies. The influence of mistuning on friction dampinghas been studied earlier by Griffin and Sinha (1985) or Lin and Mignolet (1996) and the per-formance of the damping technology discussed here as applied to blisks may be influenced bythis phenomenon.Considering the lumped parameter model of figure 1, mistuning was introduced as randomperturbations in the blades’ stiffness k b and k b as it is often the case since the disk is assumedto be symmetric. These perturbations lead to changes in the natural frequencies of the blades(cantilevered).A frequency response is shown in figure 11, the mistuning standard deviation is about 1.2%of the cantilevered blade’s nominal natural frequency. The resonant responses of the blades aredisplayed for the tuned and the mistuned case with or without ring. First, we can notice thetypical feature of mistuned responses which is the break of symmetry which leads to a separationof the resonance peak (with or without rim damping). Then, the vibratory amplitudes areincreased in the mistuned case with respect to the tuned case; however, the damping is stillefficient in presence of mistuning. 12
10 212 214 216 218 220 222 224 226 2280.511.522.533.544.555.5 x 10 −3 Excitation frequency [Hz] V i b r a ti on a m p lit ud e [ m ] (a) Tuned bladed disk
210 212 214 216 218 220 222 224 226 2280.511.522.533.544.555.5 x 10 −3 Excitation frequency [Hz] V i b r a ti on a m p lit ud e [ m ] (b) Mistuned bladed disk σ = 1 . Figure 11: Frequency responses with and without mistuning; (- - -) : linear responses, (—–) :nonlinear responses.
The results of these numerical simulations clearly highlight the possibilities and limits of thefriction ring damping strategy. In essence, the dissipation mechanism of friction is efficient onlyif some relative motion between the two bodies (the disk and the ring) in contact occurs. Asopposed to under-platform dampers, the rings are not directly in contact with the blades towhich the external forcing applies. As a consequence, the coupling between the blades and thedisk which conditions the energy transfer from the blades to the disk is an essential criterion. Ithas been demonstrated that when the coupling is weak (that is when little energy get transferredto the disk), the ring damping is less efficient. On the other hand, when this coupling issufficiently strong the rings are efficient. Finally, it was shown that the contact parameters(normal loads or friction coefficient) are also significant in the friction damping performance.The qualitative results given by these simulations are quite similar to those observed with under-platform damper; the alternation between slipping and sticking states results in frequency shiftwhich offsets the resonant frequency from its initial value ( i.e. non damped), and an optimumcan be reached for a given excitation configuration in the ring’s mass.
A strategy of damping for Integrally Bladed Disks (blisks) using a friction split ring was inves-tigated. A procedure of analysis using a multi-Harmonic Balance Method and an
AFT methodwas described and applied for calculating the steady-state nonlinear response of a system underperiodic excitation.Simulations on the lumped-parameter model presented lead to qualitative results in termsof energy dissipation with respect to several configurations. Some guidelines on the use of thistechnology, its limits and performances, were highlighted. We show that the coupling ratiobetween the disk and the blades was a critical parameter of the damping power of such rings.13 cknowledgment
Thanks go to Snecma for their technical and financial supports.