THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS. 345 E. 47th St., New York, N.Y

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1 THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS 345 E. 47th St., New York, N.Y The Society shall not be responsible for statements or opinions advanced in papers or discussion at meetings of the Society or of its Divisions or Sections, m or printed in its publications. Discussion is printed only if the paper is published in an ASME Journal. Papers are available from ASME for 15 months after the meeting. Printed in U.S.A. Copyright 1994 by ASME 94-GT-293 A NUMERICAL METHOD FOR THE PREDICTION OF BLADED DISK FORCED RESPONSE Marc Berthillier, Marc Dhainaut, Franck Burgaud, and Vincent Gamier Snecma Moissy Cramayel, France ABSTRACT [M] Structural mass matrix [M S I Structural mass matrix for 1 3 t A numerical method has been developed to predict the forced [mp ] Modal mass matrix for (3 t response of bladed disks due to a wake excitation from upstream blade rows. The structure is modelled by a 3D finite element Expanded Fourier matrix [E]* mesh of a bladed disk segment. Using cyclic symmetry, this Complex conjugate of [E] model provides a modal base for the rotating structure. The [Qp, ] Mode shape matrix for f aerodynamic damping of the vibratory modes and the excitation {f9 (t)} Gust response unsteady aerodynamic forces pressures on the blades due to the propagation of upstream flow f defects are computed separately using the same 3D unsteady { f s (t)} Gust response unsteady aerodynamic forces for Euler analysis software. A modal response solution of the blade numberj aeromechanical system is then performed. { f s } Fourier component of { f g (t)} for harmonic V This analytical methodology has been used to study the forced { fm(t)} response of an experimental high pressure compressor blisk. The Motion dependent unsteady aerodynamic forces results are analysed and compared with actual rig tests. {.ç (t)} Traveling wave representation of {f" (t)} for Q, { f (ç2 )} Centrifugal forces at speed of rotation S2 NOMENCLATURE { f} Steady aerodynamic forces {q(t)} Physical displacements [C] Structural damping matrix (Co I Structural damping matrix for (3t f qj (t) 1 Physical displacements for sector number j ' [c p I Modal damping matrix for (3 {q (t)} Traveling wave representation of the [A ] Aeroelastic matrix for fi t displacements [a ] Modal aeroelastic matrix for (3 {q^, (t)} Traveling wave displacements for I 3 [K] Total stiffness matrix for rotation speed S2 { } Modal coordinates for fi t [K ] Total stiffness matrix for rotation speed S2 and li t Isr ; {q^t } Mode shape j for (3 t [k b ] Modal stiffness matrix for rotation speed S2 and (3, N Number of blades [KE] pclassical stiffness matrix for Q=0 and 13 t Nw Number of wakes Nh [Kc (6)]^ Geometric stiffness matrix for (3t Number of harmonics [KS] it Supplementary centrifugal stiffness matrix for 0 1 Presented at the International Gas Turbine and Aeroengine Congress and Exposition The Hague, Netherlands June 13-16, 1994 This paper has been accepted for publication in the Transactions of the ASME Discussion of it will be accepted at ASME Headquarters until September 30, 1994

2 (3 1 Interblade phase angle = Nl S2 Rotation speed in rad/s U) Eigen frequency of the structure 6 Steady stress distribution e,j Excitability measure m3 Modal mass for modej 4 Critical damping ratio INTRODUCTION It is well known that high frequency fatigue is a major cause of failure in axial compressor components. It occurs when vibration takes place at or near resonant conditions of operation. In the design process, Campbell diagrams are used to predict the occurrence of the resonant conditions in the operating range of the compressor. Unfortunately, it is not always possible to avoid resonant conditions in some stages. Therefore, based on prior experiences, an estimation of the response and the dynamic stress is usually made to determine the acceptability of a blade design. This approach may lead to erroneous conclusions and it becomes necessary for the designer to have a tool that can predict more accurately the forced response and the dynamic stress of the components. A large experimental effort is devoted to the subject of blade forced response. Although these experiments are very useful to assess and explain the effects of parameters such as steady blade loading, shape and amplitude of aerodynamic forcing functions, reduced frequency (see for example Fleeter (1992)), they cannot provide quantitative answers in real life cases. This is why (apart from issues of design cycle time and cost) accurate predictions of blade forced response in the jet engine industrial environment can only be achieved through the use of numerical methods. A substantial part of the numerical analyses of bladed disk forced response reported in the literature concentrate on the structural point of view. Irretier and Omprakash (1991) studied the effects of transient phenomena in the forced response but with a very simple excitation. Wildheim (1979) found all the resonant conditions for a rotating periodic structure excited by upstream flow defects. On the aerodynamic side, many methods (i.e. flow model and numerical algorithm), of varying degrees of complexity, have been proposed to compute the motion dependent and gust response aerodynamic forces. These have been recently reviewed by Verdon. A complete definition of the resonant vibratory response of the engine bladed disks includes a coupled characterization of the structural dynamics, of the aeromechanical damping and of the external excitation. An early application of a 2.5D unsteady Euler code to perform a parametric study of a stator blade row forced response was given by Joubert and Ronchetti (1989). More recently, rather precise aeromechanical studies have been conducted by Chiang and Kielb (1992) and Murthy and Stefko (1992), with simple 2D aerodynamical computations. Such an approach has been used by Tatakis and Stockton (1993) to study the influence of airfoil geometric and aerodynamic parameters on forced response. This paper presents a full 3D numerical approach of the dynamic response of bladed disks. As the dominant source of excitation can be attributed to the presence of a nonuniform inlet flow, we will restrict our study to a wake excitation that can be created by a variety of sources including support struts and stator blade rows. Aerodynamic damping is computed via an Euler 3D code. In addition, an equivalent viscous modal damping can be taken into account. The structure is supposed to be perfectly tuned and cyclic symmetry can be used to determine the dynamic behavior of the structure starting from a finite element model of one sector of the bladed disk. An application of this procedure has been made for an experimental high pressure compressor blisk. The computations have been fitted to actual rig tests. THEORY Dynamic equations of motion The general aeroelastic response of a bladed disk is given by the following equations : [M]{q'(t)}+[C]{q'(t)}+[K]{q(t)} ={fs(t)}+{f, (t)} (1) where the vectors { q ( t ) } and 8 {f (t )} are the displacements and forces for all the sectors of the stage. If the upstream flow defects responsible for the excitation have a cyclic symmetry of order Nit, (e.g. wakes from Nw identical and equally spaced struts), the exciting forces { f a (t)} may be expressed as a Fourier series of Nh harmonics, all multiples of N. Thus, for blade j : (Vn, N^(i n)] (2) lf5(t)1=y,4 {f8}e where v = N w p with p = 1, Nh In the following, the notation Re(-) will be omitted. As we suppose that all the sectors of the bladed disk are identical, we can take advantage of the traveling wave representation of the displacement (see Crawley 1988 ). This can be written as : {q(t)} = [E]{q,(t)} (3) where [E] is the expanded Fourier matrix with: [Ell]... [EJN] i 1 [E] = and [Ek,]= diag( e j-lxk-f) [ENI]... [ENN ] Substituting (3) into equation (1) and premultiplying by [E] - ' with [E]-' = _!_[E] where [E] * is the complex conjugate of N [E]:

3 [E]-' [M][E]{qo(t)}+[E]_ ' [C][E]{qe(t)}+[E]-' [K][E]{ge(t)}= [E]- '{ fg(t)}+[e]- '{f (t)} (4) As [M], [K], [C] are block circulant matrices, the matrices of equation (4) are block diagonal (see Davis 1979). For exemple, [E]-'[M][E] = diag[mp,] and, {qp}={...{qp1},...} l = -D/2...,+D/2 D = N if N even D = N-1 ifnodd The development of the right hand term [E]- '{ fg(t)} shows that a single interblade phase angle 0, contained between [-7c, it] is excited by the harmonic v and is given by: (3r = d with I = v [N] 1 e,+ D] (5) 1-2 This relation may be represented graphically by: v We may wish to reduce the size of the different systems more, by projecting into a modal base, obtained from the undamped homogeneous form of (6). The displacement {qp, (t)} can be expressed as : {q9, ( t)} = [QR, ]{ X 9, }e 'v n r (7) where [Q,^, J is the mode shape matrix and {X 1 } the modal coordinates. Assuming linearity, we can express the motion dependent forces as: {J (t)}=[a,,]{xe,}ervn, Substituting (7) and (8) into (6) and premultiplying by the transpose conjugate of [Q,] results in the modal equations : ( (v 92)2[ma,]+iv c4c,]+[k1]){x1}= [a ] + {fs }+[ap ]{X,,} (9) Where [inn,] and [kni ] are the diagonal modal matrices of the structure for the selected eigenmodes. [CO, Iis the structural modal damping matrix [a,, ] is the modal aeroelastic matrix. (8) The resolution of (9) gives the amplitude of the different modes for the interblade phase angle (3 1. The total response of the structure is obtained by adding the different contributions for the different interblade phase angles excited. The stresses are obtained on any point of the structure using the modal stresses distributions. -^.4 D/2 I Mechanical model A finite element mesh is made for one blade and one sector of the disk. In order to reduce the time spent on that task an automatic generation of mesh is possible. A static analysis can be performed for (3 o with the following general equation: The system of equations (4) can then be reduced to Nh separated systems of equations of smaller size, each one corresponding to an interblade phase angle 03, : [M9,]t4e^(t)1 + ^C9,]149,( t)i + ^K9,^lg9t( t)j = { f8 }enn, +If9r(t)I (6) Of course if any dissymmetry was present in the structure, or if friction forces between the blades were added, some coupling between the different interblade phase angles would have to be taken into account. ([KE] +[Kc(6)]+[K5]]po {qr0 (t)} = {f(q 2 )} + {.fas} where [KE], [Ko(6)], and [Ks] are respectively the classical, geometric, and supplementary centrifugal stiffness matrices (for a more complete description, see Ferraris, Henry, Lalanne, and Trompette, 1983). The stress distribution 6 is initially unknown, thus an iterative procedure is used to perform the static solution. Once 6 is known, the stiffness matrix [Kn,] of the structure for the rotation speed f and for any iii can be constructed. For all the interblade phase angles excited by the gust induced forces, the following dynamic system of equations can be solved. 3

4 1 w 2 [M9, ]+[KK, ])Jgat I = {0} Thus the mode shapes {q p, } and all the terms of the matrices [Q] and [k,] can be determined. Unsteady Aerodynamic model This model provides an estimate of the flow forces acting on the blades. The first step is to determine the compressor operating points corresponding to the frequency crossings of interest. For each of these cases, we then have to perform a steady computation to initialize the flowfield, followed by two unsteady computations, one to determine the motion dependent unsteady aerodynamic forces { f'" (t) }, the other to determine the gust response unsteady aerodynamic forces { f g (t) }. All three computations are done using the same 3D unsteady Euler code (Gerolymos (1992)). This solver numerically integrates the 3D unsteady Euler equations discretized in finite volume formulation, using Runge Kutta schemes. A single interblade channel is considered for the computations, making use of the chorochronic periodicity of the flow field perturbed by harmonic disturbances imposed at the boundaries (i.e. blade motion and upstream vorticity). Convergence of the unsteady computation towards time and space periodicity is estimated using residuals on the first harmonic of pressure at the pitchwise boundaries. A periodicity residual of 1% is considered satisfactory. To compute the motion dependent forces {f'"(t)}, we use mode shapes determined by the mechanical model, interpolated from the structural mesh on to the aerodynamic mesh around the blade. From this is derived a distribution of movement for the aerodynamic mesh, which conforms with the blade vibration. The unsteady Euler equations are integrated in time on the moving grid, with a fixed time lag (corresponding to the interblade phase angle of the vibratory mode) imposed at the permeable pitchwise boundaries. Time integration usually covers a span of 10 vibration periods which is sufficient to establish spatial and time periodicity, as assessed by the periodicity residuals. As the vibration amplitude is small with respect to the blade chord, only the first harmonic of the unsteady pressures on the blades are of interest. Those computations are used to build the matrix [A,,] for each interblade phase angle 13, and thus to obtain the modal aeroelastic matrix after multiplication by [Q 1 ] To determine the gust response forces {p8 (t)}, we need an estimate of the flow defects present at the upstream boundary of the domain. In the case of wakes being shed by upstream blade rows, this is usually obtained through correlations giving a spatial total pressure distribution (depth and width) as a function of the upstream blade row loss coefficient, solidity and distance from its trailing edge. This total pressure distribution is then Fourier transformed in the tangential direction, at each radial station of the computational grid. Usually, only the spatial harmonic corresponding to the excitation being studied is used to specify the upstream boundary condition of the computation, since in the case of small perturbation amplitude, it is the main driver of the gust response harmonic of interest. It is useful to recognize this fact, because numerically resolving the (unneeded) higher spatial harmonics present in the original wake shape would require a much finer grid than necessary, resulting in a direct computing cost penalty. In the application considered in this paper however, the grid used could perfectly resolve all the harmonics necessary to reconstruct the original signal, so they were all kept. The Euler equations are integrated in time until convergence to periodicity is reached, which usually requires around 10 periods of wake passing. The unsteady pressures on the blades are then Fourier transformed and the first five harmonics are interpolated from the aerodynamic mesh on to the structural mesh, thus providing the gust response unsteady aerodynamic force for the forced response mechanical computation. APPLICATION Description of experiment The goal of this test series was to compare the resonant response of a blisk rotor, composed of 72 blades, shown in figure 1, with that found for rotors of conventional technology. Only the results obtained on the blisk rotor will be presented here and the experimental values of blade damping and stress compared with the numerical prediction. The compressor used (figure 2) is composed of three blade rows (IGVs, rotor and stator), and is representative of the rear stages in civil HP compressors. A section of the compressor is represented on figure 3. The aerodynamic forcing function is provided by the wakes of cylinders inserted vertically in the flowpath and equally spaced around the circumference. The number of cylinders is variable and can be adjusted in order to excite the blisk in a resonant condition, for the first bending and first torsion modes, with several engine harmonics. Also, three different cylinder diameters were tried. Since existing correlations for wake shedding are tailored for blade profiles and are not valid for cylinders, the flow defect due to the cylinder wake was measured immediately upstream of the rotor at four heights in the annulus : 20%, 40%, 60% and 80%. The experimental data was fitted with analytical functions for Fourier transformation, subsequent discretization and reintroduction at the upstream flow boundary. An example of a data fit is given in figure 4. Note that the fit was performed at the nominal operating point and with one cylinder in the annulus, and expressed as a percentage of total pressure loss. The same fit was then used for all other flow conditions and number of cylinders. For evaluation of blade forced response, strain gages were placed on seven blades; all results given subsequently (unless otherwise stated) are an average of the results of the individual blades. The Campbell diagram of the blisk is given in figure 5. In the following, we will concentrate on the resonant conditions 1T/2252. The experiment itself was comparatively standard. The compressor map was first measured and is given in figure 6. When looking for resonant conditions, the compressor was revved without changing the throttle setting, which defines an operating line. This line and the operating point corresponding to the 1T/2252 crossing are also shown on figure 6. Gage data was recorded continuously then analysed off-line to yield blade response and damping. 4

5 Mechanical model A finite element mesh of the blisk has been performed for one sector (figure 7) to predict stage frequencies and blade motions using cyclic symmetry. The mesh density on the blade has been adapted for stress predictions. The interblade phase angles being taken into account in the computations are given by equation (5). If we consider the resonance 1T/2252, we have to study separately the Fourier order 1=-22, +28, +6 respectively for the first three harmonics of the excitation force. As we can see on the Campbell diagram, there is no mode excited by the frequencies 4452 and 6652 for the rotation speed S2 chosen here. Thus we will only consider in this case the first harmonic of the excitation force associated with the Fourier order 1=-22. For this Fourier order, the eigenvalues and eigenmodes analysis show very little coupling between blade and disk motion for the first three modes. The mode shape of the mode IT is represented on figure 8. For the first three modes, a good correlation on the frequencies has been obtained. As the relative gap on the frequencies between the lt/1b and 1T/2B modes is respectively 247% and 177%, we can neglect in the computations the dynamic coupling between the IT mode and its neighbouring modes. Although the program calculates general problems, the system (9) to be solved is reduced here to a single degree of freedom equation for the IT mode. Aerodynamical model As mentioned earlier, the unsteady 3D Euler code uses H-type meshes. For this application, a mesh size of 120(axial)x21(radial)x25(tangential) was used in order to adequately resolve the unsteady aerodynamic phenomena present in the flowfield, especially close to the leading edge. Figure 9 gives the mesh section at 90% blade height. A steady computation was performed at the operating point corresponding to the crossing of interest. Since the Euler computation cannot reproduce exactly the compressor map, it was preferred to reproduce mass flow and rotational speed (for accurate upstream relative velocity) and allow a slight descrepancy on the pressure ratio. The steady computation was used to initialize two unsteady computations ; one to determine the aerodynamic damping on the modes excited by the wakes, the other to determine the gust response unsteady aerodynamic forces due to the wakes of 22 upstream cylinders. Of course, a certain amount of mistuning between blades was observed in the experiment, but the computations were performed at only one vibration frequency / rotational speed since the effect of varying this parameter over such a small range was deemed to be second order. The frequency used for the first torsion mode was 5082 Hz which corresponds to a reduced frequency of 1.1, based on blade tip chord. The corresponding rotational speed was RPM, for interaction of the IT mode with the 22 upstream wakes. 15 periods of the unsteady phenomenon were simulated, both for flutter and gust response. Unsteady convergence was adequate after 10 periods but it was prefered to push on for another 5 periods to check that convergence was continuing to improve. CPU time for each of these computations was roughly two hours on a CRAY YMP computer - an implicit residual smoothing technique added to the four step Runge Kutta scheme allowed the use of large CFL numbers and thus reduced the number of iterations per period. Results and comparison As the structure studied here is a blisk, there is no energy dissipation at the root of the blades. The only sources of damping are the internal friction in the material and the aerodynamic damping due to the aerodynamic motion dependent forces. As the former is very small for conventional steel materials, it will be neglected in the following study. We can thus compare the computed aerodynamic damping for the 1T/2252 resonant condition with the total damping measured at that point during the test and assumed here to be only aerodynamical. The aerodynamical damping is computed by solving the eigenvalue problem associated with the homogeneous form of equation (9). The critical damping ratio is = 0.54 %. An estimate of the experimental damping is given by the frequency bandwidth method applied to the frequency response of a strain gage obtained from the experimental Campbell analysis shown in figure 10. The critical damping ratio obtained is E, = 0.36%. The agreement between calculations and experiments, although not excellent, can be considered as satisfactory if we consider the lack of precision associated with the method we used to determine the experimental damping. The excitation forces can be analysed using several characteristic numbers. The first one is the force magnitude : 11 f 8II ={ f s }. Here at the crossing point 1 T/22-Q, vf^ I1f811=0.011 dan. The second one is the excitability measure for harmonic v and { }T {} I mode j defined as e _ Tr ltwhere {qq, } is the mode shape j for the phase angle (31. e,, is a real number between 0 and 1, and is independant of the mode shape normalization. It is a measure of the compatibility of the force distribution with the mode shape. For our case study, e2n.=0.004 ; the excitation forces are mainly located on the lower part of the blade (see figures 11a & 11b). If we want to compare the severity of different excitations for i}t{fs}^ different modes, we can use the ratio, where mj is the modal mass for mode j. The normalization of each mode shape is taken to induce a maximum stress equal to 1 on the blade. After the separate analysis of aerodynamical damping and excitation forces, we performed the complete resolution of equation (9). The stresses obtained are compared to the m; 2

6 measurements made with six different strain gages placed on different blades. The discrepancy between computations and measurements is of the order of the magnitude of the blade to blade variation. It is due to the mistuning, the lack of precision on the strain gage locations, and the incertaintees of the measurements. An illustration is given on figure 12 for the strain gage location shown in figure 13. Only the extremum measured values are represented. The difference in frequency between the different peaks is mainly due to blade mistuning. For the computations, the frequency of the model has been updated on the mean of the experimental distribution. To draw the numerical curve, we reasonably extended, on a narrow frequency band, the unsteady aerodynamic pressures computed for the speed of rotation and blade frequency of the resonant condition 1T/2252. The difference between maximum and minimum experimental stresses is rather large (40%). As our model does not take into account the blade to blade mistuning, it is significant to consider the average experimental stress. The numerical result and the average experimental stress level were found to match perfectly. The good agreement between experiments and calculations on the dynamic response implies that the estimate of the gust response unsteady forces is also of the right order of magnitude. Overall, the numerical method can be considered to give satisfactory agreement with the data for the case studied, at reasonable cost. CONCLUSION We have presented an aeromechanical method to predict the forced response of bladed disks to aerodynamic excitation from upstream wakes. The method is fully three dimensional, based on a finite element model of the bladed disk coupled with a timemarching non-linear Euler solver. This degree of modelling is probably the most elaborate which can be used in an industrial environment (where reasonably fast and trouble free methods are essential) for the next couple of years. The current method has been thoroughly checked, and used in this study to predict the aerodynamic damping, gust excitation and forced response of the 1T/2252 crossing of an HP compressor blisk. The predictions are borne out by experimental results on the configuration studied : orders of magnitude are well predicted and are within blade to blade dispersion. It is therefore thought that this method is adequate for selecting or improving blade configurations with respect to aeromechanics. The method is currently being used to predict the blade response at other crossings of the configuration described here and of fans subjected to inlet or downstream distortion. Crawley, E.F. : "Aeroelastic Formulation for Tuned and Mistuned Rotors" AGARD Manual on Aeroelasticity. in Axial-Flow Turbomachines, vol 2, Structural Dynamics and Aeroelasticity, Davis, P.J.: Circulant Matrices, 1979, Wiley-Interscience. Ferraris, G., Henry, R., Lalanne, M., and Trompette, P. - Frequencies and mode shapes of rotating bladed axisymmetric structures. Application to a jet engine. Advanced vibrations of bladed disc assemblies, ASME, 1983, 10 p. Fleeter, S.. "Forced Response Unsteady Aerodynamic Experiments". AIAA Paper Gerolymos, G.A.: "Advances in the Numerical Integration of the 3-D Euler Equations in Vibrating Cascades". ASME Paper 92- GT-170. Gerolymos, G.A.: "Coupled 3-D Aeroelastic Stability Analysis of Bladed Disks". ASME Paper 92-GT-171. Irretier, H. and Omprakash, V. : "Numerical Analysis of the Transient Responses of Bladed Disks". ASME 1991 DE-Vol 36, Machinery Dynamics and Element Vibrations. Joubert, H. and Ronchetti, V. "Aerodynamic Study of Forced Vibrations on Stator Rows of Axial Compressors". AGARD CP 468 : "Unsteady Aerodynamic Phenomena in Turbomachines", August Murthy, D.V.,and Stefko, G.L. "FREPS : A Forced Response Prediction System for Turbomachinery Blade Rows". AIAA Paper Tatakis, S. and Stockton, R.: "Structural and Aerodynamic Influences to Airfoil Forced Response" ASME Paper 93-GT Verdon, J.M. : "Review of Unsteady Aerodynamic Methods for Turbomachinery Aeroelastic and Aeroacoustic Applications". AIAA Journal, Vol. 31, N 2, pp , February Wildheim, S.J. : "Excitation of Rotationnally Periodic Structures". ASME Journal of Applied Mechanics Vol 46, pp , ACKNOWLEDGEMENTS The authors would like to acknowledge the contribution of Marc Loubet for running a thorough experiment and providing high quality data, which is essential for the correlation and validation of this analytical method. REFERENCES Chiang, H.D. and Kielb, R.E. : "An Analysis System for Blade Forced Response" ASME Paper 92-GT-172.

7 j! C. I.. s a ^ Figure 1: Blisk geometry Figure 2 : Test rig FREQ. (Hz) 0852 (RD1) Figure 3 : Section of the test rig wake disturbance PT/PREF 1.01 i b 0.ss ji 7052(RDE) 2sM 630 2T 57f2 2F \JiI' IT 22f tangential spacing data -a- analytical model 1 F 9Q nuuu S2 (tr/mn) Figure 4 : Comparison of the wake modeling and data Figure 5 : Campbell diagram 7

8 J X Surge line Operating point 1T/ eY N n Dsrd Figure 6 : Compressor map Figure 7 : Finite element mesh of a blisk sector Figure 8 : Mode shape on the blade for the IT mode Figure 10 : Experimental Campbell analysis for the 1T/22) crossing Figure 9 : Aerodynamic mesh at 90% span 8

9 2.Poo leading edge VALOUR trailing edge `000 trailing edge V AL E U R.E _ _ = as I a 3. S _ '., 44.6 leading edge sga;.. Figure 1 la: Module of the excitation forces suction surface Figure 11 b : Module of the excitation forces pressure surface Comp sveu - Muupsuess Min exp wens... 1 a 0.8 a N a 0.6 I 0 Z Rotational Speed (rpm) Figure 12 : Comparison between computational data stress and experimental data stress Figure 13 : Strain gage lopation 9